I. H. BILLICK, M. SCHULZ, AND G. H. WEISS
2496
Quasi-Equilibrium Sedimentation Experiments with Rotor Deceleration
by Irwin H. Billick, Michael Schulz, National Bureau of Standards, Washington, D . C.
and George H. Weiss National Institutes of Health, Bethesda, Maryland
(Received December 2 , 1966)
This paper presents an analysis of the rectangular approximation to the Lamm equation suitable for analyzing experiments with variable rotor speeds. The theory is applied to the case specified by deceleration described by w 2 = wo2 exp( - A T ) . It is shown that s / D for a two-component system can be determined experimentally by a measurement of concentration difference across the cell at a specified time after the beginning of the experiment. The resulting experiment offers the possibility of considerable savings in time over conventional sedimentation equilibrium experiments. Another experiment involving a quasi-equilibrium state is slower than the rate of approach to equilibrium for conventional sedimentation equilibrium experiments under comparable conditions.
The phenomenon of rotor deceleration was first recell base and meniscus. Although this difference does marked on in connection with the magnetically susnot itself tend to a constant under conditions of rotor deceleration, it will be shown that when the difference pended ultracentrifuge.'r2 Although such decelerais multiplied by a suitable function of time, the resulting tion is a nuisance when the ultracentrifuge is used for quantity does indeed tend to an equilibrium value. ordinary sedimentation equilibrium runs, it has been The theory developed in this paper has been checked suggested that rotor deceleration might be used to against very accurate machine calculations of soluspeed up equilibrium sedimentation experiments.2 tions to the Lamm equation4J under the condition of In order to be able to use rotor deceleration for experimental purposes, it is necessary to have an exact rotor deceleration, and has been found to be in good or approximate solution to the Lamm equation for agreement. Thus far, theoretical analyses pertaining to nonconvariable rotor frequency. Kone of the published solutions t o the Lamm equation, or the rectangular stant rotor speeds have been limited to treatments of approximation thereto, is suitable for the analysis of runs at constant speeds followed by instantaneous equilibrium experiments under the condition of rotor changes to a different, constant ~ p e e d . ~ Gehatia8 J deceleration. It is the purpose of this paper to present a formal solution t o the rectangular approxima(1) J. W.Beams, R. D. Boyle, and P. E. Hexner, Rev. Sci. Instr., 32, 645 (1961). tion to the Lamm equation suitable for the analysis of (2) J. W. Beams, D. 11. Spitzer, and J. P. Wade, ibid., 33, 151 experiments analogous to equilibrium runs. (1962). The theory to be presented pertains only to the two(3) J. E. van Holde and R. L. Baldwin, J . Phys. Chem., 62, 734 component system in which the sedimentation coef(1958). ficient is independent of solute concentration. We (4) hl. Dishon, G. H. Weiss, and D. A. Yphantis, Biopolymers, 4, 449 (1966). shall study the approach to equilibrium by means of (5) I. H. Billick, M. Dishon, G. H. Weiss, and D. A. Yphantis, to be an analog of the van Holde-Baldwin ~ a r a m e t e r , ~ published. and show that a theoretical reduction in running time (6) P.E.Hexner, L. E. Radford, and J. W. Beams, Proc. Natl. Acad. Sci. U.S., 47, 1848 (1961); cf. also P. E. Hexner, Thesis, University is possible by means of rotor deceleration. The exof Virginia, 1962. perimentally observed quantity that will be discussed (7) 9. Klenin, H. Fujita, and R. A. Albright, J . Phys. Chem., 7 0 , in this paper is the difference in concentration between 946 (1966). The Journal of Phyeical Chemistry
SEDIMENTATION EXPERIMENTS WITH ROTORDECELERATION
has presented the Faxen approximation for the case of changing rotor speeds, We will consider a formal solution which is, in principle, valid for all time, and for any pattern of nonconstant speed. Details will be given in the present article for an exponentially decreasinn- time-dependent rotor speed
2497
Equation 5 can be reduced to an infinite set of ordinary differential equations by assuming a series solution of the form
+ 5Bn(t)Cn(y,p)
e(y,t) = BO({) e x p ( y / ~ )
n=l
(9)
where the B,({) are functions to be determined and the C,(y,p) are the functions
w 2 ( r ) = wo2exp(--Xr)
Let us consider the rectangular approximation to the Lamm equation, which can be written for the present case as (sin a ny
+ 2anp cos
P
ny)
(10)
These functions satisfy the boundary condition of eq 8 as well as the normalization condition where f? is the normalized concentration, f?(y,r) = c(y,r)/co, co being the initial uniform concentration, and the remaining symbols are defined by
llcn(Y,p)cm(Y,fi) exp(-y/P)dy
=
6nm
(11)
where ,a,
= 1 for m = n and = 0 for m # n. When @ = a, a constant the expansion of eq 9 reduces to
where Tb is the radius of the base, rmis the radius of the meniscus, D is the diffusion constant, and s is the sedimentation coefficient. In order to simplify eq 2, we divide through by e-” and define a new time variable l by
the solution obtained through a separation of variables. Notice, however, that when p is a function of {, the variables are not separated in eq 9 since the C n depend both on y and on p. The function Bo({) in eq 9 can be calculated by noting that l l c n ( y , a ) dy = 0 (n
3
1)
(12)
and that conservation of mass requires that
(4) thereby transforming eq 2 to
where r is to be expressed in terms of f in p((). The function @ ( I ) is just the instantaneous value of a. Equation 5 is to be solved as an initial value problem ( i e . , f?(y,O) = 1) together with the boundary conditions
p($f’
bY
=
e
(y = 0, 1)
(8)
m
dCGmn(f)Bn ( m f: 0)
(16)
n=l
where (as derived in eq 17) (8) M. Gehatia, AFML-TR-64-377, Wright-Patterson Air Force Base, Ohio, 1965.
Volume 71, Number 8 July 1967
I. H. BILLICK, M. SCHULZ, AND G. H. WEISS
2498
Fn(i-) = Jlycn(l/>~)dY=
8n
