A mathematical model for structure-function relations in hemoglobin

J. Mol. Biol. (1972) 72, 163-197

A Mathematical Model for Structure-Function Relations in Hemoglobin Department of Chemistry, Harvard University Cambridge, iWas. 02138, U.S.A. (Received 3 January 1972, return&

by editor for revision and received in revised form 14 July 1972)

Xay

15,

A mathematical model is presented for utilizing the structural features of hemoglobin to determine its functional properties. A formulation containing the essential features of the Perutz mechanism is outlined; i.e. there are two quaternary conformations of the tetramer, two tertiary structures for each chain, and coupling terms arising from the interchain salt bridges. A diagrammatic representation is used to determine the contributions of individual structures to a generating function from which the equilibrium properties are obtained. It is shown that the model parameters have well-defined physical significance which determines their range of possible values. Consideration is given to the effect on the oxygenation curves of pH, 2,3-diphosphoglycerate, ionic strength and dissociation. Various modified, mutant and mixed-state hemoglobins are discussed. Limitations of the model found by comparison with the available data are described and the need for additional experiments is pointed out. Possible modifications and extensions of the model are suggested and it is indicated how to alter the specific assumptions within the general thermodynamic scheme to conform t,o new structural or functional information.

1. Introduction The mechanism of the co-operative binding of oxygen by hemoglobin has been of interest for more than sixty years (Bohr, 1904; Hill, 1910). An almost overwhelming body of experimental data, most of which are summarized in comprehensive reviews (Wyman, 1948; Gibson, 1959; Rossi Fanelli, Antonini & Caputo, 1964; Antonini, 1965; Riggs, 1965; Antonini & Brunori, 1970), has been obtained concerning both equilibrium and kinetic effects. Interpretations of these data have in recent years focused primarily on two phenomenological models. The first, formulated by Koshlancl, Nemethy & Filmer (1966), who elaborated a proposal by Pauling (1935), is a sequential model defined in terms of the conformations and interactions of the individual chains of the hemoglobin tetramer; the second, introduced by Monod, Wyman & Changeux (1965), is defined in terms of two conformations of the entire molecule. Attempts to distinguish between these two models on the basis oft he available information have not been successful, although a number of illuminating papers relating certain experimental

results

to one model

or the other

have

been

published

(Ogawa

& McConnell,

1967; Ogawa, McConnell & Horwitz, 1968; Ogawa & Shulman, 1971; Edelstein, 1971; Minton, 1971). However, even if it had been possible to show that one of these models is approximately correct, the molecular nature of the co-operative mechanism would 163

164

A. SZABO

AND

M. KRRPLUS

remain undetermined. For that, there is required a detailed description of the essential alterations in electron structure induced by oxygen binding and of the resulting atomic displacements with their energy or free-energy correlat’es. An important step in developing such a stereochemical mechanism has been taken in a pair of recent papers by Perutz (1970aJ). The formulation presented there is based on the classic high-resolution X-ray studies of various vertebrate hemoglobins (Perutz, Muirhead, Cox & Goaman, 1968; Muirhead & Greer, 1970; Bolton $ Perutz, 1971; Perutz et al., 1969; Moffat, 1971a) and of invertebrate hemoglobins (Hendrickson & Love, 1971; Huber, Epp & Formanek, 1969,1970), as well as on information and proposals from X-ray studies of hemes and related compounds (Hoard, 1968), and from electron spin resonance spin label (McConnell, 1971) and nuclear magnetic resonance chemical shift (Ogawa & Shulman, 1971) measurements on hemes, myoglobin and hemoglobin. The essential elements in the Perutz scheme are (a) oxy and deoxy quaternary conformations of different stability for the hemoglobin tetramer; (b) two tertiary structures for each individual chain, one corresponding to the unliganded and the other to the liganded species, and (c) the presence of interchain salt bridges (some with ionizable protons) which provide a mechanism for coupling the alterations in tertiary structure due to oxygen binding with a change in the relative stability of the two quaternary structures. Perutz suggested that his scheme is able to account qualitatively for the sigmoidal oxygenation curve and for the effects on it of changes in pH and the concentration of 2,3diphosphoglycerate. However, a quantitative interpretation of the available equilibrium and kinetic data was not attempted. It is the primary purpose of this paper to demonstrate how specific structural assumptions can be used to formulate a thermodynamic model for the equilibrium properties of hemoglobin. To provide a realistic illustration of the approach, we utilize the elements of the Perutz scheme listed above and test the results by comparison with equilibrium data for the oxygen binding of native hemoglobin and of various mutant or modified forms. Consideration is given to the effect of pH, salt concentration, 2,3-diphosphoglycerate and dissociation. Although many important details concerning the electronic and geometric changes occurring on oxygen binding do not appear in the present form of model, the structural features that are included make it possible to identify certain of the model parameters with the free-energy changes of specific processes (e.g. ionization of Bohr protons, breaking of salt bridges). The results of the formulation provide a framework for bridging the gap between the structural and the phenomenological descriptions of hemoglobin. They demonstrate, also, that there are certain limitations in the specific form of the model, as well as in the available experimental data, which can serve as a focus for future investigations. The section Formulation, which follows, presents the assumptions of the model and shows how to use a generating function, which is a macroscopic analog of the grand canonical partition function, for expressing the stereochemical and thermodynamic features in quantitative terms. A diagrammatic representation is found to be useful for classifying the contributing structures and for evaluating the corresponding terms in the generating function. The method outlined is general and can be used to modify the model as more information becomes available, or to construct a different model if new structural or functional data so dictate. The section Application of the Model describes how the parameters of the model were determined and presents the results obtained. The limitations and implications of the model are discussed in the section Concluding Discussion.

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

165

2. Formulation In this section we demonstrate how to define a specific model by introducing assumptions and thermodynamic parameters involved and then formulating generating function that corresponds to it.

the the

(a) Assumptions The essential elements of the model can be specified most simply in terms of a set of assumptions. (1) The hemoglobin tetramer has two quaternary structures, the oxy form (0) and the deoxy form (D) with an intrinsic free-energy difference -RT In Q, defined such that for Q < 1, the oxy form is more stable than the deoxy form; i.e. B’,,--F.

= - RT In Q.

(2) Each a and /3 chain has two tertiary structures, the liganded form (L) and the unliganded form (U), with the respective individual free-energy differences -RT In Ka and -RT In KB in the tetramer defined to include that of 0, binding; i.e. FLa-FUn

= -RT

In Kc,

FLB--FUB = -RT

In KB.

The tertiary structure of the chains is assumed not to affect the free-energy difference RT In Q of the quaternary structures. (3) In the deoxy quaternary structure, a number of interchain and intrachain salt bridges between protonated nitrogens and ionized carboxyl groups can exist; in the oxy quaternary structure, there are no interchain salt bridges but the intrachain salt bridges can exist. (i) Each a chain in its unliganded form can have two salt bridges with free energy of binding -RT In S,,. and -RT In SE, to the other a chain. Each /5?chain in its unliganded form can have a salt bridge with free energy of binding -RT In S,, to one of the a chains (i.e. /I1 to as and BZ to al) and one internal salt bridge with free energy of binding -RT InSi. In the liganded form of the a and fi chains, the salt bridges defined as originating in the liganded chain are broken; e.g. if the a1 chain is in the liganded form but as and pa are in the unliganded form, the salt bridges from a1 to aa are broken, but not the salt bridges from a2 to al and from /& to ai. To simplify applications of the model in accord with the limitations on the available data, we assume for most of this paper that S,,. = S,“,. = S,, = St = 1.9,since large free-energy differences among the various salt bridges are not expected. For the interpretation of certain of the experimental data (seeApplication of the Model, section (f)), this restriction is lifted and the effect of inequivalent salt bridges is considered. (ii) Two types of salt bridges have NH; groups with ionizable protons; one of these is an a-a’ interchain salt bridge (Sz,) and the other is the j3 intrachain salt bridge, 6’7. The intrinsic free-energy change of neutralization (i.e. for absorbing an OH- by the NH: group when not involved in a salt bridge) is given by -RT In Ha and -RT In HB, respectively. In the absence of the ionizable proton, the corresponding salt bridge does not form. (4) In both the oxy and deoxy tetramer certain very low-probability structures are neglected; these include the liganded tertiary structure for the individual chains in the

166

A. SZABO

AND

RI. KARPLUS

absence of bound ligand and structures with broken salt bridges in cases where the tertiary and quaternary conformations permit their existence. (b) Relation to Perutz scheme Each of the above assumptions is consistent with the conclusions drawn by Perutz (1970a,b) from stereochemical and related information, The existence of at least two quaternary forms is clear from X-ray data, as are changes in tertiary structure on oxygen binding. Since the details of the mechanism of tertiary structural change are not considered in the present form of the model, only the postulated effects on the listed salt bridges are included. The salt bridges considered are a slightly simplified representation of those suggested by Perutz as having an essential role in the conformational changes. Thus, although the model incorporates the physical ideas proposed by Peru& it is not necessarily equivalent to his stereochemical mechanism; that is, the model contains some assumptions which are an oversimplification (e.g. salt bridges of equal strengths) and some for which the structural evidence is not unequivocal. In the unliganded tertiary structure with the hemoglobin molecule in the deoxy conformation, the imidazole group of the C-terminal histidine of the /I2 chain (146fl,), which binds a proton that is expected to be ionizable in the usual alkaline pH range (free pK, r 6.0) forms a salt bridge to the carboxyl group of aspartate (948,) in the same chain. In addition, the carboxyl group of the C-terminal histidine (14615,)forms a salt bridge with the al-chain lysine (40~~) NH: group. Because of the shift in the position of residues occurring when the penultimate tyrosine (1458,) is expelled from its “pocket” in the transition from the unliganded to the liganded tertiary structure, both salt bridges are broken. Further, in the unliganded & chain of a hemoglobin molecule with the oxy quaternary structure, the position of the a1 and p, chains is such that the interchain salt bridge cannot form, though the ,f32intrachain salt bridge is apparently unaffected. For the salt bridges between the a chains, the situation is somewhat more complicated. In the deoxy conformation with the a2 chain having the unliganded tertiary structure, there is one salt bridge from the guanidinium group (N,H: ) of the terminal as-chain arginine (141~~) to the carboxyl group of the ai aspartate (126a,); the guanidinium proton is not expected to ionize (pK, G 12.5) in the usual alkaline pH range. In addition, there is a less well-defined salt bridge in which a number of groups are involved; this is approximated in the model by the int*eraction of t’he carboxyl group of the C-terminal a,-chain arginine (141~~) and the N-terminal valine NH: group (lul), which has a proton that is ionizable (free pK, E 9.6) in the alkaline range. It is possible also, in terms of the observed distances, that the imidazole group of the a1 histidine (122a,) may be involved in a more complicated multigroup interaction, including an intrachain salt bridge to the carboxyl group of the a1 aspartate (126a,), analogous to the jlZ intrachain salt bridge; this is neglected in the present treatment, though its inclusion may be important in a more complete interpretation of the effect of pH (see below). As in the /3, chain, only the unliganded tertiary form of the a2 chain can have these salt bridges, since structural changes in going to the liganded form shift the residues involved so that the stereochemical requirements for the salt bridge are not satisfied. Further, it is assumed, though the evidence appears less clear, that in the oxy quaternary structure the relative al--as chain posit,ions are such that it is not possible to have the aZ-a1 interchain salt bridges.

