Light Scattering in Protein Solutions

Light Scattering in Protein Solutions

BY PAUL DOTY Department of Chemistry. Gibbs Memorial Laboratory. Harvard University. Cambridge. Massachusetts AND

JOHN T . EDSALL University Laboratory of Physical Chemistry Related to Medicine and Public Health. Harvard University. Boston. Massachusetts

CONTENTS I . General Introduction: Two-Component Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 . Scattering by a Dilute Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page 37 37 39

2 . Scattering by a n Ideal Macromolecular Solution ...................... 3 . Scattering by Liquids and Two-Component Systems . . . . . . . . . . . . . . . . . . 4. Anisotropy and Depolarization of Scattered Light . . . . . . . . . . . . . . . . . . . . 5 . Polydisperse Solutes and the Average Molecular Weight . . . . . . . . . . . . . . . 6 . Molecular Weights of Proteins Determined by Light Scattering . . . . . . . . 7. Specific Refractive Index Increments of Proteins in Solution . . I1. Multicomponent Systems of Small Molecules: Effects of Net Cha Ionic Strength on Turbidity in Protein Solutions ...................... 1. New Phenomena Arising in Multicomponent Systems . . . . . . . . . . 2 . Definition of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Thermodynamic Fluctuation Theory for Multicomponent Systems . . . . . . 4 . Application to a Two-Component System . . . . . . . . . . . . . . . . . . . . . . 5 . Systems of Three and More Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . Experimental Data for Bovine Serum Albumin . . . . . . . . . . . . . . . . . . . . . . . 7. Interaction Effects in r-Globulin Solutions, and Between .c-Globulin and .............................. Serum Albumin . . . . . . . . . . . 8. Studies on t h e System Bovine Albumin-Urea-Water . . . . . . . . . . . . I11. Internal Interference and the Determination of Size . . . . . . . . . . . . . . . . . . . . . 1. Introductory Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Spherical, Rod-like and Coiled Macromolecules: The Dissymmetry Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . The Extrapolation Method and the Transmission Method . . . . . . . . . . . . . 4 . Experimental Investigations of Solutions Exhibiting Internal Interference 5 . The Mie Theory for Spherical Particles: Application . . . . . . . . . . . . . . . . . . IV . Study of Reactions Involving Macromolecules by Light Scattering . . . . . . . . . 1 . Insulin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Ovalbumin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Reaction between Human Serum Albumin and Mercurials . . . . . . . . 4. Antigen-Sntibody Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

41 46 48 50 54

55 55 56 60 61 62 65

70

72 73 73 74 77 81 85 90 91 92 93 95

36

PAUL DOTY AND JOHN T. EDSALL

V. Light Scattering in Dilute Solutions of Charged Macromolecules.. . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Interference Theory and the Concentration Dependence of Dissymmetry 3. Interpretation of the Interaction Constant in Terms of the Hard Sphere Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page

4. Experiments with Bovine Serum Albumin a t Low Ionic Strength and

96 96 98

103

104 107 107 1. Clarification of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Turbidity by Transmission Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3. Measurements of Reduced Intensity of Scattering.. . . . . . . . . . . . . . . . . . . 109 4. Absolute Calibration and the Refraction Correct.ions. . . . . . . . . . . . . . . . . 111 5. The Use of Working Standards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 . Depolarisation Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7. Specific Refractive Index Increments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8. Measurements on Colored Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 116 List of Principal Symbols Employed.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI. Experimental Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Although the general phenomenon of light scattering has been well understood for several decades, it is only within the last few years that its utility in the study of macromolecules has been clearly established. As a consequence it is now possible to determine the size, weight, and activity coefficient of macromolecules in solution by making relatively simple optical measurements. The method is both supplementary and complementary to other physical methods of investigating the properties of macromolecules and therefore offers both a means of measuring constants otherwise obtainable and of providing completely new information. A particularly attractive aspect of light scattering methods lies in the speed with which the necessary observations can be made. Thus, on the one hand this permits investigations that would otherwise require prohibitive lengths of time; on the other hand this feature offers completely new opportunities in the study of macromolecular kinetics. This presentation is concerned primarily with the application of the light scattering method to the characterization and interaction of proteins in solution; however, it is not possible to develop the proper perspective without some reference to work on other macromolecular systems, particularly polymer solutions and dispersions of spherical particles. Consequently, our development proceeds on a broader base than may be necessary in the future when more comprehensive and diverse studies on proteins have been carried out. The exposition of the theory and corresponding experimental investigation falls naturally into three major sections. After an introduction dealing with the scattering of gases and liquids, solutions containing macromolecules whose dimensions are small compared with the wave-

LIGHT SCATTERING IN PROTEIN SOLUTIONS

37

length of light are treated in detail as two-component systems. It will be seen that this treatment provides the basis for the molecular weight determination of proteins under favorable conditions. However, a really adequate analysis of the scattering from protein solutions requires a rigorous treatment of multicomponent systems. This matter is taken up next and followed by a summary of available experimental results. The third section takes into account the effect of internal interference arising when the macromolecules are not small compared to the wavelength of the incident light. These considerations provide the means of determining the size and shape of macromolecules. The next two sections deal briefly with the unusual behavior found in solutions of highly charged proteins a t low ionic strength and with preliminary investigations of rates of macromolecular reactions. In the final section experimental methods are discussed in some detail. A full discussion of relevant electromagnetic theory is given, for example, in treatises by Born (1933) and Stratton (1941). A comprehensive treatment of the classical work on light scattering is given by Cabannes (1929). Some specialized aspects are found in the book by Elhagavantam (1942). The following reviews of the more recent work on scattering from solutions are recommended for a fuller discussion of the theory and the results obtained in nonprotein systems: Zimm, Stein and Doty (1945)) van de Hulst (1946), Doty (1947), Mark (1948), Oster (1948, 1950a) and Edsall and Dandliker* (1951).

I. GENERALINTRODUCTION: TWO-COMPONENT SYSTEMS 1. Scattering by a Dilute Gas A beam of light is transmitted through a homogeneous, nonabsorbing medium without alteration. If, however, nonabsorbing particles differing from the medium with respect to refractive index are placed in the medium a disturbance of the wave motion results. Depending upon the size and arrangement of the particles this disturbance can be described as reflection, diffraction or scattering of the incident light. Providing the particles are small and do not exist in a crystal-like array only scattering will result. The scattered light (Rayleigh scattering) will be unchanged in frequency? a.nd it is completely characterized by its intensity relative to the incident beam, its angular distribution and its state

* Several passages in the review by Edsall and Dandliker are identical with portions of the present review, but we have here covered in detail a large number of topics which are only referred to briefly by Edsall and Dandliker. t The intensity of Raman scattering in which the frequency is changed can be safely neglected since its intensity is weaker than Rayleigh scattering by several orders of magnitude.

38

PAUL DOTY AND JOHN T. EDSALL

of polarization. From the measurement of these quantities we wish to learn as much as possible about the nature and arrangement of the particles causing the disturbance in the incident radiation. The quantitative treatment of light scattering begins with the work of Lord Rayleigh (1871) who investigated the scattering from a dilute gas composed of small, isotropic molecules. When such molecules are placed in the oscillating electric field of the light wave the negative electrons and positive nuclei are set into forced oscillation exactly out of phase with each other becoming thereby weak secondary sources of radiation. The intrinsic ability of the molecule t o redirect the incident light in the scattering process is characterized by the polarizability, a, which is the ratio of the dipole moments induced to the electric field. Rayleigh showed that the intensity of light, is, scattered at an angle e from the incident beam, relative t o the incident intensity, l o , is given by the following expression when the incident light is unpolarized.

The wavelength of light in vacuum, the distance from the molecules to the detector and the number of molecules per unit volume (1 cc.) are denoted by Xa, r, and v, respectively. For practical use the polarizability must be replaced by observable quantities ; this is readily accomplished by using the well-known relation between the polarizability and the dielectric constant of the gas, 4, and the dielectric constant of the medium in which the particles reside, €0, which in this case in a vacuum and E,, = 1. Moreover, when there is no absorption, E can be replaced by the square of the refractive index, n, providing both of these quantities are measured a t the same frequency, i.e. the same wavelength. Thus, ~ - E a=-----

4a v

O

-n2-l-n-l 4n v

2n v

(2)

From equations (1) and (2) it follows that

Consequently, i t is possible to determine the number of molecules per unit volume from readily observable quantities. The validity of this relation is best demonstrated by the accuracy with which Avogadro's number, No, can be evaluated from it. For example, Ewing (1926) has found a value of 6.03 & 0.07 X loz3from scattering measurements on organic vapors. Alternatively, the molecular weight, M , of the gas

LIGHT SCATTERING I N PROTEIN SOLUTIONS

39

molecule can be determined if the weight concentration, c, is known, since v can be replaced by No c / M in equation (3). It appears resonable that a relation similar to equation (3) might be expected to hold for macromolecules in dilute solution where the solvent replaces the empty space between the gas molecules. I n the following sections it will be shown that a similar relation does exist as a limiting case, but to obtain a more general result, a different analysis of the problem is required. Before passing on t o these considerations it is necessary to digress briefly on the characterization of the scattered light. Instead of focusing attention on the intensity of light scattered from the incident beam it is often convenient to consider the diminution of the incident beam due to scattering upon transversing a distance, 1, in the scattering medium. The natural logarithm of the fractional decrease in the transmitted intensity, I , serves well for this kind of measurement and is known as the turbidity. The turbidity is equivalent to the extinction coefficient for absorbing systems, i.e., I = Ioe-rz. The relation between T and is is found by integrating the latter over the surface of a sphere of radius r . The result is

The symbol Ro denotes the value of the ratio isr2/Io when the incident light is unpolarized and both components of the scattered light are measured. This is the quantity generally measured in light scattering investigations and is known as the reduced intensity or Rayleigh ratio. At a given wavelength ROis an intrinsic property of the scattering system. Its use is preferable t o evaluating the turbidity from scattering data, as has been frequently done, because Re refers to the actual quantity measured: the use of turbidity is best restricted to transmission measurements. The distinction is even more compelling when the angular intensity distribution is unsymmetrical, as is found in many macromolecular solutions, for then the expression corresponding to equation (3) is valid only a t e = 0. 2. Scattering by a n Ideal Macromolecular Solution

By making simplifying assumptions similar to those made in the previous section for an ideal gas we can obtain the basic limiting relation between the scattering as characterized by Re and the molecular weight of the dissolved macromolecule. We assume again that the molecules are isotropic, small compared to the wavelength of light, and that the solution is ideal in the sense that the solute molecules are randomly disposed : this state will always be sufficiently approximated at high

40

PAUL DOTY AND JOHN T. EDSALL

dilution. Finally, we recognize that more light is scattered from a dilute solution than from the solvent if the solute molecules are large and those of the solvent are small; and we assume that the scattering from the solute molecules alone can be obtained if the scattering from the solvent is subtracted from that of the solution. With these assumptions it can then be demonstrated (e.g., see Debye, 1947) that the essential form of equation (1) is preserved. I n equation (2) E and eo take on the meaning of the dielectric constant of the solution and the solvent, respectively. In substituting the refractive index for the dielectric constant the following relation is used because the refractive index increment, (n - no)/c, is an intrinsic constant of most dilute solutions, dependent only on wavelength.

+

The use of (n no) in place of 2no in the next to the last expression of equation (6) would be more exact but the approximation involved in using the latter is negligible even at moderate concentration. As a consequence of these considerations equation ( 3 ) takes the following forms :

These are the desired limiting relations. The constants K and H are characteristic of a given solution and once determined do not have to be re-evaluated. A measurement of the scattering and the concentration a t sufficient dilution then permits a determination of the molecular weight of the solute. In the following section it is shown that the result given here, equation ( 6 ) , is completely analogous t o the van% Hoff expression for osmotic pressure, ie., P = cRT/M where P , R , and T represent the osmotic pressure, the molar gas constant, and the absolute temperature, respectively. Without recourse to further theoretical considerations it has been found quite practical to implement these limiting expressions for reduced intensity and osmotic pressure by plotting the experimental data as K c / R g oor P/cRT against c. The extrapolation to zero concentration appears to be linear if the measurements themselves have not been made at too high concentrations. The intercept a t c = 0 is then equal to the reciprocal of the molecular weight.

LIGHT SCATTERING IN PROTEIN SOLUTIONS

41

We have now sketched the development of the basic relation between the scattering of light and the molecular weight of dissolved macromolecules. Before discussing specific data we shall examine the more general theoretical background with a view to eliminating the assumptions of ideal solutions and isotropic molecules that restrict the application of the treatment just outlined.

3. Scattering by Liquids and Two-Component Systems In the preceding treatment the positions of the molecules of the gas or solute were assumed to be random with respect to one another. For liquids and nonideal solutions, however, the thermal movements of the scattering units are not independent and the total intensity of the scattered light cannot be obtained by summing the intensities scattered from individual molecules. In general destructive interference occurs with the result that there is a considerable decrease in the intensity of scattered light over that expected from independent scattering centers. When randomness and disorder are absent as in perfect crystals destructive interference is complete and no light is scattered. The case of liquids is intermediate between that of gases and crystals: the decrease in the intensity of the scattered light brought about by the destructive interference due to the “local order” in the liquid state is about fifty fold. The direct calculation of this very pronounced diminution is a formidable problem and has been circumvented by treating the problem as one in fluctuations (Smoluchowski, 1908). From this point of view the scattering is considered to depend upon the irregular spacing of the scattering centers. When the molecules are regularly spaced as in a perfect crystal the scattering from one volume element is always exactly canceled by the scattering from a similar volume element appropriately located. For example, one may focus attention upon two cubic volume elements in the liquids, each 100 A. on a side, so situated with respect to the incident beam and the observer that complete interference results when each contains exactly the same number of scattering elements. Indeed the time-average population of these two elements is identical but Brownian motion causes the population of each to vary continuously. Thus at a given instant one may contain 6345 molecules and the other 6285: the scattering from the additional 60 molecules in the one element mill persist while that of the others is canceled. The scattering from a liquid is thus a residual effect which becomes larger as the fluctuations of density within volume elements increase. Optically the inhomogeneity produced by fluctuations is registered as a deviation of the dielectric constant from its mean value. If an analysis along these lines

42

PAUL DOTY A N D JOHN T . EDSALL

is carried out retaining the assumption of isotropic scattering centers the following result is obtained for the turbidity of a liquid: T =

8a3 7 V(Ae)' 3x0

(7)

where the last term represents the mean square fluctuation in dielectric constant occurring in an element of volume V . Since the fluctuation in dielectric constant -(A€) is intimately -related to the fluctuation in density ( A p ) , that is (A€)' = ( a ~ / a p ) ~ ~ ( Aa~ straightforward p)~, transformation leads t o the expression:

where p and correspond t o the density and isothermal compressibility and T O is the specific designation for the turbidity of a pure liquid. The compressibility enters into this expression because it gives a measure of the ease with which a change in applied pressure alters the density of the liquid. The more readily compressible the liquid, the greater the average fluctuations in density, from one small volume element to another. This development was extended by Einstein (1910) to include twocomponent liquid mixtures, again under the assumption th a t the molecules mere isotropic and small compared with the wavelength. The dielectric constant of a typical volume element undergoes random alterations, not only due to fluctuations in density, but also in concentration of one component with respect t o the other. Thus the mixture should scatter more than the average of the two pure liquids which make it u p ; other things being equal this additional scattering will be greater the greater the difference in dielectric constant between the two components. .The contribution of this effect to the mean square fluctuation of the dielectric constant is simply related to the mean square fluctuation in concentration and this in turn depends on the concentration dependence of the chemical potential or free energy of dilution of the solvent, since it is this quantity which governs the process by which a fluctuation is brought about. Quantitatively the relations are expressed in this way:

-

(Ac)' = -

RTVIC

aaP (- a , ) 1

VNo

LIGHT SCATTERING I N PROTEIN SOLUTIONS

43

where c = concentration of solute in grams per milliliter of solution A P , = the free energy of dilution PI = partial molar volume of solvent The substitution of equations (9) into equation (7) and the replacement of API by (-PPl), where P represents the osmotic pressure, leads t o the following expression for the turbidity due t o the concentration fluctuations: 7’-

RTc

8r3 3x04

No

(3

(10)

For ideal solutions dP/& is equal t o R T / M ; hence the scattering is proportional t o the molecular weight of the solute. This result, which was also obtained in the first section, is now seen from the point of view of fluctuation theory to be a consequence of the fact th a t for a given weight concentration the mean square fluctuation in the concentration is directly proportional to the molecular weight of the solute. To adapt this treatment t o the case of nonideal solutions use is made of the expression: P/cRT

=

l/M

+ Bc

(11)

which has been shown to have wide applicability in fitting the experimental results of osmotic pressure obtained in dilute solution and, moreover, has a firm theoretical foundation. If this expression is substituted into equation (10) one obtains with the aid of equation ( 5 ) and the definition of K and H given in equation (6) :

Kc/Rgo = l / M HC/T= 1/M

+ 2Bc + 2Bc

Equations (11) and (12) revert to those of ideal solutions when B = 0. This analysis forms the basis for the earlier statement th a t eqbation (6) is the counterpart in light scattering of van’t Hoff’s equation in osmotic pressure. To recapitulate, we have seen that the scattering from a solution composed of isotropic molecules results from two independent types of fluctuations involving respectively local variations of density and of concentration. The sum of these as given in equations (8) and (10) accounts for the total light scattered by such solutions. Since our interest lies in the solute molecules v e wish to consider only the scattering due t o concentration fluctuations. This is readily accomplished b y subtracting from the scattering of the solution the scattering of the solvent and, thereby, obtaining data suitable for use in equation (12). I n discussing

44

PAUL DOTY AND JOHN T. EDSALL

the scattering from solutions, we shall always assume that the scattering of the solvent has been subtracted. Having thus established the basis for the interpretation of light scattering investigations of two-component systems, we turn briefly to the meaning of the constant B appearing in equations (11) and (12). It may be recalled that the deviations from ideal behavior found in real gases at low pressures may be characterized by a constant known as the second virial coefficient, B'. The pressure-volume relation for n moles of a real gas in terms of the second virial coefficient is expressed as

m(;) P V = 1 + B'(+)

Since c

=

n M / V , this is readily transformed into P 1 B' - - = M + F C cRT

Equation (13b) becomes identical with equation (11) if B is set equal to B'M2 and if the gas pressure of a one-component system is identified with the osmotic pressure of a two-component system. For our present purpose such an identification is permissible and the quantity B, which may be called the interaction constant, can be interpreted along lines already shown to be useful in understanding the second virial coefficient of gases. The simplest and most fruitful procedure has been to consider B' as a measure of the volume about the center of one molecule that cannot on the average be occupied by the center of another molecule. Thus if the molecules of the gas or the solute can be assumed to be hard, elastic spheres having no net attraction or repulsion, this volume, known as the excluded volume, is equal to four times the molecular volume. If there is a net attraction, the molecules are on the average found closer to each other, and the value of B is correspondingly smaller. Larger values of B can result conversely from the existence of a net repulsion among the molecules or from the steric consequences of an asymmetric or expanded shape. The relative contributions to B of steric and energetic effects are readily determined by measuring the temperature dependence of B ; the temperature dependence is due to the energetic effects and can be used to measure the heat of dilution of the system. When B is equal to zero, intermolecular attraction is exactly compensating for the steric or space-filling properties of the molecules. Greater intermolecular attraction leads to negative values of B , but the accessibb range is quite limited because of the onset of phase separation. When the solute molecules are as compact and rigid as globular proteins, steric effects alone can make only a small positive contribution to the value of B because the excluded

