402
ROBERT L. BALDWIN [CONTRIBUTION FROM
THE
1701. 76
DEPARTMENT OF BIOCHEMISTRY, UNIVERSlTY O F
OXFORD]
Boundary Spreading in Sedimentation Velocity Experiments. 11. The Correction of Sedimentation Coefficient Distributions for the Dependence of Sedimentation Coefficient on Concentration1s2 BY ROBERT L. BALD WIN^ RECEIVED AUGUST5, 1953
A method is presented for correcting a distribution of sedimentation coefficient for the dependence of sedimentation coefi. cient on concentration. This is, in general, a n important correction and cannot be neglected. The method is applicable to uncorrected distributions obtained by the method of Signer and Gross, when diffusion is negligible, or by the method of Baldwin and Williams, when diffusion is not negligible. A considerable saving in time is achieved by the use of this correction as compared t o extrapolating the uncorrected distributions t o infinite dilution. The method requires knowledge of how the sedimentation coefficients depend on concentration.
Introduction Several problems in chemistry and biochemistry await the development of a rapid and accurate method for determining the distribution of mass in a system. For example, the size and heterogeneity of the fragments produced by enzyme action on proteins and polysaccharides are of great interest to the biochemist, and, in polymer chemistry, the question of how the distribution of molecular weight affects the measured properties of a polymer must continually be faced. Two instruments-both of them originally developed by Svedberg and his co-workers4-have been used to characterize mass heterogeneity; the equilibrium centrifuge, which yields a distribution of molecular weight, and the velocity ultracentrifuge, which gives a distribution of sedimentation coefficient. A distribution of molecular weight is more readily interpreted than one of sedimentation coefficient but the latter may be obtained without any u priori assumptions as to the nature of the distribution5-8 and this is extremely difficult in the case of data from the equilibrium centrif~ge.~-” Also the velocity ultracentrifuge experiment takes but a few hours in contrast to the situation with the equilibrium centrifuge, where two weeks may be required for a system to reach equilibrium. The boundary gradient curves obtained with the velocity ultracentrifuge may be transformed directly into distributions of sedimentation coefficient, provided that the spread of the boundary has not been affected significantly either by diffusion or by the dependence of sedimentation coefficient on concentration. Signer and Gross6 gave the following expression for transforming a boundary gradient (1) Presented a t t h e Symposium on Macromolecules, 13th International Congress of Pure and Applied Chemistry, Uppsala, Sweden, August, 1953. (2) Based on a thesis submitted t o the University of Oxford in partial fulfillment of the requirements for the degree of Doctor of Philosophy. (3) Rhodes Scholar. Department of Chemistry, University of %‘isconsin. (4) See T. Svedberg and K. 0. Pederseu, “The Ultracentrifuge,” Oxford University Press, Oxford, 1940. (5) R. Signer and H. Gross, Hela. Chim. Acta, 17, 726 (1934). (6) R . L. Baldwin and J. W. Williams, THISJOURNAL, 72, 4325 (19 50). (7) J. W. Williams, R . L. Baldwin, W. M . Saundersand P. G. Squire, i b i d . , 74, 1542 (1962). (8) L. J. Gosting, ibid., 74, 1548 11952). (9) M. Welea, 2, P h y s . Colloid Chcm., 62, 236 (lQ48).
(10)M. Wales, tbid., 55, 146 (1951). 0 1) R.f . QoldbrFU, bQbd., 09, ?B4 (1963).
curve into a distribution of sedimentation coefficient under these conditions (la)
In this expression, w is the angular speed of revolution, t is the time from the start of sedimentation, co is the total concentration of the original solution, xo is the distance from the center of rotation to the meniscus, x is the distance from the center of rotation to a specified point in the boundary and dcldx is the concentration gradient a t this point, The weight fraction of material in the original solution ds is given by of sedimentation coefficient s to s g (s)ds. l2 When the contribution of diffusion to the boundary spread is not negligible, g(s) still may be obtained by extrapolation of an “apparent distribution” g*(s) against l / t to infinite time.6--8p14 Similarly, when the dependence of sedimentation co-
+
(12) I n general, t h e temperature and speed of rotation vary during sedimentation velocity measurements. This may be allowed for explicitly by rewriting ( l a ) as
q is t h e solvent viscosity.
