Determination of Molecular Weight from the Interference Pattern of Sedimentation Equilibrium in the Ultracentrifuge G. M. Nazarian Department of Chemistry, San Fernando Valley State College, Northridge, Calf. 91324 In the past, the equation of sedimentation equilibrium for determining the molecular weight of an ideal solute in a centrifuge has been expressed either in terms of absolute concentration measurable by ultraviolet absorption or concentration gradient measurable by schlieren optics. No useful expression has been available, however, in terms of finite differences of concentration alone, the uantities measurable with extreme accuracy by inter erence optics. This paper shows that such an expression can be obtained if one conside r s concentration differ e nces between Iocations (r,, rd) for which rn'2- r,2 is constant, where r is radial distance from the axis of rotation. This concentration difference is shown to vary exponentially with radius squared, enabling the determination of molecular weight from a series of such measurements. Extensions to polydisperse and nonideal systems are also described. The procedures involved are illustrated by application of the method to experimental data obtained using ribonuclease as the solute.
7
INRECENTYEARS, the technique of molecular weight determination in the ultracentrifuge has seen considerable improvement as a result of theoretical and experimental advances ( I ) , one of which has been the introduction of Rayleigh optics. However, the mathematical equations available for the interpretation of low-speed sedimentation equilibrium experiments have been geared to ultraviolet absorption and schlieren optics which yield data on absolute concentrations and concentration gradients, respectively. The interference fringes obtained with the Rayleigh system, on the other hand, provide very accurate information regarding differences in solute concentration at different locations in an ultracentrifuge cell and equations for making direct use of such data have not been available. This has necessitated a separate calibration experiment to determine the initial solute concentration in terms of fringes by layering solvent over solution in a synthetic boundary cell and observing the fringe shift ( I , 2). Another possibility is to apply an equation involving concentration gradient derived for use with schlieren optics such as that developed by Van Holde and Baldwin (3) in their Method 11, but getting the necessary data from an interference pattern requires the less precise operation of numerical differentiation (1). Because of these disadvantages which arise in attempting to exploit the potential accuracy of interference optics, Yphantis (4, LaBar (3, and Charlwood (6) have introduced experimental modifications in the equilibrium run to circumvent the diffi(1) J. M. Creeth and R. H. Pain, Progr. Biophys. Mol. Biol., 17, 217 (1967). (2) K . E. Van Holde, Fractions, No. 1 (1967), Beckman Instruments, Inc., Palo Alto, Calif. (3) K. E. Van Holde and R. L. Baldwin, J . Phys. Chem., 62, 734 (1958). (4) D. A . Yphantis, Biochem., 3,297 (1964). (5) F. E. La Bar, Proc. Natl. Acad. Sci. US.,54, 3 1 (1965). (6) P. A. Charlwood, J . Polym. Sci., Part C, No. 16 (pt 3), 1717 (1967).
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ANALYTICAL CHEMISTRY
culties, while Chervenka (7) has suggested a way out by coupling the information from both the schlieren and Rayleigh patterns. Recently, Richards, Teller, and Schachman (8) have described further techniques for the labeling of fringes in the determination of molecular weight by the low-speed sedimentation equilibrium method including the introduction of a computer program. It is the purpose of the present paper to indicate a new method based on the fact that the equation of sedimentation equilibrium for an ideal solute can be cast rigorously into a form which does not involve absolute concentration requiring fringe labeling or derivatives of concentration requiring the measurement of fringe differences at infinitesimal separations in the cell. Instead, the result involvesfinite differences at arbitrarily large separations and these are exactly the data which can be obtained accurately from interference patterns. A preliminary study of the complications which arise in the application of the method to polydisperse and nonideal systems is also presented. THEORY
Ideal Monodisperse Systems. The derivation of the result is based on a property of the exponential function first exploited by Guggenheim ( 9 ) for the calculation of rate constants of first-order reactions. In that case, the independent variable is time, whereas in this case it is a distance squared but the mathematics goes through in the same way. The equilibrium concentration distribution of an uncharged solute having molecular weight A4 and forming an ideal solution is given by the standard equation c(q) = c(0)eAM4
(1)
in which A = (1 - ijp)w2/2RTand q = r2, where r is radial distance from the axis of rotation, 6 is the partial specific volume of the solute, p is the density of the solution, w is angular speed of rotation, R is the molar gas constant, and T is absolute temperature. The usual centrifuge cell is located a certain distance from the axis so that the significance of c(0) is purely mathematical; it is the concentration arrived at when the exponential curve of the distribution in the actual solution is extrapolated back to the origin. Applying the Guggenheim approach to Equation 1 , we have c(q
+ Q) - c(q> = [4Q>- c(0)18Mg
(2)
For fixed Q, the concentration difference in Equation 2 varies exponentially with q just as the absolute concentration of Equation 1. It should be emphasized that Equation 2 does not imply numerical differentiation; the result is exact for any (7) C. H. Chervenka, ANAL.CHEM., 38, 356 (1966). (8) E. G. Richards, D. C. Teller, and H. K. Schachman, Biochem., 7, 1054 (1968). (9) E. A . Guggenheim, Phil. Mug., 7, ( 2 ) 538 (1962).
