734
K. E. VANHOLDEAND R. L. BALDWIN
Vol. 62
RAPID ATTAINMENT OF SEDIMENTATION EQUILIBRIUM’ BY E(. E. VANHOLDEAND R. L. BALDWIN Contribution f r o m the Departments of Biochemistry, Chemistry and Dairy and Food Industries, the University of Wisconsin, and the Department of Chemistry, University of Wisconsin, Milwaukee Received February 17, 1068
The theory of Mason and Weaver has been used to predict the approach of a system to sedimentation equilibrium when the concentration is initially uniform. There is a simple equation which relates the time required to reach equilibrium, within any desired approximation, to the diffusion coefficient, to the depth of solution and to a parameter a. For small values of a this relation reduces to Weaver’s rule. I n addition to predicting the length of the transient period, the equation also can be used to measure, ap&roximately, the diffusion coefficient. Since the time required is proportional to the square of the depth of solution, equi 1 rium is reached quickly when a short column of solution is used. With a 3 mm. column equilibrium 1s reached in 14 hours with ribonuclease and in 3.5 hours with sucrose, while with a 1 mm. column equilibrium is reached in half an hour with sucrose. Equations are derived for computing results from 3 and 1 mm. columns. The effects of concentration dependence are considered for two-component systems and, for multicomponent systems, the effects of heterogenelty in M , B and diKerentia1 refractive increment R are considered. With 3 mm. columns one can measure two average molecular weights (M)l and ( M ) I / which ~ reduce to M , and M , if all solutes have the same 0 and R. The ex erimental results with sucrose show good agreement with expectation when 3 mm. columns are used but there appears toe! a small systematic error with 1 mm. columns. Interpretation of the results with ribo_nucleasedepends somewhat on the value-chosen for v. If ii is taken to be 0.695 ml./g. (Harrington and Schellman), then (M), = 1.365 f 0.007 (s.d.) X lo4 and ( M ) % = 1.382 f 0.029 (s.d.) X lo4, in good agreement with the known molecular weight of 1.3683 X lo4.
I n the determination of molecular weights, particularly of macromolecular substances, the sedimentation equilibrium method possesses distinct advantages. The molecular weight range which can be studied is very broad, only small quantities of material are required, and the theoretical foundation of the method has been thoroughly developed. A prime disadvantage, however, is the time which has been considered necessary for the attainment of equilibrium. A technique which requires one to two weeks for each experiment is not likely to be used for routine analysis, even though methods have been developed for carrying out several experiments simultaneously.2~3These considerations have led us to re-examine the factors upon which the time t o reach equilibrium depends, with the aim of shortening this time and of using the data from the transient period to establish when equilibrium has been attained. Since the simplest method of shortening the transient period is to reduce the depth of solution, we have performed experiments in which 3 and 1 mm. columns of solution are used. Solutions of sucrose in water were used t o test the agreement between theory and experiment because one can use known data (for the molecular weight, partial specific volume and dependence on concentration of the activity coefficient, density and refractive increment) to predict the form of the schlieren curves directly, without extrapolation to zero concentration. Solutions of ribonuclease were studied in order to gain familiarity with the problems of measuring the mplecular weight of a macromolecular solute. The molecular weight of ribonuclease is known from chemical studies4 and ( 1 ) This research waa supported in part by grants from the National Institutes of Health (RG 4912), from the Continental Can Company and from the Rockefeller Foundation. Published with the approval of the Director of the Wisconsin Agriculkral Experiment Station. (2) M. Wales, P . G. Sulzer and L. C. William, Nat. Bur. Standards (U.S.) Report No. 2057, 1952. (3) Rotors and cells for performing simultaneous experiments are available froni Spinco Division, Beckman Instruments, Palo Alto, California. (4) C. H . W. Hirs, W. H. Stein and S. Moore, J . Biol. Chem., 221, 151 (1956).
preparations of good purity are commercially available. I n order to use the results with sucrose for testing agreement between theory and experiment, it was necessary to derive equations relating the experimentally measured quantities to properties of the sucrose solutions. This was done for two methods of calculation applicable t o experiments with 3 mm. columns of solution, and for one method applicable when 1 mm. columns are used. I n order to interpret the results when these methods are used with a macromolecular solute, we also determined the kind of average molecular weight which is obtained when account is taken of heterogeneity in partial specific volume, in differential refractive increment and in molecular weight. Prediction of the Time Required to Reach Equilibrium.-The first estimate of the time required for the establishment of sedimentation equilibrium was made by Weaver.6 This was based on the analysis, by Mason and Weaver,6 of the sedimentation of a single solute in a rectangular cell in the gravitational field. The case of experimental interest is that of sedimentation in a sector-shaped cell in a centrifugal field, but it has been shown recently by Yphantis and Waugh7 that the equations of Mason and Weaver approximate quite closely the actual concentration distribution in an ultracentrifuge cell, at least under the experimental conditions generally used. Also Vinograd, et aZ.,* and Fujitaghave forthcoming solutions to the differential equation which are in the form of a series expansion about the equation for a rectangular cell and constant field. Hence we may use the equations of Mason and Weaver rather than Archibald’slo more complex relations for sedimentation in the ultracentrifuge. By solving the differential equation for the sedi(5) W. Weaves, Phvs. Reu., 27, 499 (1926). (6) M . Mason and W. Weaves, ibbd., 23, 412 (1924). (7) D . A. Tphrtnt.is and D . F. Waugh, THIB JOURNAL, 60,623 (1956). (8) Referred to B y R . A. Pasternak, G. WI. Nararian and J . R. Vinograd, Nature (London), 179, 92 (1957). (9) H. Fujita, private communication. (10) W. J. Archibald, Ann. N . Y . Acad. Sci., 43, 211 (1942).
RAPIDATTAINMENT OF SEDIMENTATION EQUILIBRIUM
June, 1958
735
mentation of an ideal solute in the gravitational field, Mason and Weaver arrived at equation (1) for the concentration as a function of position and time
T,(t)[sin mry
+ 2 ~ m cos a mry]
(1)
where Tm(t) =
1 6 a 2 x e - ( a m 2 s 2 + 1 / 4 a ) ~ / P ~ [l
[l
( -l)me-l/Za],
+4 ~ ~ ? n ~ a ~ ] ~
('4 When this equation is applied to sedimentation in the ultracentrifuge, the parameters may be defined as y =
(T
- a)/@- a)
(3)
(where r - a is the distance from the meniscus to the point r , and b - a is the length of the solution column). a =
RT
A[( 1
- op)W2l(b
- a)
'
(b Al(1
ture, M the molecular weight, 17 the partial specific volume, p the solution density, w the angular velocity of the rotor, P = (b+a)/2, and D is the diffusion coefficient of the solute. I n the derivation of equation 1, it is assumed that the concentration is initially uniform throughout the cell. Vinograd, et al.,s have considered the saving in time which may be achieved by starting with a step distribution of concentration. It will be noted that in equation 1 the first term represents the time-independent, equilibrium concentration distribution, while the second term is a transient one, approaching zero a t infinite time. Weaver pointed out that in the special case where a < < 1 the transient term should sensibly vanish for times greater than t = 2P. This represents twice the time required for the solute to sediment from the meniscus to the bottom of the cell, and it is this criterion which has been used in the past to estimate the time required for sedimentation equilibrium experiments. However, the condition that a<
