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[5] Light-scattering measurements
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LIGHT-SCATTERING MEASUREMENTS
147
[5] Light-Scattering M e a s u r e m e n t s By M. BIER Introduction When a colloidal solution is illuminated by a parallel beam of white light, a faint bluish light can be laterally observed. This phenomenon, caused by the scattering of light, is commonly known as the Tyndall effect. 1 The scattering of light is, however, not limited to colloidal solutions, where it is the most pronounced, but is observable in all transparent media, be it a gas, a pure liquid, a solution, or a crystal. Lord Rayleigh 2 formulated in 1871 the fundamental laws of the scattering of light by calculating the polarizability of individual gaseous molecules placed in the oscillating electromagnetic field of a light beam. It is the polarized molecules that then act as sources of secondary radiation, re-emitting the energy of excitation, thus giving rise to the scattered light. Various authors have further contributed to the theories of light scattering, and several reviews adequately cover the field2 -s The intensity of the scattered light depends on a number of measurable quantities and can be expressed as a function of the number of centers of scattering (i.e., molecules) per unit volume. Its quantitative measurement can thus be used for the determination of Avogadro's number] or it can be applied to the determination of molecular weights, if a value for Avogadro's number is adopted. The first applications of light scattering to the determination of particle weights in colloidal systems seem to be due to Smirnov and Bazenov 8 and Putzeys and Brosteaux. 9 Despite this long history it is only since the simplification of the theories by Debye 1° that the light-scattering method has become a practical tool for the study of macromoleeular systems. For the full characterization of the scattering of a nonabsorbing solution we have to know the relative intensity of the scattered light with re, J. Tyndall, Proc. Roy. Soc. 17, 223 (1869). 2 j. W. Strutt (Lord Rayleigh), Phil. Mag. [4] 41, 107, 274, 447 (1871). a S. Bhagavantam, "Scattering of Light and the Raman Effect." Chemical Publishing Co., Brooklyn, 1942. 4 H. Mark, in "Frontiers in C h e m i s t r y " (Burk and Grummitt, eds.), Vol. 5: Chemical Architecture, p. 121. Interscienee Publishers, New York, 1948. 5 G. Oster, Chem. Revs. 43, 319 (1948). J. T. F,dsall and W. B. Dandliker, Fortschr. chem. Forsch. 2, 1 (1951). 7 j . Cabannes, Ann. phys. [9] 15, 5 (1921). 8 L. V. Smirnov and N. M. Bazenov, Colloid J. (U.S.S.R.) 1, 89 (1935). 9 p. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935). t0 l'. Debye, J. Phys. & Colloid Chem. 51, 18 (1947).
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TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
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spect to the incident beam, its angular distribution, and its depolarization. Furthermore, the optical relationship between solvent and solute has to be determined and is usually expressed in terms of the specific refractive index increment, ( n - no)/C. From the proper combination of these values we can derive not only the molecular weight but also data on the size and shape of the dissolved macromolecules as well as information on the thermodynamic properties of the system. Owing to the great rapidity with which optical measurements can be carried out, light scattering offers also unique possibilities of following the kinetics of macromolecular reactions involving a change in the size or shape of the dissolved particles. ~1-14 Theory A detailed discussion of the theories of light scattering is far beyond the scope of this article, which shall be limited only to the presentation of the equations of immediate use in the evaluation of light-scattering data. In the consideration of the equations attention should be given to the limitations imposed on the systems to which they apply. The turbidity, T, of a system is defined by the equation I = Ioe -t~
(1)
and expresses the exponential loss of the intensity of light, I, on passage through any nonabsorbing medium of path length I. In appearance it is analogous to Beer's law, defining the loss of light due to specific absorption of light by " c o l o r e d " media. In practice, however, it is distinguished by two important facts. First, the absolute value of r is for most systems considerably lower than the corresponding extinction coefficient of Beer's law for usual "colored" solutions. Second, the energy lost by the transmitted beam is not transformed into heat as in colored solutions, but is immediately re-emitted, with the same wavelength as the incident light, in all directions, the scattering molecules acting as secondary sources of radiation. Whereas the above equation gives the total loss of light intensity in the transmitted beam, the Rayleigh ratio Ro - i°r2
(2)
I0 11 M. Bier and F. F. Nord, Proc. Natl. Acad. Sci. U.S. $§~ 17 (1949). 1~ G. Oster, J. Colloid Sc/. 2, 291 (1947). 13 j . D. Ferry, S. Shulman, K. Gutfreund, and S. Katz, J. Am. Chem. Soc. 74, 5709 (1952). 14 S. Katz, S. Shulman, I. Tinoeo, Jr., I. H. Billick, K. Gutfreund, and J. D. Ferry, Arch. Biochem. and Biophys. 47, 165 (1953).
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LIGHT-SCATTERING MEASUREMENTS
149
defines the intensity, i0, of the scattered light measured at the angle 0 and distance r as a function of the intensity, I0, of the incident light. It is the Rayleigh ratio, sometimes also referred to as the reduced intensity of the scattered light, that is usually measured. The relation between T and R~ is given by 7 = s/./~R0 = 16/t~Rg0 (3) In macromolecular solutions, the solvent is assumed to be continuous, and only the additional scattering, due to the solute molecules, is considered in the above equations. The molecular weight, M, of the solute is then calculated from either the turbidity or the Rayleigh ratio by means of the following simple equations:
Hc
l
-
211
(4)
and
Kc (1 R~
+ cos" 0)
=~
1
(5)
or
Kc Rg0
1 -
M
(6)
These equations apply only to infinitely dilute solutions of isotropic and dielectric molecules of relatively small size as compared to the wavelength of light (<)~/20). The terms H and K group together all the pertinent optical constants for the solute-solvent system at a particular wavelength of light: H=
3NoXo 4
2rr'no' (n c n--~°)2 K=
(s)
NoXo~
Equations 4 to 6 presuppose that the incident light is unpolarized, which then can be considered as the superposition of two polarized beams of light, oscillating at right angles to each other and incoherent in phase. The intensity of the scattered light, due to the beam of light having an electrical vector oscillating in a plane perpendicular to the plane defined by the incident beam and the direction of observation, will be the same at all angles of observation, whereas the angular distribution of the scattered light produced by the horizontally polarized light will be a
150
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
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function of cos 2 0. The addition of the two, results in the angular distribution of the scattered light, (1 + cos 2 0), indicated in equation 5. As a consequence, the scattered light at 90 ° should be fully polarized in the vertical direction, and the intensity of scattering in the forward direction (0 - 90 °) is symmetrical to that in the backward direction (90 - 180°). The similarity of equations 4 and 6 to van't Hoff's limiting equation for osmotic pressure, P, P c RT - M (9) is not purely fortuitous, as was shown by Einstein 15 in his analysis of the scattering in more concentrated systems. In condensed systems, the total observed scattering is less than would be obtained by the summation of the scattering from each individual molecule, owing to the destructive interference of the scattered rays. As an extreme case, in an ideal crystal, in which all the molecules would be arranged in a perfect lattice without thermal vibration, any of the volume elements of the crystal would contain exactly the same number of scattering centers, each polarized exactly with the same amplitude and in the same phase, when exposed to incident radiation. As a consequence, for each scattering molecule another one could be found, placed at such a distance as to cause the mutual interference of the secondary radiation. Such a crystal would scatter no light. The observable scattering in real systems is therefore due to the local random fluctuation in the density of the system. Thus, the same number of molecules will scatter much more light in gaseous state than in liquid state, and the liquid will scatter much more than a crystal. The excess scattering of a solution, in which we are interested here, is, by analogy, due to the local fluctuation ill the concentration of the solute molecules, i.e., in the osmotic pressure. This finds its expression in the following equation: Hc _ O(P/RT) (10) 7
Oc
Van't Hoff's equation (9) expresses the osmotic behavior of highly dilute solutions of low molecular weight compounds. For most colloidal solutions, however, it fails at very low concentrations, and the following empirical equation expresses the dependence of the pressure on concentration: P c R - T -- -M -~ Bc~ (11) 1~ A. Einstein, Ann. Physik 83, 1275 (1910).
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LIGHT-SCATTERING MEASUREMENTS
151
Inserting equations 10 and 11 in equations 4 and 6, we obtain Kc
Hc
1
Rg0
T
M
+ 2Be
(12)
These expressions are the ones usually employed in the determination of molecular weights. It can be seen that by plotting either Kc/R~o or H c / r versus concentration a straight line will be obtained, the intercept with the ordinate corresponding to the reciprocal of the molecular weight, while the half-slope, B, is identical with the slope of the osmotic pressure equation. Its meaning will be discussed later on. Although there is thus a great similarity between light-scattering and osmotic pressure measurements, and data obtained by the two methods are frequently compared, there arises an important difference in the results, when the colloidal solutions are not monodisperse, but contain molecules of different weights. In such a case, osmotic pressure measurements give a so-called number average molecular weight, M,~-
Zm,M~
Zm~
(13)
where mi is the molar concentration of the macromolecular component, i, and Mi its molecular weight. By light-scattering measurements, however, a different average is obtained, namely the weight-average molecular weight, Z m i M i2 M w - Y.miM~ (14) The difference is due to the fact that osmotic pressure depends only on the number of particles, whereas the light scattering depends also on the weight of the particles. As a consequence, in polydispersed solutions lightscattering measurements will always yield a higher average molecular weight, and this method is particularly sensitive to dust particles, or other giant-sized impurities. Before going into the question of further data derivable from lightscattering measurements, we have first to extend the theory so as to be able to determine the molecular weight in two of the cases so far excluded by our restrictions, relating to the isotropy and size of the molecules. In the above equations it was assumed that the molecules are optically perfectly symmetrical, i.e., that their polarization is perfectly parallel to that of the exciting beam of light. For most molecules, however, a more appropriate model is that of an ellipsoid of polarization, i.e., they have their own preferential directions of polarization. The plane of polarization of the induced secondary radiation will therefore be slightly inclined with
152
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
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respect to that of the primary radiation, and the light scattered at 90 ° will have a small horizontally polarized component besides the normal vertically polarized light. The scattering of such a system is therefore higher than that which would be expected from only the fluctuation in concentration, as it also contains the scattering due to fluctuation in orientation of the anisotropic molecules. For the calculation of molecular weights the latter factor has to be subtracted from the total measured light scattering. The correction factor for Rg0 is (6 - 7p)/(6 ~ 6p), and for r it is ( 6 - 7p)/(6-]-3p), whereby the depolarization ratio, p, is defined as the ratio of the horizontal to the vertical component of the scattered light at 90 ° with unpolarized incident light. 1~ In most macromolecular systems this correction is rather small, p amounting to about 0.004 to 0.04. This is true also in all proteins TM so far studied, although they may be quite asymmetric as in the case of the tobacco mosaic virus. Although this simplifies the use of the method for the determination of molecular weights, it is possibly unfortunate that it precludes fuller use of measurements of depolarization to the study of the shape of the molecules. More information on the actual size and shape of the molecules can be gained from the study of the angular distribution of scattered light from molecules of size comparable to the wavelength of light. Particles which are small compared to the wavelength of light can be treated as point sources of radiation. Particles approaching the dimensions of the wavelength of light (in the particular solvent, i.e., X.ol~o~t= X...... :n,olvont) will, however, have many scattering elements. Thus, interference will occur with a resulting decrease in the observed intensity of the scattering. It was repeatedly shown that the interference will be much greater in the backward direction than in the forward direction, causing a dissymmetry of the angular distribution of the scattering. Equation 5 has therefore to be multiplied by factor P(0) to give the correct angular distribution of the intensity of scattered light: K_c (1 + cos 20)P(O) Re
=
1
(15)
In Table I are listed the values of the functions P(0) for the three usual macromolecular models, namely, for spherical molecules, for rodlike particles, and for randomly coiled linear polymers, in terms of a parameter, x, for the first two and as a function of ~v/X for the coils. The le Cabannes, J., "La Diffusion mol6culaire de la Lumi~re." Presses Universitaires de France, Paris, 1929. 16~ E. P. Geiduschek, Or. Polymer Sci. 13, 408 (1954).