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

167

All of the above statements apply with the obvious alteration of subscripts to the /%-A~ t%-a2, and al-a2 interactions. (c) Generating function The behavior of the system as a function of O2 partial pressure, OH- or H+ concentration, and other variables can be det,ermined by use of a generating function, which is a macroscopic analog of the grand canonical partition function and is simply related to the binding potential introduced by Wyman (1965,1967,1968). An alternative approach, which yields identical results by use of somewhat more tedious algebra, is to solve the set of equations for the simultaneous equilibria. We consider non-interacting tetrameric hemoglobin molecules as absorbers for 0, and OH- and choose the oxy quaternary structure with the a and /3 chains in the unliganded tertiary structure (no 0, or OH- bound) as the reference state. The generating function S is then composed of a sum of terms of the form hipjqij that correspond to having i 0, and j OH - bound (i = 0,l. . . N, and j = 0,l. . . N,, where N, and N, are the total number of binding sites for 0, and OH-, respectively). The symbols h and p represent the activities of 0, and OH-, respectively; the customary units of 0, partial pressure (mmHg) and OH- concentration (moles/liter) are used. The quantity hfpjqu is proportional to the probability of obtaining the intermediate state with i 0, and j OH- bound, relative to that of the reference state; that is, hipjqrl is equal to that probability at unit concentration of the reference state. Thus qi, is the equilibrium constant for the reaction Hb+i

O,+j

OH- s Hb(O,)i(OH-)j.

To obtain the qu, it is necessary to sum the relative probabilities of all structures having i 0, and j OH- bound (see below). In the corresponding microscopic formulation, the grand partition function would be expressed as the same sum with X and ~1 equal to absolute activities and qir of the form 4ij = Ktj!liq~~ where qA and q,, are the canonical partition functions for individual molecules of 0, and OH-, respectively, and K,, is the required statistical factor. By the appropriate choice of standard states, the connection between the microscopic and macroscopic formulation can be made (Heck, 1971). For convenience we separate the complete generating function into separate parts for the oxy ( EO) and deoxy (E,,) conformers and write s = 5,(A, p)+E&,p) From equation (l), the fractional (yo,) is given by

; X’pfq& 3 x/Jq;+ 2 ; (1) *=o ,=o i=o j-0 saturation of the hemoglobin tetramer by oxygen =

and similarly the fractional number of OH- bound (i.e. protons released) is

168

A. SZABO

AND

M. KARPLUS

To enumerate all of the possible states and to determine the factors q,,, we introduce diagrams similar to those drawn by Perutz (197Ou,b) for the contributing structures. The diagrams are not essential to the formulation, but they are useful because they clearly demonstrate the structural assumptions made, provide simple pictorial representation of the intermediate structures which occur during oxygenation, and illustrate the physical content of the generating function. By associating specific features of the diagrams with the structural elements involved in the assumptions of the model, it is possible to formulate simple rules for the qr, corresponding to each diagram. The total number of diagrams is reduced by defining each one as representing a given number of 0, bound, a given quaternary structure for the tetramer, a given combination of tertiary structures for the chains (i.e. a given number of a and /3 chains in the liganded or unliganded form), and a sum over all states with respect to the number of OH- bound (H+ released). This corresponds to writing S in the form (4) with an individual

diagram or group of diagrams representing the terms

Figures 1 to 3 illustrate the elements of the diagrams. The quaternary conformation is indicated by using a drawing without a border for the deoxy form and one with a border for the oxy form; the tertiary structure of each chain is shown by having a square for the unliganded form and a circle for the liganded form. The interchain salt bridges are represented by lines connecting the chains involved; the intrachain /3 salt bridges are shown as arcs on the unliganded /3 squares. Crosses and filled circles,

(a)

(bl

FIQ. 1. Hemoglobin tetramer with all subunits having (a) deoxy quaternary and (b) oxy quaternary conformation

(a)

the unliganded tertiary (see text for details).

structure;

(b)

FIG. 2. Hemoglobin tetramer with all subunits having the liganded tertiary quaternary and (b) oxy quaternary conformation (see text for details).

structure;

(a) deoxy

STRUCTURE-FUNCTION

IN

169

HEMOGLOBIN

L

(a)

(b)

(cl

FIG. 3. Hemoglobin tetramer in intermediate states; (a) deoxy quaternary conformation, with a2 having the liganded tertiary structure and ai, ji, pa the unliganded tertiary structure, (b) deoxy quaternary conformation with fll having the liganded tertiary structure and ai, ae, ,Ts the unliganded tertiary structure and (0) oxy quaternary conformation with ai, /11having the liganded tertiary structure and ae, & the unliganded tertiary structure (see text for details).

respectively, are used for the a and /3 chain OH- adsorption sites (ionizable protons). The broken salt bridges involving ionizable protons are drawn as free lines; the other broken salt bridges are not shown. Figure l(a) and (b) correspond t,o the deoxy (a) and oxy (b) quaternary conformations with all chains having the unliganded tertiary structure. Thus, all salt bridges are present in Figure l(a) and only the intrachain ,8 salt bridges in Figure l(b). Figure 2(a) and (b) represent deoxy and oxy quaternary structures with all chains in the liganded tertiary form. Consequently, there are no intra- or interchain salt bridges. In Figure 3 are shown some examples of intermediate structures. Figure 3(a) is a deoxy quaternary diagram with the a2 chain in the liganded tertiary form and the other chains in the unliganded form; all salt bridges are indicated as present except for the two aZ-al salt bridges governed by the a2 chain. Figure 3(b) is the corresponding deoxy quaternary structure with the ,L$ chain in the liganded form; all salt bridges are present except for the /&-a2 salt bridge and the internal fll bridge, both of which are governed by the & tertiary structure. An oxy quaternary diagram is given in Figure 3(c); here all salt bridges except the internal pZ salt bridge are broken, even though only the a1 and & chains are in the liganded tertiary form. In evaluat-ing the diagrams it is important to remember that they are not meant to specify which a or /I chains are in the liganded or unliganded form; that is, they specify only the number of chains of each type. Thus, for example, Figure 3(a) represents both the structure with the a2 chain in the liganded form (a,, pl, flZ unliganded) and that with the a1 chain in the liganded form ( a2, PI, & unliganded). To illustrate the correspondence between an individual diagram and the contribution of the sum over j to the generating function E (see equat’ion (4)) we consider the diagram shown in Figure 3(a). It represents the expression

170

A.

SZABO

ANI)

M. KARPLUS

The over-all factor QP gives the stability of the deoxy quaternary struct’ure with all possible salt bridges relative to the oxy quaternary structure. The factor X(K”/P) arises from the binding of one oxygen to an CLchain with the concomitant breaking of the two al-a2 salt bridges in going to the liganded tertiary form; the factor of 2 is the statistical weight arising from the fact that either of the two a chains can bind one oxygen molecule. The terms in the sum in square brackets involving increasing powers of p correspond to the binding of OH- to the different possible sets of a and p chains. If the OH- binds to the unliganded a-chain valine involved in a salt bridge there is a factor Ha/S; if the salt bridge is already broken because the chain is in the liganded tertiary structure, there is simply a factor H a. For t#he ,6 chains, which are both in the unliganded form, binding OH- to either breaks a salt bridge and the factor is HBIS; the various factors of 2 appearing in the expression are statistical weights that take account of the two equivalent ,6 chains. The complete formula for E can now be obtained by drawing all of the diagrams for the deoxy and oxy quaternary structures; they are shown in Figures 4 and 5, respectively. As is evident from the example of Figure 3(a), the diagrams are drawn with

1 r7 (0)

02

P9 (b)

(41 x

(Ii

(cl $”

9

(21 FIU. 4. Dimgrammatic representation of the oxygenation of the hemoglobin tetramer with deoxy quaternary conformation (see text for details).

all ionizable protons present although they represent the sums indicated in equation (4) over structures with different numbers of OH- groups absorbed. The contribution to 5, of each diagram in Figure 4 is given by the formula fQS6X (f)“o+”

(1 +/LH”)~‘=( 1 +pHB)h”( 1 +pHa/S)2--h’( 1 +pHB/S)2-ha,

where i is the number of 0, bound, f is the statistical factor equal to the number of ways the structure can be realized (e.g. f = 2 for Fig. 3(a)); ha is the number of free crosses, -x (e.g. h” = 1 for Fig. 3(a)); hS is the number of free dots, -0 (e.g. hB = 1 for Fig. 3(b)); m is the number of a chains which are circles (e.g. m = 1 in Fig. 3(a)); i-m is then the number of /3 chains which are circles (e.g. i-m = 0 in Fig. 3(a)). The contribution of each diagram in Figure 5 to go can be written

STRUCTURE-FUNCTION

FIQ. 5. Diagramm&ic quaternary conformation

IN

lil

HEMOGLOBIN

representation of the oxygenation (see text for details).

of the hemoglobin

tetramer

with oxy

with the symbols having the same significance as for 3,. Summing the contributions to 5, and 8, by applying the rules to all of the diagrams in Figures 4 and 5, we obt,ain &(A,

p) = QP(1 +/JP/S)2(1+@P/S)2

1 +A [

K”(l+@? As-(1 +/&F/S) x

L l+h

1 2

P(1

+@P,

2

S2( 1 +/JHfl/S) I ’

Kb(l +/JiP)

9,(X, I*) = (1+pHy(I +pHfl/S)2[1+hKy 1+A -~-41 SPH41S) 1

2

1

(sa)

(51))

.

The relation of 9, as given by equation (5), to phenomenological models is discussed briefly in Application of the Model, section (i). A generalization of this approach to the construction of generating functions for other sets of specific structural assumptions will be presented elsewhere.

