[7] Measurement of translational and rotational diffusion coefficients by laser light scattering

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

119

Furthermore, careful sedimentation equilibrium measurements of ribonuclease in 6 M GuHC128 yielded an appropriate molecular weight (13,700 _ 300) using $' = 0.706. Thus, several factors including economy, sample size, solution conditions, and accuracy may dictate the choice of using the density gradient method. 'SF. J. Reithel, J. E. Robbins, and G. Gorin, Arch. Biochem. Biophys. 108, 409 (1964). 11. E. Marker, C. A. Nelson, and C. Tanford, Biochemistry 3, 281 (1964). 12. F. J. Reithel and J. D. Sakura, J. Phys. Chem. 67, 2497 (1963). 13. F. J. Reithel, J. E. Robbins, and G. Gorin, Arch. Biochem. Biophys. 108, 409 (1964). 14. K. 0. Pedersen, Biochem. J. 30, 961 (1936). 15. R. M. Metrione, A. G. Neves, and J. S. Fruton, Biochemistry 5, 1597 (1966). 16. R. M. Metrione, Y. Okuda, and G. F. Fairclough, Jr., Biochemistry 9, 2427 (1970). 17. L. M. Krausz and R. R. Becker, J. Biol. Chem. 243, 4606 (1968). b Pycnometric determination (see Ref. 7 above).

[7]

M e a s u r e m e n t of T r a n s l a t i o n a l a n d R o t a t i o n a l D i f f u s i o n Coefficients by Laser Light Scattering B y STUART B. DUBIN

Light scattering techniques have long provided powerful methods for the determination of macromolecular weight, size, and shape. These studies rely upon the accurate determination of the intensity of the light scattered by solutions of macromolecul~s and have been described in a previous volume in this series? Because the lifetimes of the random fluctuations in dielectric constant which produce this scattering are so long compared with the period of the incident light, the spectrum of the scattered light is far too narrow to have been measured before the recent advent of laser light sources, light-mixing spectroscopic techniques, and high resolution Fabry-Perot interferometers. The scattering has thus been referred to as "inelastic" or, more properly, "quasielastic." As early as 1926 Mandel'shtam 2 recognized that the translational diffusion coefficient (DT) of maeromolecules could be obtained from the spectrum of the light they scatter. Lack of spatial coherence and monochromaticity in conventional light sources rendered such experiments im1M. Bier, see Vol. IV, p. 147. 2L. I. Mandel'shtam, Zh. Russ. Fiz.-Khim. Obshch. 58, 381 (1926).

120

MOLECULAR WEIGHT DETERMINATIONS

[7]

possible, however. It is the purpose of the present work to describe how the laser's spatial coherence and high power have allowed the methods of light-mixing spectroscopy to make determination of DT a straightforward procedure. In addition, use of a single ]requency laser (not required for light-mixing spectroscopy), in conjunction with a highresolution Fabry-Perot interferometer, allows the determination of the rotational diffusion coefficient (DR) of small maeromoleeules, such as enzymes. Since the signal-to-noise ratio obtainable in such experiments is the primary factor in determining their feasibility, adequate theoretical and descriptive background is presented for a general understanding of the scattering process, the operation of the appropriate spectrometers, and the interpretation of the observed spectra. Several applications of these techniques to studies in enzymology are discussed in detail, ineluding molecular weight determination, absolute size and shape of enzymes in solution, and observation of eonformational changes during protein denaturation. The problems of sample polydispersity and multiple scattering are also discussed. Review of Light-Scattering Principles The two principal sources 3 of quasielastie light scattering in enzyme solutions are fluctuations in the medium's local polarizability (~) due to fluctuations in concentration and optical isotropy. These fluctuations arise because the maeromoleeules are in constant thermal agitation and hence both translate from place to place and rotate. This section treats of the intensity, spatial coherence, and temporal coherence of the light. scattered by these fluctuations and demonstrates that DT and D~ can be obtained from the spectrum of the scattered light. The description is therefore one which emphasizes the dynamic origins of the scattering, essential to the understanding of the spectrometers used in determining the diffusion coefficients. This emphasis on the temporal aspects of the scattering is usually lacking in analyses concerned primarily with the intensity of the scattered light.

Spectrum of the Scattered Light Consider a polarized radiation incident upon a solution of macromolecules, as shown in Fig. 1. The medium is described by a polarizability per unit volume, a(r,t), which is in general a tensor, reflecting the fact that the molecules may be optically anisotropic. The observer sees the superposition of the light scattered by each volume element 3Only the excess scattering of the macromolecules over that, of the solvent is considered.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

121

:FIG. 1. The scattering geometry.

in the illuminated volume. The scattered light consists chiefly of a strong component having the same polarization as the incident light. This arises from fluctuations in a(r,t) about its average value due to concentration fluctuations. In fact, by Fourier analysis of these fluctuations in terms of plane waves, it is readily shown ~ that light scattered at an angle 0 from the incident beam is the result of a "Bragg reflection" of the incident light by a fluctuation in polarizability whose wavelength (X~) satisfies the Bragg Law: (Xo/n) = 2Xf s i n ( 0 / 2 )

(1)

where Xo is the wavelength of the incident light in vacuum and n is the solution index of refraction. This result indicates that the waveveetor of the incident light in the medium (k = 2=n/X0) and that of the scattered light (Iks] = ]k I) conserve wavevector with the wavevector of the fluctuation in dielectric constant (K = 2~rf,kr) which produced the scattering¢ 4 p. Debye, Phys. Rev. Lett. 14, 783 (1965). hA. Einstein, Ann. Phys. 33, 1275 (1910).

122

MOLECULAR WEIGHT DETERMINATIONS

[7]

k FIG. 2. Geometric representation of the scattering process.

This in turn (Fig. 2) is simply the law of conservation of momentum as applied to light scattering. It is thus seen that the scattering process is a mapping of the fluctuation in polarizability of wavevector K, 8~(K,t), onto the amplitude of the electric field of the scattered light observed at a point R and time t, E~(B,t). Hence, E~(B,t) is proportional to ei~0t 8a(K,t) where o)o is angular frequency of the incident light. It is the fluctuations in the solute concentration C which almost entirely produce the fluctuations in the solution polarizability: ~(K,t)

=(~)~C(K,t).

(2)

For a dilute solution of macromolecules the fluctuations in concentration at any point r in the solution obey the translational diffusion equation6: O [~C(r,t)] =

Ot

DTV2[~C(r,t)],

(3)

where DT is the translational diffusion coefficient. Solving this equation for a fluctuation in concentration having wavevector K which appears at t = 0, one finds that the fluctuation decays exponentially to zero at a rate rT given by

FT = DTK 2.

(4)

This exponential decay of electric field implies a Lorentzian shape for 6 The diffusion equation, along with a comprehensive discussion of classical methods appropriate to enzymology for determining translational diffusion coefficients is given by H. K. Schachman (see Vol. IV, p. 32).

[7]

T R A N S L A T I O N AAND L ROTATIONAL DIFFUSION

123

the spectrum of the scattered light. This power spectrum is given in normalized form as 7 ST(,) =

.(~ _ ~0)2 + (r~/2~)2

(5)

where vo = (~0/27r). The half-width at half-height of this spectrum is (r~/2v). Thus, measurements of this half-width, the scattering angle, and the index of refraction allow determination of DT. Typical values for Dr for enzymes are in the range of 5 × 10-T cm2/sec to 10 X 10-7 cm~-/sec. This then implies values of (FT/2=) (for back scattering) in the range of about 5 kHz to about 10 kHz. Since the frequency vo of visible light is around 5 × 1014 Hz, it is seen that the extraordinary resolving power (vo/AV) of about 5 X 1011 is required to see even the crudest features of the spectrum of the light scattered by concentration fluctuations in enzyme solutions. Yet the very best grating spectrographs have resolving powers of less than 106, and even the finest spherical Fabry-Perot interferometers available today do not have values for this parameter of more than about l0 s. However, the extremely high resolving power inherent in the optical mixing spectrometer is more than sufficient to resolve this spectrum, and was first applied for this purpose by Dubin, Lunacek, and Benedeks in 1967. A portion of that light scattered with the same polarization as the incident light arises due to anisotropy scattering from the molecules if their optical polarizability is not a scalar. For small macromolecules such as enzymes, this scattering is very small and is completely masked by concentration fluctuation scattering. However, any depolarized scattered light arises solely from the anisotropy effect. The complete form of the spectrum of the depolarized light has been presented by Pecora 9 and Caroli and Parodi. 1° Their analyses assume that the molecules are in rotational diffusion about a spatially fixed center of mass, a very good approximation for small macromolecules such as enzymes whose rotational relaxation times are far shorter than the time required to diffuse a distance equal to the wavelength of the scattering concentration fluctuation, ~. In other words, the probability distribution for the 7N. A. Clark, J. H. Lunacek, and G. B. Benedek, Amer. J. Phys. 38, 575 (1970). This reference also contains an excellent presentation of the theory of optical mixing spectroscopy as applied to the study of diffusing molecules. A Brownian motion approach is emphasized. 8S. B. Dubin, J. H. Lunacek, and G. B. Benedek, Proc. Nat. Acad. Sci. U.S. 57, 1164 (1967). lZ. Pecora, J. Chem. Phys. 40, 1604 (1964). lo C. Caroli and O. Parodi, Proc. Phys. Soc. London (At. Mol. Phys.) 2, 1229 (1969).

124

MOLECULAR WEIGHT DETERMINATIONS

[7]

molecular orientation evolves (rapidly) in time according to a diffusion equation on the surface of a sphere 9 while the center of mass of the enzyme is quasistationary as its probability distribution evolves (slowly) according to the translational diffusion equation, Eq. (3). The "rotational diffusion equation ''9 is completely analogous to the translational diffusion equation, with

0

Ot p ( a t

- - •o,t)

= Dr~V]p(at -

ao,t)

(6)

where A~ is the Laplacian on the surface of a sphere, p(fh - f20,t) is the probability that if the orientation of the enzyme is within df~ about the solid angle f~o at time t = 0, its orientation will be within df~ about ~t at time t, and DR is the rotational diffusion coefficient. In general, the spectrum of the anisotropy scattering from enzyme solutions is a complex combination of several terms. ~° However, the analysis is much simplified if the enzyme can be represented as a uniform ellipsoid of revolution, so that its optical axes are coincident with its inertial principal axes, and two of its three rotational diffusion coefficients are equal and the corresponding components of the optical polarizability are also equal. In this case the spectrum of the depolarized scattered light is given in normalized form as ~°

s~(~)

=

. (~ _

~0)2 + (r~/2~)~

whereg, 10 FR = 6DR.

(8)

Here, DR is the rotational diffusion coefficient for motion of the ellipsoid about either of its two equal axes. The spectrum of the depolarized scattered light is thus seen to be Lorentzian as in the concentration fluctuation case, but now with half-width of (rR/27r)= (3D~/~r). Since a typical value of DR for small proteins is around l0 × 106/sec, ~1 the halfwidth of the anisotropy spectrum is about the same value, namely, around 10 MHz. Thus, a resolving power of about 5 × 107 is required to discern this spectrum, well within the capabilities of the modern spherical Fabry-Perot interferometer. In this case, however, as shall be described in the section on the spectrometers, a single-frequency laser is required, not simply a laser with a uniform wavefront (i.e., a "uniphase" laser), as is the case with optical-mixing spectroscopy. The single-frequency laser and spherical Fabry-Perot interferometer tech~1R. B. Setlow and E. C. Pollard, "Molecular Biophysics," p. 106. Addison-Wesley, Reading, Massachusetts, 1962.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

125

nique has been recently used by Dubin, Clark, and Benedek 12 to determine the rotational diffusion coefficient of lysozyme, the first such determination in this fashion. It must be observed that the intensity of the depolarized light scattered by enzymes is grossly inadequate for a reasonable signal-to-noise ratio in the mixing technique. 13 It is therefore appropriate to discuss in the next section the intensity of the light which is scattered by concentration fluctuation and anisotropy scattering contributions, which will be shown, in the discussion of the spectrometers, to be crucial in determining the feasibility of the experiment under consideration.

Intensity of the Scattered Light The intensity I~ of the light scattered due to concentration fluctuations by a dilute solution of small, noninteracting macromolecules is given as 14

. [2V sin ~ ~ \ Is = J0 , ) ®

(9)

where Io is the intensity of the incident polarized radiation, V is the illuminated volume, ~ is the angle between k~ and Era, and 6t is the "Rayleigh ratio," 61 =

k' ( 1 0 n y CM oc]

(10)

No

Here, M is the enzyme molecular weight, C the enzyme concentration, k the wavevector of the incident light in the medium, No Avogadro's number, and ~n/~C the "refractive index increment." The value of this last parameter is, to a reasonable approximation, remarkably independent of the particular macromolecule considered, and thus the ratio of the intensity of the light scattered by one enzyme solution to that of the light scattered by another is essentially the ratio of their respective products of M × C. If Eq. (9) is integrated over all angles, one obtains as the ratio of the total scattered power (Ps t) to the incident power (Po) the expression Pst P0

-

167r (IlL 3

(11)

~2S. B. Dubin, N. A. Clark, and G. B. Benedek, J. Chem. Phys. 54, 5158 (1971). ~'~S. B. Dubin, "Quasielastic Light Scattering from Macromolecules." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970. 14C. Tanford, "Physical Chemistry of Macromolecules," Chapter 5. Wiley, New York, 1961. This reference presents an excellent summary of classical light scattering formulas and applications.

