D. J. WINZOR, J. P. LOKE,AND L. W. NICHOL
4492
Quantitative Studies of Rapid Polymerization .Equilibria by Gel Filtration. The a-Average Elution Volume of a Reaction Boundary
by D. J. Winzor, Joan P. Loke, C.S.I.R.O. Wheat Research Unit, North Ryde, New South Wales, Australia
and L. W. Nichol Russell Grimwade School of Biochemistry, Unirersity of Melbourne, Parkville, Victoria, Australia (Received J u n e 81, 1967)
Frontal gel filtration data on polymerizing systems comprising monomer in rapidly established equilibrium with a single higher polymer (nA e A,) are in principle amenable to quantitative interpretation in terms of stoichiometry and equilibrium constant, since the weight-average elution volume may be determined readily. However, use of gel filtration data for this purpose also requires values of the monomer and polymer elution volumes, quantities which cannot be measured directly from elution profiles reflecting the coexistence of the two species in appreciable amounts. The present study describes the derivation and application of a procedure which permits estimation of these elution volumes from such data. An expression is derived which relates the experimentally determinable zaverage elution volume of the descending reaction boundary to the z-average elution volume of the equilibrium mixture. Combination of this relationship with that relating the weight- and z-average elution volumes of the system, together with the logarithmic form of the equation defining the association equilibrium constant, forms the basis of a reiterative procedure for determining the value of n and the association constant. Its application to gel filtration data on a-chymotrypsin in acetate-chloride buffer, pH 3.86, 10.20, yields values of 2 and 7 l./g for the respective quantities.
Frontal gel filtration of a system in which monomer and a single higher polymer are in rapidly established equilibrium (nA e A,) yields a nonenantiographic elution profile consisting of reaction boundaries on both advancing and trailing sides.' A weight-average elution volume, V,, may be determined directly from either side and related to the plateau (applied) concentration. The value of n and the association constant, K , , are therefore derivable from these data provided the individual elution volumes of monomer and polymer are also available. In experiments where both coexist in appreciable amounts, there is no defined position in the observed reaction boundaries corresponding to either of these quantities. However, if the zaverage elution volume of the system, V,, can be determined, this difficulty is circunivented by the coupling The Journal of Physical Chemistry
of z- and weight-average data.2 This communication presents the derivation of the relationship between V , and the corresponding experimental quantity, Vz*, obtained from the second moment of the reaction boundary. In addition, the use of this relationship for determining and employing V , in the estimation of n and the association equilibrium constant is discussed and illustrated with a revised treatment of earlier gel filtration data on a-chymotryp~in.~
(1) G. A. Gilbert, Proc. Roy. Soc. (London), A250, 377 (1959). (2) A. J. Sophianopolous and K. E. van Holde, J . Biol. Chem., 239, 2516 (1964). (3) D. J. Winsor and H. A. Scheraga, J . Phys. Chem., 68, 338 (1964).
STOICHIOMETRY AND ASSOCIATION CONSTANTS FROM GELFILTRATION
4493
+
V12a10 Vn2nK,(al0)"-
Theory z-Average Elution Volume of a Reaction Boundary. For the purpose of illustration we shall consider a spread reaction boundary on the trailing side, from which it is possible to determine weight- and z-average elution volumes of the reaction boundary using eq l a and lb, respectively. In these expressions concentrations are where
Y Jo
Jo
expressed on a weight basis, and the upper limit of integration in each case, co, is the applied or plateau concentration of solute. These weight- and z-average elut,ion volumes of the reaction boundary must be related to those referring to the composition of the applied solution; the latter are defined as
+ cnoVn) -- (al"V1 + nanOVn) ( 2 4 + Go) (a? + nu,") ( C ~ O+ V ~c,0Vn2) ~ V ~nunoVn2) ~ - (U~O+ = + c,OV,) (aloV1+ na,OV,) (2b)
V,
(CloVi
=(CI0
Ti,
-
=
+
~ ' o ( n 2 K , a l n - 1 / ( l n2Knaln-1)) dal (6b)
It is c1ea.r from eq 6 that V , and V,* become identical when K , is zero, which corresponds to the case of a pure solute. In the treatment of reaction boundaries, involving concentration gradients of both species, the estimation of V , from the experimentally obtained V,* requires evaluation of the integral, Y . Analytical solutions to the integral for values of n from 2 to 6 inclusive are shown in Table I ; solutions are also available for higher values of n. The integral may thus be evaluated for any specified set of values of n and K,. To facilitate a generalized graphical representation of Y , it is convenient to introduce the parameters ai, the weight ratio of monomer t o applied concentration, and Kn', the association constant on a weight scale, which are related to the previously defined quantities by eq 7 and 8; M1 is the molecular weight of monomer.