3. Application

of the Model

The utilization of the model for describing the equilibrium data available for oxygen binding by hemoglobin is considered in this section. The first step in the application is the evaluation of the model parameters. In the simplified formulation described above, there are a total of six parameters; they are Q, S, Ku, Kfi, Ha and Ha. It is important to note that five of these (i.e. S, K”, Kfl, Ha, Hfl) correspond to rather well-delineated chemical processes and are therefore restricted in their ranges of reasonable values. Only Q, which is concerned with as yet unspecified hemoglobin tetramer alterations, is essentially unrestricted. In what follows, we first indicate the ranges for S, K”, Kfl, Ha, HE, and then describe their quantitative evaluation, as well as that of Q, by use of the human hemoglobin 0, binding measurements of Roughton & Lyster (1965) ; these are still considered to be the best data that include a sufficient variation in oxygen partial pressure for a meaningful fit. The contributions of the

172

A. SZABO

AND

M. KARPLUS

diagrams as a function of system variables are shown and the resulting co-operativity of the oxygen binding is discussed. The model is then used for a description of the Bohr effect, of modified and mutant hemoglobins, and of the effect of variations in 2,3-diphosphoglycerate, salt and hemoglobin concentration (dissociation). (a) Rangefor parameter values The parameter S is defined such that RT In S is the additional free energy required for oxygenation when a given salt bridge is present. Although the importance of salt bridges (-NH; . . . CO,--- ion pairs) as an element in protein structure has long been discussed (Mirsky & Pauling, 1936; Eyring & Stearn, 1939; Jacobsen & LinderstromLang, 1949; Lumry & Biltonen, 1969), specific evidence concerning their presence is sparse. In fact, available X-ray diffraction results indicate that salt bridges are rather rare in proteins, though the spatial relationships of a small number of amino and carboxyl groups that appear to have a special role demonstrate that they do exist (Perutz, 1970a,b; Sigler, Blow, Matthews & Henderson, 1968; Blow, Birktoft & Hartley, 1969). As to the effective thermodynamic properties of such salt bridges, no detailed theoretical models beyond the original ion-pair treatment of Bjerrum (Harned & Owen, 1958; Davidson, 1962) seem to have been developed. However, there are now studies of model systems in low dielectric constant solvents by spectroscopic (Hudson, Scott & Vinogradov, 1969) and electron spin resonance techniques (Atherton & Weissman, 1961; Hirota, 1967) that yield free energies between 2 and 8 kcal. Thus, a wide range of salt bridge free energies appears possible, though for the conditions existing in hemoglobin and other proteins, a value nearer the lower limit (2 kcal., S-30) is to be expected (Fersht, 1972). Further analysis would be very useful here, particularly to separate enthalpic from entropic contributions so as to permit evaluation of the effect of temperature on oxygen binding (Wyman, 1964). For the 0, binding constants Ka and KB, the obvious points of reference are the free chain results for the a chains and the non-co-operative tetramer /I, (HbH) for the fl chains. In the commonly used units of inverse partial pressure of 0, (mm Hg) - l, the measured values are (Ka)chaln= (KQain E 2.0 with KB perhaps larger than K” (Rossi Fanelli et al., 1964). On incorporating the free chains into the hemoglobin tetramer, it is not unexpected that the binding constants will change. However, Ka and Kb values differing from (Ka)chainand (Kfl)chainby more than an order of magnitude would not be reasonable. Moreover, it would imply that what happens to the individual chains on oxygen binding in the hemoglobin tetramer is so different from the monomer behavior that reference to it in terms of Ku and KB values does not yield a useful model. To consider the appropriate values for Ha and Hfl, a knowledge of the nature of the protonated nitrogens is required. As described above, it is clear from the X-ray structure and other data that in the p chain a histidine is involved. For the imidazole group of free histidine, the pK, value is known to be near 6, while in proteins the corresponding pK,values range from 5.5 to 7.4 (Meister, 1965; Orrtung, 1969). Thus, log HB = 14-pK, should be in the range 6.6 to 8.5. As to Ha, the situation is more complicated since the salt bridge is less well-defined (see above). If the valine suggested by Perutz as having the ionizable proton is the essential element, a pK, in the range 7.5 to 8.3 might be expected, in contrast to the free valine value of 9.6. Thus, for log Ha, we have the range 5.7 to 6.5.

STRUCTURE-FUNCTION

IN

173

HEMOGLOBIN

(b) Fit of model to hemoglobin data The ideal experimental data for determining numerical values of the model parameters would be a set of oxygenation curves obtained for a large range of oxygen partial pressures by varying pH and keeping other conditions (e.g. ionic strength, diphosphoglycerate concentration) constant. Unfortunately, such measurements do not exist in the vast hemoglobin literature. Of the available data, those of Roughton & Lyster (1965) for the oxygenation at pH 7 and pH 9.1 appear most suitable for our purpose since they include a sufficient range of oxygen partial pressures to provide satisfactory values for the asymptotic slopes. This is particularly important at pH 7 where results for 0, partial pressures on the order of 196 mm Hg are needed. However, it must be noted that the two sets of measurements made by Roughton & Lyster differ not only in the selected pH but also in the ionic strengths of the solutions; i.e. the pH 7 values correspond to an ionic strength of 1.2, while that at pH 9.1 was only O-1. From the results of others (Rossi Fanelli et al., 1964), it is clear that such a difference in ionic strength can in itself be sufficient to produce a significant change in the oxygenation curve. To indicate the accuracy of the results that can be obtained with the present model and the range of parameter values that is found, we consider several fitting procedures. (i) All parameters except HB were obtained by fitting the Roughton-Lyster pH 7 and 9.1 curves simultaneously with a least-squares procedure; log HB was somewhat arbitrarily fixed at 7.8, near the free amino-acid value for the imidazole group and in the middle of its observed range in proteins. (ii) All parameters except H” and HB were varied and the pH 7 and the pH 9-l curves were fitted separately by a least-squares procedure (for the pH 9.1 curve, S was fixed at the pH 7 value); for log Hfl the same value as in part (i) was used and log Ha was set equal to 6, which is in the middle of the range for valine in proteins. (iii) The same procedure as in (i) was used, except that Ku and KB were set equal to the free-chain value, (Ka)chafn = (KB)chain = 2 mm Hg-I. The parameters obtained by these different procedures are listed in Table 1. Although they represent only a sample of the possibilities, they give a reasonable indication of the nature of the results. Figure 6 shows the oxygenation curves calculated at pH 7 and 9.1 with the set of parameters obtained from fit (i); the experimental data of Roughton & Lyster (1965) are indicated by (0) at pH 7 and ( x ) at pH 9.1. Within TABLE

1

Hemoglobin model parameters Parametert

Q S Ku K4 log log

(mm Hg-I) (mm Hg-I) H” HE

Fit (i) pH 7 and 9.1

Fit (ii)l PH 7

Fit (ii), pH 9.1

Fit (iii) PH 7

4.9 x 1O-5 39.3 4.69 5.33

5.9~10-~ 31.5 4.0

3.5 x 10-S (31.5) 7.2

8.7 x 10-e 30.8

(36; (7.6)

6.43 (7.8)

The values given in parentheses t For definition,

see text.

g; (7.6) were chosen independent

of the fitting

(2) (2)

procedure

(see text).

174

A. SZABO

AND

M. KARPLUS

11 04 08 'og PO,

!2

I6

FIG. 6. Hemoglobin oxygenation curve at pH 7 and 9.1; the calculated values from fit (i) are ahown by solid curves and the experimental values by ( 0) for pH 7 and ( x ) for pH 9.1.

the error of such a plot, the curve is indistinguishable from the experimental results; quantitatively, the agreement is as good as that obtained by Roughton & Lyster with the Adair equation. Correspondingly good results are obtained from fit (ii) at the respective pH values; in fact, at pH 7, the fit (ii) results are slightly better than those of fit (i). Figure 7 gives the oxygenation curves at pH 7 and 9.1 from fit (iii) and compares them with the Roughton & Lyster results. Here there is a significant difference at high 0, partial pressures, in that the calculated asymptotic slope is larger than that determined from the measurements.

FIQ. 7. Hemoglobin oxygenation curve at pH 7 and 9.1; the calculated values from fit (iii) are shown by solid curves (-) and the experimental results by ( 0) for pH 7 and ( x ) for pH 9.1.

It is clear from the results in Table 1 that for the mathematical model assumed here, unique values of the parameters are not obtained by fitting the oxygenation curves; that is, since the accuracy of the measurements is not absolute and the fit is meaningful only in terms of a least-squares error criterion, a range of values for each member of the parameter set is possible. However, the form of the generating function is such that there is significant coupling among the parameters. Moreover, it was found in a larger number of trial calculations that Q always converged in the range 1O-4 to 10e5 and S in the range 30 to 45 (RT In S values of 2 to 2.3 kcal./mole), independent of very different choices for the initial guesses. Correspondingly, K" and KB could not be varied by more than a factor of three or four from the free-chain values without

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

17.5

destroying the possibility of a fit. The Ha and Hfi values have not been explored as thoroughly (see also section (d) below) but here again only a limited range appears to be possible. Thus, it is of some significance that excellent fits to the oxygenation curves are obtained for parameter values that are reasonable in terms of the interactions that they represent (section (a) above). (c) Contributions of diagrams

and cmzformations

It is of interest to determine what hemoglobin structures have an important role in the mathematical model. In Figure 8(a) for pH 7 and Figure 8(b) for pH 9.1, we plot the fractional contributions of the species Hb(O,), (n = 0,l. . .4) as a function of the saturation parameter (yo,). It is clear that the dominant contributions come from the limiting species Hb and/or Hb(O,), for all values of (y o,). Hemoglobin molecules

02

04

06

(a) FIG.