126

MOLECULAR WEIGHT DETERMINATIONS

[7]

where L is the length of the illuminated region. The expression (16~r/3)6t is called the "turbidity" and is seen from Eq. ( l l ) to be the attenuation per unit length suffered by the incident beam. It is 6t, however, which is usually tabulated. In order to calculate the signal-to-noise ratio obtainable in the spectral determinations, it is necessary to know (R for the solution under investigation. Because of the relative constancy of the refraction index increment (very close to 0.19 ml/g for proteins), it is seen that 6{ varies essentially as the product M × C. Hence, it is sufficient to calculate its value for one enzyme solution to be used as a reference. It is convenient to choose a 1% solution of lysozyme (molecular weight of 14,600). In aqueous solution, using the common 6328A laser line, Eq. (10) then indicates that the Rayleigh ratio for a 1% lysozyme solution is = 20 × 10-6 cm -1.

(12)

It is interesting to note that such a solution of lysozyme is a scatterer comparable to the "intense" scatterers among the common pure liquids, 15 as shown in Table I. However, from the point of view of mixing spectroscopy, such liquids are weak scatterers in that the signal-to-noise ratio obtainable is not nearly so favorable as in the case of the very large molecules, such as viruses. Nevertheless, the observed values of signal-tonoise are adequate for satisfactory results, la In order to determine D~ for enzymes by the light scattering techniques, it is essential that the molecules be optically anisotropic, i.e., that TABLE I RAYLEIGH RATIOS OF REPRESENTATIVE LIQUIDSa (Rb Liquid

(in u n i t s of 10 - s c m -1)

Water Ether Carbon tetrachloride Benzene Toluene Carbon disulfide

0.6 2.6 3.6 6.2 9.0 30

I. L. Fabelinskii, Usp. Fiz. Nauk 63, 355 (1957). English translation: AEC Translation 3973, Part I, Advan. Phys. Sci. 63, 474 (1957). b All values corrected to 6328 A. ~SA general review of light scattered by pure liquids is given by I. L. Fabelinskii, Usp. Fiz. Nauk 63, 355 (1957). English translation: AEC Translation 3973, Part I, Advan. Phys. Sci. 63, 474 (1957). This reference discusses the origins of light scattering in liquids and gives expressions for the anisotropy scattering contribution.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

127

not all of the principal polarizabilities aj be equal. This situation can arise from the two distinct origins: (1) either the macromolecule is actually intrinsically anisotropic (i.e., has different indexes of refraction along its various axes) and/or (2) the molecule possesses ]orm anisotropy, which is produced by immersing a molecule of index m in a solvent of index no (m ve no). Excepting the trivial case of a spherical molecule, this latter source produces anisotropy because the electric field of the incident light induces surface charges on the molecule which incline the induced polarization slightly away from the direction of the incident electric field. Contrary to common misconception, TM it is not necessary that the molecules be long compared with the wavelength of the exciting radiation for this effect to be significant, but rather only that the axial ratio of the molecule be reasonably large 17 (say 3/1 or greater). The essential results regarding the intensity of anisotropy scattering are summarized here. Let the plane wave

E(z,t)

= :~E~e ~(k~-'~ot) + ?)E~e "k~-~°t)

be incident upon the molecule, as shown in Fig. 3. An observer in the scattering plane (~ = 90 °) would, for isotropic molecules, of course

x,/

9E /

_5

Z

k

zI

/

Y

~O

/bserver Iz

FIa. 3. Geometry for the observation of light scattered by anisotropic molecules. ~ W. F. H. M. Mommaerts, see Vol. IV, p. 170. ~'tI. C. van de Hulst, "Light Scattering by Small Molecules," pp. 70-73. Wiley, New York, 1957.

128

MOLECULAR WEIGHT DETERMINATIONS

[7]

observe no scattered light polarized along ~. However, for anisotropic molecules (i.e., not all aj are equal) which are reasonably small compared with the wavelength of the incident light (quite an appropriate assumption for enzymes), the ratio of the intensity of the light scattered with polarization along the z axis to that polarized along the x axis is given by ~s I~ = (E~ + E~)[(1/15)(A --/3)] I~ E~[(1/5)A + (2/15)~] + E~[(1/15)(h -- f~)]

(13)

where zX = ~

+ a ~ + a~~

and = aloe2 -[- O~10~3 -']- a2Ot3

Expressions are now defined for the depolarization ratio for three particular cases: I . / I ~ = pu

E~ = Eu

(14)

I z / I x = or

Ex ~ O, Eu = 0

(15)

I z / I , = oh

E~ = O, Eu ~ 0

(16)

From the value of (I~/I~) given in Eq. (13) and the definitions in Eqs. (14-16), it is clear that 2(a -

P"-

t~)

4A+~ 5-#

P~ -

35

ph=

1

+

2#

(17) (18) (19)

The most frequently quoted values in the literature are those of 0~. However, combining Eq. (17) and Eq. (18) one obtains p~ = ( ~ ) / ( 1

- (~))

(20)

Since pu is usually quite small (~0.01), Eq. (20) indicates that p~ = pu/2

(21)

is an adequate approximation (error ~ 1 ~ ) for all practical cases. This last result is presented because nearly all literature values for the depolarization ratio are for pu, whereas with the advent of lasers with polarized outputs, it is now most convenient to measure pv. Equations (17) and (18) display the ratio of the intensity of light scattered with electric field polarized along 2 to that with electric field ~aSee reference cited in footnote 17, p. 80.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

129

polarized along ~. In each case, the ratio goes to zero if A = fl, that is, if all components of the polarizability tensor are the same. Even if the incident light is unpolarized, the scattered light is completely polarized if the molecules are optically isotropic. However, as displayed in Eqs. (17) and (18), the depolarization ratios do not indicate the origins of the contributions to I~ and I,. It is straightforward to reexpress these results in such a way as to show that, when the molecule is anisotropically polarizable, light scattered with electric field polarized in the same direction as that of the incident light contains contributions both from concentration fluctuation scattering and anisotropy scattering. If I c is that component of the scattered light due to concentration fluctuations, and I# that part due to anisotropy scattering, then a3

p=

=

•

(22)

and

p~ -

(23)

Consider a plane-polarized incident light source with Ey = 0. Then, since anisotropy scattering is very weak indeed for proteins, Eq. (23) indicates that that portion of the scattered light polarized like the incident light i s overwhelmingly due to concentration fluctuation scattering, and hence spectral analysis of that portion of the scattered light [Eqs. (4) and (5)] yields Dv without error. Using an analyzer, such as a Glan-Thompson prism, one may then observe the depolarized scattered light [numerator of Eq. (23)] and hence observe the pure anisotropy scattering from which DR may then be deduced [Eqs. (7) and (8)]. It is obvious, therefore, that the value of the depolarization ratio pv of the molecule under consideration must be reasonably large in order to observe the anisotropy scattering, and one would hope to use literature values for this parameter (or pu, as is more commonly the case) to predict the feasibility of a given experiment. In the case of bovine serum albumin (BSA), for example, pv1~ was reported 2°,21 typically 19All literature values of p~ are converted to p,, via Eq. (21). M. Halwer, G. C. Nutting, and B. A. Brice, J. Amer. Chem. Soc. 73, 2786 (1951). ~lj. Edsall, H. Edelhoch, R. Lontie, and P. Morrison, J. Amer. Chem. Soc. 72, 4641 (1950).

130

MOLECULAR WEIGHT DETERMINATIONS

[7]

as about 0.01. However, Geiduschek 22 determined pv for BSA as less than 0.0001, and this observation has been recently confirmed in an entirely different fashion. '-~ Geiduschek described in detail the considerations which make accurate depolarization measurements difficult, and brought much doubt upon the validity of all such measurements in the literature. He pointed out that fluorescence, optical activity, imperfect polarizers and analyzers, detector anisotropy, finite acceptance solid angle, and multiple scattering can all contribute to improper measurement of the depolarization ratios. It is quite clear that strains in cell glass can make the glass itself somewhat birefringent. In general, it is reasonable to say that measurement of the very small depolarization ratios which arise in macromolecular solutions is difficult, especially when one considers the fact that the solvents, especially organic solvents, may themselves have very substantial anisotropy scattering. The importance of sample purity cannot be overemphasized, particularly in regard to the presence of large size contamination such as dust. Doty and Stein ~3 reported that "only negligible amounts of suspended material could be detected by low-angle examination of the solutions irradiated with a parallel beam of light in a dark room." It is extremely difficult to define "negligible" quantitatively when one is discussing depolarization ratios of the' order of 10-'-' or less. The problem of large particulate contamination is particularly troublesome at the high salt concentrations required for some macromolecular solutionsY, ~4 In light of the above discussion, one is forced to sort through the depolarization literature very carefully in order to get an idea of what reasonable depolarization ratios are. It is noteworthy that since the index of refraction m of protein molecules 2~ is typically close to 1.60, form anisotropy alone can imply a substantial value of pv for asymmetric protein molecules in aqueous solution (in which case the solvent index, no, is 1.33). The same assumptions under which Eqs. (7) and (8) are written then imply that the principal polarizabilities of the protein molecule, represented as an ellipsoid of revolution, are given as ~7

aj

~

!(/l! Lj+

m 2 ~o - - 1

*'~E. P. Geiduschek, J. Polym. Sci. 13, 408 (1954). ~3p. Doty and S. Stein, J. Polym. Sci. 3, 763 (1948). 24A. Wada, personal communication (1969). ~P. Putzeys and J. Brosteaux, Bull. Soc. Chim. Biol. 18, 1681 (1936).

(24)

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

131

25l 2O

D5

~o

~o

5

0

~ 2

I 3

L 4

(alb)~

FIG. 4. Depolarization ratio (p~) as a function of axial ratio for form anisotropy.

where v is the unhydrate volume, and Lj is a form factor dependent upon the axial ratio of the molecule. Even the so-called "globular" proteins are known to be reasonably asymmetric from hydrodynamic studies, "6 values of the axial ratio for the prolate ellipsoid of revolution hydrodynamically equivalent to the molecule being 3/1 to 5/1 typically.26 In Fig. 4, pv, as determined by combining Eqs. (18) and (24), is plotted as a function of the axial ratio, ( a / b ) . It is seen that for ( a / b ) = 5/1 p~ is slightly less than 0.002. Despite this, BSA, with an axial ratio ~7 of about 5/1, has an accurately measured depolarization ratio 1~,~2 of less than about 0.0001. One can conclude only that the intrinsic anisotropy of serum albumin is such that the intrinsic polarizability of the molecule is less along the major axis than along the minor axis, and that intrinsic and form anisotropies tend to cancel one another. This conclusion has been reached in the case of various proteins by interpreting flow birefringence data. 2s One therefore cannot claim depolarization ratios predicted on the basis of form anisotropy are a lower limit for the de2eSee reference cited in footnote 14, pp. 359, 395. 27See reference cited in footnote 14, p. 359. ~ J . Edsall and J. Foster, J. Amer. Chem. Soc. 70, 1860 (1948).

132

MOLECULAR WEIGHT DETERMINATIONS

[7]

polarization ratios. Rather, one can only use such values as a guideline in evaluating literature values. It is unreasonable to assume, however, that form and intrinsic anisotropy will always be opposed to one another, and that even in those cases when tlmy are, that they will so effectively cancel as they do in the case of serum albumin. Indeed, serum albumin seems to have one of the lowest anisotropies of the common proteins. -~s It is thus concluded that values of pv of 1 or 2 × 10- :~ are not at all unreasonable for protein molecules. A recent determination of pv for lysozyme, using a novel technique of high reliability, 1~ gives a value for p, "of 0.0014 _+ 0.0001, quite consistent with the above conclusion. But in general the evident reliability of the available experimental literature is not adequate to render precise values, although the experimentally determined values are generally no larger than 0.01. In general, pu for most proteins 29 is found, experimentally, to be around 5 × 10-3 . In those cases in which p~ is sometimes found to be substantially larger, there is usually a wide discrepancy among various measurements in the literature. For example, Halwer, Nutting, and Brice 2° find pu for ovalbumin (MW = 45,000) to be 0.024, which seems encouragingly large, yet Putzeys and Brosteaux 3° measure pu for ovalbumin to be only 0.004. In view of the similar discrepancy in the case of serum albumin, which has already been discussed, it appears that the latter result is probably correct. In general, then, a value of p~ --0.005 seems reasonable for protein solutions. The value for pv is then about half as large. It will be helpful to consider the specific case of lysozyme, for which p, is known accurately 1~ as 0.0014 _ 0.0001. If the Rayleigh ratio of the solution studied is ~, then the effective Rayleigh ratio for just the depolarized light is pv × ~ (incident light is assumed to be vertically polarized). In the case of a 1% lysozyme solution, for example, (~ is about 20 × 10-6 cm -1 [Eq. (12)]. Hence, pv x (~ is only about 0.03 × 10-6 cm -~, which means that the depolarized component is only about 5% as intense as the light scattered by water itself, for which ~ is about 0.6 × 10-~ cm -~ (see Table I). Since water is not considered a strong scatterer from an experimental viewpoint, it is obvious that measuring depolarization ratios of dilute aqueous protein solutions is an elaborate and very difficult experiment, and the confusion in the literature is not hard to understand. This problem is obviously reduced for higher protein concentrations or larger proteins, or both, but then problems of 2~p. Dory and J. Edsall, Advan. Protein Chem. 6, 35 (1951). P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

133

multiple scattering become apparent. The work of Putzeys and Brosteauxf ° for example, shows a very strong dependence of pu on protein concentration, which may well be due to multiple scattering effects. However, for reasonable concentrations of relatively small macromolecules such as enzymes this effect is probably not great. Dubin, Clark, and Benedek 1~ measured p,, for a 10% BSA solution about an order of magnitude smaller than p,, for a 15% lysozyme solution, despite the fact that the Rayleigh ratio for the former is three times greater than that of the latter. It is nevertheless generally desirable (and feasible) to so arrange the scattering geometry to leave the shortest possible path for the scattered light to exit the scattering cell. This reduces the possibility of multiple scattering events such as shown in Fig. 5, which conserve wavevector and hence are allowed. Since the turbidity of enzyme solutions is not large (only 3.3 X 10-4 cm-1 for a 1% lysozyme solution, or, in other words, only about 3/100 of one percent of the incident power is lost in traversing a centimeter path length), multiple scattering effects need not be significant for reasonable concentrations of enzymes. Although the measurement of depolarization ratios, as discussed here, is obviously difficult and the literature must be viewed somewhat cautiously, it is indeed still quite possible to obtain useful information from the spectrum of the depolarized scattered light, as is discussed in the applications section. This possibility results from the fact that although the depolarized scattered light cannot be distinguished easily from stray light and other complications which apparently plague intensity measurements, such a distinction may well be achieved in regard to the spectrum of the depolarized scattered light and the spectrum of the

2

k

"2

FIa. 5. Multiple scattering event.