(cW1
Subscripts 1 and n refer to monomer and polymer, respectively, and the individual elution volumes, V I and V,, are assumed to be independent of composition; the a notation refers to molarities as distinct from weight concentrations. Gilbert' has established the identity of V,* and V,. The corresponding z-average quantities are not generally identical, but a relationship between V,* and V , may be derived for the selected model (nA e A,). From eq 6, 7, and 9 of ref 1, eq l b may be rewritten as
l"(Vl
+ n2V,Knaln-l)da1 (3)
which becomes
(7)
Substitution for K , in eq 6b, and a change of variable from al to ai yield
aio may assume any value between 0 and 1, while the corresponding limiting values of Y are a?cO/M1 and 0. Equation 9 may then be written
Y = aiocoF/M1 (10) where the parameter F varies between the limits 1 and 0 and, from eq 9, is a function of n and a?. The dependence of F upon a1° for values of n from 2 to 8 inclusive is shown in Table 11. The data were obtained by the method of trapezoidal integration employing a CDC 3200 c ~ m p u t e r ;particular ~ solutions tested for (4) We are grateful to Dr. G. E. Hibberd, Bread Research Institute of Australia, for programming these integrals.
Volume 71, Number IS December 1967
D. J. WINZOR, J. P. LOKE,AND L. W. NICHOL
4494
Table I1 : The Parameter F as a Function of n and a10
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
n = 3
n = 2
1.00 0.91
1.00 0.81
0.34
0.74 0.68 0.63 0.59 0.54 0.51 0.47 0.43 0.40 0.36 0.33 0.29 0.25 0.22 0.18 0.14 0.09 0.05 0.00
0.78 0.73 0.68 0.63 0.58 0.54 0.5C 0.45 0.41 0.37 0.32 0.28 0.23 0.19 0.14 0.10 0.05 0.00
n = 4
1.00 0.72 0.65 0.60 0.55 0.51 0.48 0.45 0.41 0.38 0.35 0.32 0.29 0.26 0.23 0.20 0.17 0.13 0.09 0.05 0.00
n = 5
1.00 0.65 0.58 0.53 0.49 0.46 0.43 0.40 0.37 0.35 0.32 0.29 0.27 0.24 0.21 0.19 0.16 0.12 0.09 0.05 0.00
cyl0
n=6
n = 7
n = 8
1.00 0.59 0.53 0.48 0.45 0.42 0.39 0.36 0.34 0.32 0.29 0.27 0.25 0.22 0.20 0.17 0.15 0.12 0.08 0.05 0.00
1.00 0.54 0.48 0.44 0.41 0.38 0.36 0.33 0.31 0.29 0.27 0.25 0.23 0.21 0.19 0.17 0.14 0.11 0.08 0.05
1.00 0.50 0.44 0.41 0.38 0.35 0.33 0.31 0.29 0.27 0.25 0.23 0.22 0.20 0.18 0.16 0.13 0.11 0.08 0.04 0.00
0.00
cases n = 2-6 agreed with those obtained using the analytical expressions (Table I). The relationship between V , and V,* in terms of a? is obtained by combining eq 6,7, and 10
Vz
=
Vz*
+ [(VI - Vn)2aloF/Vwl
(11)
Thus V , may be computed from V,* and V , provided V1 and V n are known, since a1° follows from eq 7, and F from Table 11. A comparison of V , and V,* is The Journal of Physical Chemistry
shown in Figure 1 for the cases n = 2, 3, 4, 6, and 8, using in each instance values of VI and V , derived from the molecular weight calibration plot for Sephadex G-100 reported by Andrews? for the purpose of illustration a monomer of molecular weight 25,000 was chosen. The ordinate (V, - V,*)/V, is a reduced parameter independent of column characteristics, while the range of values of a? from 0 to 1 encompasses all possible combinations of K,' and co. For each model the relative deviation of V,* from V , is zero at the two extremes of alo and increases to a maximum in the vicinity of cq0 = 0.4. Although this maximum deviation is barely of experimental significance for n = 2 and less than 6% for all n tested, it should be stressed that larger differences would be obtained with a model in which (VI - V,)/Vl was increased. However, smaller rather than larger values of the latter quantity are likely to be encountered in gel filtration unless factors other than molecular exclusion are governing migration. Evaluation of V1, V,, n, and K,' from V,* and V,. An expression relating V , and V wmay also be obtained by combining eq 2a and 2b (cf. ref 2) to give vz =
(VI
+ Vn) - (VlVn/Vw)
(12)
Equations 11 and 12 suggest a procedure of successive approximations to analyze data from a series of experiments with different values of 8, the applied concentration of solute. In view of the relatively minor differences between V , and V,* (Figure l), the latter may ( 5 ) P. Andrews, Biochem. J., 91, 222 (1965).