(y,,)

08

100

CL,>

8. Fractional contribution of species Hb(O,), at (a) pH 7 and (b) pH 9.1; fit (i) was used.

02

04

06

08

ib)

(n = 0,l . . . 4) as & function

of saturation

with one or three ligands are important in the low and high (yo,) range, respectively. The species Hb(O,), is relatively unimportant throughout, with a maximum contribution of less than 0.1 at (yo,) g l/2. The general behavior is similar at pH 7 and at pH 9.1. However, there are quantitative differences, the most marked of which concerns the relative contributions of Hb(0,) and Hb(O,),; this change with pH is a consequence of the fact that at high pH the internal /3 chain salt bridge is broken because the imidazole proton is ionized. Figure 9 shows the fractional contributions of the most significant structures in terms of the diagrams of Figures 4 and 5, respectively; the results given correspond to pH 7. For deoxy, the structure with no 0, bound (diagram (0) in Fig. 4) and for oxy, tJhe structure with four 0, bound (diagram (4) in Fig. 5) are most important. In addition, the diagrams with only one oxygen bound (i.e. diagrams (la) plus (lb) in Fig. 4) are significant for the deoxy quaternary structure and, correspondingly, the diagrams with three 0, bound (diagrams (3a) plus (3b) in Fig. 5) are significant for the

176

A. SZABO

AND

M. KARPLUS

FIU. 9. Fractional contributions of diagrams at pH 7 as a function of the saturation (yo,). Labels (0), (la), (lb) correspond to diagrams in Fig. 4 and (a), (3a), (3b) refer to diagrams in Fig. 6.

oxy quaternary structure. Thus, structures with zero or one oxygen bound have the deoxy conformation, while those with three or four oxygen bounds have the oxy conformation. The structures with two bound oxygen, each of which individually never contributes more than one per cent, can have either the deoxy or oxy conformation. It must be remembered that in Figures 4 and 5, each of the diagrams represents a sum of terms which correspond to different degrees of ionization of the Bohr protons and concomitantly broken salt bridges. Figures 10 and 11 illustrate this point by showing the dominant contributions to the limiting deoxy and oxy structures, respectively. In Figure 10(a) and (b) there are given the main components of the deoxy conformation with no 0, bound at pH 7 and 9.1, respectively. It is seen that at pH 7, the structure with all ionizable protons bound is dominant, while at pH 9.1 all the p-chain protons have ionized and structures with no or only one a-chain proton make significant contributions. In the oxy conformation with four oxygens bound shown in Figure 11(a) and (b) for pH 7 and 9.1, respectively, the situation is different because

(a)

CO731

(b)

(0221

Pm. 10. Major text for details).

+ (0.23)

+ (0 23)

+ (0 45)

contributions

to unliganded

deoxy

conformation

(a) pH 7 and (b) pH 9-l (see

STRUCTURE-FUNCTION

ib)

IN

t

(0 05)

FIG. 11. Major contributions text for details).

to fully

ligrtnded

HEMOGLOBIN

0

0

0

0

177

(0 95)

oxy conformation

(a) pH 7 and (t-1) pH 9.1 (sea

Fra. 12. Fraction

of oxy quaternary

structure

as 8 function

of at PH 7.

the ionizable protons cannot be stabilized by salt bridge formation. Thus, even at pH 7, the most important structure has both ,8 protons ionized and the structure with all four protons makes a negligible contribution; at pH 9.1 only the structure with no ionizable protons is important. The quantitative features of Figures 10 and 11, which are, of course, dependent on the exact values chosen for Ha and H6, are not necessarily significant; however, the qualitative results are likely to be valid. Although the assumptions of the model are such that the fraction of the chains having the liganded tertiary structure is equal to (yo,>, no such relation is required for the quaternary conformation of the molecule as a whole. Figure 12 shows the fraction of the molecules having the oxy quaternary conformation as a function of (yo,). It is evident from the Figure that, except at very low and high oxygen pressures (see the inserts in Fig. 12), the relation is almost linear. Thus, high accuracy is required to differentiate the behavior of probes sensitive to tertiary structural changes as a function of (y& from those sensitive to the quaternary conformation (McConnell, 1971). (d) Bohr effect One aspect of hemoglobin oxygenation that is of both physiological and theoretical importance is its dependence on the pH of the medium, the so-called Bohr effect. The 12

178

A. SZABO

AND

M. KARPLUS

plots for pH 7 and 9.1 in Figure 6 show that the mathematical model provides a generally correct description for the alkaline pH region; the acid Bohr effect, which occurs for pH values less than 7, has not been included because the structural information concerning its origin is less well-delined. In considering Figure 6, as pointed out in section (b) above, it must be remembered that the difference in the solution composition used by Roughton & Lyster (1965) for the two sets of measurements means that the apparent variation with pH may include other effects as well. Although the change in pH is certainly the dominant factor, it would be helpful to have pure pH variation data available for a determination of the model parameters. An essential element of the present model is that the Bohr effect results from a coupling of the proton ionization to the oxygen affinity through the salt bridges which can be formed only if the protons are present. This structural mechanism, which of necessity leads to an effect of pH on co-operativity (see section (e) below), avoids the ad hoc assumption of earlier discussions (Wyman, 1939,1948; Saroff, 1970) that there are different ionization constants for the NH; groups in the deoxy and oxy forms; e.g. in his classic paper, Wyman described the alkaline Bohr effect by introducing the proton ionization constants pK, = 7.93 and pK, = 6.68 for deoxy and oxy horse hemoglobin, respectively. Since it is customary in Bohr effect studies to plot the oxygen partial pressure (P,,J at which the tetramer is half saturated (i.e. from equation (2), (yo,) = (Q/4 = l/2) as a function of pH, we present such a curve obtained from fit (i) in Figure 13. The result for the range pH 7 and pH 10 corresponds to the form given in

I.

6

FIG 13. Calculated Bohr effect for hemoglobin: saturation (log Pllz) verse pH.

the log of the oxygen

partial

pressure at half

the literature (Antonini, Wyman, Rossi Fanelli & Caputo, 1962); that is, there is an almost linear behavior in the log-log plot for pH 7 to pH ~8.5 and the value of P,,, is independent of pH for pH > 9.5. In the latter range, the Bohr protons have all ionized and only the salt bridges involving “non-ionizable” protons remain. Of course, these other salt bridges will also give up their protons at sufficiently high pH, but this effect is not included though it could be without difficulty. As to quantitative comparisons with experimental results, the calculated values approximate the maximum Bohr effect that is observed in human hemoglobin; i.e. log P,,, z 1-O at pH 7 and log P,,, E -0.1 at pH 10. A more detailed comparison is complicated by the change of more than one significant variable in most available measurements, in particular the

STRUCTURE-FUNCTION

IN

179

HEMOGLOBIN

variation of ionic strength (Antonini et al., 1962; Antonini et al., 1963) (see section (g) below). There is sn often-quoted linear relation between the number of protons “liberated” and the number of oxygen molecules bound (Wyman, 1948). To make this explicit, we plot in Figure 14 the values of (yn+) = 1-(yen-), where (yOH-) = (j/4) from

Fm. 14. Fraction of protons liberated as a function curve A pH 7, curve B pH 8 and curve C pH 9.

of the fraotional

saturation

by oxygen;

equation (3), verSuS (yO,) at three pH values. It is seen that an essentially linear relation obtains with a slope that is nearly the same for pH 7 and 8. At pH 9, a small, but significant, non-linearity is evident with the greatest curvature occurring at high (y,,) values. The slope at pH 7 corresponds to O-5 proton liberated per oxygen bound, in essential agreement with some experimental results (Kilmartin & Wootton, 1970) though slightly higher values (4.6) have also been reported (Antonini, 1965; Antonini et al., 1963). (e) Co-operativdty of oxygen binding Although the sigmoid nature of the hemoglobin oxygenation curve (Figs 6 and 7) is indicative of the co-operative nature of the phenomenon, a quantitative determination of the co-operativity is more difficult to achieve. Most commonly, experimental values of (yo,) as a function of the Oa partial pressure, Pal, are used to determine the magnitude of the constant n required to fit the measured results by the Hill equation :

Specifically, a plot of log {(yo,)/(l-(y&)} versus log P,, is made following the procedure of Wyman (1964) and the slope of the straight line portion in the neighborhood of P,,, is used as ‘(the” Hill constant n for the system being examined. Wyman has defined what corresponds to the “apparent” differential free energy of interaction, (dAF/d(y,,)), and shown how to determine this quantity from the value of n at the degree of saturation corresponding to the measurements; that is, dAF -a(Y,,)

RT(n-1) = n(Yo*Xl -
A. SZABO

180

AND

M. KARPLUS

An alternative procedure is to determine the apparent free energy of interaction, over the saturation curve. One way of *Pa,,, which is the integral of (dAF/d(y,‘)) obtaining A Fi,, is to fit the measured values by the Adair (192&b) equation: K, J’c* +2K, K, PZ1+3K, K, K, P;, +4K, K, K, K, PA, 4(1 +K, PC* +K, K, P:,+K, K, K, Pi, +K, K, K, K, Pi,)’ where the equilibrium constant K, is that for the reaction (Yo,> =

(7)

Hb(O,),:

1+O, e WO,),. The apparent free energy of interaction, AF,,,, corresponds to the equilibrium constant for the reaction HbO,+Hb(O&

eHb+Hb(O,),

relative to that obtained for no interaction (i.e. if the equilibrium constants K, are equal except for statistical factors) and is given by (Whitehead, 1970):

(8) where 16 is the statistical factor corresponding to the four possible sites for 0, binding involved in K, and the four possible sites from which 0, can be released in K,. The quantitative relation of the apparent free energy AF,,, to the true free energy of interaction for the present model with inequivalent sites will be considered subsequently. The difficulty of using a procedure based on equation (7), rather than equation (6), for the analysis of experimental data is that accurate values of K, and K, can be obtained only if measurements are made in the region where the curve of (yo,) versus Poehas its initial and final asymptotic slope, respectively. A prime reason for our use of the data of Roughton & Lyster (1965) is that they made a special effort to reach the asymptotic domains. For a tetrameric system, such as hemoglobin, there is no unique relation, in general, between the Hill constant n evaluated at P,,, and the ratio (KJK,). Consequently, we have evaluated both parameters in the present model calculations as a function of the variables that are of interest. As we see below, the ratio (K4/K1) is much more sensitive to the system variables than is n; part of this is simply a consequence of the fact that n appears as an exponent in equation (6) so that a small change in n can represent a larger alteration of the system than a corresponding change in the ratio K&G. In Table 2 we present the values of the Adair constants obtained by Roughton & Lyster and the corresponding constants resulting from tit (i) to their data. In addition we include the values of n at P,,,, (K4/Kl), and AF,,, computed as indicated above. It is clear that there is excellent agreement for the pH 7 results, but not as good agreement for those at pH 9.1. This is true in spite of the fact that the oxygenation curves (Fig. 6) at both pH values are accurately obtained from the calculated constants; to demonstrate this, we compare the results from our fit with those of Roughton & Lyster (see Note added in proof). This indicates that the values of the individual K, are very sensitive and may be less reliable than stated by Roughton & Lyster. Such a concern is reinforced by the apparent increase in co-operativity as measured by their K,/K,, with increasing pH. It is difficult to understand how this can occur at higher pH, particularly when combined with the fact that the pH 9.1 measurements

STRUCTURE-FUNCTION

IN

181

HEMOGLOBIN

correspond to a lower ionic strength than those at pH 7 and that such a change is expected to decrease co-operativity (section (g) below). There is the additional problem that, in contrast to RJR,, n decreasesin going from pH 7 to pH 9.1; whether this apparent inconsistency can be explained by the fact that K,/K, and n measure different aspects of co-operativity (see above) has not yet been investigated. We have taken the Roughton-Lyster results for pH 9.1 and found that the value of K, is very sensitive to the weight attached to the high Po, measurements in the least-squares fitting procedure. If all measurements are given equal weight, the result is K, G 147: such a K, yields a ratio (K4/K1) = 4.5, which is in the “expected” direction from that at pH 7 (see Table 2). As the weight of the high Po, results is increased, K, approaches the Roughton-Lyster value of 2. TABLE 2 Adair

constants for Roughton-Lyster

results

PH 7 Exp. K1 (mm Hg-‘) Kz (mm Hg-I) K3 (mm Hg-‘) K4 (mm Hg-‘) f&lK~ AB,,, (kcal./mole) n

0.049 0.0427 0.221 0.320 6.49 2.11 3.0

pH 9.1 Calc.