134

MOLECULAR WEIGHT DETERMINATIONS

[7]

spurious light, since these two spectra will be markedly different in width.

Spatial Coherence of the Scattered Light In this section the experimental requirements and restrictions in focusing the incident light are discussed in terms of its effect on the spectrum of the scattered light and size of the region of spatial coherence of the scattered light. By this latter phrase is meant the region in the far field over which the electric field of the scattered light does not change its phase. It is only the power contained in such a region which contributes toward the signal-to-noise ratio in an optical mixing spectrometer, 31 and hence it is desirable to make this region as large as possible without resultant distortion of the observed spectrum. For "filter" spectroscopy, such as using a Fabry-Perot interferometer, the signal-to-noise ratio depends only on the total collected scattered light, not on the power per "coherence area." In addition, since the spectrum of the anisotropy scattering is independent of K [Eqs. (7) and (8)], one can focus the incident beam as he pleases and use essentially any reasonable collection optics without spectral distortion. Hence, the discussion below concerns primary optical mixing spectroscopy, as applied to measurements of DT. Observation plane, P

\

Scattering plane

. ~ R..~. ........ ................. ......... ........... ~ ' "

.......... RU

...-"

ko

Scattering cell

';'!

Uniphose laser

FIa. 6. Geometry for determination of coherence area in the far field of the scattered light. 31A. T. Forrester, J. Opt. Soc. Amer. 51, 253 (1961).

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

135

Consider the experimental situation depicted in Fig. 6. A laser of beam diameter a is used to illuminate a cell containing a macromolecular solution. The length of the illuminated volume is L. An observer studies the scattering at point 1~ located in the plane defined by the wavevectors of the light which have ~ = 90 ° (the horizontal plane). Consider the case in which the scattering angle, 0, is 90 °. The observer at R then sees the illuminated volume as a rectangle of length L and height a. The scattered light falls on the observation plane, the phototube, for example. The solution is illuminated by the plane wave output of a uniphase laser, that is, one whose output wavefront has a uniform phase for its electric field. This is obviously required to ensure the maximum power per region of coherence in the scattered light. Such a laser is often called "single mode" in the advertising literature because only a single transverse mode runs in the laser cavity. However, many axial modes are typically permitted to run in such a laser, and it is therefore definitely not a single ]requency laser. Only the uniphase requirement (indicated TEMoo) is essential for a mixing spectrometer, whereas in the case of anisotropy scattering in proteins a true single frequency laser is required for the Fabry-Perot interferometer. Because the laser is uniphase, the scattering volume is coherently illuminated. However, as an observer at R moves from point 1 to point 2, the phase of the electric field of the scattered light will change. This, of course, is due to the difference in path length of the light scattered by various portions of the incident beam. In fact, it is well known from physical optics 32 that if the distance from 1 to 2 is approximately (X/L)R, the phase of the electric field will change sign: X/L is called the "diffraction angle" of the beam. Similarly, if the observer moves from 1 to 3, a distance of approximately (X/a)R, the electric field will again change sign. Hence, within an area of size A .~ X2R2/aL the phase of the electric field of the scattered light will be the same. Such an area is called the "coherence area" of the scattered light, Acoh, and £2/aL is called the "coherence solid angle," gtcoh. It is immediately apparent that the coherence area of the scattered light is a function of the scattering angle, 0. This obtains because the apparent source size is a function of 0. For an observer in the horizontal plane, the apparent length of the illuminated region as seen at the point R is l = L sin 0 + aleos 0 I. Hence, one has that the coherence solid angle for any scattering angle 0 is given by ~2L. D. Landau and E. M. Lifshitz, "The Classical Theory of Fields," p. 165. Addison-Wesley, Reading, Massachusetts, 1962.

136

MOLECULAR WEIGHT DETERMINATIONS ~2 ~Coh ~

a[L sin 0 + alcos 8t]

[7]

(25)

and the coherence area is then ~2k2

Acoh = R2~]Coh ~ a[L sin 8 + a]cos 01]

(26)

Now, setting ~ = 90 ° for an observer in the horizontal plane (the usual configuration) and using Eq. (9), one obtains as the ratio of the power scattered into a coherence solid angle to that of the incident beam P~ °h(~c°h) Po

2L(R

a[L sin 0 + aIcos 8[]

(27)

Equation (27) has two distinct regions of interest. The usual experimental situation is L >> a. Hence, for any reasonably large scattering angle, one sees that p~oh is independent of the beam length L and varies inversely as the beam diameter, a. The other region of interest is that of very small (or very large) angles. In this case p~oh varies as L / a 2. It is thus readily apparent that appreciably more power per coherence solid angle can be obtained at small or large scattering angles, a desirable result for the operation of optical mixing spectrometers. However, making the beam dimensions small introduces a spread in the wavevectors making up the incident beam 3~ (as indicated in Fig. 7):

Ak:~ ~ 1/a Aky ..~ 1/a 5k, = 1/L.

(28) (29) (30)

The original assumption of plane wave illumination implies Akx = ~kv = n

I

Fla. 7. Spread in waveveetor of incident light due to finite beam dimensions. See reference cited in footnote 32, p. 164.

[7]

TRANSLATIONAL

AND

ROTATIONAL

DIFFUSION

137

0. The beam length L and diameter a are usually sufficiently large that no difficulties are encountered due to the implications of Eqs. (28)-(30). This becomes progressively less the case as the scattering angle is decreased, particularly for small beam diameters. Assuming the only uncertainty in K arises from uncertainty in k due to the finite beam diameter, one has from Eqs. (1) and (28) that (_~_)

(l/a) 2k sin(0/2)

(31)

If one states, for example, that the maximum acceptable spread in K is 1%, Eq. (31) sets a criterion for ~ and a: a sin(0/2) ~ (50/k)

(32)

For the 6328 A laser line in aqueous solution, k = nko = 1.33(2:r/6.328 × 10-5 cm). Thus, Eq. (32) requires a sin(0/2) ~ 3.8 X 10-t cm

(33)

Equation (33) is indeed restrictive. It is immediately seen that focusing the beam to its diffraction limit is in general never permissible if one requires a 1% definition in the wavevector of the fluctuation being studied. As one proceeds to the forward direction, the minimum size to which the beam may be focused progressively increases until, for example, at t~ = 2 °, it is amin = 0.2 mm. This is actually quite large, yet one cannot decrease the beam diameter without an unacceptable loss in definition of K. Hence, even though Eq. (26) implies that the region of coherence in the scattered light can be made arbitrarily large by decreasing the beam diameter and studying the scattered light at small angles, such an experimental procedure must be done within the restrictions imposedby Eq. (33). It is possible that a 1% definition requirement on K may be too stringent under certain circumstances. It is important to note, for example, that a spread in K has only a secondorder effect on F~ and hence Dr [Eq. (4)] because values for K both below and above the mean value are accepted, tending to cancel the effect. Nevertheless, after a maximum value for (AK/K) is decided upon, Eq. (31) must be obeyed for that particular restriction. The problems can be completely circumvented if it is experimentally feasible to study the back-scattered light (t~ ~ 180°). In this case Eq. (33) indicates that the beam diameter must be no smaller than a couple of microns for the restriction (AK/K) < 1% to be met. For red light, this is only a few times the diffraction limit. Thus, one can focus essentially to the diffraction limit without an unacceptable spread in K if the scattered light is studied in the backward direction. This is also

138

MOLECULAR WEIGHT DETERMIN&TIONS

[7]

significant since the coherence arei~s become very large in the extreme backward direction [Eq. (26)], indicating improved signal-to-noise ratio for the optical mixing spectrometer. Even assuming that the beam diameter is sufficiently large that there is not a large spread in K due to a spread in k, the wavevector of the incident light, it is still possible to have poor definition in K. As Eq. (1) indicates, K = 2k sin(O~2). Hence, even if k is well defined, K may have an uncertainty due to acceptance angle spread given by

(AK/K) = [½ ctn(0/2)]A0

(34)

For large angles, Eq. (34) indicates that nearly an arbitrarily large A6 (acceptance angle of observer) is permissible since ctn (0/2) --> 0 as 0--> 180 °. Even for small angles, the restriction at first glance does not appear great, since even if A0 is about 0.1 ° (a reasonably large aperture), 0 can be as small as 5 ° without (AK/K) exceeding 1%. However, even an aperture of 0.1 ° will not collect an entire coherence area at 5 ° scattering angle for a well focused beam (a ~ 0.1 ram), let alone a beam focused nearer the diffraction limit. Hence, the dual problems of spread in observed K values due to finite beam diameter and finite acceptance angle must be traded off against collecting an entire coherence area. It is once again stressed that these considerations apply only to determining Dr from the (K-dependent) concentration fluctuation spectrum using optical mixing spectroscopy (which requires high scattered power per coherence area). Determining D , from the (K-independent) anisotropy scattering spectrum using a Fabry-Perot interferometer (whose obtainable signal-to-noise ratio depends only on the scattered power) is not subject to these problems. Operation of the Spectrometers

The "Self-Beating" Optical Mixing Spectrometer The light scattering observed in enzyme solutions has been discussed in the preceding section as arising from thermally generated concentration fluctuations and optical isotropy fluctuations. Such fluctuations may be regarded as arising at some time t and then decaying to zero exponentially [Eqs. (4) and (8)]. Hence, the intensity of this scattered light will rise to some value, decay and rise again, and so on. Since the fluctuations in ~ which produce the scattering are random variables, the intensity I(t) of the light they scatter is also a random variable, and might appear as in Fig. 8. Since the photocurrent i(t) is proportional to the intensity of the light falling on the surface of a phototube, i(t) will have an appearance mirroring I(t). Since it is actually the power spee-

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

139

? i(t)

t--~" Fie. 8. Short time fluctuations in the intensity of the scattered light.

trum of the fluctuations themselves that is being sought in order to determine the diffusion coefficients, it would appear that the spectrum of the photocurrent should contain essentially the same information as the spectrum of the scattered light. This is indeed the case, and if the spectrum of the incident light is Lorentzian of half-width (F/2=) [see Eqs. (4) and (5) and (7) and (8)], then the spectrum of the fluctuations in the photocurrent S~(v) is given in normalized form as 13

'i°2~(~')+ ~rlio__ 2n[ v~ +(2F/2~)(2F/2~)2J] + Geio (2r/2 ) ] i02~(, ) + 1~ io2 ~ + (2r/2~)~J + Veio

(for n _> 1)

(35)

(for n <: 1)

(36)

where n is the number of coherence areas collected [see Eq. (26)], io the average photocurrent, e the charge of the electron, and G the gain of the photomultiplier tube. Since the second term in Eqs. (35) and (36) contains r, it is referred to as the "signal" term ($(,)), while the third term is called the "noise," ( ~ ( , ) ) , since it contains no useful information and arises from the "shot effect," namely, that the electronic charge is not continuous but rather discrete. This shot effect spectrum is flat from zero to many GHz and acts as a broad background beneath the signal. Finally, the first term in Eqs. (35) and (36) is a delta function arising from the fact that the photocurrent has a nonzero average value. These remarks are summarized as

S~(v) =

io2~(~) + ($(v)) + (9~(~))

(37)

which is displayed in Fig. 9. Whereas the spectrum of the light incident on the phototube is Lorentzian of half-width (p/2~r) centered at the optical frequency (Vo) [see Eqs. (5) and (7)], the signal portion of the spectrum of the photocurrent is a Lorentzian twice as wide, but now centered at dc. In other words, the problem of the light scattered by concentration and isotropy fluctuations has been solved by shifting the spectrum to a lower frequency (here, to dc), where conventional audio-

140

MOLECULAR WEIGHT DETERMINATIONS

io2 8(v}

i

<$(v) > =

I

_ ~-

2F

~<

@ io~" . v2

t

[7]

\ 2~ /

\2rr

/ < ~ ' (v) > = Ge~o

y

~

Fia. 9. The power spectrum of the photocurrent.

frequency spectrum analyzers can resolve the spectrum. This remarkably simple, so-called "self-beating" spectrometer was first developed for light scattering by Ford and Benedek, 34 drawing on the earlier work of Forrester21 The name "self-beating" derives from the fact that the phototube has been used as a nonlinear device to mix the scattered light with itself, much as the galena chip and "cat's whisker" are used in the familiar crystal set to demodulate the information from the carrier frequency. An alternative method is to mix the scattered light with an intense "local oscillator," a portion of the incident laser beam itself, for example. This is referred to as "superheterodyne optical spectroscopy, ''35~37 and is quite analogous to the method of detection used in the ordinary "superhet" radio. The heterodyne technique gives essentially the same results as the self-beating spectrometer except the spectrum of the photocurrent now is Lorentzian of width (r/27r) and still '4N. C. Ford, Jr., and G. B. Benedek, Phys. Rev. Lett. 15, 649 (1965). 3~j. B. Lastovka, "Light Mixing Spectroscopy and the Spectrum of Light Scattered by Thermal Fluctuations in Liquids." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1967. = G. B. Benedek, in "Polarization, Matter, and Radiation," p. 49. Presses Universitaire de France, Paris, 1969. 3~It. Z. Cummins and It. L. Swinney, in "Progress in Optics" (E. Wolf, ed.), Vol. 8, North Holland, Amsterdam, 1970.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