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LIGHT-SCATTERING MEASUREMENTS
153
TABLE I PARTICLE SCATTERING FUNCTIONS FOR THE THREE MACROMOLECULAR MODELS P(O) x"
Spheres
Coils
Rods
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2,2 2,3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6
0.998 0,992 0.982 0.968 0.951 0. 930 0.906 0. 879 0.849 0.816 0.782 0.745 0.707 0.668 0.628 0. 587 0. 547 0. 506 0. 466 0.427 0. 388 0. 351 0.316 0. 282 0.249 0. 219 0. 191 0. 165 0.141 0. 119
1.000 0. 986 0.971 0.949 0.922 0. 890 0. 855 0. 817 0.777 0.736 0. 694 0.653 0.612 0.573 0.535 0. 500 0. 466 0. 434 0. 405 0. 377
0. 999 0. 996 0.990 0. 983 0.973 0. 961 0. 948 0. 932 0.916 0.897
0. 329
0. 627
0. 287
0. 583
0. 252
0.54'3
0. 223
0. 506
0. 198
0.473 0.443 0.417 0.395
0. 857 0.813 0. 767 0.719 0. 672
'~ For random coils = x/a:. p ~ r ~ m e t e r , x, is For spheres
D x = 2v~sin
(16)
For rods
x = 2~ L s i n 0
(17)
F o r coils
R2 0 8 r~ ~ sin2 2_ x = .~
(18)
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TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
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The critical dimensions of the models are the diameter, D, of the sphere, the length, L, of the rod, and the root mean square, R, of the distance between the two ends of the random coil. The parameter R is also the largest distance within the coil in its most probable configuration and can be approximated by R2 = Na 2 1 - p l+p
(19)
where N is the number of segments in the chain, a their distance, and p the cosine of the angle between two successive segments. ~7 The above equation assumes a free rotation of the various segments. Experimental results indicate, however, that the size of the coils is usually larger and can be better expressed by R2 = N a ~1 - p 1 ~ c o s ¢
(20)
1 ~ p 1 - cos where the rotation of each segment is assumed to be restricted to an angle ~.~8 With Table I the values of P(O) can be found for any scattering angle or particle size. It is a fortunate characteristic of this function, P(0), that it is sufficient to determine the intensity of the scattering at any two angles for the calculation of the particle size. The two symmetrical angles of 45 ° and 135 ° are usually chosen, and the ratio of scattering intensities at the two angles is referred to as the dissymmetry of scattering, z. In Table II are listed the values of this dissymmetry in function of the critical particle size. Also, in the same table are given the corresponding values of the reciprocal of P(90), which is the correction factor by which the calculated molecular weight has to be multiplied to correct for the interference. The above tables apply only to systems at infinite dilution. The particle size and correction factor 1/P(90), is therefore calculated from the so-called intrinsic dissymmetry, i.e., the dissymmetry extrapolated to zero concentration. Although not strictly valid, the same correction factor is applied also to the slope, 2B, of the conventional light-scattering plots. The shortcoming of the above method is that it depends on the proper selection of the macromolecular model, although the dissymmetry for most systems is rather low, where there is no great difference between the 17W. Kuhn, KoUoid-Z. 68, 2 (1934). is p. j. Flory, "Principles of Polymer Chemistry." Cornell University Press, Ithaca, N.Y., 1953.
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155
TABLE II DISSYMMETRY AND CORRECTION FACTORS AS A FUNCTION OF PARTICLE SIZE Spheres
D/x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60
l)issym- Correction metry factor (z) [1/P(90)] 1.002 1.01o 1.02t 1.037 1.05s 1.08~ 1.11~ 1.15s 1.202 1.257 1.32o 1.394 1.48, 1.582 1.699 1.837 2.000 2.192 2.41, 2.691 3.01~ 3.417 3.90~ 4.51~ 5.29~
1.002 1.01o 1.013 1.026 1.04o 1.059 1.08, 1.107 1.13s 1.173 1.214 1.26o 1.31a 1.373 1.441 1.517 1.604 1.702 1.81~ 1.94, 2.08~ 2.25, 2.44o 2.650 2.91~
Coils
Rods
1)issym- Correction merry factor (z) [1/P(90)]
Dissym- Correction merry factor (z) [1/P(90)]
1.00 1.012 1.025 1.04o 1.062 1.09o 1.12a 1.162 1.205 1.255 1.31o 1.37o 1.436 1.50s 1.582 1.663 1.74s 1.83s 1.93o 2.02o 2.12~ 2.22~ 2.328 2.42s 2.53s 2.64o 2.74~ 2.84s 2.95, 3.05~
1.00 1.005 1.017 1.02s 1.04s 1.064 1.08s 1.115 1.14s 1.183 1.223 1.26s 1.317 1.37, 1.43o 1.49~ 1.563 1.63~ 1.72o 1.80~ 1.89s 1.99~ 2.090 2.20~ 2.32~ 2.44: 2.57~ 2.70~ 2.84: 2.99~
1.00 1.005 1.011 1.02o 1.03, 1.045 1.061 1.08, 1.102 1.127 1.154 1.183 1.21o 1.25, 1.28s 1.32T 1.37o 1.414 1.46o 1.50~ 1.55s 1.60~ 1.65~ 1.70s 1.75~ 1.80~ 1.84o 1.89~ 1.93a 1.97~
1.00 1.004 1.00s 1.014 1.022 1.032 1.043 1.057 1.072 1.09o 1.10~ 1.131 1.154 1.18o 1.20~ 1.237 1.27o 1.304 1.34~ 1.38o 1.42~ 1.46~ 1.51o 1.55s 1.6(b 1.659 1.71~ 1.76~ 1.827 1.88s
three models. For larger particles the method fails to give any indication of the actual shape of the particles. This can be obtained by a more complete determination of the angular distribution of the scattered light.. In this mei hod 19 the observed angular i~ltensities of the scattering are simultaneously graphically extrapolated to zero angle and zero concentration, whereby the effects of interference disappear, as P(0) is always l For 1~B. |1. Zimm, J. Chem. Phys. 16, 1093, 1099 (1948).
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TECHI~IQUESFOR CHARACTERIZATION OF PROTEINS
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still larger particles the more complex theory of Mie 2° has to be applied. ~1 Light-scattering measurements can also yield valuable data on the thermodynamics of solvent-solute interaction. I t is sufficient to recall equation 10 and the fact that, according to classical thermodynamics, PIT"l = - A F 1
(21)
where 171 is the partial molal volume of the solvent, and AP~ is its partial molal free energy. The latter thus contains an e n t r o p y term and a heat content term, both referring to the mixing of the solvent and solute. These terms are to be found in the slope, 2B, of equation 12, and the heat content term can be determined by measuring the slope at different temperatures. T h e interpretation of the results in solutions of randomly coiled polymers has received considerable attention from various authors ~s,~2-24 but is beyond the scope of this article. I t will suffice to mention the following qualitative conclusions. In a good solvent, i.e., where there is a pronounced energetic interaction between the solute and solvent, the slope 2B will be large. In a poor solvent, however, i.e., in systems where the solvent and solute have but small affinity, the slope will be small, zero, or even slightly negative. Negative slopes are indicative of approaching phase separation, i.e., solute precipitation, the occurrence of which prevents strongly negative slopes. ~,25,26 Light scattering results in mixed solvents or solvent-precipitant mixtures employed in polymer fractionation should, however, be interpreted with caution, owing to the possible preferential absorption of one of the solvents. 27 In solutions of charged macromolecules and proteins in particular, the large effects on light scattering of the electrostatic forces surrounding them have also to be considered. Scatchard's theoretical t r e a t m e n t of such systems ~s was applied b y him and co-workers to a detailed s t u d y of the osmotic pressure of bovine serum albumin 29 and can be adapted *oG. Mie, Ann. Physik 25, 377 (1908). ~ A. S. Kenyon and V. K. LaMer, J. Colloid Sci. 4, 163 (1949). ~ M. L. Huggins, J. Am. Chem. Soc. 64, 1712 (1942); Ann. N.Y. Acad. Sci. 45, 1 (1942). 2ap. j. Flory, J. Chem. Phys. 10, 51 (1942); 13, 453 (1943); 17, 1347 (1949). 24A. R. Miller, "The Theory of Solutions of High Polymers." Oxford University Press, New York, 1948. 36F. F. Nord, M. Bier, and S. N. Timasheff, J. Am. Chem. Soc. 73, 289 (1951). 26S. N. Timasheff, M. Bier, and F. F. Nord, J. Phys. & Colloid Chem. 55, 1134 (1949). 27R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 687 (1946). 38G. Scatchard, J. Am. Chem. Soc. 68, 2315 (1946). 32G. Seatehard, A. C. Batchelder, and A. Brown, J. Am. Chem. Soc. 68, 2320 (1946).