LIGHT SCATTERING IN PROTEIN SOLUTIONS

45

volume is only a few times greater than the molecular volume. However, when the protein molecules possess a considerable net charge, the resulting repulsion causes the molecules to remain at a greater average distance from each other, with the consequence that the volume about each molecule which is unlikely to be occupied by the center of another molecule becomes large. The value of B increases accordingly, the observed effects being governed largely by the magnitude of the charge and the ionic strength: the quantitative relations have been studied by osmotic pressure, the most thorough study being that of Scatchard, Batchelder and Brown (1946) on serum albumin. The same phenomena have also been studied more recently by light scattering, which permits extension of the measurements to very low protein concentrations even when the net charge on the protein is high and the ionic strength very low (see Sect. I1 and V). Upon passing from compact molecules of nearly spherical shape t o asymmetric molecules we find that the excluded volume increases to values considerably in excess of four times the molecular volume found for spheres. Particular attention has been given t o the case of rigid rods (Zimm, 1946; Onsager, 1949) where it has been shown that the excluded volume is equal to the product of the diameter of the rod and the square of its length. Consequently the interaction constant for rigid rod-like molecules depends strongly on the length and in general has values many times that for hard spheres of the same molecular weight. The magnitude of these effects in experimental investigations depends upon the product of the interaction constant and the concentration (see equation 12). If only the steric contribution to B is considered, it is found to be so small in the case of compact macromolecules such as typical protein moleciiles that it makes a significant contribution only at relatively high concentrations of perhaps 10% or more. With rod-shaped macromolecules, however, B is larger and correspondingly its effect becomes apparent a t much lower concentrations. The value of B is, of course, increased still further if repelling forces operate between the particles. An extremely interesting use has been made of the values of B determined from solutions of tobacco mosaic virus, a rod-shaped macromolecule, by light scattering studies. This arose in the study of the problem of the spontaneous separation of a solution of tobacco mosaic virus into two phases, an upper isotropic one and a lower anisotropic one, when the virus concentration exceeds a critical value. Onsager (1949) has presented a general theory for such a phenomenon showing that it is a common property of all solutions of long, rigid macromolecules and explaining this behavior in terms of the value of the excluded volume as reflected in the interaction constant. Oster (1950b) has successfully applied this

46

PAUL DOTY AND JOHN T. EDSALL

theory to explain many of the details of the phase separation in tobacco mosaic virus solutions, The interpretation of the interaction constant has received considerable attention in the field of high polymers. The volume of space occupied by a long, coiled molecule is much larger than its molecular volume, in many cases by a factor of a hundred or more. Consequently, even in the absence of repulsive forces the excluded volume of such a molecule is very large. One of the principal theoretical developments in recent years has involved a series of attempts to calculate the value of B for such cases. These efforts, due particularly t o Huggins (1942) and Flory (1942, 1944, 1945), have provided the basis for a practical theory of high polymer solutions* although only limited success has been attained in the evaluation of B in dilute solutions. I n current work attention is being given to the interpretation of B along the lines of a hard sphere model mentioned above (Flory, 1949; Doty and Steiner, 1950a). 4. Anisotropy and Depolarization of Scattered Light We now turn to the disposal of the restriction thus far imposed, namely, that the scattering molecules must be isotropic. The electric moment induced in an isotropic molecule is parallel to the electric field vector (assumed to lie in the vertical direction), and as a consequence the light scattered in the plane perpendicular to this vector (the horizontal plane) is completely polarized in the vertical direction. Most molecules, however, are anisotropic and are conveniently characterized by a polarizability ellipsoid, the three principal axes of which correspond to different values of the polarizability. When the irradiated molecule does not have one of these three axes parallel to the electric field vector the induced moment will be inclined and the light scattered in the horizontal plane will exhibit a small horizontal component in addition to the vertical one. When depolarization occurs more light is scattered than in the case of equivalent isotropic molecules. Since the molecular weight is related only to that part of the scattering due t o fluctuations in concentration, this additional source of scattered light must be considered quantitatively in order that proper correction can be made. Such a quantitative treatment for binary systems has been carried out by Cans (1923) by including in the earlier treatment of Einstein the effect of fluctuations in anisotropy of volume elements. On this basis it is possible to determine the factor by which the intensity of scattering a t any angle, or the turbidity, must be multiplied in order to eliminate the contribution due to the anisotropy of the molecules. Cabannes (1929) has shown that the correction factor

* Similar approaches t o the problem were developed by Munster (1946, 1948) and Schulz (1947a, b, c) in Germany, and in England by Miller (1948).

LIGHT SCATTERING I N PROTEIN SOLUTIONS

47

involves only the depolarization ratio, pu, which is the ratio of the horizontal t o the vertical components of the scattered light a t 90" when the incident light is unpolarized. The correction factor to be applied to RSois (6 - 7p)/(6 6p) and to T is (3 - 7p)/(6 7p). In some liquids the contribution of the fluctuations in anisotropy to the over-all scattering is large, the value of pd for benzene being above 0.4, and that for carbon disulfide 0.6 or higher. With gases and dissolved macromolecules, however, the values of pu are only of the order of magnitude of 0.01, and the correction involved generally amounts to only a few per cent. Examples of values obtained for some proteins and other macromolecules are given in Table I. Care must be exercised that the contribution to the depolarization arising from the solvent is properly eliminated in deriving the value of the depolarization ratio for the solute particle alone. Moreover, secondary scattering, which is negligible in liquids of low turbidity, may have a serious effect here. Secondary scattering is most disturbing when the solution has a high turbidity, and when the initially scattered light traverses a long path before emerging from the solution. Therefore, it can be diminished by narrowing the aperture of the incident beam, by diminishing the concentration of solute (and hence the turbidity), and by designing the scattering cell so that the path of the scattered light in the liquid is short. The work of Lontie (1944) on the dependence of the observed value of pu on the beam width and protein concentration for hemocyanin of Helix pomatia serves as a particulqrly striking illustration of the importance of such precautions. A discussion of the additional depolarization effects that occur when the particle is large enough to produce internal interference will be found in Sect. 111. It is clear from the data in Table I that all proteins yet studied in solution give low depolarization values. There is no correlation with the geometrical shape of the protein : the very asymmetrical molecules, myosin and tobacco mosaic virus, give lower pu values than many of the corpuscular proteins. This is what might be expected from theory: the pu value for a dielectric particle, such as a protein molecule, depends almost entirely on its optical anisotropy, scarcely at all on its geometrical shape. (Conducting particles, such as metallic sols, behave quite differently, but we are not concerned with them here.) A very useful discussion has been given by Lotmar (19384. In general it appears that the optical anisotropy of protein molecules is low, as might be expected from their compact structure, with many chemical bonds of similar strength oriented in various directions. This is qualitatively in accord with the rather low birefringence of most protein crystals, but quantitative data on such crystals are almost entirely lack-

+

+

48

PAUL DOTY AND JOHN T. EDSALL

ing. Streaming birefringence of proteins in solution also gives relevant information, but conclusions must be drawn with caution, since the observed effects are due to a superposition of form birefringence and intrinsic birefringence (see for instance Edsall and Foster, 1948). A discussion of the additional depolarization effects that occur when the particle is large enough to produce internal interference will be found in Sect. 111. TABLE I Depohrization Ratios, pu, for Scattered Light at go”, with Unpolarized Incident Light ’ Data for Proteins and Certain Other Macromolecules Substance Myosin (actomyosin) Myogen Hemoglobin Casein Gelatin Amandin Hemocyanin (Homarus) Hemocyanin (Sepia) Hemocyanin (He&) Horse Serum Albumin Bovine Serum Albumin Pig Serum Albumin Hernocyanin (Helix) Ovalbumin 8-Lactoglobulin Lysosyme Bovine serum albumin Bovine serum albumin Tobacco mosaic virus (in water) Tobacco mosaic virus (in neutral buffer) Polystyrene (in methyl ethyl ketone) Polyvinyl chloride (in dioxane)

P”

.0154 .0096 .0097 .011 .006 .0095 .0065 .0065 .0065 .022 ,022 .022 .0040 .024 .021 .030 .027 .02 .0074 ,012 ,040 ,015

Observer Lotmar (1938b) Lotmar (1938b) Lotmar (1938b) Lotmar (1938b) Lotmar (1938b) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Putzeys and Brosteaux (1941) Lontie (1944) Halwer, Nutting and Brice (1950) Halwer, Nutting and Brice (1950) Halwer, Nutting and Brice (1950) Halwer, Nutting and Brice (1950) Edsall, Edelhoch et al. (1950) Doty and Stein (1948) Doty and Stein (1948) Doty and Stein (1948) Doty and Stein (1948)

The values of Putzeys and Brosteaux, and of Lontie, are extrapolated to zero protein concentration. Secondary scattering sometimes leads to falsely high pu values at higher concentrations (see text). Doty and Stein have also made depolarization measurements on tobacco mosaic virus using incident light which is vertically or horizontally polarized; they discuss the theoretical significance of such measurements in some detail (see also Doty, 1948 and compare Oster, 1950b). Most of the measurements reported here were made a t Xo = 5780, 5460 or 4358 A. The influence of wavelength on pu appears to be very small, although it has a large effect on secondary scattering (Lontie, 1944).

5. Polydisperse Solutes and the Average Molecular Weight

I f the molecules of the macromolecular solute are not of identical weight, it becomes important to consider the type of average value that

LIGHT SCATTERING I N PROTEIN SOLUTIONS

49

is obtained by the light scattering method. It is well known that the osmotic pressure method, depending on a colligative property of the solution, responds equally to all solute molecules regardless of their weight, assuming they are nondiffusible-that is, unable t o pass the membrane. Such measurements, therefore, lead to what is known as a number average molecular weight given by the formula (14) Here ci denotes the weight concentration of the i’th component (in g./cc.); mi its molar concentration; and the summation extends over all the nondiffusible components. Because of the intimate relation between the osmotic pressure and the intensity of scattered light (Sect. I, 3) it might be expected that the same kind of average would be obtained in this case. That the analogy is erroneous is readily seen by considering an ideal solution: the intensity of scattering from each molecule depends directly upon the weight of the molecule in accordance with equation (6). This dependence upon the weight of each molecule rather than only upon whether it exists or not, leads to the definition of the weight average molecular weight, which is given by the expression:

The first demonstration that the average obtained is the weight average was given by Zimm and Doty (1944). Earlier indications that higher moments of the molecular weight distribution curve could also be obtained are not valid (Brinkman and Hermans, 1949; Kirkwood and Goldberg, 1950; Stockmayer, 1950). The dependence of light scattering results on the weight average has the consequence that minute quantities of particles having quite high effective molecular weights, such as dust and other foreign, colloidal suspended matter, will seriously affect the results obtained. The technical problems that this presents are treated in the final section. Light scattering represents the only method besides that of sedimentation equilibrium of obtaining weight average molecular weights. Strictly speaking, the weight average is obtained from light scattering measurements only if each solute species has the same value of the refractive index increment. This condition is generally satisfied in investigations of high polymers, but the distinction may be of some importance in protein mixtures, since the refractive index increment may differ some-

50

PAUL DOTY A N D JOHN T. EDSALL

what from one protein t o another (see e.g., Armstrong, Budka, Morrison and Hasson, 1947). 6. Molecular Weights of Proteins Determined by Light Scattering Comparatively few data on molecular weights are yet available, and only a few of the most recent have been determined on an absolute basis. The pioneer work in the determination of molecular weights of proteins was carried out by Putzeys and Brosteaux (1935). This work was continued and amplified during subsequent years, and the results were presented in detail by Putzeys and Brosteaux (1941). Their molecular weight data are summarized in Table 11. Amandin was taken as a standard protein, its molecular weight being assumed as 330,000 from sedimentation, diffusion and viscosity measurements, and the molecular weights of the other proteins were standardized with reference t o it. These authors found t h a t the turbidity/concentration ratio was a function of the concentration. For all the proteins they studied it could be described b y a n equation which in our notation may be written:

where (R90/Kc)o is the limiting value of the molecular weight calculated at zero concentration, and c is the concentration of protein in g./cc. The values of b ranged from 1.93 for amandin to approximately 2 for the albumins and 3.2 t o 4 for hemocyanins. The range of c studied was from 0.008 t o 0.080 for serum albumin, and somewhat lower for the proteins of higher molecular weight. The significance of the slope factor b is still obscure, in particular, its dependence on the square root of the protein concentration. For the present i t must be taken as a n empirical parameter. Most of the proteins studied by Putzeys and Brosteaux were observed near their isoelectric points so that it might be expected, in view of the discussion in Section 11, G, that the values of b would be nearly zero. However, the extrapolation t o zero protein concentration from their experiments appears quite unambiguous. The molecular weights given b y Putzeys and Brosteaux in Table I1 are in very satisfactory agreement with other data obtained from sedimentation, diffusion, viscosity, osmotic pressure and other methods; for tabulation of these data see for instance Svedberg and Pedersen (1940) and Cohn and Edsall (1943). Bucher (1947) studied a preparation of the crystalline mercury compound of the enzyme enolase, prepared in Warburg’s laboratory. He obtained a molecular weight of 66,000, using edestin as a standard protein and assuming its molecular weight t o be 300,000. The value

LIGHT SCATTERING

IN PROTEIN

51

SOLUTIONS

obtained by light scattering for enolase agreed well with the molecular weight calculated from the mercury content (one gram atom Hg per mol enolase). Campbell, Blaker and Pardee (1948) studied preparations of a rabbit antibody against para-azo-phenylarsonic acid. They obtained a molecular weight of 140,000 by osmotic pressure and 158,000 by light scattering. They calibrated their instruments against carbon disulfide, which was taken as a standard material, its reduced intensity being taken as for light of wavelength 5460 A. 4.4X TABLE I1 Molecular Weights of Proteins Determined by Light Scattering

-

Molecular Weight in Thousands from Scattering a t XO Calibration Protein 5780 A. 5461 A. method Observers

Ovalbumin Horse serum albumin Bovine serum albumin Pig serum albumin Hemocyanins : Palinurus vulgaris Homarus vulgaris ( A ) H . vulgaris ( B ) Sepia o$kinulis Helix pomutia Excelsin (Amandin) Edestin. Prunus seed globulins: P . Aviuma 1'. Cerasusa 1'. Domestics" Yeast enolase Rabbit serum antibody

(anti-pazophenylarsonic)

Ovalbumin @-Lactoglobulin Lysosyme Bovine serum albuminBovine serum albuminb Tobacco mosaic virus Bushy stunt virus Influenza virus

38.2 76.5 77.7 72.1

37.6 72.2 76.6 71.9

A A A A

1 1 1 1

461 630 733 3210 6340 28 1 (330) 335

464 617 689 31.50 276 (330)

1 1 1 1 1 1

-

A A A A A A A A

2 2

28&3 16 295 290 -

66 158

A A A B C

3 3 3 4 5

-

45.7 35.7 14.8 73 77 40,000 9,000 322,000

D D D D E F G G

6 6

-

-

-

Denotes preparations which were somewhat unstable and showed gate on standing. b Value determined at wavelength Xo = 4358 A.

ct

6 6

7

8 9

9

perceptible tendency to aggre-

52

PAUL DOTY AND JOHN T. EDSALL

Calibration Methods:

TABLE 11.--Continued

A. Molecular weights given relative to amandin as a standard protein which is taken as having a molecular weight of 330,000. B. Calibration with reference to edestin as a standard protein; assumed molecular weight of edestin 300,000. C. Calibration with reference to pure carbon disulfide as a standard. D. Calibration made on a n absolute scale; for details of calibration, see Brice, Halwer and Speiser (1950). Molecular weights reported in this reference were determined also at 4358 A. with results in close agreement with those at 5461 A. E. Calibration with reference t o pure benzene as a standard scatterer; reduced intensity of benzene assumed as 49 X a t 4358 A. F. Reduced intensity measurements a t 90" calibrated by direct turbidity measurements and corrected for angular dissymmetry of scattering. See further discussion of these data in Section 111-4. G. Determination by direct turbidity measurements. Observers: 1. Putzeys and Brosteaux (1941). 2. Beeckmans and Lontie (1946).

3. 4. 5. 6. 7. 8. 9.

Putzeys and Beeckmans (1946). Bucher (1947). Campbell, Blaker and Pardee (1948). Halwer, Nutting and Brice (1951). Edsall, Edelhoch, Lontie, and Morrison (1950). Oster, Doty and Zimm (1947). Oster (1946, 1948).

Recently Halwer, Nutting and Brice (1951) have reported particularly careful determinations, on an absolute basis, of the molecular weights of four proteins : /3-lactoglobulin, ovalbumin, lysozyme and bovine serum albumin (see Table 11). The instrument which they employed had been calibrated with particular care by Brice, Halwer, and Speiser (1950) (for the methods used, see Sect. VI, 4). Their reported molecular weights agree very well for the most part with the best values reported in the literature. The value 48,300 for ovalbumin is appreciably higher than most of the values obtained by other methods which range, in general, from about 40,000 to 46,000: (see Cohn and Edsall, 1943, and the references cited by Halwer, Nutting and Brice). All of their studies were carried out in dilute sodium chloride or phosphate buffer solutions, generally a t or near the isoelectric point of the protein. Lysozyme, however, was studied a t pH 6.2, where it carries a considerable positive charge, its isoelectric point being 10.5-11.0." Some of the data obtained, at various protein concentrations, are shown

* See the review on egg proteins by H. I,.Fevold in this volume, p. 187ff.

53

LfGfiT SCATTERfNG I N PROTEIN SOLUTIONS

in Fig. 1. It is notable that all of the slopes ( B values) for the proteins shown in this figure are slightly negative, suggesting predominantly attractive interactions between the protein molecules. For serum albumin the slopes were zero or slightly positive. A further discussion of the significance of the interaction constant B is given below in Sect. 11. Halwer, Nutting and Brice reported that their samples of bovine serum albumin underwent a slow change with time on storage in the dry

0

0.4

0.0

c

x

102

I.2

(g.lcc.1

FIG.1. Reduced intensity-concentration data for several proteins (Halwer, Nutting and Brice, 1951). 1.Lysozyme; 2. p-lactoglobulin; 3.Ovalbumin (preliminaryresults).

state even at low temperature. Also several of their samples, even when first studied, gave weight average molecular weights considerably higher than the number average figure of 69,000 obtained from the very careful osmotic pressure measurements of Scatchard, Batchelder and Brown (1946). One sample gave a value of 99,000 by light scattering when first measured, and this had increased to 237,000 a t the end of 18 months, although an osmotic determination at the end of this interval gave a number average of only 104,000. Some of the aggregated matter,

54

PAUL DOTY AND JOHN T. EDSALL

responsible for this change with time, could be removed by high speed centrifuging. These studies show the sensitivity of light scattering measurements to the presence of aggregates which would be almost undetected by osmotic pressure measurements. They emphasize the need for extreme care in preparing solutions before light scattering measurements are made, and also the value of light scattering as a technique for detecting changes which might otherwise pass unobserved. It should be added that not all serum albumin preparations undergo such alterations with time. Studies on bovine serum albumin have been carried out in the laboratories of both of the present authors. Different preparations were studied in the two laboratories, over long periods of time. I n neither case was any significant change in weight average molecular weight observed in solutions made up from samples of the same lot of dry material over a period of 18 months or more. The cause of the aggregation phenomena observed by Halwer, Nutting and Brice in their preparations is still not altogether clear. The moisture content of the preparations may be a relevant factor. Further studies on serum albumin by light scattering are reported below in Sects. I1 and V.