141

centered at zero frequency if the spectrum of the incident light is Lorentzian of width (r/2=). In general, self-beating spectroscopy is more readily performed because problems of scattered light-local oscillator wavefront mismatch are not present, 3~ nor does one have to contend with noise present on the strong local oscillator? 5 On the other hand, heterodyning is often unavoidable. For example, at very small angles it is difficult to remove all light elastically scattered from windows, etc. In addition, the predetection signal-to-noise ratio (see below) is a factor of four larger in the heterodyne method25 Generally speaking, the ease of applying the self-beating technique outweighs the increased signal-to-noise available from the heterodyne methods and the latter will not be discussed further here. The cited literature contains a complete description. Equations (35)-(37) are displayed in Fig. 9, from which it is clear that if ($(v)) is always small compared with (9~) one will have difficulty in discerning the signal. In fact, the ratio (8 (0) )/(9~) will be useful to give an idea of the signal's detectability, and has come to be called the "predetection" signal-to-noise ratio, 3° (SIG/NOISE)p~, because it does not refer to the instrumental bandwidth before detection, or how long one accumulates the information contained in this bandwidth. One can say somewhat arbitrarily that a predetection signal-tonoise ratio of about 1 is desirable as a minimum value from an experimental viewpoint although experiments involving much smaller values are still feasible. 13 It is interesting to point out that a value of 1 for the predetection signal-to-noise ratio corresponds to one photoelectron per coherence area being ejected from the photocathode of the photomultiplier tube per correlation time, r = 1/r. 3~ If the acceptance angle AO is increased, more coherence areas are collected and io increases as n. However, because of the presence of n in the second term of Eq. (35), (SIG/NOISE)pRE is seen to be independent of n unless n is less than 1, in which case (SIG/NOISE)pRE decreases. The point here is that simply "opening up" the acceptance angle does not improve (SIG/NOISE)pRE but rather io must he increased in some other fashion, such as increasing the incident laser power Po or the phototube's quantum efficiency. Alternatively, one need only increase the size of the coherence angle, which results in more photocurrent for a given value of n. On the other hand, the ratio (~(0))/(9~) does not indicate the actual signal-to-noise as it is usually defined. One may now define the "post-detection" signal-to-noise ratio, 3~,36 (SIG/NOISE)~,os~. This requires the introduction of experimental parameters and hence is not so intrinsic a quantity as (SIG/NOISE)eRF. It does indicate, however,

142

MOLECULAR WEIGHT DETERMINATIONS

[7]

the actual detectability of the signal, and therefore is most useful. This ratio is defined as the average value of the signal, (8(v)), divided by the rms value of the fluctuations in the signal and shot noise terms, that is: (SIG(v)/NOISE)eosT = ((~[$(v) (S(v)} + ~(v)])~) in

(38)

Experimentally, one observes a finite section of the spectrum of the photocurrent of width vl. If the power in vl, is averaged over a time T, then the postdetection signal-to-noise ratio is given by 3~ (SIG(,)/NOISE)~osT = ($(,)) + ( u ( , ) ) "

~

(39)

where v2 = 1 / T and is called the "post-detection bandwidth." If a frequency-dependent predetection signal-to-noise ratio is defined as (SIG(~)/NOISE)pRE- (~(~)) ($(v)) then (S1G(~)/NOISE)PosT = [_(SIG(v)/NOISE)enE + The interesting implications of this result have been pointed out by Lastovka 3~ and Benedek. 36 If the predetection signal-to-noise ratio is high, the postdetection ratio becomes independent of all parameters except vl and v2. In such cases, it serves no advantage to increase Po, Acoh, etc., a rather unintuitive result. The experimental setup of a self-beating spectrometer is now described along with a discussion of realizable signal-to-noise ratios obtainable for enzyme solutions. The apparatus is diagrammed in Fig. 10. The laser must have uniphase output (TEMoo) to ensure the spatial coherence required for mixing, as already discussed. The focusing lens renders a beam in the cell of diameter a and length L. If d is the initial diameter of the beam as it leaves the laser, and ] is the focal length of the lens, then 3s a ~ (f) x

(41)

L =

(42)

~

~8M. Born and E. Wolf, "Principles of Optics," pp. 415, 441. Pergamon Press, Oxford, 1965.

[7]

143

TRANSLATIONAL AND ROTATIONAL DIFFUSION

.|.

~, f.\

d

~- L ~

Focusing lens

~

\

Scattering cell

\

\ \

~

\ ~..,.~ Collecting lens

~Fq

Stop PMT

"Wove analyzer" r

Recorder

Post detection bandwidth

Power detector

Predetection bandwidth

FIG. 10. Block diagram of self-beating spectrometer.

where, within this approximation, X can be taken as the wavelength of light in the medium. To ensure that no stray light is seen by the collecting lens, an aperture is placed immediately after the cell. The scattered light then falls on a collecting lens which maps all parallel light scattered at an angle 0 onto the photomultiplier tube. An aperture before the P M T ensures an acceptance solid angle which complies with the requirements of Eq. (34). The photocurrent is then analyzed by the spectrometer which in block form consists of a predetection filter of width vl and a power detector, and then is displayed on the strip chart recorder after being accumulated for a time T = 1/v~. It is readily shown 13 that the predetection signal-to-noise ratio is given as ((

nv

~

P0~

\ 8~r~hcDw] sin2(8/2) (SIG/NOISE) pRE =

M

(43)

(~Y} sin e q-[cos el

(Concentration fluctuation scattering) n3,

X2

÷,co (Anisotropy scattering)

(44)

0,

144

MOLECULAR WEIGHT DETERMINATIONS

[~]

where ~ is the quantum efficiency of the phototube, h is Planck's constant, c is the speed of light in vacuum, and n is the solution refractive index. The results of Eqs. (43) and (44) are worthy of comment. It is seen immediately from Eq. (43) that the predetection signal-to-noise ratio in a concentration fluctuation experiment is independent o] the wavelength o] the incident light. That is, the familiar (1/~) 4 dependence of Rayleigh ratio [Eq. (10)] is completely canceled by the X4 present in Eq. (43). Thus, one's intuitive feeling that the (l/X) 4 dependence of the intensity of Rayleigh scattering would imply improved signal-tonoise for shorter wavelengths is not valid in the case of concentration fluctuations. The physical origins of this result are clear. The predetection signalto-noise ratio is a measure of the number of photoelectrons ejected per correlation time ~- - 1/F. Since r decreases as X2 for concentration fluctuations, the value of (SIG/NOISE)p~E also decreases as X". Similarly, since the number of photons in a beam of given power decreases as the wavelength decreases, the number of photoelectrons also decreases as X. Finally, although the coherence solid angle ~co~, [Eq. (25)] is seen to be independent of X upon applying Eqs. (41) and (42), the length of the beam, L, over which scattered light is focused decreases as [Eq. (42)]. Hence, four factors of X appear in the numerator of (SIG/ NOISE)pRE completely canceling the (l/X) 4 dependence of ~. However, the value of the correlation time (l/r) increases as sin~(0/2), so that the (SIG/NOISE)p~E can still be enhanced at small angles, and the K 2 dependence of the spectrum of the scattered light does more good than harm from an experimental point of view. Since FR (anisotropy scattering) is independent of K, Eq. (44) reveals that in this type of scattering, one does obtain at least a (1/£ ~) of enhancement of the signal-to-noise ratio as X is decreased. However, since 1~ is independent of angle, there is no pronounced (SIG/NOISE)p~E improvement in the small angle regime except through the fact that the coherence areas become large. This is indicated by the factor of [(]/d)sin 0 + lcos 0]] in the denominator of Eqs. (43) and (44). This factor is a symmetric function of the scattering angle, and it is thus seen that, owing to coherence area enlargement alone, the signal-to-noise ratio increases ]or both small and large values o] 0, having a minimum at 0 = 90 °. In the case of anisotropy scattering, for example, where there is no sin2(0/2) dependence of the spectrum, the same signal-to-noise ratio can be obtained at an angle 0 and the supplementary angle 180 ° - 0. Since stray light is peaked in the forward direction, it might therefore be desirable to study the spectrum for large angles under

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

145

these conditions. Even in the case of concentration fluctuation scattering, it might be desirable to study the spectrum at large angles to eliminate problems of lack of definition in K as previously discussed. Of course, in the case of concentration fluctuations, small-angle scattering has the additional advantage that FT becomes small, and this is not the case for large angles. Although there thus does not seem to be much reason to use short wavelength incident light, there is in reality much reason to do so for reasons not related to the (l/X) ~ dependence of Rayleigh scattering. Presently, phototubes are available with a quantum efficiency, ~,, several times larger in the blue than in the red, for example. Similarly, high power short wavelength lasers are becoming increasingly common and inexpensive. Hence, indirect reasons may well motivate the use of short wavelength incident light. One can now demonstrate that reasonable values of (SIG/NOISE)eRE may be obtained in enzyme studies using presently available commercial equipment. Since self-beating spectroscopy is very difficult for scattering angles less than about 5 ° due to stray light problems, the calculation will be made for angles larger than this value. Hence, the focused beam must be about 0.1 mm in diameter or larger [see Eq. (33)], which then implies a value of (]/d) of about 160 [see Eq. (41)]. It will again be convenient to use a 1% lysozyme solution as a reference for which (~ is 20 × 10-Gem -1 [Eq. (12)]. These values, along with the other quantities needed to determine (SIG/NOISE)~RE from Eq. (43) are summarized in Table II, and lead to the following result: (SIG/NOISE) eRE=

(215) sin2(6/2)[160 sin 8 + [cos 61]

(45)

Equation (45) indicates that a value for the predetection signal-tonoise ratio in excess of 1 is easily obtained at any scattering angle. It also indicates that although very large values for this parameter may be obtained by going to small scattering angles (due primarily to the narrowing of the spectrum, not to coherence area enlargement since 6 is restricted t(~ be greater than 5°), large values may also be obtained for back scattering (6 = 180 °) where the coherence area enlargement is pronounced, and where (SIG/NOISE)pRE is 215. This is to be contrasted with the value of (SIG/NOISE)p~E at, say, 90 °, which is only 2.7. In any case, it is clear that determining DT for enzymes by selfbeating spectroscopy is quite feasible, especially upon recalling that lysozyme is a relatively small enzyme (MW = 14,600) and has a relatively large diffusion coefficient, both characteristics which reduce (SIG/NOISE)eaE [Eq. (43)]. On the other hand, these results indicate

146

MOLECULAR WEIGHT

DETERMINATIONS

[7]

TABLE II PHYSICAL PARAMETERS REQUIRED TO EVALUATE THE PREDETECTION SIGNAL-TO-NOISE RATIO FOR A 1 % LYSOZYME SOLUTION

nw -- 1.33~ ~, = 0.053b (f/d) = 160 (R = 20 X 10-6 cm-~ ~ Po = 50 mW d D = 10.6 X 10-7 cm2/sec~ X = Xo/nw = 4750 ,~1 For convenience, the solution index is treated as that of water. b Derived from cathode radiant given in specification sheet, 7265 PMT (RCA). c Equation (12). d Incident power available from a Spectra-Physics He-Ne Model 125 laser. ' S. B. Dubin, "Quasielastic Light Scattering from Macromolecules." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970. / ~ 0 = 6328/~. the difficulty of using optical-mixing techniques to measure the anisotropy spectrum of the light scattered by proteins. This is seen by dividing Eq. (44) by Eq. (43) which gives as the ratio of the predetection signalto-noise ratio obtainable in determining the anisotropy spectrum to that in determining the concentration fluctuation spectrum: ANISOTROPY

(SIG/NOISE)pR~

87r2 (DT~ ~ sin2(0/2 )

(46)

\ ~ - ' , ~ / J-~ ~ J x ~ / P R E

Since DT is expected to be typically around 10 × 10 -; cm2/sec and DR around l0 × 106/sec, Eq. (46) gives the ratio as about 1 × 10 -6. Since the anticipated predetection signal-to-noise ratio for, say, lysozyme is about 200 for extreme back-scattering (thereby taking m a x i m u m advantage of coherence area enlargement), Eq. (46) then indicates that for 0 = 180 ° the predetection signal-to-noise ratio in the determination of the anisotropy spectrum can be expected to be around 2 × 10 -4, a formidable obstacle indeed, but not necessarily an insurmountable one. Recently lasers have become available with power outputs approaching 1 W in the blue region of the visible spectrum. This increase in power, plus the increase in the quantum efficiency ~/, about a factor of 3 better i n the blue than in the red for an S-20 photocathode, raises the anticipated predetection signal-to-noise ratio to about 2 × 10-~. This is still unacceptable and even this low value assumes an experimental arrangement m a y be achieved which allows extreme back-scattered light ( 6 , ~ 180 °) to be observed to obtain m a x i m u m enhancement of the coherence solid angle. I t should be noted that observation of the ex-

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

147

treme ]orward-scattered light (~ ~ 0 °) is generally ruled out by stray light and by the presence of light scattered by large particulate contamination such as dust, aggregates, or the small percentage of denatured protein unavoidably present in the scattering cell. Of course, the post-detection signal-to-noise ratio [Eq. (40)] can be significantly better than the present value of about 2 × 10-2 for the predetection signal-to-noise ratio. Nevertheless, it would appear that optical-mixing spectroscopy is, in the present a state-of-the-art, not ideally suited to observation of the anisotropy spectrum. 1~ The capabilities of the spherical FabryPerot interferometer (SFP) were therefore suggested 13 and applied 12 to this problem. The SFP is described in a following section. It is noted that the problems described above in discerning the anisotropy spectrum of small proteins are in large measure not present for the much larger molecules, such as viruses. This arises because viruses have a molecular weight typically 3 orders of magnitude larger than enzymes and much smaller values for the rotational diffusion coefficient. Thus, Wada, Suda, Tsuda, and Soda 39 were able to measure DR for tobacco mosaic virus by using heterodyne mixing spectroscopy. This experiment has also been performed by self-beating spectroscopy13 yielding a value for DR in excellent agreement with a calculation of the rotational frictional coefficient of a cylinder4° of TMV's dimensions. No such experiments have as yet been performed on enzymes for the reasons outlined above. The self-beating optical spectrometer will now be described in detail in terms of the apparatus and alignment. Optical Alignment and Detection. Since (SIG/NOISE)p~E is not enhanced or reduced by collecting more than one coherence area (so long as at least one is collected), the relatively simple collection system indicated in Fig. 11 is adequate for most experiments. (For scattering angles less than about 30 ° , or greater than about 170 ° , an alternative scheme is presented later.) The laser is focused into the scattering cell in accordance with the requirements of Eq. (33). Stop No. 1 immediately in front of laser serves to block the large amount of stray light emitted from the laser; a narrow-band transmission filter would serve as well or better. It is suggested that a laser with an optical power of at least 10 mW be employed (recall that the calculations of the previous section are based On 50 mW). Such lasers are usually sufficiently long that problems with "mode-hopping" are not encountered. Many companies presently manufacture appropriate lasers; these include Spectra~A. Wada, N. Suda, T. Tsuda, and K. Soda, J. Chem. Phys. 50, 31 (1969). ~S. Broersma, ,1. Chem. Phys. 32, 1626 (1960).