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LIGHT-SCATTERII~I'G MEASUREMENTS
157
directly to the light scattering of protein solutions. 6,~°,3~In the present discussion we shall restrict ourselves to the most common case encountered in practice, i.e., to protein solutions in the presence of a relatively large concentration of salts. For the t r e a t m e n t of d a t a on salt-free solutions of isoionic proteins, the reader is referred to the original contributions. 3~ Accordingly, the slope 2B of equation 12 contains three terms:
2B -
1000/Z2 MS ~-~ + ~2~
~am') ~-33m/
(22)
T h e factor 1000 arises from the conversion of molal concentrations to weight concentrations, which in light-scattering measurements are usually expressed in grams per cubic centimeters. The first term, Z 2 / 2 m M 2, is the most important, as it expresses the dependence of the light-scattering intensity on the valence, Z, of the protein and the molal concentration, m, of the salt. In osmotic pressure experiments this t e r m is a direct consequence of the unequal distribution of the diffusible salts across the membrane as a consequence of D o n n a n ' s equilibrium. The dependence of the term on the reciprocal of the square of the molecular weight of the protein, M, is only apparent, as the valence of the protein, Z, is given b y h M , where h is the n u m b e r of hydrogen or hydroxyl ions bound per gram of protein. T h e t e r m thus represents the ratio of charge to mass for the given protein and would require the slope to be zero at the isoionic point (Z = 0) and increase symmetrically from b o t h sides of it. Therefore, the t u r b i d i t y of a protein solution of finite concentration should be the highest at the isoionic point, and, of course, the osmotic pressure is the lowest at the same point. In practice, however, osmotic pressure determinations 29 and light-scattering measurements of serum albumin 32 as well as on egg albumin 3°,~1 have shown t h a t the minimum slope is shifted toward more acid values from the isoionic point as a consequence of the other terms of equation 22. Increasing salt concentration should also result in an increase in t u r b i d i t y of a protein solution as a result of the decrease in slope 2B. The second term of equation 22 represents the variation in the 3°F. F. Nord and M. Bier, irt "Handbuch der K~ltetechnik" (Plank, ed.), Vol. 9: Biochemische Grundlagen der Lebensmittelfrischhaltung, p. 84. Springer-Verlag, Berlin, 1952. 31M. Bier, Dissertation, Fordham University, New York, 1950. 31~W. B. Dandliker, J. Am. Chem. Soc. 76, 6036 (1954); J. G. Kirkwood and S. N. Timasheff, Arch. Biochem. and Biophys. 65, in press (1956). ~: J. T. Edsall, H. Edelhoch, R. Lontie, and P. R. Morrison, J. Ant. Chem. Soc. 72, 4641 (1950).
158
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[~]
logarithm of the activity coefficient, % of the protein with the change in protein concentration: ~22 = 0 In ~protoin/Omp~ote~n (23) The term has a meaning similar to that of the slope 2B in solutions of uncharged macromolecules, as it represents the deviation of the system from the behavior of ideal solutions (having an activity coefficient -y = 1). The third term results from the three-component nature of such protein solutions, namely, the presence of solvent, protein, and salt. It represents the effect of the concentration of salt on the activity coefficient of the protein. It is usually of minor influence in comparison to the first two. Instruments The direct determination of the turbidity by measuring the attenuation of the intensity of light on passage through a cell of known length is rarely applicable, as the turbidity of most colloidal solutions is of the order of magnitude of r = 10-3 cm. -1. Excessively long cells would have to be used in order to measure it with sufficient accuracy. In practice, however, it is possible to determine with much greater accuracy the intensity of the light scattered laterally in appropriately constructed instruments. The instruments should provide for the determination of: 1. The intensity of the light scattered at 90 ° as compared to the intensity of the incident light (Rg0). 2. The angular distribution of the intensity of scattered light, or at least the dissymmetry, z, i.e., the ratio of intensities at 45 and 135°. 3. The depolarization of the scattered light, p. A number of instruments have been constructed for this purpose. Some of the early instruments were based on the visual comparison of the intensities of the incident and scattered light. 4 With the development of phototubes and particularly photomultip]ier tubes they became outdated. Two commercial instruments are available. The Phoenix ~3 instrument has been described in detail with particular attention given to its absolute calibration. 34 The Aminco 3~instrument utilizes a more compact design and has a particularly efficient amplification of the current emitted by the photomultiplier tube. Both instruments utilize a rectangular cell for the determination of the light scattered at 90 ° and a semioctagonal cell for the dissymmetry measurements. They measure the intensity 33 Phoenix Precision I n s t r u m e n t Co., Philadelphia, Pa. 34 B. A. Brice, M. Halwer, and R. Speiser, J. Opt. Soc. Amer. 40, 768 (1950). 3~ American I n s t r u m e n t Co., Silver Springs, Md.
[5]
LIGHT-SCATTERING MEASUREMENTS
159
either of the scattered light or of the incident light, but they do not measure directly the ratio iO/I. For the determination of the entire lightscattering envelope an instrument has been designed which utilizes a potentiometric bridge circuit, directly giving the above ratio. TMThe solution is contained in a small thin-walled conical glass bulb, and to minimize the reflections of light from the glass itself the bulb is immersed in a liquid having an index of refraction similar to that of the inside of the cell. A light-scattering instrument of very simple construction and application, presented in Fig. 1, has been utilized with satisfaction26 The light from a mercury arc lamp (General Electric AH-4) is focused by a simple optical system on the ceflter of the cell retaining the solution to be h
@ T
D
FIG. 1. Light-scattering photometer, a6 A, light source and housing; B, optical tube; C, semioctagonal light-scattering cell and housing; D, photomultiplier search unit; E, self-generating photocell; F, galvanometer; G, d.c. amplifier and measuring unit.
examined. The optical system is placed in an optical tube, which contains light filters to isolate the desired band of the mercury arc spectrum (Wratten filters No. 2A and C5 for X = 4358 A.). The housing for the cells is cylindrical, the cells being centered by a system of semicircular double recesses. It is fitted with a double bottom for temperature regulation by water circulation. The heavy wall of the housing possesses radial apertures at angles of 45 ° , 90 ° , 135 ° , and 180° to the incident beam of light. Into these circular apertures slides a short metal tube connected to the window of the search unit, which it supports. The search unit is part of an electronic photomultiplier photometer (Photovolt Corp., New York, New York). Thus, either the intensity of the transmitted light or the scattering intensity at the three symmetrical angles can be measured. The window of the search unit is equipped with a photographic shutter and the above-mentioned metal tube. This can be provided with a polarizer and contains a reversed collimating system, holding a lens (f = 5 cm.) and a screen with a central pin-hole, placed in 3~ M. Bier and F. F. Nord, Rev. Sci. Instr. 20, 752 (1949).
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TECHNIQUESFOR CHARACTERIZATIONOF PROTEINS
[15]
the focal plane of the lens. This system is useful for more accurate delimitation of the angle of scattering. The intensity of the primary beam of light is continually checked by a self-generating photoelement. The line voltage to the mercury arc lamp is regulated by a constant-voltage transformer. The lamp housing is water-cooled. A rotating plate with circular apertures of varying diameter, is interposed between the light source and the optical tube, permitting an easy adjustment of the intensity of the primary beam. Semioctagonal
j "
:vi-
,I,I
3v
@~
~600V 2_
Fro. 2. Potentiometric circuit) 7A, spot galvanometer; GM Laboratories, 570-301; B, C, tubes IC5-GT; D, Amperite 2 HT-I ballast tube; E, potentiometer; F, phototube 929; G, photomultiplier tube IP21. cells are used for measurements of dissymmetry. To avoid reflections from the back of the cell with consequent increase in measured dissymmetry, a coating of a dull black glass cement, fused on the outer surface of the back wall, was found to be indispensable2 6 Although the above apparatus permits a continuous check on the intensity of the primary light source, it does not directly compare its intensity with that of the scattered light. This was accomplished in a simple modification of the same apparatus by introducing a mirror deflecting part of the light incident to the cell and the following potentiometric bridge circuit (Fig. 2). The circuit 37 permits accurate measurements of the scattered light over a wide range of intensities. The current ~7Suggested by Mr. S. E. Krewer of Photovolt Corp., New York.
[~]
LIGHT-SCATTERING MEASUREMENTS
161
from the photomultiplier tube, exposed to the scattered light, is balanced by that from the phototube, on which part of the incident beam of light is focused. Comparatively low voltages across the dynodes of the photomultiplier tube provide for good reproducibility of readings and absence of fatigue. The balanced d.c. amplifier is matched for drift-free performance and operates at low current loads. The null detector galvanometer has a sensitivity of 0.02 t~amp./div, and fast response. The power is supplied by dry cell batteries which are well insulated and shielded. One of the most important problems with all instruments is that of their absolute calibration. Debye L° has standardized his instrument with a solution of polystyrene in toluene, the absolute turbidity of which was determined by photographic means, comparing its scattering intensity to the intensity of the primary beam, attenuated by a known factor through multiple reflection on glass plates. This sample of polystyrene was made available in dry form to several laboratories, which utilized it as their primary standard, although its turbidity was consequently re-evaluated 37a as being r = 3.50 X 10-3 cm. -~ at 0.5% concentration in toluene. 34 Other authors have preferred to utilize pure liquids of known turbidity for calibration, for example, benzene '9,3' or carbon disulfide. 4° The calibration of the apparatus with such primary standards is not unambiguous, however, as it does not take into account the varying amount of stray light present in every instrument. The best defined molecular weight of a standard sample of polystyrene is probably that. reported by Frank and Mark. 4°~ The sample was the object of a collaborative study by a number of laboratories, employing osmotic pressure, light scattering, sedimentation and viscosity measurements. In calibrating the apparatus there are also other factors to be considered.~9.34 Thus, the divergent beam of scattered light, coming from the scattering center in the cell, will diverge even more on passage from the medium of high index of refraction in the cell to the outside air. The actual scattering volume and the solid angle of scattering measured will vary in function of the index of refraction of the liquid in the cell. The most important correction is that due to the spreading of the angle of 37~ T h e molecular weight of 37,000 reported for egg albumin 11 was recalculated on the basis of the corrected value of the p r i m a r y s t a n d a r d and was found to be in good agreement with t h a t of other measurements. 38 38 M. Halwer, G. C. Nutting, and B. A. Brice, J. Am. Chem. Soc. 73, 2786 (1951). .~9 C. I. Carr a n d B. H. Zimm, J. Chem. Phys. 18, 1616 (1950); M. Halwer, G. C. Nutting, and B. A. Brice, J. Chem. Phys. 21, 1425 (1953); B. A. Bricc and M. Halwer, J. Opt. Soc. Amer. 44, 340 (1954). 40 R. H. Blaker, R. M. Badger, and T. S. Gilmann, J. Phys. & Colloid Che~7. 53, 794 (1949). 40~ H. P. F r a n k and H. F. Mark, J. Polymer Sci. 17, I (1955).
162
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
scattering, and the observed scattering intensity has to be corrected by multiplying it with the factor. =
-
-
-
r
n
n 2
(24)
where b is the distance between the scattering center and the wall of the cell, r the distance between the scattering center and the measuring phototube, and n the index of refraction of the cell content. The value of the correction factor, Cn, can vary between the limits of 1 and n 2, depending on whether the phototube is adjacent to the wall of the cell (or immersed in the scattering medium) or whether the distance r is much larger than b (as is mostly the case). A more critical discussion of this correction factor with slightly different results was given by Hermans and Levinson. 41 Other corrections also necessary for a rigorous calibration of the instrument are due to the reduced volume, secondary scattering, reflections from the glass walls of the cell, etc., but are of comparatively minor importance. 34 Unfortunately, the above primary standards used for the calibration of the apparatus are mostly not sufficiently stable to be used as working standards for the daily checking of the apparatus. Secondary standards, accurately calibrated, are therefore indispensable. Debye has suggested the use of a polystyrene solution in tributylacetyl citrate2 ° Solid blocks of plastics have also been utilized ~9 as well as opal glass, blocks of magnesium oxide, 33,39 or colloidal silica. 42 In depolarization measurements the response of the phototube to the plane of polarization of light has to be tested. Some tubes are not equally sensitive to the vertically and horizontally polarized light, and, if necessary, the readings have to be appropriately corrected. Refractometers
The implementation of light-scattering theories requires the deterruination of the specific refractive index increments, a term contained in the constants H or K of equations 7 and 8, for every solvent-solute system. Fortunately, in the relatively dilute solutions mostly employed, the specific increment is independent of concentration, and measurements at one concentration only will suffice without need for extrapolation to infinite dilution. It is expressed as (n - n0)/c, where the concentration, as usual in light-scattering measurements, is given in grams per cubic 41 j. j. Hermans and S. Levinson, J. Opt. Soc. Amer. 41, 460 (1951). 42 A. Oth, J. Oth, and V. Desreux, J. Polymer Sci. 10, 551 (1953); G. Oster, J. Polymer Sci. 9, 525 (1952); J. Kraut and W. B. Dandliker, J. Polymer Sci. 18, 563 (1955).
