Sickle Cell Hemoglobin Polymerization

SICKLE CELL HEMOGLOBIN POLYMERIZATION

.

By WILLIAM A EATON and JAMES HOFRICHTER Laboratory of Chernlcal Physics. National Institute of Diabetes and Digestive and Kidney Diseases. National Institutes of Health. Bethesda. Maryland 20892

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Structure of Hemoglobin S Molecule and Gel . . . . . . . . . . . . . . . A . Structure of Hemoglobin S Molecule . . . . . . . . . . . . . . . . . B . Aggregated Forms of Hemoglobin S . . . . . . . . . . . . . . . . . . C . Structure of Hemoglobin S Polymer . . . . . . . . . . . . . . . . . . 111. Thermodynamics of Hemoglobin S Polymerization . . . . . . . . . . . . . A . Sedimentation Studies and Nonideality . . . . . . . . . . . . . . . . B . Effect of Temperature and Solution Conditions on Deoxyhemoglobin S Polymerization . . . . . . . . . . . . . . . . . . C . Control of Polymerization by Oxygen . . . . . . . . . . . . . . . . . D. Polymerization of Mixtures of Hemoglobin S with Other Hemoglobins . . . . . . . . . . . . . . . . . . . . . . . . . IV. Kinetics and Mechanism of Hemoglobin S Polymerization . . . . . . . . . . A . Principal Results on Kinetics of Polymer Formation . . . . . . . . . . . B . Kinetic and Thermodynamic Equations of Double-Nucleation Mechanism . . . . . . . . . . . . . . . . . . . . C. Comparison of Theory and Experiment . . . . . . . . . . . . . . . . D . Effect of Shear on Kinetics of Polymerization . . . . . . . . . . . . . . E . Areas for Future Study. . . . . . . . . . . . . . . . . . . . . . . . V. Intracellular Polymerization and Rheology . . . . . . . . . . . . . . . . A . Equilibrium Measurements of Intracellular Polymerization . . . . . . . . B . Kinetics of Intracellular Polymerization . . . . . . . . . . . . . . . . C . Intracellular Polymerization Kinetics at Partial Saturation . . . . . . . . D . Rheology of Gels and Sickle Cells . . . . . . . . . . . . . . . . . . . VI . Comments on Pathophysiology and Strategies for Therapy . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 67 69 80 86 111 112 121 130

143 157 162 175 186 199

203 205 206 222 238 245 253 262

I . INTRODUCTION The polymerization of sickle cell hemoglobin has probably become the best understood of all protein self-assembly systems. From a biochemical point of view. the polymerization process is extremely simple. involving only the reversible aggregation of hemoglobin molecules . T h e identity of the structural and functional properties of sickle cell hemoglobin (hemoglobin S) in dilute solution as compared with those of normal human hemoglobin (hemoglobin A). has also permitted the vast knowledge of the parent molecule to be exploited in understanding the polymeriza63 ADVANCES IN PROTEIN CHEMISTRY . Vol. 40

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WILLIAM A. EATON AND.JAMES HOFRICHTEK

tion process. Despite its biochemical simplicity, hemoglobin S polymerization has proved to be an extremely interesting and complex physical process which has been studied by almost every physical technique of protein chemistry. As such it has served as a paradigm for the study of protein self-assembly. The successes in describing t.he polymerization process provide several elegant examples of what has been learned from these studies. T h e determination of the complex three-dimensional structure of the polymer has been a major achievement of structural biology, and has stimulated the development of new techniques in structure determination of protein assemblies. The use of other naturally occurring hemoglobin mutants, containing single amino acid replacements, has provided information which has been important in elucidating the polymer structure long before such problems could be addressed by protein engineering. The thermodynamic studies have shown how the control of polymerization by oxygen binding can be easily understood in terms of quaternary conformational equilibria using a simple extension of the two-state allosteric model. They have also demonstrated that it is possible to provide a rigorous description of protein aggregation even at the very high concentrations found in red cells (>0.3 g/cm3).The kinetics of polymerization, perhaps the most fascinating aspect of the process, have produced one of the best-characterized examples of a nucleation-controlled aggregation. The kinetics exhibit an unusual time course and the highest dependence of a rate on concentration (up to 50th power) that has been observed for any process in solution. By using a novel laser photolysis, light-scattering technique, it has been possible to carry out detailed rapid kinetic studies on single red cells. The kinetic studies have also provided one of the few examples in which stochastic fluctuations are used to measure the rate of a process. Hemoglobin S polymerization is the pathological process that is responsible for sickle cell disease. The intimate relation between polymerization and the disease has been a constant stimulus to research in this field. A basic assumption of sickle cell disease research has been that partial inhibition of polymerization will decrease clinical severity, and that a “cure” will result from complete inhibition. Understanding the polymerization process in detail is, therefore, not only important for understanding the pathophysiology of sickle cell disease, but is critical to the major problem of developing a specific therapy that could be used in the ‘treatment o f patients. The kinetic and thermodynamic studies have played a major role by providing relevant and sensitive assays for potential therapeutic agents. The kinetic studies have also been important in designing strategies for a specific therapy. In particular, the

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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discovery of the enormous concentration dependence of the rate of polymerization suggested decreasing the intracellular hemoglobin concentration as a new approach to the treatment of sickle cell disease. The purpose of this article is to describe our current understanding of the physics and physical chemistry of sickle cell hemoglobin polymerization in solutions and in red cells. Only a very brief discussion of the relation between the polymerization process and sickle cell disease is given, as we have presented a much more extensive description of this subject elsewhere (Eaton and Hofrichter, 1987). A broader treatment of sickle cell disease, including genetic and clinical aspects, can be found in two excellent books (Serjeant, 1985; Bunn and Forget, 1986) and a recent review (Schechter et al., 1987). This article is divided into five major sections. In Section I1 we discuss the structure of the hemoglobin S molecule, the structure of the various aggregated forms of hemoglobin S, and the structural analysis of the polymers. Section I11 is concerned with the thermodynamics of hemoglobin S polymerization, and includes a description of the nonideal behavior of concentrated hemoglobin S solutions and the effect of physiologically relevant variables, especially oxygen, and the presence of non-S hemoglobins on the polymerization process. Section IV is devoted to kinetic studies on solutions of purified hemoglobin S. In this section there is a summary of the major kinetic results and a detailed description of the double nucleation mechanism that has been successful in explaining the kinetics. In Section V we utilize the results of the thermodynamic and kinetic studies of solutions to explain various properties of cells, including morphological and rheological properties. Finally, in Section VI we present a very brief description of the impact that the polymerization studies have had on our understanding of the pathophysiology and therapy of sickle cell disease. We shall see that a large part of the success in understanding the polymerization process can be attributed to the application of many different physical methods, including electron microscopy, single-crystal X-ray diffraction, X-ray fiber diffraction, nuclear magnetic resonance, linear and circular dichroism spectroscopies, linear birefringence, elastic and quasi-elastic light scattering, microspectrophometry, analytical ultracentrifugation, calorimetry, laser photolysis and temperature jump kinetics, viscometry, and a variety of rheological methods. Before beginning, it is useful to present a few essential facts for readers who are unfamiliar with hemoglobin S or sickle cell disease. I n hemoglobin S, the mutation from GTG to GAG in the triplet coding for the sixth position from the N-terminus of the /3 polypeptide chain results in the replacement of a negatively charged glutamate with the neutral,

66

WILLIAM A. EA'I'ON A N U J A M E S HOFHICHTER

FIG. 1 . 1 . Scanning electron micrographs of sickle red cells. [From White (1974).] All cells (B-I,) are deoxygcnated except for the cell in A, which is oxygenated and has the normal biconcave disk shape. (SccWhire (1974) fur further description of the niicrographs and the abbreviations therein.]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

67

hydrophobic residue, valine. This residue is on the molecular surface, and does not alter the functional properties of hemoglobin S in dilute solution. The abnormal behavior of this mutant molecule is not evident until it is concentrated to levels close to those found in red cells (>0.2 g/cm3).On deoxygenation at these high concentrations, hemoglobin S aggregates after a delay period into a viscous or solidlike gel composed of long multistranded helical polymers. T h e polymers aggregate in bundles or domains of various sizes, distorting the red cell into a wide variety of bizarre shapes. Some of these are shown in Fig. 1.1. If deoxygenation is carried out slowly, many cells assume the shape of a sickle, from which the disease received its name. On reoxygenation, the polymers disassemble without a delay period, and the cells usually resume their normal biconcave disk shape. Although normal red cells have diameters that are larger than the capillaries of the microcirculation, they are quite flexible and readily distort in passing through. Polymerization of hemoglobin S produces a much more rigid cell that may not be able to traverse the narrow vessels of the microcirculation. This decreased flexibility can lead to a transient or permanent blockage of a microvessel and therefore a decreased oxygen supply to the surrounding tissues. The resulting organ damage is a major cause of the morbidity and mortality of the disease, and is presumably responsible for the extremely painful episode known as a sickle cell crisis. Because of the damage to the red cell membrane that results from the many sickling-unsickling cycles, sickle red cells are more fragile than normal cells and are more readily removed from the circulation. The decreased survival of sickle red cells results in a marked anemia, but the consequences of this anemia for the patient are not nearly as serious as those resulting from the repetitive microinfarcts caused by vaso-occlusion in almost every organ of the body. 11. STRUCTURE OF HEMOGLOBIN S MOLECULEAND GEL

By about 1960 it had been clearly established that the primary structures of hemoglobins A and S differed by the substitution of a valine for the normally occurring glutamate at the p 6 position (Pauling et al., 1949; Ingram, 1959).' From this point, structural studies on hemoglobin S

' The results of early chemical studies were somewhat confusing. The difference of 0.22 pH units in the isoelectric points of both the carbonmonoxy and deoxy forms suggested that hemoglobin S carried 2 to 4 more net positive charges than hemoglobin A (Pauling et al., 1949). Shortly thereafter,however, Pauling et al. (1950), in aone-paragraph note with no details, reported that they could detect no difference in the number of acidic

68

WILLIAM A . EATON AND.JAMES HOFKICHTEK

have focused on two primary questions. What is the effect of the valine substitution on the protein conformation? What is the structure of the aggregated hemoglobin S that forms inside sickle cells? T h e structure of the hemoglobin S molecule has been studied by single crystal X-ray diffraction, and by circular dichroism and proton magnetic resonance spectroscopies, while the structure of the hernoglobin S aggregate has been investigated by X-ray diffraction, electron microscopy, optical microscopy, polarized absorption spectroscopy, and copoly~nerizationstudies with other mutant hemoglobins. Our discussion of the structure in this section is divided into three parts. In Section 11,A we discuss the structure of the hemoglobin S molecule. We shall see that the three-dimensional structure of the molecule in the crystal is found to be identical to that of normal hemoglobin, except that there is a displacement of the A helix in one of the /3 subunits of the tetramer which is induced by its involvement in an intermolecular contact. The spectroscopic studies of the molecule in solution show that the valine substitution produces only very small changes in the conformation, which are most probably localized to the /36 region. This conclusion is consistent with the finding that the functional properties of the hemoglobin S molecule in dilute solution are normal, with no differences in either the therniodynamics or kinetics of ligand binding. The determination of the structure of the hemoglobin S aggregate, described in Sections II,B and II,C, has proved to be a much more complicated and difficult problem. It has provided a fascinating challenge to structural biochemists and is still an evolving subject. T h e fundamental structural unit of the aggregate that forms in gels and in sickled cells is a polymer that looks like a long cylindrical fiber. The fiber consists of 14 intertwined helical strands, which can be subdivided into 7 pairs ~~

~

or basic amino acid residues. Pauling et al. (1950) therefore suggested that there may be sinall differences in the number of neutral residues, which could cause a difference in folding of the polypeptide chain and thereby alter the ionization ronstants of acidic o r basic groups (a very novel idea i n 19.50). Subsequent amino acid analyses of the whole molecule also failed to detect any differences in the number of acidic or basic amino acid residues (Schroeder ul d.,19.50; Huisman el ul., 1955). T h e question of the chemical differenre hetween hemoglobins A and S was not resolved until Ingram (1956, 1959) developed thc much more sensitive technique of pcptide “fingerprinting,” and showed that in hemoglobin S a glutamate is replaced by a valine in the sixth position from the N-terminus in each p chain. Since a glutanlate residue normally has a negatively charged carboxyl group at the isoelectrir pH (- 7) of hemoglobin (Antonini and Brunori, 197 1). while valine has a neutral side chain, this chemical difference immediately explained the increased net positive rharge observed for hemoglobin S in the classic electrophoresis experiments of Pauling ct al. (1949).

SICKLE CELL HEMOGLOBIN POLYMERIZATION

69

of strands. Each pair of strands has a structure very similar to the double strand formed in the deoxyhemoglobin S crystal. T h e fibers can be found in a variety of spatial configurations, which appear to depend both on their rate of formation and on the shear forces to which the sample has been subjected. In Section II,B we describe the low-resolution features of the several different aggregated forms of hemoglobin S, which include gels, fiber bundles, and crystals, while in Section II,C we describe the structural analysis of the individual hemoglobin S polymers. A . Structure of Hemoglobin S Molecule The most complete and important information on the structure of the hemoglobin S molecule has come from single-crystal X-ray diffraction studies. The determination of the high-resolution, three-dimensional structure of horse methemoglobin showed that the /36 glutamate is on the molecular surface (Fig. 11.l), suggesting that the valine substitution at this position would not have a significant effect on the protein conformation (Perutz and Lehmann, 1968). This view was supported by preliminary X-ray studies on hemoglobin S. The oxygenated forms of hemoglobins A and S crystallize in 2.8 M potassium phosphate buffer to produce crystals with X-ray diffraction patterns having identical unit cell dimensions and the same relative intensities for the reflections (Perutz et al., 1951). The major advance in the X-ray work came with the determination of the crystal structure of deoxyhemoglobin S, first at 5 8, resolution (Wishner et al., 1975) and then at 3.0 8, resolution (Wishner et al., 1976; Love et al., 1978, 1979; Padlan and Love, 1985a,b). These studies have been important in two respects. First, they have produced a complete three-dimensional structure of the deoxyhemoglobin S molecule, and second, as we shall discuss in Section II,C, they have given a detailed picture of key intermolecular contacts which are very similar to intermolecular contacts in the polymer. The deoxyhemoglobin S crystal structure at 3.0 8, resolution has been extensively refined, permitting a detailed comparison to be made with the structure of deoxyhemoglobin A (Padlan and Love, 1985a). To understand this comparison, it is important to point out certain aspects of the arrangement of the hemoglobin molecules in the unit cell of the crystal (Fig. 11.2). The asymmetric unit (the asymmetric unit is the structure from which the entire unit cell may be generated by the rotational and translational symmetry operations of the crystal) contains two complete, tetrameric molecules so that there are four different structures for the a subunits, and four different structures for the /3 subunits. T h e

70

WILLIAM A. EATON AND-JAMES HOFRICHTER

FIG. 11.1. Schematic structure of hemoglobin molecule. (0)Kesidues which inhibit or promote hernoglobin S polymerization. (0)Residues that have been tested in copolymcrization studies but have no effect. [Modified from Dickerson and Geis ( 1 969) with permission of Irving Geis.]

structures of chemically identical subunits differ in detail because each of the eight subunits of' the asymmetric unit makes different intermolecular contacts with its neighboring molecules in the crystal. These intermolecular contacts may alter the conformation of the residues directly involved in the contact relative to the conformation of the molecule free in solution, and could change the conformation of nearby residues as well. T h e eight subunits are labeled l a , , la2,I @ ] , 1&, 2a,, 2a2, 2&, and 2&, where the first number designates molecule 1 or molecule 2 of the asymmetric unit (Fig. 11,2), arid the subscript designates the dimer that is rotated by the 2-fold symmetry axis of the molecule.* In horse nrcthcnioglohin the molecular 2-foldaxis coincides with a 2-fold symmetry axis of the crystal, so that the detailed structure of the a Ip,dimer is required by the crystal symmetry to be identical to that of the aPPPdimer (Perutz, 1970).

SICKLE CELL HEMOGLOBIN POLYMERIZATION

+b

?

71

+b

?

FIG. 11.2. Unit cell of deoxyhemoglobin S crystal. The (Y carbon backbones of the four molecules of the unit cell are shown, with (-) the p subunits and (-) the a subunits. (0)The location of the a carbon of the p6 residue. The b axis is a 2-fold screw axis that interchanges molecules 1 and 2 with molecules 3 and 4. (Courtesy of E. A. Padlan and W. E. Love.)

72

WILLIAM A. EATON AND .JAMES HOFRICHTER

In the deoxyhenioglobin A crystal, on the other hand, there is only one molecule in the asymmetric unit, and therefore two structures for each subunit. In the initial X-ray refinement, however, the structure of each type of subunit was treated as identical (Fermi, 1975; Fermi et al., 1984), resulting in one set of coordinates for the a subunit arid one set for the /3 subunit. Comparisons of each of the eight deoxyhemoglobin S subunits with the symnietry-averaged subunits of deoxyhemoglobin A show that the biggest structural change is a shift of the A helices of t w o of the four /3 subunits of deoxyhemoglobin S in a hingelike motion (Fig. 11.3b). T h e shift results in a 5 A displacement of the a carbons of the N-terminal valine and a narrowing of the 2,3-diphosphoglycerate (DPG) binding p0cket.j 'This change occurs in the Ip2 and the 2 P p subunits which are involved in an intermolecular contact with the E and F helices of p subunits in neighboring molecules (2pI and lp,, respectively, Fig. 11.2). No change is observed in the EF region of lp, , but a significant change in conformation is observed for Asp-73 of 2P1, which is nioie intimately involved in the intermolecular contact than Asp-73 of lp,. Since no change in the A helices is observed in the lp, and 2/3, subunits, the displacement observed in the I & and 2 P p subunits presumably results from the formation of the intermolecular contacts in the crystal, and is not a direct result of the substitution of valine at p6. Indeed, a comparison of the p6 region of the lp, and 2p1subunits with deoxyhemoglobin A shows no significant differences (Fig. 11.4). Aside from the changes at the p6 contact region, it is not possible to assign any significance to other dif'ferences in atomic coordinates between deoxyhemoglobins A and S, given that the crystal structure is only determined to a resolution of 3.0 A. When the structures of the deoxyhemoglobin S and A tetramers are compared, the root-mean-square deviation in the positions of homologous residues which are not involved in intermolecular contacts in the crystal is 0.9 A for all atoms and only 0.55 A for the main chain atoms, while comparison of' the two deoxyhemoglobin S tetramers of the asymmetric unit yields corresponding values of 1.2 and 0.7 A (Padlan and Love, l985a). -1hat is, the structures of the two deoxyhemoglobin S molecules in the asymmetric unit are found to differ more than either one of them dif'fers from the structure of deoxyhemoglobin A. DPG was added to the crystals after they were already formed. The narrowing of the binding pocket from the shift of the P-subunit A helices in both tetramers could explain the back of Dl'G biriding in rhe crystal (Padlari and Love, 1985a).

SICKLE CELL HEMOGLOBIN POLYMERIZATION

a

73

P6

FIG. 11.3. Comparison of a carbon backbones of p subunits of deoxyhemoglobins A and S. (-) Deaxyhemoglobin S and (-) deoxyhemoglobin A. (0)T h e location of the a carbon of the p6 residue. (a) Comparison of the structures of the lp, subunit of deoxyhemoglobin S arid the p subunit ofdeoxyhemoglobin A. The p6 residue of the lpIsubunit does not participate in an intermolecular contact in the crystal lattice (Fig. 11.2). T h e two structures are identical to within the errors in the data. (b) Comparison of the structures of the 1p2subunit of deoxyhemoglobin S and the /3 subunit of deoxyhemoglobin A. T h e 1p2subunit of' deoxyhemoglobin S contains the p6 residue that participates in an intermolecular contact in the crystal. The only significant difference in the two structures is a shift in the A helix. (Courtesy of E. A. Padlan and W. E. Love.)

74

WILLIAM A. EATON AND JAMES HOFRICHTEK

b

FIG. 11.4. Comparison ot'deoxyhemoglobins A arid S in PS region. (a) Ip: versus PA and (b) 2py vcrsus PA.Residues of the A helix are shown tor the P subunit of deoxyhcmoglobin A (-) and the IPI and 2PI subunits of deoxyhemoglobin S (-) after a leastsquares superposition of the a carbon backbone. The lP1and 2PI subunits of deoxyhemoglobin S are the /3 suhunits in which the PS residue does riot participate in an intermolecular contact (see Fig. 11.2). (Courtesy of F.. A. Padlan and W. E. Love.)

An important result of the X-ray studies, then, is that the substitution of the valine at /36 for glutamate is not observed to have a significant effect on the protein conformation in the crystal, and is therefore not expected to alter the conformation in solution. Comparison of the optical rotatory dispersion and circular dichroism spectra of hemoglobins A and S in solution supports this conclusion. Measurements have been made in the wavelength range 200 to 700 nm in both the deoxy and the oxy forms, and no differences could be detected for the intact tetramers (Li and Johnson, 1969; Yip et al., 1974; Fronticelli, 1978) o r the isolated

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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/3 subunits (Fronticelli, 1978). These results show that there are no large

differences in protein conformation, but the measurements are not sufficiently sensitive to detect changes involving only a few amino acid residues. Changes are observed in incomplete molecules, such as the heme-free p subunit or the p(1-55) peptide. In these molecules the valine substitution produces substantial differences in secondary structure (Fronticelli and Gold, 1976; Fronticelli, 1978). For example, the ps(1-55) peptide in aqueous solution is calculated from the far ultraviolet circular dichroism to contain less a helix and more p conformation (10% a helix and 30% p conformation) than the PApeptide (20%a helix and 5% p conformation). With the addition of methanol to 90% by volume, the p conformation disappears and the a helical contents are calculated to be about 50% for the Ps(1-55) peptide and about 70% for the PA(1-55) peptide. In the X-ray structure of the hemoglobin A and S tetramers 80% of the first 55 p residues are helical and there is no /3 conformation. It is interesting in this regard that an increase in /3 conformation for the ps(1-55) peptide might be expected under some solvent conditions from empirical consideration based on correlations of amino acid sequence and protein secondary structure (Chou, 1974). The most detailed structural comparison between hemoglobin S and hemoglobin A in solution is potentially available from measurements of nuclear magnetic resonance spectra. Comparisons have been carried out in those regions of the spectrum where resonance lines from single protons or small groups of protons are resolvable (Fung et al., 1975; Russu and Ho, 1980, 1982). In deoxyhemoglobin, which has 4 unpaired electrons on the iron, there are no differences in the frequencies of the 25 (per a@ dimer) paramagnetically shifted protons (Fung et al., 1975). These resonances have not yet been assigned, but the paramagnetic shift indicates that they belong to the heme or to residues close to the iron, suggesting that the structures of the two molecules near the heme are identical in the deoxy form. The proton magnetic resonance spectra of the carbonmonoxy and oxy forms, which are diamagnetic, show a number of resonances that are shifted upfield from the ring currents of the porphyrin and aromatic residues. Of these only the y-methyl protons of the a62(E11) and P68(E11) valines have been assigned. These ringcurrent shifted protons show identical spectra for hemoglobins A and S, indicating that there is also no effect of the valine substitution in the region of the protein close to the heme groups in the liganded form. Measurements of exchangable protons show that the spectra of protons involved in hydrogen bonds at the a@interface and at the interface between ap dimers are the same as well (Fung et al., 1975). The only differences in the nuclear magnetic resonance spectra of

76

WILLIAM A. EATON AND JAMES HOFRICHTEK

hemoglobins A arid S that have been observed are for the histidine protons. The proton bound to the C-2 carbon of the imidazole ring is sufficiently deshielded by the two adjacent nitrogens that these resonances are shifted downfield from most of the aromatic protons, and the resonance from individual protons can be observed. The C-4 protons of the imidazole ring are also sometimes observable as single proton resonances. These resonances are sensitive to the local electronic environment, and the chemical shifts differ by about 0.3- 1.0 ppm for the protonated and unprotonated imidazole ring. By measuring the spectra as a function of pH, the pK values for individual histidines have been deand Ho, 1982). termined (Fig. 11.5) (RUSSU There are 38 histidines in the hemoglobin tetramer, 20 in the a subunits and 18 in the p subunits. Since the molecule has a 2-fold axis of symmetry 19 C-2 protons are potentially observable in the spectrum. In the deoxy form 10 resonances are titratable, that is, exhibit monotonic shifts with pH, while 1 1 resonances are titratable in the carbon monoxide complex (Russu and Ho, 1982). Resonances may not be titratable because they are buried in the center of the protein and are not accessible to solvent, because they are broadened as a result of being firmly anchored in the structure and thereby have the same slow correlation time as the entire molecule, or because they are shifted from paramagnetic and ring current effects. Only 5 of the 10 observed histidine resonances have been assigned in the deoxy form (p2, pl16, pl17, p143, and p146), and only 2 in the carbonmonoxy form (p2 and p146). In the case of the carbon monoxide complex, only the p2 histidine shows a pK change of more than 0.1 unit (ApK = 0.13), while 2 other resonances show 0.09 unit pK shifts (p146 and an unassigned resonance). The p2 histidine is the closest of all of the histidines in the molecule to the negatively charged carboxylates of the p6 glutamate (-13 A), but a quantitative analysis of the electrostatic ef'f'ect predicts that the valine substitution will only decrease the pK of the p2 histidine by about 0.01 units (Matthew, 1978). Comparison of the results for deoxyhemoglobins A and S shows that there are more differences, but their significance is not clear. In addition to the C-2 resonance of the p2 histidine, 5 other resonances show differences of pK between 0.09 and 0.28 units, including the C-2 resonance of the pi46 histidine of deoxyhemoglobin S (ApK = 0.28) (Fig. 11.5). The interpretation is complicated by the fact that, except for 62, these same resonances show alterations in the longitudinal relaxation times only on increasing the temperature or concentration (Russu and Ho, 1980), conditions which favor the aggregation of hemoglobin S. This result suggests that the altered pK values may not result from confor-

77

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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FIG.11.5. Proton magnetic resonance titration of histidines in deoxyhemoglobins A and S. (a and b) 1H NMR spectra (250-MHz) of 0.1 g/cm3 solutions in 0.1 M Bis-Tris buffer at 27°C. (c) Titration curves for C-2 resonances of p 2 histidine of deoxyhemoglobin S [points ( X ) and solid curve (-)I and deoxyhemoglobin A [dashed curve (---), points not shown]. (d) Titration curves for C-2 resonances of p146 histidine of deoxyhemoglobin S [points ( x ) and solid curve (-)I and deoxyhemoglohin A [dashed curve (---), points not shown]. [From Russu and Ho (1982).]

mational changes in the isolated molecule, but from aggregation (Russu and Ho, 1980, 1982). Once complete assignments are made, more extensive measurements might be used to map regions on the molecular surface that are involved in intermolecular contacts in the initial aggregation steps. A comparison of the functional properties of hemoglobins S and A provides another very important and sensitive test of the effect of the /36

78

WILLIAM A. E A l O N A N D J A M E S HOPRICIITER

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In x FIG. 11.6. Comparison of oxygen-binding curves for hemoglobins A and S at 25°C in 0.15 M potassium phosphate buffer (pH 7.2). The fractional saturation (7)is plotted versus the natural logarithm o i the oxygen pressure in torr (In x). (0) Hemoglobin S at 0.15 g/ C I I I ~ ,( X ) hemoglobin A at 0.16 g/cml, and (0)hernoglobin A at 0.35 g/cmg.IFrorn Gill et nl. (1979).]

mutation. When oxygen binding is studied at concentrations which are too dilute for polymerization to occur, not only are the binding curves superimposable (Fig. 11.6) (Allen and Wyman, 1954; Kossi-Bernardi et al., 1975a,b; Gill et al., 1979), but the effects of pH, DPG, and CO, on the overall affinity (i.e., p50s) are also found to be identical (Bunn, 1972). In a related experiment the rate of dissociation of tetramers into dimers is found to be approximately the same for hemoglobins A and S (Ip et al., 1976). The kinetics of ligand binding and dissociation have also been carefully compared and show n o differences (Pennelly and Noble, 1978) (Fig. 11.7). These functional studies argue strongly that the j36 substitution has no widespread effect on the protein conformation. We shall see in Sections I11 and V that the finding of normal functional properties for the unaggregdted hemoglobin S molecule considerably simplifies the analysis of experiments on oxygen binding to hernoglobin S gels and to sickle cells. There is one property of' hemoglobin S that differs significantly from hemoglobin A in dilute solution. The oxygen and carbon monoxide complexes of hemoglobin S precipitate much more rapidly than the corresponding complexes of hemoglobin A on mechanical shaking (Asakura et al., 1973, 1974a,b; Roth et al., 1975). The deoxyhemoglobin S molecule, on the other hand, is relatively stable. The instability of the liganded molecule presumably results, at least in part, from an increased

79

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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9

PH

FIG. 11.7. Comparison of kinetics of ligand binding and dissociation for hemoglobins A and S. All measurements were made at 20°C at a total anion concentration of 0.05 M [see Pennelly and Noble (1978) for details of buffers and concentrations]. (0)Stripped hemoHemoglobin S and (m) globin A and (0)stripped hemoglobin A plus 0.1 mM DPG. (0) hemoglobin S plus 0.1 mM DPC. (a) Rate of oxygen dissociation (k), (b) rate of oxygen dissociation from the fully liganded molecule (h), (c) rate of carbon monoxide binding ( l ' ) , (d) rate of carbon monoxide dissociation from fully liganded molecule ( I i ) , and (e) rate of carbon monoxide binding to triliganded molecule ( 1 ; ) . [From Pennelly and Noble (1978).]

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WILLIAM A . EATON A N D JAMES HOFKICHTER

rate of unfolding at the air-water interface (“surface denaturation”) (Asakura et al., 197413). This instability of oxy- and carbonmonoxyhemoglobin S can be of practical importance in that the exposure of the liganded protein to excessive stirring must be minimized while preparing samples. H . Agyregated Form of HPmoglobin S

Hemoglobin S can aggregate into a number of macroscopic forms. The gel is the highly viscous, semisolid material obtained by deoxygenating a concentrated oxyhemoglobin S solution, or heating an already deoxygenated solution in the absence of continuous stirring. The gel is believed to be the physiologically relevant form because it has the structural characteristics of the material that forms inside deoxygenated sickle cells. In both the gel and in sickled cells the basic structural unit is a fiber having a diameter of about 21 nm and a variable length. Two other aggregated forms of hemoglobin S are important because they have played a major role in the structural analysis of the fibers. One is the three-dimensional single crystal, from which the X-ray structure of the deoxyhemoglobin S molecule was determined (Section I17A),and the second is the fiber bundle, a paracrystalline form obtained from stirred solutions that has been important for obtaining high-resolution electron micrographs. In this section, we describe some general properties of the various aggregated forms, and in the following section (Section I1,C) we present the detailed structural analysis of the individual polymer. T h e rheological properties of gels are discussed in Section V7D. It is instructive to begin the discussion of this topic with some historical perspective. The first evidence that deoxyhemoglobin S assembles into some type of ordered structure was the observation of birefringence in sickled red cells (Sherman, 1940) (Figs. 11.8 and 11.9). Measurements showing that the wavelength dependence of the birefringence is very similar for sickled cells and deoxyhemoglobin A single crystals, together with the observation of a lower solubility for deoxyhenioglobin S compared to deoxyhemoglobin A in concentrated phosphate buffers, led to the proposal that deoxyhemoglobin S is actually crystallizing within the sickled red cells (PerutL and Mitchison, 1950; Perutz et al., 1951). At about the same time it was suggested that the ordered structure responsible for the linear birefringence is not a three-dimensional single crystal, but instead is a liquid crystalline array made up of “long thin rodlike particles which are arranged parallel and equidistant to each other” (Harris, 1950), similar to the nematic liquid crystalline arrays seen in moderately concentrated solutions of tobacco mosaic virus (Bernal and

FIG.11.8. Optical micrographs of sickle cells deoxygenated in the presence of a magnetic field. A sealed preparation of' oxygenated sickle cells was allowed to deoxygenate spontaneously at room temperature in the presence of a 10,000-gauss magnetic field over a period of about 10 hr. Many of these cells show the classic sickle shape. The field direction is vertical and is perpendicular to the long axes of most of the cells. Partial alignment occurs because of the anisotropy in the magnetic susceptibility of the polymers (Costa Ribeiro et al., 1981). (a) Cells in 430 nm light. (b) Cells in 450 nm light between crossed linear polarizers oriented at 45" to the vertical and horizontal directions.

FIG. 11.9. Optical micrograph of a gel of deoxyhemoglobin S between crossed linear polarizers oriented at 45" to the horizontal and vertical directions.

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WILLIAM A. EATON A N U J A M E S HOFKICHTEK

Fankuchen, 1941). These early characterizations were followed by a series of studies to examine the structure of sickled cells and gels more closely using electron microscopy (Bessis et al., 1958; Stetson, 1966; Murdyama, 1966; White, 1968; White and Heagan, 1970a-c; Dobler and Hertles, 1968; Bertles et al., 1970). The results of these studies supported the contention that gels and sickled cells contain arrays of polymers with diameters of 17- 19 nm rather than three-dimensional single crystals (see the review by White, 1974). A variety of arrangements of’ these polymers were observed (fiber and polymer are used interchangeably). In sickled cells these included bundles of parallel fibers with axes coinciding with the long axis of the cell, bundles of curved fibers, and completely randomly oriented fibers (Fig. 11.10). In gels formed by adding concentrated sodium phosphate to deoxyhemoglobin S solutions at 0.08-0.13 g/cm3 spherical clusters of radially oriented polymers were routinely observed (White and Heagan, 1970~).‘l’his arrangement of polymers could explain the observation using a polarizing optical microscope that some sickled cells exhibit characteristic dark crosses, the arms of which remain parallel to the polarizer and analyzer axes when the cell is rotated (Bessis et al., 1958) (Fig. 11.9). The radial distribution of polymers causes this birefringence pattern by producing a radial distribution of principal optical directions. The more recent electron microscope studies, discussed below, have been mainly concerned with elucidating the structure of the individual polymers, and there is still very little quantitative information on the organization of polymers in gels. Some insight into the variability in gel structure and the gross morphology of sickled cells has come from observations with a polarizing microscope in the course of kinetic studies. The organization of hemoglobin S polymers, both in gelled solutions and in cells, is generally found to be in the form of birefringent domains with a birefringence pattern for the gels indicating that the polymer long axes are pointing in a radial direction (Fig. 11.9). The size arid number of these domains are determined by the time scale over which polymerization occurs. T h e largest domains, formed on a time scale of hours by slowly heating a deoxyhemoglobin S solution, can be several millimeters in diameter, while domains formed by rapid chemical reduction of oxyhemoglobin S or laser photolysis of carbonnionoxyhemoglobin S in times of 100-500 msec are considerably smaller than a red cell, and are typically microns or less in diameter (see Fig. IV.14). The detailed arrangement of polymers in these domains is still not known. From measurements on large domains the linear dichroism is found to be nearly independent of the distance from the center of the domain. This observation requires that the polymers have lengths which are small compared to the radius of the

SICKLE CELL HEMOGLOBIN POLYMERIZATION

83

FIG. 11.10. (A-F) Electron micrographs of deoxygenated sickle cells showing the variation in the arrangement of fibers. [From White (1974); see also for further description of the micrographs and the abbreviations therein.]

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WILLIAM A. EATON AND JAMES HOFKICHTER

domain, since otherwise the polymerized hemoglobin would become more dilute as the distance from the center ofthe domain increases and the linear dichroism would decrease (Eaton and Hofrichter, 1981). An additional aspect of' gel structure that may influence the morphology of cells comes from the observation that polymers in cells sometimes orient in layers, with adjacent layers rotated by about 26" (Edelstein and Crepeau, 1979). This result suggests that there are specific interactions between polymers which may consist of interlocking of helical grooves in the polymer (see Section I1,C) (Edelstein and Crepeau, 1979). From the limited information on the domain structure of gels it is possible to make some interesting speculations about the morphology of sickled cells. The classic sickled appearance of red cells produced by slow deoxygenation probably results in the formation of domains which, if not limited by the small amount of hemoglobin in a single cell, would be much larger than the cell itself. The constraints of the cell membrane might permit growth of a domain in one general direction, resulting in a cell composed mainly of approximately parallel polymers. According to this description the enormous variety of shapes of deoxygenated sickle cells results, at least in part, from differences in the number of domains (Fig. I. 1 ; see also Fig. V. 18). A closely related, but physically distinct form of polymerized hemoglobin S is obtained by stirring deoxyhemoglobin S solutions while they are being heated to produce aggregation (Pumphrey and Steinhardt, 1976, 1977). Instead of a viscous gel, a freely flowing suspension of elongated aggregates (fascicles) is obtained. These fascicles are found by electron microscopy to be bundles of parallel fibers packed in square o r hexagonal arrays (Crepeau et al., 1978; Wellenis and Josephs, 1979; Carragher P t al., 1988a) (Fig. 11.1 1). The more detailed structure of the fibers in these bundles, which is discussed in the next section, is indistinguishable from the structure of the polymers found in sickled cells (Dykes et al., 1979), and is presumably the same as that formed in unstirred gels of hemoglobin S. If the stirring is continued for many hours, the bundles of fibers are replaced by macrofibers, and finally by needlelike crystals, which are true three-dimensional crystals (Wellems and Josephs, 1979; Wilson and Makinen, 1980; Wellems et al., 1981; Vassar et al., 1982; Makinen and Sigountos, 1984; Rluemke et al., 1988). Electron microscopy and low-angle X-ray scattering indicate that these crystals are most probably identical to the three-dimensional single crystals used in the determination of the deoxyhemoglobin S molecular structure (Wellenis and Josephs, 1979; Wellems P t al., 1981) (Section 11,A). Crystals have also been observed to grow from gels formed in the absence of stirring or any other shear forces after periods of' months

FIG. 11.1 1. Electron micrographs of fascicles formed by stirring a deoxyhemoglobin S solution. (a) Fascicles formed by stirring a deoxyhemoglobin S solution, fixing with glutaraldehyde, and negatively staining with phosphotungstate. [From Wellems and Josephs (1979).] (b) Cross section of fascicle obtained by stirring a deoxyhemoglobin S solution, fixing with glutaraldehyde, and staining with osmium tetroxide. (Courtesy of S. J. Edelstein.)

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WILLIAM A. EATON AND.JAMES HOFRICHTER

(Hofrichter Pt al., 1976b) or years (Magdoff-Fairchild and Chiu, 1979), and the crystals used in the X-ray study also apparently were formed subsequent to gel formation (Wishner et al., 1975).

C. Structure of Hemoglobin S PolymPr From the discussion in the preceding section, it is clear that hemoglobin S can aggregate to form a variety of structures. In spite of this apparent complexity, however, there seems to be only one predominant polymeric structure that forms in cells, in gels, arid initially in stirred solutions. This is a solid fiber having a diameter of 2 1 nm and a variable length (Josephs et al., 1976; Crepeau et al., 1978; Dykes el al., 1979; Garrell et al., 1979; Wellems and Josephs, 1979; Carragher et al., 1988a). Other structures have occasionally been observed, including a 17-nm hollow fiber (Finch et al., 1973; Ohtsuki et al., 1977), but the 21-nm fiber is by far the most frequently occurring form in both cells and lysates (Dykes et al., 197Y), and it is the only one observed in preparations from purified hemoglobin S (Wellems and Josephs, 1979). T h e average diameter of 2 1 nm has been measured from center-to-center distances in cross-sectional views (Fig, 11.1 lb), as well as from longitudinal sections after correcting for flattening of the fiber on the electron microscope grid (Crepeau et al., 1978; Garrell et al., 1979). In addition, X-ray diffraction patterns of gels, discussed further below, contain three lowresolution equatorial reflections at spacings of 22, 11, and 7.3 nm-' (Magdoff-Fairchild and Chiu, 1979), which are consistent with a square lattice of polymers separated by 22 nm, in good agreement with the results of electron microscopy. The smaller diameters of 17-19 nm reported in earlier electron microscope studies (White, 1974) were most probably underestimated because they were measured on individual fibers where the edges were not clear (Dykes et al., 1979). A very important result is the finding that the same 21-nm fiber is the predominant polymer structure under a variety of solution conditions. A systematic comparison shows that the fibers observed in negatively stained electron micrographs exhibit the same diameter, appearance, and optical diffraction patterns over a wide range of pH (6.2-7.4), temperature (17-37"C), and ionic strength (0.05- 1.0 M ) (Wellems and Josephs, 1979). T h e determination of the high-resolution structure of the 2 l-nm fiber has required original approaches to structure analysis and a combination of a variety of techniques. For the purpose of this discussion, it is convenient to divide the analysis into three stages which are concerned with increasing levels of structural detail. The first stage concerns the general

SICKLE CELL HEMOGLOBIN POLYMERIZATION

87

packing arrangement of the hemoglobin molecules in the fiber, i.e., the determination of the polymer lattice. T h e result is a model for the fiber, consisting of 14 intertwined helical strands in which the position of the center of each hemoglobin molecule is known approximately. In this model, each hemoglobin molecule is treated as a structureless sphere. The next stage in the structure analysis has involved the accumulation of evidence to show quite convincingly that the polymer contains structures which are very similar to the double strand found in the deoxyhemoglobin S crystal. This is an extremely important result, because the single-crystal X-ray structure of the double strand provides a detailed atomic picture of the intermolecular contact involving the p6 valine which can then be inferred to be nearly identical to the p6 contact in the polymer. The results of several different experiments, including optical studies on single sickle cells and X-ray studies on gels, support the conclusion that the double strand is contained in the polymer. T h e most compelling evidence comes from extensive copolymerization studies of hemoglobin S with other mutants. In these studies, the effects of single amino acid substitutions on the molecular surface can be used to “map” sites that participate in intermolecular bonding, and it is found that these results agree extremely well with predictions from the X-ray structure of the double strand. The current stage of the structural analysis is concerned with determining how the 14 strands of the polymer should be constructed from 7 double strands in order to build a detailed threedimensional model of the polymer. In the following we describe each of these three stages of the structural investigation of the 2 1-nm fiber, beginning with the electron microscope studies of the polymer lattice. A major problem encountered in examining hemoglobin S polymers by electron microscopy is that the high concentration of protein necessary for aggregation is incompatible with the dilute solutions required for obtaining high-resolution images. The best images have been obtained by lysing and negatively staining cells in one step with phosphotungstate on the microscope grid (Fig. II.12), or washing fiber bundles obtained from stirred deoxyhemoglobin S solutions with phosphotungstate (Josephs et al., 1976; Dykes et al., 1978, 1979). Figure 11.13 shows electron micrographs of three negatively stained fibers that were used in the image reconstruction analysis (Dykes et al., 1978, 1979). These fibers show a periodic variation in their apparent diameter of 2 1 to 27 nm, which, after correcting for a 15%flattening on the microscope grid, becomes 18 to 23 nm. The fine structure observed in these micrographs suggests that individual hemoglobin molecules are being resolved. A more detailed analysis of the polymer structure requires the application of two techniques which have been used in other structure deter-

88

WII,I.IAM A . EATON A N D JAMES HOFRICHTER

FIG. 11.12. Elertron micrograph of cell lysed with negative stai11. The cell was lysed on the microscope grid with 1% phosphotungstate, which has penetrated to contrast individual fibers. The cell membrane at the upper right has ruptured, showing fibers 011 the outside ot thc cell. (Courtesy of'S. J . Edelstein.)

minations to reduce the noise inherent in electron micrographs (Misell, 19'78).The first of these techniques is a real-space enhancement method which is most easily explained by considering an electron micrograph o f a sectioned crystal. In such a specimen there are a large number of identical images of the same basic structural element. This element is the unit cell of the crystal which can be used to generate the structure of the entire crystal by simple translations. Since the noise in the electron micrograph is random, the image can be enhanced by averaging the images from a large number of unit cells to produce a resultant image with an improved signal-to-noise ratio. The second technique is one which allows selective elimination of high-resolution information in order to permit the low-resolution features of the polymer to be visualized more clearly. This procedure involves a calculation of the Fourier transform of the image, which consists of a set of maxima. Since the resolution in the original image is determined by the displacement of the maxima of the Fourier transform from the origin, filtering for low-resolution data can be done by simply eliminating the high-resolution maxima and then

89

SICKLE CELL HEMOGLOBIN POLYMERIZATION

a

b

C

d

e

FIG. 11.13. Electron micrographs of fibers and two-dimensional image reconstructions. (a) Negatively stained fiber prepared by cell lysis, (b and c) negatively stained fibers prepared from stirred deoxyhemoglobin S solutions, (d) filtered image produced by an inverse Fourier transform of all of the maxima in Fig. 11.14, and (e) filtered image produced by an inverse Fourier transform of the maxima on layer lines 1 to 6 only. T h e areas used for reconstructions are enclosed. [From Dykes et al. (1979).]

performing an inverse Fourier transform to produce the filtered realspace image. The positions of the individual hemoglobin S molecules in the polymer have been determined by a novel combination of these real-space and Fourier-space reconstruction methods (Dykes et al., 1978, 1979; Edelstein, 1980, 1980-1981, 1981a).4The final results of the full threeThe procedure consisted of first obtaining a digital representation of the intensities in the image, and performing a numerical Fourier transformation that produced a complicated set of maxima in reciprocal space (Fig. 11.14). A major difficulty in solving the structure is that the lattice is complex, producing a large number of maxima with relatively low intensities, compared to a simple helical structure which would have many fewer and more intense maxima. After excluding the maxima on all but the first six layer lines (corresponding to a resolution of about 30 nm in the direction of the fiber axis), the inverse

90

WI1,I.lAM A. EATON A N D J A M E S HOFKICHTER

FIG.11.14. Numerical Fourier transform of electron micrograph of hemoglobin S fiber. [From Dykes ct nl. (1979).]

dimensional image reconstruction are shown as the polymer lattice points in Fig. 11.16 and a model of the polymer in Fig. 11.17 (Dykes et nl., 1979). The polymer has an inner core, consisting of 4 intertwined helical strands, surrounded by an outer sheath of 10 helical strands. Fourier transformation produced a filtered image in which the fiber appears to consist 01 continuous strands. 'l'he strands appear continuous because individual molecules, which have a diameter of about 5.5 nm, are not distinguishable along the fiber axis at 30 nm resolution (Fig. II.13e). In this structure the strands are clearly seen to be helical. 'I'he number and relative positions of the strands were obtained by projecting successive sections of the intensity nnto a surface representing the plane normal to the fiber axis. Each successive section was projected at an angle deterniined by its displacetileiit along the helix from the original section and the helical repeat distance measured for the fiber being reconstructed. Thc results of this prncedure at-c shown in Fig. 11.15. T h e superposition of a large amount of density data provides convincing evidence for a fiber made up of 14 strands (also called hlaments) that are hexagonally packed. Independent electron microscope evidence tor the 14-stranded structure has been obtained from patterns observed in cross sections of fibers that have been embedded and stained with tannic acid (Garrell ct nl., 1979), and also frorri the use of cross-correlation methods that employ all of the maxima of the Fourier transfnrm (Crepeau arid Edelstein, 1984). Fourier trarisfornis of this structure were calculated to be very similar tn the original Lber- transforms for the layer lines close to the equator (the horizontal section through the center o f t h e pattern in Fig. 11.14). These resiilts permitted the observed peaks of the Fourier transform to be assigned to specific maxima in the pattern predicted by helical scattering theory. T h e higher resolution peaks in the transform could then be included in the calculation of the inverse Fourier transform to recoustruct a filtered, high-resolution image of the original fiber.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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FIG. 11.15. Cross section of hemoglobin S polymer from image reconstruction of electron micrographs. [From Dykes et al. (1979).]