STOICHIOMETRY AND ASSOCIATION CONSTANTS FROM GELFILTRATION
0.06
0.04
& VZ
0.02
0 0
0.2
0.4
0.6
0.8
1.0
Figure 1. Dependence of the relative deviation of V,* from V zon a10for selected values of n, VI, and V,.
be used as an initial estimate of V , and plotted against l/V, to give approximate values of V1 and V , from the intercept and slope. In practice the range of values of at encountered is frequently narrower than that shown in Figure 1, and the plot of V,* us. 1/V, will therefore usually approach linearity even though V , # V,*. In these circumstances first estimates of V1 and V , may be obtained by least-squares calculations. By using eq 7 and Table I1 these estimates of V1 and V , may be employed in eq 11 to give a second approximation to V,. The process may be reiterated to obtain refined values of V,, VI, V,, and aIo for all probable values of n. The relevant value of n is found from eq 8 written in logarithmic form, viz.
+ n log alocO
ing those expected for a system with a monomer molecular weight of 25,000 on the column of Sephadex G-100 used by Andrews.6 V,, V,, and V,* were calculated from eq 2a, 2b, and 11, respectively. Figure 2 summarizes these theoretical data, expressed as a plot of V , os. l/Vw, the primary “experimental” data (V,*, l/V,) being indicated by circles, while the solid line (points not shown) is obtained when the correct value of V , is used. Direct substitution of V,* for V , would obviously lead to considerable error in the values of V1 and V , deduced from the slope and ordinate intercept. In this connection it should be noted that the range of at (0.9 ,lo 0.4) was chosen deliberately because the discrepancy between V , and V,* ever increases over this range (Figure 1); the difference between the slopes of the dashed and solid lines in Figure 2 should thus be maximaL6 The triangles, squares, and inverted triangles in Figure 2 represent the respective plots with the second, third, and fourth approximations to V,, obtained by application of eq 11 and 12 with n = 8. The fourth estimates of V , closely approximate to the actual value, and thus even in a case selected for extreme nonideality V1 and V , may be obtained by this method of successive approximations. Certainly it is not possible to use extent of nonlinearity of the plot as a criterion of satisfactory agreement between estimated and actual values of V,, as the four sets of data in Figure 2 do not differ greatly in this respect. However, the plot of eq 13 (Figure 3) indicates that there is little chance of stopping the reiterative procedure prematurely. The primary data and the second approximations to V , both yield physically unacceptable plots, while the third estimates are in close proximity to the theoretical
> >
d.1“
log (1 - alo)co= log K,’
4495
(13)
Only one set of values of ai’ as a function of co will yield a straight line of the required slope when log (1 alO)cO is plotted against log aloco; the value of K,‘ follows from the ordinate intercept.