[5]
LIGHT-SCATTERING MEASUREMENTS
163
centimeters. T h e difference in the index of refraction of solution and solvent, (n - no), for 1% solute concentration is for m o s t systems in the range of 0.001 to 0.002. As it is desirable to determine this value with b e t t e r t h a n 1% accuracy, i n s t r u m e n t s are required measuring An of 0.5 to 1 × 10-5. Pulfrich r e f r a c t o m e t e r s or interferometers m a y be suitable, if used with the proper precautions. 43 However, as absolute values of the indices of refraction are not required (the index of refraction of the solvent can be assumed as known), b u t only the difference of the indices between solution and solvent has to be established, differential refractometers are best suited for this purpose. The principle on which t h e y operate is simple. A narrow m o n o c h r o m a t i c b e a m of light is m a d e to traverse a prismatic t w o - c o m p a r t m e n t cell, one half of which is filled with the pure solvent, the other with the solution. T h e deviation of the b e a m of light on passing through the cell is linearly proportional to the difference in the index of refraction between the two c o m p a r t m e n t s . T h e i n s t r u m e n t and each cell h a v e to be calibrated with reference solutions of known instruments, as, for example, solutions of T1NO~44 or sucrose. 4~ Such refractometers are commercially available 33 or can be easily constructed. T h e a p p a r a t u s of Bier and Nord ~6 could be read to a b o u t 1 X 10 -8 An. T h e r e are several a d v a n t a g e s to the use of differential refractometers. Their sensitivity is of the desired order of magnitude, and t h e y are of relatively simple design. T h e y give directly in one m e a s u r e m e n t the desired value, n a m e l y the difference between the indices of refraction of two liquids. Applied to a solution and the pure solvent, t h e y are relatively insensitive to t e m p e r a t u r e , provided b o t h c o m p a r t m e n t s are at t h e r m a l equilibrium. T h e reason for this is that, although the index of refraction changes considerably with t e m p e r a t u r e , the increment for a solution changes v e r y little. I n practice the a u t h o r found it therefore a d v a n t a g e o u s to carry out all m e a s u r e m e n t s at room t e m p e r a t u r e r a t h e r t h a n in t h e r m o s t a t e d cells. For the actual values of the indices of refraction increments for the more c o m m o n proteins the readers are referred to the original literature. ~s,46-4s 43N. Bauer and K. Fajans, in "Physical Methods of Organic Chemistry" (Weissberger, ed.), 2nd ed., Vol. I, Part 2, p. 1141. Interscience Publishers, New York, 1949. 44A. E. Brodsky and N. S. Filippowa, Z. physik. Chem. B23, 399 (1933). 45C. A. Browne and F. W. Zerban, "Physical and Chemical Methods of Sug~tr Analysis," 3rd ed., Table 6. John Wiley & Sons, New York, 1941. 4GS. It. Armstrong, Jr., 1V[. J. E. Budka, K. C. Morrison, and M. ttasson, J. Am. Chem. Soc. 69, 1747 (1947). 47 G. E. Perlmann and L. G. Longsworth, J. Am. Chem. Soc. 70, 2719 (1948). 4s H. A. Barker, J. Biol. Chem. 104, 667 (1934).
164
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
Experimental Procedure The ligh~scattering method requires the measurement of scattering data in solutions of different concentrations as well as in the pure solvent. The first step is therefore the preparation of a stock solution of the substance under investigation and of the solvent. Their preparation presents problems not encountered in usual physicochemical measurements. The most stringent requirement is that of the purity of the liquids, i.e., their freedom from any large-size particles, dust, lint, etc. In protein solutions, of course, this means also the freedom from any coagulated material, denatured protein, which is subject to easy aggregation, or surfacedenatured films. The solvents are best purified by distillation, which can be carried out in sealed glass vessels, the receiving vessel being rinsed by the distillate, which is then returned to the distilling flask. In protein solutions the solvent is usually a buffer solution, and, although carefully distilled water is used for its preparation, dust particles are introduced with the salts. These buffer solutions have therefore to be purified in a similar way as the protein solutions themselves, and, as a matter of fact, salt solutions can be frequently obtained in a colloidally purer state than salt-free water, owing to their coagulating activity. There is no single foolproof procedure which can be employed for the purification of protein solutions. Rather, for each system an individually developed procedure will have to be followed, using all the possible means of purification. These include notably a combination of ultracentrifugation and filtration through Seitz filters, Selas candle filters, sintered-glass filters, etc. The purity of solutions can be attested by several methods. One is the visual observation of the solution in the light-scattering apparatus from a forward direction. Any dust reveals itself in the form of small, brilliant particles. Frequently, a better test is to be found in the measurement of the dissymmetry of scattering, which is the first to increase in the presence of large particles. With solutions possessing an intrinsic dissymmetry of scattering, a good precaution is to ascertain that a repeated cycle of purification steps does not reduce the dissymmetry. In the experience of the author, gained with hydrophylic colloids, which are difficult to purify, or with egg albumin, known for its rapid surface denaturation, the dissymmetry can best be materially reduced to a fixed value by repeated filtration through the finest-size Seitz filters; this is followed by ultracentrifugation and final filtration through sintered-glass filters, to remove any dust coming from the Seitz filter pads. Fine gels, sometimes present in such solutions, invisible and not eliminated by centrifugation, are broken by the filtration and easily centrifuged afterwards.
[5]
LIGHT-SCATTERING
MEASUREMENTS
165
A particularly tedious problem is the one encountered with proteins subject to surface denaturation, to which serum proteins are not very prone, but of which the egg albumin is a good example. For the final filtration of such solutions the author has designed the filter presented in Fig. 3. 31 The solution to be filtered is introduced into compartment A and forced by nitrogen under pressure into compartment B through the ultrafine sintered-glass filter, C. Air can first be completely expelled from the filter element by forcing through it large volumes of distilled water, which also rinses the receiving compartment with dust-free water. This compartment is protected from dust by a glass timble. If this water is left in the filter element, and the protein solution is introduced into
A
B
£
F m . 3. Sintered-glass filter. 3~
compartment A and forced through the filter, it rises in the other compartment without any foaming or even inclusion of air bubbles. The water first contained in the filter element forms a well-visible layer on top of the protein solution and protects it from surface denaturation, as there is no protein-air interface. The same filter is also useful for the filtration of volatile liquids, as no evaporation takes place. The filters require good care for preservation of their filtering ability, and concentrated nitric acid was found to be the best cleaning agent. All the glassware employed in light-scattering measurements, notably the cells and pipets used in transfer of liquids, also has to be prepared with greatest care. They should be rinsed with dust-free solvents and dried in dust-free ovens, preferably in all-glass containers. Detergents are usually employed for their cleaning. For more energetic cleaning, the author has found concentrated nitric acid superior to the standard chromic acid cleaning solution, as the latter tends to coagulate proteins and is also washed out only with great difficulty. The determination of the exact concentration of the solutions is not
166
TECHNIQUESFOR CHARACTERIZATIONOF PROTEINS
[6]
always simple. Usually, the various dilutions employed in the measurements are prepared by progressive dilutions of a stock solution in the light-scattering cell. If measurements at very low concentrations are desired, increasing a m o u n t s of the stock solution are added to the pure solvent. I n m a n y cases the author has found it preferable to prepare every dilution in a separate light-scattering cell, as it avoids gradually increasing contamination of the solutions. Where the solute is available in dry state, as is the case with most polymers, the stock solution of desired concentration is directly prepared, with care t h a t no loss of concentration occurs in the process of purification. I n the case of protein solutions the usual methods of concentration determination are Kjeldahl nitrogen, d r y weight, index of refraction, etc., measurements. These methods are not unambiguous, however, and m a y give rise to discrepancies. 49 49p. L. Kirk, Advances in Protein Chem. 3, 155 (1947).
[6] F10w Birefringence
By W. F. H. M. MOMMAERTS The investigation of flow birefringence (double refraction of flow, anisotropy of flow, streaming birefringence) has not played a great role in enzymology. M y o s i n - A T P a s e is the only well-defined enzyme which is birefringent to a marked extent and which has been the subject of considerable investigation. 1-~ Recently, however, it has become technically possible to extend this m e t h o d of observation to less asymmetric molecules ~°-13 so t h a t several enzymes m a y become accessible to such 1F. Binkley, J. Biol. Chem. 174, 385 (1948). 2 M. Dainty, A. Kleinzeller, A. S. C. Lawrence, M. Miall, J. Needham, D. M. Needham, and S. C. Shen, J. Gen. Physiol. 27, 355 (1944). 3 j. T. Edsall and J. W. Mehl, J. Biol. Chem. 133, 409 (1940). 4 A. S. C. Lawrence, J. Needham, and S. C. Shen, J. Gen. Physiol. 27, 201 (1944). W. F. H. M. Mommaerts, Arkiv Kemi~ Mineral. Geol. 19A, No. 17 (1945). 6 W. F. H. M. Mommaerts, "Muscular Contraction, A Topic in Molecular Physiology." Interscience Publishers, New York, 1950. A. yon Muralt and J. T. Edsall, J. Biol. Chem. 89j 315 (1930). 8 A. yon Muralt and J. T. Edsall, J. Biol. Chem. 89, 351 (1930). 9 M. Joly, G. Schapira, and J. C. Dreyfus, Arch. Biochem. and Biophys. 59, 165 (1956). to j. T. Edsall and J. F. Foster, J. Am. Chem. Soc. 70, 1860 (1948). xl j. T. Edsall, J. F. Foster, and H. Scheinberg, J. Am. Chem. Soc. 69, 2731 (1947). 12j. T. Edsall, C. G. Gordon, J. W. Mehl, H. Scheinberg, and D. W. Mann, Rev. Sci. Instr. 15~ 243 (1944). la j. F. Foster and J. T. Edsall, J. Am. Chem. Soc. 67, 617 (1945).