Circumference (degrees) FIG. 11.16. Surface lattice of 14-stranded hemoglobin S polymer obtained from image reconstruction of electron micrographs. The centers of each molecule are projected radially onto a cylindrical surface surrounding the polymer. The positions of each molecular center are given in cylindrical coordinates, consisting of a distance along the cylinder axis (Z), the distance from the center of the cylinder (radius), and the angular orientation of the line from the center of the cylinder to the center of the molecule in the plane perpendicular to the cylinder axis. The strands are numbered as in Fig. 11.15, and a single asymmetric unit is enclosed. [From Dykes et al. (1979).]

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WILLJAM A. EATON A N D JAMES HOFRICIITER

b

FIG.11.17. Model of hernoglobin S polymer from image reconstruction of electron micrographs. (0) Each hernoglobin molecule. (a) The ouler 10 strands, (b) the inner core of 4 strands, and ( c ) both the inner and outer strands. [From Dykes et al. (1979).]

Alternatively, the fiber can be described as consisting of a stack of identical, 6.4-nm-thick disks composed of 14 molecules, with each disk rotated by about 7" relative to the one below it. The cross section in Fig. 11.15 shows that the disk is elliptical and that there is an apparent 2-fold axis through its center. The periodic variation in the diameter of the fiber results from the elliptical cross section and the helical nature of the structure.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

93

The second stage in the structure analysis has been concerned with demonstrating that the polymer contains a structure having a packing arrangement very similar to that found in the deoxyhemoglobin S single crystal. Figure 11.18 is a projection of part of the crystal lattice, showing that the crystal can be considered to be composed of pairs of strands, called double strands (Wishner et al., 1975). T h e strand direction corresponds to the a crystal axis, with each member of the double strand containing one of the molecules of the asymmetric unit. An approximate 2-fold screw axis parallel to a relates the two strands, so that one strand is half-staggered relative to its partner in the double strand. Because of the true 2-fold screw axis relating asymmetric units (Fig. 11.2), there is a second double strand running in the opposite direction to that of the first. Each double strand is quarter-staggered relative to its neighboring double strand. The entire crystal, then, can be constructed from simple translations of pairs of antiparallel double strands. An important feature of the double strand is that the contact between the two molecules in adjacent strands (i.e., the so-called “lateral” contact) contains the p 6 valine (Fig. 11.18). Also, the contact between molecules in the same strand (the so-called “axial” contact) contains residues which are known from studies with other mutants to influence polymerization (see below). These findings, together with the early analysis of the electron microscope images indicating staggered strands in the polymer (Josephs et al., 1976), motivated the suggestion that the polymer contains a packing arrangement similar to the double strands of the single crystal (Wishner et al., 1975, 1976). Data from a variety of sources support this hypothesis. A simple, but critical test of the double-strand hypothesis can be made from the results of polarized optical absorption measurements on single sickle cells (Hofrichter et al., 1973; Hofrichter and Eaton, 1976; Eaton and Hofrichter, 1981). In the wavelength region of the intense Soret band (400-450 nm), hemes behave as nearly perfect planar absorbers of linearly polarized light (Eaton and Hochstrasser, 1967, 1968), making it possible to calculate an accurate absorption ellipsoid from the X-ray coordinates of the porphyrin atoms for the hemoglobin molecule (Fig. 11.19) and the double strand. Because the hemes are nearly parallel there is much greater absorption of light polarized parallel to the z o r y molecular axes than the x molecular axis. The x molecular axis is nearly parallel to the a axis of the double strand. By measuring the absorption of linearly polarized light parallel and perpendicular to the fiber axis (taken as the long axis of the sickled cell, Fig. II.20), the average orientation of the a axis of the double strand relative to the fiber axis is determined (see Section V,A for further discussion of this experiment). T h e average angle between the a axis of the double strand and the fiber axis

94

WILLIAM A. EATON AND JAMES HOFRICHTER

FIG. 11.18. Double strand of' deoxyhemoglobin S single crystal. Only the backbone (Y carbons and the porphins are shown. (-) /3 Subunits and (-) (Y subunits. (0) The j3G position. (Courtesy of E. A. Padlan arid W. E. I.ove.)

95

SICKLE CELL HEMOGLOBIN POLYMERIZATION

MOLECULAR PROJECTION

-

25A

-Y

ABSORPTION ELLIPSOID

FIG. 11.19. Optical absorption ellipsoid of the hemoglobin molecule. The extinction coeficient for the absorption of linearly polarized light is proportional to the distance from the origin to the surface of the ellipsoid. y is a true molecular dyad axis, while x and z are pseudo dyad axes. [From Hofrichter et al. (1973).]

is calculated from the optical data to be 6" ( + 14", - 6"),in excellent agreement with the average angle from the electron microscope structure of 9" (Eaton and Hofrichter, 1981). Although these data only provide a consistency test, it should be recognized that the optical results are unambiguous and could have potentially ruled out the double-strand hypothesis [an earlier model of the fiber structure (Murayama, 1966) was discarded on the basis of the optical results (Hofrichter et al., 1973)l. A qualitative analysis of X-ray diffraction data on gels also supports the double-strand hypothesis. The diffraction data for the gel bear a strong resemblance to the single-crystal diffraction pattern, after the crystal pattern is averaged by rotating about the a crystal axis (Fig. 11.21) (Magdoff-Fairchild et al., 1972; Magdoff-Fairchild and Chiu, 1979). If the tilting of the double strand required to form a helically twisted polymer is simulated by superimposing exposures of the crystal diffraction pattern in which the a crystal axis is rotated by 7" relative to a fixed fiber axis, the similarity is further enhanced (Fig. 11.21). This result has been used as a qualitative argument for the similarity of the asymmetric

*

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WILLIAM A. EATON AND.JAMES HOFRICHTER

a

"400

430

460400 430 ABSORBANCE

460

FIG. 11.20. (a) Optical niicrograph of deoxygenated sickle cell in polarized light and (b) polarized absorption spectra. (a) Optical micr_ographsin 430 nni light showing that there is greater ahsorption with the electric vecror (E) of the linearly polarized light perpendicular to the long axis of the cell compared with the short axis of the cell. Electron microscopy has shown that the fibers are invariably arranged with the fiber axis parallel to the long axis of such cells. (b) Polarized absorption spectra for two different cells. The polarization ratio (P) shows a wide cell-to-cellvariation due to differences in the fraction of polymerized hemoglobin (see Section V,A). [From Hofrichter rt al. (1973).]

units in the polynier and the single crystal, suggesting that the polymer also contains structures very similar to double strand^.^ 5 In evaluating these X-ray results one caveat must he considered. The similarity between the averaged crystal pattern and the gel pattern could result from the formation of microcrystals in the gel, which might mask the much niore dift'use scattering by the polymers. I n this case, only the reflections on the equator and the 3.0 nm-' nieridional reflection would derive froni the polymers, and the remaining reflections from the crystals.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

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FIG. 11.21. X-Ray diffraction patterns of deoxyhemoglobin S crystal and gel. (a) X-Ray rotation diagram of deoxyhemoglobin S single crystal, (b) optical smearing of diagram in a to simulate disordering of the crystal, and (c) X-ray pattern of deoxyhemoglobin S gel formed from a lysate. [From Magdoff-Fairchild and Chiu (1979).]

As mentioned in the introduction to this section, the most convincing support for the double-strand hypothesis comes from extensive copolymerization studies of hemoglobin S with other variants having an altered residue on the molecular surface (Bookchin et al., 1967, 1970, 1977; Bookchin and Nagel, 1973b, 1974; Nagel and Bookchin, 1978; Nagel et al., 1979, 1980; Benesch et al., 1978a, 1979, 1982). T h e basic idea of the experiments is to determine whether the site of the amino acid change in the non-S variant is involved in an intermolecular contact in the gel by measuring its effect on polymerization. In this way, the molecular surface of the hemoglobin molecule can be ‘’mapped” for intermolecular contact sites. Most of the experiments have been performed by mixing hemoglobin S (Hb S) with the variant having an alteration at site X (Hb X) and measuring the minimum gelling concentration (Nagel et al., 1980).6 In these mixtures the tetramers dissociate into (YP dimers and reassociate to produce a hybrid molecule in which one (YP dimer contains the P6 valine and the other a@ dimer contains the altered residue of the other variant. If Hb X is a @-chainvariant, the mixture contains the three tetrameric species:

The minimum gelling concentration is the total hemoglobin concentration of the mixture at which a gel first forms. It is determined by concentrating the deoxyhemoglobin mixture with dry nitrogen until the solution ceases to flow. The minimum gelling concentration is higher than the solubility because the solution must be supersaturated for polymers to form. This paint will be explained more fully in Sections 111 and IV.

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WILLIAM A. EATON AND.JAMES IIOFRICHTEK

If Hb X is an a-chain variant:

As a reference state for comparing minimum gelling concentrations and Hb S mixture is used. solubilities the Hb A

+

LYAaA PAPA

~

ffAffA

PSp

-

LYAaA

PAPS

Anticipating the results that only one of the two PS valines participates in an intermolecular contact in the polymer and that the /3 subunit which forms the P6 contact in the hybrid molecule is the Ps subunit, and using the designation from the crystal structure of p2 for the p subunit that participates in the intermolecular contact, the tetramers in the above schemes are in the configuration:

In the hybrid molecules, then, for the P-chain variants the X mutation is on the p, subunit [often referred to as the subunit “trans” to the ps (&) subunit]. For a-chain variants the X site is on the a,subunit. The analysis of these results has been extremely simple (Nagel et al., 1980). If the minimum gelling concentration for an Hb X Hb S mixture is different from that found for an Hb A Hb S mixture at the same fraction of Hb S (usually 0.4), then the site X is considered to be part of an intermolecular contact on the PI (or a,)subunit. If it is the same, the site X is not involved in an intermolecular contact. Since the mixtures contain three niolecular species this simplified analysis is dependent on several important assumptions. First, it must be assumed that the distribution of the three molecular species in the Hb X Hb S mixture is the same as it is in the Hb A Hb S mixtures. This distribution is found to be binomial for Hb S + Hb A mixtures [a 40 :60 mixture contains 16% Hb S tetramers, 36% Hb A tetramers, and 48% hybrid molecules, see Eq. (III.22)], indicating that all three tetramers have the same tetramer-dimer dissociation constants (Bunn, 1972; Bunn and McDonough, 1974). Since the variants have altered residues on the niolecular surface, they are not expected to influence the structure at the interface between QP dimers, nor to alter the tetramer-dimer dissociation constant unless there are significant changes in long-range electrostatic effects (Mrabet et al., 1986). Second, it must be assumed that neither the a,P? nor the a2PP homotetramer copolymerizes. Finally, it must be as-

+

+

+

+

SICKLE CELL HEMOGLOBIN POLYMERIZATION

99

sumed that the polymerization probability for the hybrid tetramers from both the reference molecule, Hb A, and Hb X differ significantly in the two alternative orientations of the hybrid tetramer in the polymer, so that one is always comparing results for molecules in the same orientation. These latter two assumptions are supported by two types of data. The minimum gelling concentrations for mixtures of Hb S with two Ala) and other variants at the p6 position [Hb G Makassar (p6 Glu Hb A mixtures Hb Leiden, deletion at p6] are the same as for Hb S (Nagel and Bookchin, 1975). This result suggests that the p6 position on the p, subunit does not participate in an intermolecular contact. Also, a detailed thermodynamic analysis of extensive solubility data on Hb S + Hb A mixtures, described in Section I I I , D , indicates that there is little or no incorporation of Hb A tetramers into the polymer and that the probability of the hybrid tetramer copolymerizing is about half of that for the Hb S tetramer. The probability of one-half is most readily interpreted as meaning that only one of the two p6 sites forms an intermolecular bond in the polymer, and that the hybrid molecule forms the same noncovalent bonds as the Hb S tetramer in one orientation, but is excluded completely from the polymer when oriented with the p6 Glu at the intermolecular contact site (Fig. 11.22). Since we are only concerned with the qualitative aspects of these experiments, it appears that the assumptions necessary for the simple interpretation of the data are justified, and that comparing the results for the mixtures is equivalent in a first approximation to comparing the results for the hybrid molecules alone. The most unambiguous approach, which would be free of all three of the above assumptions, would be to isolate the hybrid molecules and measure their ~olubility.~ Experiments along these lines have been performed in one study on two a-chain variants (a16 and a47, Table 11.1) which were purified after cross-linking the mixture (Benesch et al., 1982). The results of the copolymerization studies are summarized in Fig. 11.1 and are compared with the predictions from the crystal structure in Table 11.1. For a more detailed analysis Tables 11.2 and 11.3 list the residue pairs in contact within the double strand (Padlan and Love, 1985b). In addition to mapping the surface of the a 1and p1subunits from studies on 5 a-chain variants and 30 p-chain variants (Nagel et al., 1980),

+

' Isolation and preparation of concentrated solutions of the hybrids in the deoxy form is technically difficult. The oxy or carhonmonoxy forms rapidly dissociate into dimers, which will then sort with like dimers when the solution is subjected to some fractionation procedure, with the result that only the pure tetramers are formed. In principle, the hybrid could be prepared by ion-exchange chromatography or isolectric focusing near O"C, where dissociation of tetramers into dimers is very slow (Bunn, 1972), and the solubility is high (see Fig. 111.7).

100

WILLIAM A. EATON A N D JAMES HOFKICHTER

+

+

FIG. 11.22. Copolymerization of hybrid molecule iri Hb S Hb A mixtures. This schematic diagram indicates that the ap/3*PShybrid molecule can only enter the double strand of the polymer having the P S valine in the lateral contact. The orientation with the p6 glutamate in the lateral contact is excluded. (0)A valine at the PS position (hemoglobin S) and (0) a glutamate at the /36 position (henloglobin A).

Table 11.1 also contains the results of studies from 6 variants of the a2 (Renesch el ul., 1982) and p2 subunits (Nagel et al., 1980). The extensive results in Table 11.1 are quite striking. Of the 32 pchain variants studied, 8 have alterations at (7 different) sites that form intermolecular contacts within the double strand, and all 8 show altered polymerization, i.e., the minimum gelling concentration is either higher or lower than that for the reference Hb S + Hb A mixture. Of the 24 remaining p-chain variants, with replacements at sites that are not intradouble-strand contacts, 20 show no effect on polymerization. Two of the 4 that do show altered polymerization (019 and PSO) are next to a residue that forms an intra-double-strand contact (pl8 and p79, Table 11.2). The other 2 (both at p95) are next to the interface between a@ dimers, and might affect the tetramer-dimer dissociation which would alter the distribution of species in the mixture. Only one &-chainvariant has been and this has been studied at a site of contact in the double strand ((~16) shown unambiguously in cross-linking studies to have a much higher

TABLE 11.1 Effect of Single Amino Acid Substitutionr on Polymerization and Predictionsfrom the Crystal Structure of Deoxyhemoglobin S

Name of variant

p 1 (trans) substitutionsd

Deer Lodge G Makassar Leiden G San Hose Saki J Baltimore J Amiens D Ouled Rabah E G Copenhagen C Ziguinchor J Lome J Cairo I Toulouse Korle Bu J Chicago G Szuhu Pyrgos D Ibadan Agenogi Detroit N Baltimore D Punjab 0 Arab Hofu Wool w ich Hope Man hasset S Travis /32 (cis) substitutionsP C Ziguinchor C Harlem S Travis a I (trans) substitutions J Paris

I Memphis Sealy Torrant a2 (cis) substitutions8 I Sealy Stanleyville

Substitution P2(NA2) (His Arg) /36(A3) (Glu -+ Ala) p6(A3) (Glu + 0) /37(A4) (Glu + Gly) p14(A11) (Leu -+ Pro) /316(A13) (Gly -+ Asp) /317(A14) (Lys + Gln) p19(Bl) (Asn -+ Lys) /326(BS) (Glu Lys) /347(CD6) (Asp + Asn) P5S(E2) (Pro Arg) /359(E3) (Lys Asn) P65(E9) (Lys Glu) /366(E10) (Lys Glu) /373(E17) (Asp Am) /376(E20) (Ala -+ Asp) /380(EF4) (Asn -+ Lys) /383(EF7) (Gly Asp) /387(FS) (Thr + Lys) /390(F6) (Glu -+ Lys) /395(FG2) (Lys -+ Asn) /395(FG2) (Lys Glu) p121(GH4) (Glu -+ Gln) /3121(GH4) (Glu + Lys) /3126(H4) (Val Glu) /3132(H10) (Lys -+ Gln) /3136(H14) (Gly -+ Asp) /3139(H17) (Asn + Lys) /3142(H20) (Ala -+ Val) -+

-+

-+

-+

-+

Effect on polymerization"

Intradoublestrand contact*

Interdoublestrand contactc

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0

+ + + 0 0 0 0 0

+*

+ + 0 0 0 0 0 0

+ + + 0

+ +

+*

+ + 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

p58(E2) (Glu + Val) P73(E17) (Asp Asn) /3142(H20) (Ala -+ Asn)

0 0 0

0 0 0

0 0 0

Asp) orlP(Al0) (Ala a16(A14) (Lys Glu) a23(B4) (Glu + Gln) a47(CE5) (Asp -+ His) cu126(H9) (Asp -+ Asn)

0

0

+ + *J

0 0 0 0 0

a16(A14) (Lys -+ Glu) a47(CE5) (Asp His) a78(EF7) (Asn + Lys)

+ + +

-+

-+

-+

-+

-+

-+

-+ -+

-+

+ 0 + + + 0 + +

+* 0 0 0 0 0

Of

0

0 0

+ + 0 0 0

0 0

+ 0

+

0

+

0 0

0

~~

(Contsnued)

'I'ABLE11.1 (Continued)

Name of variant a I or a2 substitutions"

Sawara Oxford Winnepeg

Substitution a6(A4) (Asp -+ A h ) a15(A13) (Gly 4 Asp) a75(EF4) (Asp -+ Tyr)

Effect on polymerization"

+*

+

+*

Intradoublestrand contacth

Interdoublestrand contactc

0 0 0

0 0

+

"0 indicates that the substitution has no effect on polymerization; + indicates that the substitution inhibits p o l y m e r i d o n ; and + * indicates that polymerization is promoted. Unless stared otherwise, the effect o n polymerization is determined by comparing the minimum gelling concentration of a 40 :60 Hh A + Hh S mixture with a 40 :60 H b X + IIb S mixture. See the text for more details. *+ indicates that the site participates in an intermolecular contact within the double strand of the deoxyhemoglobin S crystal (Padlan and Love, l985b); 0 indicates that the site does not participate. T h e intra-double-strand contacts are listed in Tables 11.2 and 11.3. The spatial disposition of individual residues within the deoxyhemoglohin tetramer is given in Sack et al. (1978). < + indicates that the site participates in an intermolecular contact between double strands in the deoxyhemoglohin S crystal (Padlan and Love, 1985b); 0 indicates that the site does not participate. dThe effects on polymerization are taken from the data compiled by Nagel and coworkers (Nagel and Bookchin, 1978; Nagel st al., 1980). 'These hemoglobins also have the PG(Glu + Val) substitution and are called double mutants. For both hemoglobin C Ziguinchor (R. L. Nagel, personal communication) and hemoglobin S Travis (Moo-Penn et al., 1977), the minimum gelling concentration is the same for the pure double mutant and H b S and is the same for the Hh SX + H b A mixture as for the Hh S + Hh A mixture. These results indicate that the p58 and p142 sites d o not participate in an interniolecular contact on either the p or p2 subunits. T h e results for hemoglobin C lla,lrn, are more complex. T h e minimum gelling concentration for pure is much higher than for hemoglobin S, but in the H h C A ~ ~+~ Hb ~ , ,A, hemoglobin C mixtures containing more than 40% Hb A, the minimum gelling concentration heconies almost identical to that of Hb S + H b A mixtures (Bookctiin and Nagel, 1971). Since the hybrid tetramer is thc dominant species in these mixtures, [.he results suggest that the inhihiting effect of the p73 substitution is on the P I subunit (Minton, 1974h; Nagel and Bookchin, 1978). In the simplest analysis, one would have expected the H h C l ~ + ~H b~A l mixture to have a higher minimum gelling concentration than the Hb S + IIb A mixture. /The effect on polymerization was determined by comparing the solubilities of the crosspS tetramer and the cross-linked a A p A a A p Shybrid tetramer linked a X ~ A a A hybrid (Benesch et nl., 1982). XThe results for hemoglobins 1 and Sealy are based on a comparison of the solubilities of the cross-linked a X p S a A p Ahybrid tetranier with the cross-linked a A p A a hybrid tetramer. For hemoglobin Stanleyville the solubility for the Hb X + Hb S mixture is the same as for the IIb A + Hb S mixture, indicating that a78 is not a contact site on the a1 suhunit, hut the solubility o f the double mutant a ~ l r y v i l ' is c pmuch ~ higher than for a#: (Rhoda et al., 19x3). *These data derive from solubilities on the double niularits acp:, s o that it is not possible to know whcther the contact site is on the a , or a2 suhunit (Benesch et al., 1979; Crepeau et al., 1981). These molecules are prepared by mixing a?p$ with p;, and separation of the a c p ; tetranier formed by subunit exchange by ion-exchange chromatography (Benesch P t al., 1979).

I02

~

~

I A B L E 11.2

Lateral Contacts within Double Strand of Deoxyhemoglobin S Single Cqstal' ~

~~~~~~

~

~

Lateral contacts between 2 strands of deoxy-Hb 5 (crystal contacts between second tetramer in asymmetric unit and first tetramer after translation by 1 unit along x) Contact

v

Gly-69 Ala-70 ASP-73 Asp-79 Gly-83 Thr-84 Thr-87 Leu-88 Glu-90 Leu-9 1 Heme

v

Ala-70 ASP-73 Thr-84 Phe-85 Thr-87 Leu-88 Heme

a

1-a* Ser-49

3

Thr-4

lb

Pro-5

2 13

Val-6

Ser-9

1-P2

Thr-12

Ala-13

>

Lys-17 Glu-121 Pro-125 Val-126

7 10 2

1

3 4

2

Ala-10

1

8*

1

1

1

3

2

5'

1

Id

4 1 2 6 1

3 3 6*

2

1

From Padlan and Love (1985b). *Probable hydrogen bond. 'Favorable ion pair interaction.

dUnfavorableion pair interaction.

TABLE 11.3 Axzal Contactr urtthin Double Strand of Deoxyhemoglobtn S Single CrystalD Axial contacts in first strand of deoxy-Hb S (crystal contacts after translation by 1 unit along x) Contact

I-a I Pro- 1 14

<

Lys- 17

Gly- 16

Val- 18

1-81

>

His-1 17 ~

~

l-a2 Pro-I 14 Ala-115 Glu-116 1-/32 His-1 17 Phe-118 Lys-120

1

1

2 2

- -

Gly-119

14 7

2

~~~~

11 2

2

Phe-118

1

8b

Axial contacts in second strand of deoxy-Hb S (crystal contacts after translation by 1 unit along x)

Contact

I

2-a2 L ~ s16 Pro-114 Ala- 115 Glu-116 2 4 2 His-116 His-1 17 Phe-118 Gly-119 LYS-120

Pro-114

2-a,

Ala-115

9b

<

Gly-16

Lys-17

Val-18

8

1 5

"From Padlan and Love (1985b). bProbable hydrogen bond.

1

His-117

Phe-118

Glu-121

46

3

1

3

>

1

3b

8b 1 8

2-8 I His-116

2

SICKLE CELL HEMOGLOBIN POLYMERIZATION

105

position compared to the cross-linked solubility in the cis (a2) hybrid (Benesch et al., 1982). In summary, the results of all the copolymerization studies provide convincing support for the double-strand hypothesis (Nagel et al., 1980). The combined results from all of these techniques, then, are consistent with the hypothesis that the 21-nm fiber contains a structure very similar to the double strand of the deoxyhemoglobin S single crystal. The importance of this result cannot be overstated, since it provides us with a detailed atomic picture of the intermolecular contacts, including the key contact involving the p6 valine, which can, at most, differ only slightly from those in the fiber. Figure 11.23 shows stereo diagrams of the axial and lateral contacts within the double strand. As can be seen from these pictures and the contact listings in Tables 11.2 and 11.3, the interactions are mainly of the van der Waal’s type, with a small number of hydrogen bonds and even fewer ion-pair interactions (Padlan and Love, 1985b). Because of the approximate 2-fold screw axis parallel to the double-strand axis (the a crystal axis, Fig. 11.18) the two axial contacts are very similar, and the two lateral contacts, which contain the 06 valine, are also similar, but less so. In the axial contact of both strands, the GH corner of the a 1subunit and residues from the A and G helices and the GH corner of the PI subunit from one molecule form a contact with the GH corner of the a2subunit and the G helix and GH corner of the p2 subunit in the adjacent molecule of the strand. For both lateral contacts, the A helix of the p2 subunit of one strand forms a contact with the E and F helices, the EF corner, and the heme. In the more extensive of the two lateral contacts, residues from the H helix and GH corner of the p2 subunit are involved as well. I n the 2p2/1p1lateral contact, the p6 valine of 2pp interacts mainly with Ala-70, Phe-85, and Leu-88 of the lp, subunit, while in the 1p2/2p1lateral contact, the /36 valine interacts mainly with Ala-70, Leu-88, and Asp-73. The interaction with the negatively charged carboxylate of Asp-73 is interesting because, in hemoglobin A, this potential lateral contact would be expected to be destabilized by the electrostatic repulsion from the negatively charged carboxylate of the P6 glutamate. The lateral contacts within the double strand consist of almost exclusively @subunit residues, and the axial contacts contain mainly residues from /3 subunits as well. As a consequence, the contacts between double strands in the fiber are expected to consist of primarily a-subunit residues, as is observed in the crystal. The copolymerization studies suggest that the inter-double-strand contacts are, however, rather different from those in the crystal. Table 11.1 shows that of the eight a-chain mutants that affect polymerization, only three are at sites of inter-double-

FIG. 11.23. Stereo diagrams of lateral and axial rontacts in the double strand of the tieoxyhcmoglobin S crystal. (a) a Carbons of the 2,&/1P1 lateral contact. (-) The subunits containing the contact residues, (-) the P subunits without rontact residues, and (-) the LY subunits. The residues participating in the contact are labeled with the one-letter ainino acid rode and residue position. (b) All norihydrogen atoms of the 2/3~/1/3, lateral contact. ‘I‘he 2Pn/IP, lateral contact contains 37 atom pairs, with one potential hydrogen hand and no ion pair interactions (Table 11.2). The more extensive 1/3&!PI lateral contact involves 72 atom pairs with 4 potential hydrogen bonds and 1 favorable ion-pair I06

interaction. In both lateral contacts, the propionate oxygen of the porphyrin side chain forms a hydrogen bond with the hydroxyl of Ser-9. (c) cy Carbons of the axial contact of strand 1. (d) All nonhydrogen atonis of the strand 1 axial contact. The axial contacts in strand 1 contain 51 atom pairs with 1 potential hydrogen bond, while the axial contact of strand 2 contains 58 atomic interactions with 4 potential hydrogen bonds. In neither axial contact are there specific ion-pair interactions. In c and d, (-) the p subunits and (-) the cy subunits. (Courtesy of E. A. Padlan and W. E. Love.) 107

108

WILLIAM A. EATON AND,JAMES HOFRICHTER

strand contacts in the crystal. Also, three P-chain mutants at sites that participate in the inter-double-strand contacts in the crystal have no effect on polymerimtion. These considerations lead us to the current stage of the structural analysis, which is concerned with precisely positioning the double strands in the polymer. As a first step in such an analysis, it is necessary to find the pairing scheme for the 14 strands. Examination of the spacing between molecules along the fiber (z) axis (Fig. 11.16) shows that the strands can be divided into 7 double strands, in which adjacent strands within each double strand are half-staggered or nearly half-staggered relative to each other as occurs in the crystal double strand (Dykes et al., 1979; Rosen and Magdoff-Fairchild, 1985). Also, in the course of the electron microscope study, minor forms of the fibers were found containing 6, 10, or 12 strands, in which the strands were missing in pairs. These two observations led to the specific pairing scheme in Fig. 11.24 (Dykes et al., 1979), which also shows a molecular model for the cross section of the fiber in terms of 7 pairs of double strands. This pairing scheme is consistent with the analysis of gel diffraction patterns (with some ad.justment of the lattice points), in which a comparison is made between the observed intensity distribution at 15 A resolution and that calculated from models for the fiber (Rosen and Magdoff-Fairchild, 1985, 1988). Recent studies using an improved image reconstruction technique based on a real-space cross-correlation method (Crepeau and Edelstein, 1984) confirm the strand-pairing scheme (Fig. II.24a) (Rodgers et al., 1986, 1987). The cross-correlation approach also reveals the relative directions of the double strands (Fig. II.24b,c) (Rodgers el al., 1986).R An alternative niodel for the fiber had been proposed (Wellems and Josephs, 1979; Vassar ~t nl., 1982) based on the observation that fiber bundles convert to crystals. From the assumption that this is a conformatioiial transition of the entire macroscopic structure, and not simply the dissociation of hemoglobin molecules from the less stable fiber bundle and association to the more stable crystal, it was argued that the fibers “fuse” to form the crystal, and that such a process is unlikely to occur unless there is an even number of double strands, as is found in the crystal (Wellems and Josephs, 1979). The crystal might then form from the fibers by sniall acljusttnents of the double strands relative to each other (Wellems and Josephs, 1979). No detailed niodel for the proposed 16-stranded structure was put forth, and there has been no image reconstruction or any other quantitative analysis that favors a 1 &stranded structure. Furthermore, as mentioned above, the copolymerization studics with a-chain mutants would not support a structure in which the interdouble-strand contacts are very similar to that found in thc deoxyhernoglobin S crystal. More recently. this group has performed an image reconstruction analysis from electron micrographs of negatively stained fibers: they confirmed the 14-stranded structure, the pairing scheme, and the relative polarity of the double strands (Carragher et nl., 1988a,b).

SICKLE CELL HEMOGLOBIN POLYMERIZATION

109

a

b

,--1

C

d

FIG. 11.24. Model for pairing strands in hemoglobin S polymer from electron microscopy. (a) Calculated images of fibers from cross-correlation method, (b) arrangement of strands with molecules having the same polarity indicated by the shading, (c) schematic orientation of molecules, and (d) planar projection of space-filling model in which each residue is represented by an open circle, with the solid circles corresponding to the p subunit. T h e separation between the seven pairs is exaggerated to avoid overlap of adjacent molecules. [From Edelstein (1980, 1981a,b)and Kodgers et al. (1986).]

110

WILLIAM A. EATON AND.JAMES HOFKICHTEK

Axis

N

B FIG. 11.25. Transformation from (A) crystal double strand to (B) fiber double strand. [From Edelstcin (1981t)).]

‘rhe problem now is to construct an atomic niodel for the 14-stranded structure. This requires transforming each of the double strands from linear into helical structures (Fig. 11.25) (Edelstein, 1981b), followed by precise positioning of these helical strands relative to each other. To corn struct such a model will require considerably more information than is currently available. Help in solving this problem should come from additional copolymerization studies using a-chain mutants to define the inter-double-strand contact residues. Also, it will be extremely important to obtain better X-ray diffraction patterns (Fig. 11.26) by preparing gels

SICKLE CELL HEMOGLOBIN POLYMERIZATION

FIG. 11.26. X-Ray diffraction of deoxyhemoglobin S gel using 1.57 diation. [From Rodgers et al. (1986).]

A

111

synchrotron ra-

having greater parallel alignment of fibers or making measurements on single fiber bundles. Better aligned samples should be obtained by polymerizing under slow nucleation conditions (see Section IV). 111. THERMODYNAMICS OF HEMOGLOBIN S POLYMERIZATION

The picture that emerges from the structural studies discussed in Section I1 is that under near-physiological conditions hemoglobin S can

112

WILLIAM A. EAI‘ON A N D JAMES HOFRICHTER

aggregate into helical polymers or three-dimensional’ single crystals. Although the crystal is the thermodynamically more stable structure, polymers always form first, and in the absence of shear forces do not convert to crystals on the hour-day time scale of equilibrium measurements. The system of solution plus polymers is commonly referred to as a gel, and the overall process of forming an aligned polymer phase as gelation or simply polymerization. In this section we shall confine our discussion to the thermodynamics of gelation. The simplest thermodynamic description of a gel is that it contains only two components, hemoglobin and solvent, and that the polymer phase behaves like a crystal in equilibrium with a solution of hemoglobin monomers, as shown schematically in Fig. 111.1. In this description the collection of domains of aligned polymers is viewed as the “crystal,” and as such is treated as a separate phase with a constant composition. In the context of this discussion hemoglobin molecules of 64.5 kDa are regarded as monomers. The concentration of hemoglobin in the solution phase is, then, the solubility, and is a precise measure of the polymer stability. Because the solubility is very high, greater than 0.16 g/cm3,the solution phase is highly nonideal. This nonideality can be completely accounted for, however, by considering only the hard-sphere excluded volume contributions to the activity coefficient of the monomer. We shall see that this two-component, two-phase description of polymerization is consistent with almost all the thermodynamic data. Our discussion of the thermodynamics is divided into four topics. In Section 111,A we discuss sedimentation experiments and describe how hemoglobin activity coefficients are obtained from sedimentation equilibrium data. T h e remaining sections are primarily concerned with the influence of physiologically relevant variables on the polymerization process. Section III,B presents the results on the effect of temperature, pressure, pH, and salts on deoxyhemoglobin S solubility. In Section lII,C we discuss the control of polymerization by oxygen, and, finally, in Section II1,D we describe the thermodynamics of polymerization of mixtures of hemoglobin S with other hemoglobins. A . Sedimentation Studies and Nonideality The solution phase of a gel contains only monomers, with no clear evidence for higher aggregates. The evidence to support this conclusion has been obtained primarily from sedimentation experiments. T w o types of sedimentation studies have been carried out. The most straightforward is to centrifuge a gel of deoxyhemoglobin S at high speed, causing the high-molecular-weight polymers to sediment into a compact

SICKLE CELL HEMOGLOBIN POLYMERIZATION

113

0

M

FIG.111.1. Schematic structure of deoxyhemoglobin S gel depicting monomer-polymer equilibrium. (0)Each hemoglobin molecule. The structure of' the polymer is that determined by Edelstein and co-workers (Dykes et al., 1979). Bar, 20 nm.

pellet. This separation of a gel into solution and polymer phases by centrifugation (Allison, 1957; Bertles et al., 1970; Magdoff-Fairchild et al., 1976; Hofrichter et al., 1976a) has become a standard equilibrium assay for gelation (Hofrichter et al., 1976a). T h e concentration of hemoglobin S in the supernatant is taken as the solubility. The measured solubility is found to be independent of the total initial hemoglobin concentration, the gravitational field strength, and the time after gelation up to about 24 hr (Hofrichter et al., 1976a). T h e fact that the solubility is independent of the time between gelation and sedimentation is impor-

114

WILLIAM A . EATON A N D JAMES HOFKICHTEK

tant because it shows that the gel, while metastable, is stable for days o r longer. The finding that the solubility is independent of the total hemoglobin concentration is fully consistent with the model that gelation can be described by a simple equilibrium between a crystal and a solution of monomers. In order for even a small amount of polymer to be formed, the solution phase must be saturated with monomer, and any additional deoxyhemoglobin S adds to the polymer phase. The absence of detectable pressure dependence of the solubility is consistent with the results of dilatometric experiments which show that there is no volume cm"g polymerized hechange on polymerization [AV = 0 & 8 x moglobin (Kahn and Briehl, 1982).] Unfortunately, even high-speed sedimentation of the polymers does not cleanly separate the polymer and solution phases, since a significant fraction of the pellet volume is occupied by the monomeric solution phase. The measured concentration of hemoglobin in the pellet is, therefore, not the concentration of hemoglobin in the pure polymer phase. The concentration of deoxyhemoglobin S in the polymer phase is not known accurately, since no experiments have been designed specifically to determine its concentration. It has been approximately measured in solubility experiments by near-infrared spectrophotometry on the turbid pellets or from rather crude measurements of the pellet volume fraction, from which the concentration of hemoglobin in the pellet is calculated using mass conservation. For a fixed set of centrifugation conditions (170,000 g for 150 min), the concentration of hemoglobin in the pellet is about 0.50 to 0.55 g/cm" (Sunshine et ul., 1979b), but increases with increasing solubility, indicating that the pellet also contains monomers. From these experiments the volume fraction of the solution phase in these pellets is calculated to be 0.35 and the concentration of deoxyhemoglobin S in the pure polymer phase is 0.69 f 0.06 g/cm3 (Sunshine et ul., 1979b). We shall see later that the concentration of deoxyhemoglobin S in the polymer phase is an important number for thermodynamic analyses.q 'I Geometric considerations can be used to show that the measured coilcentration of hernoglobin in the polymer phase is consistent with the polymer structure deterniiried from the electron microscope data. Upper bounds for the volume excluded tn molecules of different sizes hy the polymer lattice described in Figs. 11.15-11.17 can be estimated by treating the polymer as a stack of polygonal plares. T h e height of each plate is taker1 as 6.0 nm (the axial spacing between molecules along the fibei- axis), and the exterior dimensions are determined hy circumscribing the axial projection of the polymer in Fig. 11.15 with a polygon spaced either 2.5 nni outside the maximum radial lattice positions or 5.0 nm outside these positions. The volume calculated using the first dimension should approximate the vulunic excluded to small molecules of diameter of about 0.5 n n ~by the

SICKLE CELL HEMOGLOBIN POLYMERIZATION

115

The concentration of monomers in the solution phase of the gel has also been measured using an analytical sedimentation technique (Briehl and Ewert, 1973; Briehl, 1978). In this experiment the polymer is first pelleted at high speed and the rotor is then slowed to permit the free boundary produced by the sedimenting monomers to diffuse. T h e Schlieren peak produced by the diffused boundary can then be integrated to obtain the solution phase concentration. The results from these experiments are identical to those obtained using the preparative sedimentation technique. More importantly, the sedimentation velocity profile contains only a single peak having a sedimentation coefficient which is very similar to that of hemoglobin tetramers. This result argues against the possibility that there is a significant amount of aggregated hemoglobin S in the solution phase of the gel. In another type of sedimentation experiment, a deoxyhemoglobin S solution, at a concentration at or below the solubility, is spun at a much slower speed until equilibrium is reached. At equilibrium, the opposing forces of sedimentation and diffusion result in a concentration gradient, with the concentration increasing monotonically from the meniscus toward the bottom of the cell, but then changing abruptly at the boundary of the much more concentrated polymer phase. In this experiment, the pellet results from the polymers that have formed and sedimented in the course of the centrifugation, which requires several days. T h e results of the experiment are most conveniently expressed as the apparent modetermined as a function of hemoglobin concentralecular weight, Mappr tion. Figure 111.2 shows data for deoxy- and carbonmonoxyhemoglobin S (Williams, 1973). T h e carbon monoxide complex of hemoglobin S is known to be soluble under these conditions up to 0.5 g/cm3 (Briehl and Ewert, 1974; Briehl and Salhany, 1975). At about 0.14 g/cm3 there is a discontinuity in the concentration caused by the pellet boundary. hemoglobin molecules in the polymer lattice. This volume is calculated to be approximately 1.1 cm 9/g polymerized hemoglobin, equivalent to a concentration of polymerized hemoglobin of 0.9 g/cm3. The volume calculated using the larger dimension should approximate the volume excluded to spheres with a diameter of 5.0 nm (i.e., other hemoglobin molecules). This volume is considerably larger, about 1.55 crn3/g, equivalent to a concentration of polymerized hemoglobin of about 0.65 g/cm3. The actual volume will decrease if a fraction of the surface of the polymers is in contact in a crystalline or quasicrystalline array because the regions of contact could not be approached by hemoglobin molecules in the solution phase. For a square lattice similar to that shown in Fig. 11.1 Ib, the fraction of the polymer surface in contact is 0.3, yielding a concentration of polymerized hemoglobin of 0.71 g/cm3, essentially identical to the measured value of 0.69 2 0.06 g/cm3.

116

WILLIAM A. EATON AND ,JAMES HOFRICHTER

I

I

I

1

I

I

I

FIG.111.2. Sedimentation equilibrium data for hemoglobin S ('LO"<:, 0.1 M NaCI, 0.1 M sodium phosphate, I mM EDTA). T h e apparent weight average molerular weight of' hemoglobin is plotted versus the local concentration in the centrifuge cell at equilibrium. ).( Data for c.arbotirnotioxyIicn~ogl~~in S and (A)data for deoxyhemoglobin S. T h e arrow a t 6 4 K M W indicates the tetramer molecular weight. 'l'he vertical line at 0.14 g/cmJ indirates the pellet boundary for deoxyhemoglobin S (gel point). [From Williams (197q.l

The sedimentation equilibrium experiment provides important information on both the extent of aggregation and the monomer activity coefficient for deoxyhemoglobin S in the solution phase. The apparent molecular weight is defined by Map,,= M l ( 1

+ d In y l d In c)

(111.1)

where M is the true molecular weight, and y is the hemoglobin activity coefhcient (Tanford, 1961; Cantor and Schinimel, 1980). I n the limit of infinite dilution y = 1 and M = MdPP.At finite concentrations Mappis lower than M because d In y l d In c is greater than zero and increases with increasing concentration. In the absence of any aggregation, the determination of MAPP as a function of concentration results in the determination of the monomer molecular weight, the value of' d In y l d In c for the monomer as a function of concentration, and, by integration, the value of In y as a function of concentration. The identity of the MAP,, versus concentration data in Fig. 111.2 for deoxy- and carbonmonoxyhemoglobins S indicates that there is no significant aggregation of deoxyhemoglobiri S in the solution phase above the pellet boundary (Wil-

117

SICKLE CELL HEMOGLOBIN POLYMERIZATION

liams, 1973). This is important evidence for a monomeric solutioncrystal type equilibrium.l o Additional evidence for a simple two-phase system comes from measurements of quasi-elastic light scattering (Kam and Hofrichter, 1986). In these experiments, the measured fluctuations in the scattered intensity arising from the motion of the scattering molecules are used to calculate a translational diffusion coefficient from the decay in the autocorrelation function (Fig. III.3).11 In the absence of polymers, the mono1" When the equilibrium schlieren patterns are examined more closely, the sedimentation pattern appears to be more coniplex than would be expected for a simple two-phase system (Briehl, 1978). An abrupt discontinuity in the radial concentration profile is observed just above the pellet. More importantly, the monomer concentration just above this discontinuity is markedly lower than the solution phase concentration measured immediately after the polymers were pelleted. Two explanations for these phenomena have been proposed (Briehl, 1978). The first is that, when polymers are densely packed for long periods of time, a phase transformation takes place and a more stable aggregated phase of deoxyhemoglobin S is formed. This idea is supported by electron microscopic studies which show that densely packed polymers may convert to crystals within 12-24 hr (see Section 11,B). This idea is also consistent with the observation that, in the presence of inositol hexaphosphate, the final aggregate could not be completely removed by cooling to O"C, and a lower solubility was obtained on sedimenting the gel produced by repolymerizing the incompletely melted sample. The sharp increase in concentration at the pellet boundary could be explained as arising from the rough surface of the packed, crystalline phase. The primary evidence against this explanation is the observation that, in the absence of inositol hexaphosphate, incompletely melted samples repolymerized to yield the same solubility. The second possible explanation is that the phase diagram is significantly more complex and that additional species, such as randomly oriented polymers or loosely packed arrays of aligned polymers (see Minton, 1974a; Briehl and Herzfeld, 1979; Herzfeld and Briehl, 1981a,b), can be formed in the solution phase and are not sedimented when the polymers are first packed at high speed. The sharp increase in concentration at the pellet boundary would then be attributed to the formation of isotropic polymers at concentrations greater than a critical concentration (cagg). Two observations argue against this second proposal. First, there is no evidence for heterogeneity in the sedimentation velocity profiles obtained on the superuatant. A single seditnenting boundary could only be observed if the polymers were in rapid equilibrium with monomers, an unlikely possibility because of the nucleation barrier for polymer formation (see Section IV). Second, the turbidity of gelled samples disappears at the solubility measured in the high-speed sedimentation experiment (Ross ct al., 1977), arguing that the concentration of highly aggregated material must become negligible at this concentration. Ii The normalized intensity autocorrelation function is defined as C ( t ) = (1(7)1(7 t))/ ( Z ( T ) ~ ) (Berne and Pecora, 1976). If the scattering arises from a solution which contains mobile, spherical sratterers, which scatter a time-averaged intensity proportional to A, and immobile scatierers, which scatter an intensity proportional to B, then the normalized autocorrelation function can be written as C ( t ) = A* exp( - 2q2Dt) 2AB exp( - f D 1 ) + B 2 . If B << A, the measured intensity fluctuations arise only from interference of the scattered fields from the mobile scatteren, called homodyning, and the first term dominates the

+

+

118

WILLIAM A. EATON A N D .JAMES HOFKICHTER

u b-

3

a

0

50

100

150

0

50

100

150

TIME (psec)

FIG. 111.3. Norrtialized intensity autocorrelation functions from quasi-elastic lightscattering measurements at 33°C for (a) a solution of oxyhenioglobiri S and (b) a gel of deoxyhemoglobin S. T h e circles are the data, and the curves are fitted exponentials. [From Karn and Hofrichter (1986).]

mer diffusion coefficients for oxyhemoglobin S and deoxyhemoglobin S are identical. At higher temperatures, where polymerization occurs for deoxyhemoglobin S, the apparent diffusion coefficient is almost exactly a factor of 2 smaller for the gels of deoxyhemoglobin S. This factor of 2 arises from the interference between the scattered electromagnetic field of the nondiffusing polymers and the diffusing monomers (heterodyning).ll The finding of an identical diffusion coeflicient for the diffusing molecules of a gel and oxyhemoglobin S (Fig. 111.4) makes two important points (Kam and Hofrichter, 1986). First, there is no detectable aggregation of monomers in the solution phase of the gel. Second, the polymers are probably not uniformly distributed in the gel. If the polymers were uniformly distributed, they would provide an obstacle that is estimated to decrease the diffusion coefficient of the monomers by about 1596, well outside the error of the measurements. The most likely explanation for this result is that the polymers are locally packed in dense arrays which permit only a small fraction of monomers in their interstices (see footnote 9 on pp. 114- 115). There has been extensive analysis of the sedimentation equilibrium

decay of the autocordation function. I f B >> A, the observed decay is dominated by interference between the held scattered by the mobile scatterers and that scattered by the immobile scatterers, called heterodyning, and the second term dominates. Introduction of an intense, s t a h scattered field would herefore be expected to reduce the rate of decay of the autocorrelation function from 2y2D to Q'D, i.e., by a factor of 2.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

10

20

119

30

TEMPERATURE ("C) FIG.111.4. The apparent diffusion coefficient as a function of temperature for solutions of oxyhemoglobin S and solutions and gels of deoxyhemoglobin S determined from decay of the autocorrelation function in quasi-elastic light-scattering experiments. (0) A 0.255g/cm3 solution of oxyhemoglobin S and (0)a 0.255-g/cm3sample of deoxyhemoglobin S at successively lower temperatures after first gelling the sample. The discontinuity in the data between 12 and 9°C results from the disappearance of polymer. (0)The diffusion coefficients measured in the presence of polymer. These diffusion coefficients have been multiplied by a factor of two as a heterodyne correction. [From Kam and Hofrichter (19SS).]

data for the determination of the monomer activity coefficient y . T h e data can be represented by (111.2) where we have replaced d In yld In c of Eq. (111.1) by a power series in the concentration (Tanford, 1961; Minton, 1977; Ross and Minton, 1977; Ross et al., 1977). The coefficients, B I + ] ,are determined by the interaction potential between solute molecules and can be calculated from statistical mechanics. The simplest case is one in which the solute molecules behave like hard spheres, that is, the interaction potential is infinitely repulsive at center-to-center intermolecular distances of less than a molecular diameter, and is zero at greater distances. In this case, the sole contribution to the activity coefficient is this so-called excluded volume effect, and the interaction coefficients can be expressed in terms of the volume of the hard sphere, V (Minton, 1977, 1983; Ross and Minton, 1977; Ross et al., 1978). These are

120

WILLIAM A. EATON AND JAMES HOFRICHTER

7

0 7

X

Y

60 40 20

c

c

F

B

-

0

0

0.1

0.2

0.4

0.3

0

0.1

0.2

0.3

0.4

Concentration (g/cm3)

FIG.111.5. Apparent weight average molecular weight versus concentration from sedinieritatiori equilibrium experiments. (a) Hb A and IlbCO A. (b) Hb S and HbCO S. (0)HbCO A and IlbCO S and (0)deoxyHb A and deoxyHb S. T h e curves are obtained by fitting the data to obtain the molecular weight (M) and the excluded volume (V) using Eqs. (III.2)and (111.3). These are the sedimentation data at 20°C (0.1 M phosphate, pH 7) used by Ross e t a / . ( I9713). The best-tit values obtained in a simultaneous fit to six data sets (only four are shown) arc V = 0.79 t 0.02 cmVg and M = 64,600 c 700. (Courtesy of A. P. Minton.)