7. Specific Refractive I n d e x Increments of Proteins in Solution Accurate determination of the specific refractive index increment ( d n l d c ) of the solute is fully as important as the measurement of turbidity in the determination of molecular weights from light scattering. I n fact, since equations (6) and (12) involve ( d n / d c ) 2 , a small percentage error in this increment produces twice as large a n error in the calculated molecular weight as does the same percentage error in the turbidity measurement. Some of the most careful recent determinations of these increments for protein solutions are listed in Table 111. There are still important discrepancies, of the order of nearly 2%, between the values given for ovalbumin, /3-lactoglobulin, and bovine serum albumin b y Perlmann and Longsworth (1948) and b y Halwer, Nutting and Brice (1951). The discrepancies presumably arise from differences in the determination of the protein concentrations. The molecular weights given in Table I1 are derived using the refractive increments of Halwer, Nutting and Brice ; use of the values of Perlmann and Longsworth would diminish the calculated molecular weights by about 4 %. Various relations have been proposed for calculating the relation between the refractive index of a solution and the refractive indices of the pure constituents of which it is composed (for discussion, see Putzeys and Brosteaux, 1936, and Heller, 1945). No direct measurement of the refractive index of a pure anhydrous protein has, to our knowledge, been achieved; and for obvious technical reasons such a measurement would

55

LIGHT SCATTERING IN PROTEIN SOLUTIONS

be very difficult. From the calculations of Putzeys and Brosteaux, and of Heller, from data on protein solutions, the refractive indices of pure proteins should be fairly close to 1.60. This question, and its implications for the molecular weights calculated from light scattering, are further considered in Sect. 111. TABLE I11 Refractive Index Increments of Certain Proteins (c in g./ml. a t 25O) Protein Human serum albumin Human serum 7-globulin Human serum 7-globulin @-Lipoprotein @-Lactoglobulin Ovalbumin Bovine serum albumin Human serum albumin @-Lactoglobulin Bovine serum albumin Ovalbumin Lysozyme

dn/dc a t wavelength 5780 A. 5460 A. 4358 A. 0.186 0.188 0.1875 0.171 0,1842 0.1851 0.1869 0.1854

0.1890

0.1960

0.1856 0.1865 0.1883 0.1868 0.1822 0.1854 0.1820 0.1888

0.1926 0.1935 0.1954 0.1938 0,1890 0.1924 0.1883 0.1955

Observer 1 1 2 1 2 2 2 2 3 3 3 3

(1) Armstrong, Budka, Morrison and Hasson (1947). (2) Perlmann and Longsworth (1948). (3) Halwer, Nutting and Brice (1951). Most of the measurements were made in dilute sodium chloride, ( 0 . 1 4 . 3 M ) , pH between 5 and 7. I n general, dn/dc diminishes with rising temperature: (for data see Perlmann and Longsworth). For ovalbumin, serum albumin, and @-lactoglobulin,the values at 0" are higher than at 25" by 0.002-.003, but human 7-globulin is reported to show no change with temperature. For variation with wavelength, Perlmann and Longsworth find that the data for all the proteins studied by them are well fitted by the equation taking the value of the increment at 5780 A. as a point of reference: (dn/dc)x, = (dn/dc)mo(0.940 2.00 X 1 0 6 / x o s j which holds well throughout the visible region. The same equation probably gives a good approximation for most other proteins.

+

11. MULTICOMPONENT SYSTEMS OF SMALLMOLECULES: EFFECTS OF NET CHARGEAND OF IONICSTRENGTH ON TURBIDITY IN PROTEIN SOLUTIONS 1. New Phenomena Arising in Multicomponent Systems The previous discussion applies strictly only to two-component systems, or to systems containing polydispersed solutes for which the interaction coefficient ( B ) and specific refractive index increments are essentially independent of molecular weight. When three or more components

56

PAUL DOTY AND JOHN T. EDSALL

are present, and when strong interactions between them occur, new phenomena arise. Such effects were strikingly revealed by the studies of Ewart, Roe, Debye and McCartney (1946). They studied, among other materials, a polystyrene sample, which in pure benzene gave a limiting value of Kc/R90 = 3 X 10-6 a t c = 0, but in a benzene-methanol mixture containing 15% methanol by volume gave a limiting value of Kc/Rgo near 1.4 X Moreover, the interaction constant, B , was numerically much smaller in the latter medium than in the former. Benzene is, of course, a good solvent for polystyrene, whereas methanol is not a solvent and acts as a precipitant a t sufficiently high concentration in benzene-methanol mixtures. These authors explained their results in terms of selective adsorption of benzene by the polystyrene molecules, so that the solvent composition in the neighborhood of a large solute molecule is different from that in the bulk of the solution. They showed that this would lead to a change in the value of R W / Ca t infinite dilution, if the two components of the solvent had different refractive indices. If the refractive index of the two-component solvent medium was essentially independent of composition-as was true, for example, for a butanone-isopropanol mixture-then the limiting value of Kc/RgO was found to be independent of the composition of the solvent, even though one component of the solvent (in this case isopropanol) tended to act as a precipitant. By contrast it should be noted that the limiting value of the osmotic pressure/concentration ratio is independent of the composition of the solvent in any of these cases. The general correctness of the picture given by Ewart, Roe, Debye and McCartney is clear. However, a more general thermodynamic theory of light scattering in multicomponent systems, originally outlined by Zernike many years ago, has recently been formulated by several authors. It is of particular importance in its application to protein solutions, because proteins are multivalent acids and bases which can acquire high net charge by the addition of hydrogen or hydroxyl ions to the solution, and can also bind other anions, cations, and neutral molecules very strongly. Effects of net charge and of ionic strength in multicomponent systems containing proteins are therefore very marked. The definition of the components in a system containing charged proteins involves certain complexities which must be carefully considered. Much of the detailed discussion in the following part is taken directly from Edsall, Edelhoch, Lontie and Morrison (1950). 2. Definition of Components

We shall follow Scatchard (1946) in the definition of components, denoting the solvent as component 1, the protein-if only one is present-

LIGHT SCATTERING IN PROTEIN SOLUTIONS

57

as component 2, and the salt with diffusible ions-if only one is presentas component 3. Other protein components, if present, may be denoted by higher even numbers, and other diffusible components by higher odd numbers.* All components are taken as electrically neutral; hence we distinguish between the protein ion and the protein component. The valence (2,) of the protein ion is defined as the mean net proton charge per protein ion, ZZbeing zero for the isoionic protein. It is thus equal to the mols of “bound acid” (ZZpositive) or “bound base ” (2,negative) per mol of protein, determined by a titration with the hydrogen or glass electrode, with suitable corrections (see for instance Cohn and Edsall, 1943, Chapter 20; Tanford, 1950). An albumin, soluble in water in the absence of salt, may be taken as approximately isoionic (2, = 0) after thorough electrodialysis has removed all diffusible ions except H+ and OH- from the so1ution.t If a neutral salt is added to such a solution, it remains isoionic by definition, although the salt addition may cause the electrophoretic mobility to become positive or negative, due to selective binding of cations or anions by the protein. The value of 2 2 is adjusted to positive or negative values by addition of strong acids or bases. We shall consider later the effects of binding of other ions by the protein, which may make the effective net charge of the protein ion considerably different from ZZ. The specification of molecular weight of the protein, in the definition of the protein component, is largely a matter of convenience. What is essential is that the mass and chemical nature of the protein component (or components) added t o the system should be definitely specified. The value of Zz must, of course, always be expressed so that it is stoichiometrically correct; the mean number of protons bound to, or removed * The term “diffusible” in this connection, denotes ability to pass through a membrane impermeable to molecules as large as typical proteins. t If the only ions present in this solution are H+, OH- and protein ions, it is clear that the mean net charge on the protein cannot be emctly zero unless the isoionic point happens to coincide with the pH of neutrality. Thus, if the pH of the electrodialyzed solution is 5, we have [Hf] = 10-6 M , [OH-] negligible, and the protein must carry a small negative charge to balance the excess of H+ over OH- ions. For a serum albumin solution, concentration 7 g./L (1O-l M ) this requires that 22,for the electroinstead of zero. The difference is well dialyzed solution, be -0.1 ( = - 10-5/10-4), within the usual experimental error, but it can be corrected for if necessary. The correction obviously becomes more important, the more the isoionic point of the protein deviates from the pH of neutrality and the more dilute the protein solution. Scatchard and Black (1949) define an isoionic material as one which gives no noncolloidal ions other than hydrogen and hydroxyl. Thus, by their dehition, electrodialyzed albumin is isoionic when Zzmz = (OH-) - (H+), m2 being the molar concentration of protein ion; while we have defined it as isoionic when Zz = 0. In practice, the difference between the two definitions is generally negligible.

58

PAUL DOTY AND JOHN T . EDSALL

from, the isoionic protein, per gram protein, must obviously depend only on the stoichiometric composition of the system, not on the assumed molecular weight. For instance, Edsall, Edelhoch, Lontie and Morrison (1950) assumed the value of 69,000 for the molecular weight of their serum albumin preparations. The value of Z2 which they employed, therefore, represented the mols of H+ ion bound or removed, per gram protein, multiplied by the factor 69,000. The protein component, like all the other components, is so defined as to be electrically neutral. If the protein is being added to the system as an ion of valence ZZ,we must add a t the same time some diffusible ions of charge opposite in sign to 2 2 , or remove some ions of the same sign of charge as ZZ,or do both of these things, in such a way that the total increment of net charge is zero. Certain amounts of the diffusible ions in the system are thus assigned to the protein component, the amounts being positive if the ion in question is added, negative if it is removed, when the protein ion is added to the system. The protein component might be defined so as t o contain one mole of protein ion and Z2/Zi moles of the diffusible ion of opposite charge t o This is perhaps the 2 2 , where Zi is the valence of this diffusible ion. most obvious definition, but it has the serious disadvantage that the addition of one mole of protein component involves adding 1 (Zz/Zi) moles of ions to the system ; the resulting effect on the chemical potential of the solvent would be chiefly due to the diffusible ions added, not to the protein ion which primarily concerns us. Therefore, following Scatchard (1946), we choose another definition, which involves the net addition of only one mole of ions to the system per mole of protein component. If only two diffusible ions are present, and both are univalent, this requires adding Z2/2 moles of the diffusible ion with sign of charge opposite to that of the protein, and removing 2 4 2 moles of the diffusible ion with the same sign of charge as Z z , when we add one mole of protein ion to the system. Thus the net addition of moles of diffusible ions is zero, and the solution remains electrically neutral after the protein component is added. Consider the specific case of serum albumin at a concentration of molar (approximately 7 g./l.), in a solution to which 20 X moles/l. of hydrochloric acid has been added, so that Z2 = +20; 0.010 mol/l. of sodium chloride has also been added t o the solution. Thus the total concentration of diffusible ions in mols per liter is (Na+) = .010; (Cl-) = .012. (The concentration of free hydrogen and hydroxyl ions is assumed to be negligible in comparison.) The first definition discussed in the preceding paragraph would include in component 2, per liter solution, 0.0001 mols protein ion (valence +20) and 0.0020 mols chloride

+

LIGHT SCATTERING IN PROTEIN SOLUTIONS

59

ion. Component 3 (sodium chloride) would then contain .010 mols of both Na+ and C1- ion. By the second definition, which we shall employ in the following discussion, component 2 includes 0.0001 mole protein ion, 0.001 mole C1- ion, and minus 0.001 mols Na+ ion. Hence, by this definition, component 3 consists of 0.011 mols of both Na+ and C1- ion. Obviously, when we add all the constituents of all the components together, the resulting sum must give correctly the actual composition of the system. It is clearly meaningless t o ask whether (for example) any individual chloride ion, chosen a t random, belongs t o component 2 or component 3; but it is essential that the sum of the number of chloride ions assigned to these components, by any definition, should equal the number of chloride ions actually present. Consider first the simple case in which there is only one protein component, and the mean valence of the protein ion is ZZ. Assume the solution to contain in addition to H+ and OH- ions (at negligible concentrations), only one other salt, containing only one kind of cation and one kind of anion (although one mol of salt m.ay contain two or more mols of either cation or anion or both). Let m2 be the molar concentration of protein; then Zzmz is its equivalent concentration as cation or anion, with the appropriate sign. Then the algebraic sum of the total molar concentrations in the solution of other cations and anions, multiplied by their valences, must be equal numerically, and opposite in sign, to Zzmz. Some of these ions must be assigned to the protein component in order to satisfy the conditions for this component stated above. Let Z, be the valence of the cations of the added salt, Z, that of the anions. Let vzC be the number of moles of diffusible cations per mol of protein component, and vza the number of moles of diffusible anions. These numbers may be calculated as follows. T o fulfill the requirement that there shall be one mol of ions added to the system per mol of protein component, we must have: v2c = - v z e (17) and to fulfill the requirement that the protein component is electrically neutral, the relation must hold that: ZcVZe

+

ZaVZa

combining with (17) above, we have:

=

- 2 2

(18)

60

PAUL DOTY AND JOHN T. EDSALL

Thus, if the total molal concentration of diffusible cation in the solution is m,, and the total molal concentration of anion is ma, and if one mol of the salt (component 3 ) contains vQc mols of cation and va0 mols of anion, then the molal concentration of component 3 is:

These equations imply nothing concerning the nature of the forces between the protein io? and the other ions in solution; they serve simply to define the components stoichiometrically. Other definitions of the components could of course be employed. From the definition given here, it is apparent (equation 19a) that either V Z , or vza must be negative, except when both are zero. This system of definition is useful only when m3 is fairly large compared to vzcm2(or vzamz); some of the possible alternative definitions are discussed by Scatchard (1946). If another salt, containing a cation X+ and an anion Y-, is added to the system, then the ions X+ and Y- are taken as forming a new component (component 5 according to our conventions). If they are present in equivalent concentrations in the solution, these ions are not considered as forming part of the protein component.

3. Thermodynamic Fluctuation Theory for Multicomponent Systems In a multicomponent system the mean square value of the fluctuation in refractive index which determines the total turbidity, is a function of the concentrations, and refractive index increments, of each of the components of the system. It also involves cross terms, involving the correlation between the fluctuations of the concentrations of different components. For a given pair of components, i and j, the cross term is zero if the chemical potential of i is unaffected by a variation of the mass of j in the system; but if the chemical potentials are not independent in this manner the cross terms do not vanish. The fundamental conceptions involved in the application of fluctuation theory to such systems were first advanced by Zernike (1915, 1918). Recently they have been further developed independently by Brinkman and Hermans (1949), Kirkwood and Goldberg (1950), and Stockmayer (1950). The fundamental equations of all these authors lead to the same results. The general equation, for the reduced intensity, R ~ oof, systems a t constant pressure and temperature, may be written :

(m2,

LIGHT SCATTERING I N PROTEIN SOLUTIONS

61

where

The summation is taken over all but one of the components. Generally it is most convenient to omit the solvent from the summation; to compensate for this omission the complete equation should include a term for the turbidity of the pure solvent, arising from density fluctuations in it. This term is generally small, in systems containing large molecules, and for brevity it is omitted from (21). However, in practice, we have generally determined the turbidity of the pure solvent and subtracted it from that of the solution. !Pi denotes the molar refractive increment of component i ; that is, An per mol of solute per liter of solution. The a In a, terms aij, in the determinant [aijl denote the coefficients aij = -

a In a,

= aji; here the a's denote activities, and the m's denote concentraami tions in mol/l. The term Aij, in the summation in the numerator denotes the cofactor of the term ai, in the determinant IaijI; that is, the determinant derived from jaijl by striking out the row and column in which the term aij occurs, and multiplying the resulting determinant of low-er order by 1 if i j is even, and by - 1 if i j is odd. Our equation (21) differs from Stockmayer's (1950) only in that we have employed activities rather than chemical potentials and the volume V does not appear, since the concentrations are here expressed in volume units.

+

+

+

4. Application to a Two-Component System For a two-component system, the summation in (21) involves only a In a2 By definition J and Aij = 1. component 2. Hence Iaijl = a22 = am2 of the activity (az)and the activity coefficient (YZ): In Hence,

a2 =

In mz

+ In yz = In mz + P Z

62

PAUL DOTY AND JOHN T. EDSALL

Here \E2 = an/am2. Rearranging (24))we may then write:

Since cz as :

= _ m 2_ M 2 this ~

1000

gives the interaction constant B in equation (12)

where p 2 2 0 is the limiting value of 0 2 2 at low values of m2. The derivation given here shows that for a two-component system the limiting slope of the curves for Kc2/Rguis directly proportional to the effect of change in the concentration of component two on the logarithm of its own activity coefficient ( B E ) . This coefficient is by equation (26) directly proportional to the interaction constant B , the significance of which has been discussed a t length in Sect. I, 3. The treatment given here shows how the previous discussion may be directly correlated with the activity coefficient and its derivatives. 5 . Systems of Three and More Components

For multicomponent systems in general, we shall follow Scatchard's (1946) formulation of the expressions for the activities of the components and their derivatives with respect to the masses of the components-that is, the coefficients which enter into equation (21). We shall use the subscript K to denote any component made up of small molecules or ions, and i t o denote a small ion which is a constituent of one or more components, but is not itself a component." Again denoting the protein as component 2, we have: In a2 In

a K

=

In m2

+ 2,vZ2 In m, + In ( Z J V J ~+J v 2 m ) + P Z + PK

+

/j2

= In m2 Zv2, = Z v K z In m, =

+

In (BJvJzmJ v2,m2>

Z V K ~

Here, by definition: P2

PK

=

In

y 2

= ZivKi

+

/jh

(27) (28)

In y s

I n the special case of a three-component system, in which component 3 is a salt composed of one anion and one cation, me have p3 = 2 In y3. * This notation should not be confused with our previous use of the symbols i and j to denote components.

LIGHT SCATTERING IN PROTEIN SOLUTIONS

63

From these relations, we derive the coefficients employed in equation

apz

apK

(21), denoting -as Pz2, __ amz amz

apz

as PZR,and so forth. amK

= __

Here J denotes any diffusible component other than K . For the three-component system containing one salt (component 3) with two ions, the summation denoted by Bi includes only one cation and one anion, and vaC = vSa = 1. From (19a) and (19b) we have vzc = - v Z a = -Z,/(Z, - Za)> which becomes -Zz/2 if the ions denoted by c and a are both univalent. Then, using equation (20), we have:

m, = m3 - -zz mz 2

zz m z ma = m3 4-2

(34) (35)

From (31) we obtain:

Here the factor

E

is defined as:

The value of e lies always between zero and unity, and it may be taken as unity* when Zzmz << 2m3.

* I n t h e extreme case when t h e only diffusible ions present are those required t o balance the net charge on the protein, e becomes equal t o zero. Under these circumstances, the application of our equations would become meaningless and a different, definition of components should be adopted. (Systems in which t approaches zero are discussed in Sect. V.) The general treatment adopted in the present section implicitly assumes t h a t E is not very far from unity.

64

PAUL DOTY AND JOHN T. EDSALL

From (32), (34) and (35) we obtain: V2c

V2a

a23=-+-+1923= m, ma

-=+ Z2Zm2

P23

Finally for the effect of variation in the log of the activity of component

3 with variations in its own molarity we have, from (28), (34) and ( 3 5 ) :

The value of Pa3 is determined independently, from measurements of electromotive force, freezing point, or vapor pressure, in solutions of the pure salt. Applied to the three-component system just discussed, equation (21) becomes: %lo

=

- 2*2*3a23 a22a33 - a 2

K'(Q~~a33

+ *32a22)

(40)

For protein systems, the first term in the numerator of (40) is generally much the largest, and the second and third may often be neglected. This point will be considered in terms of the system water-serum albuminsodium chloride. The molar refractive increment, q2,of serum albumin, per liter solution, is, from the data of Perlmann and Longsworth (1948) and of Armstrong, Budka, Morrison and Hasson (1947)) equal to 12.9 for the sodium D line, assuming a molecular weight of 69,000. The data for sodium chloride solutions, a t the same wavelength when extrapolated to Thus *z2 is greater than 2\E2!P3 infinite dilution, give * 3 = 9.5 X by a factor of 680, and exceeds * 2 by a factor of nearly 2 X l o 6 . On the other hand, we must consider the relative magnitude of the coefficients a22, a Z 3and a 3 3 . A critical evaluation of these terms for the case of serum albumin solutions in sodium chloride and sodium thiocyanate solutions (Edsall, Edelhoch, Lontie and Morrison, 1950) shows that the second and third terms in the numerator of equation (40) are ordinarily less than 2% of the first term. Hence, it is justifiable as a good approximation t o neglect the two latter terms and the equation then becomes

We shall consider here only the limiting condition in which 122m21 = 1 to a close approximation, and az3 p 2 3 . Then (for

<< 2m3,hence e

LIGHT SCATTERING IN PROTEIN SOLUTIONS

65

notation compare equation (24)) :

Thus, under these limiting conditions, the slope 2B of the curve for Kcz/R90 as a function of c2 becomes identical with the slope of the curve for osmotic pressure divided by concentration of protein as a function of c2, given by Scatchard, Batchelder and Brown (1946; see their equation 9). The interaction constant which multiplies c2 in equation (43) is seen by this analysis to be subdivisible into three terms. The first of these, which is proportional to Z22/2m3,may be called the "Donnan term," since it is mathematically identical in form with the term arising from the Donnan equilibrium distribution across a membrane in an osmotic pressure experiment (Scatchard, Batchelder and Brown, 1946, and Wagner 1949). Since this term is proportional to ( Z 2 / M 2 ) 2it depends only on the ratio of valence to mass for the protein, and is independent of its molecular weight. Being inversely proportional to ms, this term is largest at low salt concentrations. If all activity coefficients are unity, it is the only term which contributes to B in a three-component system. The second term in the parenthesis, involving ' 2 2 , is formally identical with the B value given in equation (26) for a two-component system. However, it must be noted that since the definition of the protein component changes with the valence of the protein in a three-component system, ' 2 2 must be regarded as a function of 2 2 ; it is, of course, also a function of the salt concentration and other variables. The third term in the parenthesis of equation (43) involves the influence of the protein on the activity coefficient of the salt or the thermodynamically equivalent influence of the salt on the protein, given by the term &a. Note that the contribution of this term to the slope since this factor appears as /hs2, is always negative. Often, however, this term is very small compared to the other two.