Exp.

0.0516 0.0233 0.3532 0.3577 6.95 2.76 3.05

0.240 4.064 0.732 1.992 8.3 2.85 2.5

Celc. - .- ___~~0.4254 0.8862 2.81 1.22 2.9 2.24 2.63

In Figure 15(a) and (b), we show the calculated values of K,/K, and n as a function of pH. In correspondence with the results in Table 2, it is seen that K,/K, drops with pH, leading to K,/K, = 6.95 at pH 7 and K,/K, = 2.0 at pH 10. By contrast, the value of n over this pH range changes only from n = 3.0 at pH 7 to n = 2.4 at pH 10. It is to be noted that these calculations are made at constant values for all the parameters; i.e. the pH dependence is intrinsic in the model (see section (d) above) and is not introduced by assuming that a parameter, such as Q, varies with pH as has been suggested in other treatments (Edelstein, 1971; Rubin & Changeux, 1966). Although the calculated variation in n with pH is in agreement with the RoughtonLyster measurements, data obtained by Antonini et al. (1962,1963) suggest that n is essentially constant over the range pH 5 to 10; it is not evident from their published data whether a variation in n of 0.4, the predicted change between pH 7 and 9.1, is within experimental error. Further studies on this question are clearly needed since some coupling between co-operativity (as measured by K,/K, or n) and the Bohr effect (as measured by P,,, versus pH) is inherent in the model. However, the coupling is not complete because, of the six interchain salt bridges, only two are involved in the Bohr effect, and concomitantly, of the four salt bridges involved in the Bohr effect, only two are interchain. There is also a possible change in the model that appears to be consistent with the X-ray results (Perutz, 197Oa,b) and would reduce the coupling even further. This involves alteration of assumption 3(ii) (Formulation, section (a)) to take account of the possibility that the a-a’ salt bridge involving the ionizable proton has an intrachain component.

182

A.

SZABO

AND

M. KARPLUS

(b)

,------

302622

6

Fra. 16. Co-opemtivity

,__\_ 7

a PH

aa & function

9

IO

of pH: (a) K,/K,

and (b) n.

(f) Modified and mutant hemoglobins A great variety of modified and mutant hemoglobins have been studied (Perutz BE Lehmann, 1968; Perutz, 1971). In many of these, the difference between the special hemoglobin and the normal molecule (HbA) is such that several, if not all, of the parameters included in the present model might be altered in a manner that is difficult to specify from our knowledge of the structural changes involved. We restrict our consideration, therefore, to a few species in which it is possible to make reasonable guesses as to what parameters are affected and how their magnitudes are changed. Of course, the resulting formulation represents at best an oversimplification, as does the mathematical model itself.

(i) C-tewninal amino acids The simplest cases to consider are the modified hemoglobins in which the C-terminal amino acid has been removed from the a chains, the /3 chains, or both; these are the des-His (1468), des-Arg (141a) and des-His (146/3)+des-Arg (141a) species discussed by a number of workers (Antonini, 1965; Kilmartin & Wootton, 1970; Kilmartin & Hewitt, 1971). From the description of the model in the Formulation section, it is clear that, as a first approximation, the effect of these deletions is to be introduced by altering the salt bridges in which the specific residues are involved. Figure 16(a), (b), and (c) shows the assumptions made in the calculations. The des-His (1468) hemoglobin in its limiting deoxy quaternary struoture (Fig. 16(a)) has no internal p salt bridges, but Greer has pointed out (Peru& 197Ou,b), that the ,f?to a salt bridges are replaced by weaker salt bridges involving the carboxyl group of tyrosine (145/3). A corresponding formulation of the des-Arg (141~1)hemoglobin (Fig. 16(b)) does not

STRUCTURE-FUNCTION

IN

(a) FIG. 16. Diagrammatia (1468), (b) des-arginine quaternary conformation

HEMOCLOBIN

(b)

183

Cc)

representation of C-terminal modified hemoglobins: (a) des-histidine (141a) and (c) des-His (146fl) + des-Arg (141a). The unliganded deoxy is shown; weak salt bridges are indicated by dotted lines (see text).

have the a-a’ salt bridge involving the guanidinium group of the missing Arg (14Ia), but does include a weaker a-a’ salt bridge with the penultimate tyrosine (140a) carboxyl group replacing that of Arg (141a). As in the original formulation of the salt bridges (Formulation section), so here in the mutants, the situation for those between a chains is less clear than for the other salt bridges. Finally, the des-His (146j?)+des-Arg (141a) hemoglobin shown in Figure 16(c) combines the changes described for the des-His (146j3) and the des-Arg (141a) compounds. Introduction of the “weak” and absent salt bridges into the mathematical model is made by a straightforward modification of the development outlined in the Formulation section. A corresponding diagrammatic prescription for obtaining the generating function from the diagrams is developed to take account of the presence of two types of salt bridges, the standard one S and a weaker form 8. With S set equal to half of S, we obtain the results given in Table 3, which also includes the normal hemoglobin values for comparison. The columns headed S,,#, Sz ,, SBlr, SF, in accord with notation of the preceding section, list the values of the individual salt bridge parameters appropriate for each of the modified molecules; that is, for a normal, weak, or absent salt bridge, the parameter value is 39*3,19*7 or 1, respectively. It is seen that both the

TABLE

3

Calculakd~ and experimental$ Hill constants, ajinities and Bohr effects for normal, C-terminal modi$ed and mixed state hemoglobins P 1/a

12

f&i HbA(pH 7) 39.3 HbA(pH 9.1) 39.3 Des-His (1468) 39.3 Des-Arg (141a) 1 Des-Arg-His 1 a; (CN-)&(K, = 0) 1 a.J?;(CN-)(& = 0) 39.3

- AbP,talApH§ Calc. Obs.

s,“,,

s,,

8,”

Calc.

Obs.

Calc.

Obs.

39.3 39.3

39.3 39.3

39.3 39.3

3.06 2.63

3.0 2.6

9.2 0.95

9.2 0.96

0.53

39.3

19.7

2.33

2.5

39.3 19.7

1.1 0.55

0.1

39.3

39.3

1

1

1.77

2.0 1.1 1.2 1

0.27 0.45

39.3

1.5 1.4 1.0

2.85 0.92 0.28

2.8

19.7 19.7 1

1 39.3 1

1.21 1.8

1.5 1.5

0.53 0.45

0.51

-

0.35 0.30

(O,-

0.1) 0.5

0.6

t The parameters used are given in Table 1 (fit (i) ) exoept where indicated. f For HbA, n and Pill are taken from Roughton & Lyster (1966); the mixed state hemoglobins are compared with the results of Brunori et al. (1970). The rest of the data is from Kilmartin & Hewitt (1971). 0 --log Pi/z/ApH = (log Pi/&s r - (log P&H s.

184

A. SZABO

AND

M. KARPLUS

des-His (146g) and des-Arg (141a) hemoglobins have reduced co-operativity (as measured by K,/K, or by n), increased oxygen affinity (as measured by P,,,) and a smaller Bohr effect (as measured by the change in P,,, with pH) relative to normal hemoglobin; for des-His (1468) fdes-Arg (lala), n s 1 and the Bohr effect is almost completely inhibited. The model calculations have the correct behavior except for the small change obtained on removing the C-terminal amino acids from both chains relative to that with only Arg (141a) missing. One factor of importance here may be that, in des-His (1468) +des-Arg (141a), 82% of molecules with no 0, bound (P,, = 0) are calculated to have the deoxy conformation, while Perutz (1970aJ) finds that this species crystallizes in the oxy conformation in the absence of oxygen. This suggests that refinements of the calculations by use of different S and 8 parameters for the salt bridges might be appropriate. (ii)

Mixed state hemoglobins

A variety of modified hemoglobins exist in which either the two ccchains or the two /3 chains have the iron in the ferric (Fe3+) form. These have been made by chemical oxidation in the presence of a specific ligand (e.g. Ni, CN-, F-) (Brunori, Amiconi, Antonini, Wyman & Winterhalter, 1970) or without one (the ligand is then presumably Hz0 of OH - depending on the pH) (Banerjee & Cassoly, 1969). In all cases, there is a decrease in the value of n to near unity (the al/l2 forms appear to have n s l-1 to 1.2 and the a& forms have n g 1). However, there is considerable difference among the modified species as to the Bohr effect and oxygen afinity. It is likely that one important element in the observed differences is the spin state of the

(a)

(b)

FIG. 17. Diagrammatic representation of mixed state hemoglobins: (a) a&, and (b) &f12. The unliganded deoxy quaternary conformation is shown; the ferric chains are indicated by a plus sign and only the existing salt bridges are drawn.

ferric ion, which is known to be high spin (S = 512) for ligands such as F- and H,O and low spin (S = l/2) for ligands such as CN- and N; . Considering the low-spin case first, we assume that the Fe3 + is essentially in the heme plane (Hoard, 1968) and that, therefore, the oxidized chains have the liganded tertiary structure. This implies, in the simplest approximation, that the salt bridges originating in the oxidized chains are broken and leads to diagrams of the form shown in Figure 17(a) and (b) for the limiting deoxy conformations of the (a&)iowspin and the (ai/?2)lowspin molecules, respectively, This is equivalent to assuming that the low-spin mixed-state hemoglobins have structures and properties identical with the corresponding doubly oxygenated species. Formulation of the generating function for this situation and calculation with