size Q. Since absolute fringe number J is related to weight concentration according to (3) where a is the cell thickness, k the specific refraction increment, and A the wavelength of the light, Equation 2 shows that the fringe shift, AQJ = J(q Q) - J(q)-Le., the number of fringes crossed in traversing the interference pattern from q to q Q is given by
+
+
ak ApJ = - [c(Q) - c(0)]eAMq
x
(4)
Choosing any two points q1 and q2, this yields In
J(qz
+ Q) - J ( q z )
while differentiation gives d- = l n AQJ
AM
(6)
4 These results are to be compared with
and
Equation 7 is due to Lamm (IO) and is intended for use with schlieren optics; Equation 5 shows that Lamm's equation remains exactly valid when the derivatives are replaced by differences measured between locations whose q-values differ by the same amount, Q. Equation 8 is the basic expression used for the most precise determination of M from Rayleigh interference records; deciding on the absolute value for J is the main experimental difficulty (I). However, Equation 6 shows that measurements of J are unnecessary and that the basic expression is still valid when J i s replaced by AQJ. In practice it is desirable to use a large number of pairs of points to average out experimental errors. If measurements of AQJare made for (ql,q1+ Q), . . . ,(qn,qn Q,) . . . ,(q2v,q N Q), then a graph of log A Q J us. q will be a straight line and knowing A , the molecular weight can be calculated from the slope, since from Equation 4 we have
+
+
log ApJ = AMq/2.303
solution if no loss of solute or solvent has occurred during the experiment. Ideal Polydisperse Systems. With the usual assumptions that all solutes have the same k and also the same 0 and therefore A , it is known ( I ) that the apparent molecular weight obtained by applying Equation 8 to any point in the cell corresponds to the weight average molecular weight at that point-Le., M,(q) = 2 Mge(q)/2ci(q),while Equation 7 leads to the z-average molecular weight-Le., M&) = 2Mc 2cc(q)/2 Mccc(4). To determine the meaning of the apparent molecular weight obtained by the method proposed here under the same assumptions, we return to Equations 4 and 6 writing the total fringe shift as the sum of the contributions from all solute species
+ log ak7[c(Q) - c(O)]
(9)
In addition, the intercept yields c( Q) - c(0) or c(0) [exp(AMQ) - 11which in conjunction with the value of AMfrom the slope leads to c(0) and hence the absolute concentration distribution. In particular, the average concentration of solute actually present at equilibrium can be determined from
If we were to choose Q sufficiently near zero, exp(AMtQ) - 1 would approach AMiQ and M u p p would become M,(q). However, our purpose is to reduce the relative error in the measurement of fringe shifts by working with sufficiently large values of AQJ,which means Q might be chosen as large as possible-Le., spanning about one half the total available range. Therefore let us examine the situationwhen Q = QmUz= ( q b - qJ2. As to the magnitude of AMc, we set C = 2 ci and find from Equations 1 and 10 the known (3) result: (40 - q,)AM, = [c(qh) - c(q,)]/c. In the usual low-speed experiment, the conditions are chosen such that the 1. latter concentration ratio is near unity, so ( q b - q,)AM, Therefore AM,Qm,, '12 and it is not satisfactory to approximate exp(AMiQ) - 1 by AMiQ. To obtain a form involving only alternate powers of AM