This corresponds to rewriting the expres-
sion for the sedimentation coefficient, s = In E/ xo w V , as szo = In
F’
:/
E d t and assumes that there is no significant error in setting stqt =
swpo.
If the sedimentation coefficient depends on concentration, the
expression s = In
”- /
wzt
xo
gives a n average s which will vary with 1
because the concentration decreases with time. Alberty’J has discussed methods of calculating s which take this into account. At the highest concentration used in the experiments reported here, s changed by 2.5% from t h e beginning t o the end of t h e centrifuge run. I n general, conditions where this effect must be allowed for are not suitable for boundary-spreading measurements, The integral in (lb) is obtained by numerical integration and is first measured from an arbitrary time origin t o ; t h e complete integral is found by plotting In X , , / X O (where xp denotes t h e position of the maximum gradient) against
.6”^‘
(xp = xo) t o find
6’
wz
uz Eodt and extrapolating t o the meniscus qt
P d 1 . For simplicity, the speed and tempera?t
ture will be treated as constant throughout this paper, with the understanding t h a t any variation may be allowed for in the manner indicated here. When t h e refractive increment, An (the difference in refractive index between t h e original solution and dialysate in equilibrium with it) is directly proportional t o co and the proportionality constant is the same for all species present, (dc/dr)/co may be replaced by (dn/dz)/An in equations l a and l b . Even when t h e proportionality conntwtr? are not the same, doing this gives a well-defined distribution.’ (13) R . A. Uberty, THm JOURNAL, T 6 , 191 (1964). (14) R,t.Baldrrfn, Biochrm. J., to be publinkrd.
Jan. 20, 1954
SEDLIENTATION COEFFICIENT DISTRIBUTIONS : CONCENTRATION DEPENDENCE 403
boundary position, x = x&w’~, at a fixed value of u2t, dx/ds is found to be u2xt and g(s) = g(x) (dxlds) = (dc/dx)e2swztw2xt/co. This is identical with (la) since (x/xo)* = e2sW’t. When the sedimentation coefficient depends upon concentration, this interpretation of the boundary gradient curves requires considerable modification. The problem will be split into two parts by assuming for the moment that dc/dx still measures the amount of the species vanishing a t x . The effect of the dependence of s upon c may then be’described as a sharpening of the boundary caused by a change in concentration from zero at the far left of the boundary to coe-2Sw2t a t the far right. Theory (This will be referred to as the boundary sharpening To begin with, the situation in the ultracentrifuge dect.) The equation giving the position a t which will be described for the case when diffusion is a species vanishes in the boundary is still x = xoeswzL, negligible and s does not depend upon c. Sedi- but now s is the sedimentation coefficient at cr, cX mentation is observed b y studying the boundary being the total concentration of all species at the which leaves the meniscus, moving outward from plane where this species vanishes. cx changes, of the center of rotation. (For convenience, this course, throughout the boundary and is different direction will be called to the right.) The velocity at every plane where a new species vanishes. The concentration at x is known (since diffusion of sedimentation is given by the field strength, w2x, and the sedimentation coefficient, s: dx/dt = is negligible, c must be zero a t xoand so cZ is simply s d x . This expression may be integrated to show (dc/dx)dx) ; if the dependence of sedimentation that the solute molecules originally a t the meniscus, coeficient of a species upon cx is known, its sediXO, will have sedimented after a time t to a position given by x = xoesw*t. Consequently this gives the mentation coefficient a t infinite dilution can be position of the boundary formed b y species of calculated from s and c,. Then the correction for sedimentation coefficient s; just as all molecules the dependence of s on c is simply once more a matof this (and other) species were to the right of xo ter of changing variables in a distribution function ; a t the start of the