148

MOLECULAR WEIGHT DETERMINATIONS

[7]

First surface mirror

~__~.,,t- ' I

d

I

""" /

Autocollimating telescope

/ acosino lens i i Scat!ering cell

~ Stop~ 3

L ~

Light trap

--

Stop # 4

--,--

Stop#l

Collecting lens

< ~"~ J ~k .

film N.D.

filter # 3

Silicon solar cell

~ p Loser

~,.~'-J~ "B"

Stop # S

MT

FI~. 11. Block diagram of optical setup for concentration fluctuation scattering.

Physics (Models 124 and 125 being particularly attractive for the present applications) and Jodon Optical (Model HN-50). If a large initial investment in a laser can be accepted, one of the currently available high-power ( ~ 1 W) argon-ion lasers is to be recommended, such as the Coherent Radiation Model 52A. Before being focused, the laser light is rotated to a convenient angle by a front surface deflecting mirror. The optical bench containing all optics between points A and B in Fig. 11 is then rotated until the incident beam executes the path indicated in the figure. Despite careful cleaning, a small amount of light is elastically scattered by both the mirror and the focusing lens, and stops Nos. 2 and 3 serve to prevent this light from entering the scattering cell. In all experiments standard fluorescence cells can be used to hold the sample (Lui Scientific Inst. Corp., New York, New York). These are made of near-ultraviolet glass and allow a path length of 2 cm. In all cases, this length is sufficient to collect scattered light over the entire focused region of the incident beam. After exiting the cell, most of the transmitted beam is reflected into a light trap by a No. 3 metallic film neutral density filter. A small portion (10-3) of the transmitted light is allowed to fall on a silicon solar cell the output of which is used to monitor the laser power. Since the photocurrent io is proportional to the laser power, any drifts in Po

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

149

cause distortions in the observed spectrum which can be corrected via Eqs. (35) and (36) if the laser power is monitored. If desired, the solar cell output may be used to control a laser power servo to obviate this difficulty entirely. 13 Parallel light scattered at an angle 6 relative to the incident beam is focused by the collecting lens onto the surface of an RCA 7265 photomultiplier tube. The S-20 photosurface of this tube offers a quantum efficiency ~, of about 5% at the 6328/~ laser line and about a factor of 2 to 3 higher in the blue-green region. This is a high-gain, stable, reliable, and relatively inexpensive PMT. Stop No. 4 prevents stray light, particularly that scattered at the entrance and exit faces by the incident beam, from reaching the phototube. It is seen from Fig. 11 that the apparent scattering angle in this geometry is ~'. One readily obtains ~ from ~' by Snell's law if the index of refraction of the sample being studied is known. This is easily measured for each sample with a refractometer, such as the Bausch and Lomb Abbe unit. The apparent scattering angle, ~', is measured with any desired accuracy by the following technique. An autocollimating telescope (for example, Model D657 of Davidson Optronics, West Covina, California) is aligned parallel to the optical axis of the collecting lens, which is arranged to have stop No. 5 (the P M T aperture) at its focus. This alignment is achieved by focusing the collimator at infinity and then moving it until stop No. 5 is centered on the cross hairs of the telescope. The scattering cell is removed and replaced by a front surface mirror mounted on a rotating table calibrated in degrees and readable to the desired accuracy. The table is rotated until autocollimation is achieved with the alignment telescope. This renders the mirror perpendicular to the optical axis of the alignment telescope. The table is then rotated until it sends the incident laser beam back through .the small aperture in stop No. 3. The apparent scattering angle ~' can then be read off the calibrated table. An aperture, stop No. 5, is placed in front of the phototube to restrict the angular acceptance, A~. Lastovka 35 has shown that, if an acceptance angle A~ is collected about the mean angle ~, then the observed spectrum is broadened, as given by FoBs = F

[

1~-

(47)

if the spectral width of the scattered light varies as K 2. For the experimental apparatus indicated in Fig. 11 it is easy to arrange (A0/O) tO be quite small, certainly less than 10-2. Indeed the acceptance angle

150

MOLECULAR WEIGHT DETERMINATIONS

[7]

can always be made large enough, in this arrangement, to accept sufficient scattered light to swamp phototube dark current without spectral lineshape distortion. The scattering geometry depicted in Fig. l l can be used with no difficulty for scattering angles as small as about 30 ° and as large as about 170 ° . If one tries to study the scattering for angles smaller than about 30 ° with this technique, problems with stray light become increasingly severe. The optical arrangement, however, can be easily modified to allow studying the spectrum of the scattered light in the vicinity of 0 = 2 ° or even smaller, and this is outlined in Fig. 12. The imaging lens is used to image the entire scattering cell onto stop No. 2 immediately in front of the photomultiplier tube. The image is enlarged by the magnification of the lens, M. A small aperture (stop No. 1) restricts the acceptance angle of the imaging lens, and hence a sharp image of the cell appears at the focal plane of the PMT, and stop No. 2 is used to block the well-defined images of the spots caused by

Incident laser beam i~ I

Spot formed

as laser exits cell

Spot formedas laser enterscell

! I

,,J \~,~ ~

~

Scatteringcell

Stop # I

imaginglens

~ .~.

Image of spot Stop~ 2

Imageof spot

Photomultiplier FIO. 12. Block diagram of optical setup for small angle scattering.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

151

the laser entering and exiting the scattering cell. Since the magnification of the lens is M, one has A B = M L sin 0

and may then solve Eq.

(48)

(48) for the scattering angle, to obtain 0 = sin -1 ~

The angle is thus easily determined by measuring the distance between the spot images on the P M T focal plane, and the eell length. With this system, angles between 10 and 3 ° are easily obtained, and the principal source of Stray light--the spots on the cell at laser entrance and exit faces--is eliminated. Stops Nos. 1 and 2 combined can still be arranged to give a (z~0/0) 2 value better than 10-2 in all eases. It is important to add at this point that light scattered by particles of dust can become significant when studying the spectrum of light scattered by macromolecules at small angles since the light scattered by large particles (e.g., "dust") is peaked in the forward direction. This problem can be overcome by surrounding that portion of the collection optics which includes the scattering cell and phototube with a plastic box onto which the dust can be precipitated. A toy van de Graaff generator, such as available from Edmund Scientific Corp., Barrington, New Jersey, is excellent for charging up the box, and the dust will usually be precipitated in an hour or so. An alternative method is to pass filtered air through the plastic box until the dust-laden air is removed. In any event, the elimination of air-borne dust is essential when working at small scattering angles. E l e c t r o n i c D e t e c t i o n . The spectrum of the fluctuations in the photocurrent may be analyzed by essentially the same techniques first employed by Ford and Benedek. 34 The equipment is block diagrammed in Fig. 13. The photomultiplier tube 41 must be mounted in an appropriately wired base. This can be easily wired by the experimenter, or commercially wired units are available (for example, from Pacific Photometrics, Berkeley, California, or EG and G, Salem, Massaehuse~;ts, and many others). It is unnecessary to cool the phototube for studying enzymes at reasonable concentration ( ~ 1 % ) and with reasonable laser power (~>10 mW). The P M T will require an operating voltage of typically 2000 V, and appropriate power supplies are readily available. To avoid high-frequency roll-off due to cable capacitance, an emitter fol41A discussion of photomultiplier tubes is given by L. Brand and B. Witholt in Vol. XI, p. 776.

152

[7]

MOLECUL4R WEIGHT DETERMINATIONS

T--I i_:

Counter

'mP I

~[.~

Wave ]

meter

Time averoger

Analog squarer

Strip

~1~

,i

,. ,

Istrp chart recorder

FIc. 13. Electronics of self-beating spectrometer.

lower preamplifier with low output impedance must be located at the phototube. The low output impedance of this device then allows long cable length (several feet) between preamp and wave analyzer. The fluctuations in the photocurrent are then analyzed either by a General Radio 1900A wave analyzer 4'2 (for spectra up to about 2 kHz wide), or by a Hewlett-Packard 310-A 43 for wider spectra. The several adjustable bandwidths (2vl) available on these analyzers (3, 10, and 50 Hz for the GR 1900 A, and 200, 1000, and 3000 Hz for the HP 310-A) General Radio Co., West Concord, Massachusetts. 43Hewlett-Packard, Palo Alto, California.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

153

allow choosing the largest possible value for this parameter and hence obtaining the largest value of (SIG/NOISE)PosT [Eq. (39)] without distorting the observed spectra. Lastovka 35 has shown that because vl is finite, the observed spectrum is still Lorentzian but broadened by 100(vlTr/I')2%. Because of the range of bandwidths available in these wave analyzers, this quantity is easily rendered negligibly small. Because the detector in the wave analyzer delivers an output proportional to the voltage contained in the bandwidth 2Vl, the output is squared (after being averaged for a time T = 1/v2 by an RC filter) before being displayed on the strip chart recorder. One thereby records the power spectrum of the fluctuations in the photocurrent. This is a particular convenience in the case of (SIG/NOISE)pRE < 1, since it allows one to "buck out" the shot noise (9Z(v)> and enlarge the display of ($(v)). Since S~(v) = io~(v) + ($(v)> + (9Z(v)), as given by Eq. (37), it is seen that even if one "bucks off" the residual level from a square-root (voltage) spectrum, the strip chart recorder displays V'($(v)> + <9Z(v)>- ~v/ = Geio [Eqs. (35) and (36)] can be made sufficiently large to be measured on the same wave analyzer scale as the experimental runs. One then records the system response to the shot noise and thereby calibrates the frequency response to within a small fraction of a percent. It is of course desirable that this response be "fiat," and this condition can be achieved by trimming the frequency response of the P M T preamplifier appropriately. If this is not done, the calibration chart of the system response, made in the system described above, must be used to correct the observed spectrum point by point. Ordinarily this correction is so small as to be negligible. However, in the regime of very poor predetection signal-to-noise ratio, most of the displayed spectrum is shot noise, (9Z(v)). Hence, a system nonlinearity of, say, 4%, could affect the signal term 8% or more for cases in which (SIG/NOISE)pRE

154

MOLECULAR WEIGHT DETERMINATIONS

[7]

<1. For example, if the shot noise level is 100 units and (8(0)) only 50 units, a 4% amplitude nonlinearity in the wave analyzer at , = 0 is six units in the total display of the power spectruM, or 12% of (8(0)). This problem obviously worsens for v ~ 0, since ($(v)) then decreases. This consideration amply justifies the effort expended in the wave analyzer amplitude response calibration described here. Since an actual experimental run may take, typically, an hour or so, it is important to comment on system stability. It has been observed 13 that if the temperature in the experimental area remains reasonably constant (--~0.2°), then temperature dependent shifts in the phototube's quantum efficiency are negligible. In addition, drifts in the associated electronics described here are also negligible. Thus, it is not difficult to make experimental runs lasting even substantially more than 1 hour. It is also observed in Fig. 13 that the dc photocurrent, io, is monitored during an experimental run. By "bucking out" most of io and enlarging the remaining portion to full-scale deflection on a strip chart recorder, very slight changes in io can be observed. Since io is proportional to the incident laser power, Po (which is also monitored as previously mentioned), the two should vary together if Po drifts, and either can be used to correct the observed spectrum via Eqs. (35) and (36). However, it is still desirable to monitor both of these quantities since io is also a very sensitive check on the purity of the protein sample being studied (sample filtering and cell cleaning and filling techniques are described later). It will be found that the photocurrent will drift if observed immediately after cell filling while a few troublesome dust particles settle out, and then will stablilize. It has been found 13 that if io drifts more than about 0.2% during the course of an experimental run, reproducible results are difficult to obtain, and hence if io does not stabilize to this value or better within a reasonable period of time, it will probably be necessary to refill the scattering cell and thus begin again. One cannot overemphasize the utility of just "looking" into the scattering cell. Any substantial particulate contamination can be readily detected by looking for asymmetry in the forward-scattered light. It is also helpful to look into the scattering cell (at the unfocused beam) using a low power ( ~ 2 0 × ) microscope. Even small amounts of particulate matter may be seen as bright spots immersed in the beam which will properly appear "soft" and "diffuse" in enzyme solutions. The actual experimental spectrum, Si(v), is obtained by slowly sweeping the center frequency about which the bandwidth vl is located. The entire region of interest may then be plotted on the strip chart

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

155

recorder, extending from as close to zero frequency as possible out to several half-widths. The signal portion of the photocurrent is given in unnormalized form as



(50)

= ~ + (2r~/2~) 2

This equation reflects two characteristics of the spectrum: its value at zero frequency and its half-width. A computer program may be written which will adjust these two parameters to yield a least-squares best fit to the data. Alternatively, by plotting (]/<$'(v)>) versus v2 as shown in Fig. 14, and putting the best straight line through these data, one obtains a line of intercept b and slope m, where % / b / m (2FT/27r). Hence, one obtains DT from Eq. (4). Of course, the values for DT so obtained must be corrected to standard conditions 6 (20 °, viscosity of water) before they may be compared with other values or employed in the Svedberg equation for molecular weight determination, etc. An alternative to the "slow sweep" technique described here is to sweep through the desired frequency range very quickly and store the output of the wave analyzer in a "CAT" (computer of average transients). Sweeps are repeated a sufficient number of times for an adequate resulting signal-to-noise value. This method does not improve the observed signal-to-noise over that of the slow sweep technique, but does have the advantage of being relatively immune to long-term drifts in =

~y

t

I

[

I

=mx+b

I

FIG. 14. Linear plot of observed

I

I

spectrum.