Applications Demonstration of the Reiterative Procedure with a Theoretical System. The validity of obtaining VI, V,, n, and K,’ from V,* and V , by a series of reiterative steps using eq 11-13 is demonstrated by application of the method to a, selected set of data for a simulated monomer-octomer system. A value of 8 for n was chosen for this purpose to provide a more critical test of the procedure, since greater discrepancies between V , and V,* may be expected with such a system than with one involving a smaller value of n (Figure 1). K,’ was arbitrarily chosen as unity, and six initial solute concentrations (8) were selected so that the values of the weight fraction of monomer were equally spaced in the range 0.9 aIo 0.4. Vl and V swere taken as 140 and (38ml, respectively, values approximat-
>
>
Figure 2. Estimation of VI and V , from weight- and z-average data for a simulated monomer-octomer system: 0, primary data (V.’); A, I, and ‘I,second, third, and fourth approximations, respectively, to V,. The solid line represents the theoretical V. us. 1/V, plot.
(6) Alternatively values of a 1 0 between 0.1 and 0.4 could have been taken, whereupon the slope of the V,* va. l / V wplot would have been correspondingly smaller than that obtained with values of V..
Volume 7 1 , Number 13 December 1967
D. J. WINZOR, J. P. LOKE,AND L. W. NICHOL
4496
I
-
12.8 1 3 T
-
P
I
I
L
lag
"40"
Figure 3. Plot of eq 13 using (YPvalues obtained with estimates of VI and V, from Figure 2. The symbols have the same significance as in Figure 2, and the solid line is the theoretical plot for the simulated system (n = 8, K,' = I).
plot, indicated by the solid line (slope 8, intercept 0); values of V1and V , calculated from these estimates were 139 and 69 ml, respectively, differing by 1 ml in each instance from the corresponding theoretical values, viz., 140 and 68. Fourth approximations coincided with theoretical values. An initial choice of an incorrect value of n should be detected readily since it would appear from Figure 3 that good estimates of V1 and V , are prerequisites of a linear log-log plot, which in turn defines n. This point is exemplified in Figure 3, from which the first and second approximations to V , are obviously unacceptable, but the third estimates are barely distinguishable from the actual data. This difference between the second and third approximations is even more dramatic when it is recalled that for a fixed combination of V1 and V , the correction term in eq 11 is smallest for n = 8, since F decreases with increase of n (Table 11). Evaluation of V1 and V , for a Stable Monomi-Polymer System. The gel filtration pattern obtained with a monomer-polymer system in which there is no interconversion of species during the experiment may be considered as the summation of two pure solute patterns, for each of which the relationships V,* = V , = V , = V1 (or V,) hold (see discussion of eq 6). It follows that V,* obtained by performing the integration in eq l b across both boundaries (from c = 0 to c = 8 )is identical with V,. Thus a plot of V,* us. l/Vw with data from a series of experiments with different proportions of monomer and polymer theoretically yields the elution volumes of the observed boundaries. A stable system therefore provides a convenient test of the experimental limitations of the present method for determining the elution volumes of monomer and polymer by The Journal of Physical Chemistry
Figure 4. Estimation of monomer and polymer elution - volumes from frontal gel filtration experiments on polydisperse ovalbumin solutions: ( a ) the linear plot from which VI and V , were calculated; ( b ) the relationship of the calculated values to two of the trailing elution profiles (derivative plots).
combining weight- and z-average data; for this purpose the aluminium-dependent polymerization of ovalbumin' a t pH 5 proved a suitable model system. Frontal gel filtration experiments were performed on a 17 X 1.3 cm column of Sephadex G-100 equilibrated with acetate-chloride buffer,pH 5,I0.12, an automated biuret method being used for the continuous analysis of the column effluent, which was maintained a t a flow rate of 17 ml/hr. The concentration of ovalbumin in each experiment was 0.72% (0.16 mM), while that of aluminium, added as alum, varied from 0.05 to 0.80 mM. V , and V,* were calculated from trailing elution profiles using eq l a and lb, respectively. Figure 4a presents the relevant data for the estimation of V1 and V,, least-squares calculations leading to values differing only slightly from those inferred directly from derivative plots of the elution profiles (Figure 4b). Furthermore, the discrepancies are only 3-5%, and errors of this magnitude must be expected in data based on secondmonient calculations (eq lb) , which magnify experimental uncertainty. The agreement between the observed and calculated value is certainly sufficient to be considered a satisfactory test of the method's potential for providing reasonable estimates of monomer and polymer elution volumes in systems where direct measures are not available. Estimation of n and K,' f r o m Gel Filtration Data on a-Chymotrypsin. Finally we wish to illustrate the method with experimental data on a rapidly polymerizing system. Figure 5a presents a plot of V,* vs. l/Vw for a-chymotrypsin in I 0.20 acetate-chloride buffer, (7) J. A. Gordon and M. Ottesen, Biochim. Biophys. Acta, 75, 453 (1965).