B,

=

8V

B , = 35.30V4

Bs = 15V2

B,

=

47.4V5

B4 = 24.48V'

R7

=

65.9V"

(111.3)

With the above values for the coefficients, B k c l , it is possible to fit the sedimentation equilibrium data with only two parameters, M and V. T h e results of such fitting are shown in Fig. 111.5 for the deoxy and carbonmonoxy forms of hemoglobins A and s. T h e values of M and V are found to be the same to within a few percent for all four molecules, and to be independent of temperature between 2 and 37°C (Ross et al., 1978). Nearly identical values are obtained from osmotic pressure data obtained over 60 years ago (Adair, 1928; Ross and Minton, 1977). T h e best fit value of M = 64,600 2 700 is in excellent agreement with the value based on the chemical composition of 64,500, and the value of V = 0.79 & 0.02 cmYg is close to the partial specific volume of 0.75 cm"g (Cantor and Schimmel, 1980).1p In publications prior to Minton (19133), the expressions for the interaction coefficients wer'c in error (A. P. Minton, personal cornnitmication), resulting in a best fit value for V of 0.92 cmY/g(Ross ct nl., 19713). T h e values of y were, however, unaffected since virtually identical values for y are obtained using the earlier expressions for the interaction coefficients with V = 0.92 cm3/gas with the present coefficients with V = 0.79 cni3//g; the

SICKLE CELL HEMOGLOBIN POLYMERIZATION

121

It appears, then, that the simple hard-sphere approximation can account quantitatively for the observed nonideality in hemoglobin solutions. The excellent agreement between experiment and this simple theoretical description most probably results from the fact that the activity coefficients have been determined at neutral pH, close to the isoelectric point of hemoglobin (pH,, = 7.2 at 4°C) (Winterhalter and Colosimo, 1971), and at relatively high ionic strength, so that long-range electrostatic interactions are damped out (Ross and Minton, 1977; Ross et al., 1978; Minton, 1983). The slightly higher value for the volume excluded to other hemoglobin molecules compared to the partial specific volume, which is the volume excluded to solvent, may be due to the crevices between subunits that can admit water molecules (Minton, 1980). Integration of the expression for d In yld In c in Eq. (111.2) results in 6

In y

= k= 1

B,+,c~

(111.4)

Figure 111.6 shows a plot of log y and d log yld log c versus c calculated from Eqs. (III.2)-(III.4). At the hemoglobin concentrations found inside red cells, the activity coefficient becomes quite large, and is about 70 at 0.35 g/cm3. In the following sections, we shall see that the large activity coefficients play a major role in determining the effect of composition and temperature on the equilibrium properties of gels. B . Effect of Temperature and Solution Conditions on Deoxyhemoglobin S Polymerization

For thermodynamic and kinetic studies, the most important characteristic of hemoglobin S polymerization is that a gel can be prepared by heating a liquid solution at the appropriate concentration and “melted” by cooling. This behavior is readily understood from the observed dependence of the deoxyhemoglobin S solubility on temperature shown in Fig. III.7a. The solubility under these solution conditions decreases from about 0.32 g/cm3 at 0°C to a minimum of 0.16 g/cm3 at about 35”C, and then increases again. Thus, at total concentrations below 0.16 g/cm3, deoxyhemoglobin S solutions remain liquid between 0 and data are equally well fit with either expression. Notice also that the interaction coefficients, B , , are here defined as the coefficients of the concentration in the expansion of In y [Eq. (111.4)], whereas in previous papers (Ross et al., 1978; Sunshine et al., 1979b. 1982) they were defined as the coefficients of the concentration in the expansion of d In yid In c in the expression for the molecular weight.

I22

WILLIAM A. EATON A N D JAMES HOFKICH'I'ER

20

15

10

5

0.2

0.1

0

0 0.5

0.4

0.3

Concentration (g/cm3) FIG. 111.6. IIemoglobin activity coefficient. T h e decadic logarithm of the activity coefficient [ y (-)I and the derivative [ d log y / d log c (---)I are plotted versus the concentration (c). The curves are drawn using Eqs. (111.2)-(111.4) with V = 0.79 g/cm3 obtained from the fits to the sedimentation equilibrium data in Fig. 111.5. Temperature (T) 50

40 I

30 ~

'

0

10

20 I

I

'

I

'

6.5

b

6

5.5

5

0.1

0

10

20

30

Temperature ("C)

40

50

0.0032

0.0034

0.0036

4.5

l/f(K)

FIG. 111.7. Tcmpcratur-e dependence of deoxyhemoglobin S solubility. (a) T h e solubility in 0.15 M potassium phosphate and 0.05 M sodium dithionite, pH 7.15, is plotted versus temperature. The curve through the data is a leasl-squares fit using the empirical equation c, = 0.319 - 0.008837' + 0.000125T2, with the temperature (T) in degrees centigrade (data o f Koss el al., 1977). (b) The natural logarithm of the equilibrium consranr, ralculated from the solubility data in Fig. III.7a using Eq. (111.7), is plotted versus the reciprocal of the absolute temperature. T h e curve through the data is the least-squares fit using Eq. (111.12) with AHo = 106.0 2 0.5 kcal/mol, AC, = - 342.2 2 1.5 cal/mol-deg, and the constant of integration = 1165.8 2 5.1.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

123

45°C; at concentrations above 0.32 g/cm3, the solution contains poly-

mers at all temperatures; and between 0.16 and 0.32 g/cm3, the solution undergoes reversible polymerization and depolymerization on heating and cooling. The fraction of polymerized hemoglobin, xp, at any temperature and total concentration, co, can be calculated from the solubility data (Fig. 111.8)using the mass conservation relation: xp = (1 -

(I11.5)

- c,/cp)

C$/C(I)/(l

where c, is the solubility and cp (= 0.69 g/cm3)is the concentration of hemoglobin in the polymer phase. To obtain thermodynamic parameters from the solubility data we consider the equilibrium: Hb (solution)

+ nH,O

(solution) = Hb(H,O), (polymer)

which indicates that the polymer phase is formed with n moles of water per mole of hemoglobin. The value of n is estimated to be approximately 2500 from the relation

l

I

r

,

,

,

r

,

,

,

,

'

b

0

10

20

30

Temperature

40

50

("C)

FIG. 111.8. Fraction of polymerized hemoglobin S as a function of temperature. The fraction polymerized was calculated from the solubility data in Fig. I11.7a using the mass conservation Eq. (111.5) with cp = 0.69 g/cni3. The curves "ere calculated with total concentrations ( c o ) varying from (a) 0.20 gkm3 to (b) 0.45 g/cm3 at 0.05 g/cm3 intervals.

124

WILLIAM A. EATON A N D .JAMES HOFRICHTER

where v is the specific volume of hemoglobin in the polymer phase, is the ratio of hemoglobin and taken to be 0.75 cm3/g, (MWHi,/MW,,?~~) water molecular weights ( = 64,500/18), and the density of water has been taken as unity. The equilibrium constant, K , for this process is (111.7)

where uHhis the activity of the hemoglobin monomer and u ~ , , is ~ ) the water activity in the solution phase. T h e activity of hemoglobin is simply the product of the solubility and the activity coefficient (aHh = ylc,), evaluated at the solubility from Eqs. (111.3) and (111.4). T h e activity of water can be obtained from the hemoglobin activity through the GibbsDuhem relation: Y H dPHt, ~

-t ~

di.L~~,o= 0

1 1 ~ 0

(1 11.8)

Ignoring the salt as a component implicitly assumes that both the number of moles of unbound salt per mole of water and the number of moles of bound salt per mole of hemoglobin are the same in the solution and polymer phases. Expressing the chemical potentials, p, in terms of activities (dp = RT d In a), and substituting for the mole ratio, v H ~ / v H ~ in ~ , terms of measured quantities, Eq. (1 11.8) becomes

Figure 111.9 shows a plot of and uHZoversus hemoglobin concentration calculated from Eq. (111.9). Although the water activity is close to ~ ~ ) " a significant unity over the entire concentration range, ( u ~ ~ , makes contribution to the equilibrium coristant at high concentrations because n ( = 2500) is so large. (Notice that for n = 0, K would be a simple solubility product for a single component crystal.) The enthalpy change, A H , for polymerization can be obtained from the temperature dependence of the equilibrium constant using the van't Hoff relation: d111 K -

d( l/T)

- --A H R

(I1l.lO)

Figure III.7b shows a plot of In K versus 1/T. T h e large deviation from linearity indicates that AH is strongly temperature dependent. Assuming that AH is linear in the temperature, i.e.,

AH = A H o

+ AC,T

(111.11)

SICKLE CELL HEMOGLOBIN POLYMERIZATION

125

Concentration (g/crn3) FIG. 111.9. Activity of water as a function of hemoglobin concentration. T h e activity of water (---)I and the quantity (aHyo)~1 (-) with n = 2500 were calculated from the Gibbs-Duhem relation using Eq. (111.9),with v = 0.75 g/cm3, cp = 0.69 g/cm3, and V = 0.79 cm3/g.

where AC, is the heat capacity change, the integrated form of Eq. (111.10) becomes

AH0 I n K = -RT

+ AC R

T

+ constant

(I11.12)

The data in Fig. III.7b are reasonably well fit by Eq. (111.12) with AHo = 106 kcal/mol and AC, = - 342 cal/mol-deg. The large decrease in the apparent A H with temperature from the van’t Hoff analysis of the solubility data is confirmed by direct calorimetric measurements. Figure 111.10 shows the results of two types of calorimetric experiments (Ross et al., 1975, 1977). In the first (Fig. III.lOa), the temperature of a solution is slowly increased at a constant rate and the heat flux is measured as a function of temperature. A narrow peak of energy absorption is observed which is much sharper than would be predicted from the equilibrium curves in Fig. 111.8 for the fraction of polymerized hemoglobin versus temperature. This occurs because of the unusual kinetic behavior of polymerization. Polymer formation does not start immediately once the solubility is exceeded, but is preceded by a marked delay followed by the explosive autocatalytic appearance of polymer. This behavior is clearly seen in the calorimetric experiments carried out at constant temperature, in which a liquid sample, initially at

126

WILLIAM A . E A T O N A N D JAMES HOFRICIITER

a

b

f

100 pw

1

-.i I

20

I

-

I

8

22 24 TEMPERATUREPC)

26

1

\ I

I

2 3 TIME (HOURS)

FIG. 111.10. Calorimetric energy absorption curves. (a) Heat flux as a function of temperature into a 0.23-g/cmS deoxyhemoglobin S sample which was heated from 0°C at 4"C/hr. (b) Heat flux as a function of time into a 0.23-g/cmSdeoxyhemoglobin S sample after removing it from an ice bath and inserting it into the calorimeter at T = 19.6"C. (From Ross et al. (1975).]

O"C, is inserted into the calorimeter at a temperature where the total concentration exceeds the solubility (Fig. 111. lob). The onset temperature in the scanning experiment and the delay time in the isothermal experiment are very sensitive to the heating rate and temperature, respectively, and can be related by a simple empirical theory (Ross et al., 1975). The kinetics of polymerization are fully discussed in Section IV. The heat absorption at the same temperature measured by the scanning and isothermal calorimetric methods are the same to within about 15% (Ross et al., 1975). Figure 111.11 shows the enthalpy change as a function of temperature obtained from scanning calorimetric measurements, the more sensitive of the two methods (Ross et al., 1977). T h e best straight line through the calorimetric data gives AH, = 71.9 kcall mol and ACp = - 234 cal/mol-deg. The resulting values for AH are in fair agreement with those obtained from the solubility data, but are lower by 0-3 kcal/rnol. This difference suggests that it may be necessary to include some additional process in the thermodynamic analysis, such as proton or other ion binding. Table 111.1 summarizes the thermodynamic data in the temperature range 15-45°C. T h e free energy change is relatively constant, decreasing by only 0.6 kcal/mol as the temperature is increased from 15 to 45°C. In contrast, there is a large decrease in both the enthalpy and

127

SICKLE CELL HEMOGLOBIN POLYMERIZATION

Temperature ("C)

0

10

280

20

290

40

30

300

Temperature

310

50

320

(K)

FIG. 111.11. Enthalpy change for hemoglobin S polymerization as a function of temperature. (0)Data from scanning calorimetric experiments (Ross et al., 1977). T h e AHs have been recalculated using c,, = 0.69 g/cm' (instead of the previously used 0.49 g/cm3) to determine the number of moles of hemoglobin polymerized. Also, the temperature is taken as the average temperature of the energy absorption curve. (-) A least-squares fit using Eq. (111.11) with AH, = 71.9 2 3.2 kcal/mol and ACp = - 234 2 11 callmol-deg. (---) The AH calculated from the van't Hoff analysis of the solubility data in Fig. III.7b.

TABLE 111.1 Thermodynamic Parameters for Deoxyhemoglobin S Gelation" Temperacs ture ("C) (g/cm3)

ys

(aHTO)n

KC (M-')

AG (kcal/ mol)

AHb (kcal/ mol)

TAS (kcal/ mol)

ACP (cal/moldeg)

7.4 (4.4) 4.0 (2.0) 0.6 (-0.3) -2.9

10.5 (7.5) 7.4 (5.4) 4.1 (3.2) 0.8

-342' (-234)"

~~

15

0.214

6.96

0.707

220

-3.1

25

0.176

4.40

0.785

315

-3.4

35

0.162

3.80

0.810

368

-3.5

45

0.174

4.30

0.789

323

-3.7

"Conditions: 0.15 M potassium phosphate, 0.05 M sodium dithionite, pH 7.15. bFrom van't Hoff analysis of solubility data; values in parentheses from calorimetry. 'Calculated from Eq. (111.7) with n = 2500 using the data in Fig. 111.7. dFrom van't Hoff analysis. 'From calorimetry.

128

WILI.IAM A. EATON A N D .JAMES EIOFRICHTER

entropy changes (7- 11 kcal/mol) as a result of the large change in heat capacity accompanying polymerization.13 The positive enthalpy and entropy changes and the negative heat capacity change for polymer formation are characteristic of protein aggregation driven by hydrophobic interactions (Ross and Subramanian, 1981). This result would predict that the intermolecular contacts in the polymer are composed of van der Waals interactions rather than hydrogen bonds or ion-pair bonds. T h e structural data are consistent with this interpretation. In the deoxyhemoglobin S crystal, where the lateral and axial contacts within the double strand are very similar to those in the polymer, the intermolecular interactions are mainly of the van der Waals type ('Tables 11.2 and 11.3) (Padlan and Love, 1985b). T h e crystal data also suggest that there are no large conformational changes on polymerization or crystallization, in keeping with the relatively small values of the observed enthalpy and entropy changes. Thus far, we have only discussed the solubility data under a single set of solution conditions: 0.15 M potassium phosphate, 0.05 M sodium dithionite, pH 7.1. These conditions were chosen to maintain a constant pH in the presence of dithionite, which is necessary for long-term storage of samples, and because a large number of early studies on hemoglobin mixtures were carried out in this buffer. Fortunately, the solubility is very similar in a range of buffers. Since the initial solubility studies in the phosphate buffer (Ross et al., 1975, 1977; Magdoff-Fairchild et al., 1976; Hofrichter ut al., 1976a), there have been a number of investigations of deoxyhemoglobin S using a variety of solution conditions. Of particular interest are the effects of the physiological variables pH and DPG. As shown in Figure 111.12, the solubility changes very little between pH 6.0 and 7.0, and then increases sharply at more acid and alkaline pH values (Goldberg et al., 1977), presumably because of changes in the ionization state of histidines. 'I'he minimum in the solubility-pH profile is at about pH 6.5, not far from the isoelectric pH of deoxyhemoglobin S, suggesting that a major effect of higher or lower pH is to destabilize the polymer through an increase in the net electrostatic repulsion between molecules. The pH dependence may also arise from the titration of histidines that form ion-pair or hydrogen bonds in 13 Measurements of the deoxyhemoglobin S solubility with respect to the solid phase obtained in stirred solutions give similar results (Jones and Steinhardt, 1979). Since the solid phase formed by stirring changes with time from bundles of fibers to single crystals (Wellems and Josephs, 1979), it is not clear whether crystallization is complete in these experiments. The solubilities are about 30% lower than those found for gels and the enthalpy change is about 6 kral/mol in the range 15-30"C. but the data are not sufficiently precise to evaluate a heat capacity change.

129

SICKLE CELL HEMOGLOBIN POLYMERIZATION

a

0.30 -

E . h

Is)

v

0.25

t d

a

-

rn

0

52

0.20-

a a

:

3

a

a

(. a

0.15

I

I

I

I

0.05-

b I

I

1

o.2 0.18

c

0

1

2

3

EFFECTOR/Hb (Molar Ratio) FIG.111.12. Effect of pH, DPG, and IHP on deoxyhemoglobin S solubility. (a) Effect of [From Goldberg et al. pH on solubility in 0.06 M sodium phosphate buffer at 25°C (0). (1977).] (b) Effect of pH on solubility at 20°C in 0.05 M Bis-Tris, 0.1 M NaCl (a),plus 10 mM DPG (A),plus 10 mM IHP (H).[From Briehl (1978).] (c) Effect of DPG (0and 0 ) and IHP (Aand A) on solubility of native (0 and A) and cross-linked (0and A) deoxyhemoglobin S in 0.1 M Bis-Tris buffer, pH 6.8, at 30°C. [From Poillon et al. (1986a).] T h e cross-link is between Lys-82pI and Lys-82p2 with bis(3,5-dibromosalicyl)adipate which fills the cleft between the p subunits. The small effect of DPG and I H P on the solubility of the cross-linked molecules suggests that the binding of the effectors in this cleft is responsible for the decrease in the solubility of the native molecule.

130

WILI.IAM A. EATON AND.JAMES HOFRICHTER

intermolecular contacts. So far, the structural data have given no indication as to which residues are involved. T h e only histidines in the lateral or axial contacts of the crystal double strand that form hydrogen bonds do so with lysines (His 0116-Lys pl6 and His p l l 7 - L ~p17 ~ of strand 2, Table 11.3), so that increasing the pH would stabilize the contact and lower the solubility, which is the opposite of what is observed. The data in Fig. 111.12 show that DPG decreases the solubility by about 15% when added to stripped hemoglobin, while the effect of inositol hexaphosphate (IHP) is much larger, almost halving the solubility in the pH range 6-7 (Briehl, 1978; Poillon ct al., 1985, 1986a,b). T h e X-ray data do not provide an explanation for these effects, since DPG was added to the crystal preparation, but was not detected in the electron density map (Padlan and Love, 1985a). The effect of ionic strength depends on the particular salt used. An extensive study of salt effects shows that, for the sodium and potassium salts of monovalent anions, the solubility increases with increasing concentration in the range 0-0.3 M , with slopes, dc,/d[salt], varying from about 0.05 g/cm3/M for KCl and NaCl to 0.4 g/cm3/Mfor NaSCN (PoilIon and Bertles, 1979). These results suggest that there are favorable electrostatic interactions between molecules and that these are being damped by the increasing ionic strength. They also suggest that there is some dissociation of ions from the protein on polymerization. The effect of potassium phosphate is particularly interesting. T h e solubility increases slightly between 0 and 0.3 it4 (Rookchin and Nagel, 1973b), but then decreases exponentially with increasing phosphate to about lo-’ g/cmJ at 1.8 M (Poillon and Bertles, 1979; Adachi and Asakura, 1979a). Over this range of salt concentrations, a birefringent gel is formed with approximately the same kinetic progress curve, exhibiting a marked delay prior to the appearance of polymer (Adachi and Asakura, 1979a). In 1.8 M phosphate the solubility of deoxyhemoglobin A is about 0.02 g/cm3and, surprisingly, also exhibits a delay time (Adachi and Asakura, 1979b), suggesting that a polymer with a structure very similar to that of deoxyhemoglobin S is being formed. A possible structural rationalization is that the negatively charged p6 glutamate of deoxyhemoglobin A is being neutralized at the high cation concentration, removing the unfavorable electrostatic interaction with the p73 aspartate in the acceptor site for p6 (Table 11.2).

C. Control of Polymerization by Oxygen The physiological variable to which hemoglobin S polymerization is most sensitive is oxygen, and it is therefore essential to understand in detail how ligand binding influences the thermodynamics of polymeriza-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

131

tion. In the context of the two-phase model, a complete thermodynamic description is provided by the dependence of the solubility on oxygen saturation and the oxygen binding curves for the solution and polymer phases. For a two-phase system, any two of these three measurable properties can be used to calculate the third. For example, the solubility can be directly calculated from the ligand binding curves of the two phases. We shall see that the results are consistent with a two-phase model for the gel over the entire range of oxygen pressures (Fig. 111.13). There is a very simple molecular interpretation of the results. The control of polymerization by oxygen can be nearly quantitatively explained by a model in which all hemoglobin molecules having the deoxy(T) quater-

H

FIG. 111.13. Schematic structure of hemoglobin S gel at partial saturation with oxygen.

(0) Deoxyhemoglobin molecules and (0)molecules with one or more oxygens bound. Bar, 20 nm. [From Sunshine et al. (1982).]

132

WILLIAM A. EATON A N D .JAMES HOFKI<:HTEH

nary structure polymerize with equal probability, independent of the number of oxygen molecules bound, while molecules having the oxy(R) quaternary structure do not copolymerize at all. In Section V we shall show that the gel data can be used to explain a variety of experiments on the effect of oxygen on morphological cell sickling and intracellular polymerization. Although it has been known for 40 years that fully deoxygenated hemoglobin S gels while fully oxygenated hemoglobin S does not (Harris, 1950), the solubilities and binding curves have not been determined until relatively recently. The reason is that it is technically quite difficult to make precise measurements on concentrated solutions and gels at partial saturation with oxygen. The most straightforward measurements are the determination of the solubility as a function of solution phase saturation. In these experiments, the polymers in a partially saturated gel are sedimented by high-speed centrifugation, and the composition of the supernatant is determined from spectrophotometry in the near-infrared, where oxy- and deoxyhemoglobin have characteristic charge-transfer optical absorption bands (Eaton et aE., 1978; Eaton and Hofrichter, 1981) (Fig. 111.14). A major technical difficulty in these experiments is that the sample composition is not stable because of extensive oxidation 2.0I

2.0

I

b

I.5 0

1.o

? z

0.5

800

900

1200

0

WAVELENGTH (nrn)

FIG. 111.14. Near-infrared absorption spectra. (a) Ungellcd 0.32-g/cm3sample at 0°C. 45% saturated with oxygen. (b) Solution phase of same sample (58% saturated) after gelation and sedimentation of polymers at 23.5"C. (---) 'The spectra of the pure components (HbO,, Hh, and MetHb) that were used in a least-squares synthesis of the observed (obs) spectruni (-). The sums of the component spectra [also (---)I superimpose on the observed spectra. [From Sunshine el al. (1982).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

133

c

0.150

0.5 1.o SOLUTION FRACTIONAL SATURATION

FIG. 111.15. Hemoglobin S solubility as a function of solution phase saturation with oxygen and carbon monoxide. (0)Oxygen at 23.5"C (Sunshine et ol., 1982) and (0)carbon monoxide at 25°C (Hofrichter, 1979). (---) Obtained from a least-squares fit with the empirical function c, (g/cm5) = 0.183 + 0.0924y, 0.0980ya + 0.235y!5. (-) Calculated from Eq. (111.18) using the observed solution and polymer binding curves. [Lower (-....)I Calculated from Eq. (111.18) for the hypothetical case that the polymer affinity is the same as solution T state molecules [Eq. (III.20)]. [Upper (..*)I T h e hypothetical case that the polymer binds no oxygen at all pressures. (---) T h e result calculated from Eq. (111.18) without the term [(l/cp)- v ] / [ ( l / c ,-) vl.

+

to methemoglobin. Consequently, an enzyme system was used to reduce methemoglobin, and the spectral measurements were made as a function of time because of slow oxygen consumption associated with the reduction. Figure III.15 shows the solubility data for oxygen (Sunshine et al., 1982) and carbon monoxide (Hofrichter, 1979) at room temperature. At high oxygen saturations the solubility increases sharply and is greater than 0.4 g/cm3 at 90% saturation. As mentioned earlier, the solubility of fully saturated carbonmonoxyhemoglobin S is greater than 0.5 g/cm3 (Briehl and Ewert, 1974). Two other techniques have been used to obtain solubilities (Gill et al., 1980). In one, the oxygen pressure was increased until the aligned polymer phase, which was detected by the presence of linear birefringence, disappeared. The solubility at this saturation is simply the total hemoglobin concentration of the sample. In the second method, the relative fraction of polymerized hemoglobin was calculated from the magnitude

134

WILLIAM A. EATON AND JAMES HOFRICHTER

of the linear birefringence (Ross el al., 1975) at a given oxygen pressure compared to the magnitude at zero oxygen pressure where the fraction has been measured in a sedimentation experiment (Gill et al., 1980). The solubilities obtained from these two birefringence experiments are somewhat lower (about 10% lower at 85% saturation) and much less precise than those determined from sedimentation experiments. Kinetic effects, in which the polymer has not yet melted to its equilibrium value as the oxygen pressure is increased, or deviations from the assumed theoretical relation between birefringence and fraction of polymerized hemoglobin, might be responsible for the lower values. There are as yet no precise measurements of the solubility as a function of oxygen saturation at different temperatures (see Gill et al., 1980). There are, however, data for carbon monoxide, where the sample composition is stable indefinitely (Hofrichter, 1979). These data show that the solubility versus saturation curves are approximately parallel, i.e., they are displaced at all saturations by the difference in the solubility at zero carbon monoxide saturation. Because of the identical solubility versus saturation curves of oxygen and carbon monoxide at room temperature (Fig. III.15), it is probably safe to assume that the solubility versus saturation curves for oxygen are also displaced in the same way. We can, therefore, calculate the solubility and fraction of polymerized hemoglobin at any temperature and any oxygen saturation by combining the empirical functions that have been used to fit each curve (see legends to Figs. 111.7 and 111.15).The resulting equation is cs(g/cm3)= 0.321 - 0.00883T

+ 0.000125T2

+ 0 . 0 9 2 4 ~+~ ~0.0980~~:+ 0 . 2 3 5 ~ : ~ (111.13)

where yS is the fractional saturation of the solution phase with oxygen and 1 is the temperature in degrees Celsius. This equation will be extremely valuable in describing polymerization experiments on intact red cells in Section V. Figure 111.16 shows the fraction of polymerized hemoglobin as a function of solution phase oxygen saturation at 37°C for various total hemoglobin concentrations calculated from Eqs. (111.13) and (111.5). At a total concentration of 0.45 g/cm3, which is near the highest values observed for red cells, polymer is present up to a solution phase fractional saturation of 0.95. This result is significant for the kinetics of polymerization in vivo, and will be discussed in Section VI. To complete the thermodynamic description, we need to know the solution and polymer phase binding curves. The solution phase binding curve of hemoglobin S has been measured using thin-layer techniques a1 0.15 g/cmS,which isjust below the solubility at zero saturation of 0.18 g/cm'3.The curve is identical to that of hemoglobin A (Fig. 11.6) (Gill et

SICKLE CELL HEMOGLOBIN POLYMERIZATION

135

' 9 t 1

.-

0.5

c

0

c

rr V

L

0

0

0.5

1

Fractional Saturation

FIG. 111.16. Fraction of' polymerized hemoglobin S as a function of solution phase oxygen saturation at 37°C. Calculated from Eqs. (111.5) and (111.13) with cp = 0.69 g/cm3. The curves were calculated with total concentrations (co) varying from (a) 0.20 g/cm3 to (b) 0.45 g/cm3at 0.05 g/cm3 intervals.

al., 1979). Furthermore, at higher concentrations the binding curves are identical at high saturations where no polymer is present (Gill et al., 1979; Pumphrey et al., 1979). These findings suggest that the solution phase of a gel binds oxygen normally, and in the subsequent analysis we shall see that this assumption is consistent with all of the data. The most interesting and difficult quantity to measure is the polymer binding curve. Because the solution phase is also present in the pellet obtained by high-speed sedimentation, it is not possible to separate the polymer phase cleanly and to determine its composition by ordinary absorption spectroscopy. Linear dichroism, which is the difference in light absorption for light linearly polarized in mutually perpendicular directions (Hofrichter and Eaton, 1976; Hofrichter, 1979; Eaton and Hofrichter, 1981), has proved to be a powerful technique for this purpose. Polarized absorption studies on well-oriented polymers in single sickled cells showed that the polymers exhibit a large linear dichroism (Hofrichter et al., 1973; Eaton and Hofrichter, 1981). The extinction coefficient for light polarized perpendicular to the polymer axis is about 4 times the extinction coefficient for light polarized parallel to the polymer axis (Eaton and Hofrichter, 1981). Since solution phase molecules are randomly oriented, they absorb light of all polarizations equally, and, therefore, make no contribution to the linear dichroism of a gel.

136

WILLIAM A. EATON A N D JAMES HOFRICHTER

Frc. 111.17. Optical micrograph of hemoglobin S gel partially saturated with oxygen viewed through crossed linear polarizers. The dark regions correspond to regions where the long axis of the polymers is parallel to one of the polarizers. The large domain of polymers in the center of the picture (0.2 mm in diameter) was formed by focusing an argon ion laser to about 0.02 mm in diaincter and heating the solution from 0°C at about l"C/hr. The heating of t.he laser results in nucleatiou and growth of a single domain of polymers sufficiently large for measurements before smaller adjacent domains form by spontaiieous nucleation. [From Sunshine et al. (1982).]

137

SICKLE CELL HEMOGLOBIN POLYMERIZATION

A major technical difficulty in the measurement of linear dichroism on gels is the rapid preparation of sufficiently large regions of wellaligned polymers. This was accomplished by nucleating a single domain of polymers by heating a small region with a focused argon ion laser (Fig. 111.17) (Sunshine et al., 1982). Figure 111.18 shows an example of

0.010 g

B

T 8

c

0.005

F; A

I 51

0

v)

5

0 0

E

0 -

WAVELENGTH (nm)

FIG. 111.18. Polarized absorption and linear dichroism spectra of a hemoglobin S gel partially saturated with oxygen. (a) Polarized absorption spectra with light polarized parallel (OD,,)or perpendicular (OD,) to the long axis of the polymers. (b) Isotropic spectrum (-) calculated from the data in a using the relation OD,5u= 1/3(20D, OD,,),and leastsquares synthesis from deoxyhemoglobin (Hb) and oxyhemoglobin (HbOn) spectra (---). (c) Observed (obs) linear dichroism (-) of sample in a and least-squares synthesis from deoxyhemoglobin linear dichroism (Hb LD) spectrum (---) and oxyhemoglobin solution (HbOs s o h . abs.) spectrum (---). (d) Comparison of oxyhemoglobin solution (HbOn soh. abs.) spectrum (---) and spectrum (-) obtained by subtracting the deoxyhemoglobin linear dichroism (Hb LD) component from the observed (obs) linear dichroism spectrum in c. [From Sunshine et al. (1982).]

+

138

WILLIAM A. EATON AND JAMES HOFRICHTER

the polarized absorption and linear dichroism spectra of a gel at partial saturation with oxygen measured with a microspectrophotometer. T h e polarized absorption spectrum is used to determine the total saturation of the gel (solution plus polymer), while the linear dichroism measures only the polymer phase saturation. ‘The solution saturation corresponding to each polymer saturation is obtained from sedimention results on a gel of the same total concentration and saturation. Finally, the oxygen pressure associated with each polymer saturation is obtained from the solution phase binding curve. Figure 111.19 shows the polymer binding curve, the solution binding curve, and the binding curve of a gel. The polymer has a much lower affinity than the solution phase molecules, and, unlike the solution phase molecules, the polymer binds oxygen noncooperatively u p to the highest measured saturation of 0.14. Before discussing a molecular interpretation of the polymer binding curve it is important to show that all of the results are thermodynamically self-consistent. The data analysis assumes a two-phase model in which the gel consists of a solution phase containing hemoglobin monomers that exhibit a normal cooperative binding curve and a polymer phase of constant total hemoglobin concentration, c,, = 0.69 g/cm3. This model is consistent with the data according to two critical tests. First, the measured gel binding curve agrees with the curve calculated from the solution and polymer phase binding curves (Fig. 111.19).T h e gel binding curve can be calculated from the oxygen mass conservation relation: Y L

= ypxp

+ ys(1

-

x,,)

(111.14)

where y, is the total saturation of the gel and yp is the polymer saturation given by (111.15) and ys is the saturation of the solution phase molecules. Any function which accurately describes the solution phase binding curve (ys versus pressure) may be employed, such as the two-state allosteric model saturation function (Monod et al., 1965): ys

=

LK,p(l L(1

+ K T ~ +) K,p(l ~ + K@)‘ + KTP)4 + ( 1 + KRP)4

(I11.16)

where L ( = T o / R o )is the population ratio of zero-liganded T- and Rstate molecules, and KT and KKare their intrinsic association constants.

139

SICKLE CELL HEMOGLOBIN POLYMERIZATION

I

I

-0.8

'

-1.2

'/ I

I

I

I

/

Polymer

-1

2

1

0

Log Oxygen Pressure (torr) FIG. 111.19. Solution, polymer, and gel binding curves. The solution (0)and gel binding (0and A) data were measured in 0.15 M potassium phosphate buffer (pH 7.2) at 25°C (Gill etal., 1980). The polymer ( + ) saturations were measured in 0.15 M potassium phosphate buffer (pH 7.0) at 23.5%. This buffer also contained 0.02 M glucose 6-phosphate, which was slowly converted to glucose by the methemoglobin reductase system in the course of the measurements. (Inset) The polymer binding curve on an expanded scale. The straight line through the polymer (0)data was obtained from a least-squares fit with a slope = 0.97 0.09 and intercept = -2.19 2 0.10. The curve through the solution (0)data was obtained from a least-squares fit using the allosteric saturation function [Eq. (111.16)] with L = T J R , = 60,500, K , = 1.47 ton-1, and KT = 0.0160 torr-1. The curve through the gel binding (0 and A) data was calculated from Eq. (111.14). (0) The gel binding data obtained from oxygenation experiments and (A) the gel binding data obtained from deoxygenation experiments (Gill et al., 1980). The hysteresis results from the kinetics of polymerization and depolymerization. [From Sunshine et al. (1982).]

*

The second test of the analysis in terms of a two-phase model is more interesting, and uses a thermodynamic relation between the solubility data and the binding curves that derives from the Gibbs-Duhem relation for each phase (Wyman, 1964; Minton, 1976a; Gill et al., 1980; Benedict et al., 1981): VHb d P H b

+

+

v02 d/-bpv H 2 0 d P H 2 0

==

(111.17)

140

WILLIAM A. EATON A N D JAMES HOFRICHTER

where v, is the number of moles of component i and p, is the chemical potential of‘component i. ‘The working equation (Minton, 1976b; Benedict et al., 1981; Sunshine et al., 1982) is In

(I,=

In np

+4

I”b’

Y. - Y

1 - [(l/c,

-

z,)hcs

- v)]

d In

p

(111.18)

where a, = y ~ is, the solution phase hemoglobin activity at the oxygen pressure p, u: = y!c: is the activity at zero oxygen pressure, and v is the partial specific volume taken as 0.75 cm3/g for both solution and polymer phases. The term (l/c,, - v)/(l/cs - v) corresponds to the number of moles of water per mole of hemoglobin in the polymer phase divided by the number of‘moles of water per mole of hemoglobin in the solution phase. This term becomes increasingly significant as the oxygen pressure is raised and the solubility becomes high. (Notice that for a single component crystal it is zero.) After substituting for yF from Eq. (111.16) and for y,, from Eq. (111.15), Eq. (111.18) can be solved iteratively by numerical integration until the calculated solubility versus pressure curve equals the input solubility versus pressure curve. T h e first approximation to the solubility versus pressure curve is obtained by evaluating the integral in Eq. (111.18) with the term (l/c,, - v)/(l/cs - v) set equal to zero. Using the saturation at each oxygen pressure from the solution binding curve, the calculated solubility versus saturation curve is compared with the observed curve in Fig. 111.15. The agreement is excellent, and the small differences are within the experimental uncertainties of the measurements and the parameters used in the calculations. The agreement can also be assessed by varying the binding constant, Kp, in solving Eq. (111.18) to get the best least-squares fit to the solubility data. When this is done, KPis found to be 0.0069 torr-I, in excellent agreement with the value of K , = 0.0059 0.0015 torr-’ determined from the polymer binding curve. From Eq. (111.18) it can be seen that given any two of the three functional relations-solubility versus pressure curve, solution binding curve, and polymer binding curve-the third can be calculated. The self-consistency among the four data sets [solubility versus saturation and polymer saturation versus solution saturation curves (Sunshine et nl., 1982) and the solution and gel binding curves (Gill et al., 1980)] provides convincing evidence that the two-phase model, in which the solution phase molecules have a normal cooperative binding curve, is an adequate basis for the thermodynamic analysis of the effect of oxygen on polymerimtion. To give a molecular interpretation to the results it is useful to consider

*

SICKLE CELL HEMOGLOBIN POLYMERIZATION

141

the two-state allosteric model (Monod et al., 1965). This model provides an excellent first-order description of cooperativity in hemoglobin and can rationalize a large body of data on hemoglobin structure-function relations (Perutz, 1970; Szabo and Karplus, 1972; Shulman et al., 1975; Edelstein, 1975; Baldwin and Chothia, 1979; Perutz et. al., 1987). According to the two-state allosteric model, hemoglobin exists in two affinity states, called T and R. The T state corresponds to the quaternary structure of fully deoxygenated hemoglobin, while the R state corresponds to the quaternary structure of fully oxygenated hemoglobin. The central features of this model are that binding within each quaternary structure is noncooperative, and that the sigmoid binding curve (solution binding curve of Fig. 111.19) results from the increase in the fraction of the high-affinity R-state molecules as the average fractional saturation increases. The conceptually simplest model for the effect of oxygen on polymerization is a straightforward extension of the two-state allosteric model (Sunshine et al., 1982). Because of the thermodynamic relation between solubility and oxygen binding expressed by Eq. (111.18), there are two equivalent descriptions of the model. In terms of ligand binding the model postulates that the polymer binds oxygen noncooperatively with an affinity that is the same as that of solution phase T-state molecules. In terms of a solubility analysis, the model postulates that R-state molecules cannot polymerize at all, while T-state molecules polymerize with equal probability, independent of the number of oxygen molecules bound.I4 Both descriptions assume that oxygenation has no effect on the intermolecular interactions in the polymer, which, in terms of binding, postulates that there is no positive o r negative cooperativity. T h e binding curve for the polymer, then, should be noncooperative with the same affinity as solution T-state molecules. Complete exclusion of R-state molecules is consistent with the structural data. There is a large difference in the arrangement of the subunits in the T and R quaternary structures (Baldwin and Chothia, 1979), and model building studies show that the R structure cannot fit into the double strand of the deoxyhemoglobin S crystal (Padlan and Love, 198513). The model can immediately explain the major features of the polymer binding curve. The polymer exhibits a noncooperative binding curve up to the highest measured saturation of 0.14, and has an affinity which is It is interesting that both methemoglobin S and nitrosylhemoglobin S aggregate in the presence of IHP, which shifts the R-T equilibrium in favor of the T quaternary structure (Briehl and Ewert, 1974; Briehl and Salhany, 1975). Birefringent gels are formed, but there are as yet no data on the detailed structure of the aggregated material.

142

WILLIAM A. EATON A N D JAMES HOFRICH'IER

very similar to that of solution phase T-state molecules. T h e polymer affinity is only 3 times smaller than the T-state affinity (K, = 0.016 torr-I, KP = 0.0059 torr-'), while the T-state affinity is 90 times smaller than the K-state afinity (K, = 1.47 torr-I). A thermodynamically equivalent comparison of the data with predictions of the model can be made using the solubility data. T h e solubility can be calculated in terms of polymerization probabilities of the individual molecular species from the relation (Minton, 1976a, 1977; Benedict et al., 1981; Sunshine et al., 1982):

(I I I . 19) where x, is the mole fraction of molecular species i, e, is its probability of polymerization relative to zero-liganded hemoglobin S, and the water activities are calculated from Eq. (111.9).For a perfectly T-state polymer, e, is unity for all T-state species and Eq. (111.19)becomes

l/c 4

(nc~lr~c~)(aH,olaR,o)" =

r=O

XT,

where xT, is the mole fraction of T-state molecules in solution with i oxygen molecules bound. T h e solubility for a perfectly T-state polymer is calculated from Eq. (111.20) [or equivalently from Eq. (111.18)with Kp = KT in Eq. (III.15)] to be lower than the observed values (Fig. III.15), indicating that liganded T-state molecules copolymerize with a lower probability than fully deoxygenated hemoglobin S . If we now turn t o the more realistic case of noncooperative binding by the polymer with an affinity, Kp, different from that of the 'r state, the

The equivalence of Eq. (111.21)and Eq. (111.18)can be observed by recognizing that the term on the right-hand side of Eq. (111.21)contains the ratio of binding polynomials for the solution and the polymer (QlL9p) (Wyman, 1964; Hess and Szabo, 1979), and that the fractional satura-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

143

tion is related to the binding polynomial by 4y = d In Q / d In p . T h e term (Kp/KT)i is the probability that a T-state molecule with i oxygen molecules bound will polymerize relative to one with zero oxygen molecules bound. Using the relative affinities obtained from the fits to the data in Fig. 111.19, this probability is 0.37 for singly liganded T-state molecules, 0.37* for doubly liganded T-state molecules, and so on. The decreased affinity of the polymer for oxygen compared to solution T-state molecules (or, alternatively, the decreased probability of polymerization for liganded T-state molecules) indicates that the intermolecular contacts in the polymer alter the T-state structure to some extent. One possibility is that small tertiary conformational changes associated with ligand binding to the T state become less favorable as a result of the constraints imposed by the intermolecular contacts (Sunshine et al., 1982). For example, the acceptor site for the p6 region in the lateral contact of the double strand contains residues of the E and F helices, and small conformational changes are known to occur in this part of the molecule on ligand binding to the T state (Anderson, 1975; Liddington et al., 1988). From the preceding discussion it is clear that the control of polymerization by oxygen is relatively well understood. To extend the analysis further will require considerably more experimental data. Precise data are only available for a single set of solution conditions, and it would be important to have comparable information on the solubility and polymer binding curve at different temperatures and at various concentrations of DPG, protons, and other physiologically relevant solution components. It would also be interesting to extend the polymer binding curve to higher saturations. Such experiments are necessary to establish whether there is any significant heterogeneity resulting from the large number of structurally inequivalent hemes in the polymer (28 in the structure of Dykes et al., 1979), or whether there is any cooperativity resulting from intermolecular interactions in the polymer.