156

MOLECV*,A~ WEIGHT DETERMINATIONS

[7]

laser power, phototube gain, etc. This fast-sweep method has been employed by Rimai, Hickmott, Cole, and Ca rew in a study of the thermal denaturation of ribonuelease. 4. It may be desirable to study a particular sample for only a very short period of time, since it may be unstable at room temperature or be undergoing some time-dependent conformational change. Since the realizable signal-to-noise ratio, (SIG/NOISE)Posr, depends on the data "accumulation time" T = 1/v2, it is necessary only to have information on &(v) for this period of time. Hence, the fluctuations in the photocurrent can be recorded on an endless-loop magnetic tape for the desired value of T, & (v) then being determined by slowly sweeping the wave analyzer through the range of frequency of interest, measuring the output of the tape recorder. Thus, the sample need remain stable only for the period T, which would rarely exceed 60 seconds. Similarly, several such loops could be made, each corresponding to a later time during some time-dependent effect, such as thermal denaturation. The tape recorder must obviously possess a relatively flat frequency response over the range of values of v necessary to determine the spectral width accurately. Commercial units employing the frequency-modulation recording system would be appropriate for this application. It must be pointed out that the response of the recorder would have to be carefully determined, as well as its stability, but the possibility of employing such a device for observing relatively slow conformational changes in enzymes is very attractive indeed. It is appropriate at this point to indicate that there exists another method to analyze the fluctuations in the photocurrent which is conceptually somewhat different from the spectral analysis discussed so far. Consider the definition of the autocorrelation function, R~(r) of the photocurrent i(t) :

Ri(r) = r---+ lim.~ - ~1

f_-

i(t)i(t + r)dl

(51)

It is readily shown 45 that the autocorrelation function of the photocurrent and the power spectrum of the photocurrent are simply the Fourier transforms of one another. Hence, one obtains from Eqs. (35) and (36) the correlation function of the photocurrent as

Ri(r)

~io~ + io~ e_~rt, I + Geio~(r)

(n > 1)

(52)

(io ~ + io=e-=rl*l + GeioS(r)

(n <_ 1)

(53)

4~L. Rimai, J. T. tgickmott, T. Cole, and E. B. Carew, Biophys. J. 10, 20 (1970). ~5C. Kittel, "Elementary Statistical Physics," p. 136. Wiley, New York. 1958.

[7]

TRANSLATIONAL

AND

ROTATIONAL

DIFFUSION

157

, , ~ j ~ G ei o 8 ('r)

F

r

t

2

-2F

Ivl

)>=I i° e i 2

-2F Irl

L~-e

~>1

~"": io2

Fla. 15. The correlation function of the photocurrent. The correlation function is displayed in Fig. 15, where ( $ ( r ) ) and (~(r)> are defined as the "signal" and the "noise" in analogy with the spectral case. I t has been shown 46 that the signal-to-noise ratio is the same for either spectral or autocorrelation function determinations. T h a t is, if Si (v) is studied for n discrete values of v simultaneously, and Ri (r) is studied for n discrete values of T simultaneously, then S~ (v) and R~ (r) are determined with the same accuracy for a given period of observation, T = l/v2. Obviously, one would like to obtain St (v) or Ri (T) for as many discrete values of their arguments as possible in simultaneous observation, rather than "mapping" them out one point at a time. To obtain full advantage of the correlation function technique, a digital autocorrelator must be employed. A suitable device is not presently available commercially, although several have been constructed and successfully; employed by individuals 47 in studies of biological macromolecules. With the advent of digital integrated circuits, one can expect the early appearance of an. appropriate commercial unit at a reasonable price. For the moment, multichannel wave analyzers are already available which are suitable for enzyme studies, 48 and these may expect to 4~v. Degiorgio and J. Lastovka, Phys. Rev. A 4, 2033 (1971). 47R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, and A. R. Peacocke, Nature (London) 227, 242 (1970). 'SFederal Scientific Corp., New York, New York.

158

MOLECULAR WEIGHT DETERMINATIONS

[7]

find increasing application. In addition, analog autocorrelators 49 may be purchased, although these are decidedly less satisfactory than the digital units. This type of correlator has been employed in a study of polymers in solution2 ° Such devices, however, like the multichannel wave analyzers, represent a far more substantial initial investment than the single-channel wave analyzers already described21 The Spherical Fabry-Perot Inter]erometer

In the discussion of the self-beating spectrometer, it was shown that the realizable signal-to-noise ratio of that technique is inadequate to observe the anisotropy scattering in protein solutions. It is shown in the present section that the spherical F a b r y - P e r o t interferometer (SFP), used in conjunction with a stabilized, single-frequency laser, is ideal for this application. The fiat F a b r y - P e r o t interferometer was divised at the end of the last century by Fabry, Perot, and Boulouch, 5~ and has been discussed extensively in its modern application elsewhere23 In the range of interest appropriate to the study of the anisotropy scattering in enzyme solutions, the recently developed spherical F a b r y - P e r o t interferometer 54,~5 offers enhanced light gathering ability and greater ease of alignment and stability, and hence is to be preferred. The following discussion ~6 concerns the SFP only. The SFP is an optical resonant cavity as shown in Fig. 16. I t consists of two spherical mirrors of spacing and radius of curvature R. The ratio of the intensity of the light transmitted by such a filter (It) to that incident upon it (L) is given as sin -~-j J Ii where if, the "free spectral range," is c - 4nR

(54)

(55)

°~Princeton Applied Research, Inc., Princeton, New Jersey. ~°N. C. Ford, Jr., W. Lee, and F. E. Karasz, J. Chem. Phys. 50, 3098 (1969). By "single channel" it is meant that S,(p) may be studied for only a single value of v at a time. ~2p. M. Duffieux, Appl. Opt. 8, 329 (1969). ~ K. W. Meissner, J. Opt. Soc. Amer. 31, 405 (1941). ~P. Connes, Rev. Optique Theor. Instrum. 35, 37 (1956). ~P. Connes, J. Phys. Radium 19, 262 (1958). ~This summary is taken from T. Greytak, "Spectrum of Light Scattered from Thermal Fluctuations in Gases." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1967.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

159

MIRRORSCATTERINCELL G MIRROR SINGLE MODE

I

HeFREQUENCYNeLASER-STABILIZED

0

I

I k/2 PLATE

PINHOLE SPATIAL FILTER

GLAN-THOMPSON PRISM STRIP-CHART

J/'~'/~'k RECORDER J/'\~'/~'k

RECORDER

~--~ ~ L~J

HERICALREoROTTER F'SPATBERRF;

FWOTOM 130ULTIPLIER FIG. 16. The spherical Fabry-Perot interferometer and associated equipment. From S. B. Dubin, N. A. Clark, and G. B. Benedek, J. Chem. Phys. 54, 5158 (1971). and F~, the "finesse," is

-

2(1

-- r)

(56)

Here c is the speed of light in vacuum, r the reflectivity of the mirrors, and n the index of refraction of the medium between the mirrors. Clearly, if t h e index n can be "swept," Eq. (54) indicates t h a t the transmission frequency of the interferometer can also be varied, and hence the SFP can be made a tunable filter. This sweeping of n is readily achieved by initially evacuating the can surrounding the mirrors and allowing gas from a reservoir slowly to leak in. The variation of n can be made reasonably linear in time by allowing the gas to flow into the SFP can through a restricted orifice. 57 When n is adjusted so t h a t a particular center frequency ,o m a y be transmitted, the filter will also transmit neighboring frequencies since the cavity has a finite resolving power due to the finite reflectivity of the mirrors used. In other words, the filter has an "instrumental profile" of half-width at half-height2 s 5~D. H. Rank and J. N. Shearer, J. Opt. Soc. Amer. 46, 463 (1956). Note that the symbol ~ is used here in analogy with the predetection bandwidth of the self-beating spectrometer [see Eq. (39) and Fig. 10].

160

MOLECULAR WEIGHT DETERMINATIONS

~1 -

2F~

[7]

(57)

Since the SFP is a resonant cavity, this profile is Lorentzian in shape. In practice, values of F~ of around 50 may be obtained using commercially available mirrors or even complete interferometers. The output spectrum of the SFP is the convolution of the instrumental profile, SsFp(v), with the spectrum of the incident light. In the present ease, this latter spectrum is SR(V), which is also Lorentzian [Eq. (7)]. The measured spectrum is then SovT(~) = SSFP(~) ® SR(~) =

/ ~~ S SFe(u)SR(u ' ' -- v)dp'

(58) (59)

In the present case, in which both the instrumental profile and the spectrum of the scattered light are Lorentzian, the output spectrum is also Lorentzian but with half-width at half-height of ForT = FR + pl 27r 2r

(60)

Thus, one must simply subtract the instrumental half-width from the measured half-width to obtain FR and hence DR [Eq. (8)]. It remains to demonstrate that the interferometer has sufficient resolving power to discern the spectrum of the scattered light and that the realizable signal-to-noise ratio is adequate. In order that the tails of successive transmission peaks do not suffer appreciable overlap with one another, it is desirable to make the free spectral range much larger than the width of the spectrum to be studied. Since (ra/2~) is expected to be in the range of 10 MHz, a value for ] of about 500 MHz would be appropriate. Since a finesse Fr [Eq. (56)] of 50 is commonly obtainable, an instrumental half-width (vl) of 5 MHz [Eq. (57)] is achievable, and hence the desired spectral width may be clearly resolved. In actual practice even better resolving powers are obtained22 It is thus seen that from a resolving power standpoint alone the SFP is well suited to the study of the anisotropy scattering in enzyme solutions. Lastovka 35 has made a detailed study of the relative merits of the self-beating and Fabry-Perot spectrometers from a signal-to-noise point of view. Assuming that each spectrometer views only a single coherence area, he obtained the remarkable result that the ratio of the minimum power in the scattered light to achieve a (SIG/NOISE)PosT [Eq. (38)] of unity in each case is

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

pSELF-BEAT rain psF.e --rmn

= ~

=

4

~1 ~" F2

161 (61)

where T is the accumulation time [T = 1/v~ in Eq. (40)]. This ratio obviously diverges for large T and the SFP is shown to be inherently much more sensitive than the self-beating spectrometer. For example, vl is expected to be in the range of 5 MHz. Taking T as 1 second, one finds from Eq. (61) that about 4000 times less power is required by the SFP than by the self-beating spectrometer to achieve the same output signal-to-noise ratio (SIG/NOISE)posT of one. Returning to the problem of a 1% lysozyme solution as a representative example, it was shown that a value of p~ =1.4 × 10-3 plus the parameters given in Table II implied a (SIG/NOISE)pRE of 2 × 10-4 [Eq. (44)]. Setting Vl = 5 MHz and T = 1 second, Eq. (40) then yields (SIG/NOISE)PosT of about 0.4, still quite unsatisfactory. (It is noteworthy that such a signal-to-noise ratio would make the observed spectrum look something like Fig. 8, where I ( t ) and t are read as Sour(v) and v, respectively. Needless to say, this is an unacceptable output spectrum for quantitative results.) From the above discussion, this same output signal-to-noise ratio could be obtained with only 1/4000 as much laser power if the SFP were employed. Since the output signalto-noise obtainable with the SFP is proportional to the square root of the incident power, ~9 the given value (50 mW) implies a value of (SIG/NOISE)PosT of about 30, nearly two orders of magnitude greater than in the beating case. Such a value has been confirmed experimentally. 12 Hence, the SFP is seen to possess the desired characteristics of high resolving power and high signal-to-noise capabilities. It might appear that the feasible (SIG/NOISE)posT obtainable with the SFP has been somewhat underestimated above because Eq. (61) was derived assuming only a single coherence area is collected. This restriction yields a value for (SIG/NOISE)pRE for the mixing spectrometer which is not improved by collecting more coherence areas, but is not required in the use of the SFP29 However, it is generally not possible to open up the acceptance solid angle arbitrarily in the SFP because the finesse, which has been assumed to be reflectivity-limited [Eq. (56)], will then be degraded due to nonsphericity of the mirrors. In fact, very small apertures must be used in general to approach the ~'This simply reflects the fact that the number of pulses per unit time (N) in the photoeurrent is proportional to the power falling on the photoeathode. Thus, since the average photoeurrent divided by the fluctuations in the photoeurrent is proportional to N/V'N, this quantity is also proportional to Ps/x,/P.~ = ~¢"~.