STOICHIOMETRY AND ASSOCIATION CONSTANTS FROM GELFILTRATION
+1
I c
Y
8 -1 d
-2 -2
-1 Log q c o
0
Figure 5. Evaluation of n and K,‘ for a-chymotrypsin in acetate-chloride buffer, pH 3.86, Z 0.20: (a) the linear plot from which VI and V , were calculated; (b) the plot of eq 13 with values of al0calculated from V,, VI, and V,.
pH 3.86, the weight-average data having been reported previously in Figure 2 of ref 3; since the theoretical treatment outlined above does not consider effects of physical interactions in gel f i l t r a t i ~ n , ~experiments ,~ with concentrations greater than 3 mg/ml have not been included. Least-squares calculations lead to values of 26.9 and 23.6 ml for the first approximations to V1 and V,, respectively, and the plot of eq 13 (Figure 5b) indicates a value of 2 for n. As predicted from Figure 1 the correction factor (eq 11) required to obtain a second estimate of V , is very small, values of (Vl V,JSFa?/V, ranging from 0.06 to 0.10 ml. An adjustment of 0.1 ml to each value of V,* changes the estimated monoiner and polymer elution volumes to 27.6 and 23.0 mi, respectively, but the 2.5% differences between these arid the corresponding initial estimates are not considered to be significant experimentally in the light of Figure 4b. The line in Figure 5b is drawn through the point (log (1 - aP)co, log aPc”), the mean of the experimental data, with a slope of 2. (Leastsquares calculations yield a value of 2.1.) From the slope it is concluded that a-chymotrypsin is a nionomerdimer systeni under these conditions, with an association equilibrium constant (from the ordinate intercept) of 7 L/g, but it is stressed that a value of 5 l./g, obtained with the revised estimates of V,, is just as consistent with the experimental results. Weight-average molecular weight data on the same system were interpreted in terms of a monomer-dimer system with an association constant of 4 l./g.3
Discussion The most important theoretical contribution of the present study is undoubtedly the demonstration that second-moment ctilculations across a reaction boundary define a quantity which may be related to the x-average
4497
elution volume of a rapidly reacting monomer-polymer system. It should be noted that eq 6 and 11 are readily adapted to apply to all forms of frontal transport data, their application to moving boundary electroplioresis, for example, merely requiring the substitution of velocities for elution volumes.lOrll With the same substitution these relationships would also apply to sedimentation velocity, but it should be remembered that the second moment is required for the weight-average velocity in this case12 and thus an even higher moment would be required for the z-average velocity of the reaction boundary. As has been shown in the previous section the derivation of eq 11 has extended considerably the quantitative potential of gel filtration by providing a general procedure for determining monomer and polymer elution volumes, but we do not wish to imply that this is the sole or best possible use of this relationship. However, the present application has now made possible the quantitative study of the last of the simpler rapid equilibria by gel filtration, since methods for studying reactions of the type A B Cg113-15 and A B e C DI8have already been established. The use of gel filtration for determining n and K,’ is certainly not a new concept, as Ackers and Thompson17 have already employed such data to obtain the dissociation constant for human hemoglobin, a system for which experimental conditions exist for the direct estimation of V1and Vin. In addition, they also demonstrated a method of estimating the stoichiometry from the position of the minimum in the derivative plot of the trailing elution profile for systems with n > 2. However, inspection of this procedure also reveals that values of V1 and V , are prerequisites of its application, a point emphasized previously by Nichol and Bethune18 in their use of the analogous expression to define the sedimentation coefficient of the polymer species for selected values of n. The present procedure is thus applicable more generally in that it defines VI and V ,
+
+
+
(8) D. J. Winsor and L. W. Nichol, Biochim. Biophys. Acta, 104, 1 (1965). (9) L. W. Nichol, A. G. Ogston, and D. J. Winsor, Arch. Biochern. Bwphys., in press. (10) G. A. Gilbert, Nature, 210, 299 (1966). (11) L. W. Nichol, A. G. Ogston, and D. J. Winzor, J . Phys. Chem., 71, 726 (1967). (12) R. J. Goldberg, ibid., 57, 194 (1953). (13) L. W. Nichol and D. J. Winzor, ibid., 68,2455 (1964). (14) L. W. Nichol and D. J. Winzor, Biochim. Bhphys. Acta, 94, 591 (1965). (15) L. W. Nichol and A. G. Ogston, Proc. Roy. SOC. (London), B163, 343 (1965). (16) L. W. Nichol and A. G. Ogston, ibid., B167, 164 (1967). (17) G. K. Ackers and T. E. Thompson, Proc. Natl. Acad. Sei. U . S., 53, 342 (1965). (18) L. W. Nichol and J. L. Bethune, Nature, 198, 880 (1965).