D. Polymerization of Mixtures of Hemoglobin S with Other Hemoglobins So far, we have discussed the effects of temperature, pH, allosteric effectors, salts, and oxygen on the thermodynamics of hemoglobin S polymerization. We conclude our discussion of equilibrium studies with a description of the influence of non-S hemoglobins on the polymerization process. There have been two principal motivations for the study of hemoglobin S polymerization in mixtures with other hemoglobins. First, hemoglobin S frequently occurs in red cells in combination with a large fraction of other hemoglobins, and increased proportions of normal

144

WILLIAM

A. E AT ON A N D J A M E S HOFKICHTEK

hemoglobin (hemoglobin A), fetal hemoglobin (hemoglobin F), and hemoglobin C result in decreased clinical severity (Serjeant, 1985; Bunn and Forget, 1986). Second, as already discussed in Section II,C, polymerization studies on mixtures have made a major contribution to the elucidation of the polymer structure, Considerable information on the location of intermolecular contacts in the polymer has been obtained from the qualitative analysis of minimum gelling concentration experiments in mixtures of hemoglobin S with mutant hemoglobins having a single amino acid substitution in either the a or /3 subunits. For these studies it is important to have a thermodynamic description in order to make correct inferences concerning the structure. The discussion in this section will be limited to studies on completely deoxygenated mixtures of hemoglobin S with hemoglobins A, A,, C, and F, since these are the physiologically most important mixtures and the ones for which the most extensive thermodynamic data are available. We shall see that the results are consistent with what is known about the interniolecular contacts in the polymer. Many of the structural interpretations of the studies presented here have been anticipated in the discussion of Section I1,C. The clinical relevance of both the thermodynamic and kinetic studies on mixtures will be discussed in Section VI. The fraction of hemoglobin S in red cells differs considerably among the various heterozygotes and double heterozygotes. T h e most common situation is sickle trait, the benign heterozygous state, in which there is 30-40% hemoglobin S and 70-60% hemoglobin A (Wells and Itano, 1951; Huisman, 1977; Brittenham, 1977; Set-jeant, 1985). There are also two relatively common double heterozygous conditions: SC disease and sickle p+-thalassemia. In SC disease, the red cell contains 50-60% hemoglobin S and 50-40% hemoglobin C ( p 6 Glu +- Lys) (Huisman, 1977; Bannerman et al., 1979).In sickle p+-thalassemia, the presence of the thalassemia results in decreased production of PA subunits and the red cell contains 20-30% hernoglobin A and 80-70% hemoglobin S (Serjeant, 1985). A rare, but important double heterozygous condition is sickle cell disease with hereditary persistence of fetal hemoglobin, in which the red cells contain 20-35% hemoglobin F and 80-65% hemoglobin S (Weatherall and Clegg, 1972; Serjeant, 1985). As discussed earlier (Section II,C), hemoglobin dissociates along the interface between the aP dimers related by the exact 2-fold axis (the a,& and a&, interfaces). Dimers containing unlike p subunits can then reassociate to form so-called hybrid molecules. The dissociation constant is sufficiently small that there is a negligible equilibrium concentration of dimers at the high concentrations employed in polymerization experiments (Smith and Ackers, 1985). If all the tetranieric species have

SICKLE CELL HEMOGLOBIN POLYMERIZATION

145

the same dissociation constant, then a binomial distribution will be present at equilibrium. For example, in a mixture containing mole fraction X of hemoglobin A and (1 - X) of hemoglobin S, the mole fraction of the individual molecular species will be XI = X(a&f) = (1 - X)*

= X(a2pSpA) = 2X(1 x3 = X ( a 4 $ ) = xz x 2

-

X)

(111.22)

A distribution very close to binomial has been observed for oxyhemoglobin S mixed with oxyhemoglobin A, C, or F (Bunn, 1972; Bunn and McDonough, 1974). Such a distribution is expected for these mixtures from functional and structural studies. The oxygen binding curves for hemoglobins A, S, and C in dilute solution are identical (Bunn, 1972). Since dimers are present in significant amounts, the identity of the binding curves indicates that the tetramer-dimer dissociation constants are also the same. T h e three-dimensional structure of the interface between dimers is the same in a&$ and anP$molecules (Padlan and Love, 1985a), and therefore should also be the same in the a2ps/3Ahybrid molecule, suggesting equal dissociation constants for all three molecules. In hemoglobin C the replacement of glutamate by lysine at the p6 position would not be expected to significantly alter its dissociation properties, since this residue position is far from the a l p 2interface. The amino acid sequence of the y chain differs from the PAchain in 39 of the 146 residues, but none are in the a l p Binterface in the oxy quaternary structure (Frier and Perutz, 1977). A binomial distribution is also expected for oxygenated mixtures of hemoglobins S and A2. T h e sequence of the 6 chain differs from the PAchain in 10 residues (Bunn and Forget, 1986). However, none of these differences occurs in the alp2interface. In all of these mixtures, differences of net charge of the p subunits might lead to small differences in dissociation constants (Williams and Kim, 1975; Mrabet et al., 1986; Bunn, 1987), resulting in deviations from a binomial distribution. In the analysis that follows we shall assume that the binomial distribution is exact. The presence or absence of hybrid molecules depends on whether the mixture is prepared with oxygenated or deoxygenated hemoglobins. The rate constant for dissociation of oxygenated tetramers into dimers at room temperature is about 1 sec-1 (Smith and Ackers, 1985). As a result, an equilibrium distribution of molecular species is obtained within seconds after preparing a mixture of oxygenated hemoglobins. Once the oxygen is removed, however, dissociation of tetramers into

146

WILLIAM A. EATON ANDJAMES I-IOFRICHTER

dimers is slowed enormously. The half-time for dissociation of deoxy tetramers at room temperature is 1-10 hr and about 1 hr at 37°C (Bunn and McDonough, 1974; Ip and Ackers, 1977; Smith and Ackers, 1985). Because of these kinetics, it is possible to carry out two types of experiments for each mixture. In the first, the two hemoglobins are mixed in the deoxygenated state, polymerized, and then centrifuged to separate the solution and polymer phases. If the temperature is kept low enough and polymerization and sedimentation carried out sufficiently rapidly, no significant formation of hybrid tetramers takes place. In the second case, the oxygenated solutions are mixed, followed by deoxygenation, polymerization, and sedimentation. If polymerization and centrifugation are performed quickly, the distribution of species in the total sample is fixed (kinetically “frozen”) at the binomial distribution prior to polymerization, and the distribution of species in the solution phase is no longer binomial. In the simplest thermodynamic model for polymerization in mixtures there are only two phases and a single polymer structure. T h e solution phase consists of hemoglobin monomers having only hard sphereexcluded volume interactions, so that the activity coefficients for all the species are the same, and the copolymers of the S and non-S hemoglobin molecules have the same structure as the pure deoxyhemoglobin S polymer. The important parameters for relating the thermodynamics of polymerization to structural properties are the relative copolymerization probabilities of the individual molecular species in the mixture. In the ideal experiment, the concentration of the individual species in both the solution and polymer phases would be measured over a range of compositions. The copolymerization probabilities relative to the a2/3$ molecule, e , , could then be evaluated from (Minton, 197413, 1975, 1976a, 1977)

f;

= x , e , / c x,e, I

(111.23)

where f i is the mole fraction of species i in the polymer phase, x, is the mole fraction of species z in the solution phase, arid e, is the copolymerbation probability. Equation (111.23) [and Eq. (I11.19)] results from equating the chemical potentials of each species in the two phases, assuming that there is only a single repeating lattice site in the polymer and that the relative copolymerization probabilities are independent of the polymer composition. Unfortunately, it has not yet been possible to carry out experiments in which the fractions of the individual molecular species in both the solution and pure polymer phases are accurately measured. Conse-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

147

quently, the copolymerization probabilities have been obtained indirectly by modeling experiments containing much less information. The most frequently performed experiment has been one in which there are only three measured quantities: the total hemoglobin concentration, co , the total mole fraction of the non-S hemoglobin, X , and the solubility, c,, which is the total hemoglobin concentration (sum of all molecular species) in the supernatant obtained after sedimenting the polymers. In these experiments, there are several effects which determine the solubility and therefore the extent of polymerization according to the simple copolymerization model (Sunshine et al., 1979b). These include the formation of hybrid molecules containing a ps subunit and a non-S /3 subunit, the copolymerization probability of the non-S hemoglobin and the hybrid molecule, the large activity coefficient, and the total concentration of all hemoglobin species in the sample. T h e solubility is expected to depend on the total concentration of the sample because of two effects. First, the species which have the lowest tendency to polymerize collect in the solution phase and, as a result, their concentration as well as the total concentration of the solution phase increases with increasing total hemoglobin concentration. This effect is partially compensated by the increase in the activity coefficient of the polymerizing species in the solution phase which results from the increase in total hemoglobin concentration in the solution phase. Remember that the activity coefficient, which arises solely from excluded volume effects, increases dramatically as the total concentration of hemoglobin in the solution phase increases (Fig. 111.6). The copolymerization probabilities can be evaluated in a straightforward fashion from the results of the solubility measurements and are given in Tables 111.2 and 111.3. For the case in which no hybrids are present Eq. (111.19) becomes (Sunshine et al., 1979b, 1982; Benedict et al., 1981)

+

( ~ ~ ~ ~ l ~ ~ c ~ ) (=a 1/C ~ , xiei ~ l a=~l/(x, ~ , o ) x3e3) ,

(111.24)

molecules where x1 and x3 are the mole fractions of the a2p$and a2/3i)on-S in the solution phase, and e3 is the copolymerization probability of the non-S molecules. After introducing the mass conservation relations for each species in the solution and polymer phases:

(I11.25) for i = 1, 3. Equations (111.23)-(111.25) can be solved for e, (Sunshine et al., 1979b, 1982) (see footnotes to Table 111.2). T h e results for the S + F, S + A, and S +A, mixtures are very similar, with e3 values that are

148

WILLIAM A. E A T O N A N D JAMES HOFRICHTER

'TABLE

111.2

Copolymeriration Probabilities of a 2/3 gall-' Molecules ( e 3) in Unhybndized Mixture.
Mixture

S + F

S + A

S

+ A2

S

+ BSA

TemperaLure ("C)

Solubility

0.18-0.54

20 2.5 30

-0.01 2 0.04 -0.15 f 0.06

0.09 0.46-0.59 0.26-0.59

20 25 30

-0.05 & 0.05 -0.01 t 0.03

0.17-0.59

30

-0.17 -+ 0.12

0.09

20

Xb

0.15

0.49-0.63

es

Composition 0.14 0.16 -0.14 (0.08 0.12 0.38 0.20 (0.24 -0.10 (0.14 0.14

-+

0.03

t 0.12 2 0.07

Reference e

f

* 0.0.5)"

g

* 0.22

.t

2 0.02

0.12 0.05)" k 0.13 t 0.04)h f 0.03

+2

g

e

R g

g g P

"In these mixtures, the hemoglobins are first deoxygenated and then mixed to prevent formation of hybrid molecules containing one /3 subunit from each hemoglobin. This is most conveniently accomplished by adding methemoglobin S to the non-S deoxyhemoglobin, and then reducing the methemoglobin S to dcoxyhemoglobin S with the sodium dithionite (Goldberg rt al., 1977). "This is the range of inole fractions of the non-S hemoglobin in the total sample. (This is the copolymerization probability for the non-S hemoglobin molecules and is calculated from the total mole fraction of non-S hemoglobin, X , the total hemoglobin concentration in the sample, co, and the solubility, es, using the relation that derives from Eqs. (111.23)-(111.25): e3 =

cfr[(c,- cpr)(cp - co) - x(cp- c,)col (cy- cs)(co - c , ) c p - xc;r(cp- c,)co

where ci' is the solubility of pure deoxyhenioglobin S and c p is the concentration of the polymer phase, taken as 0.69 g/cni3. T h e quantity r is defined as

which from Eq. (111.4) and Eq. (111.9) becomes

"his is the copolymerization probability for the nan-S hemoglobin molecule and is calculated from the total mole fraction of the non-S henloglobin, X , the total hemoglobin concentration, ro, the solubility, e , , and the mole fraction of the non-S hemoglobin in the supernatant, x:,, using the relation that derives from Eqs. (111.23) and (111.25):

SICKLE CELL HEMOGLOBIN POLYMERIZATION

149

either zero or small negative numbers (Table 111.2). A negative value for e3 results when the measured solubility is higher than that calculated for no copolymerization (e3 = 0) of the non-S hemoglobin. These data would suggest, then, that there is little or no copolymerization of the non-S hemoglobin when hybrid formation is prevented. This result simplifies the analysis of the solubility data for mixtures in which hybrid molecules are formed. Assuming e3 = 0, Eq. (111.24) becomes (rsc,/rpcs”)(a”,o/a“H20)” = 14x1 +

x2e2)

(II I. 26)

where x2 is the mole fraction of the hybrid molecules in the solution phase. Equations (111.22),(111.23),(111.25), and (111.26) can be solved to give an equation from which e2 can be calculated from the measured quantities by numerical analysis (Sunshine et al., 1979b, 1982) (see footnotes to Table 111.3). T h e results of this analysis are given in Table 111.3. For S + F and S + A2 mixtures, e2 is calculated to be zero o r a small positive number, suggesting little o r no copolymerization of the a2PSy and a2Ps6hybrid molecules. T h e excellent agreement between the theory with e2 = 0 and experiment for S + F mixtures can be seen in Fig. II1.20a, where the solubility at a constant total hemoglobin concentration is plotted versus the total mole fraction of hemoglobin F. For S A mixtures, the values of e2 in the different studies range from 0.3 to 0.5 (Fig. III.20b), and for S C mixtures from 0.4 to 0.6. A careful comA and S + C 50:50 mixtures, in which the A and C parison of S

+

+

+

e3 =

(1

-

X3”l

- X b p - LAC0 - ( 6 0 - CACp -

X3[(1 -

xa)(cp - CO)CS -

(1 -

(1 -

x)(Cp

-

xs)(cp - co)csl

cs)C01

“From Behe and Englander (1979) (0.25 M phosphate, “excess” sodium dithionite, pH 6.9). The values of e3 reported by these authors (called “gelling coefficients”) are about 0.1 lower than those calculated here, because they used the monomer density of 1.33 g/cm3 instead of the concentration of polymerized monomers (cp = 0.69 g/cm3) in their mass conservation calculations (Eqs. 13c and 13d of their Appendix 1). fFrom Goldberg et al. (1977) (0.06 M sodium phosphate, 0.02 M sodium dithionite, pH 7.0). gFrom Benesch et al. (1980) (0.1 M potassium phosphate, 0.05 M sodium dithionite, pH 7.3). “Calculated from Eq. (111.23)using the supernatant mole fraction, x3, and the polymer mole fraction, f s , given by the authors from their analysis of the pellets obtained by sedimentation.

a 2

I

Y

L6

90.0 7 LO'O

OF 06 06

OF'O 90.0 90'0

2

z

Y

I

1

Y 1

LF ZZ L6 LF

I Z ' O T 9P.0 ZO'O 5 SF'O

OF L5

PZ'O 7 59'0

L0'0

T

PO'O

80'0 5 OP'O

LO'O

7

OP'O

I S'0-PI '0 9L.O-82'0 OL'O-ZI'O OL'O-ZZ'O 8L'O-6 1'0 6S'0-6P'O

zv+s 3

+s

1

I

B

1 a

Y

Z1.O 5 PE'O 60'0 5 LZ'O 90'0 T 8E'O

01'0 T GI'O

OF

52 E'FZ 02

L6

19'0-51'0

09'0-PP'O 65.0- 50.0 01'0

6L'O-XZ'O

v+s

I

i B

I

J

a

LO'O 7 50'0 50'0 T 60'0 PO'O 90'0 90'0 i00'0 FO'O 7 00'0

05

1 '9Z 5Z

G'SZ Z'ZZ OZ

LS'O-61'0 6P'O-SI'O O9'0-SP'O BF'O-60'0

OF'O-51 .O 51.0

d + S

0s I

X31II3IHdOH S3NVf CINV NOLV3 'V NVl771M

SICKLE CELL HEMOGLOBIN POLYMERIZATION

151

hemoglobins were purified from an AC heterozygote, gave the same values for e2 within the experimental error [e2 (S A) = 0.34 k 0.01, e2 (S C) = 0.36 0.04 (Bunn et al., 1982; Bookchin and Balazs, 1986)]. If only one of the p6 sites in each tetramer is involved in forming an intermolecular bond in the polymer, e2 should be 0.5 for the a2pspAand

+

+

*

hybrid molecule. For given values of X, co, and cs, the equation can be solved numerically to obtain e p . dThiscopolymerization probability for the hybrid molecule with e3 assumed to be zero is calculated from the total mole fraction of non-S hemoglobin in the sample, X, the total hemoglobin concentration, cg, the solubility, c,, and the mole fraction of the non-S hemoglobin in the supernatant, x. It is calculated from the relation which derives from Eqs. (111.22), (111.23), and (111.25): e2 =

f’rcc, - C O N 1 - 2xics

(1

-

2f)tx(cP -

+ X’(Cp -

~ 0 ) ~ X2(cp s

-

Cs)COl

CACOI

where fis the fraction of the non-S 0 chains in the polymer, given by the mass conservation relation:

f=

X(c, -

C&o

(Lo

- x(c, -

CdC,

CJCp

‘Behe and Englander (1979) (0.25 M phosphate, “excess” sodium dithionite, pH 6.9). The values of e 2 reported by these authors (called “gelling coefficients”) are about 0.1 lower than those calculated here because they used the monomer density of 1.33 g/cm3 instead of the concentration of polymerized monomers ( c , = 0.69 g/cm3) in their mass conservation calculations (Eqs. 13c and 13d of their Appendix 1). /From Sunshine et al. (l979b) (0.15 M potassium phosphate, 0.05 M sodium dithionite, pH 7.1). gFrom Goldberg et al. (1977) (0.06 M sodium phosphate, 0.02 M sodium dithionite, pH 7.0). *These data show a clear increase of e p with total mole fraction of deoxyhemoglobin F, starting with an eZ of approximately zero. ‘From Benesch et al. (1980) (0.1 M potassium phosphate, 0.05 M sodium dithionite, pH 7.3). ]Calculated from Eq. (111.23) using the supernatant mole fraction, x, and the polymer mole fraction, f, given by the authors from their analysis of the pellets obtained by sedimentation. RFromCheetham et al. (1979) (0.03 M sodium phosphate, 0.02 M potassium phosphate, 0.2 M sodium chloride, pH 7.0). ’From Bunn et al. (1982) (0.15 M potassium phosphate, 0.055 M sodium dithionite, pH 7.1). “These are pure hybrid molecules in which the aPs dimer is covalently cross-linked to either the ay dimer, the as dimer, or the ap” dimer (Benesch el al., 1980). T h e polymerization probabilities were calculated from ceT/c,.

152

WILLIAM A. EATON AND JAMES HOFRICHTER

0.3

-

-x a3 -

0.3

0.25

I-

8

0.2

0.1 5 0I

0.2

0.4

Fraction Hb F

0.60

0.2

0.4

0.1 5

0.6

Fraction Hb A

FIG.111.20. Effect of deoxyhemoglobins F and A on solubility in hybridized mixtures at 23.5"C. (a) The solubility is plotted versus the total mole fraction of deoxyhemoglobin F at a constant total hemoglobin concentration of 0.287 g/cniy (0).[Data from Table 4 of' Sunshine et al. (197W4.1 T h e curve is theoretical where it is assumed that only a2/3$ molecules polymerize (el = I , e2 = eg = 0). It is calculated from co(cp - c , ) X p + 2co(c, - c,,)X + (c, - ca)(c, - cP r) = 0,which derives from Eqs. (111.22), (111.25),and (III.26)(with e2 = 0). r contains activity coefficients and water activities and is calculated from the equation in footnote c to Table 111.2. (b) T h e solubility is plotted versus the total mole fraction of deoxyhemoglobin A at constant total hemoglobin concentrations of 0.27 1 g/cm3 (0)and 0.308 g/cm3 (0). [Data from Table 5 of Sunshine el al. (1979b).] The curve is theoretical where it is assumed that there is no copolymerization of a2/3$molecules, and ap/3SPAmolecules copolymerize with a probability of0.38. It is calculated using the equation in footnote c to 'Ikble 111.3,which derives from Eqs. (111.22).(111.23), (111.25), and (111.26). T h e theoretical calculations differ from those of'Sunshine etal. (1979b) in that the water in the polymer phase has now been included in the thermodynamic analysis (Sunshine et al., 1982).

cu2PSPchybrids, assuming e,

= 0 for the ayPC and a2P$molecules. This value results from the fact that the hybrid molecule can polymerize in only one orientation, while a& can polymerize with equal probability in two orientations (Minton, 1974b). Considering all sources of experimental uncertainties in the quantities that enter into the calculation of the copolynierization probabilities, we can tentatively conclude from the solubility data that e2 is not significantly different from 0.5 for both S + A and S + C mixtures. Furthermore, in S + C mixtures there is no temperature dependence to e2, which is expected for a structural model in which the intermolecular bonds in the SS polymer are the same as in the SC copolymer (Bunn et al., 1982). The copolymerization probabilities can be more directly obtained from measurements of the composition of the supernatant after sedimentation of the polymers. Several types of measurements have been used to determine the fraction of each hemoglobin, including electro-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

153

phoresis (Goldberg et al., 1977; Benesch et al., 1980), radioactive labeling of one of the hemoglobins (Behe and Englander, 1979; Bunn et al., 1982), and amino acid composition (Benesch et al., 1980). Tables 111.2 and 111.3 summarize the results of these studies in terms of copolymerization probabilities that are calculated from Eqs. (111.22), (111.23), and (111.25) (see footnotes to Tables 111.2 and 111.3). T h e activity coefficients do not enter into these calculations, so that the copolymerization probabilities only depend on the four measured quantities: the total hemoglobin concentration, the total mole fraction of hemoglobin S, the solubility, and the mole fraction of hemoglobin S in the supernatant. T h e composition of the polymer is calculated from mass conservation assuming a concentration of 0.69 g/cm3 for the pure polymer phase. T h e best agreement with the results of the solubility analysis is obtained in the composition study at low total mole fractions of the non-S hemoglobin (Behe and Englander, 1979). In these experiments, the incorporation of bovine serum albumin into the polymer was measured. It was found to be small (e3 = 0.14 ? 0.03), and identical to that observed for the a,y, and a,@ molecules in unhybridized mixtures of S + A and S + F hemoglobins (Table 111.2). This result can be interpreted as indicating no copolymerization of the non-S molecules, with the positive values of the copolymerization probabilities being attributed to a small systematic error in the measurements. The values of the copolymerization psobabilihybrid molecules also agree well with the ties for the a2PSyand a2pSpA solubility data. The copolymerization probabilities calculated from the composition data at higher fractions of non-S hemoglobin generally have much larger standard errors, reflecting the intrinsic technical difficulties in making a precise determination of both the solution and polymer phase compositions. Nevertheless, there appears to be an increased incorporation of all molecular species that show little o r no copolymerization according to the analysis of the solubility data and the composition data at low fractions of the non-S hemoglobin (Tables 111.2 and 111.3). In the case of the hybridized S + F mixtures, there is a single data set that shows a clear increase with increasing fraction of the non-S hemoglobin; the copolymerization probability for the hybrid molecule is initially low, and then increases sharply above a deoxyhemoglobin F fraction of 0.4 (Benesch et al., 1980). The composition results suggest that, at high fractions of the non-S hemoglobin, the copolymerization process in mixtures may be somewhat more complex than described by the simple copolymerization model. T h e excellent agreement between the simple theory and the solubility data at the high fractions of the non-S hemoglobin (Fig. 111.20) might, then, result from the fortuitous cancellation of com-

154

WII.LIAM A. EATON A N D JAMES HOFRICHTER

peting effects. For example, there may be aggregation of the various species in the solution phase (Benesch et al., 1980), raising the solubility, while interactions in the polymer could increase the copolymerization probability for the non-S hemoglobin, thereby lowering the solubility. Additional evidence for copolymerization of hybrid molecules comes from the study of purified hybrid molecules obtained by covalently cross-linking aPSdimers with ay, a6, or CUP*dimers (Benesch et al., 1980). The important qualitative result is the observation that these hybrid molecules appear to form gels, suggesting that they should copolymerize with molecules. Quantitative analysis of the solubilities (Table 111.3) gives polymerization probabilities that are similar to those calculated from the solubility data on mixtures. To assess the significance of these results further it would be worthwhile to investigate the structure of the gels formed from these cross-linked hybrids by electron microscopy. The apparent discrepancies between the theoretical analysis of the solubility data and the results of composition measurements at high frac0.45

1

h

E

P 0) Y

0.15

0

0.5 Fraction Hb For Hb A,

1 0

0.5 1 Fraction Hb F or Hb A,

0

FIG. 111.21. Effect ofdeoxyhemoglobins F or AS(Hb F o r H b A?) on polymerization in hybridized mixtures at 37°C. (a) T h e solubility is plotted versus the total mole fraction of deoxyhenioglobin F or A,. (-) T h e solubility in the limit of zero polymer. This curve divides the plot into two regions. At total hemoglobin concentrations below the curve no polynier can form, while at total concentrations above the curve polymer can form. It is calculated from ( 1 - X)*c, - 1': r = 0, which derives from Eqs. (111.22) and (111.26) (with en = 0). r is calculated lrom the equation givcn in footnote c to Tdb:able 111.2. (---) T h e solubility curves at various constant total henioglobin concentrations at 0.05 g/cni'
155

SICKLE CELL HEMOGLOBIN POLYMERIZATION

tions of non-S hemoglobins show that our understanding of mixtures is still incomplete. Nevertheless, the simple copolymerization model has important practical utility. Because the theory gives an accurate description of the observed solubility, it gives us a means of calculating the solubility and fraction of hemoglobin polymerized for a given fraction of non-S hemoglobin and total hemoglobin concentration over a wide range of compositions. Figure 111.21 shows the results for S + F mixtures and Fig. 111.22 for S + A mixtures. The solubility results for S A, mixtures are sufficiently close to those of S F mixtures (Tables 111.2 and 111.3) that they are treated as identical in Fig. 111.21. Similarly, the calculated solubilities for S C mixtures are treated as identical to those of S + A mixtures. This should be valid u p to a deoxyhemoglobin C fraction of about 0.6 (Bunn et al., 1982). At higher fractions the solubility is lower, and, unlike deoxyhemoglobin A, pure deoxyhemoglobin

+

+

+

0.45

1

. h

E0

0

Y

0.15

0

0.5

1 0

0.5

1

0

Fraction Hb A or Hb C

Fraction Hb A or Hb C

FIG. 111.22. Effect of deoxyhemoglobins A or C (Hb A or H b C ) on polymerization in hybridized mixtures with deoxyhemoglobin S at 37°C. (a) T h e solubility is plotted versus the total mole fraction ofdeoxyhemoglobin A or C . (-)The solubility in the limit of zero polymer. This curve divides the plot into two regions. At total hemoglobin concentrations below the curve no polymer can form, while at total concentrations above the curve polymer can form. I t is calculated from (1 - X)' c, 2X(1 - X)c,e, - cf = 0, which derives from Eqs. (111.22) and (111.26). r is calculated from the equation given in footnote c to Table 111.2 and e2 is taken as 0.4 (el = 1 , eS = 0). (---) The solubility curves at various constant total hemoglobin concentrations at 0.05 g/cm3 increments between (A) 0.25 g/cm3 and (B) 0.45 g/cm$ calculated from the equation in footnote c to Table 111.3. T h e solubilities for S + C mixtures are expected to be lower above a deoxyhemoglobin C fraction of about 0.6. (.....) The solubility curve for S + C mixtures at zero polymer fraction above 0.6 (Bunn et al., 1982). Deoxyhemoglobin C alone forms crystals with a solubility of 0.31 g/cm:' (Bunn et al., 1982). (b) T h e fraction of hemoglobin polymerized is plotted versus the total mole fraction of deoxyhemoglobin A or C at constant total hemoglobin concentrations between (A) 0.25 g/cmYand (B) 0.45 g k m 3at increments of 0.05 g/cmg. The fraction polymerized is calculated from the total hemoglobin concentration and the solubilities in a using Eq. (111.5).

+

156

WILLIAM A. EATON AND-JAMES HOPRIGHTER

C crystallizes under these solution conditions, with a solubility at 37°C of about 0.3 1 g/cmY. To make any further progress on the copolymerization problem will require considerably more structural information on the polymer phase in mixtures than is currently available, as well as much more accurate composition data. One way of improving the quality of' the composition data would be to distinguish between the different hemoglobins in the polymer phase using linear dichroism and some kind of optical labeling. Solubility studies suggest that hemoglobin S in which nickel(I1) has been substituted for iron has very similar polymerization properties (Alston et al., 1982, 1984). Cobalt(I1)-substituted hemoglobin S could be even more useful in this regard. The structure of cobalt-substituted deoxyhemoglobin A is known to be identical to that of unsubstituted hemoglobin (Fermi et al., 1982), and one would therefore expect that cobaltsubstituted deoxyhemoglobin S would form identical polymers with the same solubility. Because the cobalt and iron molecules have different optical absorption spectra, the incorporation of a non-S hemoglobin could be determined with high accuracy by measuring the linear dichroism of the copolymer, as was done in the case of oxygen binding to the polymer (Fig. 111.18). There are as yet no equilibrium studies on mixtures of hernoglobin S with other hemoglobins at partial saturation with oxygen. Such measurements would be difficult and tedious because of time-dependent changes in the sample as a result of methemoglobin formation (Sunshine et al., 1982) as well as time-dependent changes in the distribution of molecular species from slow tetramer-dimer dissociation at low saturation~.'~ We can, however, combine the thermodynamic results for the polymerization of' pure hemoglobin S as a function of oxygen pressure with those on the polymerization of mixtures at zero oxygen pressure to estimate theoretically the properties of mixtures at partial saturation. To do so, we first make the simplifying assumption that all molecular species have the same oxygen binding properties in both the solution and polymer phases. Hemoglobin F is known to have a higher affinity in the solution phase than hemoglobin S, so this assumption introduces some uncertainty in the results for S + F mixtures. For mixtures, Eq. (111.21) becomes 15 A tew solubility measurements using the sedimentation technique have heen carried out on a lysate of trait blood at 37°C (Noguchi et al., 1981). No methenioglobin reducing system was appdrelllly employed in these experiments, and no description of the saturation measurements was given. It is therefore not possible to assess the influence on these data of methemoglohin formation, which is rapid under these experiiriental conditions.

157

SICKLE CELL HEMOGLOBIN POLYMERIZATION

4

1

1

In deriving Eq. (111.27) we have assumed, as before, that the non-S homotetramer does not copolymerize (i.e., e3 = 0). For S + F and S + A2 mixtures, the hybrid molecule is assumed not to copolymerize at all oxygen pressures (i.e., ep = 0), while for S A and S + C mixtures, the hybrid molecules copolymerize with a probability of e2 = 0.4 at zero pressure. Unlike the previous calculations, where we assumed the composition to be kinetically “frozen” with respect to dissociation of tetramers into dimers and reassociation to form hybrid tetramers, we assume that the gel is at equilibrium. That is, a binomial distribution of homo- and hybrid tetramers exists in the solution phase at all oxygen pressures and extents of polymerization. After incorporating mass conservation and Eq. (111.23) for S A and S C mixtures, Eq. (111.27) can be solved numerically to give the solubility at any given oxygen pressure, fraction of non-S hemoglobin (X), and total hemoglobin concentration. T h e solubility may then be used to calculate the fraction polymerized and the total fractional saturation of the gel. The results of these calculations are shown in Fig. 111.23 for S + F and S + A, mixtures and in Fig. 111.24 for S + A and S + C mixtures.Ifi

+

+

+

IV. KINETICSAND MECHANISM OF HEMOGLOBIN S POLYMERIZATION

The most intriguing aspect of the physical chemistry of hemoglobin S is the kinetics of polymer formation. In addition to providing almost all of the experimental information on the molecular mechanism of assembly, kinetic studies on hemoglobin S polymerization have been important for several other reasons. We have already seen in Section 111 that a knowledge of the kinetics has been important in interpreting experiments aimed at obtaining true equilibrium parameters. Later we shall l6 The results of these calculations differ somewhat from those reported by others (Noguchi, 1984; Brittenham et al., 1985; Schechter et al., 1987, 1988), despite the fact that these authors also utilize the results and equations of Sunshine et al. (1979b, 1982). However, none of their calculations are sufficiently documented for us to reproduce them, which is necessary to determine the origin of the differences.

0.45

z. 0,

Y

$ 0.3 3 0

0

0.15 0

1 0

0.5 Fractional Saturation

50 100 0 Oxygen Pressure (torr)

0.5 Fractional Saturation

0.5 FractionalSaturation

0

1

1

FIG.111.23. Effect of hemoglobins F and As on polymerization at partial saturation arid 37°C. (a) Solubility a t zero polymer fraction as a function of saturation o f the solution with oxygen at various mole fractions of hemoglobin F or A, between (A) X = 0 and (R) X = 0.7 in increments of 0. I . (b) Effect of total hemoglobin concentration on the solubility at Zero polymer niole fractions of hemoglobin F or A2 of (A) X = 0.3 and (B) X = 0.6. (-) fraction and (---) total hernoglobin concentrations between 0.25 and 0.45 g/cmS in 0.05 g/cm3 increments. For (C) X = 0, there is 110 effect of total concentration, since the additional hernoglobin simply adds to the polymer phase. The total concentration is given by for zero fraction the intersection of the dashed curve (---) with the continuous curve (-) polymerized. l'he solubilities in a arid b are calculated from Eq. (111.27) with e2 = 0 and xI = ( 1 - x)", where x is the m()le fraction of hemoglobin F or A, in the solution phase, i.e., ( 1 - x)*c,% - r: r = 0 with Z = L ( l + Kpfi)4/[L(l KTp),' + (1 + KRp)'] and x = X/( 1 - xp), since in Eq. (111.23),f? = f9 = 0. r is calculated from the equation given in footnote c to Table 111.2. (c) Total saturation of gel (y,) of total concentration ( 9 ) )0.35 g/cni:i between (A) X = 0 and (B) X = 0.7 as a function of oxygen pressure at various inole fractions o f hemoglobin F or A,. T h e total saturation is calculated from Eqs. (111.14)(111.16) using the solubilities as calculated fox- b. (d) Fraction polymerized as a function of the total saluraliori of the gel at various niole fractions of tietrioglobin F and A? between (A) X = 0 and (U) X = 0.7 tor a total hemoglobin concentration (c,,) of 0.35 g/cm3. In all o f t h e above calculations, the values of the parameters were K r = 0.00825 tom-I, K R = 0.949 t w r I, K p = 0.003 torr I, L = 3.1 x lo5,and cp = 0.165 g/cnis. 'These parameters were chosen to simulate near physiological conditions and are discussed in Section V,A.

+

159

SICKLE CELL HEMOGLOBIN POLYMERIZATION

0.45

E. h

0)

v

2.

5 0.3 3 3 -

$

0.15

0

0

0.5 1 0 Fractional Saturation

Fractional Saturation

50 100 0 Oxygen Pressure (tor?)

0.5 FractionalSaturation

0.5

1

1

FIG.111.24. Effect of hemoglobins A and C on polymerization at partial saturation and 37°C. (a-d) Details are the same as in the legend to Fig. 111.23, except that the solubility is calculated from [( 1 - x)' + 2x( 1 - x)e2]c,Z - c: r = 0, where x is calculated using Eq. (111.23) and the mass conservation relation for the non-S beta chains, x ( l - xp) + xpf/2 = X , as the solution (with the positive root) to the equation [ ( I - xp)(2e2- I)]x' + [ I - (2er - l)X + xi,(eQ- I)]x - X = 0.

see that measurements of the delay time in single red cells provide the most accurate and sensitive data on intracellular polymerization (Section V,B). Finally, kinetic studies have played a central role in understanding the pathophysiology of' sickle cell disease and the design of strategies for therapy (Hofrichter et al., 1974b; Eaton et al., 1976a). This topic is discussed in Section VI, and has been developed more extensively in a recent article (Eaton and Hofrichter, 1987).

160

WILLIAM A. EATON A N D JAMES HOFRICHIEK

The unique features of the polymerization kinetics include an unusual autocatalytic time course, an enormous dependence of the rate of polymerization on concentration and temperature, and stochastic variations in the time at which polymerization occurs in small volumes. A typical progress curve for polymerization of an initially polymer-free solution is shown in Fig. IV.l. There is a marked delay period during which no aggregation is detected no matter what the physical technique. Following the delay, there is an explosive, autocatalytic formation of polymer (Hofrichter et al., 1974a,b; Steinhardt and Malfa, 1974; Malfa and Steinhardt, 1974; Moffat and Gibson, 1974).T h e delay time is extraordinarily sensitive to solution conditions. In particular, the delay time depends reciprocally on the hernoglobin S concentration to a very high power. This power is not constant and varies with the hemoglobin S concentration from about 15 at 0.35 g/cm" (Ferrone et d., 1980, 1985a) to as high as 35-50 at 0.25 g/cmY(Hofrichter et al., 1974h, 1976a,b; Sunshine et al., 1979b). An important consequence of the delay period is that it permits samples of hemoglobin S, which will form gels containing significant amounts of polymer at equilibrium, to exist as metastable, liquid solutions for long periods of time. Another striking feature of the kinetics is that the delay time shows stochastic fluctuations in experiments on Time

Delay Time

Tenth Time

FIG. 1V.I. Time course of hemoglobin S polymerization. This is a typical progress curve showing that there is a delay period prior to the appearance of polymer. The delay time has usually been defined as the inrersection of the Inaximum slope of the progress curve with the time axis arid is a convenient measure of a characteristic time for the reaction. The tenth time is defined as the time required to reach 10% of the maximum signal amplitude. These two characteristic times are the same to within the reproducibility of' most experiments.

161

SICKLE CELL HEMOGLOBIN POLYMERIZATION

HOMOGENEOUS NUCLEATION 0

&

8 = & =

.......

-

7

@

.......

7

critical

HETEROGENEOUS NUCLEATION

"UCleUS

FIG.IV.2. The double-nucleation model. (0)A hemoglobin molecule. T h e critical nucleus is defined as the aggregate at equilibrium with the lowest concentration (or activity, in the case of the homogeneous nucleus). As indicated by the relative lengths of the arrows, monomer addition to prenuclear aggregates is thermodynamically unfavorable, while monomer addition to the critical nucleus and all subsequent aggregates is thermodynamically favorable. There are two pathways for nucleation. The first polymer in any given solution volume forms by homogeneous nucleation. Nucleation of additional polymers, called heterogeneous nucleation, can then take place on the surface of this polymer. As polymers grow, the formation of additional surface area increases the probability of heterogeneous nucleation relative to homogeneous nucleation. The nuclei are assumed to grow as close packed structures having the same intermolecular bonds that are found in the infinite polymer. [From Ferrone el al. (3985b).]

small sample volumes (10-lo to cm3) when the average delay time is longer than a few seconds (Ferrone et al., 1980, 1985a,b; Hofrichter, 1986). These findings have been explained by the novel nucleation mechanism shown schematically in Fig. IV.2 (Ferrone et al., 1980, 1985b). The mechanism postulates that the first polymer in a given solution volume forms via a simple homogeneous nucleation mechanism. This polymer grows by addition of monomers to the ends. T h e lateral surface of the growing polymer may also serve as a template for the nucleation of new polymers. The primary nucleation of polymers in the solution phase without any template is called homogeneous nucleation, while the secondary nucleation of polymers on the surface of existing ones is called heterogeneous nucleation. The extra stabilizing interactions between the nucleus and the polymer surface cause heterogeneous nucleation to be more probable than homogeneous nucleation. During polymerization, the available surface is continuously increasing with time, and, as a result, the rate of heterogeneous nucleation also increases, providing a mechanism for the autocatalysis that is manifested as an apparent delay in the kinetic progress curves. The mechanism also provides a natural

162

WILLIAM A. EATON AND JAMES HOFRICHTER

explanation for the high concentration dependence of the delay time. Assuming that the nuclei are in constant equilibrium with monomer, the rates of both homogeneous and heterogeneous nucleation depend on the hemoglobin S concentration to a power which is the number of molecules contained in the nuclei. Statistical thermodynamic considerations indicate that these nucleus sizes should increase with decreasing concentration, and may become quite large, as is observed. Finally, the irreproducibility of the delay time observed in small sample volumes is accounted for by the mechanism as resulting from stochastic variations in the time at which the homogeneous nucleus for a single polymer molecule is formed. Orice nucleated, this polymer initiates the formation of an entire domain of polymers via the heterogeneous nucleation pathway (Ferrone et al., 1980, 1985b; Hofrichter, 1986). Our major objective in this section is to describe the principal kinetic data on polymer assembly and to show how these results can be quantitatively explained by the double-nucleation mechanism. I n Section IV,A we briefly describe the methods employed in the kinetic studies and summarize the data that are used in fitting for the parameters of the model. Section IV,B presents the kinetic and thermodynamic equations of the mechanism, and Section IV,C uses these equations to fit the data and assess the quantitative success of the model. In Section IV,D we discuss the effect of shear on the kinetics of polymerization. Finally, in Section IV,E, we discuss the limitations of the double-nucleation model and point to future areas for research. A. Principal Results on Kinetics of Polymer Formation

When comparing methods for measuring the kinetics of polymerization, the principal variables are the method of initiating polymerization and the method of detection. Polymerization can be initiated by any method which can produce a solution of higher concentration than its equilibrium solubility. This has been most frequently accomplished either by removing ligand or by heating an already deoxygenated solution. The most frequently performed experiment has been one in which a completely deoxygenated solution is heated from 0°C to a temperature at which the solubility is exceeded (see Fig. 111.7’). The temperature increase is usually achieved by manually transferring the sample from an ice bath to the thermostatted instrument employed for the measurement of the kinetic progress curves. A large variety of techniques have been used to monitor the polymerization process. These include linear birefringence (Hofrichter et al., 1974a,b, 1976b; Gill et al., 1980; Basak et al., 1988), turbidity (Moffat and Gibson, 1974; Hofrichter rt al., 1976a;

SICKLE CELL HEMOGLOBIN POLYMERIZATION

163

Pumphrey and Steinhardt, 1977; Noguchi and Schechter, 1977; Sunshine et al., 1978, 1979a,b; Elbaum et al., 1978; Adachi and Asakura, 1978, 1979a, 1980, 1982, 1983; Adachi et al., 1979, 1980a,b; Hofrichter, 1979; Goldberg et al., 1981, 1982; Bunn et al., 1982; Wenger and Balcerzak, 1984), light scattering (Pumphrey and Steinhardt, 1976; Hofrichter et al,, 1978; Briehl and Christoph, 1987; Madonia et al., 1983; Hofrichter, 1986; Basak et al., 1988), viscosity (Malfa and Steinhardt, 1974; Harris and Bensusan, 1975; Kowalczykowski and Steinhardt, 1977; Fieschko et al., 1978; Behe and Englander, 1978, 1979; Briehl, 1982; Danish and Harris, 1983), water proton magnetic resonance linewidths (Eaton et al., 197613) and transverse relaxation times (Waterman and Cottam, 1976; Cottam et al., 1977, 1978; Shibata et al., 1977; Waterman et al., 1979), and electron paramagnetic resonance (Thiyagarajan and Johnson, 1983). Figure I V.3 compares progress curves obtained with the birefringence and calorimetric techniques, while Fig. IV.4 compares the progress

0.5

I

2

FIG. IV.3. Time course of polymerization from calorimetric and optical birefringence measurements. The fractional extent of polymerization (f) is plotted versus time. Curve 1 results from measuring the change in optical birefringence (Le., the light transmitted between crossed polarizers) of a deoxyhemoglobin S gel following a decrease in temperature from 20 to 2°C. Curves 2 and 3 result from measuring the heat absorption and optical birefringence, respectively, after heating the same sample from 0 to 20°C. T h e composition of the sample was 0.23 g/cm3 deoxyhemoglobin S dialyzed against 0.25 M potassium phosphate (pH 6.9) and containing 0.05 M sodium dithionite. [From Hofrichter et al. (1974b).]

164

WILLIAM A. EATON AND JAMES HOFRICHTER

10

,

20

50

,

1

1

*

1

100 TIME (minutes)

FIG. 1V.4. Comparison of kinetic progress curves for deoxyhemoglobin S polymerization in temperature jump experiments using three different monitoring techniques. Measiirements were made at (d-f) 16.4"C and (a-c) 20.O"Con a 0.267-glcin' deoxyhemoglobin S sample dialyzed against a 0.15 M potassium phosphate buffer (p1-I 7.35),containing 0.05 M sodium dithionite. (m) Data from the measurement of the width of the water proton magnetic resoiiance line, (A)data from the measurement of the optical birefringence, and (0)data from the measurement of the turbidity (the increase in optical density at 1090 nm). Bar, 20% oftotal signal change. [From Eaton et nl. (1976t1).]

curves from measurements with the turbidity, NMR linewidth, and birefringence techniques. Representative curves from 90" light-scattering and viscosity experiments are shown in Figs. IV.5 and IV.6. 'The progress curves measured by all the techniques are similar, exhibiting a delay period, followed by the explosive formation of polymer. In Figs. IV.3 and IV.4 the measurements were performed on samples of identical Concentration at the same temperature. A more extensive comparison of delay tinies is shown in Fig. IV.7. These results show that the delay time measured by all the techniques is approximately the same. T h e determination of the detailed relation between the signal amplitude and the concentration of polymerized hemoglobin remains an outstanding problem in interpreting kinetic experiments. For the calorimetric measurements it is reasonable to assume that the fraction of the total heat absorption nieasures the fractional extent of polymerization. In the birefringence technique, the light transmitted between crossed linear polarizers is measured: T h e appearance of birefringence means

SICKLE CELL HEMOGLOBIN POLYMERIZATION

0

165

I

-3

-6

0

250

500

TIME (seconds) FIG. IV.5. Kinetic progress curves measured by 90" light scattering at 1064 nm. A Brice-Phoenix light-scattering photometer was modified to use a neodymium : YAG laser as a source and a silicon photovoltaic diode to measure the scattered light. The sample holder of the photometer was redesigned to permit scattering measurements to be performed on hemoglobin S samples sealed in the quartz electron paramagnetic resonance (EPR) tubes described by Hofrichter et al. (1976a). A 4 x microscope objective was used to focus the scattering volume onto the photodiode. An identical diode was used to measure the intensity of the input beam after deflection off a beam splitter. The logarithm of the ratio of the two photocurrents was generated by an analog log ratio module and recorded using a digital oscilloscope. The sample concentration was 0.236 g/cmY.The enlarged points (0)were fitted to the function I ( t ) = A[cosh(Bt) - 11, and the fits are shown (---). The fitting parameters were (a) A = 7.3 X lo-', B = 0.1085 at T = 25.4"C and (b) A = 2.6 X lo-', B = 0.0417 at T = 22.2"C.

that anisotropically oriented polymers are formed. In the simplest case, where the orientation of the polymers is independent of their concentration, the concentration of aligned, polymerized hemoglobin is proportional to the square root of the intensity of the transmitted light (Ross et al., 1975). The observed progress curves indicate a more complex situation. There is a slow increase in birefringence following the initial rapid rise which suggests that the alignment of polymers is continuing to increase once polymerization is complete. This could result either from a slow reorientation of the polymers, or from the transfer of monomers from less aligned to more aligned polymers.

166

WILLIAM A. EATON A N D JAMES HOFKICHTER

600

a 0

Y

500 400

>-

= 300 I-

0 0

' 200

E

too 0

150

300

450

600

750

9(

SECONDS FIG.1V.G. l'irne course of' deoxyhemoglohin S poly~iierizationmeasured by viscosity at different shear rates. The apparent viscosity as a function of' time at three diff'erent shear rates [(a) 76.8 sec-', (b) 38.4 sec-I, and (c) 19.2 seccl] measured with a cone-plate viscometer is shown following an increase in temperature from 2 to 25°C for a 0.25-g/cm3 deoxytietriog1ot)in S hemolysate. [From Hal ris arid Bensusan ( 1975).]