6 . Experimental Data for Bovine Serum Albumin The molecular weight data for the serum albumins have already been discussed in Sect. I, 6, and summarized in Table 11. A systematic study of the effect of variation in the valence ( 2 2 ) of the protein, and of the ionic strength, on the value of the interaction constant B , has been

66

PAUL DOTY AND JOHN T. EDSALL

carried out by Edsall, Edelhoch, Lontie and Morrison (1950). Most of their work covered a range of Z 2 from +20 to -20, corresponding t o p H values between 4.2 and 7.4. I n different series of measurements, the added salt (component 3) was sodium chloride, sodium thiocyanate, or calcium chloride. The range of ionic strengths studied was from 0.003 to 0.183 in most experiments; a few studies were made a t 2 2 = +25 and to at very low ionic strengths A systematic study of osmotic pressures of bovine serum albumin solutions had already been carried out, over the same range of 2, values,

7’

0

600-

X

N U

E

400200

o 0.15 NaCl

-

o 015 N a C l 0 0.18 NaGl

-(Scatchord) OSMOTIC PRESSURE

0 -

1

t20

+I0

0

z,

-10

-20

FIG. 2. Comparison of the interaction constant B (equations 17 and 18) for bovine serum albumin, from osmotic pressure and from light scat,tering measurements. Slopes are expressed as BM2*/1000,where M z is the molecular weight of the protein. The abscissa gives Z f , the net proton charge per molecule of albumin. Solid line from osmotic pressure data of Scatchard, Batchelder and Brown. Points from light scattering measurements (Edsall, Edelhoch, Lontie and Morrison, 1950).

in sodium chloride solutions, a t ionic strengths near 0.15 (Scatchard, Batchelder and Brown, 1946). Thus a comparison between osmotic pressure and light scattering data could be made under these conditions. Figure 2 shows the values so obtained for the interaction constant B by the two methods. The values obtained from osmotic pressure, and from light scattering are in generally good accord, as the argument given above mould predict. The values from light scattering generally lie a little below those from osmotic pressure, but the differences are hardly to be considered significant, especially as two different preparations of bovine serum albumin were employed in the two series of studies.

LIGHT SCATTERING IN PROTEIN SOLUTIONS

67

I n light scattering studies, where only a single solution is studied a t a time and there is no need of establishing equilibrium between two phases across a membrane, it is relatively easy to make reproducible measurements a t high values of net charge on the protein and at very low ionic strengths-conditions under which osmotic pressure readings tend to be very unstable and lacking in reproducibility. Figure 3 shows values for Kc2/Rg0in serum albumin, when Z 2 = +25 and the ionic strength ranges or less, to 0.15. If the Donnan term, involving Z22/2m3,were from

0

0

ODOl

0.002 Concentration in g,/cc.

I

0.003

FIG.3. Values of K c z / R ~ as o a function of cp, for bovine serum albumin with a net proton charge of +25, at several different concentrations of sodium chloride.

the only term contributing t o the interaction constant, we should expect a very large value of B a t extremely low ionic strengths, which would decrease in proportion as the ionic strength (in this case practically equal to m 3 ) increases. Qualitatively this is just what is observed. The limiting slope (2BMZ2/1000)in the absence of added salt is of the order it falls rapidly as salt is added,* of lo5at the lowest ionic strength * The increase of turbidity which occurs on adding salt, corresponding to a transition from a point on the top curve to one on the bottom curve of Fig. 3, occurs with extreme rapidity. The final steady state is attained certainly in less than a minute, the time taken for mixing albumin with salt solution and taking a reading in the light scattering apparatus.

68

PAUL DOTY AND JOHN T. EDSALL

and becomes practically zero at 0.15 M sodium chloride; however in all cases, the extrapolated value of the molecular weight at c2 = 0 is the same within the limits of error. At such high values of 2 2 , and at ionic strength below 0.001, measurements of scattering as a function of angle should reveal a dissymmetry less than unity, an effect which is discussed in Sect. V. However, the studies of Edsall, Edelhoch et al. were confined, for the most part, to ionic strengths of 0.003 and above, where these special effects do not appear. The variation of the interaction constant with the valence ( 2 2 ) of the protein, and the ionic strength is shown in the lower half of Fig. 4, where data are plotted for albumin in sodium chloride at four different ionic strengths. These curves are roughly of the parabolic form that would be expected from the Donnan term alone, but it is clear that other terms are important. The absolute magnitudes of the slopes are less, often much less, than would be predicted from the Donnan term, so that other effects of opposite sign must enter in. The minimum of the curve is found to lie not a t 2 2 = 0, but always at positive values of 2 2 . Moreover, as the ionic strength of the sodium chloride increases, the minimum shifts progressively over to the left in Fig. 4. It must be noted that 2, is calculated only in terms of the protons bound to or removed from the isoionic albumin molecule, and takes no account of the binding of other ions. If it is assumed that chloride ions are also bound, the progressive shift of the minimum with increasing ionic strength is readily interpretable. There is now a large mass of evidence from quite independent measurements that chloride ions are bound by serum albumin; other ions such as thiocyanate are even more strongly bound. The best quantitative evidence for such binding is probably that given by Scatchard, Scheinberg and Armstrong (1950). They have described their data in terms of the assumption that an albumin molecule contains 40 sites capable of binding C1- or CNS- ions; 10 of these are assumed to be all alike, each with an intrinsic association constant ( K ) of 44 for C1-, or 1000 for CNS-; the other 30 bind much more weakly, with K values of 1.1 for C1-, and 25 for CNS-. These intrinsic binding constants must be corrected for electrostatic effects arising from the variable net charge on the protein; Scatchard, Scheinberg and Armstrong found that they could be described by a simple spherical model for the albumin ion, using the Debye-Huckel theory. Taking their values for chloride binding at different ionic strengths, we may calculate a new quantity, the total net charge on the albumin, denoted as 2 2 * , which is equal to Zz - V, where V is the number of chloride ions bound per mole of albumin. When the slopes are plotted against Z2* as in the upper section of Fig. 4, the minima in the different curves are all brought nearly into coincidence and

LIGHT SCATTERING IN PROTEIN SOLUTIONS

69

6000'

4000

-

L)

Q X

2

(YN

2000NaCl

r/2

0.003

0 0 0.010 8

0.033

o 0.183

0>

+20

+I0

0

-10

-20

70

PAUL DOTY AND JOHN T. EDSALL

they lie very nearly a t Zz* = 0. Conversely me could have used the displacements of the minima in the curves of the lower section of Fig. 4 as a measure of the number of chloride ions bound. Much greater shifts are found in thiocyanate than in chloride solutions, and here again the light scattering data give results in accord with the calculated number of ions bound by the protein, as deduced from the entirely independent measurements of Scatchard, Scheinberg and Armstrong. Thus, light scattering measurements are a useful tool for the study of specific ion binding by proteins. Studies on calcium chloride solutions indicated considerable binding of calcium by serum albumin, especially a t negative Zz values, such as are found in the physiological p H range. 7 . Interaction Efects in r-Globulin Solutions, and between r-GlobuEin and Serum Albumin Most of the effects of charge and ionic strength, as already described for serum albumin, must be in their general features characteristic of other soluble proteins. Studies of a human 7-globulin preparation, however, by Lontie and Morrison (1947) revealed certain phenomena not apparent in albumin solutions. While albumin is-except perhaps a t p H values distinctly acid to its isoelectric point-electrophoretically homogeneous, all r-globulin preparations hitherto prepared contain a large number of components of differing isoelectric points, distributed more or less symmetrically about a mean value near p H 7. Thus, there is no true isoelectric point, but rather a broad isoelectric zone, for the preparation as a whole. These phenomena have been particularly well characterized by the work of Alberty (1948). Acid t o p H 6, almost all the molecules carry a positive net charge; alkaline to pH 8.5, almost all are negatively charged. Under these conditions light scattering measurements give results qualitatively similar to those in albumin solution; values of KcIR90, when plotted as a function of c, yielded large positive slopes a t lorn ionic strength, the slope decreasing approximately to zero a t ionic strengths between 0.1 and 0.3 (sodium chloride). I n the isoelectric zone, however, especially between pH 6.5 and 7.8, the slopes become moderately large and negative a t low ionic strength. This is most naturally explained as due to electrostatic interactions between positively and negatively charged r-globulin molecules, leading to association. In a preparation containing so many electrophoretic components, it is still uncertain whether the association reactions involved most of the molecules in the system, or only a small fraction of them. The preparation studied had been dialyzed for two days, and the insoluble euglobulins removed; however it had not been electrodialyzed, and, therefore, cannot be considered as composed entirely of pseudoglobulin. The association

LIGHT SCATTERING I N PROTEIN SOLUTIONS

71

may not involve formation of definite complexes, but only increased probability of approach between protein ions of opposite charge. In any case the phenomena observed at low ionic strength are rapidly reversed by addition of salt. The weight average molecular weight of this preparation was well above 200,000, whereas the number average of such preparations has been found to be 156,000 (Oncley, Scatchard and Brown, 1947). The occurrence of markedly negative B values at low ionic strength, from light scattering measurements, may serve in general as an important

=or

1-12

0.001

0.003 0.0I 0.03 0.10

0.30

01

0

1.0

2x)

3.0

FIG.5. Reduced intensity-concentration data for mixtures of serum albumin and 7-globulin at pH 5.6 and varying ionic strengths. Albumin-globulin ratio 3 : l by weight. For infinite dilution the ordinate values give molecular weights directly. The arrow indicates the weight average molecular weight. Measurements of R. Lontie and P. R. Morrison.

clue to the presence of interacting positively and negatively charged protein molecules in the system. Electrostatic interactions between serum albumin and 7-globulin were strikingly revealed in an experiment of Lontie and Morrison, who mixed these two proteins in a weight ratio of 3: 1 at pH 5.6, the albumin molecules at this pII being negatively, and the 7-globulin molecules all positively charged. Each protein preparation alone, at this pH value and at low ionic strength gave a positive value of the interaction constant B ; but the curve for the solution of the two together, as a function of total protein concentration, gave a decidedly negative slope at low ionic strength. As the ionic strength increased, the slopes of the curves

72

PAUL DOTY AND JOHN T. EDSALL

diminished and in 0.3 M sodium chloride the system behaved almost like an ideal solution. The results of the experiment are plotted in Fig. 5 ; note that the ordinate of this figure shows the reduced intensity concentration ratio directly, rather than its reciprocal. It has already been remarked (Sect. I, 3) that markedly negative values of B, such as those which would characterize the data of Fig. 5 at low ionic strength, indicate incipient phase separation. The albuminy-globulin solutions studied here did not form precipitates on standing in water at pH 5.6, even at low ionic strength; but precipitation in such systems was readily brought about by addition of ethanol at low temperature, in amounts quite insufficient to precipitate either albumin or y-globulin alone (Cohn, Gurd, Surgenor et al., 1950). Thus light scattering may be used for the detection of incipient phase separation in protein solutions, as in other systems, and may prove a useful guide in exploring the possible conditions under which a protein fractionation is to be carried out.

8. Studies on the System Bovine Serum Albumin-Urea-Water Recently this interesting multicomponent system was studied by Doty and Katz (1950). At pH 5, corresponding closely to the isoelectric point of the albumin, the curves for Kc/Rgo extrapolated to the same limiting value at zero protein concentration for all values of the urea concentration. Acid or alkaline to the isoelectric point, however, the extrapolated values were not the same. The presence of high urea concentrations caused positive displacements at low pH and negative displacements at high pH, indicating preferential adsorption of water a t low pH and urea at high pH. The amount of preferential adsorption became very large when the urea molality was in the vicinity of 8 M . At this urea concentration and a t pH values of 3 or 8, it appeared that several thousand molecules of water and of urea, respectively, were preferentially adsorbed per molecule of albumin. It has been commonly believed that the action of concentrated urea caused the folded protein chain of a corpuscular protein molecule to unfold into an elongated shape. It is, indeed, true that serum albumin shows a large increase in specific viscosity and a marked decrease in diffusion constant in such solutions. Doty and Katz, however, could find no evidence of significant angular dissymmetry in the scattered light from such solutions ; hence, the dimensions of serum albumin molecules in urea were still small compared with the wavelength of the light. In view of this and the large adsorption of solvent it appears that the principal change undergone by the serum albumin molecule in concentrated urea solutions is that of approximately isotropic swelling.

LIGHT SCATTERING I N P ROT EIN SOLUTIONS

73

111. INTERNAL INTERFERENCE AND THE DETERMINATION OF SIZE 1. Introductory Considerations

Our treatment up to this point has been restricted to scattering by particles that are small compared with the wavelength of light. This restriction was imposed in order that the scattering from individual molecules could be treated as radiation from point sources. This is a valid procedure when the particles are quite small, but when their largest dimension exceeds approximately one-twentieth of the wavelength of light the consequences of a spatial distribution of scattering elements must be considered. Thus, the path length transversed by light scattered from different parts of the same molecule will no longer be the same;

FIG.6. Illus1,ration of the dependence of destructive interference on the scattering angle. Note the larger path length difference for light scattered in the backward angle (AB BC - AC) than in the forward angle (AB BD - AD).

+

+

consequently, optical interference will occur. This point is of major importance because it leads to a diminution of the intensity of the scattered light, the effect increasing with the scattering angle, 0, as illustrated in Fig. 6. Since the molecular weight relations (equation 12) do not take this internal interference into account, it is obvious that some correction must be applied to R9,,in order to preserve the usefulness of the relation. On the other hand the amount of interference is directly’ related to the distribution of matter in the particle and hence is a function of the size and shape of the particle. As a result an examination of the angular variation of the intensity of scattered light will provide, if the particles are large enough, a measure of their absolute dimensions. It will be seen in the following parts of this section that it is relatively easy to deal with internal interference if one neglects the distortion of the electric field of the incident light beam by the scattering particle due to the difference between its refractive index and that of the medium. This neglect becomes increasingly serious as the size of the particle and its refractive index relative to the medium increases. The error introduced by this approximation will be most pronounced when the particle has the

74

PAUL DOTY AND JOHN T. EDSALL

most compact geometrical form, that of a sphere. Fortunately, the problem of scattering from isotropic spheres of arbitrary size and refractive index has been exactly solved by Mie (1908) and, therefore, the accuracy of the approximate treatment can be assessed in the case where the maximum error can occur. Detailed consideration of this matter will be found in part 4 of this section. 2. Spherical, Rod-like and Coiled Macromolecules: The Dissymmetry Method

It has been seen that the effect of internal interference which we wish to compute in a quantitative fashion has its origin in the phase shifts encountered by rays scattered from different parts of the same molecule. If we assume that the electric field of the incident beam is not significantly distorted by the different refractive index of the particle, this computation is purely a geometrical task involving the tabulation of the decreased intensity of scattered light that will result for the interference of rays from every pair of volume elements in the scattering particle for every value of the angle 0. This problem was solved long ago in connection with X-ray scattering (Debye, 1915; see e.g., James, 1948) : if the diminution relative to the intensity of scattering in the absence of internal interference at the angle 0 is denoted by P ( 0 ) the general solution is given by

where the double summation extends over all pairs of scattering elements, i a n d j : k = 2?r/X, X being the wavelength of light in the solution, s = 2 sin 0/2, and ri, is the distance between the ith and j t h scattering element. If this expression can be brought into a simple, explicit form for molecular shapes in which we are interested, it will provide the solution to the problems posed by the existence of the interference effect. Thus the value of Rsocan be multiplied by the reciprocal of P(90) in order to restore the validity of the molecular weight relation; and the value of a characteristic dimension can be found by fitting the observed angular intensity distribution with the expression for P ( 0 ) . The derivation of an expression for P ( 0 ) , which may be called the particle scattering factor, for $he case of spheres was obtained first by Rayleigh (1911) and later by Cans (1925). When this factor is introduced into the Rayleigh equation (equation 6) the result is known as the cos2 0). The partiRayleigh-Gans scattering law: Rs = K c M P ( 0 ) (I cle scattering factor for rod-like particles was obtained by Neugebauer

+

LIGHT SCATTERING IN PROTEIN SOLUTIONS

75

(1943) and that for randomly-coiled, chain moIecuIes by Debye (1947, see also Zimm, Stein and Doty, 1945). These expressions are:

ksD 'sin'\x coil

ksL

2

P(6) = - [e-2 2 2

where D = diameter of sphere, L = length of rod and R = root mean square of the distance between ends of the random coil. A plot of these particle scattering factors is given in Fig. 7. It is pertinent t o note that

I

(fiFOR

COIL)

FIG.7. Particle scattering factors for spheres, rods, monodisperse and polydisperse randomly kinked coils.

when 8 = 0 the particle scattering factor is unity because there is no phase shift in the direction of the incident beam. This same problem presents itself in low-angle X-ray investigations: in this connection particle scattering factors have been derived for ellipsoids of revolution (Guinier, 1939, 1943; Shull and Roess, 1947; and Roess and Shull, 1947), cylinders of revolution (Fournet and Guinier, 1950), discs of negligible

76

PAUL DOTY A N D JOHN T. EDSALL

thickness (Kratky and Porod, 1949). However, almost all other forms will require two or more parameters to characterize the particle and their evaluation is then not unique. For example, ellipsoids of revolution mill require the specification of the length of both axes and assignment of values to them from light-scattering data will involve the same ambiguity that arises, for example, in the interpretation of intrinsic viscosity. Using hypergeometric functions Roess and Shull (1947) derived an expression for P(O),for a system containing ellipsoids of revolution, with the ellipsoid axes distributed at random. Recently Debye (personal communication) has derived an equivalent formula. If the ellipsoid semiaxes are a, a, and c, and if p is defined by the relation, p = (c2 - a2)/a2, then Debye's formula is: n

m

- 4 4725 H ( 1 + 9 +3 y + $ ) + .

....

(45d)

This equation holds for all values of p from rods ( p = m ) to discs ( p = -1). Numerical data derived from (45d) are conveniently represented by the procedure of Zimm (1948b) in which the reciprocal relative intensity, l y e ) , is plotted against (ksa)2-or, for a specific wavelength and a particular molecule, against sin2(O/2). For prolate ellipsoids of axial ratio 5.1 ( p = 25), the limiting slope, dP-1(0)/d(ksa)2is 1.87. If the axial ratio is 10, the limiting slope is 6.86; if the axial ratio is 20,the slope is 26.86. Detailed computations show that the curves so plotted remain nearly linear up to fairly high values of (ksa)2-the higher terms in the summation rapidly become large as ksa increases, especially for p values of 100 or more, but the net result of summing positive and negative terms deviates surprisingly little from the simpler linear relation. Debye has also derived the first terms in the expansion of P(0) for ellipsoids with three different semi-axes, a, b, and c :

p(e) = 1 -

--

5

+5[(T)2+(!!)2+&&)j) 2 a2 - b2

If a = b, this reduces to the expansion of (45d).