LIGHT-SCATTERING MEASUREMENTS
147
[5] Light-Scattering M e a s u r e m e n t s By M. BIER Introduction When a colloidal solution is illuminated by a parallel beam of white light, a faint bluish light can be laterally observed. This phenomenon, caused by the scattering of light, is commonly known as the Tyndall effect. 1 The scattering of light is, however, not limited to colloidal solutions, where it is the most pronounced, but is observable in all transparent media, be it a gas, a pure liquid, a solution, or a crystal. Lord Rayleigh 2 formulated in 1871 the fundamental laws of the scattering of light by calculating the polarizability of individual gaseous molecules placed in the oscillating electromagnetic field of a light beam. It is the polarized molecules that then act as sources of secondary radiation, re-emitting the energy of excitation, thus giving rise to the scattered light. Various authors have further contributed to the theories of light scattering, and several reviews adequately cover the field2 -s The intensity of the scattered light depends on a number of measurable quantities and can be expressed as a function of the number of centers of scattering (i.e., molecules) per unit volume. Its quantitative measurement can thus be used for the determination of Avogadro's number] or it can be applied to the determination of molecular weights, if a value for Avogadro's number is adopted. The first applications of light scattering to the determination of particle weights in colloidal systems seem to be due to Smirnov and Bazenov 8 and Putzeys and Brosteaux. 9 Despite this long history it is only since the simplification of the theories by Debye 1° that the light-scattering method has become a practical tool for the study of macromoleeular systems. For the full characterization of the scattering of a nonabsorbing solution we have to know the relative intensity of the scattered light with re, J. Tyndall, Proc. Roy. Soc. 17, 223 (1869). 2 j. W. Strutt (Lord Rayleigh), Phil. Mag. [4] 41, 107, 274, 447 (1871). a S. Bhagavantam, "Scattering of Light and the Raman Effect." Chemical Publishing Co., Brooklyn, 1942. 4 H. Mark, in "Frontiers in C h e m i s t r y " (Burk and Grummitt, eds.), Vol. 5: Chemical Architecture, p. 121. Interscienee Publishers, New York, 1948. 5 G. Oster, Chem. Revs. 43, 319 (1948). J. T. F,dsall and W. B. Dandliker, Fortschr. chem. Forsch. 2, 1 (1951). 7 j . Cabannes, Ann. phys. [9] 15, 5 (1921). 8 L. V. Smirnov and N. M. Bazenov, Colloid J. (U.S.S.R.) 1, 89 (1935). 9 p. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935). t0 l'. Debye, J. Phys. & Colloid Chem. 51, 18 (1947).
148
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
spect to the incident beam, its angular distribution, and its depolarization. Furthermore, the optical relationship between solvent and solute has to be determined and is usually expressed in terms of the specific refractive index increment, ( n - no)/C. From the proper combination of these values we can derive not only the molecular weight but also data on the size and shape of the dissolved macromolecules as well as information on the thermodynamic properties of the system. Owing to the great rapidity with which optical measurements can be carried out, light scattering offers also unique possibilities of following the kinetics of macromolecular reactions involving a change in the size or shape of the dissolved particles. ~1-14 Theory A detailed discussion of the theories of light scattering is far beyond the scope of this article, which shall be limited only to the presentation of the equations of immediate use in the evaluation of light-scattering data. In the consideration of the equations attention should be given to the limitations imposed on the systems to which they apply. The turbidity, T, of a system is defined by the equation I = Ioe -t~
(1)
and expresses the exponential loss of the intensity of light, I, on passage through any nonabsorbing medium of path length I. In appearance it is analogous to Beer's law, defining the loss of light due to specific absorption of light by " c o l o r e d " media. In practice, however, it is distinguished by two important facts. First, the absolute value of r is for most systems considerably lower than the corresponding extinction coefficient of Beer's law for usual "colored" solutions. Second, the energy lost by the transmitted beam is not transformed into heat as in colored solutions, but is immediately re-emitted, with the same wavelength as the incident light, in all directions, the scattering molecules acting as secondary sources of radiation. Whereas the above equation gives the total loss of light intensity in the transmitted beam, the Rayleigh ratio Ro - i°r2
(2)
I0 11 M. Bier and F. F. Nord, Proc. Natl. Acad. Sci. U.S. $§~ 17 (1949). 1~ G. Oster, J. Colloid Sc/. 2, 291 (1947). 13 j . D. Ferry, S. Shulman, K. Gutfreund, and S. Katz, J. Am. Chem. Soc. 74, 5709 (1952). 14 S. Katz, S. Shulman, I. Tinoeo, Jr., I. H. Billick, K. Gutfreund, and J. D. Ferry, Arch. Biochem. and Biophys. 47, 165 (1953).
[5]
LIGHT-SCATTERING MEASUREMENTS
149
defines the intensity, i0, of the scattered light measured at the angle 0 and distance r as a function of the intensity, I0, of the incident light. It is the Rayleigh ratio, sometimes also referred to as the reduced intensity of the scattered light, that is usually measured. The relation between T and R~ is given by 7 = s/./~R0 = 16/t~Rg0 (3) In macromolecular solutions, the solvent is assumed to be continuous, and only the additional scattering, due to the solute molecules, is considered in the above equations. The molecular weight, M, of the solute is then calculated from either the turbidity or the Rayleigh ratio by means of the following simple equations:
Hc
l
-
211
(4)
and
Kc (1 R~
+ cos" 0)
=~
1
(5)
or
Kc Rg0
1 -
M
(6)
These equations apply only to infinitely dilute solutions of isotropic and dielectric molecules of relatively small size as compared to the wavelength of light (<)~/20). The terms H and K group together all the pertinent optical constants for the solute-solvent system at a particular wavelength of light: H=
3NoXo 4
2rr'no' (n c n--~°)2 K=
(s)
NoXo~
Equations 4 to 6 presuppose that the incident light is unpolarized, which then can be considered as the superposition of two polarized beams of light, oscillating at right angles to each other and incoherent in phase. The intensity of the scattered light, due to the beam of light having an electrical vector oscillating in a plane perpendicular to the plane defined by the incident beam and the direction of observation, will be the same at all angles of observation, whereas the angular distribution of the scattered light produced by the horizontally polarized light will be a
150
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
function of cos 2 0. The addition of the two, results in the angular distribution of the scattered light, (1 + cos 2 0), indicated in equation 5. As a consequence, the scattered light at 90 ° should be fully polarized in the vertical direction, and the intensity of scattering in the forward direction (0 - 90 °) is symmetrical to that in the backward direction (90 - 180°). The similarity of equations 4 and 6 to van't Hoff's limiting equation for osmotic pressure, P, P c RT - M (9) is not purely fortuitous, as was shown by Einstein 15 in his analysis of the scattering in more concentrated systems. In condensed systems, the total observed scattering is less than would be obtained by the summation of the scattering from each individual molecule, owing to the destructive interference of the scattered rays. As an extreme case, in an ideal crystal, in which all the molecules would be arranged in a perfect lattice without thermal vibration, any of the volume elements of the crystal would contain exactly the same number of scattering centers, each polarized exactly with the same amplitude and in the same phase, when exposed to incident radiation. As a consequence, for each scattering molecule another one could be found, placed at such a distance as to cause the mutual interference of the secondary radiation. Such a crystal would scatter no light. The observable scattering in real systems is therefore due to the local random fluctuation in the density of the system. Thus, the same number of molecules will scatter much more light in gaseous state than in liquid state, and the liquid will scatter much more than a crystal. The excess scattering of a solution, in which we are interested here, is, by analogy, due to the local fluctuation ill the concentration of the solute molecules, i.e., in the osmotic pressure. This finds its expression in the following equation: Hc _ O(P/RT) (10) 7
Oc
Van't Hoff's equation (9) expresses the osmotic behavior of highly dilute solutions of low molecular weight compounds. For most colloidal solutions, however, it fails at very low concentrations, and the following empirical equation expresses the dependence of the pressure on concentration: P c R - T -- -M -~ Bc~ (11) 1~ A. Einstein, Ann. Physik 83, 1275 (1910).
[5]
LIGHT-SCATTERING MEASUREMENTS
151
Inserting equations 10 and 11 in equations 4 and 6, we obtain Kc
Hc
1
Rg0
T
M
+ 2Be
(12)
These expressions are the ones usually employed in the determination of molecular weights. It can be seen that by plotting either Kc/R~o or H c / r versus concentration a straight line will be obtained, the intercept with the ordinate corresponding to the reciprocal of the molecular weight, while the half-slope, B, is identical with the slope of the osmotic pressure equation. Its meaning will be discussed later on. Although there is thus a great similarity between light-scattering and osmotic pressure measurements, and data obtained by the two methods are frequently compared, there arises an important difference in the results, when the colloidal solutions are not monodisperse, but contain molecules of different weights. In such a case, osmotic pressure measurements give a so-called number average molecular weight, M,~-
Zm,M~
Zm~
(13)
where mi is the molar concentration of the macromolecular component, i, and Mi its molecular weight. By light-scattering measurements, however, a different average is obtained, namely the weight-average molecular weight, Z m i M i2 M w - Y.miM~ (14) The difference is due to the fact that osmotic pressure depends only on the number of particles, whereas the light scattering depends also on the weight of the particles. As a consequence, in polydispersed solutions lightscattering measurements will always yield a higher average molecular weight, and this method is particularly sensitive to dust particles, or other giant-sized impurities. Before going into the question of further data derivable from lightscattering measurements, we have first to extend the theory so as to be able to determine the molecular weight in two of the cases so far excluded by our restrictions, relating to the isotropy and size of the molecules. In the above equations it was assumed that the molecules are optically perfectly symmetrical, i.e., that their polarization is perfectly parallel to that of the exciting beam of light. For most molecules, however, a more appropriate model is that of an ellipsoid of polarization, i.e., they have their own preferential directions of polarization. The plane of polarization of the induced secondary radiation will therefore be slightly inclined with
152
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
respect to that of the primary radiation, and the light scattered at 90 ° will have a small horizontally polarized component besides the normal vertically polarized light. The scattering of such a system is therefore higher than that which would be expected from only the fluctuation in concentration, as it also contains the scattering due to fluctuation in orientation of the anisotropic molecules. For the calculation of molecular weights the latter factor has to be subtracted from the total measured light scattering. The correction factor for Rg0 is (6 - 7p)/(6 ~ 6p), and for r it is ( 6 - 7p)/(6-]-3p), whereby the depolarization ratio, p, is defined as the ratio of the horizontal to the vertical component of the scattered light at 90 ° with unpolarized incident light. 1~ In most macromolecular systems this correction is rather small, p amounting to about 0.004 to 0.04. This is true also in all proteins TM so far studied, although they may be quite asymmetric as in the case of the tobacco mosaic virus. Although this simplifies the use of the method for the determination of molecular weights, it is possibly unfortunate that it precludes fuller use of measurements of depolarization to the study of the shape of the molecules. More information on the actual size and shape of the molecules can be gained from the study of the angular distribution of scattered light from molecules of size comparable to the wavelength of light. Particles which are small compared to the wavelength of light can be treated as point sources of radiation. Particles approaching the dimensions of the wavelength of light (in the particular solvent, i.e., X.ol~o~t= X...... :n,olvont) will, however, have many scattering elements. Thus, interference will occur with a resulting decrease in the observed intensity of the scattering. It was repeatedly shown that the interference will be much greater in the backward direction than in the forward direction, causing a dissymmetry of the angular distribution of the scattering. Equation 5 has therefore to be multiplied by factor P(0) to give the correct angular distribution of the intensity of scattered light: K_c (1 + cos 20)P(O) Re
=
1
(15)
In Table I are listed the values of the functions P(0) for the three usual macromolecular models, namely, for spherical molecules, for rodlike particles, and for randomly coiled linear polymers, in terms of a parameter, x, for the first two and as a function of ~v/X for the coils. The le Cabannes, J., "La Diffusion mol6culaire de la Lumi~re." Presses Universitaires de France, Paris, 1929. 16~ E. P. Geiduschek, Or. Polymer Sci. 13, 408 (1954).