There is no quantitative theory to describe the NMR results. Two effects, however, are believed to contribute to the increased water proton linewidth on polymer formation. One contribution results from the increased correlation time of the water fraction that is tightly bound to polymerized protein molecules, and the second from inhomogeneities in the local magnetic field caused by the formation of polymer domains having magnetic susceptibilities which differ from the bulk solution, as well as by the anisotropy in the magnetic susceptibility of aligned polymers (Thompson et al., 1975; Eaton et al., 1976b). Finally, no quantitative analysis has been developed for either the viscosity or the lightscattering and turbidity measurements. T h e detailed interpretation of the signal change for both techniques is complicated by the presence of maxima in the progress curves (Figs. IV.5 and IV.6), and by the fact that the polymers form as dense aligned domains in a highly concentrated solution. In spite of these uncertainties, when the measured signals are plotted as the fractional extent of the signal change versus time, the initial portion of the progress curves is very similar for all the techniques. This result suggests that the fractional extent of polymer formation is,

167

SICKLE CELL HEMOGLOBIN POLYMERIZATION

a

FIG. IV.7. Comparison of the temperature dependence of the reciprocal delay time using proton magnetic resonance (0),birefringence (A),and turbidity (A) techniques [From Eaton et nl. (1976b).] (See legend to Fig. IV.4.)

to a first approximation, linearly proportional to the fractional change in the signal amplitude (or square root of the signal amplitude in the case of the birefringence measurements). A linear relation is predicted for light scattering from a dilute suspension of rods which are long compared to the wavelength (Berne, 1974). To examine the shape of the initial part of the progress curves in more detail, the fractional change in signal amplitude for the light-scattering data is plotted on a logarithmic scale in Fig. IV.5. The initial appearance of polymer is found to be exponential, as indicated by the linear region of the semilog plot over the range of amplitudes from about to about lo-'. This is an important result, since, as we shall see in Section IV,B, the observation of an exponential portion to the progress curve is one of the most straightforward predictions of the double-nucleation mechanism.

168

WILLIAM A . EAI'ON A N D .JAMES H O F R I C H I E K

An important step in the development of the kinetic mechanism resulted from studies of the polymerization kinetics as a function of variables which alter the solubility of hemoglobin S. The solubility and the delay time were found to be related in a very simple way: the delay time depended on the supersaturation of the solution, where the supersaturation is defined as the ratio of the initial hemoglobin S concentration to the equilibrium solubility, CJC, (Hofrichter et nl., 1976a). Figure IV.8 shows the results of experiments in which the delay time and solubility were measured under identical conditions on the same sample (Hofrichter et nl., 1976a). The solubility was varied by changing the temperature, the pH, the concentration of urea, or the fractional saturation with carbon monoxide. In each case, the reciprocal delay time was found to be proportional to the solubility raised to a power similar to what was found for the concentration dependence. T h e results could be summarized by the approximate empirical expression, called the "supersaturation equation" (Hofrichter et ul., 1976a): 1

td

= X(C"/C,)"

(IV. 1)

This result suggested that a preequilibrium exists for the rate-limiting step in the mechanism, and motivated the assumption of equilibrium nucleation in the formulation of the double-nucleation model. At a more detailed level, Eq. (IV.1) implies that the intermolecular bonding in the nuclei is very similar to the bonding in the infinite polymer. We shall see in Section IV,B that the introduction of these two features considerably simplifies the mathematical treatment of the mechanism. Up to now, we have only discussed experiments in which the temperature-jump technique is used to initiate polymerization. This method has the advantages of simplicity and compatibility with all the known techniques for measuring polymer formation. Furthermore, the measurement c;in be repeated by cooling the sample to 0°C to disassemble the polymers. A critical limitation of the temperature-jump technique is that it is not applicable to red cells or to deoxyhemoglobin solutions having the concentrations found in red cells. At least several seconds are required for thermal equilibration, so that the shortest delay times which can be accurately measured are of the order of 10 sec. At 37°C a 10-sec delay time corresponds to a concentration of only 0.23 g/cm3, which is much lower than the concentrations of 0.25 to 0.5 g/cmgfound in sickle red cells (Section V,B). An additional constraint is imposed by the solubility of deoxyhemoglobin S in near-physiological buffers. It is not possible to prepare stable, polymer-free solutions at concentrations higher than about 0.3 g/cm3,because this is the solubility at 0°C (Fig. 111.7).

169

SICKLE CELL HEMOGLOBIN POLYMERIZATION

1.o

I

I

I

0.1

0.01

0.001

I 7 .O

I

I

I

I

c '

I

I

I

1.3

1.4

1.5

n=33

0.1 r

0.01 r

0.001 r

o.ooo1

1.2

I

I

1.1

I

1.2

I

1.3

I

1.4

SUPERSATURATION RATIO FIG. IV.8. Dependence of the reciprocal delay time on the supersaturation ratio. The supersaturation is here defined as the ratio of the total hemoglobin concentration to the equilibrium solubility, measured as the hemoglobin concentration in the supernatant resulting from sedimentation of the polymers. The supersaturation was changed by changing (a) the total concentration (C) and the temperature (T), (b) the saturation with carbon monoxide, (c) the pH, or (d) the concentration of urea at nondenaturing concentrations. Both scales are logarithmic. [From Hofrichter ct at. (1976a1.1

170

WILLIAM A. EATON AND JAMES HOFKICHTER

To investigate the kinetics of polymerization at higher concentrations,

it is necessary to use a method capable of initiating polymerization much

more rapidly. One elegant technique takes advantage of the fact that the carbon monoxide complex of hemoglobin is readily dissociated by light (Ferrone et al., 1978, 1980, 1985a; Antonirii et al., 1978; Coletta et al., 1982, 1988; Hofrichter, 1986; Mozzarelli el al., 1986, 1987; San Biagio et al., 1989; Basak et al., 1988). Experiments with pulsed lasers have shown that complete photodissociation can be achieved in less than 1 p e c (Sawicki and Gibson, 1976; Greene et al., 1978; Martin et d., 1983), and that the change from the R to the T quaternary conformation necessary for polymerization takes place in about 20 psec at room temperature (Hofrichter et al., 1983; Murray el al., 1988). Using a continuous argon ion laser, complete photodissociation can be achieved in about 1 nisec and maintained indefinitely (Ferrone et al., 1980, 1985a). By focusing the laser to a small spot (3-20 p m ) inside a thin layer of solution (2-4 pm), the heat generated by absorption of the laser light is minimized. Heat conduction out of the illuminated volume is very efficient because of the large surface-to-volume ratio. At the power densities nec-

~

a

~~

HbCO

HbCO \

\

\

\

HbCO

\

\ \ \ \ \ \ \

b

HbCO

Hb+CO

HbCO

Photolysis

A

HbCO

HbCO

HbCO

CO Recombination

-<\w

FIG. IV.9. Sequence of events associated with the photolysis of a thin layer of a concentrated carbonmorioxyheinoglobin (HbCO) S solution (Ferrone et ul., 1985a). (a) Thin layer of HbCO S prior to photolysis. (b) Illumination with the laser is triggered by opening a mechanical shutter. (c) The high concentration of free CO generated by photodissociation is derreased by diffusion out of the illuminated volume. (d) Deoxyheinoglobin (Hb) S polymerizes, increasing the scattering of the laser light. Diffusion of CO and polymerization may occur concurrently. (e) Extinction of the laser results in recombination of CO, diffiisioii of CO back into the photolyzed volume, and depolymerizatiori.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

171

essary to obtain immediate and full photolysis the temperature rises are measured to be less than 5°C (Ferrone et al., 1985a). Figure IV.9 shows a schematic picture of the sequence of events occurring in the laser photolysis experiment. There are several useful features to this experiment. Carbonmonoxyhemoglobin S is soluble up to concentrations of 0.5 g/cm3 (Briehl and Ewert, 1974), so that it is possible to work with samples having the entire range of intracellular concentrations. Polymer formation is monitored by measuring the light scattered by the photolysis beam, so that it is not necessary to overlap a second monitoring beam with the photolysis beam. When the laser is turned off, the carbon monoxide recombines and the polymers disassemble rapidly, allowing the experiment to be repeated indefinitely. Finally, the volume of the solution can be as small as 30 fl(3 X l o - " cm3), which permits similar measurements to be carried out on single red cells (Section V,B) (Coletta et al., 1982). The most complete set of kinetic data has been obtained using a combination of the laser photolysis-light-scattering and temperaturejump-turbidity techniques. A systematic study was carried out using 13

40m 40m 8oDl,opl ;m 20

c

@

3 I-

20

O O

1opoo

0 0

40POO

OO

4,WO

OO

2opoo

40

OO

20

40° 0

400

TIME (seconds) FIG.IV. 10. Polymerization progress curves measured by turbidity in temperature-jump experiments. (a-d) The change in optical density at 1090 nm following a temperature increase from 0 to 25°C for different concentrations of deoxyhemoglobin S dialyzed against 0.15 M potassium phosphate (pH 7.35) and containing 0.05 M sodium dithionite. The sample concentrations were (a) 3.95 mM (0.255 g/cm9), (h) 3.66 mM (0.236 g/cmg), (c) 3.52 mM (0.227 g/cm3),and (d) 3.38 mM (0.218 g/cm3).(e-h) T h e initial portion of the progress curves from a-d. T h e points (0)are the data, and the continuous curve (-) is the least-squares fit using the operational form of Eq. (IV.8): OD(t) = ODo + A,(OD,,. OD,)[cosh(Bt) - I], where OD,,, is the maximum optical density of the progress curve, and Ar, B, and ODo are ad.justable parameters. [From Ferrone et al. (1985a).]

172

WILLIAM A. EATON A N D JAMES HOFRICHTEW

different hemoglobin S concentrations between 0.2 and 0.4 g/cm3 at temperatures ranging from 5 to 50°C (Ferrone et al., 1985a). Kepresentative progress curves from this study are shown in Figs. IV.10 and IV. 11. Figure IV.l I shows that in the laser photolysis experiments the onset of polymerization becomes more gradual in rapidly polymerizing samples, but that the process still exhibits a clear delay, even when polymerization is complete in 50 msec. Figure IV. 12 shows a log-log plot of the tenth time as a function of concentration at three different temperatures. [The tenth time is the time required to reach one-tenth of the maximum signal amplitude; because the progress curve rises so sharply, the tenth time and delay time are very similar (Fig. IV.l).] The tenth time increases from a few milliseconds to over 10,000 sec with a decrease in the hemoglobin S concentration of only about a factor of 2. The concentration dependence of the tenth time is not constant and increases significantly with decreasing hemoglobin concentration. In the range

0

0.07

0

0.1

0

0.2

0.4

0

5

TIME (seconds)

FIG.1V. 1 I . Polymerization progress curves measured by light scattering in laser photolysis experiments. (a-d) The change in scattered intensity of an argon ion laser at 514 iini S [dialyzed agaiiisL 0.15 M potassiuiri following photolysis of carbonmo~~oxyhemoglobin phosphate (pH 7.35) and containing 0.05 M sodium dithionitel. T h e sample concentrations and temperatures were (a) 5.79 m M (0.373 g/cm3) at 24.4"C. (b) 5.14 mM (0.33 I at 24.9"C, (c) 4.81 rnM (0.310 g/cm3)at 26.I0C, and (d) 4.17 iiiM (0.269 g/cm3) at 25.6% (e-h) The initial portion of the progress curves from a-d. T h e points (0)are the is lhe least-squares fit using the operational form of data, and the contiriuous curve (-) Eq. (1V.8): l ( t ) = lo + A,[cosh(Bt) - I], where lois the scattered intensity prior to the onset of polymerization, and A, and B are adjustable parameters. [From Ferrone et al. ( 1 985a).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

173

CONCENTRATION (g/cm3)

0.3 r\

U

0.4

4

aJ

v

z

W

2

r

c

Z iL1 t

0

0

1

-2 I

0.5

0.6

I

0.7

0.8

LOG CONCENTRATION (mM) FIG.IV. 12. Concentration dependence of the tenth time from temperature-jump (A, 0, and 0) and laser photolysis (A, 0 ,and M) experiments. Data are shown at three different temperatures: 15°C (0 and M), 25°C (0 and O ) ,and 35°C (Aand A). [From Ferrone et al. (1985a).]

0.3-0.4 g/cm3 the tenth time is inversely proportional to about the fifteenth power of the initial hemoglobin S concentration, while in the range 0.2-0.3 g/cm3 the dependence increases to about the thirty-fifth power. We shall see in Sections IV,B and IV,C that this increase in the concentration dependence with decreasing concentration reflects an increase in the size of both the homogeneous and heterogeneous nuclei. One of the most striking and initially surprising results from the laser photolysis measurements was found in experiments on the reproducibility of the kinetic progress curves (Ferrone et al., 1980, 1985a; Hofrichter, 1986). Figure IV.13 shows the results from a series of experiments on a sample in which the photolyzed volume was only 80 fl (8 x lo-" cm3) and the mean delay time was altered by changing the temperature. Representative progress curves are shown in Fig. IV. 13a-c, while the observed distributions of tenth times are shown as histograms in Fig. IV.13d-f. At the highest temperature (29"C), where the mean tenth time is 0.4 sec, the progress curves are virtually superimposable and the variability in the tenth time is extremely small. At 19"C, the progress curves are no longer superimposable, and there is an increase in the variability of the tenth time. T h e most dramatic result is observed at the lowest temperature (14"C), where the variability is enormous. The

174

WILLIAM A. EATON A N D JAMES HOFKICH'IEK

n

rn

e 0

>

W

0

0.5

1

0

5

10 0

100

200

100

200

TIME (seconds)

0

0.5

1 0

5

10 0

TENTH TIME (seconds) FIG. IV. 13. Repetitive measurement of kirictic progress curves for hemoglobin S polymerization in lascr photolysis experiments. The laser is turned off' for a sufficient period of time between experiments for the complete recombination of carbon monoxide and depolymerization of the illuminated volume. Representative curves for a 4.5 inM (0.29 g/cms) sample with an illuminated volume of 80 p m Yare shown at 29.2"C (a), 18.9"C(b), and 14.3"C(c). (d-f) The corresponding distribution of tenth times. [From Hofrichter (1986).]

mean tenth time is 100 sec, with a range of 20 to more than 250 sec. I n spite of this large variability in the tenth time, once polymerization begins progress curves have very similar shapes from the point where polymerization is observable. The interpretation of these experiments is aided by the observation that there is a marked decrease in the density of polymer domains with increasing delay time. This is demonstrated in Fig. IV.14, which shows optical micrographs of gels formed with different delay times in temperature-jump experiments. I n the laser photolysis experiments, only a single domain was observed in the photolyzed volume when the delay time became extremely variable, while for the more rapidly poly-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

175

FIG.IV. 14. Optical micrographs of gels between crossed polarizers at 440 nm formed at different rates (Christoph et al., 1990). Deoxyhemoglobin S solutions at different concentrations were heated from 3 to 23°C. The concentrations and delay times were (a) 0.234 g/cm3 and 11.000 sec, (b) 0.257 g/cm3 and 150 sec, and (c) 0.274 g/cm3 and 50 sec. The image size in a-c is 0.6 x 0.6 mm'.

merizing samples with reproducible delay times, no individual domains could be observed (Ferrone et al., 1980, 1985a; Hofrichter, 1986). These findings suggested that the large variability in the delay time results from stochastic fluctuations of a single molecular event which is responsible for initiating polymerization of an entire domain. According to the double-nucleation mechanism, this initiating event is the homogeneous nucleation of a single polymer molecule. The observation of the stochastic variation in the homogeneous nucleation process is made possible by the very large amplification resulting from the autocatalytic heterogeneous nucleation pathway for polymer formation. When the number of domains formed in the observation volume is large, as in the rapidly polymerizing samples in the photolysis experiments or in the much larger volumes of the temperature-jump experiments, then the number of homogeneous nucleation events is large, and reproducible delay times are observed. B . Kinetic and Thermodynamic Equations of Doubk-Nucleation Mechanism In Section IV,A we presented a summary of the principal results of kinetic experiments on the assembly of hemoglobin S polymers and briefly described how the double-nucleation mechanism shown in Fig. IV.2 could provide a qualitative explanation of the data. In this section, we present the kinetic and thermodynamic equations that describe the mechanism, and in Section IV,C w e show that the mechanism also explains the data quantitatively. The basic idea of the nucleation processes for the assembly of hemo-

1’76

WILLIAM A. EATON A N D JAMES HOFRICHTER

globin S polymers is that aggregates grow by a sequential addition of monomers which is initially thermodynamically unfavorable, but becomes increasingly favorable as the size of the aggregate increases, i.e., M + M % M, M M2 % M ,

+

+

M M+, M,. M M,* % M I * + ,

M

+

+ M,

%

Mm

As indicated by the relative lengths of the arrows, once the aggregate reaches a certain size (containing i* molecules), addition of monomer becomes thermodynamically favorable, and monomers continue to add to form long polymers. T h e aggregate containing i* monomers is the thermodynamically least stable species and is called the critical nucleus or simply the nucleus. The continuous decrease in the free-energy change for monomer addition qs the aggregate size increases results from competing thermodynamic forces. Aggregation is favored by the formation of intermolecular bonds, but it is opposed by the net loss of motional freedom of the monomers on incorporation into an aggregate. There is a net loss in entropy because the entropy associated with center-of-mass and torsional vibrations in the aggregate only partially compensates for the loss of translational and rotational entropy (Steinberg and Scheraga, 1963). As the aggregate grows there is an increase in its stability from the increase in the number of bonds per molecule, from 1/2 bond per molecule in a dimer to a maximum of 4.1 bonds per monomer in the infinite polymer (Fig. IV.2). Eventually, the stabilizing effect of intermolecular bond formation becomes larger than the destabilizing effect of the loss of motional freedom, and addition of monomer becomes thermodynamically favorable. This description applies to both homogeneous and heterogeneous nucleation. In the case of heterogeneous nucleation there are additional stabilizing interactions from the bonds between the aggregate and the polymer surface (Fig. IV.2). As a result, the size of the heterogeneous nucleus is smaller than the size of the homogeneous nucleus. The kinetic equations that describe the mechanism are straightforward. The rate of polymer formation is given by (Ferrone et al., 1980, 1985b)

SICKLE CELL HEMOGLOBIN POLYMERIZATION

177

k=j*+l

= (%)CC,*

+ (k+y)cc,*

(IV.2)

where c, is the concentration of polymers containing k monomers. T h e first term in Eq. (IV.2) represents the rate of formation of polymers via the homogeneous nucleation pathway and the second term represents the rate of polymer formation via heterogeneous nucleation. A polymer is considered as any size aggregate containing more than i* monomers in the case of homogeneous nucleation, or more than j* monomers in the case of heterogeneous nucleation. T h e number concentration of polymers is designated as p, c is the free monomer concentration, c,* is the concentration of homogeneous nuclei of size i*, 6,. is the concentration of heterogeneous nuclei of size j*, k + is the concentration-independent bimolecular rate constant for monomer addition, y is the monomer activity coefficient, y,. is the activity coefficient of the nucleus, and y,.+ is the activity coefficient of the activated complex for the addition of monomer to a nucleus. Anticipating the large excluded volume effects, the activity coefficients have been included in the apparent rate constants (the quantities in parentheses), as required by transition state theory (Hill, 1960). Activity coefficients for the j*-mer and activated complex j* + l-mer do not appear, because these species are not free in solution, but are attached to a polymer which is treated thermodynamically like a crystal. Also, in carrying out the summation, the term arising from the formation of an i*-mer by loss of a monomer from an i* + l-mer is neglected. To proceed further, we need to know how the nucleation rates depend on monomer concentration. This is done most simply by assuming that both the homogeneous and heterogeneous nuclei are in constant equilibrium with monomer (Eaton and Hofrichter, 1978; Ferrone et al., 1980, 198510). The equilibrium approximation introduces a great simplification into the mathematical treatment of nucleation, and is suggested by the finding in temperature-jump experiments that the delay time is proportional to a high power of the solubility [Eq. (IV.l) and Fig. IV.81. The concentration of homogeneous nuclei at equilibrium is given by c,. = (yc)l*K,,/y,,

(IV.3)

where K,* is the equilibrium constant for forming a critical nucleus of size z* from i* monomers. To obtain an expression for the concentration

178

WILLIAM A . EATON ANL),JAMES IIOFRICHTER

of heterogeneous nuclei it is convenient to treat the hypothetical pathway in which aj*-mer forms in solution and then attaches to the polymer surface. For identical sites with no interaction between them, the fi-actional occupancy of' nucleation sites on the polymer surface (i.e., sites that bindj*-mers), 8, is given by (IV.4)

where c,. is the concentration of attached j*-mers (i.e., heterogeneous nuclei), cl*, is the total concentration of sites for binding a j*-mer, ZiJ* is the equilibrium constant for attaching a j*-mer to the polymer surface, and y;. and cl. are the activity coeflicient and concentration of the unattached j*-mers free in solution. 'The total concentration of nucleation sites can be approximated as the concentration of polymerized hemoglobin, co - c, where cg is the total initial monomer concentration, multiplied by a scale factor, 4, which gives the number of nucleation sites per polymerized monomer. By assuming a low fractional occupancy, the denominator in Eq. (IV.4) can be set equal to unity, and, replacing y;.c.i. by ( y c ) ~ * K ~the * , equilibrium concentration of heterogeneous nuclei becomes c,* =

K;*K,*+(co -

C)(YC)I*

(IV.5)

Substituting Eqs. (IV.3) and (IV.5) into (IV.2) gives the rate of polymer formation as

wheref(c) is the rate of homogeneous nucleation and g ( c ) is the rate of heterogeneous nucleation per concentration of polymerized monomer. Equation (IV.6) shows an interesting dependence of the nucleation rates on monomer concentration. For both homogeneous and heterogeneous nucleation the rate of polymer formation depends on a power which is the size of the nucleus plus one monomer. This concentration dependence is also obtained from a steady-state treatment of nucleation (Hofrichter et al., 1974b). T h e co - c term is responsible for the autocatalytic effect of heterogeneous nucleation. At t = 0, co - c = 0, and the heterogeneous nucleation rate is zero. As polymerization proceeds, the magnitude of co - c increases, and the heterogeneous nucleation term becomes increasingly more important.

179

SICKLE CELL HEMOGLOBIN POLYMERIZATION

The number concentration of polymers, p , has not yet been measured in kinetic experiments. The measurable quantity is the concentration of polymerized hemoglobin, co - c, which we define as A . The rate of formation of polymerized hemoglobin is given by (Oosawa and Asakura, 1975; Eaton and Hofrichter, 1978)

d_A dt

dc

-- =

dt

’(

dt

k=,*+l

kck

+

5

k=j*+l

kck)

+ i*f(c) + j * A g ( c )

+ i*f(c) + j * A g(c)

= (k+yc -

k-)p

= k+(yc -

rScs)p+ i*f(c) + j * A g(c)

(IV.7)

where i*f(c) and j * A g(c) are the rates of incorporation of monomer into homogeneous and heterogeneous nuclei, respectively, k - is the concentration-independent rate constant for the dissociation of monomers from polymers, and, from the equilibrium condition, k - = k+y,t,.In Eq. (IV.7) both k + and k - are assumed to be independent of polymer size. Also, the ratios of polymer activity coefficients that appear in the rate constants for the addition and subtraction of monomers to polymer have been set equal to unity. These approximations simplify the subsequent mathematical development, but should not have a major effect on the results. Equations (IV.6) and (IV.7) are the principal rate equations that describe the double-nucleation mechanism. These coupled differential equations are nonlinear, and can only be solved by numerical integration. In the limit of a small amount of polymer formation, however, they can be linearized and analytically integrated to give (Bishop and Ferrone, 1984)’’ I7 Expanding Eq. (IV.6) in a Taylor series about c = co and retaining only first-order terms results in (Bishop and Ferrone, 1984)

dP = f(co) + P A , dt

P

g(co) - f ’ ( ~ o )

wheref’(c0) is the concentration derivative off’(c) evaluated at c c near co, both A and p are small, so that Eq. (IV.7) becomes dA -= dt

CUP+

8A

+ i*f(co),

Differentiating and substituting for dfldt gives

8 = j*g(co)

-

= co. Also,

i*f‘(co)

for values of

180

WILLIAM A . EATON AND JAMES HOFRICHTER

A(t)

A[cosh(Bt) - 11

=

=

=

$ A@

B2At2

for Bt >> 1 for Bt << 1

(IV.8)

where (IV.9) (IV.10) (IV.11) with (IV.12) (IV.13)

d In co

(IV.14)

Equation (IV.8) has a very simple form. It contains only two parameters, A and B, which are functions of both the homogeneous and heterogeneous nucleation rates. In the regime where heterogeneous nucleation dominates [i.e., g(co) >> f ' ( c o ) ] A is simply the ratio of the homogeneous to heterogeneous nucleation rates, while B 2 is the product of the heterogeneous nucleation and growth rates. The combined parameter B'A is useful because it depends on the product of the growth and homogeneous nucleation rates, but is independent of the heterogeneous nucleation rate over the entire range of validity of Eq. (IV.8). The characteristics of Eq. (IV.8) are shown in Fig. IV.15. As indicated ~~~

~~~

~

~

~

~~~~~~

which is a linear differential equation. For the initial condition in which no polymer. is present 1i.e.. A(t = 0 ) = 0 and (dAldt),=,,= z*f(s,)], the solution for g(co) > J ' ( c 0 ) is (Hofrichter, 1986)

whew S' = a@ I11

+ PI4

general, g(c0) >> J '(co), so that this equation simplifies to Eq. (1V.H).

SICKLE CELL HEMOGLOBIN POLYMERIZATION

181

by its limiting behavior, the function begins as a parabola, and then becomes an exponential. The parabolic limit corresponds to the very early part of the polymerization process where only homogeneous nucleation is occurring. [It should be noted that A = IB2At2 is the solution to the linearized equations for the case in which there is only homogeneous nucleation (Eaton and Hofrichter, 1978).1 Figure IV.15 shows that the delay period results from small values of A , i.e., from a low rate of homogeneous nucleation compared to heterogeneous nucleation [Eq. (IV.9)], and that the time of appearance of polymer is much more sensitive to the parameter B than to A . T h e relation between the tenth time and the parameters A and B is, for Btlllo>> 1, (IV.15)

where A(a) is the concentration of polymerized hemoglobin at equilibrium. The dependence of the parameters A and B on the initial hemoglobin S concentration is more complex than is apparent from Eqs. (IV.9)-

0

10

20

Bt

FIG.IV. 15. Behavior of integrated rate expression of the linearized differential equations describing the double-nucleation mechanism. The functions (a) A[cosh(Bt) - l ] and (b) log{A[cosh(Bt) - I]} are plotted for values of A of 10-1, and decreasing from left to right. [From Ferrone et al. (1985a).]

182

WILLIAM A. EATON A N D J A M E S HOFKICHTER

(IV. 14). The activity coefficients are large and highly concentration dependent because of large excluded volume effects. Statistical thermodynamic considerations predict that the sizes of the nuclei, i* and j * , also vary with concentration, and, because the equilibrium constants depend on nucleus size, they are concentration dependent as well. T h e monomer activity coefficients and their concentration dependence are derived from experimental measurements and are quite accurate up to about 0.35 g/cm:' (see discussion in Section III,A and Figs. 111.5 and 111.6).The activity coefficient for the activated complex, yt.+ however, can only be obtained theoretically. Figure IV. 16 shows the activity coefficients calculated from scaled particle theory (Minton, 1981, 1983). In this calculation, the activity coefficient is assumed to arise solely from the volume excluded to the aggregate from monomers. T h e

CONCENTRATION (g/cm3)

IZ

k! C,

LL LL

30 -

3""..........."' ...-.~ _ _ _ _ _ _ _ . ~ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ . . . - . . - 4

5

6

CONCENTRATION (mM) Fic:. IV. 16. Activated complex activity cocffirients as a function of monomer coricentration. (-) 'Ihe artivity Coefficients calculated from scaled particle theory (Minton, 1981, 1983) for activated coinplexes containing 6, 16,26, and 36 monomers (obtained by adding a monorner to rritical homogeneous nuclei containing 5, 1.5, 25, and 35 monomers, respectively). Only the contribution to the activity ot' the activated complex, assumed to be spherical, from the volume excluded to it by monomers is considered. (---) T h e activity coefficient for the activated complex, having the size calculated in Fig. 1V. 19 (plus one monomer) from the fits to thc kinetic data in Fig. 1V.17. Above about 4 mM, this activity coefficicnt is relatively cnnstant because the iiicrcased excluded volume from the increased moriomer concentration is compensated by the decrease in the size of the homogeneous nucleus. (---) The monomer activity coefficient calculated froni Eq. (111.4).I From Ferrone et al. (1985b).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

183

calculated activity coefficients are enormous, e.g., a 15-mer is calculated to have an activity coefficient of about 1OO ' in a 0.25 g/cm3 solution of monomers. Since there are no experimental data to test their validity, the activated complex activity coefficients remain a major source of uncertainty in the numerical analysis. The concentration dependence of the nucleus sizes and equilibrium constants can be obtained from an approximate statistical thermodynamic treatment of nucleus formation (Ferrone et al., 1980, 1985b; Hill, 1987). In this treatment, the nuclei are assumed to grow as close-packed spherical structures in which all intermolecular bonds are equivalent and identical to those of the infinite polymer. The chemical potential of the monomer is partitioned into contributions from translational, rotational, and internal degrees of freedom, while the chemical potential of the aggregate also contains contributions from intermolecular bonding and intermolecular (center-of-mass and torsional) vibrations. In the case of homogeneous nucleation, the nucleus size, i*, is given by (Ferrone et al., 1985b) (IV.16) and the equilibrium constant, K,,, is given by In K,.

=

(4RT

+ G1ppc)lni*

+ i! In 2 + (6,

-

(z* - 1)ln ysc, - In p

- I)ppc

(IV.17)

In Eqs. (IV.16) and (IV.17) S is the activity supersaturation (= ~ o c o / y s c s ) , pupc is the contribution to the chemical potential of the nucleus from

intermolecular bonds, 8 , and 6, are parameters that describe the fraction of intermolecular bonds in the nucleus relative to the infinite polymer, and p is the ratio of the polymer density to the monomer density. T h e only free parameter in Eqs. (IV.16) and (IV.17) is ppc,6, and 6, being determined from structural considerations and p from experimental measurements (Table IV. 1). Equation IV.16 shows that the nucleus size is predicted to decrease as the activity supersaturation increases, that is, as the concentration increases or the solubility decreases. An increase in the total concentration alters In S,while a decrease in the solubility alters both In S and ppc.The range of activity supersaturations corresponding to the 0.2-0.4 g/cm3 concentration range of the kinetic experiments is about 2 to 250, predicting that the nucleus size will decrease by almost a factor of 10.

184

WILLIAM A. EATON AND .JAMES HOFRICHTER ?rABLE 1 v. 1 Puruineter.c nf Double-Nucleation Model"

Parameters 81'

6p c

Pd

pL(kcal/mol)' pLpv (kcal/mol)/ u E (cm-')g log k, (M-1 sec-')h E , (kcallmol) for k, log I$ pccu I (kcal/mol)/ p ccu 2 (kcal/niol)I j,,, = fldfl11 X2'

Calculated from polymer structure 1.29 2 0.04 0.84 ? 0.06 0.55

Obtained from fits to kinetic datab 15°C

25°C

35°C

-

-

-

-

-

*

-8.6 0.2 -26.4 0.12 6.3 2 0.2 18 t: 5 -3.8 k 0.6 0.70 2 0.10 -9.7 t 0.6 14

-8.0 k 0.2 -25.4 -

5.6 f 0.2 -

-3.6 t 0.6 0.39 2 0.09 -7.8 ? 0.6 20 23

17

*

-9.0 0.2 -27.4 6.5 2 0.1 -3.3 2 0.4 0.79 2 0.08 -10.5 k 0.4 13 6

aThese are the pdrdnleterS of Eqs. (IV.16)-(Iv.19) from Ferrone el al. (1985b). "These parameters are obtained from fits to the log B versus log co and log B'A versus log c g data of Fig. 1V.17. (81 and S p are parameters of the function: S(i) = 1

-

8,Ini

82

t

i

-- -

where 6 ( i ) is the fraction of intermolecular contacts present in an infinite polymer that are formed in an aggregate of size t . d p is the ratio of the polymer density to the density of the monomer. L p p cis the contribution to the chemical potential of a monomer in an infinite polymer from intermolecular bonds. T h e value at 25°C was obtained from fits to the data of Fig. IV. 17, while the values at 15 and 35°C were calculated from Eq. (IV.21) using the value of v E = 0.12 cm-' calculated from the 25°C data. f p p v is the contribution to the chemical potential of a monomer in an aggregate from center-of-mass and torsional vihrations and is calculated from Eq. (1V.20).In Eq. (IV.20) the translational chemical potential, p s T , of a monomer in solution in its standard state of 1 mM is given by (Hill, 1960) psT =

- f R T In(%mk?'/h') + RT In(N/l/o) 19 kcal/mol at 25°C

= -

and the monomer rotatiorial chemical potential, psR, is given by (Hill, 1960)

where the moment of inertia is calculated using the formula for spheres, I T h e activity, YJra.is calculated from Eqs. (111.4) and (111.13)(with y a 0 ) .

=

(2/5)mr2.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

185

In the case of heterogeneous nucleation, the nucleus size, j * , is given by (Ferrone et al., 1985b)

i* =

-

j* = -

pcca2 - 81PPC pCcul- RT In S ~IPPC

j* < j m a x

+ 3RT = i * + - In1 S

RT In S

(IV.18)

j * >j m a x

The overall equilibrium constant for heterogeneous nucleus formation is a product of the equilibrium constants for forming aj*-mer in solution from j * monomers (KJ*)and attaching the j*-mer to a polymer surface (KJ.), and is given by RT In KJ.K,, = -(pUc(:u2 - 6,ppc)lnj* - j*RT In yc -Pcc@n - (61 - 62)pPc

RT In KJ*K,, = (GIpPc + 3RT)ln j * - (6, -

3RT(1 + lnjII,'lJ -

j* 82)ppc

PICrnl*


(1v.19)

j* > j,,,

where pCcis the contribution to the chemical potential of the heterogeneous nucleus per unit area of contact with the polymer, and wI and u2 are parameters that describe the dependence of the contact area on the size of the nucleus. Once the heterogeneous nucleus has reached a certain size, the contact area no longer increases with the addition of more

g v E is the frequency of the Einstein solid calculated from Eq. (IV.21) using the fitted value of ypcat 25°C. h k + is the rate constant for the addition of monomers to aggregates of all sizes greater than the critical nucleus. '4 is the fraction of polymerized monomers that serve as attachment sites for the heterogeneous nucleus. ]These parameters describe the bonding free energy between the heterogeneous aggregate and the polymer surface, where ylccis the chemical potential per unit surface area of contact, and u Iand (TZ are parameters of the function, u ( j ) , the contact area between the heterogeneous aggregate and the polymer surface given by

The value of,j,,ax[= -ue/uI, calculated by setting du(j)/dj = 01 is the aggregate size for which no further bonds are formed with the polymer surface on monomer addition. kCalculated according to Bevington (1969).

186

WILLIAM A. EATON AND,JAMES HOFRICHTEH

monomers. The size at which the bonding of the heterogeneous nucleus to the polymer saturates is designated j,,,. As in the case of homogeneous nucleation, the size of the heterogeneous nucleus is predicted to decrease with increasing activity supersaturation.

C. Compurison of Theory and Experiment Using the equations developed in Section IV,B, we can now quantita-

tively assess the success of the double-nucleation mechanism. T h e first major requirement of the mechanism is that the integrated rate expression {A = A[cosh(Bt) - 13, Eq. (IV.8)) fit the kinetic progress curves. Only the initial 10-15% of the progress curve can be considered, because the linearization of the rate equations is only valid for small concentrations of polymerized hemoglobin. Comparison of the data and the fitted curves in Figs. IV.10 and IV.11 shows that the cosh function provides an excellent fit to the data, and readily accounts for the delay period and the sharper onset of polymerization as the delay times become longer. A more demanding test of Eq. (IV.8) is shown in Fig. IV.5, in which the progress curve from a more sensitive light-scattering experiment is found to be exponential {the limit in which A[cosh(Bt) - 11 = 1/2ALL}over more than 3 decades in the fractional extent of polymer formation. This result has also been obtained by others (Briehl and Christoph, 1987). I t is an important result, for it clearly confirms one of the basic predictions of the mechanism, namely, that the delay period is only apparent, and that the time required for the first observation of polymer is a function of the sensitivity of the detection method (see Fig. IV.15). The second major requirement of the double-nucleation mechanism is that the fits to the kinetic progress curves be obtained with physically plausible values for the parameters of the model. T h e ability of Eq. (IV.8) to fit the kinetic progress curves considerably simplifies the problem of obtaining values for the model parameters. T h e progress curve at each initial concentration and temperature is characterized by just two parameters, A and B of Eq. (IV.8). To obtain values for the model parameters, then, the dependence of A and B on the initial hemoglobin S concentration can be fit using Eqs. (IV.g)-(IV.14). T h e combined parameter B*A is used instead of A. This is done because A, which describes the shape of the progress curve, is not accurately measured in some of the experiments (Ferrone et al., 1985a). In rapidly polymerizing samples, where the shape of the progress curve approaches the parabolic limit of' the cosh function (A = l/2B2At2),B2A is much better determined than A. Figure IV.17 shows log-log plots of B and B'A versus concentration together with the fits from the double-nucleation model using Eqs.

187

SICKLE CELL HEMOGLOBIN POLYMERIZATION

CONCENTRATION (g/cm3) 0.4

0.3 2 -

0.4

0.3

a

f 1

0

aJ

0 -

v)

V

Q

s c3

-2

-4

-

0.5

0.6

0.7

0.8 0.5

LOG CONCENTRATION

0.7

0.6

-

-5

-

-10

0.8

(mM)

CONCENTRATION ( g/cm3) 0.3

0.4

0.3

0.4

-

N

f I

0

aJ

ffl

V

Q

s c3

U a,

c

f

U

T

N

a3 c3

s

FIG. IV. 17. Comparison of experimental data and theoretical curves calculated from the double-nucleation model. The points are experimental and the curves were calculated using Eqs. (IV.lO)-(IV.l4) and (IV.l5)-(IV.l9), varying (a and b) five parameters for the and D) and data at 25°C (0and 0 )and (c and d) four parameters for the data at 15°C (0 35°C (A and A).The data were obtained using both the temperature-jump (0,0,and A) and laser photolysis (0,D, and A) techniques (Ferrone et al., 1985a). [From Ferrone et al. (1985b).J

188

WILLIAM A. EATON AND JAMES HOFRICHlER

(IV.g)-(IV. 14) and (IV.16)-(IV.19). These equations contain a total of eight model parameters. Three of these (a,, S p , p ) are evaluated from structural considerations, and the five remaining parameters (k, , pLpf:, 4, p(:(;u,,p(:c:uy) are varied freely to obtain the best least-squares fit to the concentration dependence of B and R 2 A at 25°C (Ferrone et nl., 1985b). The data at 15 and 35°C are then fit with only four parameters (k+, 4, pcccr,,kc(:u2) using the value of prCat 25°C and the statistical thermodynaniic expression for the temperature dependence of ppc: (Ferrone et al., l985b): Prc:

+

PPV

=

PSI.

+

PSR

+

pSR

(Iv .2 0)

+ RT In YIC,

or pp(: = ST

+ RT

In yIc,

+ 6RT ln(kTlhvE)

(IV.21)

Equation (IV.20) simply equates the chemical potential of the monomer in the polymer to the chemical potential of the monomer in solution, where the chemical potential of the monomer in the polymer has been partitioned into contributions from intermolecular bonding, ppc,and intermolecular vibrations, ppv, and the standard state chemical potential of the monomer in solution has been partitioned into contributions from translation, psT,and rotation, pSK(the contribution from internal degrees of freedom is assumed to be the same for the monomer in the polymer and free in solution). The translational and rotational chemical potentials, ps, and psR,can be calculated from the ideal gas equations for a spherical particle. In Eq. (IV.21) the vibrational chemical potential has been approximated with the expression for an Einstein solid. I n the Einstein model, the molecules of the solid behave like threedimensional harmonic oscillators and all vibrate with the same frequency (Hill, 1960). This frequency, v F ,can be calculated from the 25°C data to be 0.12 cm-1. For the 15 and 35°C data, then, the value of B 2 A is determined by a single adjustable parameter, the monomer addition rate constant, k+ [see Eqs. (IV.ll), (IV.12), (IV.I6), and (IV.17)]. Comparison of the theoretical curves with the experimental data in Fig. IV. 17 shows that the model can provide excellent fits to the data for B , and adequate fits to the data for B 2 A . An impressive result of the analysis is that the nonlinear dependence of log B 2 A on log co (Fig. IV. 17d) can be fit with a single adjustable parameter, k, . The values of all of the parameters of the model are summarized in Table lV.l.We may now ask whether these values are physically plausible. The monomer addition rate constant k, is comparable to what is found for other protein association reactions (Koren and Hammes,

SICKLE CELL HEMOGLOBIN POLYMERIZATION

189

1976). Both the magnitude of the rate constant and the activation energy of 18 kcal/mol indicate that monomer addition does not take place by a purely diffusion-limited process. T h e parameter 4, which is the fraction of polymerized monomers that can serve as attachment sites for the to heterogeneous nucleus, is of the order of This value is consistent with the assumption of the model that 4 be small in order that there be no interaction between sites. T h e plausibility of the value of ppc ( = - 8.6 kcal/mol at 25°C) can be assessed to some extent by consideration of the Einstein frequency, vE = 0.12 cm-I, calculated from Eq. (IV.21). There are no independent estimates of lattice frequencies for protein polymers o r crystals, but the calculated Einstein frequency would be expected to be lower than the lowest intramolecular vibrational modes. This is indeed the case. From a normal mode analysis of pancreatic trypsin inhibitor, the lowest frequency mode for this 6-kDa protein is about 5 cm-l (Go et al., 1983; Brooks and Karplus, 1983; Levitt et al., 1985). The value of -8.6 kcal/mol for ppcat 25°C corresponds to a vibrational chemical potential of - 26 kcal/mol [Eq. (IV.20)], indicating that about 75% of the 36-kcal/mol free energy increase from the loss of 6 deg of translational and rotational freedom on incorporating a monomer into a polymer is compensated by the free energy decrease associated with the gain of 6 deg of vibrational freedom of the polymerized monomer. The remaining terms to be discussed are pccuIand pccu2,where ,uCc is the free energy per unit area of contact for the heterogeneous nucleus, and u1and upare parameters that describe how the area depends on nucleus size. The ratio --v2/u,gives the aggregate size (imax) for which the addition of monomers results in no further contacts with the polymer surface. This value (Table IV. 1) is between 13 and 20, which is consistent with models that can be built for the interaction of a closepacked aggregate with the polymer surface. At 25°C the contribution to the free energy of heterogeneous nucleation for a 15-mer from bonding between the nucleus and the polymer surface is - 1 kcal/mol per monomer. In contrast, after correcting for the missing contacts in the nucleus compared to the infinite polymer, the intermolecular bonding within the heterogeneous nucleus is about -6 kcallmol per monomer (Ferrone el al., 1985b). The much smaller free energy for binding of monomers in the aggregate to the polymer surface compared to the binding between monomers within the aggregate is consistent with the basic physical picture of the double-nucleation model. The preceding discussion shows that a major success of the doublenucleation mechanism is its ability to fit the data over the entire experimentally accessible range of concentrations and temperatures with physically reasonable values for the five independent parameters of the

190

WII.LIAM A. EATON AND JAMES HOFRICHTER

model. We can now use these model parameters to examine some of the interesting properties of the homogeneous and heterogeneous nucleation processes. Figure IV. 18 shows how the free energy of formation of an aggregate depends on its size at various monomer concentrations for both homogeneous and heterogeneous aggregation. For homogeneous nucleation, the free energy passes through a maximum (the activity of the aggregate passes through a minimum) at all experimentally accessible monomer concentrations, while, for heterogeneous nucleation, there is a maximum in the free energy at mononier concentrations less than 5.5 mM (0.35 g/cm3). ‘I‘he aggregate size at the free energy maximum is what we have defined as the critical nucleus, and it is seen that the size of the critical nucleus decreases with increasing monomer concentration for both homogeneous and heterogeneous nucleation. Figure IV. 19 shows the detailed dependence of the nucleus sizes on monomer concentration. For homogeneous nucleation, the critical nucleus de-

a

b I L

,

I

I

10

20

30

I

20

10

0

10

20

40

30

AGGREGATE SIZE

50

n

0

40

AGGREGATE SIZE

FIG.IV. 18. Equilibrium concentrations at 25°C of (a) homogeneous and (b) heterogeneous aggregates as a function of size. I n a, the free cncrgy of forming a homogeneous aggregate of size i ( - K?’ In yzcc)is plotted as a function of size for a series of concentrations in steps of 0.5 mM. and is ralculated from Eq. (A3.11) of Ferrone et al. (1985b). In b, the free energy of forming a heterogeneous aggregate of size j ( - KT In c,) is plotted as a function of size and is calculated from Eq. (A3.21) of Ferrone ct al. (l985b) for a concentration of nucleation sites [r,b(co r ) ] equal to I niM. T h e critical homogeneous nucleus is the aggregate having the lowest activity [y,c,(largest value of - RT 1x1 y,c,)l,while the critical heterogeneous nucleus is the aggregate having the lowest concentration [c, (largest value of - RT In c,)]. The parameters used in these calculations wcrc obtained from the fits to the kinetic data in Fig. IV.17, which are summarized in Tahle IV.1. [From Ferrone rt al. ( 1985b).] ~

51

SICKLE CELL HEMOGLOBIN POLYMERIZATION

3

4

5

191

6

Concentration ( m M )

FIG.IV.19. Calculation of critical nucleus sizes as a function of monomer concentration from model parameters. The sizes of the homogeneous [i* (O)] and heterogeneous nuclei [ j * (a)]are calculated as a function of the initial monomer concentration at 25°C from Eqs. (IV.16) and (IV.18) using the parameters in Table IV.l. [From Ferrone et al. (1985b).]

creases from a size of about 18 at 3.5 mM (0.23 g/cm3) to 3 at 6.3 mM (0.41 g/cm3), while the heterogeneous nucleus decreases from 19 at 3.5 mM (0.23 g/cm3) to less than 1 at 5.5 mM (0.35 g/cm3). (A nucleus size of less than one means that the free energy barrier to polymer formation has disappeared and monomer addition to the polymer surface and all subsequent monomer addition steps are thermodynamically favorable.) These decreases in nucleus size with increasing monomer concentration are responsible for the nonlinear dependence of the rate parameters on monomer concentration in log-log plots (tllloin Fig. IV.12; B and B2A in Fig. IV.17).IR Figure IV.18 also shows that the heights of the free energy barriers (or, equivalently, the equilibrium concentrations of critical nuclei) are extremely concentration dependent. This dependence is manifested as an enormous sensitivity of the nucleation rates to the free monomer concentration, which is shown in Fig. IV.20. Between 3.3 mM (0.23 g/cm3) and 6.0 mM (0.39 g/cm3)the homogeneous nucleation rate increases by a factor of 1015and the heterogeneous nucleation rate by a factor of lolo. The heterogeneous nucleation rate depends on the concentration of polymerized monomer. To evaluate the relative contributions of homogeneous and heterogeneous nucleation, it is instructive to integrate the rate equations numerically [Eqs. (IV.6),(IV.7),(IV.l6)-(IV.l9)]. Because no interactions between aggregates are included in the rate equations, Attempts to fit the data with constant values for the nucleus sizes produced a curvature opposite to that observed (Ferrone et al., 1985b).