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

185

the pH 7, pH 9.1 parameters (fit (i) of Table 1) leads to the results given in Table 3 with the experimental data for ul(CN-)pa and a.$i(CN-). For the high spin forms, (al /3Jhigh Spinand (a,/?: )hlghSpinwith an H,O or F - ligand, the measured oxygen affinity is intermediate between the low spin and normal species (e.g. ai/32 and a&l with H,O as the ligand have I’,,, values of 4 and 9 mm Hg, respectively, at pH 7) and a large Bohr effect is present. This suggests that for the high-spin case, in which the ferric iron is expected to be out of the heme plane (4.3 A) as observed in met-hemoglobin (Perutz, 1970a,6) but less so that in high-spin ferrous (Iv09 A), the tertiary structure is intermediate between that of unliganded and liganded ferrous chains in the hemoglobin tetramer. Although it is possible to adjust the model parameters to represent this case, we feel it is not useful to do so atI this stage of analysis. An interesting question concerns the apparent difference between the calculated and experimental n-values for the low-spin mixed-state hemoglobins; that is, t#he model gives n = 1.0 for al(CN-)& and n = 1.8 for cQ~(CN-), while the experimental values are 1.2 and 1.0, respectively. Additional information is provided by the paramagnetic chemical shift observations of Ogawa & Shulman (1971) that in the al (CN- )/I, form oxygenation of the /? chains affects the nuclear magnetic resonance spectrum of the a chains, while there is no effect on the /I-chain resonance spectrum from oxygenation of a,/3,f (CN-). The simplest explanation, as pointed out by Ogawa t Shulman (1971), is that CL;(CN-)/I, undergoes a change in quaternary conformation from the deoxy to the oxy form on oxygenation, while a& (CN- ) does not because it is in the oxy form independent of whether or not oxygen is bound. From such an interpretation, it is a,f (CN-)/I, w h’lc h can have n > 1, while CC.$~(CN-) is expected to have n = 1, in agreement with experiment, The recent experiments of Ogata & McConnell (1971) on the binding of a spin-labeled triphosphate to the mixed-state hemoglobins provide independent evidence that it is u.$,(CN-) which is in the oxy conformation, while al(CN-),6, represents a mixture of oxy and deoxy conformers in the absence of oxygen. However, the mixed-state CL&?:(CN-) molecule with a spin label on the t3chains shows an effect of a-chain oxygenation on the spin-label spectrum (Ogawa et al., 1968); this suggests that a$: (CN-) does undergo a change in quaternarv conformation on oxygenation, unless a tertiary structural change in one chain is assumed to affect the spin label in the other. From the above discussion, it is clear that the present model gives results that are in agreement with the experimental data concerning the low P,,, and n values, as well as the Bohr effect and affinities (P,,,), for the low-spin mixed-state hemoglobins. There is, however, an apparent disagreement concerning the ratio of deoxy to oxy quaternary conformers at zero oxygen tension ((yc,) = 0) and the resulting ratio of n values for ai(CN-)/3, and a&(CN-); that is, the model yields the result that, at is essentially all in the oxy quaternary structure and y$ )= 0, (4 Bzhw spin a2 z l0WSpinis 50% deoxy, while the measurements of Ogata 81 McConnell (1972) indicate that ai(CN-)/3, is about 60% in the deoxy quaternary structure, while aJ3,‘(CN-) is almost all in the oxy form at (yc,) = 0. The model results are a consequence of the fact that, with salt bridges of equal strength, it is the a chains that have the primary role in the co-operative behavior; that is, there are four a--a’ interchain salt bridges relative to only two /3-a interchain salt bridges. By making the salt bridges of unequal strengths (i.e. increasing S,, by ~1 kcal./mole and decreasing Sz, and S,” by ~1 kcal./mole) it is possible at pH 7 to obtain satisfactory results for

186

A. SZABO

AND

M. KARPLUS

normal hemoglobin and the experimental ordering of the n values for the mixed-state species. A change in the intrachain versus interchain character of Sz, (see section (d) above) would have a similar effect. It is important to note that the significance of the mixed-state species is that they provide tetramers with two “ligands”. As pointed out in section (c) above (see Fig. 8), the intermediates Hb(O,), make a very small contribution in the oxygenation of normal hemoglobin because of its high co-operativity. Thus, the mixed-state hemoglobins can, in principle, serve the role of testing models in a range not accessible by measurements on the normal molecules. The difficulty in their use for such tests is that one has to make comparisons with respect to quantitative details that could easily be significantly affected by differences between the mixed-state and normal hemoglobins other than those included in the present formulation or that of other simplified models (e.g. Ogata & McConnell, 1972). Further discussion of this question will be presented in a subsequent paper. A number of mutant hemoglobins (HbM) have the mixed-state form a,‘& or a$: as a result of substitution of the proximal (FS) or distal (E7) histidine by tyrosine (Perutz & Lehmann, 1968; Udem, Ranney, Bunn & Pisciotta, 1970; Greer, 1971a). The properties of these compounds, relative to normal hemoglobin and to the artificial mixed-state compounds considered above, are such that a detailed treatment taking account of specific structural changes appears to be required. (iii) Interchain contacts There are a number of mutants (Perutz & Lehmann, 1968) in which the a& and asp1 contacts are modified by a substitution in either the j? chain (e.g. Hb Chesapeake) or the a chain (e.g. Hb Yakima). These mutants have an increased oxygen atity, a diminished Hill constant (1.0 < n < 15)f, and a rather normal Bohr effect relative to ordinary hemoglobin (HbA). One mutant (e.g. Hb Kansas), in which the /I chains are modified at the alp2 and a& contacts, has a decreased oxygen affinity, a diminished value of n (n g 1.3 at pH 7) and the Bohr effect is, if anything, somewhat greater than normal. As a first approximation, it seems appropriate in the present model to discuss these mutants in terms of a change in the parameter Q, which determines the intrinsic relative stability of the two quaternary conformations. The alteration of the Q parameters for these mutants is based on the suggestion from X-ray studies that the interactions determining the oxy vers’susdeoxy stability are concentrated at the aI/& and a,& interfaces (Peru@ 197Oo,b). Figure 18(a), (b), and (c) show plots of log P,,,, K,/K, and of n at pH 7 verse --log Q, similar to those given by Rubin & Changeux (1966) and by Edelstein (1971). It is evident that the various increased oxygen atSty mutants listed above require a decrease in Q(Q M 5 x lo-*) relative to HbA (Q = 4.9 x1O-5); i.e. a relative stabilization of the oxy quaternary structure. Correspondingly, for the decreased 0, affinity of hemoglobin Kansas, an increase in Q is needed (Q M 10-l). Figure 19 shows log P,,, as a function of pH for Q = 10-l and 10-O, in comparison with the HbA result. It is seen that the Bohr effect is present over the entire range of Q values, although that for Q = 10-O is reduced from that calculated at Q = 4.9 x 10V5 (HbA). The rather weak dependence of n on pH is preserved for all Q values examined; however, there is an indication that at the largest t In a recent study, Nagel. Gibson & Jenkins (1971) find ra = 2.2 for Hb Capetown, in disagreement with earlier work. Since, in addition, they find that the oxygen atiity is increased less than that of Hb Chesapeake, which has n s 1.3, their results are consistent with the model.

STRUCTURE-FUNCTION

FIG. 18. Properties (log J’d (b) k/K1

of model as a function and (~1 n.

IN

HEMOGLOBIN

of the quaternary

FIQ. 19. Bohr effect as a function of the quaternary stability curve B, Q = 4.9 x lo-& (HbA); and curve C, Q = 10-Q.

stability

187

parameter

parameter

Q: (a) affinity

Q: curve A, Q = 10-l;

Q value (Q = 10-l), the maximum n is shifted to higher pH (i.e. n = l-4 at pH 7 and n = l-6 at pH 9). In both types of mutants with modifications at the alpa and asp1 interfaces, it has been noted (Perutz & Lehmann, 1968; Bonaventura t Riggs, 1968; Bunn, 1970) that there is a concomitant effect on the tetramer 7t dimer dissociation equilibrium of the protein in the oxy conformation. In the high-affinity mutants (Hb Chesapeake) a decrease in dissociation is observed, while in the low-affinity mutant (Hb Kansas) there is an increase in dissociation relative to normal HbA. Such a relation between

188

A. SZABO

AND

RI. KARPLUS

oxygen affinity and oxy tetramer dissociation is consistent with the model but is not a necessary consequence, since increase in affinity and decrease in dissociation are coupled uniquely only if one assumes that the mutations alter nothing but the oxy tetramer stability and ignores any concentration dependence of the phenomenon. More detailed quantitative data are needed for a complete analysis. The measured increase in the dissociation constant of Hb Kansas by two orders of magnitude (Bonaventura & Riggs, 1968) relative to HbA is of interest in this regard. Although the above codification of the ai& a2)?1 interface mutants in terms of a variation in Q appears to have some success, it is most likely that the actual situation is more complicated (Ogata & McConnell, 1972). Certainly, the apparent variation in the free-energy difference between the deoxy and oxy conformer by more than 10 kcal./mole (Q g 10-l to Q g 10-O) suggests that the magnitude of the perturbation resulting from the amino-acid substitution is large enough to alter other parameters as well. This is also made probable by the extensive structural alterations relative to hemoglobin observed in the X-ray results available for certain mutants, such as hemoglobin Kansas (Greer, 1971b). One question concerns the possibility of changes in the individual chain binding constants (K” and K8) due to displacement of the heme and distortion of the positions of the various amino-acid residues. Some information on this point can be obtained by accurate measurements in the mutants of K, at high pH, where K, + KaKB/2(Ka+KB) z K/4 for Ku N Kfl. In the isolated chains of Hb Chesapeake (Bunn, 1970), there is only a slight change in oxygen affinity, but this does not necessarily imply that the binding constants of the chains in the tetramer are unaltered by the mutation. (g) Other effects (2,3-diphmphoglycerate, ionic strength and dissociation) Although certain basic elements of hemoglobin co-operativity have been related to the mathematical model in the previous paragraphs, there are many additional effects to be considered. Of these, we concern ourselves here only with 2,3diphosphoglycerate, ionic strength, and dissociation, all of which are implicitly encompassed by the model. As we show below, the effect of DPGt and dissociation can be treated approximately in terms of the simplest assumptions; the effect of ionic strength, by contrast, appears to be more complicated and will require additional theoretical and experimental analysis . (i) 2,3-diphosphoglycerate The primary effect of DPG is to lower the oxygen affinity of hemoglobin (Benesch & Benesch, 1969; Benesch, Benesch & Yu, 1968,1969; Benesch, Benesch, Renthal & Gratzer, 1971; Tyuma, Shimizu & Imai, 1971; Gibson, 1970; Bailey, Beetlestone t Irvine, 1970; Tomita & Riggs, 1971); e.g. at pH 7.3 with a “physiological” concentration of O-1 M-NaCl, the value of log P,,, for hemoglobin (6 x 10s5 M) without DPG is 057 and that with DPG (4.0 x lo-* M) is 0.95 (Benesch et al., 1971), the magnitude of the change being dependent on both the salt and the DPG concentration. As to the action of DPG on other aspects of hemoglobin oxygenation, the experimental results are not completely unequivocal. There is some indication that the Hill coefllcient is increased by DPG (e.g. the result of Tyuma et al. (1971) is that TZchanges from 2.52 to 3.02 on the addition of DPG) and that there is a considerable rise in the apparent free energy of co-operativity as a result of a decrease in the first Adair constant K, in the t Abbreviation

used: DPG, 2,3-diphoaphoglymrate.