- . .. . . .

LIGHT SCATTERING I N PROTEIN SOLUTIONS

77

Let us turn now to the actual use of the particle-scattering factors given in equation (45). From Fig. 7 it is seen that the scattering factors decrease monotonically from a value of unity; because of this it is possible to use the intensity of scattering from any two angles to determine the dimension of the particle giving rise to this particular degree of interference. It has become fairly standard practice to choose the angles of 45 and 135" for measurement and the ratio of these intensities is known as the dissymmetry, z. As an example of these relations consider a rodlike macromolecule having a length equal to 0.416X. The argument x will then have values of 1.000, 1.848, and 2.414 at 0 = 45, 90, 135", respectively. From Fig. 7 the corresponding values of P(0) are found t o be 0.895, 0.708, and 0.582. Consequently z = 0.895/0.582 = 1.538, and the correction factor for Rgo is 1/P(90) = 1/0.708 = 1.412. Proceedilig in this fashion the dimension relative to X and the correction factor can be calculated as a function of the dissymmetry for each of the particle shapes. The results are shown in Fig. 8. At finite concentrations the intensity of the scattered light is generally altered due to solute interaction as characterized by the constant B as well as by internal interference effects. These dual effects cannot be rigorously resolved but to a good approximation it can be shown (Zimm, 1948) that c

K -P(90) Rso

=

1

+ 2BP(90)~

Therefore, when B is not negligible the value of P(90) is determined from the value of z, extrapolated to zero concentration, this correction factor is used to multiply the observed values of Kc/R90 which in turn are plotted as a function of concentration as in Sect. I. The slope of the resulting line is interpreted as indicated in equation (46). This procedure for obtaining the molecular weight, interaction constant, and dimension is known as the dissymmetry method. Two other methods of obtaining the same information from light scattering have been proposed. 3. The Extrapolation Method and the Transmission Method The more important of the other methods is based upon the fact that the effects of internal interference vanish a t 0 = 0 and, therefore, if scattering data are obtained over as large a range of scattering angle as possible, an extrapolation to zero angle is feasible (Zimm, 1948a). Such a procedure obviously makes the widest use of the scattering phenomenon and offers the greatest potential accuracy as well as a certain independence from assumptions of particle shape. An analysis of this method by Zimm has shown that the data are best plotted as Kc/Re against kc where k is an arbitrary constant chosen to spread out the sin2 (0/2)

+

78

P A U L DOTY A N D JOHN T. EDSALL

L 0.

0

0

D -

x

0

0

1

FIG.8

Correction factors and particle dimensions as a function of the dissymmetry (R450/R1360).

LIGHT SCATTERING I N PROTEIN SOLUTIONS

79

data. Smoothed lines are drawn through the points at constant angle and constant concentration thereby forming a grid. Each of the lines is then extrapolated to zero angle or zero concentration and these extrapolated points are then joined into two lines that should meet at the same intercept of Kc/Rs at zero concentration. The intercept will be Kc/Ro and its reciprocal is equal to the molecular weight if the incident light is vertically polarized. If unpolarized incident light is used, the molecular weight is half the reciprocal of the intercept, due to the inclusion of the cos2 0). The' value of B is obtained from the slope of the zero term (1 angle line and the dimension can be calculated from the ratio of the slope

+

O

20

.40

.$o sin*

.I30

I.'O

:1.

I

+ 100 c

FIG.9. Reciprocal reduced intensity and turbidity plots for tobacco mosaic virus in 0.1 M phosphate buffer.

of the zero concentration line to the intercept. A plot of this type is shown in Fig. 9 for a partially aggregated.sample of tobacco mosaic virus in 0.1M phosphate buffer. The reciprocal of the intercept shows the molecular weight to be 51,000,000: the zero angle line has zero slope indicating that B = 0. The length is calculated to be 2650 A (Doty and Steiner, 1950b). The additional effort and care required in the extrapolation method raises the question as to the conditions under which it is markedly superior to the dissymmetry method and when it may be equally satisfactory to use the latter method. The extrapolation method always has some advantage in that it presents a more complete picture of the scattering. For example, it may indicate the presence of a second component of large size by showing a rapid increase of scattering at low angles. This may be an impurity that has not been removed or it may be a com-

80

PAUL DOTY AND JOHN T. EDSALL

ponent whose presence is important. In either event this conclusion could only have been obtained by the dissymmetry method in an indirect manner or would have been overlooked. On the other hand, the advantage of the extrapolation method with its independence of molecular weight assignment on the choice of particle shape exists only in the range of very large particles. It can be seen in Fig. 8 that below dissymmetry values of about 1.4 the correction factor curves corresponding to the different particle forms are essentially coincident. It is in this range of x = 1 to 1.4 that the simplicity and speed offered by the dissymmetry method recommend its use. Although this range appears limited it probably includes all but the most asymmetric proteins. In Section I it was pointed out that the measurement of the reduced intensity R,, was equivalent to measuring the turbidity for small particles. The reduced intensity was favored because the turbidity of solutions of such particles is generally too small for practical measurement. With larger particles this may not be the case, but internal interference may then occur and the question arises of how to deal with it when making only measurements of transmission. The solution of this problem (Doty and Steiner, 1950b) lies in making measurements of the turbidity a t different wavelengths. It is shown that as the interference increases the exponential wavelength dependence decreases from a limiting value of 4 and in general plays the role of the dissymmetry in that method. Thus graphs similar to Figs. 7 and 8 can be constructed and from them the correction factor and dimension can be determined from the wavelength dependence of the turbidity. Although this method requires only a precision spectrophotometer, in practice it is not capable of as high accuracy as either of the other methods. The transmission method just described is analogous to the dissymmetry method in many respects. The counterpart of the extrapolation method also exists. The fundamental parameter in the analysis of interference is the quantity x of equation (45) which depends on both B and D/X. In the extrapolation method one attempts to reach x = 0 (where interference disappears) by extrapolating to 0 = 0. Alternatively one may alter the wavelength and thereby extrapolate to D/X = 0. This approach has been explored with respect to transmission methods by Cashin and Debye (1949) but appears to be severely restricted in application by the relatively short range of wavelength values in which the solution is nonabsorbing. However, when internal interference is small, a direct extrapolation of the turbidity against is practical since the region of Rayleigh scattering is then either within or near the range of accessible wavelengths. The precise implementation of this procedure is usually limited by the necessity of having refractive index increment

LIGHT SCATTERING I N PROTEIN SOLUTIONS

81

data at all the wavelengths measured. The application of this method to influenza virus is discussed in the next section. As indicated before the matter of depolarization requires additional treatment when internal interference exists. We shall consider here only the case of transversely scattered light. The scattered light can be resolved into a vertical ( V ) and a horizontal ( H ) component. The ratio of these two components can be determined under three different, but not unrelated conditions: those in which the incident light is unpolarized, vertically polarized, and horizontally polarized. Denoting these conditions by subscript initials, the following depolarization ratios may be defined:

In macromolecular solutions the V , component is always many times more intense than any other: indeed, for small isotropic spheres this is the only component in the transversely scattered light. With small anisotropic particles the three other components are found with relatively low but equal intensities. With larger particles, however, internal interference will cause a diminution of the intensity which is more effective in reducing H , and v h components than the H h component. Consequently, ph assumes values of less than unity and is, therefore, related t o the size of the scattering particle. The value of pv is related t o the anisotropy of the scattering particles in the same way as previously discussed for p u . Indeed, when ph = 1, pu = 2 p v . However, when Ph # 1, the value of pu depends upon both p,, and Ph. But only the contribution of p v results from the anisotropy of the molecule and it is only this contribution that should be used in the Cabannes correction. This provision is properly made by a substitution of 2 p v for p u in the correction factors given in Sect. I, 4 (Zimm, Stein, and Doty, 1945). For a further discussion of the relation of the depolarization ratios to the properties of macromolecules the reader is referred to the literature. Sinclair and LaMer (1949) discuss the depolarization of isotropic spheres in terms of the Mie theory; considerable data on macromolecules are given by Lotmar (1938b); a detailed discussion and review is given by Doty (1948) and studies on tobacco mosaic virus are reported by Doty and Stein (1948).

4. Experimental Investigations of Solutions Exhibiting Internal Interference Although there are undoubtedly many common proteins which give rise t o internal interference in solution, such as the larger serum globulins and fibrinogen, only a few investigations have been published and these

82

PAUL DOTY AND JOHN T. EDSALL

are in most cases only preliminary in nature. The viruses, in particular, offer a rich field for studies based on interference methods because in nearly all cases the sizes are in the proper range. However, no exhaustive studies have yet been published although fairly complete examinations of tobacco mosaic virus and influenza virus have been made. Tobacco mosaic virus can be prepared in fairly homogeneous form, and it has been well studied by different physical methods. For these reasons it was chosen for the first investigation of light scattering from rod-like macromolecules (Oster, Doty, and Zimm, 1947). This virus is known t o have the form of rods of uniform thickness of 150 A. The uniformity of its length is still debated but the sample used in this investigation had a relatively narrow length distribution with a weight 1.0

1.61

0

I

I

I

2

I

CONC.(g./CC.

FIG.

I

3

4 X

I

5

I

6

lo4)

10. Dissymmetry measurements for tobacco mosaic virus in aqueous and buffer solutions.

average of 2710 A. The dissymmetry measurements in water and buffer solutions are shown in Fig. 10 from which a limiting dissymmetry of 1.94 was obtained a t the angles of 49" and 1 3 1 O . From plots similar t o Fig. 7 for these particular angles, one finds that this dissymmetry corresponds to L/X = 0.66. Since A = 5461/1.33 = 4090 A., L = 2700 A. This compares well with the weight average determined from the electron micrograph and a value of 2600 A. derived from the intrinsic viscosity. The limiting value of Kc/Rsowas found to be 1.51 x 10+. The correction factor, 1/P(90), evaluated from the limiting dissymmetry was 1.66. It follows that the molecular weight of this sample was 40,000,000, likewise in good agreement with the electron micrograph value and other measurements. Recently a less homogeneous sample of tobacco mosaic virus has been investigated in buffer solution (see Fig. 9) by all three light scattering

LIGHT SCATTERING IN PROTEIN SOLUTIONS

83

m3h& (DotY and Steiner, 1950). The values obtained spread Over a range of 10% but this was understandable in view of the absolute errors involved and the probable heterogeneity of the sample. In 1946 Oster published turbidity measurements on solutions of influenza virus (PR 8 strain) and bushy stunt virus. The measurements extended up to XO = 10,000 A., and in this limit interference effects were essentially absent for the bushy stunt virus but probably caused an error of 10% ’ for the influenza virus. Molecular weights of 322 and 9.0 million respectively were obtained. These results are in good agreement with other investigations. It was also shown in this paper that the turbidity of virus solut’ions without added salt went through an abrupt discontinuity at the isoelectric point and that this provided a sensitive way to determine the isoelectric point. This behavior is consistent with that discussed on pages 65-70. Later (Oster, 1948) dissymmetry data were published for the same influenza virus from which a diameter of 1280 A. was determined compared with a value of only 970 A. on the basis of the foregoing molecular weight measurement together with density data. Two reasons may be advanced to explain this disagreement. One arises from the fact that the particle diameter is too large for interference to be neglected even at l o= 10,000 A. Even a t this wavelength a correction factor of 1.1 would be necessary, thereby increasing the value of the molecular weight derived from the turbidity data. On the other hand, the particle diameter is overestimated when dissymmetry data are interpreted according to the Rayleigh-Gans approximation because the particle size is so large. This point is discussed further in Part 5 of this section. Reports of other investigations are much less complete. Dissymmetry measurements have been made on an actomyosin preparation by Jordan and Oster (1948). The results were interpreted as indicating that the actomyosin becomes more highly coiled upon the addition of adenosine triphosphate. Recently Mommaerts has studied the light scattering of purified myosin solutions. These were prepared by Mommaerts and Parrish (1951), and were homogeneous in the ultracentrifuge. The molecular weight from light scattering was about 850,000, in very close agreement with the measurements of Portzehl (1950) in H. H. Weber’s laboratory a t Tiibingen, which gave 858,000 from sedimentation and diffusion, and 840,000 by osmotic pressure. The length of the molecule obtained from angular dissymmetry measurements by Mommaerts (1950) was about 1500 A., while Portzehl obtained a greater length, 2000-2400 A. The explanation of t,his discrepancy is not yet clear. Portzehl, Schramm, and Weber (1950) have studied the turbidity of

84

PAUL DOTY AND JOHN T. EDSALII

actomyosin, and found it to be greater than that of myosin (L-Myosin in their terminology) by a factor of nearly 10. The turbidity of actomyosin solutions a t a given concentration was also found by Mommaerts to be much higher than for myosin; and the actomyosin solutions showed much greater angular dissymmetry of scattering. The effect of ATP on dissymmetry was found by Mommaerts to be variable, but it always reduced the total turbidity, in agreement with the concept that ATP dissociates the actomyosin into actin and myosin. These studies will be discussed in more detail in a review by Weber and Portzehl in the succeeding volume of this series. Light scattering investigations have been made on a rod-like particle from human red-cells (Dandliker, Moskowitz, Zimm and Calvin, 1950) and on a pathological human serum globulin (Edsall and Dandliker, 1950). In both cases complete angular intensity curves between 20 and 144' were determined. In view of the polydispersity of the samples, the agreement between the light scattering results and the ultracentrifuge studies was satisfactory. In the case of the pathological globulin the angular dependence data could be accounted for equally well by thin rods 563 A. long or by spheres 414 A. in diameter. The spherical model is excluded because it would require a molecular weight of 30 million. The observed molecular weight, 1,160,000, is consistent with the rod model since its diameter can be appropriately chosen. The frequently occurring difficulty of finding satisfactory solvents for fibrous proteins is an obstacle to investigations in this direction. However, some progress is being made in studies of tropomyosin, gelatin, polypeptides and synthetic polymeric electrolytes in the Gibbs Memorial Laboratory. Perhaps the scattering from solutions of denatured proteins can be interpreted in terms of the coil model. With regard to other chainlike molecules of biological interest mention should be made of sodium thymonucleate. Smith and Sheffer (1950) have investigated several preparations and note the extreme practical difficulties of cleaning the solutions. The molecular weights of three different samples in dilute salt solution were greater than 3.7 X lo6. The dissymmetries observed were approximately 4.0 (X = 5460 A.). Somewhat smaller values have been found by Oster (1950a, b) and Bunce and Doty (1950) for a sample prepared by Dr. D. 0. Jordan, Nottingham University. It, would appear from these preliminary data that the sodium thymonucleate molecule is considerably larger than ultracentrifuge and diffusion measurements indicate and that it is gently coiled, the chain stiffness being comparable to that of cellulose derivatives. Most studies of chain molecules have been directed toward high polymers and many extremely interesting results have been obtained

LIGHT SCATTERING IN PROTEIN SOLUTIONS

85

(Ewart, Roe, Debye and McCartney, 1946; Doty, Affens, and Zimm, 1946; Debye and Bueche, 1948; Zimm, 1948; Outer, Carr, and Zimm, 1950) but the results are too extensive t o review here. 5. The Mie Theory for Spherical Particles: Applications Let us return now t o a consideration of the consequences, thus far neglected, that result from the distortion of the field of the incident beam by the particle. It is to be expected that this distortion will depend on the ratio of the refractive index of the particle t o that of the medium, a quantity denoted by m. In the limiting case where m = 1 there would be no distortion, but this would be of little interest because no scattering would occur. For larger values of m it does not appear possible to evaluate the effect of the distortion alone. Fortunately, however, an exact solution has been carried out for isotropic spheres of any size and relative refractive index (Mie, 1908). Therefore, a comparison with Mie’s results permits a precise assessment of the error introduced by use of the approximate solution used here for the case of spherical particles. The exact solution of the problem for other particle shapes is prohibitively difficult. The solution for spheres, though limited, is particularly valuable because it corresponds to the case where the maximum error might arise due t o the use of the approximate solution. In Mie’s treatment the total radiation field of the spherical particle is represented by dipoles, quadripoles, and higher poles having the center of the sphere as their origin. Their amplitudes and phases are derived exactly from solving Maxwell’s equations in this particular context. The general solution of the problem is obtained in rather complicated form in terms of the relative refractive index, m,and the quantity CY which is equal to 2 r / X times the radius. The comparison falls into two separate parts. One involves the possible error occurring in the absence of internal interference, that is for very small particles or at angles close to 0 = O”, due only t o the distortion related to the relative refractive index. The other concerns errors that may follow from this same effect in the particle scattering factor (equation 45a) for spheres. In other words the Rayleigh-Gans scattering law may be in error particularly for larger spherical particles such as influenza virus and an estimate of this error is highly desirable. Consider first the case of small spherical particles with relative refractive index m. The expression for R ~from o Mie’s theory is

86

PAUL DOTY AND JOHN T. EDSALL

If this is equated to equation (Ga) we have after rearrangement

where the refractive index and density of the particle are represented by p. If equation (49) is a precise representation of the relation between the refractive index of the solvent, solute, and solution when the solute particles are of spherical form the two expressions for Rgo are equivalent and the use of an experimental determination of Rsoin either equation (6a) or (48) will lead to the same molecular weight. Equation (49) is simply a “refractive index mixing law’’ for solutions of small spherical particles and may be verified or rejected by relatively simple refractive index studies. Heller (1945) has surveyed the relevant data and come to the conclusion that n’ as given by equation (49) deviates from observed values in proportion to (n’ - no). Now one finds that the value of n’ calculated by equation (49) from data on protein solutions is about 1.610. According to Heller this value is too large by 0.011. The use of the value of 1.599 in equation (48) leads to a molecular weight value 8% higher than that which would result if the value of 1.610 were used. Thus if Heller’s analysis is correct one must expect that the molecular weights derived for proteins by the use of equation (Ga) may be in error up to 8% depending on how closely a spherical shape is approached.. This error would be reduced if the value of m could be lowered by increasing the refractive index of the solvent; but such a procedure is seldom practical with protein solutions, and it would in any case have the disadvantage of diminishing the intensity of the observed scattering. On the other hand, it is important to emphasize that this value of the estimated error applies strictly only if the protein molecules are spherical. Although no quantitative theory exists, it is most probable that the more asymmetric the form of the molecule the less will be the distortion of the field of the incident beam and the less will be the error. Indeed, it seems unlikely that any significant error occurs from this source when equation (Ga) is applied to rod-shaped and coiled macromolecules with specific refractive index increment values comparable to those of aqueous solutions of proteins. However, even less concern is felt in this matter when the data used by Heller are examined critically, for it appears that in none of the investigations of spherical particles, such as sulfur sols, were the three refractive indicies carefully determined. Consequently it is our feeling that equation (49) has not

n‘ and

LIGHT SCATTERING IN PROTEIN SOLUTIONS

87

been proved incorrect experimentally, but that some degree of reservation must be made on the precise molecular weight determination of proteins until a thorough experimental examination of this relation has been made. Until this matter is more thoroughly investigated it appears that molecular weight determinations of proteins by light scattering methods, using equation (Ba), must be viewed with the reservation that they may be several per cent too high. One further point deserves consideration. Equation (6a) holds in general only if the particles are isotropic and if the shift in phase of the incident wave as it traverses a particle is very small. However, by a slight modification of the treatment given by Schuster and Nicholson (1928) we find for the case where p.. is zero in t h e forward direction, but where the size and refractive index are arbitrary, that as c -+ 0,

For all ordinary cases (e.g., a protein with a molecular weight of a few million or less) the second term in the bracket is negligible compared to the first and the equation reduces t o equation (6a). I n general, however, for large refracting particles the second term cannot be neglected. Physically, the first term gives the intensity scattered in phase with the incident wave, while the second term gives the component scattered with a phase difference of 90" to the incident wave. Thus a negligible phase shift means t h a t the second term can be ignored. We are indebted to Professor Bruno H. Zimm for pointing these relations out to us.