162

MOLECULAR WEIGHT DETERMINATIONS

[7]

theoretical finesse as given in Eq. (56), and thus the assumption of collecting only a single coherence area is reasonable. In Fig. 16 the block diagram of the complete SFP interferometer and associated equipment is presented. This particular apparatus was used to determine Da for lysozyme from its anisotropy spectrum. 12 The first, and perhaps most difficult to obtain, piece of equipment is the stabilized, single-frequency laser. The calculations on the signal-to-noise ratio for the SFP were based on 50 mW, yet the typical commercially available unit has an output of only about 0.1 mW (for example, the SpectraPhysics Model 119). And the stabilized, single-frequency requirement is essential since the SFP measures the spectrum of the scattered light, not that of the fluctuations in the photocurrent, as is the case for the selfBeating spectrometer. Hence, the spectrum of the light source must be narrow compared with the expected spectrum of the anisotropy scattering, that is, narrow compared With ~10 MHz. The line width of a gas laser is determined by the Doppler-broadened gain curve, and is about 2 GHz for a He-Ne laser. However, another interferometer can be placed in the laser cavity which will permit only a single axial mode to run in the cavity, and hence a single-frequency source is obtained. 6°,~1 This one axial mode is frequency stabilized by locking the laser output to the transmission peak of a reference SFP which is highly thermally stabilized21 While elaborate, this technique produced a single-frequency laser of about 10 mW optical power and stable to about 2 MHz per hour. Recently, both Spectra-Physics and Coherent Radiation have presented relatively high-power lasers with an optional etalon attachment for single frequency output. These lasers are not stabilized, however, and provision would have to be made for this stabilization, since drifts of hundreds of MHz would otherwise result. Because of the rapidly expanding utility of stabilized single-frequency lasers in applications of various kinds, it is reasonable to anticipate the appearance of a high power commercial unit in the relatively near future. The light which leaves the laser has associated with it an irregular halo which arises from reflections inside the laser tube and scattering at the exit windowY The beam is thus "cleaned" by a pinhole spatial filter, various forms of which are available commercially (Jodon Engineering Associates, Inc., Ann Arbor, Michigan, for example). By focusing the output to the diffraction limit of the primary beam, and placing a pinhole only slightly larger than this limiting diameter, any light which P. W. Smith, 1EEE g. Quantum Electron. B-l, 343 (1965). el N. A. Clark, "Inelastic Light Scattering from Thermal Fluctuations in Gases." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

163

is not focused to the diffraction limit at this point is prevented from exiting the filter. In principle, this precaution should also be taken in the operation of the self-beating spectrometer in determinations of D~. However, in this latter case one is usually not so close to marginal signal-to-noise conditions as in studying the anisotropy spectrum, and hence the spatial filter is not ordinarily necessary. The output of the particular laser described here had horizontal polarization; a half-wave plate was therefore inserted after the spatial filter to convert the polarization to vertical. The vertically polarized light then enters the scattering cell. It is obvious from the discussion of the expected signal-to-noise ratio that as much scattered light as possible must be collected. In order to enhance the intensity of collected scattered light, the laser beam in this particular arrangement is made to pass several times through the scattering cell between a pair of high-reflectivity (dielectrically coated) mirrors. The first mirror has a small portion of its coating removed to allow initial entry into the scattering cell. The amount of "beam walking" arising from these several passes is highly exaggerated in the figure. A factor of five increase in the intensity of the collected scattered light was reported with this technique. 1-~ It is most convenient to study the scattered light for 0 = 90 °. Light scattered at this angle is collected by a lens, passed through a GlanThompson prism, and then frequency analyzed by the SFP. Fortunately, several manufacturers have commercial SFP units available. These include Tropel, Inc., Fairport, New York, and Jodon Engineering Associates, Inc., Ann Arbor, Michigan. Alternatively, one may purchase the required high quality spherical mirrors (from, for example, Coherent Optics, Inc., Fairport, New York) and then construct the associated container. The latter approach is desirable in the sense that it allows picking the most convenient free spectral range (in the range of 500 MHz for the present purposes), but it has the disadvantage that constructing highly stable interferometer cavities is certainly nontrivial. 56,61 The SFP is frequency swept by varying the gas density in the container. This is most simply done by the constant rate leak of a dry gas from a large reservoir into the initially evacuated SFP can, 57 but other methods exist. The reader is cautioned that piezoelectric sweeping of the SFP by mounting one of the mirrors on a cylinder of piezoelectric material onto which linear ramp voltage is applied presents problems of nonlinear motion of the transducer, as well as hysteresis. Care would have to be exercised to ensure that the instrumental profile remained constant and that the apparent free spectral range did not vary. The Glan-Thompson prism selects which polarization of the scat-

164

MOLECULAR WEIGHT DETERMINATIONS

[7]

tered light is studied. If the prism is oriented to pass vertically polarized scattered light, the Fabry-Perot observes essentially only light scattered by concentration fluctuations, as already discussed. The spectrum of this component is typically only several kHz wide and hence appears to the Fabry-Perot essentially as a delta function, since vl is in the vicinity of several MHz. Thus, analyzing the vertically polarized scattered light gives the instrumental profile of the interferometer, namely, SsFp(v) [see Eq. (59)]. The Glan-Thompson prism is then rotated 90 ° to observe the depolarized scattered light, that is, the anisotropy scattering. It is important to note that this process has not altered the geometry of the collection optics and, therefore, the spatial distribution of the illumination of the SFP remains unchanged. This guarantees that the instrumental profile Ss~P(v) as determined by observing the polarized component of the scattered light is indeed the same instrumental profile of the spectrometer when it observes the depolarized scattered light. It is an understatement to say that the exact shape of SsFe(v) is a sensitive function of the nature of the illumination of the FabryPerot, and the above technique completely obviates any possible misinterpretation of the instrumental profile. The spectrum of the anisotropy scattering is studied with the GlanThompson prism rotated 90 ° from the vertical position. Since the anisotropy scattering is extremely weak, this 90 ° adjustment is easily made by seeking the minimum photocurrent after centering the SFP transmission at or near a peak on the vertically polarized light. The intensity of the depolarized light scattered by enzyme solutions is indeed weak. One therefore can anticipate that the dark current in the phototube may compete strongly with the signal, and this is found to be the case. 12 Cooling of PMT's can produce substantial dark current reduction, but there are alternative techniques which yield very significant reduction even at room temperature. Since much of the dark current results from spontaneous emission from the photocathode, it is desirable to make this as small as possible, while still ensuring that all the collected scattered light is seen by the PMT. Phototubes are commercially available (the I T T FW 130 series, for example) with effective photocathode diameters of less than 0.01 inch. These very small effective diameters are achieved by electronic focusing coils which allow only those photoelectrons which originate from the desired (small) region of the photocathode to proceed down the dynode chain. The dark current may be then further reduced by recognizing that some random emission arises from the first and succeeding dynodes themselves. However, the pulses observed at the last dynode produced by these random emissions are smaller than those arising from photoelectrons since they have

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TRANSLATIONAL AND ROTATIONALDIFFUSION

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experienced at least one stage less gain. In fact, using an ITT FW 130 PMT with an effective photocathode diameter of 0.1 inch, and discriminating against the output pulses to accept only 90% of those arising from the photocathode, the apparatus in Fig. 16 yielded a dark pulse rate of only 13 per second. The pulses passed by the single channel analyzer are amplified,6~ accumulated for a period T = 1/v2 by an RC filter, and displayed on the strip chart recorder. The spectrum so displayed, which is the convolution of the instrumental profile with the spectrum of the scattered light, will contain a broad background, similar in appearance to the shot effect spectrum in the self-beating technique. This arises from the anisotropy scattering of the water itself, which is many GHz wide, and the small but finite dark current. This background level is subtracted before fitting the observed spectrum to obtain rR and hence DR. The fact that bovine serum albumin has such a small value for pv may be used to good advantage in aligning the equipment diagrammed in Fig. 16. The scattering cell is filled with a BSA solution and p~ is determined by measuring the area of the spectrum of the depolarized scattered light, and comparing this value with that obtained for the polarized component. This ratio is pv by definition, and is ~ 10-4 for BSA. 12,22, Since contributions to the apparent depolarized scattered light from Glan-Thompson prism leakage and acceptance angle contributions are less than this value, 12 any measured value greater than 10-~ must arise from stray light or sample impurity. This reference point for BSA allows the investigator to perfect his techniques in cleaning the sample and filling the scattering cell, as well as in eliminating all sources of stray light. These techniques are necessary and difficult, and are obtained primarily through experience with the particular macromolecule under study.

Cleaning and Filling the Scattering Cell No amount of discussion could possibly overemphasize the requirement of absolute cleanliness for the sample cell and the enzyme solution it contains. Techniques which have been discussed for cleaning enzyme solutions preparatory to scattered light intensity measurements1 are completely appropriate in spectral studies as well and should be applied. A few general remarks regarding spectral determination studies, however, are worthy of comment here. 12,13 Because large particulate contamination such as dust has a very 63Modular electronics for pulse discrimination,shaping, and amplification are available from EG and G, Inc., Salem, Massachusetts.

.166

MOLECULAR WEIGHT DETERMINATIONS

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small diffusion coefficient, one might expect the light scattered by such material to have an extremely narrow spectrum and hence be unimportant in the interpretation of the observed spectrum. 63 This argument fails for both the self-beating and SFP techniques of determining diffusion coefficients. In the former case, the spectrum of the photocurrent is that of the incident light convolved with itself, 34 but now centered at dc. Thus, if the spectrum of the incident light is Lorentzian, the spectrum of the signal term in the photocurrent is also Lorentzian but twice as wide, as previously discussed. If, on the other hand, the incident light contains in addition to the light scattered by the enzymes an intense component arising from dust scattering, the spectrum of the photocurrent will contain three terms~3: the self-beat spectrum of the contamination scattering (generally negligibly narrow); the self-beat spectrum of the light scattered by the enzymes (containing the desired signal term); the cross-beat spectrum of the two components in the scattered light, which is approximately one half as wide as the desired self-beat signal of the light scattered by the enzymes themselves. The presence of this cross-beat term can produce significant line-shape distortion and render it impossible to obtain accurate results. The situation is no better in the case of the SFP. Because the output of the SFP is the convolution of the spectrum of the incident light with the instrumental profile [Eq. (59)], even a monochromatic illumination presents a contribution to the output spectrum as wide as the instrumental profile itself. Hence, line shape distortion cannot be avoided if contamination is present. There are thus no special reasons why sample contamination can be more readily tolerated in spectral measurements than in those of intensity. All the traditional cautions and restrictions employed in determining the latter 1 still apply in studies of the former. The standard Teflon-stoppered spectrophotometric cuvettes (Lux Scientific Instrument Co., New York) suggested as scattering cells have been cleaned with good success in the following manner for studies on lysozyme.12,13 The cells are first cleaned in chromic acid, rinsed, and then ultrasonically cleaned in glacial acetic acid for 15 minutes. Four liters (approximately 500 cell volumes) of filtered distilled water are then forced through the cell in a closed system under pressure. The Teflonstoppered cells are ideal for this because Teflon or polyethylene hoses may be tightly fitted into the stopper, which itself fits very tightly into the cell. Large size filters (14 cm in diameter) are employed to ensure a high flow rate (about 250 ml/minute). The 0.22 t~ Millipore Corp. 6~M. J. French, J. C. Angus, and A. G. Walton, Science 163, 345 (1969).

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TRANSLATIONAL AND ROTATIONAL DIFFUSION

167

(Bedford, Massachusetts) cellulose ester filters are ideal for this purpose. Without opening the cell to air, filtered dry nitrogen is then admitted to displace the remaining water and dry the cell. When precleaned in this fashion, the scattering cells are found to be spectroscopically clean. When filled with distilled water, for example, only the Brillouin scattering and anisotropy scattering from the water itself are visible. Filtering the enzyme sample into the precleaned cell presents many complex problems which depend on the nature of the particular sample under'investigation. Surface denaturation forbids the Millipore filtration of many proteins, although this technique has been successful for lysozymeY ,13 Bier 1 has presented a detailed discussion of the problem. Applications Traditionally, the promise of diffusion coefficient information has not been fully realized because of the difficulties associated with conventional determinations of D. Most of these difficulties are centered around the time required for such determinations, typically measured in days. The almost routine fashion in which DT may now be measured in about an hour or less has dramatically enhanced the desirability of measuring the diffusion coefficient. Although the procedure for determining DR presented here is somewhat more elaborate than that for DT, many alternative methods are even more elaborate. The potential of these two new techniques has already been tapped, and a few representative applications are given below.

Determination o] Protein Molecular Weights The technique of sedimentation-diffusion has long provided accurate molecular weight information. The limiting factor was usually the translational diffusion coefficient, for which an uncertainty of around 5% 84 was typical. From the Svedberg equation, one has for the molecular weight (M) :

(S)

M = ~

RT

(1 --p~)

(62)

where R is the ideal gas constant, S the sedimentation coefficient, p the solvent density, and ~ the partial specific volume of the protein. Hence, from Eq. (62), one obtains (M)Ty

( pA~ y]~/~

where A signifies the uncertainty in each parameter. The 5% typical See reference cited in footnote 14, p. 360.