experiment, so all of this species SO,the sedimentation coefficient a t infinite dilution, must be to the right of x a t timet. is substituted for s, the sedimentation coefficient a t The value of dc/& a t this plane is a direct a particular concentration in the boundary and ds measure of the amount of the species which vanishes at x (or first appears, depending upon your g(s0) = g(s) G. This correction for the boundpoint of view) because the concentration of each ary sharpening-effect is essentially the same as the species is constant to the right of the plane where it one proposed b y Jullander,16 who considered the vanishes. There is a change with time, but not problem of transforming g(s) to g ( M ) , the distribuwith distance, in the concentration of each species. tion of the molecular weight. (Because of the dilution caused by sedimentation in However, dc/dx no longer measures the amount a sector-shaped cell and a changing field, of the species vanishing a t x when the sedimentaCt = Coe-2sw’t 18 tion coefficients of the various species are dependThus the boundary gradient curve is itself a ent upon the concentration. I n fact, every species distribution curve: i t gives the concentration of a present a t a plane in the boundary changes in conparticular species versus its position in the bound- centration a t that plane if its sedimentation coary. (Although all species of lesser sedimentation efficient varies with the total concentration. This coefficient are also present at this position, they do effect, which was neglected by Jullander,15 was not contribute to dcldx.) The curve of dc/dx treated quantitatively by Johnston and OgstonZ0 vs. x is not a conventional distribution function for systems of two components; they showed that it is if the sedimentation coefficient of the slower moving because the area under the curve is c o e - 2 S W 2 t ; customary to normalize a distribution by requiring component is greater in the absence than in the the area under the curve to be unity. I n this case, presence of the leading component, there will be a if the distribution function, g(x), is defined as corresponding change in concentration of the trail(dc/dx)e2sw21/co then Jg(x)dx = 1. ing component. This may be thought of as the I n order to obtain g(s) from g(x) we need only trailing component piling up behind the leading make use of the general relation for changing boundary because i t moves more rapidly behind variables of a distribution functionlg: g(s) ds = than in front of it, with the result that its cong(x)dx. By differentiating the expression for centration is greater behind the leading boundary (16) I. Jullander. Arkiv. Kemi, Mineral., Gcol., PlA, No. S’(1945). than in front of it. (16) N . Gralen and G . Lagermalm, J . Phys. Chem., 66, 514 (1952). This effect also appears in a single boundary (17) A. Fuhlbrigge, A. Haltner, Jr., W. M. Saunders. K. van Holde, J. A. Williams, J. W. Williams, Progress Report to the Office when the substance forming the boundary is heteroof the Surgeon General, Dept. of the A r m y , November aO,1951. These geneous, The situation seems insuperably complex experiments will be published in the usual way at a later time. in a multicomponent system until one realizes (18) T. Svedberg and H. Rinde, THISJOURNAL, 46, 2877 (1924).
efficient on concentration significantly affects the boundary spread, the distribution of sedimentation coefficient may be obtained by extrapolating curves of g(s), as defined b y equation 1, to infinite dilution.’5-l7 However, this is a very time-consuming process and there is no theoretical guide for the form of the extrapolation t o infinite dilution. I n this paper, a method is presented for correcting the curve of g(s) obtained a t a single concentration for the dependence of s on c. Only the case of negligible diffusion is considered here but i t is shown how the method may be applied, when diffusion is not negligible, to the extrapolated curve of g(s) .
6:
(19) T. C. Fry, “Probability and its EngineVity Unu,” D. Vpa NODtrS&ldGO,, New York, N. Y., 19P8,p. 168,
(20) J,
(1B46)
P,johmton and A, Q, Qgaton, Tmnr, Farrrdag fie