The other source of error to be noted is that occurring in the particle scattering factor for spheres (Rayleigh-Gans scattering) because of the assumption that there is no distortion of the field of the incident beam due to the difference in refractive index between the particles and the medium. This basic assumption requires in other words that neither the phases nor the intensities of the incident and outgoing waves for any scattering element in the particle are appreciably changed by the presence of the other elements. An analysis of the Rayleigh-Gans derivation (see van de Hulst, 1946, for example) shows that this assumption amounts to the condition 2a(m - 1) <
88

P A U L DOTY AND JOHN T. EDSALL

index as a protein in water and having a radius of 1300 A. With the blue mercury line this would exhibit a dissymmetry (R45/R135)according to the Mie theory of 10.9 and according to the Rayleigh-Gans approximation a value of 5.9. Thus the use of the Rayleigh-Gans equation for the determination of the diameter of spherical protein molecules should be restricted to spherical proteins having a diameter of less than about one-fourth of the wavelength: this corresponds to a dissymmetry of about 1.5. The foregoing discussion shows that the use of the more elaborate Mie theory is unnecessary for the interpretation of the dissymmetry of protein solutions, except for the extremely large spherical macromolecules. However, the wide range of size and refractive index which the Mie theory covers has led to its application in numerous problems involving scattering by spherical particles considerably larger than macromolecules. When a is less than approximately 0.5, the scattering diminishes monotonically with increasing angle in the customary manner. In this range the particle weight can be obtained from absolute intensity measurements or from dissymmetry measurements, since a determination of the radius is equivalent to a determination of the molecular weight in the special case of spherical particles. Larger particles, however, give rise to maxima in the angular scattering curve. With increasing size a maximum first appears at low angles and moves toward higher angles as a second maximum emerges at low angles. When there is only one maximum in the radiation envelope its angular location determines the particle size, for a given value of m:with several maxima a more careful fitting of the intensity data may be required in the determination of the particle size. Instead of analyzing angular intensity data, arising from irradiation with essentially monochromatic incident light, use may be made of a rather spectacular effect resulting from the use of incident white light. If the particles are of nearly uniform size the interference effects that occur with white light result in the original intensity maxima being replaced by different colored bands that appear to repeat themselves upon examining the scattering as a function of angle. The origin of these bands is obvious if one considers that the angles at which the intensity maxima occur are functions of the ratio of the diameter of the spherical particles to the wavelength of the light in the medium. If the size of the spheres is nearly uniform, the wavelength is the only variable and the maxima and minima for red scattered light will occur a t quite different angles than those for green scattered light. When a maximum for red scattered light coincides with a minimum for green scattered light a red band will be observed. This aspect has been particularly well investigated by LaMer and collaborators (Johnson and

LIGHT SCATTERING I N PROTEIN SOLUTIONS

89

LaMer, 1947; Kenyon and LaMer, 1949) using monodisperse sulfur sols. The number of similarly colored bands in the radiation pattern of the solution is a sensitive measure of the particle size: as many as nine similarly colored bands have been found in sulfur sols. Similar effects have been observed in bacterial suspensions (Lanni and Campbell, 1948). If these effects are to be clearly observed the preparation must be reasonably monodisperse ; polydispersity leads to overlapping of the maxima and minima from particles of different sizes, and blurring of the colored bands. In addition to absolute intensity and angular intensity distribution measurements the size of spherical particles can also be determined by depolarization measurements or by transmission measurements. The interpretation of these are likewise based on the Mie theory. In the work of LaMer previously mentioned it has been shown that these several independent optical observations lead to the same value of the particle radius. I t is also of interest to note the recent investigation by Dandliker (1950) of the particle size of a very homogeneous polystyrene latex which is being used extensively as a secondary standard for calibrating electron microscope magnification (Williams and Backus, 1949). In this study the particle diameter was found to be 2720 A. both by reduced intensity measurements at different angles, extrapolated to e = 0, and by locating the angular position of the single minimum in the radiation pattern. This compares with a value of 2590 A. determined from electron micrographs. The difference of 5 % is almost within the probable experimental error of the two determinations. Finally attention should be drawn to Dandliker’s useful analysis of the effects of polydispersity in application of the Mie theory. One further digression on applications of the Mie theory to colloidal systems may be of interest. In terms of this theory the turbidity is best considered in terms of the extinction El which is the ratio of the scattering cross section to the geometric cross section, i.e., the ratio of the energy lost by the incident wave to the energy of the beam that is geometrically obstructed or intercepted by the spherical particle. The geometrical obstruction presented by a sphere is, of course, Pr2. The ratio, El (assuming that the refractive index is known) allows the determination of both particle size and concentration from transmission data. Therefore, assuming that the particles scatter independently, the turbidity is given by T = E d v (51) where v is the number of particles per cc. By measuring T as a function of A and plotting T versus k-l, one can compare the experimental curve

90

PAUL DOTY AND JOHN

T.

EDSALL

with a plot of E as a function of a. Since the maxima in both curves occur at the same value of a,the value of r is readily determined. Once r is known, a n absolute turbidity measurement suffices to find v from equation (51). A more specific application of biological interest follows from noticing in Fig. 11, where E is plotted as a function of a, that E is approximately independent of particle size for values of a greater than unity. Thus for particles corresponding t o a > 1 the turbidity is

d

a for m = 1.33 (van de Hulst, 1946). Here r being the radius of the spherical particles.

FIG.1 1 . Extinction as a function of 01

= %-/A,

approximately independent of size and depends only on the concentration. NOWmost bacteria which are approximately spherical in form have diameters of about 10,000 A. and correspond t o a value of about 10. Consequently the turbidity of a dilute suspension of bacteria can serve as a means of counting bacteria. This conclusion has been ingeniously applied b y Bonet-Maury (1948) in studying the modulations of the multiplication-time curves of various virulent bacteria as a function of the kind and amount of antibiotic added t o the culture. IV. STUDYOF REACTIONSINVOLVING MACROMOLECULES BY LIGHT

SCATTERING

Reactions involving the association or dissociation of macromolecules are particularly suitable for study by light scattering methods. The speed and precision with which the state of the system may be determined a t any moment make this a n admirable method for study of kinetic measurements. If the reaction studied attains a well-defined equilibrium light scattering is also very well adapted for the determination of the equilibrium state, since the measurement can be made without disturbing the system. The possibilities of work in this field are enormous and exploration of them has only begun. Oster (1947) has given a discussion of light scattering in polymerizing and coagulating systems which treats some of the problems encountered in this field; but the total range of types of reactions that can be studied is very broad indeed. It seems probable that light scattering measurements will be particularly useful for studying reactions in which the number of bonds formed or broken

IJGHT

SCATTERING IN PROTEIN

91

SoLurIoNs

is relatively small and the size of the molecules is large. may be given here. 1. Insulin

A few examples

It is now well known that insulin exists in several forms of different molecular weights, and that these are in a more or less mobile equilibrium with one another. A molecule of molecular weight 12,000 appears t o be a

P

n

I

0

0.2

I

0.4

0.6

I

0.8

1.0

I 1.2

1.4

I

1.6

1.8

I

2.0

2.2

GONC. g. / 100 cc.

FIG.12. Reduced intensity-concentration data for insulin solutions at different acid pH values.

fundamental unit. It can associate to form a trimer of molecular weight of 36,000 and perhaps also a tetramer of 48,000.* A good summary of the situation has been given by Tristram in the previous volume of this series (Tristram, 1949, page 132-133). As yet there are no published light scattering data on insulin. Nevertheless, the type of results that may be expected are shown in Fig. 12 from preliminary work by Doty and Rabinovitch (1949). Since there is no dissymmetry in this system, measurements of the transverse scattering

* Fredericq and Neurath (1950) have recently calculated from sedimentation and diffusion measurements that molecules of molecular weight 12,000 may dissociate into half molecules (MW 6,000).

92

P A U L DOTY AND JOHN T. EDSALL

suffice;however, below pH 2 fibril formation (Waugh, 1948) begins and this is readily detected by an increase in dissymmetry. The data shown are for a sample of crystalline beef insulin. This work is currently being repeated, because of the greater precision and accuracy now attainable. Despite this short-coming, however, the qualitative nature of the equilibrium is clear. The equilibrium shifts toward the monomer with decreasing concentration and with diminishing pH. The equilibrium constant can be calculated from the weight average molecular weight, which in turn can be determined as a function of concentration from the data in Fig. 12 subject only to the accuracy with which the value of B can be assigned. I n this case it appears that the value of B for the trimer can be determined from data at high concentration where the trimer form predominates. The value of B for the monomer will be the same (page 65). The equilibrium constant can then be calculatpd and the data analyzed to see if the existence of a dimer can be detc ed. Unfortunately the type I!? of analysis. accuracy of the present data does not justify ?? The light scattering data confirm the view that, at least in acid solution, the association and dissociation reactions of insulin are very rapid; the new equilibrium state, when the system is altered by changing the pH or salt concentration, is reached in much less time than it takes to mix the solutions and take a reading in the light scattering apparatus. 2. Ovalbumin Bier and Nord (1949) have studied the apparently spontaneous aggregation of freshly filtered, salt-free solutions of ovalbumin at pH 4.2, that is within the usually accepted stability range. Whereas the turbidity increases linearly with time, the rate of this increase, coinciding with the progress of the aggregation] was found to be particularly dependent on the concentration and temperature of the solution. In the absence of dissymmetry measurements it is not possible to decide whether this is a general condensation coagulation or whether only a relatively few molecules are uniting to form large aggregates.* Riley and Herbert (1950), from low angle X-ray diffraction studies of ovalbumin solutions, also inferred that aggregation phenomena were occurring at high albumin concentrations. Bier and Nord reported a molecular weight of 37,000 for ovalbumin but in the absence of detailed information concerning calibration and other essential matters it is not possible to assign a high degree of precision to this determination.

* Oster (personal communication) has pointed out that the approximately linear increase of turbidity with time in Bier and Nord’s experiments indicates a general “condensation coagulation.” If clumping of a few large aggregates occurred, the curve should be non-linear (see G. Oster, J . Co22oid. Sci. 2, 291 (1947)).

LIGHT SCATTERING I N P ROT E IN SOLUTIONS

93

3. The Reaction between Human Serum Albumin and Mercurials

It was discovered by Hughes (1947) that the major portion of human serum albumin undergoes a reaction with mercuric chloride which can lead to the formation of a dimer containing two albumin molecules per atom of mercury. The fraction of human serum albumin which is capable of undergoing this reaction has been designated by Hughes (1950) as mercaptalbumin, since he has produced decisive evidence that the reaction involves a sulfhydryl group in the protein molecule. Furthermore the evidence is clear that rnercaptalbumin contains only one such group per molecule. The reactions involved may be written as follows, if we designate albumin with its sulfhydryl group as Albb-SH: Alb-S-H Alb+-HgCl

+ HgClz + Alb-SH

Alb--S-Hg-S-Alb

ki

~

k--1

Alb-S-HgC1

+ H+ + C1-

Alb-S-Hg-S-Alh

+ HgC1,

k3

k-a

2Alb-S-HgCl

+ H+ + C1-

I I1

111

The kinetics of the reaction have been studied in detail by light scattering measurements, although only preliminary reports have so far appeared (Lontie, Morrison, Edelhoch and Edsall, 1948; Hughes, Straessle, Edelhoch and Edsall, 1950). The results show quite decisively (see also Hughes, 1950) that: kl>> k-l and kl>>Icz Thus reaction I1 is the rate limiting step, and the formation of the dimer, according to this reaction, is readily followed by light scattering. Reaction I11 is too rapid to measure by present light scattering techniques, being essentially complete in less than a minute a t room temperature. Measurements at three different temperatures indicated that the rate of reaction 11 is somewhat more than doubled by a 10' rise of temperature; thus there is no indication of an unusually high energy of activation. Viscosity measurements show that Alb-S-Hg-S-Alb is more asymmetric than Alb-SH, but shorter than two moles of Alb-SH joined end to end. This suggests that the two albumin molecules must be approximately in juxtaposition over a considerable area, involving a face of each monomer molecule of which the sulfhydryl group forms a part. Since reaction I11 is rapid, the mercury linkage must be readily accessible to small ions and molecules in the solvent, rather than being completely shielded by other portions of the protein. Added halide ions, or SH compounds, displace reaction I1 to the left, the effect being in

94

PAUL DOTY AND J O H N T. EDSALL

the order RSH > I- > Br- > C1-. Silver ion appears t o compete with mercury for the SH group; all other metallic ions tested were without effect. Other mercury derivatives are now being studied in this reaction. The nature of the equilibrium in these reactions is such that the maximal formation of dimer occurs when one half mole of mercury is added per mole of mercaptalbumin. When more mercury than this is added, the dimer dissociates according t o reaction I11 above. Figure 13 shows the percentage of dimer a t equilibrium as determined by light scattering measurements in a 10 per cent aqueous solution of serum

I

0

0.5

HgCldAlbumin

I .o

5

FIG.13. Equilibrium between monomer and dimer forms of human serum mercaptalbumin in the presence of varying proportions of added mercury (Hughes, 1950).

mercaptalbumin. The character of this curve is excellent evidence that mercaptalbumin contains only one sulfhydryl group per mole of protein. Straessle (1951) has employed another type of mercurial which also interacts with mercaptalbumin t o form a dimer. This molecule consists of a dioxane ring with two CH2Hg+groups projecting from carbon atoms on opposite ends of the ring. When one molecule of this compound is added t o two moles of mercaptalbumin, dimer formation a t p H 5 is extremely rapid-much faster than with mercuric chloride. At p H values above 6 the reaction is slow enough t o measure, but is still very fast as compared with the reaction with mercuric chloride. Both types of reaction are reversed by addition of cysteine, cyanide, and other reagents with a high affinity for mercury. The fact that the reaction

LIGHT SCATTERING IN PROTEIN SOLUTIONS

95

proceeds with such great speed is presumably due to the fact that the distance between the two mercury atoms in this mercurial is fairly large (about 10 A.) and the two albumin molecules can, therefore, be coupled together through this link without having to come into such close juxtaposition as when the binding takes place through a single mercury atom. 4. Antigen-Antibody Reactions

Light scattering should offer many advantages in the study of antigenantibody reactions, but only a few preliminary studies have yet been carried out. In 1938 Pope and Healey made use of opacity measurements, with a photoelectric detector, to follow the rate of the reaction between diphtheria toxin and antitoxin. They showed that the rate of increase of opacity was greater in a mixture containing equivalent amounts of toxin and antitoxin than in one which contained an excess of either. The opacity curves of toxin-antitoxin mixtures in various proportions a t equilibrium were similar to those for total protein of the precipitates in such systems. Recently Goldberg and Campbell (1951) have studied the increase in light scattering which occurs in the reaction between bovine serum albumin and purified rabbit antibody. I n most of the experiments they held the weight of albumin constant and varied the amount of added antibody. When the antibody was present in equivalent amounts or in excess of the amount of antigen, the curves for reduced intensity, as a function of time, showed a rapid increase from the very beginning of the reaction, and there was no obvious break in the curve at the point where visible particles appeared. When less than one equivalent of antibody was added, there was practically no increase in turbidity with time after mixing the components. This is what would be expected if soluble complexes were formed in the antigen excess region. All the measurements reported in this study were RgOvalues. Gitlin and Edelhoch (1951) have studied the reaction between human serum albumin and the homologous antibody from horse serum. Unlike the system studied by Goldberg and Campbell, this system shows a precipitation curve similar to that of diphtheria toxin and antitoxin; that is, the amount of precipitate is a maximum for equivalent amounts of antigen and antibody, and little or no precipitate is formed if either ant,igen or antibody is markedly in excess. The rate studies, as determined by measurements of RW as a function of time, showed that the final state was reached more rapidly in the zone of antigen excess than in the zone of antibody excess. However, the final values obtained for a definite antigen-antibody ratio appeared to be the same in the equivalence zone when the final state was approached from either zone of excess.

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PAUL DOTY AND JOHN T. EDSALL

This was true whether one reagent was added to the other all at once, or whether it was added in several successive increments. On the other hand, the reaction on the antigen excess side of the precipitation curve was rapidly and completely reversed by addition of a further excess of antigen, while in the antibody excess zone, addition of more antibody resulted in only a very slow reduction in the amount of scattered light a t 90" to the incident beam. As might have been expected for such a system, low angle scattering n-as much more sensitive t o the course of the reaction than 90" scattering. The latter was adequate t o follow the very early stages of the reaction, when the aggregates formed were still small compared to X; but the readings a t low angles continued to increase long after those at 90" had apparently reached a nearly constant plateau. Conversely addition of more antibody in the zone of excess antibody resulted in a decrease of lorn angle scattering, but very little change at 90". The measurements showed that large complexes consisting of soluble aggregates of antigen and antibody appeared t o exist over all possible ratios of antigen to antibody, and that the specific aggregates were larger in size the closer the reaction took place t o the point of maximum precipitation. Perhaps the most important uses of light scattering in the study of such reactions will be found in the detection of the very early stages of reaction, or in the study of reactions far from the equivalence zone. Inhibition of reaction due to added haptenes, and many other phenomena, should be observable much more rapidly than by other methods.

V. LIGHTSCATTERING IN DILUTE SOLUTIONS OF CHARGED MACROMOLECULES 1. Introduction

There has been relatively little study of the physical properties of protein solutions that do not contain added salts : this is especially true for pH values removed from the isoelectric point. Two contributing reasons for this situation are the feeling that proteins should be studied under conditions approximating that of their natural environment and the intrinsic difficulties encountered in measuring and interpreting the physical properties of such solutions. The former reason carries certain weight, but it is being abandoned with increasing frequency as more diverse methods are brought into the field of protein research. It does not seem unlikely that new information of a t least indirect value may result from examining the region of low ionic strengths and relatively high charge. The practical and theoretical difficulties encountered in

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97

this region are substantial. In equilibrium methods there is the problem of maintaining the unbuffered solutions unaltered over long periods of time and in the particular case of osmotic pressure measurements there are inherent difficulties such as those presented by membrane hydrolysis and the low precision attained with dilute solutions. In dynamic methods the basic theoretical problem of the kinetic effects of the electric double layer is not yet adequately solved. For example, the effect of the double layer is t o increase the specific viscosity and to diminish the sedimentation velocity but an adequate quantitative theory is lacking. The light scattering method appears to avoid many of these difficulties. The reduced intensity, extrapolated to 0 = 0 if the dissymmetry is not unity, provides essentially the same information as that obtained from osmotic pressure measurements. Both measurements lead to the molecular weight (although a different average results when there is a distribution of molecular weight species) and the interaction constant which in this region is principally a function of Z22/m3 (Stockmayer, 1950; see also Sect. 11). Moreover, when ZZ2/m3is large, it is possible with the aid of interference theory and angular intensity measurements, to obtain an alternate and in some cases a more detailed picture of the spatial distribution of protein ions in solutions than can be obtained from the determination of the reduced intensity a t zero degrees or the osmotic pressure. Before taking up the theory and application of the interference method let us examine in a qualitative fashion the anticipated behavior of solutions of charged macromolecules with respect to light scattering. The behavior of solutions at moderate ionic strength has been discussed in Section 11. In this case the protein ions exist in a medium that is rather densely populated with small positive and negative ions. Around each protein ion there will be an electric double layer containing an excess number of ions of opposite charge sufficient to equal the charge on the protein ion. The fluctuating species here is obviously the protein ion and its gegenions. This electrically neutral unit will be preserved upon dilution at constant ionic strength and it is the molecular weight of this neutral unit that is obtained from the customary interpretation of reduced intensity measurements as a function of protein concentration. Since the charge of each protein ion is locallyneutralized by its double layer, the repulsions between the protein ions would not be large: the interaction constant B would be relatively small. On the other hand consider a solution containing only the protein ions and sufficient gegenions of small valence to provide electroneutrality for the bulk solution. At moderate concentration only a fraction of the gegenions will be on the average associated with the protein ions. The

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PAUL DOTY AND JOHN T. EDSALL

inadequate shielding of the protein ions will result in repulsions strong enough t o give rise t o a large value of the interaction constant. Upon dilution more gegenions will leave the double layer around the protein ions and the interaction constant should increase correspondingly. Thus the plot of c/Rsoagainst c which is generally linear at low concentrations will now be highly curved. It is clear that upon going to infinite dilution all the gegenions will leave the protein ion and the molecular weight derived from the limiting intercept of K c / R g O will be that of the protein ion.* Generally the difference between the weight of the protein ion and the protein salt (protein ion and gegenions) will be of the order of magnitude of experimental error, but with heavy gegenions such as iodide, it could not be neglected. It is interesting t o note that the gegenions can contribute many times more to the intensity of the scattered light if they remain associated with the protein ion than if they become independent. For example, if the specific refractive increment of the gegenions is the same as that of the protein ions, the gegenions associated with the protein ion will contribute to the scattering in proportion to the fraction by which they increase the weight of the scattering unit. On the other hand their scattering will be insignificant if they maintain a n independent existence. Of course, this difference is the direct outcome of the fact that the reduced intensity leads to a weight average molecular weight. I n concluding this part we may summarize: The degree to which gegenions are associated with protein ions depends upon ionic strength and protein concentration and is reflected in the molecular weight derived by the light scattering method although the effect may be quite small. The degree of association is of predominant importance in determining the deviation from ideality as indicated by the magnitude of the interaction constant. 2. Interference Theory and the Concentration Dependence of Dissymmetry

The fluctuation theory provides a relation between the reduced intensity and the properties of the system that is valid only in the absence of interference. In Sect. 111 we have already treated the corrections that must be applied when there is interference of light scattered from different parts of the same particle, that is internal interference. Here we are interested in the interference of light scattered by different particles, t ha t is external interference. Both types of interference disappear when 8 = 0; consequently, the fluctuation theory will always be valid for scattering data properly extrapolated to zero degrees. This conclusion * This assumes that the weight of the protein ion, not the protein salt, is used in computing concentration and that d n / d c is correctly evaluated. Actually if R xere determined a t sufficiently low values of e the interference which destroys the scattering power of the expanded double layer a t 90" would not exist, and the corresponding complications would vanish.