168

MOLECULAR WEIGHT DETERMINATIONS

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uncertainty in Dr is generally the largest error by far in the study of proteins and this is dramatically improved when Dr is measured by the self-beating technique. In principle both Dr and S can be measured in the same cell (the centrifuge cell), and this procedure thus guarantees that both Dr and S are measured under identical experimental conditions. Since both DT and S can now be measured with an accuracy of 1% or better, and since ~ can be measured for proteins to better than 1/~%, Eq. (63) indicates that protein molecular weight can be measured with an accuracy of better than 2%. This places the technique of sedimentation-diffusion among the most accurate probes of protein molecular weight. In the case of lysozyme,1~,13 for example, a value of 14,500 _+ 300 is obtained, which compares quite favorably with the value 14,600, determined from the known amino acid sequence of the enzyme25 It is noteworthy that this technique is by no means restricted to the small enzymes, and actually becomes progressively more useful as the molecular weight of the particle studied becomes larger. This occurs because Dr is increasingly more difficult to obtain by classical techniques as M increases, culminating in the remarkable state of affairs that a month is not an unusual period to be required for determining the translational diffusion coefficient of a large virus by traditional methods. Hence, mixing spectroscopy has allowed the determination of coliphage and coliphage-DNA molecular weights with very high accuracy. 66 Con]ormationaI Changes in E n z y m e s

The rapid determination of DT made possible through the techniques of mixing spectroscopy allow the diffusion coefficient to be used as a probe of changes in conformation, in which application it joins such techniques as measurement of intrinsic viscosity, optical rotatory dispersion (ORD), difference spectroscopy, and sedimentation velocity. Studies of the chemical denaturation of lysozyme18,67,6s and the thermal denaturation of ribonuclease 44 have been presented. The effect of chemical denaturation on the diffusion coefficient of lysozyme is displayed13 in Fig. 17. The shape of this curve is very similar to that observed for changes in the optical rotation69,7° and inD. C. Phillips, Proc. Nat. Acad. Sci. U.S. 57, 484 (1967). S. B. Dubin, G. B. Benedek, F. C. Bancroft, and D. Freifelder, J. Mol. Biol. 54, 547 (1970). e7S. B. Dubin, G. Feher, and G. B. Benedek, Biophys. J. 9, A213 (1969). ~S. B. Dubin, G. Feher, and G. B. Benedek, submitted for publication (1972). (in press). *~K. Hamaguchi and A. Kurono, J. Biochem. 54, 111 (1963). 7oC. Tanford, R. It. Pain, and N. S. Otchin, J. Mol. Biol. 15, 489 (1966).

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169

TRANSLATIONAL AND ROTATIONAL DIFFUSION ~o

I

_.L.

°~e

C4

i i--

~1

j--j~..i~j

I ~

I

i

I

Ie

8--

E

%

7-6--

--c

5

c v

--

4- -

I % Lysozyme solution 0.1 M Sodium Acetote-Acetic ocid buffer

o 5c~ 2-

pH = 4 . 2

I

I I

i

I

L

I

L

I

2 5 4 Concentration GuCI (molar)

t

L 5

t 6

FIG. 17. Effect of chemical denaturation on the diffusion coefficient of lysozyme.

From S. B. Dubin, "Quasielastic Light Scattering from Macromolecules." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970. trinsic viscosity69 of lysozyme, and thus confirms that all three methods indeed probe the same conformational change. It must be pointed out, however, that in common with ORD and intrinsic viscosity, measurement of DT cannot reveal whether more than one species of lysozyme is present for any value of the concentration of the denaturant, quanidine hydrochloride--at least when the change in DT is so small over the range from completely native to completely denatured. This occurs because the spectrum of the light scattered by two species of molecules with not very different values of DT is still highly Lorentzian in shape, with halfwidth reflecting the m e a n diffusion coefficient of the two species present. The signal term of the self-beat spectrum of the photocurrent is the convolution of the spectrum of the scattered light with itself (but centered at dc),13,34 and this spectrum is even more nearly Lorentzian than the spectrum of the scattered light itself. This result is shown quite dramatically in Fig. 18.13 The solid line refers to the self-beat spectrum of the photocurrent expected from a mixture of equal numbers of molecules with the same molecular weight but with one species having a diffusion coefficient 50% larger than the other. The open circles indicate the best single Lorentzian fit, from which a value of D~ equal to 98% of the average diffusion coefficient is deduced. Not only is the self-beat spectrum of such a mixture seen to be accurately Lorentzian in shape, but also

170

MOLECULAR WEIGHT DETERMINATIONS

000

_.~

5i(~)

Best single Lorentzion fit

, 20

[7]

o.,

9

, 15

~

,

<~--?--"-~ I0 Frequency

i

I

i

I

I

I

5

FIG. 18. Self-beat spectrum of the light scattered by a two-component mixture (molecules of equal number density and molecular weight, but differing by having diffusion constants of D1 and 1.5D1, respectively). From S. B. Dubin, "Quasielastic Light Scattering from Macromolecules." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970. it is a p p a r e n t that the average diffusion coefficient is determined from such a spectrum with a high degree of accuracy. On the other hand, if it is known from other data, such as recent N M R studies, 71 that two species are present, then the relative numbers of each can easily be determined from the spectrum by recalling that it is an accurate measure of the average diffusion coefficient. Hence,

D ~ [NID1 Jr (1 -- NI)D~]

(64)

where D1 and D2 are the two values of D, and N1 is the fraction of molecules with diffusion coefficient D1. The two values D1 and D2 are determined by examining the spectrum of the completely native and completely denatured protein. As the ratio D2/D1 increases beyond about 1.5, the value of D r determined from the spectral width becomes progressively "pulled" below the average of (D~ + D~)/2. This is indicated for values of D2/DI from 1 to 4 as shown in Fig. 19. TM Also displayed in this figure is the normalized rms error of the best single Lorentzian fit to the self-beat spectrum, t h a t is, the percentage ratio of the rms deviation of the "best fit" Lorentzian to the value of the spectrum at v = 0. I t is ~lj. D. Glickson, C. C. McDonald, and W. D. Phillips, Contribution No. 1553, Central Research Department, E. I. du Pont Co., Wilmington, Delaware; C. C. McDonald and W. D. Phillips, Abstracts 3rd Intern. Conf. on Magnetic Resonance in Biological Systems, Warrenton, Virginia, 1968.

[7]

TRANSLATIONAL AND ROTATIONAL DIFFUSION

171

i.O

0.8

t

rBest PAveroge

0.6

o~ C~ 0.=% o~ [~ ~

0.2

o.o

i

]

,

I

i

2.5

0.5-

°-°,S

~

,'

'

io

'

'

;o

(D2/Dl)

Fro. 19. Width and rms deviation of best single Lorentzian fit to the self-beat spectrum of the light scattered by various two-component mixtures. From S. B.

Dubin, "Quasielastic Light Scattering from Macromolecules." Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1970. clear from the figure that this error does not exceed 1% until D2/D1 = 2.5, and hence D 2 / D I will have to be significantly larger than 2.5 before the self-beat spectrum will be conspicuously non-Lorentzian. In short, spectral analysis is definitely not an ideal probe of the polydispersity of a system, although it is shown that if the diffusion coefficients of the constituents can be determined separately, then the spectrum of the light scattered by a mixture may be used to determine the relative numbers of each constituent. The problem of detecting polydispersity by spectral analysis has been examined for various mixtures of spheres, rods, and conformations ~3 and it is shown that the method is generally insensitive to such mixtures.

172

MOLECULAR WEIGHT DETERMINATIONS

[7]

In addition, two detailed theoretical studies of the spectrum of the light scattered by polydisperse rods and random coils have been presented. 72,7~ Size and Shape o] E n z y m e s in Solution

For many years it has been common practice to determine the size and shape of proteins in solution by combining various physicochemical measurements such as those of diffusion and intrinsic viscosityJ 4,z5 In principle such combinations can yield the major (2a) and minor (2b) axes of the ellipsoid of revolution hydrodynamically equivalent to the protein. In addition, if the partial specific volume (~) and molecular weight (M) are also known, the degree of solvation of the protein may be determined. These techniques have proven only partially successful, however, because the accuracy of the individual measurements has not been high enough, particularly in the case of diffusion coefficients. Hence, a family of solutions is possible within the error limits of the measured parameters, and this leads to considerable uncertainty in the size, shape, and degree of solvation of the protein. The techniques described in this paper allow the determination of diffusion coefficients with rather high accuracy. For example, recent determinations of DT and Da for lysozyme have been reported by Dubin, Clark, and Benedek 12 of (10.6 ± 0.1) × 10 ; cm2/sec and (16.7 ± 0.8) × 106 per second respectively, both values being corrected to water at 20 ° . Here the translational diffusion coefficient was determined by the method of self-beating spectroscopy, and the spherical F a b r y Perot interferometer was employed to measure the rotational diffusion coefficient. Since Perrin ~,~7 has provided expressions for the translational and rotational diffusion coefficients of an ellipsoid of revolution, the above values may be used to obtain the dimensions of such an ellipsoid hydrodynamically equivalent to lysozyme in solution. Values for the major and minor axes of two such ellipsoids are thereby obtained: Prolate: Oblate:

2a = (55-+-1) A 2a = (12.5=1= 1) A

25 = ( 3 3 + 1)

25 = (55.5 4- 1)

The oblate ellipsoid is grossly incompatible with the shape of crystalline ~Y. Tagami and R. Pecora, J. Chem. Phys. 51, 3293 (1969). 7. R. Pecora and Y. Tagami, J. Chem. Phys. 51, 3298 (1969). ~4j. L. Oncley, Ann. N.Y. Acad. Sci. 41, 121 (1941); J. L. Oncley, in "Proteins, Amino Acids, and Peptides" (E. J. Cohn and J. T. Edsall, eds.), p. 543. Reinhold, New York, 1943. ~ H. A. Scheraga and L. Mandelkern, J. Am. Chem. Soc. 75, 179 (1953). ~6F. Perrin, J. Phys. Radium 5, 497 (1934). ~TF. Perrin, J. Phys. Radium 7, 1 (1936).

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TRANSLATIONAL AND ROTATIONAL DIFFUSION

173

lysozyme,TM and it could be rejected for this reason alone. However, the data contain internal evidence that the prolate form is the appropriate choice, thereby lending strong support to the belief that lysozyme retains its crystalline form even in solution. This is deduced as follows. The volume of a prolate ellipsoid with the above dimensions is 31,400 A 3, while the volume of the unsolvated molecule is 16,900 A3 as determined from ~ = (0.703 ± 0.004) ml/g 79 and the molecular weight of 14,600 determined from the X-ray diffraction map of the molecule.6~ Hence, 14,500 A8 of the hydrodynamic volume of the prolate ellipsoid model must reflect the solvation of lysozyme. It is reasonable to assume that this solvation has the density of the solvent itself, since the molecular weight of lysozyme determined by amino acid sequence is the same as the physicochemical determinations. Thus, a value for the protein's solvation of (0.60 ± 0.03) gram of solvent per gram of dry lysozyme was obtained. 12 This value compares very favorably with that of (0.50 ± 0.05) g/g presented by Steinrauf8° as determined by measuring weight loss in dried crystals. The oblate model gives a value of only 0.12 g/g, clearly inconsistent with the Steinrauf measurement. Finally, assuming that the bound solvent is excluded from the interior of the enzyme (a view supported by the results of X-ray studies), it is seen that lysozyme in solution consists of the unsolvated molecule of dimensions (48 ± 1) A by (26--+ 0.8)A, which in turn is covered by a shell of solvent 3.5 A thick. Hence, the enzyme is covered with essentially a monolayer of tightly bound water. The dimensions of the unsolvated molecule may be compared with those of lysozyme in the crystalline state which are approximately 45 A by 30 A.7s By measuring the area of the spectrum of the depolarized light scattered by lysozyme and comparing this value with that of the polarized component, a value of p~. of 0.0014 ± 0.0001 was reported for the enzyme.12 Assuming the bound solvent has the same index of refraction as water, the pertinent dimensions to be used in determining the form factor 17 (Li) in Eq. (24) are those of the unsolvated molecule, giving a value for the axial ratio (a/b) of 1.84. Thus, combining Eqs. (18) and (24) with the measured value for p,, it was determined that lysozyme must have intrinsic anisotropy in addition to its form anisotropy. Values for the index of refraction of lysozyme along its maior and minor axes were then determined as 1.62 and 1.59, respectively. The above three examples of applications of the new techniques for ~8D. M. Chipman and N. Sharon, Science 165, 454 (1969). ~A. J. Sophianopoulos, C. K. Rhodes, D. N. Holcomb, and K. E. van Holde, J. Biol. Chem. 237, 1107 (1962). 8°L. K. Steinrauf, Acta Crystallogr. 12, 77 (1959).

174

MOLECULAR WEIGHT DETERMINATIONS

[7]

d e t e r m i n i n g the diffusion coefficients of e n z y m e s are b y no m e a n s a c o m p r e h e n s i v e list, b u t r a t h e r are given as an i n d i c a t i o n of some of the m a n y uses for the i n f o r m a t i o n c o n t a i n e d in diffusion coefficients. B y m a k i n g such i n f o r m a t i o n a v a i l a b l e with g r e a t e r ease a n d w i t h higher a c c u r a c y t h a n b y t r a d i t i o n a l techniques, the u t i l i t y of such m e a s u r e m e n t s m a y be m o r e f u l l y r e a l i z e d t h a n in t h e p a s t . Acknowledgment The author acknowledges with thanks the support toward this work of the Department of Physics and Institute for Molecular Biology, California State College, Fullerton, as well as that of the California State College Fullerton Foundation. Portions of this work are based upon a Ph.D. thesis submitted by the author to the Department of Physics, Massachusetts Institute of Technology, in February, 1970. This is publication No. 5 from the Institute for Molecular Biology, California State College, Fullerton.