[5]
LIGHT-SCATTERING MEASUREMENTS
153
TABLE I PARTICLE SCATTERING FUNCTIONS FOR THE THREE MACROMOLECULAR MODELS P(O) x"
Spheres
Coils
Rods
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2,2 2,3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6
0.998 0,992 0.982 0.968 0.951 0. 930 0.906 0. 879 0.849 0.816 0.782 0.745 0.707 0.668 0.628 0. 587 0. 547 0. 506 0. 466 0.427 0. 388 0. 351 0.316 0. 282 0.249 0. 219 0. 191 0. 165 0.141 0. 119
1.000 0. 986 0.971 0.949 0.922 0. 890 0. 855 0. 817 0.777 0.736 0. 694 0.653 0.612 0.573 0.535 0. 500 0. 466 0. 434 0. 405 0. 377
0. 999 0. 996 0.990 0. 983 0.973 0. 961 0. 948 0. 932 0.916 0.897
0. 329
0. 627
0. 287
0. 583
0. 252
0.54'3
0. 223
0. 506
0. 198
0.473 0.443 0.417 0.395
0. 857 0.813 0. 767 0.719 0. 672
'~ For random coils = x/a:. p ~ r ~ m e t e r , x, is For spheres
D x = 2v~sin
(16)
For rods
x = 2~ L s i n 0
(17)
F o r coils
R2 0 8 r~ ~ sin2 2_ x = .~
(18)
154
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
The critical dimensions of the models are the diameter, D, of the sphere, the length, L, of the rod, and the root mean square, R, of the distance between the two ends of the random coil. The parameter R is also the largest distance within the coil in its most probable configuration and can be approximated by R2 = Na 2 1 - p l+p
(19)
where N is the number of segments in the chain, a their distance, and p the cosine of the angle between two successive segments. ~7 The above equation assumes a free rotation of the various segments. Experimental results indicate, however, that the size of the coils is usually larger and can be better expressed by R2 = N a ~1 - p 1 ~ c o s ¢
(20)
1 ~ p 1 - cos where the rotation of each segment is assumed to be restricted to an angle ~.~8 With Table I the values of P(O) can be found for any scattering angle or particle size. It is a fortunate characteristic of this function, P(0), that it is sufficient to determine the intensity of the scattering at any two angles for the calculation of the particle size. The two symmetrical angles of 45 ° and 135 ° are usually chosen, and the ratio of scattering intensities at the two angles is referred to as the dissymmetry of scattering, z. In Table II are listed the values of this dissymmetry in function of the critical particle size. Also, in the same table are given the corresponding values of the reciprocal of P(90), which is the correction factor by which the calculated molecular weight has to be multiplied to correct for the interference. The above tables apply only to systems at infinite dilution. The particle size and correction factor 1/P(90), is therefore calculated from the so-called intrinsic dissymmetry, i.e., the dissymmetry extrapolated to zero concentration. Although not strictly valid, the same correction factor is applied also to the slope, 2B, of the conventional light-scattering plots. The shortcoming of the above method is that it depends on the proper selection of the macromolecular model, although the dissymmetry for most systems is rather low, where there is no great difference between the 17W. Kuhn, KoUoid-Z. 68, 2 (1934). is p. j. Flory, "Principles of Polymer Chemistry." Cornell University Press, Ithaca, N.Y., 1953.
[5]
LIGHT-SCATTERING MEASUREMENTS
155
TABLE II DISSYMMETRY AND CORRECTION FACTORS AS A FUNCTION OF PARTICLE SIZE Spheres
D/x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60
l)issym- Correction metry factor (z) [1/P(90)] 1.002 1.01o 1.02t 1.037 1.05s 1.08~ 1.11~ 1.15s 1.202 1.257 1.32o 1.394 1.48, 1.582 1.699 1.837 2.000 2.192 2.41, 2.691 3.01~ 3.417 3.90~ 4.51~ 5.29~
1.002 1.01o 1.013 1.026 1.04o 1.059 1.08, 1.107 1.13s 1.173 1.214 1.26o 1.31a 1.373 1.441 1.517 1.604 1.702 1.81~ 1.94, 2.08~ 2.25, 2.44o 2.650 2.91~
Coils
Rods
1)issym- Correction merry factor (z) [1/P(90)]
Dissym- Correction merry factor (z) [1/P(90)]
1.00 1.012 1.025 1.04o 1.062 1.09o 1.12a 1.162 1.205 1.255 1.31o 1.37o 1.436 1.50s 1.582 1.663 1.74s 1.83s 1.93o 2.02o 2.12~ 2.22~ 2.328 2.42s 2.53s 2.64o 2.74~ 2.84s 2.95, 3.05~
1.00 1.005 1.017 1.02s 1.04s 1.064 1.08s 1.115 1.14s 1.183 1.223 1.26s 1.317 1.37, 1.43o 1.49~ 1.563 1.63~ 1.72o 1.80~ 1.89s 1.99~ 2.090 2.20~ 2.32~ 2.44: 2.57~ 2.70~ 2.84: 2.99~
1.00 1.005 1.011 1.02o 1.03, 1.045 1.061 1.08, 1.102 1.127 1.154 1.183 1.21o 1.25, 1.28s 1.32T 1.37o 1.414 1.46o 1.50~ 1.55s 1.60~ 1.65~ 1.70s 1.75~ 1.80~ 1.84o 1.89~ 1.93a 1.97~
1.00 1.004 1.00s 1.014 1.022 1.032 1.043 1.057 1.072 1.09o 1.10~ 1.131 1.154 1.18o 1.20~ 1.237 1.27o 1.304 1.34~ 1.38o 1.42~ 1.46~ 1.51o 1.55s 1.6(b 1.659 1.71~ 1.76~ 1.827 1.88s
three models. For larger particles the method fails to give any indication of the actual shape of the particles. This can be obtained by a more complete determination of the angular distribution of the scattered light.. In this mei hod 19 the observed angular i~ltensities of the scattering are simultaneously graphically extrapolated to zero angle and zero concentration, whereby the effects of interference disappear, as P(0) is always l For 1~B. |1. Zimm, J. Chem. Phys. 16, 1093, 1099 (1948).
156
TECHI~IQUESFOR CHARACTERIZATION OF PROTEINS
[5]
still larger particles the more complex theory of Mie 2° has to be applied. ~1 Light-scattering measurements can also yield valuable data on the thermodynamics of solvent-solute interaction. I t is sufficient to recall equation 10 and the fact that, according to classical thermodynamics, PIT"l = - A F 1
(21)
where 171 is the partial molal volume of the solvent, and AP~ is its partial molal free energy. The latter thus contains an e n t r o p y term and a heat content term, both referring to the mixing of the solvent and solute. These terms are to be found in the slope, 2B, of equation 12, and the heat content term can be determined by measuring the slope at different temperatures. T h e interpretation of the results in solutions of randomly coiled polymers has received considerable attention from various authors ~s,~2-24 but is beyond the scope of this article. I t will suffice to mention the following qualitative conclusions. In a good solvent, i.e., where there is a pronounced energetic interaction between the solute and solvent, the slope 2B will be large. In a poor solvent, however, i.e., in systems where the solvent and solute have but small affinity, the slope will be small, zero, or even slightly negative. Negative slopes are indicative of approaching phase separation, i.e., solute precipitation, the occurrence of which prevents strongly negative slopes. ~,25,26 Light scattering results in mixed solvents or solvent-precipitant mixtures employed in polymer fractionation should, however, be interpreted with caution, owing to the possible preferential absorption of one of the solvents. 27 In solutions of charged macromolecules and proteins in particular, the large effects on light scattering of the electrostatic forces surrounding them have also to be considered. Scatchard's theoretical t r e a t m e n t of such systems ~s was applied b y him and co-workers to a detailed s t u d y of the osmotic pressure of bovine serum albumin 29 and can be adapted *oG. Mie, Ann. Physik 25, 377 (1908). ~ A. S. Kenyon and V. K. LaMer, J. Colloid Sci. 4, 163 (1949). ~ M. L. Huggins, J. Am. Chem. Soc. 64, 1712 (1942); Ann. N.Y. Acad. Sci. 45, 1 (1942). 2ap. j. Flory, J. Chem. Phys. 10, 51 (1942); 13, 453 (1943); 17, 1347 (1949). 24A. R. Miller, "The Theory of Solutions of High Polymers." Oxford University Press, New York, 1948. 36F. F. Nord, M. Bier, and S. N. Timasheff, J. Am. Chem. Soc. 73, 289 (1951). 26S. N. Timasheff, M. Bier, and F. F. Nord, J. Phys. & Colloid Chem. 55, 1134 (1949). 27R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 687 (1946). 38G. Scatchard, J. Am. Chem. Soc. 68, 2315 (1946). 32G. Seatehard, A. C. Batchelder, and A. Brown, J. Am. Chem. Soc. 68, 2320 (1946).