Volume 71, Number If3 December 1967
4498
and thus removes the necessity of either finding conditions suitable for their direct estimation or assuming values of these parameters. A factor barring its application to all monomer-polymer systems is the assumed absence of intermediate polymers. I n this connection it is noted that the presence of such species in experimentally significant amounts is indicated by curvature of the V , us. 1/V, plot2 and also, in some instances, by obvious nonconformity of the trailing elution profile with that predicted' for monomer-single polymer system^.'^ A disadvantage of the present procedure is that K,' cannot be determined with very great precision owing partly to uncertainty introduced into the values of V1 and V , by experimental scatter of the weightand z-average elution volumes (Figure 4). This is evident from the above result for a-chymotrypsin, which confirms rather than contradicts the earlier estimate of K,' deduced from molecular weight datala despite an almost twofold difference between the two values. Inability to define K,' more precisely obviously renders the present method unsuitable for detecting small differences in the extent of association of two essentially identical monomer-polymer systems, e.g., a comparison of different mammalian hemoglobins or the various bovine chymotrypsins. However, from numerical computations it would appear that a similar limitation of gel filtration studies also extends to the comparison of systems in which VI and V , may be estimated directly.*O To circumvent this difficulty Gilbert2' has suggested the adoption of a layering technique of gel filtration analogous to differential velocity sedimentation, wherein one of the solutes under investigation is used to elute a column equilibrated with an identical concentration of the other solute; differences in extent
The JOUTWL~ 0.f Physical Chemistry
D. J. WINZOR,J. P. LOKE,AND L. W. NICHOL
of association are detected by a temporary increase or decrease of concentration a t the junction of the two solutions. It is not clear from ref 21 that the concentration profile will generally exhibit both a dip and a hump in this region, and that the quantity @,At',, or 8AV, in the present terminology, refers to the net effect.22 As a method of estimating the difference between Vw for two solutes, this procedure obviously eliminates the experimental uncertainty inherent in inferences from separate experiments, but the interpretation of this difference in terms of the association constants assumes implicitly the identity of V1, and also V,, for the two systems. The validity of each of these assumptions would thus need to be established by demonstrating that the quantity COAT/', is indeed zero in layering experiments under conditions where both systems exist in the pure monomer or polymer form. In the absence of such conditions a combination of the present method and the layering technique may well prove necessary, the former to (a) establish the approximate identities of Vl, V,, and n for the two systems and (b) evaluate an association constant for one solute, and the latter a t a series of concentrations to define K,' for the second solute relative to the value obtained for the other system. An extrapolated value of (0, 0) for the plot of AV, us. co would establish the equality of VI values, while agreement between estimates of K,' for the second system at the different solute concentrations (8)would signify the identity of the polymer elution volumes. (19) J. L. Bethune and P. J. Grillo, Bbchemistry, 6 , 796 (1967). (20) G. A. Gilbert, Anal. Chim. Acta, 38, 275 (1967). (21) G. A. Gilbert, Nature, 212, 296 (1966). (22) We are grateful to Dr. Gilbert for confirming this interpretation.