192

WILLIAM A. EATON AND JAMES IIOFRICHTER

0

I

I

I

1

E

a: c

-5 0

I

__---------a I I

_#---

-

-10

0

2

8 -J

-

-20 I

I

-30

I

3

1

I

4

I

I

5

I

I

6

Concentration h M )

FIG. IV.20. Calculation of nucleation rates as a funrtinn of niononier ronrentration from model parameters (Ferrone et al., IY85b).The rate of homogeneous nucleation (---) is calculated as ( k , yy,.ly,.+,)cc,.. The rate of heterogeneous nucleation (-) per millimolar polymerized mnnomer is ralculated as k , ycr,..

the numerical integrations probably become less accurate as polymerization proceeds. Nevertheless, certain qualitative features are apparent. Figure IV.2 I shows the time course of the concentration of homogeneous nuclei, the concentration of heterogeneous nuclei, and the concentration of polymerized monomer for three different initial monomer concentrations. At the highest monomer concentration (6.0 mM, 0.39 g/cmg), the numbers of polymers formed by the homogeneous and heterogeneous pathways are roughly equal as equilibrium is approached. As the monomer concentration decreases, heterogeneous nucleation becomes increasingly more probable, and completely dominates once a very small amount of polymer is formed. So far, w e have found that the double-nucleation model produces the correct shape for the initial portion of the kinetic progress curves, and that it can fit a large body of kinetic data with physically plausible values for the five adjustable parameters of the model. The final major requirement is that it account for the dramatic findings on the reproducibility of the progress curves (Fig. IV.13). It has already been pointed out in Section IV,A that the large variability in the delay time for samples with long delay times can be interpreted as resulting from stochastic fluctuations in the formation of a single homogeneous nucleus. This nucleus then triggers the polymerization of the entire observed volume via heterogeneous nucleation and growth to form a single domain of polymers (Fig. IV.14). Because a very large number of molecules polymerize via heterogeneous nucleation and growth, the single event of forming a ho-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

193

TIME (ksec) FIG. IV.2 1. Nunierical integration of the rate equations of the double-nucleation mechanism (Ferrone et al., 1985b). These are the results of numerically integrating Eqs. (IV.6) and (IV.7) after substituting for the nucleus sizes and equilibrium constants from Eqs. (lV.l6)-(IV.l9) using the parameters of Table IV.l. (a, c, and e) Plots of the concentration of polyme.rized monomer (-) as a function of time at 25°C for three different initial concentrations of monomer (6.0, 4.0, and 3.3 mM, respectively). (b, d, and f ) The concentration of polymerized monomer (-) is plotted on a logarithmic scale together with the concentration of homogeneously nucleated (.....) and heterogeneously nucleated (---) polymers.

mogeneous nucleus can be detected. When the rate of homogeneous nucleation increases, as occurs in samples with shorter delay times, the number of homogeneous nucleation events in the observation volume increases, and the delay times become much more reproducible. This description argues that once polymerization is initiated by the formation of a single homogeneous nucleus, the large number of molecules polymerized by heterogeneous nucleation and growth should result in a reproducible shape for the progress curve. T h e results in Fig. IV.13 show that, once polymerization is first detected, the progress curves have very similar shapes, particularly for the initial portion of the curves, in spite of the great variability in the tenth time. This is more clearly shown in Fig. IV.22a, in which the progress curves have been translated along the time axis to have the same tenth time. T h e same conclusion is reached by examining the variation in the parameters A and B of the individual progress curves of Fig. IV. 13. The rate parameter B , which depends only on heterogeneous nucleation and growth in this time regime [Eq. (IV.lO), g(co) >> f ' ( c o ) J ,shows a small variation with tenth time (Fig. IV.22b). In contrast, the parameter A, which depends on the rate of homogeneous nucleation [Eq. (IV.9)], shows a very large variation.

194

WILLIAM A. EATON A N D JAMES HOFKICHTER

TIME (seconds)

TENTH TIME (seconds)

TENTH TIME (seconds)

Frc:. IV.22. Comparison of kinetic progress curves for hemoglobin S polymerization in laser photolysis experiments on reproducibility. (a) The initial portions of the progress curves have been translated along the time axis so that the tenth times for each curve coincide. (b) The rate parameter B is plotted against the tenth time for all the curves of Fig. IV.ISt (c) The value of the iritcrcept on a semilog plot of each curve of Fig. IV.ISf (In A,) is plotted versus the tenth time. [From Hofrichter (1986).]

'To begin a discussion of this stochastic behavior, it is useful to compare the frequency of homogeneous nucleation events at different delay times in the numerical integrations of the kinetic equations in Fig. IV.21. These integrations give an estimate of the time and volume regimes in which stochastic behavior is expected from the double-nucleation model. For the lowest concentration of 3.3 mM, the model predicts a final concentration of homogeneous nuclei of only 2 x mM for a delay time of about lo5 sec, or about one homogeneous nucleus in cm3. This calculation predicts that stochastic behavior would be observed for samples with a mean delay time of lo5 sec in volumes of the order of cms or less. At a concentration of 4.0 mM, the model predicts a final concentration of homogeneous nuclei of 1.4 X mM for a mean delay time of about 10 sec, o r about 1 nucleus in 10-'" cm9.Finally, at the highest concentration of 6.0 m M , the model predicts a final concentration of homogeneous nuclei of about 4 x mM for a delay time of 2 msec, or about 2400 nuclei in lo-'" cms. T h e observation volume in the photolysis experiments is 0.8 x lO-"' cmy. The results of the numerical integrations of the kinetic equations of the doublenucleation model would suggest, then, that large stochastic fluctuations in the delay time be observed for samples with a mean delay time of' tens of seconds or longer, in qualitative agreement with the observations (Fig. IV.13). The observation of stochastic behavior presents the opportunity to obtain a totally independent measurement of the rate of homogeneous nucleation [f(co), Eq. (IV.12)] which can be compared with the rate derived from fitting the hulk kinetic data (Fig. IV.17). The simplest case to con-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

“ 0

50 100 150 200 NUCLEATION llME (seconds)

250

200

250

195

0 n L W

g

-10

-c

-20

z

U

0

50

100

150

TIME AFTER NUCLEATION (seconds)

0

50

100 150 200 TEMH TIME (seconds)

250

FIG. IV.23. Theoretical distribution of tenth times for the case in which a single homogeneous nucleus forms in the volume of observation. The rate of homogeneous nucleation is assumed to he a constant, and no more than one homogeneous nucleus forms in the volume of observation. (a) T h e probability per unit time that a single nucleus forms in the experimental volume at time t is plotted versus time. (b) This is the polymerization progress curve resulting from the formation of a single homogeneous nucleus at t = 0, showing that 55 sec is required for heterogeneous nucleation and growth to produce 10% of the equilibrium concentration of polymerized hemoglobin [tenth time (arrow)]. (c) T h e probability per unit time of observing a tenth time is plotted versus the (tenth) time [Eq. (IV.22)]. T h e distribution is the same as that in a, except that it is shifted along the time axis by the 55 sec required for one nucleus to produce 10% of the equilibrium concentration of polymerized hemoglobin. [From Hofrichter (1986).] (<.P-
sider is one in which only a single homogeneous nucleus forms in the volume of observation, V (Fig. IV.23). Assuming that the rate of homogeneous nucleation, &,, is constant, the probability of observing a tenth time in the interval between t and t + At is given by

P,,,, ( t , t

+ At)

=

At

<0e-60(t-r)

( t > T)

(IV.22)

196

WILLIAM A. EATON A N D JAMES HOFRICHTER

where r is the time required for heterogeneous nucleation and growth to produce one-tenth the total number of polymerized molecules at equilibrium.I9 The probability distribution predicted by Eq. (IV.22) is one in which the most probable tenth time is the shortest observed tenth time, and the probability decays from this maximum value exponentially with a relaxation rate to = NoV'(co), where N o is Avogadro's number. The measured histogram in Fig. IV.13f shows an exponential decay, and the curve through the data is the least-squares fit using Eq. (IV.22) which yields a relaxation rate to= 0.012 nucledsec, corresponding to a homogeneous nucleation ratef(co) = 2.5 X 10-lo mhllsec. The picture becomes somewhat more complex when more than one nucleation event occurs during the average tenth time. Clearly, when the number of nuclei becomes large, the progress curves become so reproducible that experimental variability determines the width of the measured distribution. Under these conditions, only a lower limit for the number of nuclei can be obtained. For example, if the delay times are all identical to within 1%, the number of nuclei must exceed 1000. In the intermediate case, the rate of homogeneous nucleation was originally estimated using a Monte Carlo procedure to model the width of the distribution of delay times (Hofrichter, 1986). A more accurate homogeneous nucleation rate can be obtained by using an analytical expression derived using the stochastic theory of chemical reactions and the theory of first passage times (Szabo, 1988).

-

-

Iq 'This equation can be readily derived for the case when homogeneous nucleation is very slow compared to heterogeneous nucleation and growth. We consider, for the moment, the problem of forming homogeneous nuclei in the absence of growth, with the rate of homogeneous nucleus formation, 50. If 60 is assumed to be time independent, then the average number of nuclei formed in time t is cot, and the probability that there are zero nuclei at time t, P o ( t ) ,satisfies the equation dPO(t)ldt= -coPo(t), i.e., P0(t) = e-k' (Feller, 1960).The total probability that the first nucleus is formed in the time interval between t and t + At, P,(t, t + At), is given by

Pi(t, t

+ At) = Po(t) - Po(t + At)

2

i&

(0'

At

If we now assume that growth of the nucleus to observable size occurs via heterogeneous nucleation and growth, the concentration of polymerized hemoglobin as a function of time, A(t), is (Ao/2)P-'i),where t l is the time at which the nucleus formed. T h e experimental observable is the time for one-tenth of the polymerized hemoglobin to form, i.e., A ( t ) = A(m)/lO, which rakes place at a time [see Eq. (IV.l5)]: t , + (1/B)ln[A(m)/5Ao] = t 1 + T . The probability of observing a given tenth time is therefore equal to the probability of observing the formation of a homogeneous nucleus at the earlier time t - T , k . , P I j j ~ ( l 1,

+ At) = P,(t - T, t + A t

- T)

= &,c-~o"-') At

SICKLE CELL HEMOGLOBIN POLYMERIZATION

197

The approach of the analytical formalism is to consider an analogous process in which bacteria that can divide at rate B are added to a welldefined volume V at a rate &, (Szabo, 1988). Addition of new bacteria to the volume corresponds to the formation of a homogeneous nucleus while the division of bacteria to form a colony is analogous with a rate to the process of the heterogeneous nucleation and growth of polymers with the rate B [Eq. (IV.10) with g(co)>>f’(cO)]to form a single polymer domain. This process can be described by the deterministic differential equation for the average number of bacteria at a time t, (n(t)):

cO,

d(n(t))/dt = B ( n ( t ) ) +

(IV.23)

50

In the stochastic formulation of the problem, the average number of bacteria is a random variable and the process is described by a differential equation for the probability of finding a precise number of bacteria at a time t, P,,(t): dP,(t)/dt

= [go

+ B(n

-

l)]Pn-l(t)-

(50

+ Bn)P,(t)

(IV.24)

The first term represents the rate of forming n bacteria from n - 1 bacteria either by adding bacteria to the volume V at a rate or dividing one of the existing (n - 1) bacteria, while the second term is the negative 1 bacteria. By identifying a critical number of the rate of forming n of bacteria, n,, with the amount of polymerized hemoglobin present at the tenth time, i.e., n, = A(m)/5Ao, the distribution of tenth times to reach n, can be derived using the theory of first passage times to give (Szabo, 1988)

+

where r ( x ) is the gamma function. Equation (IV.25) can be used to fit the distribution of tenth times for the rate toand the parameter A, with the value of A(m) obtained from solubility data and the value of B from the slope of the semilogarithmic plot of the initial portion of the individual progress curves which have the form &A0exp(Bt). Figure IV.24 shows that the distribution of tenth times is fit very well with this function. The rate of homogeneous nucleation which is determined from the distribution of tenth times can be compared with the rate extracted from the fits to the bulk kinetic progress curves. These results can also be used to calculate the monomer addition rate, k , , as well as the rate of heterogeneous nucleation, g(cO),from Eqs. (IV. 10) and (IV. 11) (Hofrichter,

198

WILLIAM A. EATON A N D JAMES HOFRICHTER

n 20 w

> U W

-

v)

15

W

10 -

Z

5-

rn 0 U

m

r 3

00

100

200

TENTH TIME (Seconds) I

I

TENTH TIME (Seconds) FIG. IV.24. Fits to tenth time distributions using the analytical expression of Szabo, Eq. (IV.25). (a) ti, = 0.010 (?0.001) sec-I, A. = 6.4 (?2.7) x lo-’ mM, (b) lo = 5.1 (k0.4) sec-I, A,, = 1.4 ( k O . 1 ) X mM. [From Szabo (1988).]

1986).‘The results are shown in Table IV.2. The difference between the rate of homogeneous nucleation obtained from the fits to the concentration dependence of the combined parameter B2A (Fig. IV.17) and that obtained from the distribution of tenth times is not surprising. T h e calfrom the fits involves the calculation of’ activity coefficulation of f(~,,) cients f’or the homogeneous nucleus from an untested approximate theory. Over the range of concentrations of the bulk kinetic measurements, the homogeneous nucleation rate is predicted to vary by a factor of lOI5 (Fig. IV.20), and the activity coefficient of the activated complex of the critical nucleus varies between 1 0 H and 10’0 (Figs. IV.16 and

199

SICKLE CELL HEMOGLOBIN POLYMERIZATION

TABLE IV.2 Comparison of Derived Rates from Bulk Kinetic Progress Curves and Tenth Time Distributionc" Tenth time distributionsb

Bulk measurementsC

Parameter

15°C

25°C

35°C

15°C

25°C

35°C

log f ( c g ) (mhf/sec)d log k , (mM-' sec-l)e log g(co)(sec-')f

-9.2 -0.3 -2.3

-5.5 0.8 -0.7

-4.0 1.6 -0.3

-11.8 2.6 -5.7

-7.2 3.3 -3.7

-6.4 3.5 -2.8

"From Hofrichter (1986). bFrom fits to the tenth time distributions using Monte Carlo simulations. [From Hofrichter (1986).] "From fits to the concentration dependence of the parameters B and B 2 A in the study by Ferrone et al. (1985b). [From Hofrichter (1986).] d f ( c g )is calculated for the tenth time distributions from lo6 SolNoV, where 5 0 is the observed rate of homogeneous nucleation in nuclei/sec, NU is Avogadro's number, and V is the volume of observation in cmS. "+ is calculated from Eq. (IV.11) from B'A obtained from the fits to the bulk progress curves and f (cg) from the tenth time distributions. f g ( c g ) , the rate of heterogeneous nucleation, is calculated from Eq. (IV. lo), neglecting f ' ( c o ) , using the observed B and the k , calculated above.

IV.20). Considering the large numbers involved in these calculations, the difference of 10' to lo9 in the homogeneous nucleation rates in Table IV.2 may be considered reasonable (even good) agreement at this stage of the development of the theory.

D. Effect of Shear on Kinetics of Polymerization The basic ideas of the double-nucleation model can be extended to rationalize a variety of kinetic experiments which have been carried out in order to provide additional information on the unusual time dependence of the progress curve for the polymerization process. One interesting class of experiments are those which characterize the dependence of the delay time and polymerization progress curves on shear stress in viscosity experiments. Prior to the development of the model, there was no good physical explanation for the existence of a delay period, and several investigators incorrectly interpreted this interval as the time required for a nucleus to form. As a result, experiments were carried out in which the sample conditions, particularly the applied shear field, were altered during the delay period. In this section, we examine the effects of shear stress on the polymerization reaction and show that they can be qualitatively understood as arising from increases in the rate of hetero-

200

WILLIAM A. EATON ANDJAMES IIOFRICHTEK

geneous nucleation, resulting from polymer breakage by the applied shear field. To expand the double-nucleation model to include the effect of shear in viscosity experiments, the fundamental rate equation [Eq. (IV.S)] must be modified to include a shear-dependent breakage rate. Each break in a polymer increases the number of polymers by one but does not alter the concentration of polymerized hemoglobin, A = co - c . If we specify the breakage rate by h(c0, cr) at the initial monomer concentration, co, and shear rate, cr, and, for the moment, ignore shear and consider breakage as a thermal process, the probability of breaking a polymer can be assumed to be directly proportional to the number of longitudinal bonds in the polymer which is, in turn, proportional to A. The breakage rate can then be treated as an additional term multiplyirig A in Eq. (IV.6) (Bishop and Ferrone, 1984). With these assumptions, Eqs. (IV.9) and (IV. 10) become

f

(en)

= -(v

3

-

+

1)

(IV.26)

where v { [ a / [ g ( r o + ) h(cn)]}”‘ is the number average degree of polymerization. ’The effect of polymer breakage is to increase the rate, B , and Lo decrease A from the values in the absence of shear [h(c,) = 01. Processes which increase the rate of polymer breakage are therefore predicted to have a number of measurable effects on the polymerization progress curves. First, the delay time will decrease because of the increase in B [the direct effect on B dominates the logarithmic effect of the decrease in A, Eq. (IV.15)J. Second, the progress curves will sharpen when compared to those observed in the absence of breakage as a result of the decrease in A. Finally, the concentration dependence of the delay time will decrease as the breakage rate increases. In the presence of shear, the rate of polymer breakage is no longer directly proportional to A, since it depends on the details of the interaction between the polymers and the shear field. Since long polymers are much more susceptible to shear breakage than short ones, the rate of breakage is highly dependent on the distribution of polymer lengths. In the most simple model, it can be assumed that the shear field simply stretches the polymer along its long axis and, as a result, weakens the

SICKLE CELL HEMOGLOBIN POLYMERIZATION

20 1

bonds between monomers along the length of the polymer.zoAccording to this model, the maximum force produced at the center of the polymer is proportional to the shear rate. The breakage rate, h(co),would therefore be expected to increase exponentially with increasing shear rate as a result of the decrease in the activation barrier for breakage. When shear-dependent breakage dominates secondary nucleation [i.e., h(co, a) >> g(co)],B, which is proportional to h(co, u)lin, is predicted to increase exponentially with increasing shear rate, and the logarithm of the delay time is predicted to decrease linearly with shear rate [Eq. (1V.15)]." At low shear the dependence of the delay time on shear is expected to decrease, and eventually disappear when h(co, a) << g(co). The effect of shear on the delay time would also be expected to decrease as the delay time decreases, because the average polymer lengths formed in the absence of shear are much smaller when B is large. A marked decrease of the delay time in the presence of shear is the clearest result from the viscometric studies (Harris and Bensusan, 1975; Pumphrey and Steinhardt, 1977; Kowalczykowski and Steinhardt, 1977; Briehl, 1982; Danish and Harris, 1983; Wenger and Balcerzak, 1984). In the most extensive study, the logarithm of the delay time was found to decrease roughly in proportion to the logarithm of the shear rate for *O The total force produced on the center of a cylindrical particle can be calculated (Batchelor, 1967) to be approximately

F = 7125 cos e

where

6

=

log(7.4r)/apu cos 0)

where 0 is the angle between the streamlines and the long axis of the cylinder, r) the solvent viscosity, p the solvent density, u the shear rate, and a the radius of the cylinder. The maximum magnitude of the component of this force which is directed along the long axis of the cylinder, for a viscosity of 0.04 poise, a cylinder radius of 10 nm, a solvent density of 1 g/cmYand a shear rate of 10 sec-l is equal to 3.1(10-2)12dyn, where 1 is the cylinder length in centimeters. Expressing this number in different terms, the total energy of the polymer decreases by 3( 10-'O)1' ergs per angstrom of increase in the length of the bond at the breakage point. If the activated complex in the breakage reaction is assumed to be extended by roughly 3 A relative to the polymer at equilibrium, the activation barrier for polymer breakage is reduced by kT for a polymer length of about G5 pm. A t a particle length of about 200 pm, the breakage rate is increased by a factor of about lo4 as a result of the quadratic dependence of the stretching force on polymer length. These predictions assume a constant polymer length and, in reality, the mean polymer length is also expected to decrease as the breakage rate increases. Consequently, the dependence of the delay time on shear will be weaker than predicted on the basis of a fixed-length distribution.

202

WILLIAM

A. EATON

A N D JAMES HOFKICHTER

shear rates ranging from 2 to 200 sec-l (Briehl, 1982). At the higher shear rates, the results are consistent with a 1.5 power dependence of the delay time on shear rate, but the dependence decreased considerably at lower shear rates. A lower dependence (-0.7 power) was also observed in experiments on lysates (Harris and Bensusan, 1975). These results imply that h(co, u)>> g(co),even at these moderate shear rates. The expectation that the shear dependence of the delay time will disappear at low shear is also consistent with the observation that there is no dependence of the polymerization kinetics on shear for shear rates varying from 0.08 to 0.2 sec-' (Kowalczykowskiand Steinhardt, 1977). No experiments have yet been carried out to test directly the predicted effect of shear on the shape of the progress curve. However, several limited studies in which the concentration and temperature dependence of the delay time have been determined in the presence of shear permit a qualitative test of the prediction that shear reduces the concentration dependence of the delay time. In one of the most interesting of these studies, both a turbidometric and a viscometric method were used to measure the concentration dependence of the delay time for identical samples (Wenger and Balcerzak, 1984). T h e concentration dependence for the sheared samples was 11.8 2 2.6 while the unsheared samples yielded a much higher value of 31.6 k 8. This result is consistent with the qualitative predictions of the double-nucleation model. The effective value of h(c,) can be estimated to be about 3 times greater than g ( c o )in this experiment. Decreased concentration and temperature dependence were found for sheared samples in other studies as well (Fieschko el al., 1978), and the temperature dependence of the delay time in the presence of shear was also found to be much smaller than that measured for data obtained by extrapolation to zero shear rates (12 kcal/mol versus 48 kcal/mol) (Hriehl, 1982). In experiments at very low shear, reported values for the concentration and temperature dependence of the delay time (Kowalczykowski and Steinhardt, 1977; Fieschko Pt al., 1978) are remarkably consistent with those observed by other techniques in the absence of shear (Hofrichter et al., l974b, 1976a,b; Ferrone et al., 1985a). 'The available data, then, are consistent with the qualitative predictions of the simple model presented above. The results argue that shear dramatically increases the net rate of secondary nucleation by breaking polymers. From the existing data, it is impossible to draw more quantitative conclusions, but it is clear that a study in which the rate o f polymerization is simultaneously characterized by viscosity and by a second observation technique, for example, turbidity or light scattering, as a function of shear rate would provide data which are essential for a quantitative analysis of the shear-dependent contributions to the poly-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

203

merization kinetics. T h e simple calculations presented above suggest that the observed effects might be explained as resulting from the breakage of individual polymers, but more complex mechanisms for sheardependent polymer breakage, such as polymer collisions or the breakage of aggregated polymers, cannot be ruled out.

E. Areas for Future Study Although much has been learned about the molecular mechanism of assembly of a hemoglobin S gel, there are still many important kinetic measurements to be made and several unsolved theoretical problems. It would be particularly interesting to carry out a complete study of the concentration dependence of the distribution of delay times caused by the stochastic fluctuations. Such a study would yield a totally independent determination of the dependence of the homogeneous nucleation rate on concentration. It would also be important to extend the measurements made with the temperature-jump technique to investigate the influence of all of the important physiological variables, particularly oxygen saturation and non-S hemoglobins, on the kinetics of gelation on the physiologically relevant time scale of tenths of seconds to tens of seconds. It has not yet been possible to carry out such experiments with oxygen, but the equivalent information can be obtained by using the laser photolysis technique to measure the kinetics of gelation following rapid partial desaturation of the carbon monoxide complex. Such measurements are in progress (San Biagio et al., 1988) (see Section V,C). A relatively unexplored area is the mechanism of formation of polymer domains. In early studies, the delay period in the absence of shear was found to be the same by all methods (Eaton et al., 1976b). We assumed that any differences were due to differences in the sensitivity of the method. More recently, experiments in which light scattering and linear birefringence are measured simultaneously have shown that the scattering appears prior to the birefringence (Basak et al., 1988). These results suggest that polymers first grow with random orientation, and then align. This could occur by the entropy-driven alignment process (Onsager, 1949; Flory, 1956), and be viewed as taking place by some kind of rotational diffusive process, or by a redistribution of monomers from the unaligned to the more stable aligned polymers (Herzfeld and Briehl, 1981b). Measurements which spatially resolve both the light scattering and birefringence should eventually clarify this issue (Basak et al., 1988). From this work it has been suggested that spherulites grow by redistributing monomers from short to long polymers, and from entangled to radially directed polymers (Basak et al., 1988).

204

WII.I.IAM A. E A l O N ANLl.JAMES IIOFRICHTER

There are also very few kinetic studies of depolymerization (Hofrichter et al., 1974b; MofFdt and Gibson, 1974; Messer et al., 1976; Harrington et al., 1977). Measurement of the rates of depolymerization are important for quantitative analysis of the behavior of sickle cells in vivo, since oxygenation in the lungs may not lead to complete depolymerization by the time the red cell enters the microcirculation (see Section VI). Depolymerization studies could also provide important mechanistic information. For example, if depolymerization takes place only from the polymer ends, then the depolymerization rates may lead to an indirect determination of the number concentration of polymers. Our theoretical discussion has focused on the most unusual feature of the kinetics-the delay period. We have seen that the double-nucleation mechanism is enormously successful in explaining the major observations concerning the initial rate of polymer formation. It should be possible to extend the mechanism in a straightforward way to explain how the delay time depends on variables other than the initial concentration, such as the fractional saturation with ligand or the effect of non-S hemoglobins. This could be done by applying the thermodynamic description for multiple molecular species in a manner similar to what was done for homogeneous nucleation alone (Sunshine et al., 197Yb). In this approach the nucleation equilibrium constants are formulated with the same statistical thermodynamic treatment that was used to explain the solubility results. Some efforts have already been made along these lines for the ef‘fect of oxygen saturation (Ferrone et al., 1986). Such a treatment will require a more rigorous approach to the question of the huge activity coefficients for the activated complex of the homogeneous nucleus. Although we have been unable to propose an alternative mechanism for the nucleation and growth of polymers, we have by no means “proved” that the double-nucleation mechanism is correct. A critical test of the mechanism would be to determine the number concentration of polymers and their length distribution at various times during the kinetic progress curve. The most direct determination of these quantities would be by time-resolved electron microscopy; such measurements have recently been carried out by sampling the solutions at various times during the delay period arid lixing the polymers with glutaraldehyde. The results show the exponential length distribution predicted by the double-nucleation mechanism, but fail to show the very high predicted Concentration dependence of the length distribution (Hriehl and Mann, 1989). Finally, we should point out that the mechanism of assembly of a gel has so far only been discussed at “low resolution,” in which hemoglobin molecules are treated as structureless spheres (Fig. IV.2). There is no

SICKLE CELL HEMOGLOBIN POLYMERIZATION

205

information on the structure of the prenuclear o r nuclear aggregates. Is the contact in the dimer the lateral or the axial contact of the double strand, or is it an inter-double strand contact? Are the prenuclear aggregates close-packed, or is the formation highly anisotropic with the double strand forming first? Are there stereospecific interactions between polymers in a domain? These are questions that will most probably be answered by a combination of nuclear magnetic resonance, Xray scattering, and electron microscopic methods. V. INTRACELLULAR POLYMERIZATION AND RHEOLOCY To understand the pathophysiology of sickle cell disease, it is necessary to characterize both the thermodynamics and the kinetics of polymerization inside sickle red cells. An implicit assumption of all of the work described in Sections 11-IV has been that polymerization inside red cells is essentially the same as in solution. It becomes critically important, then, to compare quantitatively the polymerization process in cells with that in purified solutions. To understand the circulatory abnormalities produced by intracellular polymerization, it is also necessary to characterize the rheological properties of cells which contain polymerized hemoglobin S.We have already pointed out in the discussion of Section I1 that the structure of the hemoglobin S polymer observed in electron micrographs of cells is indistinguishable from that found for gels. In this section, we discuss physical experiments on sickle cells. T h e first two parts describe the equilibrium and kinetic studies of cells. We shall see that the equilibrium properties of cells can be readily explained using the thermodynamic description of gelation and the data obtained from the solution studies. The kinetic studies of intracellular polymerization are also fully consistent with the kinetics in solutions and confirm the very important point that polymerization inside sickle red cells proceeds by the same nucleation and growth mechanism as in purified solutions. T h e conclusion from both types of studies is that there are no important cellular factors which affect polymerization that have not already been considered in the solution studies. T h e cell membrane, apart from its role in determining the intracellular hemoglobin concentration by regulating ion and water content, appears to have little or no effect on the polymerization process, and to a good first approximation the cell may be regarded as a “flexible microcuvette” for the hemoglobin solution. In the third part of this section, w e describe the results of experiments aimed at determining the role of kinetics in controlling the extent of intracellular polymerization in viva We shall see that the delay time is

206

WILLIAM

A. EATON AND.JAMES IIOFKICHTER

sufficiently long for most cells that they return to the lungs before any significant polymerization has begun (Mozzarelli et al., 198'7). Finally, in the fourth part, we discuss the rheological properties of cells and gels. We find that the rheological properties are consistent with the results of other studies on gelation and sickling, but the quantitative relations between solution and cell experiments have not yet been made because it has not been possible to make the same measurements on both systems. A major difference between solution and cellular studies is the heterogeneity in the composition of individual cells. The intracellular hemoglobin S concentration is by far the most important composition variable in determining the equilibrium and kinetic behavior of a cell. There are two major causes of differences in intracellular hemoglobin S concentrations. One is that certain cells, called F cells, contain a very high proportion of fetal hemoglobin-about 40% (Dover el ad., 1978). There is a large patient-to-patient variability in fetal hemoglobin levels, with an average of about 6% (Serjeant, 1974; Wrightstone and Huisman, 1974), corresponding to about 15% F cells. The second is the dehydration of cells associated with the loss of potassium ions due to membrane damage (Hookchin and Lew, 1983). Early studies on the fractionation of cells by sedimentation showed a variation in the intracellular hemoglobin concentration from about 0.3 to 0.5 g/cm3(Chien et al., 1970; Seakins et al., 1973). The physiological significance of this finding was not appreciated, however, until the discovery of the enormous concentration dependence of the rate of polymerization (Hofrichter et al., 1974a,b). These kinetic studies suggested that this relatively modest variation in the intracellular hemoglobin S concentration would produce changes in the rate of intracellular polymerization of several orders of magnitude in the cells of a typical patient. A major result of the cellular studies described below is the confirmation of this prediction (Coletta et al., 1982). A. Equilzbnum Measurements of' Intracellular Polymemzation Two types of experiments have frequently been used to measure intracellular polymerization in sickle cells. The first is the oxygen binding curve, in which the total saturation of hemoglobin in the cells, including both solution and polymer, is measured as a function of oxygen pressure. The binding curve of an individual cell is equivalent to the gel binding curve described in Section II1,C (Fig. 111.19).T h e whole blood binding curve is the superposition of a large set of such curves, one for each cell in the population. Because the polymer has a very low affinity for oxygen, polymer formation is observed indirectly in this experiment as a decrease in oxygen affinity of sickle blood compared to normal

SICKLE CELL HEMOGLOBIN POLYMERIZATION

207

blood, a so-called “right shift” in the oxygen binding curve (Seakins et al., 1973; May and Huehns, 1975; Winslow, 1978). The second experiment is the classic sickling experiment, in which the fraction of sickled cells is measured as a function of oxygen pressure. Visual observations of cellular deformation have been the most frequently used assay for intracellular polymerization, primarily because of their technical simplicity. We shall see that kinetics actually play a major role in this assay, and it should be classified as a quasi-equilibrium technique. A limited number of measurements have been carried out using two other methods for determining the amount of polymerized hemoglobin in cells. In the first, the average fraction of polymerized hemoglobin is measured as a function of oxygen saturation using 13C nuclear magnetic resonance spectra at natural abundance. In the second, the optical densities in polarized light of single cells at zero oxygen pressure are measured. Because of the long times required to carry out these experiments, both can be regarded as equilibrium or near-equilibrium techniques. The principal question to be discussed in this section is, To what extent is it possible to explain the experiments on cells from the results on purified hemoglobin S solutions? To address this question, it is necessary to have both a theoretical framework for analyzing cell experiments and some means of extrapolating the solution data to intracellular conditions. We begin by briefly reviewing the solution data, and then present the equations and parameters that are required to model the cell experiments. The basic data from the solution studies that are required for making comparisons with experiments on cells are (1) the oxygen binding curve of the solution phase of the gel, (2) the oxygen binding curve of the polymer phase of the gel, and (3) the solubility and fraction of hemoglobin polymerized as a function of oxygen pressure. Because of technical difficulties in working with a physiological solvent at 37”C, the solution studies were carried out in phosphate buffer at room temperature (0.15 M potassium phosphate, pH 7.0, 23.5”C) in the presence of a methemoglobin reducing system (Sunshine et al., 1982). We shall see, however, that these data are useful for a first-order description of polymerization under physiological conditions. The solution data used in subsequent calculations are shown in Fig. V. 1. For the solution phase binding curve we simply use the binding curve for normal blood under physiological conditions (Fig. V. la). This curve is well represented by the allosteric saturation function (Monod et al., 1965):

208

WILLIAM A. EATON A N D .JAMES HOFKICIITER

1

0.5

0

0

50 Oxygen Pressure (torr)

100 0

0.5

1

Fractional Saturation

FIG.V. 1. Hemoglobin S solutiou and polymer binding curves and soluhility-saturation curve under physiological conditions. (a) Solution [top (-)I and polymer [bottom (-)I binding curves. The solution curve is a least-squares fit to the oxygen binding curve of normal human blood at. 37"C, pH 7.4, and Pco2 = 40 torr (Rossi-Bernardi et al.. 1975a), using the allosteric saturation function (Eq. (V.l)] with L = 3.08 X lo', KT = 0.00825 torr-l, and KR = 0.949 torr I. A least-squares fit to the same data using the Hill equation [Eq. (V.2)] gave p.50 = 26.2 torr and n = 2.64. T h e polymei- binding curve was calculated from Eq. (V.3) with KP = 0.003 torr-I. The theoretical (---) binding curves for K (top) and T (bottom) state niolecules are also shown. (b) Solubility-saturation curve. This is the solubility versus the frartional saturation with oxygen of the solution phase of the gel. (-) Calculated from the enipirical equation [Eq. (V.5)]. (---) Calculated from the thermodynamic linkage relation [Eq. (III.18)], using the binding curves in a and a solubility at zero saturation of 0.165 g/cnig.

where y, is the fractional saturation with oxygen, p is the oxygen pressure, L is the ratio of T-state to R-state molecules at zero oxygen pressure, KT is the T-state association constant, and K , is the R-state association constant. T h e solution phase binding curve can alternatively be represented by the Hill equation:

where K , = l / p 5 0 is the apparent association constant at 50% saturation, and n is the Hill coefficient, which is a measure of'cooperativity. T h e Hill equation is useful because in many studies only the $150is reported. It is less accurate than the allosteric equation [Eq. (V.l)], but the accuracy is more than sufficient for modeling the cell experiments. In phosphate buffer, the initial portion of the polymer binding curve was found to be noncooperative (Sunshine et al., 1982; Section 111,C). We shall assume that under physiological conditions the polymer binding curve is noncooperative over the entiIe range of saturation, and use the simple binding function:

SICKLE CELL HEMOGLOBIN POLYMERIZATION

yp

=

1

209

KPP

+ Kpp

where yP is the fractional saturation of the polymer with oxygen and K , is the polymer association constant. In phosphate buffer K , was found to be 0.0059 torr-', which is 37% of the T-state binding constant. It is reasonable to assume that K , will scale to the T-state binding constant in the same way under physiological conditions ( K , = 0.00825 torr-I), and we use a value for K p of 0.003 torr-' (Fig. V.la). The last piece of information required from the solution studies is the fraction polymerized, xp,as a function of oxygen pressure. T h e fraction polymerized is calculated from the hemoglobin mass conservation relation: xp

=

1 1

- CJCa -

c,/cp

where co is the total hemoglobin S concentration, cp is the concentration of hemoglobin in the polymer phase, taken as 0.69 g/cm3 (Sunshine et al., 1979b, 1982), and c, is the solubility, Solubility measurements using carbon monoxide instead of oxygen (Hofrichter, 1979) gave essentially identical results as oxygen at room temperature (Sunshine et al., 1982) (Fig. 111.15). Measurements were not made at 37°C with oxygen because of the rapid oxidation to methemoglobin at the elevated temperature. Since the 37°C solubility-saturation curves with carbon monoxide are simply displaced downward by the difference in solubility at zero saturation (Hofrichter, 1979), it is expected that the solubility-saturation curves for oxygen are displaced in the same way.22We therefore use the empirical equation for the solubility-saturation curve, changing the solubility from 0.183 at 23.5"C to 0.165 g/cm3 at 37"C, i.e., c, =

0.165

+ 0 . 0 9 2 4 ~+~ 0.0980~,3+ 0 . 2 3 5 ~ : ~

(V.5)

This curve is shown in Fig. V. lb. Because of the thermodynamic linkage relation between binding and solubility [Eq. (111.18)],the solubility-saturation curve given by Eq. (V.5)

** Because the solubility at zero saturation is relatively insensitive to pH, ionic strength, and 2,3-DPG concentrations (Section III,B), it is difficult to justify making corrections to the zero-saturation solubility for these effects in the absence of solubility data under physiological conditions. Furthermore, in changing to physiological conditions, the solubility versus solution phase saturation curve is not expected to change significantly; for any given solution phase saturation, the allosteric model predicts that there is no more than a few percentage change in the relative populations of T-state and R-state molecules.

210

WILLIAM A. EATON ANDJAMES HOFKICHTER

should be consistent with the solution and polymer phase binding curves. The dashed curve in Fig. V. 1b is calculated from the solution and polymer phase binding curves and the zero-saturation solubility, and shows that the choice of binding and solubility curves under physiological conditions is indeed thermodynamically self-consistent. In addition to the basic data on oxygen binding and solubility in Fig. V. 1, it is necessary to know the intracellular hemoglobin S concentrations. As pointed out in the introduction to this section, a major factor in understanding experiments on cells is the wide variation in intracel-

I

I

l

l

b

0

l e

d

i

0.2

0.4

‘1

0,6

lntracellulor

Hb Concentration (g/cm3)

FIG. V.2. Representative distributions of intracellular hemoglobin concentrations from a study of 43 patients with homozygoiis sickle cell disease. Each distribution Ca-i) is for an individual patient [mean concentrations arc (a) 0.312 g l a n s , (b) 0.324 g/cm’, (c) 0.345 g/cm3, (d) 0.353 g k d , (e)0.370 g k d , (f) 0.364 g/cm3,(g) 0.373 glcm’, (h) 0.382 g/crn’, and (i) 0.387 glcm”].‘Ihe distrihutions were determined by sedimenting cells in a PercollStractan continuoms density gradient. The points (e)are rhe fraction of the total nuinbcr is the of cells at a given mean intracellular hemoglobin concentration. T h e curve (-) probability density and is obtained from the points after correcting for the unequal intervals resulting from the nonlinearity of the density gradient. T h e probability density curve is normalized, so that the area under the curve is unity. [Data of M. E. Fabry and R. I.. Nagel, from the study by Fahry et nl. (1984).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

21 1

M a r hemoglobin S concentration. The distribution of intracellular hemoglobin concentrations has been measured by sedimenting cells in a density gradient (Fabry and Nagel, 1982;Clark et al., 1982; Kaul et al., 1983a;Noguchi et al., 1983;Fabry et al., 1984). Figure V.2 shows representative distributions from a study of 43 patients with sickle cell disease (Fabry et al., 1984). There are very large variations from patient to patient. For comparing calculated and observed properties on blood samples with unknown concentration distributions, we shall make use of the average distribution shown in Fig. V.3. Figure V.4 shows the effect on the oxygen binding curve and on the fraction polymerized produced by varying the hemoglobin S concentration over the range found in cells. In these calculations the fractional saturation, y,, is given by Yl

= x p y p $-

(1 -

W.6)

Xpbs

where y,, yp, and xp are calculated from Eqs. (V.l),(V.3),and (V.4)and (V.5),respectively. T h e effect of concentration on the binding curves is

0.3

: 0.2

0"

. t

.-c 0

; 0.1

-Q

LL

0

0.2

0.3

0.4

0.5

0.6

lntracellular Hb Concentration (g/cm3) FIG. V.3. Average distribution of intracellular hemoglobin concentrations found in homozygous sickle cell disease. This is the average of the distributions for 40 patients and was calculated as the average of the distributions in Fig. 3a and b of Fabry et al. (1984). The points (0)are the fraction of the total number of cells at a given mean intracellular hemoglobin concentration. The dashed curve (---) is the probability density and is obtained from the points after correcting for the unequal intervals resulting from the nonlinearity of the density gradient. T h e probability density curve is normalized, so that the area under the curve is unity. [From Fabry et al. (1984).]

212

WILLIAM

A. EATON

AND JAMES HOFRICHTER

1

1

C

-0

H

a,

.-0

a,

.-2

=I

ii

!!0

0.5

0.5

k 2 c

ti

0 .5

LL

LL

e

s 0

0

50 1000 Oxygen Pressure (torr)

0.5

1

0

Fractional Saturation

FIG. V.4. Effect oP hemoglobin S concentration on gel binding curves and fraction polymerized under physiological conditions. (a) Gel binding curves (-). These curves are calculated from Eqs. (V. I), (V.S), (V.4), (V.5),and (V.6)using L = 3.08 X lo5,Kr = 0.00825 torr-1, KR = 0.949 torr-1 (corresponding to a p50 = 26 torr), and K , = 0.003 torr-' at 0.05 gkm3 increments in hemoglobin S concentrations (c") between (A) 0.25 g/crn3and (B)0.5g/cm3. Solution binding curve (---). At sufficiently high oxygen pressures, the solubility exceeds the total hemoglobin S concentration, and the solution and gel binding curves coincide because no polymer is present. (b) Fraction polymerized. T h e fraction of hemoglobin polymerized is plotted versus the saturation of the gel using the same equations and paramerers as in a at 0.05 g/cmg increments between (A) 0.25 g/crng and (B)0.5 g/cm9.

significant: both the gel p50 and the fraction polymerized at zero saturation increase by about a factor of 2 as the concentration increases from 0.25 to 0.5 g/cm3.Notice that there is a discontinuity in the gel binding curve which occurs at the oxygen pressure at which the total hemoglobin S concentration is equal to the solubility. Such a sharp break is difficult to observe experimentally, because kinetic effects cause both polymerization and depolymerization to become slow at concentrations close to the solubility. Moreover, in unfractionated samples the distribution of intracellular concentrations results in a broadening (see below). In the discussion that follows, we shall see that lack of information on the distribution of intracellular hemoglobin S concentrations for the particular blood sample is a major source of uncertainty in making detailed comparisons of the solution and cell results. T h e determination of the whole blood oxygen binding curve has probably been the most important equilibrium rneasurement on cells. It is important to understand this curve in detail in order to characterize both the oxygen delivery function of sickle blood (Eaton and Hofrichter, 198'7) and the intracellular polymerizatiorl process. In this experiment, the fractional saturation is now the average fractional saturation, (yt), and the mass conservation relation for oxygen [Eq. (V.S)] becomes

SICKLE CELL HEMOGLOBIN POLYMERIZATION

213

where (x,) is the average fraction of polymerized hemoglobin. Equation (V.7) has this simple form because the cell volume decreases as the intracellular hemoglobin concentration increases, with the net result that the total amount of hemoglobin per cell is approximately constant (Seakins et al., 1973). For a distribution of concentrations, (x,) is given by

where P ( c O )dc,, is the probability that the total hemoglobin S concentration in a cell lies between co and co dco, and can be calculated from the concentration distributions in Figs. V.3 and V.4.23 The value of the lower limit in the integral, c : , depends on whether the experiment is performed by oxygenating cells or by deoxygenating cells. Because of the kinetics of polymerization there is a readily measurable hysteresis in oxygen binding curves, with the deoxygenation curves shifted to the left of the oxygenation curves (Mizukami et al., 1977; Winslow, 1978; Benesch et d.,1978b). When a polymer-free cell is deoxygenated, there is a delay prior to the onset of polymerization (see Section V,C). When the cell is reoxygenated polymer should disappear without a delay. Consequently, we expect the oxygenation curve to be close to the true equilibrium binding curve. For oxygenation experiments, then, we assume that c: = c,, the equilibrium solubility, while in deoxygenation experiments, cl is greater than the equilibrium solubility because the solution must be supersaturated for polymerization to occur. T h e value of cl will depend on the rate of deoxygenation. For a step change from 100% saturation to some fixed oxygen pressure, followed by a 15min equilibration period, the value of cl is given by the concentration that corresponds to an activity supersaturation (= y:cl/ysc,) of about 3. Figure V.5 compares the calculated and observed oxygen binding curves for blood from two individual sickle cell patients. T h e hysteresis in the calculated curves is only a few torr. The p50s for the measured binding curves are 41 and 46 torr, compared to 26 torr for normal whole blood. In studies on a series of SS patients the whole-blood p50 was

+

2s Alternatively, the value of (xp)can be calculated, albeit less precisely, from the discrete summation:

wheref; is the fraction of cells containing a mean intracellular hemoglobin S concentration ( c ~ of ) F , , and&(1 - c,/c,) = 0 for c, > c,.