[~]
LIGHT-SCATTERII~I'G MEASUREMENTS
157
directly to the light scattering of protein solutions. 6,~°,3~In the present discussion we shall restrict ourselves to the most common case encountered in practice, i.e., to protein solutions in the presence of a relatively large concentration of salts. For the t r e a t m e n t of d a t a on salt-free solutions of isoionic proteins, the reader is referred to the original contributions. 3~ Accordingly, the slope 2B of equation 12 contains three terms:
2B -
1000/Z2 MS ~-~ + ~2~
~am') ~-33m/
(22)
T h e factor 1000 arises from the conversion of molal concentrations to weight concentrations, which in light-scattering measurements are usually expressed in grams per cubic centimeters. The first term, Z 2 / 2 m M 2, is the most important, as it expresses the dependence of the light-scattering intensity on the valence, Z, of the protein and the molal concentration, m, of the salt. In osmotic pressure experiments this t e r m is a direct consequence of the unequal distribution of the diffusible salts across the membrane as a consequence of D o n n a n ' s equilibrium. The dependence of the term on the reciprocal of the square of the molecular weight of the protein, M, is only apparent, as the valence of the protein, Z, is given b y h M , where h is the n u m b e r of hydrogen or hydroxyl ions bound per gram of protein. T h e t e r m thus represents the ratio of charge to mass for the given protein and would require the slope to be zero at the isoionic point (Z = 0) and increase symmetrically from b o t h sides of it. Therefore, the t u r b i d i t y of a protein solution of finite concentration should be the highest at the isoionic point, and, of course, the osmotic pressure is the lowest at the same point. In practice, however, osmotic pressure determinations 29 and light-scattering measurements of serum albumin 32 as well as on egg albumin 3°,~1 have shown t h a t the minimum slope is shifted toward more acid values from the isoionic point as a consequence of the other terms of equation 22. Increasing salt concentration should also result in an increase in t u r b i d i t y of a protein solution as a result of the decrease in slope 2B. The second term of equation 22 represents the variation in the 3°F. F. Nord and M. Bier, irt "Handbuch der K~ltetechnik" (Plank, ed.), Vol. 9: Biochemische Grundlagen der Lebensmittelfrischhaltung, p. 84. Springer-Verlag, Berlin, 1952. 31M. Bier, Dissertation, Fordham University, New York, 1950. 31~W. B. Dandliker, J. Am. Chem. Soc. 76, 6036 (1954); J. G. Kirkwood and S. N. Timasheff, Arch. Biochem. and Biophys. 65, in press (1956). ~: J. T. Edsall, H. Edelhoch, R. Lontie, and P. R. Morrison, J. Ant. Chem. Soc. 72, 4641 (1950).
158
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[~]
logarithm of the activity coefficient, % of the protein with the change in protein concentration: ~22 = 0 In ~protoin/Omp~ote~n (23) The term has a meaning similar to that of the slope 2B in solutions of uncharged macromolecules, as it represents the deviation of the system from the behavior of ideal solutions (having an activity coefficient -y = 1). The third term results from the three-component nature of such protein solutions, namely, the presence of solvent, protein, and salt. It represents the effect of the concentration of salt on the activity coefficient of the protein. It is usually of minor influence in comparison to the first two. Instruments The direct determination of the turbidity by measuring the attenuation of the intensity of light on passage through a cell of known length is rarely applicable, as the turbidity of most colloidal solutions is of the order of magnitude of r = 10-3 cm. -1. Excessively long cells would have to be used in order to measure it with sufficient accuracy. In practice, however, it is possible to determine with much greater accuracy the intensity of the light scattered laterally in appropriately constructed instruments. The instruments should provide for the determination of: 1. The intensity of the light scattered at 90 ° as compared to the intensity of the incident light (Rg0). 2. The angular distribution of the intensity of scattered light, or at least the dissymmetry, z, i.e., the ratio of intensities at 45 and 135°. 3. The depolarization of the scattered light, p. A number of instruments have been constructed for this purpose. Some of the early instruments were based on the visual comparison of the intensities of the incident and scattered light. 4 With the development of phototubes and particularly photomultip]ier tubes they became outdated. Two commercial instruments are available. The Phoenix ~3 instrument has been described in detail with particular attention given to its absolute calibration. 34 The Aminco 3~instrument utilizes a more compact design and has a particularly efficient amplification of the current emitted by the photomultiplier tube. Both instruments utilize a rectangular cell for the determination of the light scattered at 90 ° and a semioctagonal cell for the dissymmetry measurements. They measure the intensity 33 Phoenix Precision I n s t r u m e n t Co., Philadelphia, Pa. 34 B. A. Brice, M. Halwer, and R. Speiser, J. Opt. Soc. Amer. 40, 768 (1950). 3~ American I n s t r u m e n t Co., Silver Springs, Md.
[5]
LIGHT-SCATTERING MEASUREMENTS
159
either of the scattered light or of the incident light, but they do not measure directly the ratio iO/I. For the determination of the entire lightscattering envelope an instrument has been designed which utilizes a potentiometric bridge circuit, directly giving the above ratio. TMThe solution is contained in a small thin-walled conical glass bulb, and to minimize the reflections of light from the glass itself the bulb is immersed in a liquid having an index of refraction similar to that of the inside of the cell. A light-scattering instrument of very simple construction and application, presented in Fig. 1, has been utilized with satisfaction26 The light from a mercury arc lamp (General Electric AH-4) is focused by a simple optical system on the ceflter of the cell retaining the solution to be h
@ T
D
FIG. 1. Light-scattering photometer, a6 A, light source and housing; B, optical tube; C, semioctagonal light-scattering cell and housing; D, photomultiplier search unit; E, self-generating photocell; F, galvanometer; G, d.c. amplifier and measuring unit.
examined. The optical system is placed in an optical tube, which contains light filters to isolate the desired band of the mercury arc spectrum (Wratten filters No. 2A and C5 for X = 4358 A.). The housing for the cells is cylindrical, the cells being centered by a system of semicircular double recesses. It is fitted with a double bottom for temperature regulation by water circulation. The heavy wall of the housing possesses radial apertures at angles of 45 ° , 90 ° , 135 ° , and 180° to the incident beam of light. Into these circular apertures slides a short metal tube connected to the window of the search unit, which it supports. The search unit is part of an electronic photomultiplier photometer (Photovolt Corp., New York, New York). Thus, either the intensity of the transmitted light or the scattering intensity at the three symmetrical angles can be measured. The window of the search unit is equipped with a photographic shutter and the above-mentioned metal tube. This can be provided with a polarizer and contains a reversed collimating system, holding a lens (f = 5 cm.) and a screen with a central pin-hole, placed in 3~ M. Bier and F. F. Nord, Rev. Sci. Instr. 20, 752 (1949).
160
TECHNIQUESFOR CHARACTERIZATIONOF PROTEINS
[15]
the focal plane of the lens. This system is useful for more accurate delimitation of the angle of scattering. The intensity of the primary beam of light is continually checked by a self-generating photoelement. The line voltage to the mercury arc lamp is regulated by a constant-voltage transformer. The lamp housing is water-cooled. A rotating plate with circular apertures of varying diameter, is interposed between the light source and the optical tube, permitting an easy adjustment of the intensity of the primary beam. Semioctagonal
j "
:vi-
,I,I
3v
@~
~600V 2_
Fro. 2. Potentiometric circuit) 7A, spot galvanometer; GM Laboratories, 570-301; B, C, tubes IC5-GT; D, Amperite 2 HT-I ballast tube; E, potentiometer; F, phototube 929; G, photomultiplier tube IP21. cells are used for measurements of dissymmetry. To avoid reflections from the back of the cell with consequent increase in measured dissymmetry, a coating of a dull black glass cement, fused on the outer surface of the back wall, was found to be indispensable2 6 Although the above apparatus permits a continuous check on the intensity of the primary light source, it does not directly compare its intensity with that of the scattered light. This was accomplished in a simple modification of the same apparatus by introducing a mirror deflecting part of the light incident to the cell and the following potentiometric bridge circuit (Fig. 2). The circuit 37 permits accurate measurements of the scattered light over a wide range of intensities. The current ~7Suggested by Mr. S. E. Krewer of Photovolt Corp., New York.
[~]
LIGHT-SCATTERING MEASUREMENTS
161
from the photomultiplier tube, exposed to the scattered light, is balanced by that from the phototube, on which part of the incident beam of light is focused. Comparatively low voltages across the dynodes of the photomultiplier tube provide for good reproducibility of readings and absence of fatigue. The balanced d.c. amplifier is matched for drift-free performance and operates at low current loads. The null detector galvanometer has a sensitivity of 0.02 t~amp./div, and fast response. The power is supplied by dry cell batteries which are well insulated and shielded. One of the most important problems with all instruments is that of their absolute calibration. Debye L° has standardized his instrument with a solution of polystyrene in toluene, the absolute turbidity of which was determined by photographic means, comparing its scattering intensity to the intensity of the primary beam, attenuated by a known factor through multiple reflection on glass plates. This sample of polystyrene was made available in dry form to several laboratories, which utilized it as their primary standard, although its turbidity was consequently re-evaluated 37a as being r = 3.50 X 10-3 cm. -~ at 0.5% concentration in toluene. 34 Other authors have preferred to utilize pure liquids of known turbidity for calibration, for example, benzene '9,3' or carbon disulfide. 4° The calibration of the apparatus with such primary standards is not unambiguous, however, as it does not take into account the varying amount of stray light present in every instrument. The best defined molecular weight of a standard sample of polystyrene is probably that. reported by Frank and Mark. 4°~ The sample was the object of a collaborative study by a number of laboratories, employing osmotic pressure, light scattering, sedimentation and viscosity measurements. In calibrating the apparatus there are also other factors to be considered.~9.34 Thus, the divergent beam of scattered light, coming from the scattering center in the cell, will diverge even more on passage from the medium of high index of refraction in the cell to the outside air. The actual scattering volume and the solid angle of scattering measured will vary in function of the index of refraction of the liquid in the cell. The most important correction is that due to the spreading of the angle of 37~ T h e molecular weight of 37,000 reported for egg albumin 11 was recalculated on the basis of the corrected value of the p r i m a r y s t a n d a r d and was found to be in good agreement with t h a t of other measurements. 38 38 M. Halwer, G. C. Nutting, and B. A. Brice, J. Am. Chem. Soc. 73, 2786 (1951). .~9 C. I. Carr a n d B. H. Zimm, J. Chem. Phys. 18, 1616 (1950); M. Halwer, G. C. Nutting, and B. A. Brice, J. Chem. Phys. 21, 1425 (1953); B. A. Bricc and M. Halwer, J. Opt. Soc. Amer. 44, 340 (1954). 40 R. H. Blaker, R. M. Badger, and T. S. Gilmann, J. Phys. & Colloid Che~7. 53, 794 (1949). 40~ H. P. F r a n k and H. F. Mark, J. Polymer Sci. 17, I (1955).
162
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
scattering, and the observed scattering intensity has to be corrected by multiplying it with the factor. =
-
-
-
r
n
n 2
(24)
where b is the distance between the scattering center and the wall of the cell, r the distance between the scattering center and the measuring phototube, and n the index of refraction of the cell content. The value of the correction factor, Cn, can vary between the limits of 1 and n 2, depending on whether the phototube is adjacent to the wall of the cell (or immersed in the scattering medium) or whether the distance r is much larger than b (as is mostly the case). A more critical discussion of this correction factor with slightly different results was given by Hermans and Levinson. 41 Other corrections also necessary for a rigorous calibration of the instrument are due to the reduced volume, secondary scattering, reflections from the glass walls of the cell, etc., but are of comparatively minor importance. 34 Unfortunately, the above primary standards used for the calibration of the apparatus are mostly not sufficiently stable to be used as working standards for the daily checking of the apparatus. Secondary standards, accurately calibrated, are therefore indispensable. Debye has suggested the use of a polystyrene solution in tributylacetyl citrate2 ° Solid blocks of plastics have also been utilized ~9 as well as opal glass, blocks of magnesium oxide, 33,39 or colloidal silica. 42 In depolarization measurements the response of the phototube to the plane of polarization of light has to be tested. Some tubes are not equally sensitive to the vertically and horizontally polarized light, and, if necessary, the readings have to be appropriately corrected. Refractometers
The implementation of light-scattering theories requires the deterruination of the specific refractive index increments, a term contained in the constants H or K of equations 7 and 8, for every solvent-solute system. Fortunately, in the relatively dilute solutions mostly employed, the specific increment is independent of concentration, and measurements at one concentration only will suffice without need for extrapolation to infinite dilution. It is expressed as (n - n0)/c, where the concentration, as usual in light-scattering measurements, is given in grams per cubic 41 j. j. Hermans and S. Levinson, J. Opt. Soc. Amer. 41, 460 (1951). 42 A. Oth, J. Oth, and V. Desreux, J. Polymer Sci. 10, 551 (1953); G. Oster, J. Polymer Sci. 9, 525 (1952); J. Kraut and W. B. Dandliker, J. Polymer Sci. 18, 563 (1955).