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

189

presence of DPG, the last constant K, remaining approximately unaltered (Tyuma et al., 1971; Gibson, 1970); however, the values of the constants K, obtained by Tyuma et al. (1971) for normal hemoglobin at pH 7.4 are not in agreement with those of Roughton & Lyster ( 1965). As to the Bohr effect, the data of Tomita & Riggs ( 1971) for human hemoglobin indicate that it is increased somewhat by DPG, though the results of others (Benesch et al., 1968; Bailey et al., 1970) do not show a significant change. The effect of DPG is ascribed to its preferential binding to the deoxy conformer; i.e. in the presence of DPG, deoxy is assumed to be stabilized relative to the oxy conformer, resulting in a decrease in the oxygen affinity of hemoglobin. The simplest mechanism for incorporating DPG into the mathematical model is based on exclusive binding by the deoxy conformer and equilibrium between deoxy molecules with and without DPG. On the assumption that DPG binding does not alter any of the intrinsic model parameters (i.e. S, K”, KB, Ha, Ho), one can formally subsume the effect of DPG into a change in Q; that is, if Q0 is the parameter for hemoglobin in the absence of DPG (“stripped” Hb), that in the presence of DPG is Q = Qo[l+K,,,(DPG)], where (DPG) is the concentration of DPG and KDPG is the DPG binding constant of the deoxy conformer. The log P,,, value of stripped hemoglobin at pH 7.3 requires Q0 s 6.3 x10e6, which results in a value of n z 3.0. Since the measurements of Roughton & Lyster were done in the presence of DPG, the fit (i) value of Q, Q = 4.9x1O-5, can be related to Q,,. For KDPG = 6.67 x lo4 liters/mole (Benesch et al., 1971) the effective concentration of DPG in the Roughton $ Lyster experiment would have to be 10T4 moles/liter for agreement. This is a reasonable value, though no direct determination was made. The dependence of I’,,, on DPG concentration can be calculated from Figure 18(a). Noting that, over the range of interest, log P,,, varies essentially linearly as a function of log Q with a slope close to 0.27 and using the relation Q = QO[l +K,,,(DPG)], one obtains log P,,, z log C’$+O*27 log (1 +Knpa(DPG)), (9) where Pllz and PF,, are the oxygen a%nities at half saturation in the presence and absence of DPG, respectively. Figure 20 shows that equation (9) is in good agreement with the experimental data of Benesch et al. (1971). 14r

020

-II 04

08

i---L--.*-J 12 16

24

log[ltKD,,tDPG)]

FIa. 20. The variation of log Pllz with DPG concentration. The experimental CT=6.67 x 10’ are taken from Benesch et al. (1971) and the, solid line is calculated with log P$, = 0.57; (DPG) is approximated by the total DPG concentration.

points and KDPo from equation (9)

190

A. SZABO

AND

M.

KARPLUS

The nearly linear variation of log P,,, as a function of log (l+K,,,(DPG)) with slope close to 0.25 is a consequence of any model for hemoglobin of the Monod et al. (1965) type (M.W.C.) if it is assumed that DPG binds only to the deoxy quaternary structure. For the M.W.C. formulation (see section (i) below), log P,,z versus log L has a slope 0*25( 1 -c)/( 1 +c -d-) c , w h ere L is the allosterio parameter and c is the ratio of the dissociation constants for relaxed and tense forms. The maximum value of the c so that, as a result of the large co-operativity Hill constant n is 1+3(1 -&)2/(1 +1/-)2 of hemoglobin (n N 3), c 6 0.01 and the slope must lie between 0.25 and 0.27. At sufficiently large DPG concentration, we expect equation (9) to break down because binding of DPG by the oxy quaternary conformation becomes important. To a fist approximation this effect can be treated by writing

[1+JkPW ’ = ”

[l +K&(DPG)]’

where Kg,,, K&., are the DPG binding constants of the deoxy and oxy quaternary the binding to the oxy quaternary conformations, respectively. If K&,-10-2K&G, conformation must be considered for (DPG) 2 10m3M. The presence of DPG is expected to have different effects on the individual binding constants in the Adair equation (equation (7)). For the range of Q values corresponding to the DPG concentrations studied by Benesch et al. (1971), the model shows that K, is essentially constant, while K,, K, and K3 undergo significant variation; the value of K,, in particular, is most sensitive to the DPG concentration. This behavior is in qualitative accord with the observations of Tyuma et al. (1971), although there is considerable difference between their K, values and those obtained from the calculations. (ii) Ionic strength Although a variety of salts may have specific effects on oxygenation (Guidotti, 1967), there are some fairly general patterns of change in the oxygenation behavior that seem to depend on the ionic strength of the solution per se. It is only the latter which are considered in the present discussion of the mathematical model. A series of papers by Antonini et al. (1962,1963) have shown that an increase in ionic strength results in a decrease in the oxygen afinity, an increase in the Hill coefficient n at P 1,2, and a decrease in the Bohr effect. Qualitatively similar, though quantitatively somewhat different, behavior is found on variation of the ionic strength in “stripped” hemoglobin. An expected consequence in the model of increasing the ionic strength is a weakening of the salt bridges. Certain results of spin-label studies made by McConnell, Deal & Ogata (1969) can be interpreted (Moffat, 1971b) by assuming that an increase in ionic strength weakens the salt bridges involving the terminal histidine of the j3 chain. To examine this effect, we have made a model calculation in which the salt bridge parameter S was reduced to 30, three-quarters of its normal value. The effect of pH on P,,, is somewhat reduced, but the oxygen afinity is increased and the value of n is decreased; for S = 30, we obtain at pH 7, PI,, = 5.4 mm Hg and n = 2.95, relative to the values of 9.3 mm Hg and 3.05, respectively, obtained from fit (i). Thus, increasing the ionic strength of the solution must have other consequences as well. One possibility is a change in Q due to, for example, a perturbation of the interactions at the a& and azpl contacts, particularly those involving hydrogen bonds or salt’

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

191

bridges (Benesch et al., 1968). A related suggestion (Benesch et al., 1968) that negative ions (e.g. Cl-) are bound more strongly by the deoxy conformer has been made on the basis of the similarity of the effects of salt concentration and DPG. Also, the concomitant effect of salts in promoting dissociation of the oxy quaternary conformation (Antonini & Brunori, 1970) may indicate an analogy with the dissociation of the low a0inity hemoglobin mutant, Hb Kansas (see section (f) above). It is clear that additional work, both experimental and theoretical, is needed to sort out the importance of these and other factors. (iii) Dissociation Although the question of the importance of tetramer to dimer dissociation in hemoglobin oxygenation under normal conditions has a long history (Antonini et al., 1963; Edelstein, Rehmar, Olson & Gibson, 1970), it is only recently that a large difference between the dissociation constant of the deoxy and oxy hemoglobin tetramer conformations has been found in certain experiments (Kellett & Gutfreund, 1970; Kellett, 1971). Perutz (197Ou,b) has suggested that the salt bridges, which exist in the deoxy conformer, could in themselves be sufficient to account for a difference of such a magnitude. In the present model, the relative equilibrium constants for dissociation of oxy and deoxy hemoglobin involve not only the salt bridges but also the parameter Q. As a first approximation, the limiting ratio of equilibrium constants is given by &Se. With the fit (i) parameters (Table I), this quantity equals 2 x 105, in satisfactory order of magnitude agreement with experiment (Kellett, 1971). If one assumes that only the oxy form dissociates significantly, the value of Q used in the present calculations implicitly includes the effect of dissociation. A detailed model of the effect of dissociation, its relation to ionic strength, and the concomitant dependence of the oxygenation curve on hemoglobin concentration would require an extension to include dimers in equilibrium with tetramers. This involves a straightforward introduction of additional terms in the generating function. However, because of the uncertainties concerning hemoglobin dissociation and the lack of the necessary oxygenation datat, we have not included such a calculation. (h) Comparison with isolated chains Since measurements of the oxygen binding curves of isolated chains are available (Antonini & Brunori, 1970), it is of interest to consider the implications of the present model for the change in behavior of the chains on incorporation into the hemoglobin tetramer. It is known that individual Q chains, /l chains in the f14tetramer hemoglobin H, and the p-mercuribenzoate derivatives of individual Q and /l chains all have high oxygen a6lnity; the values of P,,, are 0.5 and O-4 mm Hg for the CCand /l chains, respectively, except that P,,, for /!? (p-mercuribenzoate substituted) seems to be significantly larger (Brunori, Noble, Antonini & Wyman, 1966). Also, the individual chains have a hyperbolic binding curve (n = l), and no observable Bohr effect. The isolated chain equilibrium constants [(Ka)chaln z (KB)chain E 2 mm Hg-l] are thus significantly smaller than those obtained for the chains as part of the hemoglobin tetramer according to the model (e.g. for 6t (i), Table 1, Ka = 4.69 and Ks = 5.33 mm Hg-I). Although the exact values of these parameters are uncertain, that they t The measurements by Anderson, Antonini, Brunori BEWyman (1970), of the dependence of ethyl isooyanide binding on hemoglobin oonoentration are appropriste data for such an analysis; see slso the data for sheep hemoglobin of Roughton, Otis $ Lyster (1965).