214

WILLIAM A. EATON AND JAMES IIOFRICHI'EK

1

1

.-C0

ZT

0 .-"

P

3

5m

c

iij

- 0.5

v)

20

0.5

m

C

.+

0

u

TI

LL

LL

i!

i! 0

0

0

50 100 0 50 100 Oxygen Pressure (torr) Oxygen Pressure (torr)

FIG. V.5. Whole sickle blood binding curves at 37"C, pH 7.4, Pcm, = 40 torr. (a) Data of Wiiislow (1978). The data for (---) normal blood and (0)sickle blood (1.50 = 4 1 torr). (---) The equilibrium (resaturation) curve, calculated using Eqs. (V.2). (V.3), (V.5), (V.7), and (V.8), arid (-) a slow desaturation corresponding to a step change in the oxygen pressure tollowed hy a 15-min equilibration period prior to the measurement of the fractional saturation. Calculations are shown for three diff'erent distributions of intracellular heiiioglobin S. The central pair of curves corresponds to the average coricentration distribution of Fig. V.3, while the left and right pairs correspond to the extreme individual concentration distributions of Fig. V.2a and h, respectively. For the solution phase binding curve the p.50 in Eq. (V.2) was taken as 29 torr and 7~ as 2.6. The elevated p.50 was used to account for the increased 2,S-DPG lcvels found in sickle blood (see legend to Fig. V.6). (b) Data of Kossi-Bernardi ct nl. (1975b). The desrription is otherwise the same as in a. The p50 tor sickle hlood is 46 torr.

found to range from about 33 to 45 torr (Bookchin et al., 1976; Winslow, 1978). The calculated curves in Fig. V.5 show that this variation can be readily explained as resulting from patient to patient differences in intracellular hemoglobin S concentration distributions. T h e coniparison in Fig. V.5 points out that unless both the concentration distribution and binding curve have been measured for the same sample, it is only possible to make qualitative comparisons between curves calculated using solution data and whole-blood binding curves. Fortunately, there is a rather extensive data set on fractionated cells (Seakins et al., 1973), which allow a quantitative comparison. In these experiments, cells were separated into three fractions according to their density by sedimentation. T h e p50,mean intracellular hemoglobin concentration, fraction of fetal hemoglobin, and 2,3-DPG levels were nieasured for each fraction. Figure V.6 shows a plot of the p50 at pH 7.1 versus intracellular hemoglobin S concentration for the most arid least dense fractions from 15 patients. Two calculated curves are shown, one assuming a p 5 0 of 31 torr for the solution phase, the 1 5 0 of normal

SICKLE CELL HEMOGLOBIN POLYMERIZATION

*O

70 C

b

215

c

60

v c

0

ln

P

50

40 ......................................,.................. ...................................... 3o 0.25 0.35 0.45 lntracellular Hb S Concentration (g/cm3) FIG. V.6. Oxygen affinity as a function of intracellular hemoglobin S concentration in density fractionated sickle cells. The data are from the study by Seakins et al. (1973). (A) The data for the least dense of three fractions obtained by sedimentation and (0)the data for the most dense fraction. T h e measured hemoglobin F in each fraction was used by the authors to obtain the intracellular hemoglobin S concentration from the total intracellular hemoglobin concentration. T h e experiments were carried out in a phosphate-buffered saline solution at pH 7.13. (---) Calculated from the solution data using Eqs. (V.2)-(V.6), with n = 2.6 and $50 = 31 torr for the solution phase. T h e value of 31 torr was measured on normal blood in the same buffer. (-) Calculated using the same five equations, except that the effect of the heterogeneous distribution of 2,3-DPG was accounted for with the approximate empirical relation: p50 = 4 3 - 26c0, which was substituted for l/Ksin Eq. (V.2). This equation derives from the ef€ect of 2.3-DPG on p50 at pH 7 . 4 in the concentration range of 14-28 pmol 2,3-DPG/g Hb, p50 = 25 + 0.42 [DPG] (Duhm, 1971) (the effect of 2,3-DPG is probably somewhat smaller at the lower pH of 7.13) and from the data of Seakins et al. (1973) on the relation between 2,3-DPG and intracellular hemoglobin S concentration, which is well represented by the relation [DPG] = 41 - 62c0.

blood at this pH, and the other taking into account the finding that 2,3DPG concentrations are normal in the most concentrated cells, but increase with decreasing intracellular hemoglobin concentration (Seakins et al., 1973). After including the effect of 2,3-DPG, the agreement between the observed and calculated p50s must be considered very good, but the calculated p50s are still slightly lower than the observed. This small difference could arise from small changes in the polymerization parameters at the lower pH. For example, if a zero-pressure solubility of 0.15 g/cm3 (instead of 0.165 g/cm3)and a polymer binding constant of 0.002 torr-' (instead of 0.003 torr-') are used, there is perfect agreement between the observed and calculated p50s.

2 16

WILLIAM

A. EA'I'ON AND JAMES

IIOFRICHTER

'l'he most direct measurement of the extent of intracellular polymerization has been obtained from natural abundance, carbon-13 nuclear magnetic resonance experiments (Noguchi P t al., 1980, 1983). With this technique, the fraction of hemoglobin polymerized in the sample is measured as a function of the total saturation with oxygen on decreasing the oxygen pressure. Two types of carbon- 13 proton magnetic doubleresonance techniques are used to measure the polymer fraction. T h e relative amount of polymer is measured from the integrated area of the proton-enhanced carbon- 13 spectrum, which has no contribution from solution phase molecules (Sutherland et al., 1979; Noguchi et al., 1979). 'lo obtain the absolute fraction polymerized, the integrated areas under the proton scalar decoupled carbon-13 spectrum are compared for fully oxygenated and fully deoxygenated samples. Only the solution phase hemoglobin molecules contribute to this spectrum (Sutherland et al., 1979). Figure V.7 shows the results of an experiment in which both the

0.3 -

v)

s

1

I

I

I

I

I

I

a

0

E

10

0.2

u

c

0

Ti

e

0.1

LL

0

0.2

0.3

0.4

0.5

0.6

lntracellular Hb Concentration (g/cm3)

0

0.5

1

Fractional Saturation

FIG. V.7. Concentration distribution and nuclear magnetic resonance determination of polymer fraction as a function of oxygen saturation in sickle cells [data of Noguchi et u1. (198S)l. (a) Conrenti-atiou distribution. The total intracellular hemoglobin concentration was measured in a series of fractions of differing density obtained by sedimentation in a discontinuous Stractan gradient. T h e fraction of cells has been equated to the fraction of the total hemoglobin, since the total amount of hemoglobin in each cell is approximately the same (Seakins el ul., 1973).(0)The measured values and (-) the probability density. 'l'he concentration for the densest fraction was not quoted by the authors and was calculated from the Stractan corkcentration and their hemoglobin conccntration~Stractanconcentration calibration curve. (b) Polymer fraction as a function of' oxygen saturation.).( The polymer fractions determined at 37°C using natural abundanre carbon- 13 nuclear magnetic resoiiancc techniques. (---) Calculated from Eqs. (V.2),(V.3),(V.5), (V.7).and (V.8) using the concentration distribution in a, with R = 2.6 and p50 = 29 torr for the solution phase. (-) Calculated assuming that no polynier is formed unless the hemoglobin activity is at least lhree times greater than the equilibriuni activity at that oxygen pressure. 'I'his activity supersaturation of 3.0 corresporids to a delay time of about 15 min.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

217

concentration distribution and fraction polymerized as a function of oxygen saturation were measured on the same sample (Noguchi et al., 1983). T h e concentration distribution in Fig. V.7a was used to calculate the polymer fraction versus saturation curve of Fig. V.7b from the solution data. Again, both the desaturation and equilibrium (resaturation) curves are calculated, showing a small and probably unobservable hysteresis. The agreement between the observed and calculated curves is remarkably good, and may in part be fortuitous because of the large uncertainties in the NMR determination of the fraction polymerized and the approximate nature of the calculations (Noguchi et al., 1979, 1983). Nevertheless, these experiments demonstrate in a very direct way that the fraction of polymerized hemoglobin inside cells does not differ greatly from what is expected from solution studies. The most common technique that has been used to monitor intracelM a r polymerization has been to observe changes in cell morphology as the oxygen pressure is decreased. This is an extremely important experiment to understand because it is the simplest of all methods for studying intracellular polymer formation, and has been widely used in sickle cell research (Sherman, 1940; Griggs and Harris, 1956; Harris et al., 1956; Bessis and Delpech, 1982). Figure V.8 shows a representative result for the blood of an individual from a study of seven patients (Bookchin et al., 1976). In this experiment, cells that were initially oxygenated by air were equilibrated at a given oxygen pressure for about 15 min, and then fixed in formalin. A cell was considered to be “sickled” if any “clear distortion from the shape of a biconcave disk” occurred. At each pressure the number of irreversibly sickled cells, which comprised about 15% of the average sample (R.M. Bookchin, personal communication), was subtracted from the total number of sickled cells, to give the number of “newly sickled” cells. Irreversibly sickled cells are cells that are deformed even when fully oxygenated. They usually comprise about 10-20% of the cell population, with a range among different patients of 2-50%, and have the highest intracellular hemoglobin concentration (Bertles and Milner, 1968; Seakins et al., 1973; Clark et al., 1982; Rodgers et al., 1985). This experiment can be modeled by calculating the fraction of cells having a concentration of hemoglobin S that exceeds the solubility at each oxygen pressure by a certain amount. This fraction, F ( $ ) , is given by

As discussed above, the lower limit of the integral, c:, is greater than the solubility because intracellular polymerization will not take place unless

218

WILLIAM A. EATON AND.JAMES HOFRICHTER

C

C

.-0 0 E

0 .I3

CI

L

O

0

50 100 Oxygen Pressure (torr)

0

0.5

1

E o L L

Fractional Saturation

FIG.V.8. Counting sickle cells as a function of oxygen pressure or oxygen saturation. The data are at 37”C,pH 7.35,P‘co:, = 40 torr from a single individual, but are representative results from a study of seven Iioniozygous SS patients by Bookchin et al. (1976). (a) Fraction of cells sickled as a function of oxygen pressure. T h e points (@) are the data. T h e range of pressures at which 50% of the cells sickled for the seven patients was 32-39 torr. (---) T h e solubility was calculated at each oxygen pressure from Eq. (V.2) (with p.50 = 26 torr and n = 2.6) and Eq. (V.5), and the fraction of cells with concentrations equal to or exceeding the solubility ic: = c, in Eq. (V.9)lwas calculated by interpolation of‘ the cumulative froin the directly measured concentration distributions of Figs. V.24 V.3,and V.2h, respectively (b). [Left (---)I The calculated equilibrium curve corresponding to the concentration distribution of Fig. V.2a. [Right (---)I T h e equilibrium curve corresponding to the concentration distribution of Fig. V.2h. These two curves define the approximate extremes expected for equilibrium curves. [Middle (---)I T h e equilibrium curve that corresponds to the average coricentratiol7-$istribution of Fig. V.3.These equilibrium curves should be close to “unsickling” curves, ix., the fraction of sickled cells as a function of oxygen pressure on reoxygenating a completely deoxygeiiated cell suspensinn. (-) T h e predicted sickling curves, calculatcd for the same three concentration distributions, except that c: is the concentration that corresponds to the activity supersaturation (=yocolysc,) of 3.0.(b) The curnulatives of the distributions in Figs. V.2a, V.3,arid V.2h,respectively. [(-) and (---)I The probability that the intracellular hemoglobin concentration of a cell is greater than the specified value. (c) Fraction of sickled cells as a function of the average saturation with oxygen. The points (@) are the data. The range of oxygen saturations at which 50% of the cells sickled for the seven patients was 0.39-0.54.[(---) and (-)I Calculated as in a, except that at each pressure the average saturation ((y,)) was calculated using Eqs. (V.2), (V.3),(V.5).(V.7),and (V.8).

the supersaturation is sufficient to produce a delay time less than the 15-min equilibration period. To obtain the fraction of cells as a function of oxygen saturation, the average saturation, (y,), is calculated at each pressure for which F ( p ) is calculated [Eq. (V.7)]. T h e most concentrated 15% of the cells was deleted from the distribution in calculating the fraction of sickled cells to account for the neglect of the irreversibly sickled cells in the renormalization of the experimental data.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

2 19

The calculated curves in Fig. V.8 show the results expected for the average concentration distribution of Fig. V.3 and the extremes of the concentration distributions (Fig. V.2a and V.2h). The calculations show that kinetic effects are predicted to be much larger in this experiment than in either the oxygen binding curves (Fig. V.5) or the NMR determination of the fraction polymerized (Fig. V.7). Because there is no delay in depolymerization, the curves obtained by counting sickled cells on reoxygenation (“unsickling” curves) should follow the equilibrium curves much more closely. The equilibrium curves are the dashed curves in Fig. V.8 with cl = c, in Eq. (V.9). Only preliminary experiments have been performed in which the pressure at which cells recover their normal shape on reoxygenation has been measured. T h e results, however, do show the large hysteresis predicted from the preceding analysis with the “unsickling” curve considerably right-shifted from the “sickling” curve (A. Mozzarelli, unpublished observations). Both unsickling and sickling curves have been measured using carbon monoxide and a laser photolysis technique (Mozzarelli et al., 1987); these experiments are discussed in Section V,C. The last experiment to be discussed in this section is the measurement of the optical densities in polarized light of single sickled cells at zero oxygen pressure (Hofrichter et al., 1973) (see Fig. V.9). T h e optical theory of this experiment has been discussed in detail in the original paper (Hofrichter et al., 1973), and in reviews (Hofrichter and Eaton, 1976; Eaton and Hofrichter, 1981). The quantity of interest is the polarization ratio, defined as the ratio of optical densities for light polarized perpendicular and parallel to the polymer axis. For a gel consisting of perfectly parallel polymers the polarization ratio, PR, is given by (Eaton and Hofrichter, 1981) PR

=

EIXP

+ E(l

- xp)

El&

+ 41

- xp)

(V. 10)

where E~ is the extinction coefficient for light polarized perpendicular to the polymer axis, ell is the extinction coefficient €or light polarized parallel to the polymer axis, and E is the isotropic extinction coefficient, i.e., the extinction coefficient for the randomly oriented molecules of the solution phase which is the same for both parallel and perpendicular polarizations. Equation (V.10) points out that absorption by solution phase hemoglobin molecules decreases the observed polarization ratio from that for a pure polymer phase with perfectly parallel polymers (PR = E ~ / E ~ Because ~ ) . of the variation in intracellular hemoglobin S concentrations in a cell population, a range of polarization ratios is observed.

220

WILLIAM A. EATON AND .JAMES HOFKICHTER

,

i

R,, -..-...

....

,

3

2

1

*-__

Polarization Ratio

FIG.V.9. Observed and calculated distribution of polarization ratios for sickle cells. of the distribution of polarization ratios measured at room temperature for the Soret band of 106 completely deoxygenated sickle cells Iron1 10 different patients. (---) The calculated distribution (i.e., the probability density) IP(PR)] calculated from Eqs. ( V . l l ) and (V.12) with c, = 0.183 g/cm3,cp = 0.69 g/cmS,e , = 1.33c, and E,, = 0 . 3 4 ~ (Eaton and Hofrichter, 198 I). The concentration distribution [P(co)]was obtained from a least-squares fit to the measured average concentration distribution of Fig. V.3, using a sum of two Gaussians: nl exp{-[(c,, - cl)/w1]2} + a2 exp{ - [ ( c o cZ)/w21'} with a1 = 8.38, a2 = 1.59, c l = 0.336 g/ctn3, c'2 = 0.380 g/cm9,w l = 0.040 g/cmS, and w2 = 0.151 g/crn7.The amplitudes (a1 and a2) were scaled to make P(PR) normalized. For each value of PR,toand d(c~)/d(PR)were calculated from Eq. (V.12) and its derivative. The probability density for the calculated ca, obtained from the Gaussian fit, multiplied by d(co)ld(PR), resulted in the probability density for the selected value of PR IEq.(V.l I)]. [Data of Hufrichter et al. (1973).] (0) Histogram

~

We may ask whether the distribution 01 polarization ratios can be explained from the intracellular concentration distribution and the extinction coefficients of the polymer. T h e polarization ratio distribution, P(PR), can be calculated from

P(PR) where

cg as

=

(V.1 1)

P(c,,)(dc,,/dPR)

a function of YR is from Eqs. (V.4) and (V.lO): ~ ( 1 PR) (&(I

-

E)PR

-

(EL

- E)

I)-'

(V.12)

SICKLE CELL HEMOGLOBIN POLYMERIZATION

221

The extinction coefficients can be accurately calculated from the 14stranded polymer structure (Section II,C) for the Soret region, where the heme behaves as a nearly perfect planar absorber, and are found to = 1.338 and E,, = 0.348 (Eaton and Hofrichter, 1981). be Figure V.9 shows the distribution of polarization ratios measured on cells from 10 different patients, and the polarization ratio distribution calculated from Eqs. (V.11) and (V.12)using the measured average concentration distribution of Fig. V.3. T h e calculated distribution is somewhat sharper than the observed, which could simply represent differences in the concentration distributions (see Fig. V.2).Two other effects, which have not been taken into account in the calculation, could broaden the distribution. One is that there is a decreased ordering of polymers, and therefore a lower polarization ratio, with a decreasing fraction polymerized (Sunshine et al., 1982);the second is the presence of fetal hemoglobin in F cells, which reduces the polarization ratio by lowering the amount of polymer formed at the same total hemoglobin concentration (Section HI$). Nevertheless, the good qualitative agreement between the observed and calculated distributions shows that the wide range of intracellular hemoglobin concentrations in the cell populations is the primary cause of the wide distribution of measured polarization ratios. This result is important because it shows that the absorption ellipsoid calculated from the 14-stranded polymer structure is consistent with the observed optical properties, thereby providing a critical test of the proposed structure, as discussed in Section II,C. Recently, the coupling of an optical microscope with an area detector has permitted the spatial distribution of aligned hemoglobin in individual, glutaraldehyde-fixed sickled cells to be measured directly. By comparing the difference image obtained using orthogonal linear polarizations of the input light with the averaged (isotropic) image measured with Soret illumination (400-440 nm), an image of the optical anisotropy was obtained. Integration of this signal over the entire cell surface then permitted the total amount of aligned hemoglobin in the cell to be quantitated (Mickols et al., 1985,1988;Beach et al., 1988).T h e principal result from this study is that the amount of aligned hemoglobin at a fixed rate of deoxygenation decreases with increasing number of domains in cells, even though multiple-domain cells are predicted to contain more polymerized hemoglobin. Presumably, the cancellation of the optical anisotropy from overlapping domains reduces the linear dichroism more than the larger amount of polymerized hemoglobin increases it. It was also observed that there is an increase in the fraction of cells with no optically resolvable domains as the deoxygenation rate increases, a result which is expected because of the increase in rate of homogene-

222

WII.I.IAM A. EATON A N D J A M E S HOPKICHTER

ous nucleation (see Fig. V. 18). One problem with this work is that the average percentage of aligned hemoglobin in sickled cells containing one to three domains is calculated to be only 11- 14%, compared to almost 70% for the percentage of total polynierized hemoglobin predicted from gelation studies and measured by nuclear magnetic resonance (Fig. V.7). In contrast, the optical anisotropy data discussed above on unfixed cells containing a single domain (Fig. V.9) are consistent with almost perfect alignment of all of the polymerized hemoglobin, suggesting that glutaraldehyde fixation produces a major diminution in either the amount or alignment of the polymerized hemoglobin.

B . Kinetics of Intracellular Polymerization

Prior to the development of the laser photolysis technique described in Section IV, A, the only kinetic information on intracellular polymerization came from indirect measurements that relied on changes in red cell shape or filterability. 'These studies gave apparently conflicting results, and none of them demonstrated the large range of delay times predicted from the solution studies. Furthermore, studies on the effect of red cell membrane components on the delay time for deoxyhemoglobin S polymerization in solution showed little or no effect (Goldberg et al., 1981) (Fig. V.lO). The development of the laser photolysis technique permitted the first direct measurement of intracellular polymerization (Coletta et al., 1982). As in the solution experiments (see Section IV,A), the cw argon ion laser is used both to initiate the polymerization process and as a source for monitoring polymer formation. T h e laser light rapidly creates deoxyhemoglobin S by photodissociating the carbon monoxide complex, and the appearance of polymer is detected as an increase in light scattering. T h e high intensity of a laser is required both to achieve complete photodissociation in less than about 1 msec and to detect changes in light scattering from the very small volumes of a single cell. Figure V. 11 shows a representative set of' kinetic traces from a series of measurements on over 400 cells (Coletta et al., 1982). Also shown are kinetic curves for hemoglobin S solutions and the scattering observed from normal red cells. The kinetic curves for sickle cells are more complex than those measured for solutions for at least two reasons. First, polymerization causes the cell to deform. This cellular deformation results in an irreproducible change in the light scattering, which may interfere with or totally obscure the observation of the polymerization progress curve. Second, there is much greater noise observed with sickle

i.;

223

SICKLE CELL HEMOGLOBIN POLYMERIZATION

1.o

-5 .-E

0.1

??

0.01

-

l a

Ib

C

0

0.001

0.17

0.19

L 0.17

0.19

I

0.17

I

l

0.19

l

1

1

0.21

Hb S (g/clm3)

FIG. V.10. Effect of’red cell membrane on the delay time of deoxyhemoglobin S (Hb S) polymerization. Delay times (td) were measured using the temperature-jump, turbidity technique (Section IV,A). (a-c) T h e control, deoxyhemoglobin S solutions with no added membrane (0). (a) Effect of 0.5 1 mg/ml of inside-out vesicles (0). (b) Effect of 0.60 mg/ml of inside-out vesicles to which were added actin and spectrin (0). (c) Effect of added membrane ghosts: 0.67 mg/ml (U),2.07 mg/ml (A), 3.7 mg/ml (0). T h e data were corrected by Goldberg et al. (1981) for the excluded volume effect of the membrane ghosts o n the delay time. [From Goldberg et al. (1981).]

cells compared to solutions. This noise also occurs with normal cells (Fig. V. 11) and has been called the “flicker phenomenon”; it has been interpreted as arising from Brownian motion (Parpart and Hoffman, 1956; Burton et al., 1968; Padilla et al., 1973). Brownian fluctuations in the cell thickness could modulate the interference of the light reflected from the top and bottom cell surfaces, which would appear as noise in the reflected laser light. For the most rapidly polymerizing cells the signals are most similar to those observed for solutions, exhibiting a clear delay, followed by an abrupt increase in scattering due to polymer formation. An interesting characteristic of cell progress curves is the marked decrease in the amplitude of the noise at the onset of polymerization. This could result from damping of the thickness fluctuations caused by the increase in internal viscosity of the cell accompanying polymer formation. It would appear that only a small amount of polymer formation is sufficient to reduce these fluctuations markedly. As the delay times become longer than about 0.03 sec, the abrupt change in the noise amplitude becomes more apparent and the distortion of the signal from cell sickling becomes more prominent (Fig. V.12). In cells where a polymerization pro-

224

WILLIAM A. EATON AND JAMES HOFHICHTER

..ih-"c

i-d+ t z

W

I - -

0

W

u

-

U

W

k

I-

Q

0

v)

-

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 I

1 1

1 I

1 1

1 1

1 1

1 1

1 1

I

I

I

1

1

I

I

1

1

1

1 I

I

1

1

1 I

I

1

1

I

1

1

1

l

1

1

]

1

1

1 I

i-

1 1 1 1

1

1 I

1

1

1 1

1

l

1 I

1

l

1 I

l

l

t 1

l

1 1

l

1 1

;

I

l

1 1

1 1

l 1

I

1

1

;

I

I

;

I

~

I

I 1

1

I

l

l

I

1

I

TIME

FIG. V.11. Kinetic progress curves for henioglobiri S solutioiis and sickle cells in laser photolysis experiments. The scattered light iiiteiisity is plotted versus time. (a-g) Kinetic progress curves are shown for puritied hemoglobin S solutions (a and b), 5 normal cells, and 10 sickle cells at 37°C [(a) 20 mseddiv, (b) 500 msec/div, (c) 5 rnsec/div, (d) 20 msec/div, (e) 100 msecldiv, ( f ) 250 msec/div, and (g) 5 seddiv]. T h e data are representative of experimcnts on 453 cells from 4 different patients. N o normalization has been applied to the amplitudes. The arrows indicate the onset of polymerization. Traces without arrows are for normal rells. lSee Coletta ct al. (1982) for details.] [From COlettd ct rid. (1982).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

225

.-K0

c

e u

L

Log Time (sec)

FIG. V.12. Fraction of progress curves with solution-type progress curves in laser photolysis experiments on sickle cells. The kinetic traces in laser photolysis experiments on single cells can be roughly classified into two types. The first type appears like a solution progress curve in which there is a clear delay, followed by an increase in light scattering of amplitude comparable to that found in solution. In the second type, there is a marked decrease in the amplitude of the noise, but the subsequent change in light scattering is quite variable, and is attributed to cellular deformation. The fraction of cells with solutiontype progress curves) . ( decreases approximately exponentially with delay time as indicated by the exponential curve (-) which has a l/e time of 0.4 sec. [Data from Coletta et al. (1982).]

gress curve is no longer evident, the delay time is taken as the time at which there is a sudden decrease in the amplitude of the noise. Figure V.13 shows the distribution of delay times observed in cells from four different patients with homozygous sickle cell disease. There is a very wide distribution of delay times from a few milliseconds to more than 100 sec. This is exactly what is expected from the solution studies, which predicted an enormous variation in the delay time for intracellular polymerization (Hofrichter et al., 1974b, 1976a,b; Ferrone et al., 1980, 1985a). The interpretation of the progress curves for cells with long delay times and the width of the delay time distribution have been confirmed using an improved scattering technique (Mozzarelli et al., 1987). In these experiments, polymerization is monitored using forward scattered light, so that the interference from reflections and scattering by the cell surface are reduced dramatically. Sample progress curves and the distribution of delay times are shown in Fig. V. 14.

226

WILLIAM A. EATON A N D JAMES HOFRICHTER

LOG DELAY TIME (sec)

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

30 20 10 0

20

I", 10 -I

w

0

LL

0

a w

m

f 2

0 20 10 0 20 10 0

LOG DELAY TIME (sac)

FIG.V.13. Delay time distributions for cells from four different patients (a-d) with homozygous sickle cell disease. For each distribution, the median delay time (tao) and percentage of fetal hernoglobin (% Hh F) are (a) tao = 0.060 sec, 6.8% Hh F; (b) t 5 0 = 0.060 sec, 3.8% Hb F; (c) t6,, = 0.083 sec, 8.2% Hb F; (d) tso = 0.23 sec, 13.8% Hh F. The number of cells with delay times longer than 50- 100 sec (which were not measured) were (a) 4 out of 155 cells, (b) 4 out of 105 cells, (c) 4 out of 87 cells, and (d) 6 out ot' 101 cells. These cells were counted in the calculation of the median delay time. [From Coletta et al. (1982).]

To compare the solution and cell studies more quantitatively, a concentration distribution can be calculated from the delay time distribution. The results are shown in Fig. V.15,where the intracellular hemoglobin S concentration corresponding to each delay time was calculated from the solution kinetic data. The calculated concentration distribution has a roughly Gaussian shape with a mean value of 0.32 g/cm3. This mean value is in good agreement with the mean intracellular hemoglobin S concentration of 0.32-0.36 g/cm" measured from the volume fraction of cells in whole blood, the total hemoglobin concentration, and the

227

SICKLE CELL HEMOGLOBIN POLYMERIZATION

I

I

I

I

1

LOG DELAY TIME (sec)

TIME FIG. V. 14. Delay time distribution and kinetic progress curves for sickle cells using the laser photolysis light-scattering technique with detection of the forward scattered light. (a) The distribution of delay times measured for 154 cells from a single patient. [From Mozzarelli el al. (1987).] (b-e) Samples of the progress curves obtained for individual cells in this experiment. The scattered intensity is plotted versus time in b-e. Progress curves from cells with comparable delay times have been selected for display. T h e microscope optics used in this experiment focus a 2-pm spot from an argon ion laser onto the center of the cell. The light scattered around a mask placed in the center of the back aperture of a 90X objective is collected and measured. T h e experiment is described in more detail by Mozzarelli et al. (1987). (b) Bar, 10 msec; (c) bar, 25 msec; (d) bar, 250 msec; and (e) bar, 2 sec. The arrows indicate the delay times.

228

WILLIAM A. EATON AND JAMES HOFRICHTER

LOG TENTH TIME (sec)

P

4.AL j 00.2

0.3

0.4

, _ 0.5

CONCENTRATION (g/cm3)

FIG. V. 15. Distribution of intracellular hemoglobin S concentrations calculated from delay time distribution on sickle cells. (a) Distribution of delay times. This is the sum o f t h e delay time histograms (0)for patients a-c from Fig. V.13. The histogram from patient d was not included, since the percentage of fetal hemoglobin was more than two standard deviations from the mean of 5.6 +- 3.6% found in sickle cell patients (Serjeant, 1974; Wrightstone and Iluisman, 1974). (b) Concentration dependencc of the tenth time from solution experiments. ‘The solution data (0)of Ferrone et al. (1980) at 35°C (see Fig. IV.12). (-) Drawn from the empirical function c (g/cmJ) = a exp( - bx + dr2)+ e, where the parameters a = 0.104 g/rm3, h = 0.287, d = 0.021, and c = 0.167 g/cm3 were obtained from a least-squares Iit to the data. (c) This is the histogram (0) of intracellular hemoglobin S concentrations obtained by calculating the concentration corresponding to each delay time using the empirical function in b. (---) The gaussiari function P = a exp{ - [(b - c)/d]*}, where c is the intracellular hemoglobin S concentration in g/cm3, and the parameters (I = 33.5, b = 0.316 g/crn’, and d = 0.057 g/cm3 were obtained from a least-squares fit to the histogram. The mean concentration in c is 0.32 g/cm’. [From Coletta et (11. (1982).]

229

SICKLE CELL HEMOGLOBIN POLYMERIZATION

mean fraction of fetal hemoglobin (Seakins et al., 1973; Kaul et al., 1983a; Fabry et al., 1984). We can make an even more detailed comparison of the solution and intracellular delay times by comparing the calculated distribution of intracellular hemoglobin S concentrations with the average measured distribution of total hemoglobin concentration (Fig. V. 16). T h e mean for the average measured distribution is somewhat higher than that calculated from the intracellular kinetics. There are several possible reasons for this difference, other than inaccuracies in the measurement of the intracellular concentrations. First, the patients selected for the kinetic studies might not have the most common concentration distributions, but have distributions closer to that of Fig. V.2a. Second, the solvent for the solution studies (0.15 M potassium phosphate, pH 7.0, 0.05 M sodium dithionite) has a different composition than the intracellular solvent, which could very well affect the rate parameters for the polymerization process. Finally, the presence of fetal hemoglobin influences the delay times, but not the measurement of the total hemoglobin concentration. The exchange of fetal hemoglobin for hemoglobin S markedly increases the delay time (Sunshine et al., 1978, 1979b). To understand the effect of fetal hemoglobin, it is necessary to recognize that, except in a rare double heterozygous condition (sickle cell disease with hereditary 15

0.3

n

-.m

: 0.2

10

6

v0

b 07

0,

n

5 z LL

> E

5

0.1

b I n 0 n

e

a 0

0.2

0.3

0.4

0.5

0.6

3

lntracellular Hb Concentration (g/cm3) FIG.V. 16. Comparison of calculated and observed intracellular hemoglobin concentration distributions. (0) Histogram of the intracellular hemoglobin S concentrations calculated from the delay time distribution (Fig. V.15) and (---) the average concentration distribution measured from the density gradients (Fig. V.3). The points (0)are the measured cell fractions.

230

WILLIAM

A. EATON AND JAMES

HOFRICHTER

persistence of fetal hemoglobin), it is not evenly distributed among the red cell population. It is concentrated in so-called “F cells,” which contain about 40% hemoglobin F and 60% hemoglobin S (Dover et al., 1978). These F cells are least abundant in the portion of the cell population which has the highest total intracellular hemoglobin concentration (Seakins et al., 1973). From the kinetic data on S + F mixtures, a cell with a total hemoglobin concentration of 0.35 g/cm3 and 40% hemoglobin F is predicted to have a delay time of about 10 sec (Sunshine et al., 1978), while a cell containing only hemoglobin S at 0.35 g/cmq is predicted to have a delay time of about 0.02 sec. This calculation indicates that the subpopulation of F cells is responsible for the longest cellular delay times, and can account for the lowest calculated intracellular hemoglobin S concentrations. T h e effect of fetal hemoglobin, however, can account for only part of the increase in the measured intracellular concentrations compared to those calculated from the delay times in Fig. V.16. Another interesting parameter of the kinetics to compare in solution and cell studies is the reproducibility of the delay time. In the solution studies described in Section IV, we found that, in measurements on small volumes, the delay time is highly reproducible for samples with delay times of less than about 0.1 sec. When the delay time is longer than several seconds, however, it becomes highly irreproducible (Ferrone et al., 1980). This large variability in the observed delay time results from the stochastic fluctuations in the homogeneous nucleation process (Ferrone et al., 1980, 1985b; Hofrichter, 1986). Stochastic behavior is observed because polymerization of the entire sample volume under observation is initiated by the homogeneous nucleation of a single polymer molecule (Section IV,C). In samples with short delay times a large number of homogeneous nucleation events have occurred and polymerization is observed as a highly reproducible process. Figure V. 17 shows the FIG. V.17. Keproducibility of delay times from repeated measurements on the same cell. (a-m) Earh histogram (0)is the number of measurements in which the delay time was found in a specified interval. I n earh case, the total widt.h of the histogram is approximately three times the mean delay time. The bin size in earh case is one-fifteenth the total width of the histogram. For each histogram, the mean delay time and its standard deviation are as follows: (a) 0.054 2 0.006 sec (31), (h) 0.0.51 2 0.005 sec ( l Y ) , (c) 0.12 +. 0.01 sec (30), (d) 0. IS 0.03 sec (20). (e)0.21 0.02 sec (SO), (f) 1.7 2 0.3 sec (23). (g) 1.5 f 0 . 5 s e c ( l l ) , ( h ) I . 7 ~ 0 . 5 s e r ( I . 5 ) , ( i ) 2 . 9 ~ 0 . 7 s e c ( ‘ L 7 ) , ( j )36. 1. s0e~c ( 9 ) , ( k ) 8 . 3k 5 . 7 sec (14). (I) 9.1 f 9.0 sec ( I I ) , (m) 10.8 +. 7.2 ser (8). (The number in parentheses is the number of measurements on each cell.) (n) A plot of the standard deviation divided by the nieati for each cell as a hnction of the logarithm of the niean delay time (a).[Frnm data of Coletta et al. (1982).1

*

19

20 b

I

I

i

i5 n .Ls=--K

Olo O

0.05

0.1

0.15

11:i5I

0L

I i

01

0

1.5

h

3

3

3

I

6 6

4.5O

I

10

do g o

(i

t

1 5

4-24-Au5 lk

20

0.4

0.6

0

1’ j L P 3 - l

0

OO

Olo O

0.2 I

1.5

I

3

4.5

10

I

I

10

20

30

20

3oo

c j : ’

Delay Time (sec) 5

232

WILLIAM A. EATON AND .JAMES HOFKICHTER

results of repeated experiments on the same cell. Experiments on an individual cell cannot be repeated indefinitely, and after about 10-30 cycles of polynieriLation and depolymerization the cell becomes irreversibly distorted, and further measurements are not possible. As was found with solution studies, the irreproducibility, as measured by the relative standard deviation from the mean, increases dramatically for delay times longer than a few seconds (Coletta et al., 1982). This result indicates that the rate of homogeneous nucleation relative to the rates of heterogeneous nucleation and polymer growth is the same in cells and in purified solutions. There are, then, three major experimental results on cells that agree with predictions from solution experiments: the shape of the kinetic progress curves in rapidly polymerizing cells, the wide range of delay times, and the sharp increase in the irreproducibility of the delay time at long times. The conclusion from this comparison is that polymerization inside sickle red cells proceeds by the same nucleation and growth mechanism as in purified hemoglobin S solutions (Coletta et al., 1982). This is a comforting result, in view of the large amount of research on solutions that have implicitly assumed this to be true. With the data from the laser photolysis experiments it is possible to rationalize many of the results of earlier kinetic studies based on observations of cellular deformation and changes in filterability. In these investigations, oxygenated cells are mixed with sodium dithionite, which decreases the oxygen pressure outside the cell to zero by chemical reduction. Dithionite does not enter the red cell, which can be shown by its failure to reduce intracellular methemoglobin to deoxyhemoglobin (E. M. Eaton and W. A. Eaton, unpublished observations). With rapid mixing techniques the half-time for complete deoxygenation of intracellular oxyhemoglobin can be as short as 10 msec at 37°C and 70 msec at 20°C (Rampling and Sirs, 1973). In the filterability experiments, the increased internal viscosity of the cells from polymerization impedes their ability to pass through Millipore filters. For sickle cells a marked reduction of filterability was found at 0.12 sec at 37"C, which was the shortest time investigated (Messer and Harris, 1970). This result is consistent with the finding that intracellular polymerization has taken place in over 50% of the cells by 0.12 sec (Fig. V. 15). Interpretation of the results from studies using cellular deformation as a criterion for intracellular polymerization is more difficult. A major problem with these experiments is that there is no quantitative criterion for the onset of intracellular polymerization as judged by the change in cell shape, and different authors use different criteria. l h i s is in part due to the fact that cells assume a myriad of distorted shapes. Cells hav-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

233

ing the highest intracellular concentration exhibit the least deformation (Kaul et al., 1983a), and the extent of deformation is smallest for the more rapidly deformed cells (Fig. V.18) (Harrington and Nagel, 1977). These observations may be explained by postulating that the more distorted cells (Fig. V. 18) contain fewer polymer domains. As the intracelM a r concentration increases the rate of homogeneous nucleation increases, which results in an increasing number of smaller domains, and, therefore, a less deformed cell. This hypothesis is consistent with the observation that the most rapidly polymerizing cells in the laser photolysis experiments exhibited solution-type progress curves, and that the decrease in the number of polymer domains results in greater cellular deformation and a more distorted light-scattering signal (Figs. V. 1 1 and V. 12). An interesting observation has been made in photolysis experiments with a weaker (xenon) light source in which deligation occurs in several seconds (Coletta et al., 1988). In repeated sickling and unsickling experiments, cells attached to a cover slip are observed to deform along the same axis, suggesting that weak points in the membrane determine the direction of elongation of polymers. The difference in criteria for intracellular polymerization could explain some of the apparent discrepancies in the cell deformation studies. In experiments using a rapid flow apparatus and glutaraldehyde fixation, the minimum delay time was reported as 30 If: 3 msec and the median delay time as 87 k 6 msec (Rampling and Sirs, 1973). While the median delay time is in excellent agreement with that found in the laser photolysis studies (Figs. V.13-V.15), the minimum delay time of 30 msec is significantly longer. The longer time presumably results from the fact that about 30 msec is required for 80% deoxygenation, and the delay time for the fastest polymerizing cells are lengthened because of incomplete deoxygenation. In an earlier study using similar techniques, deformation was noted in only 5 % of the cells at 120 msec, even though there was a marked decrease in filterability at this time (Messer and Harris, 1970). This apparent discrepancy might very well be due to the fact that cells with delay times of 120 msec or less have a large number of polymer domains, and hence did not satisfy the author’s criterion for “sickling and its early changes.” The most extensive morphological studies on cells utilized a flow channel in which the cells were adhered to a glass surface and the airsaturated suspending solution was replaced with a deoxygenated sodium dithionite solution. Direct visual observations were made by video tape recording or filming (Zarkowsky and Hochmuth, 1975, 1977). T h e time required for complete deoxygenation was not measured in these experi-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

235

ments, but was estimated to be 400 msec. This value was therefore subtracted from all of the observed delay times. At 25-27°C the median delay time was found to be 2.1 t 0.7 sec (Zarkowsky and Hochmuth, 1975), while the minimum delay time was generally more than 1 sec (Fig. V.19). Shorter delay times may have been observed in a significant fraction of cells, however, since the so-called “irreversibly sickled cells” were described as undergoing “extremely rapid” cell deformation, and were excluded from the data. Because they have the highest intracellular hemoglobin S concentration (Bertles and Milner, 1968; Clark et al., 1982), the delay times for irreversibly sickled cells are expected to be among the shortest ones observed. From studies of the temperature dependence (Zarkowsky and Hochmuth, 1975), the median delay time in this study can be estimated to be about 1 .O sec at 37”C, which is more than 10 times longer than the median delay times observed in the laser photolysis study (Fig. V.13). The flicker phenomenon was observed to disappear concomitant with cell deformation, suggesting that this difference does not arise from the criterion used for intracellular polymerization. The more likely explanation is that deoxygenation was slower than estimated. This would have the effect of considerably lengthening the delay times of the more rapidly polymerizing cells. Because the delay times for the fastest polymerizing cells in the flowchannel experiments are significantly lengthened compared to the values at complete deoxygenation, while deoxygenation is probably complete for the slowest polymerizing cells, the distribution of observed delay times in these experiments is considerably narrowed relative to those of the laser photolysis experiments. This has the effect of reducing the sensitivity of the median delay time to changes in the extracellular solution, and it is not possible to make quantitative comparisons between the results of the flow-channel experiments and solution studies. T h e qualitative similarities are, however, apparent in these cell experiments (Zarkowsky and Hochmuth, 1975, 1977). The effects of pH and temFIG.V.18. Optical micrographs of sickle cells formed with different rates of deoxygenation. (a) Cells formed by slow replacement of oxygen with nitrogen in a sealed chamber. [From Sherman (1940).] (b) Cells formed by rapid replacement of oxygen with nitrogen. [From Sherman (1940).] (c) Cells formed by slow deoxygenation with nitrogen at room temperature (linearly polarized 430 nm light). (d) Cells formed by mixing suspension with sodium dithionite at room temperature to rapidly remove oxygen (linearly polarized 430 nm light). (e) Cells in c between crossed polarizers (450 nm light). (f) Cells in d between crossed polarizers (450 nm light). In c-f, the electric vector of the linearly polarized light is horizontal. [(c-f) Courtesy of G . W. Christoph. See also Asakura and Mayberry (1984).]

236

140 120

WILLIAM A. EATON AND JAMES HOFRICHTER

-- a

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80

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SICKLE CELL HEMOGLOBIN POLYMERIZATION

237

perature are consistent with solution studies, in that increasing the extracellular pH between 7.0 and 7.6 resulted in a continuous increase in the median delay time (Fig. V. 19a-d). The effect of intracellular concentration was observed in two types of experiments. In the first, increasing the osmolarity of the extracellular medium markedly decreased the median delay time (Fig. V.19e-g). In the second, cells were separated according to density by centrifugation, and the median delay time decreased monotonically from the least dense to the most dense fractions (Fig. V. 19h-j). The effect of fetal hemoglobin has also been observed in a more recent study using the same flow-channel technique, where a correlation was observed between the fraction of cells with delay times longer than 3 sec and the fraction of fetal hemoglobin (Sewchand et al., 1983). We now turn to the last topic to be considered in this section-the intracellular disassembly of hemoglobin S polymers. As in the case of the solution studies (Section IV,D), there has been no systematic study on the rate of depolymerization in cells, and there is only rather sketchy information. Three different types of measurements show indirectly that depolymerization is rapid, and can occur in a few seconds or less at 37°C. First, early filtration studies showed that the low filterability of a deoxygenated red cell suspension returns to its oxygenated value within 0.12 sec (Messer and Harris, 1970). Second, in measurements on the rate of oxygenation of sickled red cells the half-time for reoxygenation is only increased to about 0.1 sec, compared to 0.07 sec for normal AA cells (Harrington et al., 1977). Finally, visual observations in the course of the laser photolysis kinetic studies (Coletta et al., 1982) showed that, following extinction of the laser, cells reformed their biconcave disk shape in a few seconds or less. From the discussion in this section, it should be clear that a number of very important experiments remain to be carried out on cells. To make a more rigorous comparison of solution and intracellular kinetics it will be necessary to carry out solution studies under more physiological soFIG. V.19. Effect of physiological variables on the delay time for sickling of SS cells following deoxygenation in a flow channel. (0) Number of cells sickling and (0-0) percentage of cells sickling. Unless stated otherwise, all measurements were performed in phosphate-buffered saline (pH 7.4), 280 mOsM at 27°C. (a-d) Effect of pH. (a) tsn = 3.2 sec, pH 7.01; (b) t 5 0 = 3.5 sec, pH 7.19; (c) tao = 4.5 sec, pH 7.40; and (d) t5" = 9.4 sec, pH 7.60. (e-g) Effect of osmolarity. (e) tan = 4.9 sec, 200 mOsM; (f) t5,, = 1.74 sec, 280 mOsM; and (g) t 5 0 = 1.45 sec, 360 mOsM. (h-j) Effect of cell density. (h) t5,, = 3.34 sec, top fraction; (i) t 5 0 = 1.82 sec, middle fraction; and (j) tin = 1.4 sec, bottom fraction. [From Zarnowsky and Hochrnuth (1975).]

238

WILLIAM A. EATON AND JAMES HOFKICHTEK

lution conditions, and to extend the measurements to hemoglobin S concentrations greater than 0.4 g/cm 3. Such measurements are technically difficult. It will also be important to make kinetic measurements on cell populations of known concentration distributions and on densityfractionated cells. Delay time measurements on cells with uniform intracellular hemoglobin concentrations, prepared either by density fractionation or with ionophores (Brugnara et al., 1985, 1986), should determine the extent to which there is heterogeneity in the solubility, apart from the effect of fetal hemoglobin. Finally, a whole class of experiments remain to be performed which simulate the kinetics of‘polymerization and . would involve variation in the rate and depolymerization in Z J ~ Z J OThese extent of’oxygenation and deoxygenation, corresponding to physiological values. The f e w experiments that have been performed show the qualitative effects of oxygen pressure and intracellular hemoglobin concentration predicted from the solution studies (Messer et al., 1976; Hahn et al., 1976). In the next section we describe experiments that begin to address these questions. C. Intracellular Polymerization Kinetics at Partial Saturation

A complete description of the kinetics of intracellular polymerization that is relevant to the physiological situation requires measurements at rates and extents of desaturation that are found in vivo. Such measurements have not yet been performed with oxygen. However, a technique in which laser photolysis is used to control the fractional saturation of hemoglobin S with carbon monoxide has been developed and used to begin such an investigation. In this section, we describe the initial results with this method. Their significance for the pathophysiology is described in Section VI. The basic idea of the experiments is to use continuous illurnination with a laser to photodissociate the carbon monoxide complex and thereby create any desired steady-state saturation by adjusting the laser power. Two types of experiments can then be performed. In one, the scattering from the laser can be used to monitor the kinetics of polymer formation (San Biagio et al., 1988).In the other, a second, more intense, laser beam is used to photodissociate completely the remaining carbon monoxide, generating deoxyhemoglobin S, and the scattered light from this beam is used to measure the kinetics of polymer formation (Mozzarelli et al., 1987). I n both experiments, careful absorption measurements are made with a xenon source to determine the fractional saturation. Much more extensive measurements have been performed using the second technique, so we shall discuss them first.