[5]
LIGHT-SCATTERING MEASUREMENTS
163
centimeters. T h e difference in the index of refraction of solution and solvent, (n - no), for 1% solute concentration is for m o s t systems in the range of 0.001 to 0.002. As it is desirable to determine this value with b e t t e r t h a n 1% accuracy, i n s t r u m e n t s are required measuring An of 0.5 to 1 × 10-5. Pulfrich r e f r a c t o m e t e r s or interferometers m a y be suitable, if used with the proper precautions. 43 However, as absolute values of the indices of refraction are not required (the index of refraction of the solvent can be assumed as known), b u t only the difference of the indices between solution and solvent has to be established, differential refractometers are best suited for this purpose. The principle on which t h e y operate is simple. A narrow m o n o c h r o m a t i c b e a m of light is m a d e to traverse a prismatic t w o - c o m p a r t m e n t cell, one half of which is filled with the pure solvent, the other with the solution. T h e deviation of the b e a m of light on passing through the cell is linearly proportional to the difference in the index of refraction between the two c o m p a r t m e n t s . T h e i n s t r u m e n t and each cell h a v e to be calibrated with reference solutions of known instruments, as, for example, solutions of T1NO~44 or sucrose. 4~ Such refractometers are commercially available 33 or can be easily constructed. T h e a p p a r a t u s of Bier and Nord ~6 could be read to a b o u t 1 X 10 -8 An. T h e r e are several a d v a n t a g e s to the use of differential refractometers. Their sensitivity is of the desired order of magnitude, and t h e y are of relatively simple design. T h e y give directly in one m e a s u r e m e n t the desired value, n a m e l y the difference between the indices of refraction of two liquids. Applied to a solution and the pure solvent, t h e y are relatively insensitive to t e m p e r a t u r e , provided b o t h c o m p a r t m e n t s are at t h e r m a l equilibrium. T h e reason for this is that, although the index of refraction changes considerably with t e m p e r a t u r e , the increment for a solution changes v e r y little. I n practice the a u t h o r found it therefore a d v a n t a g e o u s to carry out all m e a s u r e m e n t s at room t e m p e r a t u r e r a t h e r t h a n in t h e r m o s t a t e d cells. For the actual values of the indices of refraction increments for the more c o m m o n proteins the readers are referred to the original literature. ~s,46-4s 43N. Bauer and K. Fajans, in "Physical Methods of Organic Chemistry" (Weissberger, ed.), 2nd ed., Vol. I, Part 2, p. 1141. Interscience Publishers, New York, 1949. 44A. E. Brodsky and N. S. Filippowa, Z. physik. Chem. B23, 399 (1933). 45C. A. Browne and F. W. Zerban, "Physical and Chemical Methods of Sug~tr Analysis," 3rd ed., Table 6. John Wiley & Sons, New York, 1941. 4GS. It. Armstrong, Jr., 1V[. J. E. Budka, K. C. Morrison, and M. ttasson, J. Am. Chem. Soc. 69, 1747 (1947). 47 G. E. Perlmann and L. G. Longsworth, J. Am. Chem. Soc. 70, 2719 (1948). 4s H. A. Barker, J. Biol. Chem. 104, 667 (1934).
164
TECHNIQUES FOR CHARACTERIZATION OF PROTEINS
[5]
Experimental Procedure The ligh~scattering method requires the measurement of scattering data in solutions of different concentrations as well as in the pure solvent. The first step is therefore the preparation of a stock solution of the substance under investigation and of the solvent. Their preparation presents problems not encountered in usual physicochemical measurements. The most stringent requirement is that of the purity of the liquids, i.e., their freedom from any large-size particles, dust, lint, etc. In protein solutions, of course, this means also the freedom from any coagulated material, denatured protein, which is subject to easy aggregation, or surfacedenatured films. The solvents are best purified by distillation, which can be carried out in sealed glass vessels, the receiving vessel being rinsed by the distillate, which is then returned to the distilling flask. In protein solutions the solvent is usually a buffer solution, and, although carefully distilled water is used for its preparation, dust particles are introduced with the salts. These buffer solutions have therefore to be purified in a similar way as the protein solutions themselves, and, as a matter of fact, salt solutions can be frequently obtained in a colloidally purer state than salt-free water, owing to their coagulating activity. There is no single foolproof procedure which can be employed for the purification of protein solutions. Rather, for each system an individually developed procedure will have to be followed, using all the possible means of purification. These include notably a combination of ultracentrifugation and filtration through Seitz filters, Selas candle filters, sintered-glass filters, etc. The purity of solutions can be attested by several methods. One is the visual observation of the solution in the light-scattering apparatus from a forward direction. Any dust reveals itself in the form of small, brilliant particles. Frequently, a better test is to be found in the measurement of the dissymmetry of scattering, which is the first to increase in the presence of large particles. With solutions possessing an intrinsic dissymmetry of scattering, a good precaution is to ascertain that a repeated cycle of purification steps does not reduce the dissymmetry. In the experience of the author, gained with hydrophylic colloids, which are difficult to purify, or with egg albumin, known for its rapid surface denaturation, the dissymmetry can best be materially reduced to a fixed value by repeated filtration through the finest-size Seitz filters; this is followed by ultracentrifugation and final filtration through sintered-glass filters, to remove any dust coming from the Seitz filter pads. Fine gels, sometimes present in such solutions, invisible and not eliminated by centrifugation, are broken by the filtration and easily centrifuged afterwards.
[5]
LIGHT-SCATTERING
MEASUREMENTS
165
A particularly tedious problem is the one encountered with proteins subject to surface denaturation, to which serum proteins are not very prone, but of which the egg albumin is a good example. For the final filtration of such solutions the author has designed the filter presented in Fig. 3. 31 The solution to be filtered is introduced into compartment A and forced by nitrogen under pressure into compartment B through the ultrafine sintered-glass filter, C. Air can first be completely expelled from the filter element by forcing through it large volumes of distilled water, which also rinses the receiving compartment with dust-free water. This compartment is protected from dust by a glass timble. If this water is left in the filter element, and the protein solution is introduced into
A
B
£
F m . 3. Sintered-glass filter. 3~
compartment A and forced through the filter, it rises in the other compartment without any foaming or even inclusion of air bubbles. The water first contained in the filter element forms a well-visible layer on top of the protein solution and protects it from surface denaturation, as there is no protein-air interface. The same filter is also useful for the filtration of volatile liquids, as no evaporation takes place. The filters require good care for preservation of their filtering ability, and concentrated nitric acid was found to be the best cleaning agent. All the glassware employed in light-scattering measurements, notably the cells and pipets used in transfer of liquids, also has to be prepared with greatest care. They should be rinsed with dust-free solvents and dried in dust-free ovens, preferably in all-glass containers. Detergents are usually employed for their cleaning. For more energetic cleaning, the author has found concentrated nitric acid superior to the standard chromic acid cleaning solution, as the latter tends to coagulate proteins and is also washed out only with great difficulty. The determination of the exact concentration of the solutions is not
166
TECHNIQUESFOR CHARACTERIZATIONOF PROTEINS
[6]
always simple. Usually, the various dilutions employed in the measurements are prepared by progressive dilutions of a stock solution in the light-scattering cell. If measurements at very low concentrations are desired, increasing a m o u n t s of the stock solution are added to the pure solvent. I n m a n y cases the author has found it preferable to prepare every dilution in a separate light-scattering cell, as it avoids gradually increasing contamination of the solutions. Where the solute is available in dry state, as is the case with most polymers, the stock solution of desired concentration is directly prepared, with care t h a t no loss of concentration occurs in the process of purification. I n the case of protein solutions the usual methods of concentration determination are Kjeldahl nitrogen, d r y weight, index of refraction, etc., measurements. These methods are not unambiguous, however, and m a y give rise to discrepancies. 49 49p. L. Kirk, Advances in Protein Chem. 3, 155 (1947).
[6] F10w Birefringence
By W. F. H. M. MOMMAERTS The investigation of flow birefringence (double refraction of flow, anisotropy of flow, streaming birefringence) has not played a great role in enzymology. M y o s i n - A T P a s e is the only well-defined enzyme which is birefringent to a marked extent and which has been the subject of considerable investigation. 1-~ Recently, however, it has become technically possible to extend this m e t h o d of observation to less asymmetric molecules ~°-13 so t h a t several enzymes m a y become accessible to such 1F. Binkley, J. Biol. Chem. 174, 385 (1948). 2 M. Dainty, A. Kleinzeller, A. S. C. Lawrence, M. Miall, J. Needham, D. M. Needham, and S. C. Shen, J. Gen. Physiol. 27, 355 (1944). 3 j. T. Edsall and J. W. Mehl, J. Biol. Chem. 133, 409 (1940). 4 A. S. C. Lawrence, J. Needham, and S. C. Shen, J. Gen. Physiol. 27, 201 (1944). W. F. H. M. Mommaerts, Arkiv Kemi~ Mineral. Geol. 19A, No. 17 (1945). 6 W. F. H. M. Mommaerts, "Muscular Contraction, A Topic in Molecular Physiology." Interscience Publishers, New York, 1950. A. yon Muralt and J. T. Edsall, J. Biol. Chem. 89j 315 (1930). 8 A. yon Muralt and J. T. Edsall, J. Biol. Chem. 89, 351 (1930). 9 M. Joly, G. Schapira, and J. C. Dreyfus, Arch. Biochem. and Biophys. 59, 165 (1956). to j. T. Edsall and J. F. Foster, J. Am. Chem. Soc. 70, 1860 (1948). xl j. T. Edsall, J. F. Foster, and H. Scheinberg, J. Am. Chem. Soc. 69, 2731 (1947). 12j. T. Edsall, C. G. Gordon, J. W. Mehl, H. Scheinberg, and D. W. Mann, Rev. Sci. Instr. 15~ 243 (1944). la j. F. Foster and J. T. Edsall, J. Am. Chem. Soc. 67, 617 (1945).