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99

is subject to the assumption that the relative refractive index m is not large enough to produce significant effects. The validity of this assumption was discussed in Sect. 111, 5. A complete interference theory would account for the intensity of scattering at all angles including zero and would, therefore, include the results of the fluctuation theory as a special case. However, it requires an assumption or a priori knowledge of the average arrangement of the scattering units as an integral part of the theory: consequently, the fluctuation theory is preferred in the absence of external interference. The interference theory is essential, however, for the interpretation of angular intensity measurements. We shall consider now its simplest application in which it provides the same type of information that comes from a corresponding interpretation of the interaction constant B : it thus serves as an independent check in this case. In more complicated situations, however, its superiority is clear. In keeping with the treatment of internal interference the diminution of intensity of light scattered per particle is derived from knowledge of the phase differences resulting from the spatial distribution of the scattering centers, in this case the protein ions. The most convenient way of characterizing the spatial arrangement of the protein ions is by means of a radial distribution function, p ( r ) , which expresses the probability that the centers of two specified protein ions should lie at a vector distance r apart. We might imagine the two ions could be marked in some way, so that their positions in the solution could be identified. If we could then make a very large number of independent determinations of the distance between them at a series of instants chosen at random, the frequency with which distances whose magnitudes, independently of their directions, lay between r and r dr would occur would be proportional to 4 d p ( r ) d r . It is assumed that the function p ( r ) , the distribution function, is spherically symmetrical : its numerical value is based upon its being unity when all distances of separation are equally probable. In terms of the distribution function the diminution of the scattering can be expressed in a general form (Zernike and Prins, 1927) which is analogous to the particIe scattering factor for internal interference in that it denotes the loss in scattering power per particle as a result of interference :

+

The number of molecules in the volume V is denoted by N . This general form can be used in two ways. Various distribution functions can be assumed and the angular intensity distributions derived therefrom are

100

PAUL DOTY AND JOEN T. EDSALL

compared with the observed distribution: the distribution function given the best fit is accepted. Alternatively, when there is sufficient detail in the observed angular intensity distribution it can be subjected to a Fourier inversion which leads directly to the distribution function. For reasons t ha t are later apparent this latter method can seldom be applied to protein solutions and we must rely upon the former approach. Here we shall only examine the consequences of assuming the simplest possible distribution function, that is the "hard sphere model." We assume that the repulsion between the protein ions has the effect of preventing the centers from approaching closer than a distance D'. Of course, in reality there will be a distribution of distances of close approach but our approximation requires that we use only an average value. The protein ions will then behave as if they were spheres, larger than their physical size in proportion t o the magnitude of the repulsion between them. The distribution function will have the form: p ( r ) = 0 for T < D' and p(r) = constant, nearly equal to unity, for r > D'. Since the center of one protein ion will not be able to come within a distance D' of the center of another a volume of $ 5 ~ 0 is ' ~thereby excluded from occupancy by the center of the first protein ion. This volume, which is eight times the volume of a sphere of diameter D', multiplied by the number of scattering centers is known as the excluded volume, 9. The use of this distribution function for hard spheres in the foregoing general expression leads to

[

"

R~ = KCiwqe) 1 - , Q ( ~ s D ' )

I

(53)

The function 9 is one that commonly occurs in interference theory and it equal t o the square root of the particle scattering factor for spheres, that is, equation (45a). A plot of this function is given in Fig. 14. The lass factor in square brackets accounts for the diminution due to external interference in terms of the hard sphere model and was first derived by Debye (1927) for interpreting the scattering of X-rays from gases. Its use in this form for light scattering is due to Oster (1949) and Doty and Steiner (1949). The implications of the foregoing equation with regard to the angular intensity distribution is readily seen with the aid of Fig. 14. At infinite dilution 9 = 0; consequently, there is no external interference and the only observed dissymmetry is due to P(6). With increasing concentration the second quantity in the square brackets takes on finite values since the excluded volume is no longer negligible. At a given concentration the quantity to be subtracted from unity decreases with the angle 0, as indicated in Fig. 14, and consequently the intensity of scattered light will be diminished more strongly a t low than a t high angles. This effect

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101

is opposite to that of the particle scattering factor, P(B),which decreases with increasing angle. On the basis of this simple model we should then expect that the observed dissymmetry would diminish from a limiting value determined by P(t9) with increasing concentration. The rate of decrease of the dissymmetry with concentration would depend on the effective diameter, D', of the macromolecules; consequently, this feature may be used to evaluate the effective diameter. This falling off of the dissymmetry with concentration was first noted by Doty, Affens and Zimm (1946) in polymer solutions. The effect was shown to depend upon the solvent-solute interaction and a

FIG.14. Graph of the function Q = (3/x3)(sin x - x cos x) as a function of x where x = ksD'.

more quantitative theoretical analysis particularly suited to polymer molecules was given later by Zimm (1948a). More recently the adaptation of the hard sphere model to polymer solutions has been discussed (Doty and Steiner, 1950a). The alteration of the dissymmetry-concentration dependence with ionic strength is clearly evident in the data for tobacco mosaic virus shown in Fig. 10. Since no dissymmetry variation is noticed when buffer is present the effective diameter must be quite small. However, in the absence of buffer, the repulsion due t o the net charge on the molecules gives rise to a large effective diameter which is responsible for the rapid falling off in the dissymmetry. Since the virus molecule is so asymmetric there is little gain in analyzing these data quantitatively in terms of a model that assumes spherical symmetry. Before concluding the discussion on tobacco mosaic virus it is interesting to note the data shown in Fig. 15 (Oster, 1949) because it shows clearly the wavelength dependence of both internal and external interference. At zero concentration the dissymmetry is higher for X = 4360 A. than for X = 5460 A. This is the result of the dependence of P(0) on L/X and the change in the intercept in

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PAUL DOTY AND JOHN T. EDSALL

Fig. 15 is consistent with t,his difference in A. The concentration dependence of the dissymmetry, however, is greater for the blue line than for the green with the result that a t moderate concentrations a reversal has taken place. This follows from equation (53) because of the occurrence of X in the argument of the function +. As a result k is smaller, the function @ larger, and the dissymmetry-concentration dependence less at the longer wavelength. The dissymmetry data for influenza virus in aqueous solution, presumably somewhat removed from its isoelectric point, were given by Oster (1948). The size of these virus particles as determined from the limiting dissymmetry was discussed on pages 83 and 87. The particles are

0 1 -.

0

1 2 3 4 CONC. (g./cc. x lo3)

FIG. 15. Dissymmetry data for tobacco mosaic virus solutions at two different

wavelengths.

spherical with a diameter of approximately 1280 A. Oster (1949) has interpreted the data with the aid of equation (53) to indicate that the effective diameter of the virus in this particular state was 4520 A. This conclusion is probably invalid, however, because the angular intensity distribution is no longer a monotonic function of 0 for such large values of D' and hence the dissymmetry is no longer an adequate measure of the angular intensity distribution. This becomes clear upon examining Fig. 14. The value of the abscissa for a particle exhibiting an effective diameter of this size would be 6.7 and 16.1 at 8 = 45" and 135", respectively, for the wavelength used. These values occur in the region where the function oscillates and consequently their ratio does not uniquely determine the value of the argument and hence of D'. This oscillation begins a t about ICsD' = 4 where the first minimum appears. This means that with increasing values of the effective diameter a maximum appears in the angular distribution function a t about D' = 2000 A. for X = 4360 A. Consequently for effective diameters greater than this value the occurrence of maxima and minima is to be expected and a more detailed

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103

analysis of the angular intensity distribution will be necessary. Such effects would probably have been observed for the influenza virus if angular measurements had been made. On the basis of the investigations of Doty and Steiner (1951) it appears unlikely that effective diameters large enough to produce maxima and minima can be attained with typical globular proteins. However, with larger virus molecules where net charges of several hundred may be produced this effect may be expected. Maxima were observed in silver iodide sols when sufficiently charged, and the data were interpreted by means of Fourier inversion to give the radial distribution function. 3. Interpretation of the Interaction Constant in Terms of the Hard Sphere Model

It may be recalled that the discussion of the meaning of the interaction constant, B, on p. 44 dealt with the relation between B and what has now been called the excluded volume. This idea has long ago been given quantitative form by calculating the virial coefficients of a hypothetical gas composed of hard, elastic, nonattracting spheres (see, e.g., Fowler and Guggenheim, p. 289, 1939). As indicated in the earlier discussion the results for the gas which deal with pressure-volume relations are valid for the osmotic pressure-concentration or reciprocal reduced intensity-concentration relations for solutions. Consequently the value of the second virial coefficient (B’ of equation 13a) obtained for this model provides an expression for B. This is B

=

B‘/M2 = 2rDf3No/3M2

(54)

Thus the interaction constant depends directly upon the excluded volume and can be used to evaluate the effective diameter B‘. This evaluation would be quite independent from that provided by the use of equation (53) with dissymmetry data. However, at zero degrees the absolute value of the reduced intensity used in equation (53) should lead to the same value of D’ as that deduced from B because the fluctuation theory and the interference theory should lead to the same result for scattering at 0’ if the same model is adopted. The equivalence is seen if equation (12a) is rearranged to give (55)

If this is compared with equation (53) and P(0) set equal to unity as is implied in equation (12a) it is then seen that 1

a

+ 2BMc = v @(ksD’) 2BMc

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PAUL DOTY AND JOHN T. EDSALL

At 0 = 0", @ = 1. Since Q = 4?rDf3N/3 this equation becomes an identity when 2BMc << 1. When this condition is not met higher virial coefficients or the corresponding interaction constants must be employed. At low concentrations, however, the equivalence of the two methods is obvious. It appears, therefore, that in low ionic strength solutions of proteins carrying a high charge the effective diameter, or the average distance of approach, could be determined either from B, as determined from the reduced intensity as a function of concentration, or from the dissymmetry. An agreement between the results of the two methods should indicate the adequacy of the hard sphere model in very dilute protein solutions. 4. Experiments with Bovine Serum Albumin at Low Ionic Strength and High Charge For reasons that need not be discussed here it is unlikely that solutions of uncharged polymers for which P(6) = 1 will show a decrease in dissymmetry, which would require values less than unity, and indeed none has been observed. The same conclusion would hold for protein solutions unless particularly strong repulsion existed. But at low ionic strength and high protein charge the necessary repulsions would be expected. It has been shown (Doty and Steiner, 1949; 1951) that aqueous solutions of bovine serum albumin adjusted to the pH range of 3.2 to 4.2 show dissymmetries of less than unity over a part of the concentration range. In Fig. 16 the plots of K c / R S Oand R45/R,35as a function of concentration are shown for solutions of electrodialyzed bovine serum albumin at pH 5.1 and after adjustment to pH 3.2 with the minimum amount of hydrochloric acid. Let us examine these in the light of the foregoing discussion. At pH 5.1, very close to the isoelectric point, there is essentially no concentration dependence : this is to be expected since the protein ions carry practically no charge and, therefore, occupy nearly random positions in the solution. At pH 3.2 the situation is very different. The protein ions carry a net charge of about fifty protons and there is also a corresponding amount of chloride ions in the solution. The initial slope of the reduced intensity plot is very steep and if the interaction of the protein ions remained the same as the concentration varied one would expect the slope to continue and eventually to increase further a t higher concentrations. The value of D' calculated from the limiting slope is approximately 500 A. The observed decrease in slope with increasing concentration can only be interpreted as a falling off in the repulsion between the protein ions and this is reflected in a decreasing value of

LIGHT SCATTERING IN PROTEIN SOLUTIONS

105

D'. This decay of repulsive forces is undoubtedly due to the increasing concentration of gegenions (chloride ions). At higher concentrations of the protein salt a larger fraction of the chloride ions shield the high charge carried by the protein ions. Indeed at concentrations of above 1 g./100 cc. the shielding is nearly complete so that the value of B , and hence D', are nearly zero. Obviously data taken only in this range would extrapolate to an erroneous molecular weight value. Turning to the dissymmetry data one notes the initial decrease in this quantity but at still quite low concentrations it passes through a minimum and returns to

0

M l N l Y Y Y GEGENLON CONC

0 CONSTLNT QEGfNlON CONC

c

I

lo'~e/ccl

FIG. 16. Reduced intensity-concentration data and dissymmetry-concentration data for bovine serum albumin solutions a t pH 5.1 and 3.2.

values of unity at about 0.7 g./100 cc. The initial decrease is expected from equation (53)but no provision is found there for the rapid return to normal values. Moreover it appears that the return to normal values, that is unity, coincides with the flattening off of the reciprocal reduced intensity plot. The initial decrease corresponds to a value of D' of about 500 A. The remainder of the curve can only be fitted if it is assumed that D' continuously decreases. When D' reaches values of about 150 A., not much larger than the actual molecule, the dissymmetry expected from equation (53) approaches unity within experimental error. Thus both reduced intensity data and dissymmetry data for solutions of a protein salt can be interpreted in terms of the decreasing repulsion with increasing concentration brought about the increased shielding effect of the gegenions.

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PAUL DOTY A N D JOHN T. EDSALL

An interesting variation of the experiments just discussed have been reported recently b y Friend and Schulman (1951). They measured Rgo for a solution of bovine serum albumin as a function of added sodium hydroxide and sodium dodecyl sulfate. Although the concentration of albumin remained essentially constant the value of Rgo diminished with the increasing net charge created on the protein ion by the neutralization of acid groups in one case and by the adsorption of negatively charged dodecyl sulfate ions in the other. On a molar basis the detergent was nearly as effective as the base in diminishing R90. A further interesting feature of this work was the discovery that a large dissymmetry was found after the albumin-sodium dodecyl sulfate solution was passed through a sintered glass filter whereas the dissymmetry remained a t unity or less if the albumin solution and detergent solutions were mixed in the same proportion after filtering. This would indicate that the adsorption of detergent molecules makes the albumin susceptible t o denaturation a t the glass-water interface. Having seen the predominant effect of the concentration of the gegenion in determining the repulsion between protein ions it is of interest to study the behavior of protein ions as a function of concentration when the concentration of gegenions is maintained constant. This is readily achieved by diluting a solution of a protein salt with a sodium chloride solution having the same molality as the gegenions in the protein salt solution. The pH of the sodium chloride solutions is first adjusted t o that of the protein salt so that the charge on the protein ion will remain constant upon mixing. Measurements of dilutions carried out in this manner are shown in Fig. 17 together with data resulting from dilution with p H adjusted water. We see a t once that the Kc/R90 has a more normal shape in th at the initial slope remains constant a t low concentrations and then curves upward. The upward curvature of the data a t constant gegenion concentration can be adequately accounted for on the basis of the hard sphere model if the third virial coefficient is also included. (The third virial coefficient is five-eighths times the square of the second coefficient given in equation 54.) The line drawn through the data is that obtained from equation (12a) with the third virial coeEcient added, using the value of B determined from equation (54) with D’ = 170 A. From the closeness of the fit thereby obtained it appears that the net repulsion between protein ions remains constant if the gegenion concentration is maintained constant. In this particular example the protein ions exhibit a n average distance of close approach of 170 A , : however, if the measurements begin with more dilute protein solutions and correspondingly weaker sodium chloride solutions, larger effective diameters are demonstrated. The dissymmetry also varies in a more normal

107

LIGHT SCATTERING IN PROTEINSOLUTIONS

fashion in that it increases upon dilution in a linear fashion extrapolating to a value of unity. The values of D’calculated from the dissymmetryconcentration data are in good agreement with those obtained by fitting the Kc/Rsodata to the hard sphere model. Thus we may conclude that the scattering from highly charged solutions of proteins a t low ionic strength can be interpreted in terms of the hard sphere model. The scattering from solutions of polymeric electrolytes, such as polymethacrylic acid, has also been examined by light scattering (Doty and Oth,

80

7.0 6.0

5.0 K4

X105

Rvo

4 0

30 20 0 M I N I M U M GEGENION CONC 10

13 CONSTANT GEGENION CONC

,

0

FIG.17. Reduced intensity-concentration data for bovine serum albumin a t pH 3.2 a t minimum gegenion concentration (a), and a t constant gegenion concentration (b).

1950) and it has been found that many of the points discussed here have their counterpart in solutions of highly charged linear macromolecules. VI. EXPERIMENTAL METHODS

We shall not attempt here to describe all details of experimental procedure, but only to indicate some procedures and techniques used in our laboratories and to refer to a few of the most recent and most important papers. 1. Clarijkation of Liquids

It is a prime condition for any satisfactory light scattering measurement that the liquid studied should be clear and free from dust particles. Pure volatile liquids can be repeatedly distilled without boiling, in a

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PAUL DOTY AND JOHN T. EDSALL

sealed all-glass apparatus consisting of a reservoir bulb and an observation cell connected by a glass neck. Air can first be removed by pumping it off with a vacuum pump before the apparatus is sealed off. The observation cell is then cooled and the reservoir gently warmed; the liquid thus distills over into the former. The cell is then tilted, so that the liquid flows back into the reservoir, carrying most of the dust with it. The process is repeated several times until the liquid in the observation cell is clear when observed a t angles close to a strong incident beam of light. Involatile liquids, such as protein solutions, are clarified by filtration through carefully washed fine grain filter pads, or “fine” or “ultrafine” sintered glass filters; or by high speed centrifuging at 20,000 g or above; or by a combination of both procedures. Carefully distilled, dust-free water should be used in making up such solutions. Many proteins are easily denatured at surfaces, and the denatured molecules readily aggregate ;hence precautions must be taken to minimize denaturation. When successive dilutions of a protein solution are made from one container into another, it may be desirable to moisten both the pipette used for transfer and the receiving vessel with dust-free water to minimize surface denaturation. However, with serum albumin, which is not as readily denatured at surfaces as some proteins, this precaution has not been found necessary. The presence of dust particles in the liquid is most readily detected by visual examination of the forward scattered light, fairly close to the incident beam. 2. Turbidity b y Transmission Measurements The most direct method of measuring turbidity is by determination of incident and transmitted intensity, using a standard cell of known length containing the solution under study, in such an instrument as the Beckman spectrophotometer. It must be shown, however, that all the loss in transmitted intensity is due to scattering and none to true absorption. This is often difficult to prove; measurements at different wavelengths, as indicated in the preceding section, are almost essential in this connection; if the values vary approximately as the inverse fourth power of the wavelength, the evidence is good that scattering is involved, rather than absorption. The angular divergence of the transmitted beam and the aperture before the phototube must be as small as possible, so that scattered light does not contribute significantly to the measured intensity. This direct method for turbidity is only applicable when the turbidity is fairly high, so that the ratio I o / I differs from unity by a factor which is large enough to be determined accurately. Further details are given by Doty and Steiner (1950b).