2 39

SICKLE CELL HEMOGLOBIN POLYMERIZATION

The objective of the double-laser-beam experiments was to determine the fractional saturation at which polymers first form in single cells in desaturation experiments, and the saturation at which they completely disappear in resaturation experiments. Polymer can be detected with very high sensitivity by measuring the kinetics following complete photodissociation. In the absence of polymer there is a delay period, but the presence of even very small amounts of polymer, as little as O.I%, is sufficient to shorten significantly the delay time (Figs. V.20 and V.2 1).

2

b

a

L ,

I

,

,

I

,

,

e

I

100 200

0 0 In

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100 200 100 200 TIME (msec)

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_r +

100

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>

00

a 1

0

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FIG. V.20. Detection of polymer formation and disappearance in solutions of partially photolyzed carbonmonoxyhemoglobin S. (a-h) Kinetics of gelation after complete photodissociation of partially photolyzed solutions. In a-d, the saturation prior to complete photodissociation decreases. In e-h, the saturation of the gel prior to complete photodissociation increases. The presence of polymer in d, e, and f is indicated by the loss of the delay time. (i) Delay time as a function of the logarithm of the power of the preparative beam in desaturation experiments (0)and resaturation experiments (0).(j) Fractional saturation with carbon monoxide as a function of the logarithm of the power of the preparative beam in desaturation experiments (0)and resaturation experiments (0).T h e deviation from the theoretical curve for hemoglobin molecules free in solution results from the decreased affinity of the polymer. [From Mozzarelli et al. (1987).]

240

WILLIAM A. EAI‘ON AND JAMES HOFKICHIER

0.2 0.04 -

C

0 . 1 1

0,oz -

0

TIME (sec)

-

10

TIME (sec)

20

0

4

3

J

0

10

TIME (sec)

FIG.V.21. Sensitivity of kinetics of gelation to the presence of polymer. (a) Disappearance of polymer scattering following decrease in intensity of preparative laser illuminating a gelled sample. (b) Disappearance of polymer scattering on an expanded scale. The arrows indicate the times at which the kinetics of polymerization were measured with the more intense probe beam [ ( I ) 5 sec, (2) 10 sec, and (3) 20 sec]. (c) Kinetics following complete photodissociation at ( 1 ) 9 sec, (2) 10 sec, (3) 20 sec, arid (4) 30 sec after decreasing the power in the preparative beam. [From Mozzarelli et al. (1987).]

To show that this experiment accurately simulates the effects of oxygen, it was first necessary to demonstrate that polymerization at a given fractional saturation with carbon monoxide produced by continuous laser photolysis is equivalent to polymerization at the same fractional saturation with oxygen. To do so, the point at which the delay time was recovered as the fractional saturation of a gelled sample was increased (by reducing the laser power) was measured for samples of known concentration (the sample was first polymerized by exposure to an intense laser beam). The concentration of the sample at this so-called “critical saturation” is equal to the solubility. A plot of the total concentration versus the measured critical saturation should therefore be identical to a plot of the solubility measured as a function of solution phase fractional saturation. Figure V.22 shows that the solubility measured with the laser photolysis technique is the same as the solubility as a function of oxygen saturation measured in sedimentation experiments (Fig. 111.15) (Sunshine et al., 1982). This result implies that the distribution of R- and 2’state molecules is virtually the same for each case, a result which is not surprising. I n the binding of carbon monoxide to hemoglobin, cooperativity results primarily from a difference in the overall association rates, while for oxygen, cooperativity results primarily from a difference in the overall dissociation rates (Szabo, 1978). Illumination by the laser mark-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

241

0.45

0.3

0.15

0

0.5

1

FRACTIONAL SATURATION FIG. V.22. Comparison of solubility measured by laser photolysis and sedimentation methods in 0.15 M potassium phosphate (pH 7.0) at 35°C. (-) The least-squares fit to the data for partially oxygenated hemoglobin S (Fig. 111.15) (Sunshine et al., 1982), adjusted to 35°C. The points are the data for partially photolyzed carbonmonoxyhemoglobin S. (0)Recovery of delay time (see Fig. V.20;). (fl)Recovery of solution fractional saturation (see Fig. V.20j). [From Mozzarelli et al. (1987).]

edly increases the dissociation rate but has no effect on the association rate, and therefore a very small effect on the distribution of ligation states and quaternary structures (Brunori et al., 1972).24 A similar series of experiments was carried out on sickle cells (Mozzarelli et al., 1987) (Fig. V.23). Each cell was characterized by its delay time at zero saturation, the saturation at which polymer appeared in desaturation experiments as determined by the disappearance of the delay time after complete photodissociation, and the saturation at which polymer disappeared in resaturation experiments as determined by the reappearance of the delay time. Experiments were carried out at desaturation rates that differed by a factor of about 10. The saturation at which polymer first formed was always lower at the faster rate of desaturation (1 min compared to 10 min), and the saturation at which polymer disappeared was much higher than either of these. T h e results are 24 Because of its large effect on the relative dissociation rates, partial photolysis of the oxygen complex is predicted to decrease dramatically the apparent cooperativity of the oxygen binding curve. A partially saturated solution prepared by photolysis of oxyhemoglobin S at high oxygen pressure is therefore expected to have significantly different polymerization properties from one prepared at the identical fractional saturation by decreasing the oxygen pressure.

242

5 2

u -

WILLIAM A. EATON AND.]AMES HOFRICHTER

3

~

j

7

2

-

4

-

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v)

1

100 200 100 200 TIME (msec) TIME (msec)

f

100 200 100 200 TIME (msec) TIME (msec)

h

I

loo 200 TIME (msec)

i

FIG.V.23. Detection of polymer formation and disappearance in sickle cells containing parlially photolyzed carhonmonoxyhenioglobin S. (a) Kinetirs of gelatiuri after complete photodissociation of fully saturated cell. (h and c) Kinetics of gelation after complete photodissociation of increasingly photolyzed cells. (d and e) Kinetics of gelation after complete photodissociation of cells at increasing saturations. (f-j) Optical micrographs of cells. Disappearance of the delay time in c is accompanied by gross cellular deformation, while reappearance of the delay time in e is accompanied by reformation of the original biconcave disk shape. [From Mozzarelli el al. (1987).]

shown in Fig. V.24a.'= To compare these results with those in Fig. V.8c, we calculate the equivalent oxygen pressure from the measured fractional saturation using the binding curve for normal whole blood. These results are shown in Fig. V.24b. Comparison of the results shows that the oxygen pressures for 50% sickling are similar, considering that the data are for blood from different patients and the most concentrated cells were riot included in the morphological analysis. The hysteresis in the laser photolysis experiments is, however, larger than predicted in Fig. V.8c, suggesting a change in intracellular factors after sickling, such as an increase in the intracellular concentration due to potassium and water loss resulting from membrane damage (Bookchin and Lew, 1983). An important observation in the laser photolysis experiments was that the formation of polymer, as indicated by the loss of the delay time, was 25 These results cannot he directly coinpared with those of Fig. V.8c, because, in the laser photolysis experiments, the measured saturation is the saturation of the solution phase (y,), whereas in the morphological experiments, the saturation is the total cell saturation (y,), which is less because it includes the saturation of the low-affinity polymerized hemoglobin.

SICKLE CELL HEMOGLOBIN POLYMERIZATION

243

1

w

0.5

0

0.5

1

SOLUTION FRACTIONAL SATURATION

U

LL

0.5

0

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20 40 60 80 100 CALCULATED OXYGEN PRESSURE (torr)

FIG. V.24. Sickling (0and 0) and unsickling (0)curves from laser photolysis experiments. (a) Fraction of cells containing polymer versus fractional saturation with carbon monoxide of the solution phase produced by laser photolysis. (0)Data from experiments in which the saturation was decreased from 100% over a period of 1 min, (0)data for a 10-min desaturation, and (0)data from resaturation of cells. (b) Fraction of cells containing polymer versus calculated oxygen pressure. T h e pressures were calculated from the oxygen binding curve for normal whole blood. No account has been taken of the heterogeneity in the 2,3-DPG concentration, which would have the effect of shifting the curves to higher pressures, the effect increasing with decreasing pressure because the more dilute cells contain the higher 2,3-DPG concentrations. [From Mozzarelli et al. (1987).]

always accompanied by gross cellular deformation (Fig. V.24) (Mozzarelli et at., 1987). Similarly, in resaturation experiments, the disappearance of polymer, as indicated by the recovery of the delay time, was simultaneous with the reformation of the original cell shape. This is a very fortunate result, for it suggests that a change in cell shape is indeed a reliable indicator of polymer formation and disappearance. A similar conclusion was reached in rheological studies using micropipette techniques (Nash et al., 1986) (see Section V,D). The most recent application of the laser photolysis technique has been to measure the kinetics of gelation following relatively rapid partial de-

244

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A. EATON A N D JAMES

HOFRICHTER

saturation (San Biagio el al., 1988).In this experiment, the sample is continuously monitored by both absorption and light scattering, so that a profile is obtained for both the fractional saturation and the amount of polymer as a function of time. Figure V.25 shows the kind of results that are being obtained with this method. The desaturation, as monitored by absorption, is approximately exponential with a time constant of 2- 3 sec, which is comparable to desaturation times in viva. This time course is the result of a competition between photodissociation by the laser and rebinding of carbon monoxide, which has a continuously decreasing rate because of diffusion of the free carbon monoxide out o f t h e illuminated volume. There is an increase in optical density on gelation from t w o effects. First, a concentration gradient of free hemoglobin molecules is created as a result of polymerization, causing diffusion of hemoglobin molecules into the illuminated volume. This produces an increase in both the total hemoglobin and deoxyhemoglobin concentrations in the optical path. Second, there may also be a small contribution to the optical

0

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30

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Time (sec) FIG.V.25. Optical density (a) arid light-scattering (b) profiles for continuous laser illumination of the carbon monoxide complex of hemoglobin S. (a) 'Ihe optical density at 433 nm is plotted versus time for a 0.335 g/cm3 solution at 37°C containing a bicarboiiate/COp physiological buffer. The three curves (1-3) in a and b correspond to experiments with iricreasirig power of an argon ion laser at 488 nm. (b) l h e forward scattering of the laser is plotted versus time for the same laser powers as in a. [From San Biagio et al. (1988).]

SICKLE CELL HEMOGLOBIN POLYMERIZATION

245

density increase from the turbidity caused by light scattering from the polymers. At each laser power, a separate experiment can be performed which uses the forward scattering of the laser to give a much more accurate measurement of the delay time (Fig. V.25). It is clear from these initial results that the laser technique is a powerful method to study gelation kinetics at physiological rates and extents of desaturation. By performing measurements over a range of hemoglobin S concentrations and rates and extents of desaturation, it should be possible to construct an empirical theory that will be extremely useful for modeling intracellular polymerization kinetics under in vivo conditions. Measurements on cells should also be possible, but these will be technically quite difficult.

D . Rheology of Gels and Sickle Cells The most important consequence of gelation for the pathophysiology of sickle cell disease is the decrease in red cell deformability. T h e investigation of the rheological properties of sickle cells has therefore been of considerable interest for many years. Studies on hemoglobin S solutions are relatively more recent, and did not begin until after the initial description of the basic kinetic and thermodynamic properties of gels. We have already discussed the influence of shear on the kinetics of gelation in Section IV,D. In this section, we present a brief description of other rheological studies on gels, followed by a summary of the most important rheological findings on sickle cells. Shear is a force which produces deformation o r flow and is a principal determinant of the mechanical properties of the gel. T h e rate of shear determines whether a gel exhibits the rheological properties of an elastic solid, a viscous liquid, or an intermediate behavior known as viscoplasticity (Briehl, 1980, 1981a,b, 1983; Danish et al., 1987). Figures V.26 and V.27 show examples of the type of rheological behavior that is observed. Perhaps the most interesting result is that gels formed in the absence of shear exhibit solidlike behavior (Briehl, 1980, 1981a,b; Gabriel et al., 1981; Danish et al., 1987) (Fig. V.26b). At shear stresses below a critical value, called the yield stress, a gel behaves like an elastic solid, that is, the relative deformation of the gel (the strain) is linearly proportional to the stress (Fig. V.27a), and the deformation is completely reversed when the stress is relieved (Gabriel et al., 1981). T h e elastic modulus (i.e., the slope of the stress-strain plot, Fig. V.27a) depends very sensitively on the concentration of polymerized hemoglobin (Fig. V.27b). T h e gel also supports a stress indefinitely (Briehl, 1981a; Danish et al., 1987). If a sufficiently large shear stress is applied, however, these gels undergo

246

WILLIAM A. EATON A N D JAMES HOFRICHTEH

a

xb* .

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, 20

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15 (OC)

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FIG. V.26. Rheology of hemoglobin S gelation. (a) Schematic of cone-plate viscometer used for measurements. [From Briehl (198 Ih).] The sample is sheared by rotation of the inner cone relative to the plate, which is the bottom of the cup containing the sample. In the steady-state mode, the apparent viscosity is measured from the ratio of the shear stress [a(dyn/cni')I required to maintain a constant angular velocity, to the shear rate, determined from the angular velocity. A principal advantage of this type of visconieter is that the shear rate for liquids, which is the gradient in the velocity of successive layers of the sample, is the same throughout the sample because the increase in linear velocity of the cnnc relative to the plate at increasing radii is matched by the increase in the perpendicular dislarice from the cone t o the plate. Solidlike properties can be determined by stopping the motor that turns the cone, thereby stopping the shear. An elastic solid can support the stress after the niotor is stopped, while in a liquid there is complete relaxation of the stress. Partial relaxation of the stress can also be observed, indicating that the gel has both solid and viscous roniporierits. (b) Application of shearing strcss to an annealed gel at two temperatures [top (17.Y"C) and bottom (15.2OC:)I. [From Brie111 (19Hla).]T h e arrows indicate a change in motor speed [revolutions per minute (rpm)]. Half a rpm corresponds to a shear rate of only 1.9 sec-I. Shearing was stopped (first arrow) at the beginning of gel formation (resulting in complete relaxation of the stress, indicating purely liquid behavior), and the gel was allowed to anneal. After the annealing period, the rate of increase of the stress is L ~ saine K for frozen water, indicating that the gel could only undergo a small deformation. For the 17.9% (top) experiment the motor was stopped after increasing the stress to its maximum value of 245 dyn/cm2. There is no decay of the stress, indicating the solidlike behavior of the gel. At the lower temperature lbottorn (15.2"C)], the gel shows a yield stress, as indicated by the deviation from (--). At the cessation of shear, there is partial relaxation of the applied stress, indicating both viscous arid solidlike components. (c) Yield stress versus yield teinpcrature at two different deoxyhemoglobin S concentra-

247

SICKLE CELL HEMOGLOBIN POLYMERIZATION

a

b

1000

-

7

1

* I

0.02

.

0.179

E

-

C u 2.

0

0.163

0.01

0.1

0.2 SHEAR STRAIN

0.01

0.02

0.1

0.2

0.3

Hb S (g/crnJ)

FIG. V.27. Solidlike behavior of deoxyhemoglobin S gels. (a) Shear stress (u)versus shear strain (dimensionless) for deoxyhemoglobin S gels at various concentrations indicated in the figure. (b) Elastic modulus (G) versus total deoxyhemoglobin S (Hb S) concentration (GO). The elastic modulus is calculated from the slope of the plots in a. [From Gabriel etal. (1981).]

irreversible deformation (Fig. V.26b). This so-called yield stress decreases as the concentration of polymerized hemoglobin in the gel decreases, either by lowering the temperature or by decreasing the total initial hemoglobin S concentration (Fig. V.26~).Once a gel has yielded, the applied stress is no longer maintained when the shearing is stopped, but decays to a lower value (Fig. V.26b). T h e irreversible deformation above the yield stress (characteristic of a plastic) and the partial decay of the stress following cessation of shear (characteristic of a viscous material) is called viscoplastic behavior (Briehl, 1980, 198la,b, 1983). Gels formed in the presence of continuous shear show liquid behavior at the onset tions [0.227 g/cm3(0)and 0.209 g/cm’ (O)]. [From Briehl(1981a).] The yield temperature is determined as the temperature at which relaxation of the applied stress occurs on lowering the temperature of a maximally stressed gel. (d) Solidlike and viscous contributions to apparent viscosity of gel formed under continuous shear. [From Briehl (1980).] If the motor is stopped early in the development of apparent viscosity, there is complete relaxation of the applied stress, indicating liquidlike behavior. If the motor is stopped later in the progress curve, there is only partial relaxation of the stress, indicating contribution of a solidlike component to the apparent viscosity. Bar, 1 min. (e) Conversion of a gel from a solid to a viscous liquid by prolonged shearing. [From Briehl(1981a).] A gel formed in the absence of shear is subjected to prolonged shearing at a shear rate of 38 sec-I. When the motor is stopped, there is rapid and complete relaxation of the stress, indicating liquid behavior.

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WILLJAM A. EATON AND JAMES HOFRICHTER

of the viscosity increase, i.e., the stress relaxes to zero at the cessation of shear, but as more gel forms, viscoplastic behavior is observed, i.e., at the cessation of shear there is only a partial relaxation of stress (Fig. V.26d). Finally, if shearing is carried out for a long period after a viscoplastic gel or an elastic solidlike gel is formed, it can be converted into a viscous liquid (Fig. V.26e). There are as yet no companion observations by either optical or electron microscopy of the properties of the gels prepared in the rheological studies, so it is not possible to give any detailed structural explanation of the findings. The solidlike behavior of gels indicates that there are significant noncovalent interactions between fibers. ‘These interactions play a role analogous to the covalent cross-links of other types of polymer gels, such as polyacrylamide (Flory, 1953). I n the absence of the interfiber interactions, the collection of hemoglobin S fibers would always have the rheological properties of a viscous liquid. It is interesting to speculate on the structural origin of the influence of shear on gels. Shearing might control the rheological properties of a gel by fracturing the gel into a number of polymer domains. Under excessive shear, a gel may be converted into a suspension containing a large number of very small domains or fragments of domains, in which there is very little reannealing, giving it the properties of a liquid. In order to determine the relative importance of interdomain and intradomain interactions in determining the rheological properties of gels, it would be useful to explore the dependence of the yield stress on the rate of gel formation. T h e yield stress may not only depend on the concentration of polymerized hemoglobin, as was found in the experiments so far, but also on the number of polymer domains, which is determined by the rate of homogeneous nucleation. For example, consider the limiting case where a small solution volume is deoxygenated slowly to form a gel containing a single polymer domain, which could occur in a cell where gelation occurred with a relatively long delay time (a few seconds or more) and a single homogeneous nucleation event. At the yield stress both intrafiber and interfiber noncovalent, intermolecular bonds are broken. At the other extreme is a gel, having the same amount of polymerized hemoglobin, that has formed with a very short delay time (milliseconds) because of very rapid deoxygenation. In this case, there have been a large number of homogeneous nucleation events and many polymer domains form. T h e yield stress for such a gel might be much less or its viscous component much larger, because it can deform by the relative motion of domains which might be expected to have weak intermolecular interactions between them. What is the effect of intracellular gelation on the rheological properties of sickled cells? Several different types of measurements, including filterability, viscosity, and morphological studies on red cell suspensions

249

SICKLE CELL HEMOGLOBIN POLYMERIZATION

and micropipette and morphological studies on individual cells, demonstrate that sickled cells have a reduced deformability (see review by Klug et al., 1974; Chien, 1977a; Mohandas et al., 1979). Figures V.28 and V.29 summarize the key experimental findings using the various techniques. In filterability studies changes in resistance or flow through small pores

-

102

a

HbSS RBC in Ringer 145% RBCI

5

-

8v,

>

102-

W

a

h

2

10-

Hb SS RBC in Ringer 146% RBCl Hb AA RBC in Ringer 145% RBCJ

.-.-.0-0-

o-o--D!p

-0

1

Hsdened

*-b-.-•

1

I

I

30 Polycarbonate Sieve (5-pm pores)

PO, (mm Hg)

FIG. V.28. Kheology of sickle red cells. (a) Viscosity of sickle cells in Ringer's solution as a function of oxygen pressure. [From Usami et al. (1975b).] T h e cells are suspended in Ringer's solution to avoid aggregation of red cells. Measurements in plasma are complicated by a shear-dependent increase in viscosity caused by aggregation, e.g., rouleaux formation. (b) Viscosity of sickle cells in Ringer's solution as a function of shear rate at two oxygen saturations. [From Chien et al. (1976).] (c) Resistance to constant flow of normal and sickle cells as a function of oxygen pressure. [From Usami et at. (1975a).] (d) Diagram of normal (upper row) and oxygenated sickle cells (lower row) in the presence and absence of fluid stress of 125 dyn/cm2 from observations in the rheoscope, and the corresponding laser diffraction pattern observed with the ektacytometer. [From Bessis and Mohandas (1977).]

250

a

WILLIAM A . EATON A N D JAMES HOFRICNTEK

b-:

P=O

ij

L TIME (1. sec)

d

Po, (torr)

.

5~

10

20

30

40

50

Po, (torr)

FIG. V.29. Micropipette measurements on individual cells. (a) Diagram of micropipette measurements. (h) Aspirated rnernbrane tongue length ( L ) divided by pipette radius (Kp) as a function of time ( 1 ) following changes in aspiration pressure. [From Nash et al. (1986).] T h e aspiration pressures in mm H,O are given in parentheses. (c) Effective membrane rigidity (EMR) versus oxygen pressure (Pop)for unfractionated sickle cells at 23°C. [From Nash et al. (IUSS).] (d) Half time for tongue growth as a function of oxygen pressure fbr urifractionated sickle cells at 23°C. [From Nash el (11. (1986).]

are measured. Only qualitative results can be obtained with this technique because of technical problems such as blockage of the pores by a small subpopulation of cells, and the theoretical problem of interpreting the pressure-flow relations in terms of cell deformability. Filtration measurements have been useful, however, for showing that intracellular

SICKLE CELL HEMOGLOBIN POLYMERIZATION

25 1

gelation may occur rapidly (Messer and Harris, 1970). Also, in slow deoxygenation experiments there is a sharp increase in the resistance to flow at an oxygen pressure of about 80-90 torr, indicating the onset of intracellular gelation in the most concentrated cells (Klug et al., 1974; Usami et al., 1975a; Lessin et al., 1977) (Fig. V.28~). The viscosity of oxygenated sickle cell suspensions is slightly elevated compared to normal cells. The increase has been attributed to a less deformable membrane and a higher internal viscosity resulting from the increased intracellular hemoglobin concentration (Chien et al., 1970). As with normal cells, the viscosity of oxygenated SS cells decreases with increasing shear rate due to the deformation of the cells. For cells deoxygenated to an average saturation of 15%, however, there is only a barely detectable decrease in viscosity with increasing shear (Fig. V.28b) (Chien et al., 1976; Chien, 1977b). The slight decrease could arise from a small fraction of cells that contain no polymer, e.g., F cells with a low total intracellular hemoglobin concentration. The viscosity of the deoxygenated cell suspensions is higher than that of oxygenated cell suspensions, with the increase beginning at about 80 torr (Fig. V.28a) (Usami et al., 1975b), in agreement with the filterability studies. According to the viscosity measurements so far, then, sickled cells are “rigid” (Chien et al., 1976; Chien, 1977b); in the absence of a clear shear dependence there is no quantitative information on red cell deformability. Morphological studies have been performed to examine the response of sickle cells to fluid shear stress. The average change in shape of cells in a suspension can be monitored from their laser diffraction pattern on shearing in a coaxial cylinder viscometer (an ektacytometer) (Bessis and Mohandas, 1975; Groner et al., 1980), or the changes in individual cells can be observed with a microscope in the stationary layer at the center of a viscometer with counterrotating cone and plate (a rheoscope) (Schmid-Schoenbein and Wells, 1969). Normal cells, and oxygenated sickle cells having a biconcave disk shape called diskocytes, deform into ellipsoids with their long axes parallel to the flow direction (Fig. V.28d) (Bessis and Mohandas, 1977). Irreversibly sickled cells apparently exhibit no deformation, and orient with their long axes perpendicular to the flow direction, or rotate under flow (Bessis and Mohandas, 1977). Normal deformation of oxygenated irreversibly sickled cells can be brought about by reducing the concentration of the most dense cell fraction from about 0.45 to about 0.33 g/cm3 with a hypotonic medium, indicating that the decreased deformability of these cells results primarily from the high intracellular hemoglobin concentration and not the more rigid membrane (Clark et al., 1980). On deoxygenation of a suspension of density-fractionated cells in the ektacytometer, there is a

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WILLIAM A. EATON A N D JAMES HOFRICHTER

change in the diffraction pattern, which has not yet been interpreted (Bessis et d.,1982). It could result from a superposition of rotating, rigid sickled cells and deformable unsickled cells. N o measurements on the deformability of individual sickled cells under fluid shear stress have been reported (Schmid-Schoenbein, 1982). Another interesting result from ektacytometry measurements is the deformability of oxygenated normal and sickle diskocytes as a function of cell water (Gulley et al., 1982). Normal cells have an optimal intracelM a r hemoglobin concentration. If the cell swells from water gain, the increased surface-to-volume ratio decreases the deformability of the red cell because of the large resistance of the membrane to an increase in area (Evans and Lacelle, 1975). If the cell shrinks from water loss, the markedly increased viscosity of the intracellular hemoglobin (Chien et al., 1970) reduces deformability (Gulley et J.,1982). Swelling of oxygenated sickle diskocytes, on the other hand, leads to an increase in deformability, showing that they, as well as irreversibly sickled cells, are suboptimally hydrated (Gulley et al., 1982). The most informative results on the deformability of sickle cells have come from measurements on the aspiration of cells into micropipettes (Fig. V.29). The initial studies showed that much larger negative pressures are required to partially or completely aspirate oxygenated irreversibly sickled cells or oxygenated sickle diskocytes (Have11 et nl., 1978; Lacelle, 1975, 1980). On partial deoxygenation, there are increases in the negative pressure (Lacelle, 1975, 1980). Recently, more detailed and quantitative information has been obtained with the micropipette technique (Evans et ul., 1984; Nash et al., 1984, 1986). Both static and dynamic rigidities have been measured. T h e static rigidity is characterized by the change in the length of the “tongue” aspirated into the pipette with a change in negative pressure, while the dynamic rigidity is characterized by half the time required to achieve the final tongue length after initiating the pressure change (Fig. V.29a). For oxygenated cells there are only small increases in these quantities, with the largest increases for the irreversibly sickled cells and the densest cells (Evans et nl., 1984; Nash P t al., 1984). These increases are partially or completely removed by swelling the cells in hypoosmolar media, suggesting that they result from the presence of polymerized hemoglobin S. A series of important results has come from micropipette measurements as a function of oxygen pressure (Nash et d.,1986). First, cells that are “spiculated” or have a “granular” surface, suggesting the presence of a gel with multiple polymer domains, show markedly altered rheology, with monotonically increasing static and dynamic rigidities as the oxygen pressure is decreased (Fig. V.29). In contrast, diskocytes that maintain a “smooth surface” show the same static and dynamic rigidities

SICKLE CELL HEMOGLOBIN POLYMERIZATION

253

at all oxygen pressures as normal cells, in agreement with the conclusion from the kinetic studies (Mozzarelli et al., 1987). In cells showing morphological evidence of gelation, both the static rigidity and the half-time for tongue growth increase approximately exponentially with decreasing oxygen pressure (Fig. V.29). At the lowest oxygen pressures the static rigidity increases by up to a factor of 100, and the half-time for tongue growth becomes as high as 30 sec, compared to 30-200 msec for normal cells and oxygenated sickle diskocytes (Chien et al., 1978; Nash et al., 1984). Although the tongue completely retracts in most cells, indicating “essentially elastic” behavior, different tongue lengths are observed at the same pressure for decreasing and increasing pressures. Plastic-type behavior is also observed, in which, at the same threshold negative pressure, there is much more rapid tongue growth followed by formation of a bud which breaks away from the cell. Another interesting observation is that, for small initial pressure decreases, there is no tongue growth, which might correspond to the presence of a yield stress that is observed in solidlike gels (Briehl, 1981a; Nash et al., 1986) (Fig. V.26b). The micropipette technique appears to be the most promising of all of the rheological methods for studying the deformability of sickled red cells, and there are a number of obvious important experiments. First, it would be important to repeat the micropipette measurements (Nash et al., 1986) with much more precise control and measurements of the oxygen pressure. Second, it would be useful to carry out the same measurements for polymerized cells as a function of increasing oxygen pressure. In resaturation experiments, the fraction polymerized is expected to decrease monotonically until the solubility is greater than the total intracellular hemoglobin concentration. It would, therefore, be interesting to determine whether there is a monotonic decrease in both the static and dynamic rigidities until values for normal cells are reached. In desaturation experiments, small changes from the rheological properties for normal cells are not observed, because the solution must be significantly supersaturated before polymerization can occur, resulting in the formation of a considerable amount of polymer at the end of the delay period (Fig. 111.16). Finally, it would be important to carry out experiments on fractionated cells and on cells in which the same amount of polymer has formed, but at different rates. ON PATHOPHYSIOLOGY AND STRATEGIES FOR THERAPY VI. COMMENTS

As was pointed out in Section I, the principal motivation for much of the research described in this article has been to understand the pathophysiology of sickle cell disease in detail and to help in the quest for a

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WILLIAM A. EATON AND JAMES HOFRICHTER

specific treatment of patients with this disease. In this section, we present a brief summary of the major ideas that have derived from studies on the gelation of hemoglobin S. We shall see that the kinetics of gelation play a dominant role, and have suggested a novel approach to the therapy problem. A much more extensive discussion of this subject can be found elsewhere (Eaton and Hofrichter, 1987). In order to gain some perspective on the problem of pathophysiology, it is useful to review briefly the most important advances up to about 15 years ago (Castle, 1976; Conley, 1980). The initial critical observations were that in vitro deoxygenation induced sickling and that unsickling occurred on reoxygenation (Hahn and Gillespie, 1927); also, much more extensive sickling was found in venous blood compared to arterial blood, suggesting that sickling in the tissues and unsickling in the lungs were taking place in trim (Sherman, 1940). Postmortem examinations, showing apparent occlusion of vessels with sickled cells, although obviously subject to ambiguities, pointed to obstruction of the microcirculation as the cause of organ damage (Diggs, 1965, and references therein). T h e first insight into the mechanism of blockage resulted from the observation of an increased viscosity of sickle blood on deoxygenation (Ham and Castle, 1940). This was later explained by the finding that deoxygenation of concentrated hemoglobin S solutions produced a “semi-solid gel-like” state (Harris, 1950). At this point it became clear that gelation was the abnormal property of the hemoglobin S molecule (Pauling et al., 1949; Harris, 1959) that caused sickle cell disease. Moreover, a strong correlation of’disease severity with measurements of gelation, sickling, and blood viscosity for sickle trait and a number of double heterozygous conditions suggested that the differences in clinical manifestation could be explained in terms of differences in intracellular gelation (Singer and Singer, 1953; Allison, 1956; Griggs and Harris, 1956; Charache and Conley, 1964). The finding of an increased blood viscosity, together with the observation of a sharp increase in the number of sickled cells at venous oxygen tensions, led to the concept of the “vicious cycle” (Ham and Castle, 1940; Harris et al., 1956). According to this idea, an increase in blood viscosity slows blood flow in the microcirculation, initiating a cycle of increased oxygen extraction, additional sickling, and further viscosity increase, the final result being “masses of sickled erythrocytes. . . solid enough to occlude vessels and result in the ‘thrombotic’ episodes characteristic of the disease” (Harris P t d.,1956). Later measurements brought the role of bulk viscosity into question. Because the hematocrit of SS blood is low, its viscosity is very similar or even slightly lower than that of normal blood at physiological oxygen pressures (Charache and

SICKLE CELL HEMOGLOBIN POLYMERIZATION

255

Conley, 1964; Chien et al., 1976). Consequently, changes in bulk viscosity have been assumed to play only a minor part in causing vascular obstruction (Charache et al., 1982). Nevertheless, the fundamental notion of the vicious cycle, that the desaturation of sickle cells is coupled to a decreased blood flow which produces further desaturation, has remained an essential element of the pathophysiology. Emphasis on the role of individual cells in blocking the microcirculation was first suggested by experiments showing that sickled red cells could not pass through 5-pm pore filters (Jandl et al., 1961). An important advance in describing the pathogenic mechanism in terms of the behavior of individual cells came with the discovery of the highly unusual kinetics of gelation (Hofrichter et d., 1974a,b). T h e finding of a delay period prior to gelation, which could span both the capillary and venous transit times, led to the formulation of the “kinetic hypothesis.” According to this hypothesis, a comparison of the times required for gelation with the transit times in the microcirculation for individual cells is the critical consideration in determining disease severity (Hofrichter et al., 1974b; Eaton et al., 1976a). In this description, changes in either the transit time or the delay time could alter the probability of intracellular gelation within the microcirculation. Because of the enormous sensitivity of the delay time, this concept could not only provide a possible rationalization for the episodic nature of the disease, but could also explain the influence of relatively small changes in physiological variables on its clinical course (Eaton et al., 1976a). The ideas introduced in the description of the vicious cycle and the kinetic hypothesis have provided the framework for current thinking about the disease. In order to make this picture more quantitative, it is useful to consider the possible events that are thought to occur as a red cell passes through the microcirculation of a patient with sickle cell disease (Eaton and Hofrichter, 1987). These are shown schematically in Fig. VI.1. There are several possibilities for polymer-free cells emerging from the lungs. If deoxygenation proceeds to an extent that the total hemoglobin concentration is greater than the solubility, polymerization will eventually occur. The important question then becomes: what is the delay time relative to the relevant transit times? A red cell spends 1-2 sec in the arterial circulation, about 1 sec in the microcirculation, and requires about 15 sec to return to the lungs. Thus, if the delay time is longer than 15 sec, the cell can return to the lungs and be reoxygenated before any significant polymerization has begun. If it is between 1 and 15 sec, gelation will occur while the cell is in the venous circulation. Sickling in the large veins does not produce vasoocclusion, but the cell membrane may be damaged, resulting in a loss of water and a shorter delay time in subsequent

256

WILLIAM A. EATON AND .JAMES HOFRICHTEK

Arterial a+.

f

2 7 /Venous

FIG.VI.1. Possible events in the microcirculation of a patient with homozygous SS disease. A diagram of an arteriole, rapillary, and venule is shown. In a, a cell containing no polymer enters the capillary, deforms to squeeze through, and reaches the venule without polymerization occurring. In b, the delay time is longer than the capillary transit time, but the cell sickles in the venule. In c, the delay time is shorter than the capillary transit time and the cell sickles within the capillary, but escapes to the venule, while in d intracapillary sickling results in trlinsient or permanent blockage. In e and f, the cell, depicted as an irreversibly sickled cell, already contains polymerized hemoglobin in the arteriole, and may pass through the capillary (e) or produce a transicnt or permanent occlusion (f). [From Mozzarelli P! al. (1987).]

trips through the circulation. If the delay time is less than about 1 sec, gelation can occur while the cell is in one of the narrow vessels of the microcirculation. Because the cell is much less deformable, it may not be able to “squeeze” through, and may become transiently o r permanently stuck. This vasoocclusion can block the further passage of cells, leading to the “log-jam” effect that causes decreased oxygen delivery to the tissues. Cells in which polymerization has occurred in the microcirculation or the venous return may not be completely depolymerized by reoxygenation in the lungs (Eaton et d.,1976a; Winslow, 1978; Noguchi et al., 1980).Incomplete depolymerization can occur before the cell enters the microcirculation in its next trip, either because the total hemoglobin concentration is greater than the solubility at arterial pressures, or because the total hemoglobin concentration is so close to the solubility that de-

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257

polymerization is slower than the 1- to 2-sec residence time in the arterial circulation. Cells entering the microcirculation with polymerized hemoglobin will undergo very rapid polymerization because the rate of heterogeneous nucleation is enormously increased or nucleation is complete (Mozzarelli et al., 1987). In these cells there may be sufficient polymerized hemoglobin that they become stuck in the arterioles (Noguchi and Schechter, 1981). The schematic picture in Fig. VI. 1 immediately raises the question of the relative frequency of the different events. This is clearly an extremely complex problem, but some insight can be gleaned from the measurements on the fraction of cells containing polymer as a function of the saturation in the solution phase in the laser photolysis experiments discussed in Section V,C. In order to relate the data from these experiments to the physiological situation, the oxygen pressure corresponding to each saturation can be calculated from the oxygen binding curve for normal blood. Figure VI.2 shows a plot of the fraction of cells containing polymer on deoxygenation of cells that initially contained no

0

20

40

60

80

Oxygen Pressure (torr)

100

FIG. VI.2. Fraction of cells containing polymer as a function of calculated oxygen pressure determined by a double laser beam photolysis technique. T h e oxygen pressures were calculated from the measured saturations of Fig. V.24 (Mozzarelli et al., 1987) with carbon monoxide using the least-squares fit of the two-state allosteric saturation function to the binding curve of normal blood. [Unsickling (O)]T h e equilibrium data obtained in reoxygenation (ie., resaturation with carbon monoxide) experiments, [sickling (O)] data obtained from experiments where deoxygenation (desaturation) is carried out over a period of 1 min, and (---) a theoretical estimate using the double-nucleation model for deoxygenation carried out in I sec (Ferrone el al., 1986). (1) T h e average oxygen pressure found in the arteries and veins of patients with homozygous SS disease (Lonsdorfer et al., 1983). [From Eaton and Hofrichter (1987).]

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WIILIAM A. EATON A N D J A M E S HOFKICHTER

polynier (the sickling curve), and reoxygenation of cells that have polymerized (the unsickling curve). Both deoxygenation and reoxygenation were carried out in about 1 min, which is considerably longer than the in vivo times. Since depolymerization is fast, the reoxygenation curve should be relevant for in vivo considerations. The in vivo deoxygenation curve, on the other hand, is expected to be considerably shifted to lower oxygen pressures [the dotted theoretical curve in Fig. VI.2 is a rough theoretical estimate (Ferrone et al., 1986)l. At the typical arterial oxygen pressure of 85 torr, the unsickling curve in Fig. VI.2 suggests that only a few percentage of cells in this particular patient would not be depolymerized after reoxygenation in the lungs. If equilibrium were achieved, about 90% of cells would be polymerized at the oxygen pressure of 45 torr of the mixed venous blood. However, the sickling curve suggests that less than 10% of cells are sickled at venous oxygen pressure. These results point to the enornious significance of the delay period. They suggest that over 80% of cells are returning to the lungs without any significant amount of polymerization occurring because the delay time at venous oxygen pressures is longer than the venous return time. This conclusion is supported by much more extensive data in which cells sampled from the arteries and veins of sickle cell patients were fixed with glutaraldehyde and examined by optical microscopy. These studies, which included observations on about 30,000 cells from 60 different patients, showed that an average of about 10% of cells are sickled in the arterial circulation and about 20% are sickled in the veins (Jensen et al., 1960; Serjeant et al., 1973; Lonsdorfer et al., 1983). The large number of circulating sickled cells (- 10l2) indicates that blockage of niicrovessels is a rare event. T h e probability of forming a blockade which ultimately results in the destruction of the cell can be estimated from measurements of mean red cell lifetimes to be about 1 in lo4 trips of a sickled cell through the microcirculation (Eaton and Hofrichter, 1987). It is much more difficult to estimate the probability of a transient blockade, since there are no data on the duration of such events.y6Assuming an average duration of 10 sec and using measurements of peripheral resistance on an isolated rat mesentery preparation (Kaul et al., 1983a,b, 1986), the probability of a transient blockage can z6 It is pnssihle that the average duration is controlled by the oscillatory vasodilations observed using laser doppler velocinietry rneasurements on the skin of sickle cell patients (Kodgers el al., 1984). The period of these oscillations is about 8 sec. It is interesting to nnte that in the rat mesentery preparation the peripheral resistance could be restored to normal values by denervation to produce vasodilation (Kaul el al., 1986).

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259

be crudely estimated to be about 1 in lo2 trips of a sickled cell through the microcirculation. One explanation for these low probabilities could be that blockage is a cooperative event requiring participation of multiple sickled cells. The only data that bear on this point, however, show no evidence for cooperativity, since the increase in peripheral resistance is roughly proportional to (our estimate of) the fraction of sickled cells (Kaul et al., 1986). From the above discussion, a dynamic picture emerges in which the rate of blockage of the microcirculation is balanced by the rate of reopening of occluded vessels. The result is a steady state in which a certain fraction of capillaries are blocked in each tissue. The balance could be quite delicate, and small changes in either the rate of blockage or the rate of reopening could produce physiologically significant changes in the fraction of occluded capillaries. We might expect that, once this fraction exceeds a certain level, oxygen extraction from cells traversing neighboring microvessels would increase, thereby increasing the probabilities for sickling and vasoocclusion. This is a somewhat refined version of the vicious cycle in which widespread vasoocclusion results from an autocatalytic increase in the fraction of blocked capillaries caused by an increase in the probability of sickling, rather than by a slowing of blood flow due to a bulk viscosity increase. It would appear, then, that the distribution of delay times is a critical variable in determining the fraction of occluded capillaries by affecting the rate of blockage. Increasing the delay times, thereby allowing more cells to escape the microcirculation or return to the lungs before polymerization has begun, should, therefore, result in amelioration of the disease. Prior to the kinetic studies three principal methods had been suggested to inhibit gelation in vivo (Eaton and Hofrichter, 1987). One is to promote fetal hemoglobin synthesis, taking advantage of its large inhibitory effect on gelation. The second is to develop a drug that binds to the hemoglobin molecules and interferes with formation of the polymer. The third is to increase the oxygen affinity to decrease the concentration of molecules capable of polymerizing. The new approach suggested by the kinetic studies is to decrease the total intracellular hemoglobin concentration (Hofrichter et al., 1974b; Eaton et al., 1976a; Sunshine et al., 1978; Eaton and Hofrichter, 1987). How much of a decrease is necessary? An approximate answer to this question has come from studies on mixtures of hemoglobin S and other hemoglobins that are found in double heterozygous conditions that are associated with a milder clinical course. Figure VI.3 shows the effect of non-S hemoglobins on the delay time

260

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WILLIAM A. EATON A N D JAMES HOFRICHTER

c 0

20

SIC Disease AIS Trait

"

0

0.2

0.4

0.6

Fraction Hb F or Hb A

20 0

- 3 -2 - 1

0

1

2

3

Log Delay Time (sec)

FIG.VI.3. Effect of non-S hemoglobins on gelation delay times in solutions and cells. (a) Logarithm of the ratio of the delay time of the mixture to the delay time of pure

deoxyhemoglobin S at the same total hemoglobin concentration. The effect of hemoglobin C on the delay time is identical to that of hemoglobin A (Bunn et al., 1982). (0)Hb S + Hh F and (0)Hb S + Hb A. (b-d) Distribution of delay times at zero saturation for cells from a patient with (b) homozygous SS disease, (c) hemoglobin SC disease, and (d) sickle trait at 37°C. The data in h and c are taken from Coletta et al. (1982), while the data in d are from Zarkowsky and Hochmuth (1975) after using the temperature dependence of the median delay time to correct the data to 37°C. [From Sunshine et al. (1978) and Eaton and Hofrichter (1987).]

for solutions at total constant deoxyhemoglobin concentration and on the distribution of delay times for cells. The solution data can be used to estimate the necessary decrease in intracellular hemoglobin concentration (Sunshine et a/., 1978; Eaton and Hofrichter, 198'7). In the double heterozygous condition of sickle p+-thalassemia, cells contain 20-30% hemoglobin A and 80-70% hemoglobin S. In temperature-jump experiments the delay time in this mixture is increased by a factor of 30-300, compared to deoxyhemoglobin S alone. The same increase in the delay time could be achieved by a decrease in the total hemoglobin concentration of 10- 15%.This is a less severe form of sickle cell disease, so that a decrease in the intracellular hemoglobin concentration of only 10- 15% is predicted to have some therapeutic effect. In the double heterozygous condition of sickle cell with pancellular hereditary persistence of fetal hemoglobin, which is a very mild form of sickle cell disease, the hemoglobin composition is 20-35% hemoglobin F and 8 0 4 5 % hemoglobin S. The delay time for this mixture is increased by a factor of 900- 100,000, which could be achieved by a decrease in the total hemo-

SICKLE CELL HEMOGLOBIN POLYMERIZATION

26 1

globin concentration of 15-2576, suggesting that such a decrease would lead to a major therapeutic effect. Finally, in sickle trait, which is a totally benign disorder, the same analysis indicates that a decrease in the total intracellular hemoglobin concentration of about 30% would result in a ‘‘cure.’’27 These numbers suggest that decreasing the intracellular hemoglobin concentration, taking advantage of the extraordinary sensitivity of the delay time, is a viable approach to therapy. This concept has stimulated a number of investigations to find ways to decrease the intracellular concentration by swelling red cells, and to prevent cellular dehydration (Eaton and Hofrichter, 1987). One approach that has not yet been explored is to reduce the intracellular hemoglobin concentration by reducing biosynthesis (Eaton et al., 1976a), which occurs in iron-deficiency anemia. The same kind of analysis can be applied to the question of agents which bind to hemoglobin and interfere with polymer formation and/or increase oxygen affinity (Sunshine et al., 1978; Eaton and Hofrichter, 1987). It should be clear from the preceding discussion that studies on the gelation of hemoglobin S have played an important role in thinking about the problem of a specific therapy. Because there are several independent ways of inhibiting gelation there is cause for optimism, since several noncompetitive drugs could be used simultaneously. At this point it appears that there are two distinct paths to the development of drugs for sickle cell disease. One is the “rational” approach, which uses the basic information about the polymer structure, and the control of fetal globin synthesis, cell volume, 2,3-DPG concentration, and other factors which influence the gelation process, to design a drug from first principle. A second approach recognizes that very frequently the ratelimiting and most expensive step in drug development is the acquisition of toxicological information. It would therefore seem wise to also employ a semiempirical approach which takes advantage of the knowledge about the gelation process described in this article to develop rapid and sensitive assays that could be used in screening a large number of materials of known toxicity. If both approaches are pursued vigorously, it is very likely that we will witness the development of a specific treatment for sickle cell disease in the near future. 2’ it is interesting to point out that in sickle trait the delay times at zero oxygen pressure are longer than 1 sec (Fig. VI.Sb), suggesting that, with the possible exception of’ the hypertonic renal medulla where cells can osmotically shrink, sickling does not occur at all in uiuo.

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WII.I.IAM A. EA’I‘ON AND JAMES HOFRICHTER

ACKNOWLEDGMENT

We thank Allen P. Minton for helpful discussions on the thermodynamics of‘ gelation, and Eduardo A. Padlan for pictures of the molecular and crystal structure of deoxyhemoglobin S.

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