June 5, 1961 [CONTRIBUTION FROM
CALCULATION OF DIFFERENTIAL DIFFUSIONCOEFFICIENTS THE
2425
DEPARTMENT OF BIOLOGICAL CHEMISTRY, HARVARD MEDICALSCHOOL, BOSTON, MASSACHUSETTS]
A Method for Calculating Differential Diffusion Coefficients in Two Component Systems : Application to Glycine-Water and Bovine Mercaptalbumin-Buff er Systems' B Y T. E. THOMPSON* AND J. L. ONCLEY RECEIVED APRIL 21, 1960
A procedure, based on the Boltzmann method, is described for calculating the concentration dependence of the diffusion coefficient in two component systems from Rayleigh Diffusiometer data. Application of the method is made to two systems a t 20": glycine-water and bovine mercaptalbumin-buffer (sodium acetate:acetic acid, PH = 4.75, I'/2 = 0.05). Differential diffusion coefficients and the magnitude of the concentration dependence for the glycine system are in good agreement with existing literature values. The concentration dependence of D in the bovine mercaptalbumin system is in satisfactory agreement with the value obtained from a series of differential experiments in which the mean concentration of the boundary varied from 0.04 to 1.1g./dl. protein. Certain modifications of the diffusion interferometer and of the technique of boundary formation are described.
In free diffusion work on two-component systems, in which the concentration in the boundary region is recorded refractometrically, it has long been evident that the diffusion process rarely gives rise to a boundary region which may be characterized by a Gaussian probability function. This lack of ideality may be attributed to two causes; (1) a non-linear relation between refractive index and concentration, and (2) a dependence of the diffusion coefficient on concentration. The functional dependence of refractive index on concentration can be determined by independent experiment. In many systems a linear relation has been shown to hold up to relatively large solute concentrations. For example, the protein systems thus far studied show a linear dependence of refractive index on concentration up to protein concentrations in excess of 10 g./dl.3s4 On the other hand, there are often substantial deviations from linearity in electrolyte5 and dipolar ion6 systems. In systems of this type, if the refractive indexconcentration function is known, then the primary refractive index data obtained for the diffusion boundary can always be reduced to concentration data. It is customary in the investigation of concentration dependence of the diffusion coefficient to perform a series of experiments a t different mean solute concentrations. If the concentration difference in each experiment is kept small, differential diffusion coefficients corresponding to each mean concentration may then be calculated to establish the degree of concentration dependence. This procedure is time-consuming and in the case of protein work may require more material than is available. Also, if the Rayleigh Diffusiometer is used, this procedure is less than satisfactory since it is evident that the necessary information for calculating the concentration dependence is present in the data of any individual experiment. (1) Work carried out under National Institutes of Health Grant H-2127. Portions of this paper are derived from the dissertation presented by T . E . T . to the Faculty of Arts and Sciences, Harvard University in partial fulfillment of the degree of Doctor of Philosophy, 1955. (2) National Institutes of Health, Post-doctoral Fellow, 1955-1957. Present address: Department of Physiological Chemistry, The Johns Hopkins School of Medicine, Baltimore, Maryland, (3) G. E. Perlmann and L. G. Longsworth, J . A m . Ckriiz. Soc., 70,
2719 (1948). (4) R. Barer and S . Tkaczyk, S u t u r e , 173, 821 (1954). (5) L. J. Gosting, J . A m . Chem. Soc., 72, 4418 (1930). ( 6 ) M. S. Lyons and J. V. Thomas, ibid., 72, 4,506 (1950).
Apart from the theoretical interest attached to the concentration dependence of the diffusion coefficient, it is of importance in the study of proteins and other macromolecules to have a diffusion coefficient a t zero concentration which may be used with other data to give information about the size and shape of the kinetic unit of the solute. Creeth' has described an elegant method for calculating the concentration dependence of the diffusion coefficient in two component systems using the high precision data obtained with the Rayleigh Diffusiometer. Creeth's method is based on a series solution to the differential equation of diffusion obtained when a cubic relation between diffusion coefficient and concentration is assumed. It is the purpose of this paper to describe a different procedure for calculating the concentration dependence from Rayleigh diffusion data. This method, unlike that of Creeth, requires no a priori assumption regarding the nature of the dependence of the diffusion coefficient on concentration. Application of this method is made in studying the concentration dependence of the glycine-water system and of the system of bovine mercaptalbumin a t pH 4.74, in sodium acetateacetic acid buffer, I'/2 = 0.05. Theory.-Fick's Second Laws for the case of onedimensional free diffusion in a two-component system, may be written
where C is concentration, t is time, x is the vertical position coordinate in the diffusion cell and Dc is the diffusion coefficient a t concentration C. It is convenient to arrange experimental conditions such that a t t = 0, for r>O,C=
c*
x