192

A. SZABO

AND

M. KARPLUS

have to be increased over the free-chain values to fit the near-saturation data of Roughton & Lyster (1965) is unequivocal. This would suggest that there are tertiary structural changes on forming the hemoglobin tetramer which increase the individual chain oxygen affinities. Further, the structural changes appear to persist, in part, in the oxy quaternary structure. There is spectroscopic evidence that the changes involved in going from the isolated unliganded chains to the unliganded tetramer are greater than those for the liganded species (Brunori, Antonini, Wyman & Anderson, 1968). Tertiary structural alterations do not seem unlikely when the magnitude of the energies involved in forming the tetramer is considered. Since the /3 chains in the hemoglobin tetramer have an internal salt bridge involving a Bohr proton, it follows that if in hemoglobin H (or p, p-mercuribenzoate) the chains underwent the same type of tertiary structural change on binding oxygen as they do in the hemoglobin tetramer, a Bohr effect should be observed. The absence of a Bohr effect indicates that the salt bridge is not present in HbH and/or that the tertiary structural change on binding of 0, is absent or different from that occurring in HbA. This agrees with the observation (Riggs, 1961) that in HbH in the absence of oxygen, the SH group of cysteine (93/3) is as reactive as in the liganded form; the relation of the reduced reactivity of the SH group in the unliganded tertiary structure of the /3 chain in hemoglobin to the existence of the salt bridge has been discussed by Perutz (1970a,b) and others (Guidotti, 1967; Antonini & Brunori, 1969; Neer, 1970). From the diagrams in Figure 4, the unliganded a chains in the deoxy conformation of the hemoglobin tetramer have a maximum of four salt bridges connecting them together. This suggests the possibility that, although liganded a chains are monomers (Antonini, Bucci, Fronticelli, Wyman & Rossi Fanelli, 1966), the unliganded a chains might dimerize. If, as indicated above for the p chains, the individual a chains do not undergo the same structural change on ligand binding as they do in the hemoglobin tetramer, no dimerization might occur. A careful experiment concerned with this possibility would be of interest. (i) Relation to phmomedogicd

deb

A comparison of the mathematical model presented here with the models proposed by Koshland et al. (1966) (K.N.F.) and by Monod et al. (1965) (M.W.C.) for hemoglobin and other allosteric proteins is of interest. The present model assumes that there are two quaternary conformations, as was postulated by M.W.C., but each of these can be made up of monomers in a number of different combinations of tertiary structures (see Figures 4 and 5). The latter is consistent with K.N.F., but was not envisaged by M.W.C. For purposes of comparison with M.W.C., we consider the high pH limit (i.e. p -+ co corresponding to all ionizable protons dissociated) and take Ku = KB = K. Introduction of these assumptions into equations (5a) and (5b) yields a generating function of the form (to within a multiplicative constant) &w,c.(x)

= E,(h)fE,(h)

= Qs4(1+hK/S)4+(l+X)*.

(10) This equation is identical to the generating function of the M.W.C. model with the parameters L = Qf14and c = S-l. The generating function in equation (10) suggests a simple interpretation of the M.W.C. model in t,erms of the structural concepts introduced by Perutz (1970a,b). The oxy (R, relaxed) conformation is intrinsically more stable than the deoxy (T, tense) since Q < 1; however, in the absence of ligand (i.e.

STRUCTURE-FUNCTION

IN

HEMOGLOBIN

193

h = 0) the deoxy quaternary structure is stabilized by four salt bridges (i.e. S4). The affinity of the deoxy conformer is smaller than that of the oxy conformer because a salt bridge must be broken in the former when a ligand is bound (i.e. the effective binding constant is K/S). As oxygens are bound to the T conformer the tertiary structure of the monomers changes so as to break the constraining salt bridges and drive the system towards the high-a%lnity (R) conformation. Recently Ogata & McConnell (1971,1972) extended the M.W.C. model to allow for the non-equivalence of the a and /? chains by introducing new parameters c,, cg and K& Kk. The generating function implicit in their treatment. E,,,

= L(l+h~,K”,)~(l+hc~K~)~+(l+hK~)~(l+hK~)~

(11) has the same mathematical structure for fixed pH as equations (5a) and (5b). The precise relation between the present model and that of Ogata & McConnell will be discussed elsewhere. As to the relation of the present model to K.N.F., there is no simple physical correspondence in that the quaternary structural changes that have an essential role in the former are excluded in the latter. Although changes in tertiary structure on oxygen binding and pairwise interaction between monomer units are important in both models, there is a significant difference. In the K.N.F. formulation for hemoglobin, the binding of 0, to one chain has a direct effect on the binding of O2 to one or more of the other chains. In the present model, the tertiary structural change on the binding of O2 to one monomer does not directly alter the ease of 0, binding to the adjacent monomer. Considering the al-aa interaction as an example, we see that if a1 binds O,, the al-a2 salt bridges are broken but not the a2-a1 salt bridges. Since it is the latter that have to be broken when a2 binds 02, the tertiary structure of a1 has no direct effect on the binding constant of the a2 chain. Co-operativity is introduced only by the possibility of quaternary conformational change, which serves as the mechanism for transmitting the effect of the tertiary structural alteration from one monomer to another.

4. Concluding

Discussion

A mathematical model for the co-operative oxygen binding by hemoglobin is described. In its present form, the model has elements that are related to specific features of the molecular interactions in the stereochemical mechanism proposed by Perutz. Parameters are used to represent the oxygen a&-&y of the individual chains, the strength of the salt bridges, the ionization constants of the Bohr protons, and the intrinsic free energy difference of the two quaternary conformations of the tetramer. A generating function is formulated to express the relative stabilities of the structures associated with the oxy and deoxy quaternary conformation of the hemoglobin tetramer; diagrams that display the physical content of the generating function are introduced. From the model parameters, which are determined by fitting the Roughton-Lyster measurements on human hemoglobin, the relative contribution of each structure as a function of the system variables (e.g. 0, pressure, pH) is evaluated. It is shown that a satisfactory description of many, but not all, of the available equilibrium data can be obtained from the model with reasonable values for the physically meaningful parameters. The data considered include the co-operative binding curve itself, the effect of pH and DPG, and the behavior of a number of modified and mutant species. Of particular interest is the fact that the Bohr effect 13

194

A. SZABO

AND

hf. KARPLUS

follows directly from the model, without the need for special assumptions about changes with oxygenation of the pK, values of certain groups. Some experimental results, such as the effects of ionic strength,

are not given consistentfly

by the model in

its present form; possible extensions and modifications of the model are suggested t,o account for these observations. The existing structural and chemical evidence is insufficient to prove or disprove the validity of the mathematical model proposed in this paper. That both quaternary and

tertiary conformational changes occur is clear from the X-ray results of Perutz and his co-workers; what is not certain is the role they have in determining the chemical properties of hemoglobin. Moreover, since the available X-ray results for normal hemoglobin provide information concerning the structures of only the limiting species (i.e. Hb and Hb(O,),), the detailed sequence of the breaking of the salt bridges is not known; that is, it is not possible at this stage to determine whether the assumptions made to permit formulation of a specific thermodynamic model are correct. For example, the tertiary structural change induced by oxygen binding in a single subunit chain may not be sufficient to break its salt bridges, as is assumed in the model: as long as there is some energy coupling between the tertiary structure and the effective stability of the salt bridges (e.g. in terms of a shift of the intrinsic equilibrium for the penultimate tyrosine remaining in its hydrophobic pocket), a slightly generalized form of the present model would be applicable. If some or all of the salt bridges are of importance only in determining the relative stability of the two quaternary structures and the effect of tertiary structural changes on the energetics proceeds through interchain interactions not directly involving the salt bridges, a modification of the model would be required. The mathematical model thus represents at best a first step at unifying the structural and chemical properties of hemoglobin. Even in its present form, however, the model inoludes many aspects of the mechanism of oxygen binding and suggests additional experiments that are needed for a more complete understanding. We hope to concern ourselves with the further elucidation of the co-operative nature of hemoglobin oxygenation, including its structural, thermodynamic and kinetic aspects, in subsequent papers. One of the authors (M. K.) gratefully acknowledges the introduction to the complex lore of hemoglobin co-operativity provided by many stimulating discussions with Drs G. Guidotti and R. G. Shulman. Conversations with Drs J. T. Edsall, C. M. Park and R. Banerjee were also very helpful. A set of lectures given by Dr M. Perutz at M.I.T. (Autumn, 1970) served as a starting point for the examination of his proposals presented here. Unpublished data were kindly supplied by Drs R. E. Benesch and R. Benesch. Grateful acknowledgement is made by one of us (M. K.) to Aux Deux Magots for providing the hospitable environment in which a first draft of this paper was written. This work was supported in part by grants from the National Science Foundation and the U.S. Public Health Service. One of us (A. S.) is a National Research Council of Canada postgraduate scholar. REFERENCES Adair, G. Adair, G. Anderson, Anton%, Antonini,

S. (1926~). PTOC. Roy. Sot. A, 108, 627. S. (1925b). J. Biol. Chem. 63, 529. N. M., Antonini, E., Brunori, M. & Wyman, J. (1970). J. Mol. E. (1965). Phyeid. Chem. 45, 123. E. t Brunori, M. (1969). J. Bid. Chem. 244, 3909.

Biol.

47, 205.

STRUCTURE-FUNCTION Antonini, Antonini,

IN HEMOGLOBIN

E. & Bnmori, M. (1970). Ann. Rev. B&hem. 39, 743. E., Bucci, E., Fronticelli, C., Wyman, J. & Rossi Fan&,

195 A. (1966). J. Mol. Biol.

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Note added in proof: To demonstrate that the fit (i) values of the Adair constants give good agreement with the measured results at pH 7 and pH 9.1, we present below a tabulation of the experimental data obtained by Roughton t Lyster (1966) at the two pH values with the (yo,) values calculated by them with their constants and the (yo,) values from the fit(i) constants.

STRUCTURE-FUNCTION

Tabulation

0.491 0.848 1.160 1.493 1.646 4.987 6.968 7.119 8.730 9,870 11.32 14.07 17.67 22-7 26.7 33.8 62.1 76.2

197

HEMOULOBIN

of experimental data obtained by Roughton 0 Lyster (1965)

in %

PH 7 P (nUX%g)

IN


pH 9.1

Expt

Fit (i)$

RL§

0.578 1.143 l-728 1.945 2.166 16.76 23.49 32.80 45.63 66.07 64.60 78-69 86.76 92.21 94.46 96.26 97.99 98.96

O-636 1.112 1.666 2.068 2.166 14.80 21.94 31.76 46.21 66.66 66.62 78.43 87.34 92-86 94.93 96.76 98.37 98.98

0.620 1.100 1.668 2.096 2.188 16.42 22-63 32.32 46.42 56.66 66.12 77.78 86.66 92.20 94.45 96.35 98.12 98.83

P (mXI3Hg) 0.0966 0.1346 0.1660 0.1835 0.4642 0.6748 0.6816 0.8240 0.9430 1.090 1.549 2.269 18.6 20.2 21.8 23.8 30.8 -

Expt

Fit (i)$

R.L.$

1.130 1.808 2.483 2.920 16.56 26.29 32.22 46.60 53.92 63.16 78.08 86.59 99.171 99,369 99,438 99.604 99.661 -

1.230 1.881 2.605 2.934 15.20 24,27 33.20 44.92 63.62 62.50 78.96 88.53 98.90 98.99 99.06 99.14 99.34 -

1.058 l-770 2.482 2.978 16.60 26.70 31.72 46.60 63.84 62.29 78.46 88.61 99.26 99.31 99.37 99.42 99.66 -

7 Experimental values from Roughton BELyster (1966). $ Valuee calculated with K, values from “fit (i)” listed in Table 2 as “Calc.” 8 Values calculated with K, values of Roughton & Lyster (1965) listed in Table 2 85 “Exp.”