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3. Measurements of Reduced Intensity of Scattering The earlier investigators in the field all employed visual or photographic methods of measuring the scattered intensity; these are well discussed by Cabannes (1929) and Bhagavantam (1942); some of the most recent techniques of this class are also discussed by Mark (1948). The "visual turbidimeter," and visual apparatus for determination of angular dissymmetry, described by Stein and Doty (1946) may be cited as examples of effective and relatively simple designs of such instruments. Nearly all recent instruments employ photoelectric cells as light detectors. Putzeys and Brosteaux (1935) were among the first t o employ photocells in such studies. The recent development of the extremely sensitive photomultiplier tubes* has marked a major advance in light scattering technique; it is now possible to work even with very dilute protein solutions and still obtain results of satisfactory accuracy. Also a very narrow beam of incident light may be used; this permits reduction in the size of the cell which contains the scattering liquid-an important consideration in the study of proteins, which often are available only in extremely small amounts. One of the most satisfactory types of apparatus yet developed is that of Zimm (1948b). In his arrangement, the cell containing the liquid under study is a small thin-walled glass bulb, immersed in an outer cell containing a liquid of approximately the same refractive index as that in the inner cell. The scattered light is received on a photomultiplier tube, while a portion of the incident beam falls on a phototube of much lower sensitivity. The currents produced in the two tubes are balanced in a precision potentiometer; at balance, the setting of the potentiometer shows the ratio of the currents. Det,ails of the electronic circuit are given in the original publication. A similar circuit is now in use in the laboratory of one of us (J.T.E.);in this apparatus the scattering cell is rectangular, 1 cm.2 in cross section, and is adapted specifically for measurements at 90" to the incident beam. The scattering at 90" from pure at 4358 A.) is readily measured with a prebenzene (R90= 48.4 X cision of 2 or 3% with this apparatus. Brice, Halwer and Speiser (1950) have recently described in great detail the design of an apparatus adapted for measurements of scattering at 45, 90, and 135". The scattering cell is so designed that the cell contents can be viewed normally at any of these angles, or at 0". The output of the photomultiplier tube is not

* The most satisfactory photomultiplier tubes produced in America are manufactured by the Radio Corporation of America and carry the designation IP21-LB60. t This instrument is available commercially through the Phoenix Precision Instrument Company, 3803 North Fifth Street, Philadelphia, Pennsylvania.

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further amplified but read directly from a sensitive galvanometer. Experimental precautions, and the necessary corrections that must be made to the experimental measurements, are treated with great thoroughness in this paper. Particular attention is given to the important matter of absolute calibration. A somewhat different light scattering photometer with direct current amplification of 10,000 fold is currently available through the American Instrument Company, Silver Spring, Maryland. The detailed design and circuit of this instrument have not been published. Its principal advantage appears to lie in the high stability achieved in the amplification. This feature makes possible the use of very small quantities of solution. Moreover, the amplification permits the use of a very narrow incident light beam which in turn permits measurments down to very low angles. Another type of apparatus, for angular intensity distribution measurements, has been described by P. P. Debye (1946); with some modifications* it has served satisfactorily as an angular photometer. The scattered light is received on a mirror inclined at 45' to the vertical, and is thus reflected down on to a photomultiplier tube situated beneath it. The mirror is attached to a head that moves in an arc around the cell which contains the scattering liquid; by turning a graduated dial it may be set to receive scattered light at any angle to the incident beam, between about 20" and 144'. The conical scattering cell, with a capacity of 12 cc. of liquid, is similar to that used by Zimm. It is immersed in a large rectangular cell, which contains a liquid-usually dust-free water-of nearly the same refractive index as the liquid inside the cell. The mirror is also immersed in this outer liquid; thus bending of the path of the scattered light is minimized. The cell is calibrated by taking readings as a function of angle, using a very dilute fluorescein solution; the intensity of the fluorescent radiation from this solution is independent of angle. The light source, as with almost all types of apparatus now in use, is a high pressure mercury arc (General Electric H3) ; with suitable filters to isolate the blue (436 mp), green (546 mp) or yellow (579 mp) lines. In the Gibbs Memorial Laboratory a light scattering photometer has been constructed especially for kinetic studies by Dr. M. D. Stern. In this instrument the photomultiplier tube rotates at an adjustable constant speed about the cell containing the solution, and the output activates a recording galvanometer. The incident light intensity is recorded at each revolution and thus the radiation envelope in absolute terms as a function of time can be obtained. At top speed four complete envelopes are recorded per minute. * W. B. Dandliker, unpublished work.

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4. Absolute Calibration and the Refraction Corrections

Precise scattering determinations depend upon evaluating the ratio,

i e ~ ~ / I oprecise : turbidity determinations depend upon evaluating the ratio, I / I o . The rigorous demonstration that these ratios are being

precisely measured in any given case requires a thorough analysis of the geometric optics involved and a critical inspection of every aspect of the experimental arrangement that can affect the final observation. These conditions have been generally satisfied in the case of transmittance due to the wide spread development and use of spectrophotometry (Gibson and Balcom, 1947). In contrast there has been little comparable coordinated activity associated with the determination of ier2/Io. The precision attained has been approximately an order of magnitude less than in spectrophotometry : for example, values in the literature for benzene vary over a range of 15% and it now appears that this range is considerably displaced from the true value. Consequently, with the rapid ascendancy of light scattering techniques a determined effort has been made recently to establish precise methods for the evaluation of this ratio. Determination of the absolute values of reduced intensity requires precise determination of the effective volume of illuminated liquid which supplies light to the detecting system. In most experimental arrangements, the scattered light emerges from the liquid under study into air on its way to the photocell. If the walls of the scattering cell are flat and normal to the line between the scattering solution and detector, a beam of light diverging from a point in the center of the cell diverges still more widely on emerging into a medium of lower refractive index. There is also, owing to the refractive index change, a small alteration in the effective volume of liquid which serves as a source of light to the photocell. These problems were well-known to Cabannes (1929) and other earlier authors; but their study has very recently been taken up again by several authors, because of the importance of placing light scattering measurements on an absolute basis. Carr and Zimm (1950) have termed the two major corrections mentioned above the “refractive index” and the “volume” correction factors, respectively. The former is given by the factor Qn (or by Qn2: see below): & n = n [ l - x ( ,Ar )] n - 1

(57)

where n is the refractive index of the scattering liquid, Ar is the distance from the center of the cell to the face at which the light emerges, and X is

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the distance from the center of the cell to the photocell detector. It is assumed, in deriving this equation, that the medium outside the cell is of refractive index unity (air). If X >> Ar, as is true with many types of apparatus, including our own, then Qn n ; if the photocell is directly in contact with the scattering cell, or immersed in the scattering medium, Qn = 1. For a cylindrical scattering cell (circular cross section) Qn is the factor by which the observed intensity reading is multiplied t o give the true intensity which would have been determined in the absence of refraction. For a cell with a plane surface, the corresponding factor is Qn2;for a spherical cell, with the center of observed scattering at the center of the cell, the refractive index correction need not be made, since the rays all emerge normal t o the surface of the sphere. Carr deduced these relations and verified them experimentally. Brice, Halwer, and Speiser (1950) independently treated these relations; they discussed also other minor but significant corrections that must be made to obtain precise results. The “volume correction factor” is in general very small compared t o the refractive index correction; we shall not discuss it here. Both Carr and Zimm and Brice, Halwer, and Speiser have studied carefully the calibration of absolute intensity of scattering by comparison with the reflected intensity from diffusely reflecting surfaces, such as magnesium oxide or casein paint. Apparently none of these surfaces acts as an ideal diffuse reflector; there is always at least a small fraction of the light which undergoes specular reflection, and this must be allowed for in the calibration of light scattering measurements. In Debye’s laboratory at Cornell University, absolute turbidities have been determined by comparing the intensity of the scattered light with that of a small portion of the incident beam, the latter having been weakened in intensity by a known factor by means of several successive reflections (P. Debye, 1944, 1947; P. P. Debye, 1946). In concluding this discussion of absolute calibration it should be emphasized that agreement upon the turbidity of one substance alone, such as benzene, for example, does not completely solve the practical problem of calibration. Every instrument has a certain background due t o stray light, a fraction of which is included in the measurement of a standard such as benzene. The contribution which this background makes is not properly compensated by subtracting the instrument reading with the cell empty. The only rigorous solution which avoids the effect of background, which, of course, varies from one instrument to another, is to use two standard substances of the same refractive index the difference of whose turbidities is precisely known. Such standards are not yet available and consequently, the calibration of light scattering

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photometers generally depends to a small but arbitrary extent upon the light level of the background. 5. The Use of Working Standards

In making a series of light scattering measurements, it is important that the scattered intensity from the solutions under study be directly compared with that from a working standard material which has itself been calibrated in absolute terms. The ideal working standard should be stable and easy t o handle; and it should give a reduced intensity comparable t o that of the liquids with which it is t o be used. It is more convenient to use a standard which gives a depolarization factor close to zero, and gives no angular dissymmetry of scattering, so that the relation between its turbidity and its reduced intensity may be accurately given by equation 4. Many working standards have been employed, and none as yet appear completely satisfactory. We have employed pure dustfree benzene, in a sealed all-glass container connected t o a reservoir bulb for distillation to remove dust. However, the reduced intensity of benzene (RgO= 49 x 10-6 at = 4358 A.) is somewhat lower than that of most dilute protein solutions in water. We have also used toluene and p-xylene, with reduced intensities slightly greater than that of benzene. Carbon disulfide has been used by Blaker, Badger, and Gilmann (1949); they report for it a value of RW = 151 X at 4358 A. and 47.8 X 10-6 at 5461 A. This is a very convenient value for a working standard; but the samples of carbon disulfide employed by us became noticeably yellow in a relatively short time after exposure to 4358 A. radiation. Possibly this is due to impurities in the particular samples employed, since Blaker, Badger and Gilmann report one of their samples to have been stable for three years when used chiefly with light of 5461 A. Benzene, toluene, xylene and carbon disulfide all have refractive indices much higher than that of water; the readings obtained from them must, therefore, be adjusted by the refractive index correction factor, discussed above, so as to make them comparable with these obtained from aqueous solutions. All these standard liquids have large values of pa; hence it is not legitimate to use equation 5 in calculating their turbidities from the reduced intensities given above. A standard preparation of polystyrene, prepared by the Dow Chemical Company, has been distributed by P. Debye and A. M. Bueche of Cornell University. The solid polystyrene is dissolved in toluene (5 g./l.): the solution is not very stable, and must be made up afresh at frequent intervals. Values for the turbidity (or reduced intensity) of this standard solution have been determined in several laboratories and these are in good agreement: for results, see Brice, Halwer, and Speiser

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(1950), Table V I ; also Edsall, Edelhoch, Lontie and Morrison (1950), and Doty and Steiner (1950). Solid blocks of plastics, of moderate turbidity, have been used as working standards, but their properties may change slowly with time. Zimm (1948b) used a rectangular block made of polystyrene to which 5% of methyl acrylate had been added when the polymerization was half completed. This gave a convenient degree of turbidity, but the absolute value obtained was rather sensitive to temperature; Zimm stated that it was desirable to keep this standard thermostated. Brice, Halwer, and Speiser (1950) used working standards of opal glass, which was studied both as a transmitting and as a reflecting diffusor. This proved useful in practice but, particularly when used as a transmitting diffusor, gave values which had to be carefully corrected in terms of an absolute standard. 6 . Depolarization Measurements

Visual methods of determining depolarization factors (Cornu method) are well described, with references, by Lotmar (1938a, b) and Lontie (1944). Square Polaroid discs, cut so that one side of the square is parallel to the direction of the electric vector in the light transmitted by the Polaroid, are convenient for insertion in front of a photocell which views the scattered light; the factor p is then given from the readings with the square disc inserted to transmit horizontal and vertical vibrations, respectively. Since the latter component is usually much the more intense, it may be desirable in reading it to introduce a neutral filter of known transmittance with the Polaroid, so that the readings on the photocell are comparable for horizontal and vertical components. Many photomultiplier tubes show markedly different relative responses for polarized light of a given intensity, depending on the plane of the electric vector; the differences may be as great as 20% or more in some cases. This possibility must always be tested for, and the appropriate corrections applied to the measured readings, if necessary.

7. Xpeci& Refractive Index Increments In order to evaluate the quantity K or H appearing in equation (6) the refractive index difference between the solvent and solution must be accurately known. The concentration of the solution used should not exceed 1 to 2% since the increment may alter slightly in value at higher concentrations. Thus the refractive index difference to be measured is of the order of magnitude of 0.001. Since this quantity is used as the square, the difference must be measured with a precision of about 0.000005 in order to limit the error introduced by this measurement to

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1%. Such precision can only be attained with a differential refractometer. With care this precision can be obtained with the Pulfrich refractometer using a divided cell or with the interferometric refractometer. Refractometers based on the deflection of a beam passing through a divided cell have come into wide use for this purpose (P. P. Debye, 1946; Brice, Halwer, and Speiser, 1950). Since this is a standard physical measurement, reference should be made t o standard works such as Bauer and Fajans (1949) for further details.

8. Measurements on Colored Xolutions

The interpretation of light scattering measurements from colored solutions presents some special problems, the first of which concerns the adequacy of the present theory when the scattering particles also absorb. This arises because we have assumed that the frequency of the alternating electric field of the incident beam is much less than the natural frequency of the bound electrons which it induces into forced oscillation. The occurrence of absorption indicates that the incident light frequency is close t o the natural frequency of some of the electrons in the particle. This special problem is included in the treatment of scattering from spheres given by Mie (1908) in which it is shown that the only change required in the theory is that the.refractive index be replaced by the complex refractive index, (n - ik), where i = drl and k is the absorption coefficient. The dimensions of k are those of reciprocal length and the units must be the same as those used for the wavelength. The numerical value of k is, therefore, given in reciprocal Angstroms and is for all practical cases extremely small ; consequently, this change from real to complex refractive index is a negligible effect in most solutions. It reaches importance only in such systems as colloidal gold sols, carbon black suspensions and the like. It may be of importance in very strongly absorbing protein solutions such as hemoglobin but this proves to be only a hypothetical question because practical considerations appear to rule out the investigation of light scattering from such solutions, except a t wavelengths for which their absorption is small. The practical limitations in investigating colored solutions arise from purely geometrical considerations. For example, if the light scattered from a particular particle must pass through 1 cm. of solution having an extinction coefficient of 2 cm.-* before leaving the cell its intensity is reduced about a hundred fold. Consequently the scattering we wish to measure is greatly diminished in highly absorbing solutions. Moreover, the intensity of the incident light varies throughout the volume viewed and the diminution of the scattered light is not related in a linear manner to the location of the volume element with respect to the detector. An

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PAUL DOTY A N D JOHN T. EDSALL

analysis of these difficulties has been given by Putzeys and Dory (1940) and their method of correction is quite feasible if the absorption coefficient is not too large, that is, somewhat less than 1 em.-' The assignment of the limit of practical application of light scattering in an absorbing solution cannot be definite, however, because it depends upon the value of the reduced intensity of the scattered light as well as upon the absorption coefficient. The case of hemoglobin, for example, appears to be hopeless on both counts: because of its relatively low molecular weight the reduced intensity is low, while on the other hand, its absorption coefficient throughout the visible spectrum is fairly high. When a solution is only slightly colored, little difficulty is encountered since the effect of the color is eliminated upon extrapolation t o infinite dilution: the concentration dependence and hence the value assigned t o B may be significantly altered, however. To see how more extreme cases are dealt with reference should be made t o the study of sulfur sols in the ultraviolet absorption region by Kenyon and LaMer (1949).

LIST OF PRINCIPAL SYMBOLS EMPLOYED

'

aJ-Activity of component aJK-a In aj/amK B-Interaction constant (see eqs. 11; 12, 13 et seq.) c-Concentration of solute in g./cc. D-Diameter of a spherical particle (eq. 45a) D'-Collision diameter of repelling hard spheres (D'2 D) (eq. 53) H-Factor defined in eq. 6b (see Debye, 1947) H v , H*, HA-Reduced intensity of horizontal component of scattered light, when incident light is respectively unpolarized, vertically polarized, or horizontally polarized (eq. 47) Z-Intensity of transmitted light beam Zo-Intensity of incident light beam (eq. 1) &-Intensity of light scattered at angle 0 (eq. 1) k-%/X (eq. 44 et seq.) K-Factor defined in eq. 6a K'-Factor defined in eq. 21 [ = 1000 K / ( d n / d c ~ ) ~ ] L-Length of a rod-shaped particle (eq. 45b) M-Molecular weight i@,,-Number average molecular weight (eq. 14) i@,-Weight average molecular weight (eq. 15) m-Refractive index of dissolved particles relative to that of medium (m = n'/no) mJ-Molarity of J t h component N-Number of molecules

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No-Avogadro's number (6.022 X loz3) n--Refractive index of solution (or of medium in general) no -Refractive index of pure solvent %'--Refractive index of solute particle P--Osmotic pressure P(B)-Angular distribution function for scattered intensity a t angle 0 (eqs. 44, 45a, 45b, 45c). P(0) = 1 for all 0 if scattering particles are much smaller than X R--Molar gas constant R-Root mean square distance between ends of a random coil (eq. 45c) Re-Reduced intensity of scattering at angle 0 (eq. 4, etc.) Rso-Reduced intensity of scattering a t 90" to incident beam r-Radius of a spherical particle (r = 0 / 2 ) (in Sect. 111, et seq.) r-Distance from scattering particle to observer (eq. 1) s-2 sin 0/2 (eq. 44 et seq.) Vu, V,,, Vh-Reduced intensity of vertical component of scattered light, when incident light is respectively unpolarized, vertically polarized, or horizontally polarized (eq. 47) x-A parameter defined in eqs. 45a, 45b, 45c 2,-Valence of the ith ion in system Z2-Valence of protein ion z-Dissymmetry (R45/R135) GREEK LETTERS

a-Polarizibility (eqs. 1 and 2) where T is radius of spherical particle (in Sect. I11 et seq.) PJ = In yJ-where J is the Jth component PJR = 13 In y J / a m K (eq. 23b) y-Activity coefficient of J t h component (eq. 23a) E-A factor defined by eq. 37 e-Dielectric constant of medium (for optical frequencies) eqs. 2, 5, 7, 9, 10, etc. ( E = n2) eo-Dielectric constant of pure solvent (for optical frequencies) eqs. 2, 5, 7, 9, 10, etc. (eo = no2) 0-Angle at which scattering is observed, referred to forward direction of incident light beam X-Wavelength of light in the medium ( = LO/%) Xo-Wavelength of light in vucuo v-Number of particles per cc. (eqs. 2 and 3) v2,-Number of mols of ions of species i per mol of protein component (component 2) ?-Mean number of mols of small ions of a given species bound by one mol protein a = 27rr/X,

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p,-Depolarization ratio for scattered light p-Density of particles (eq. 48) p-Distribution function (in Sect. V) 7-Turbidity (eq. 4 and preceding discussion) qJ-Specific refractive index increment of component J (per mol component J per liter) @-An interference function used in eq. 53 Q-Excluded volume (eq. 53)

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