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velopments in Biochemistry, Taipei, Taiwan, Academic Sinica. Huang, R. C., and Bonner, J. (1962), Proc. Natl. Acad. Sci. U. S. 48, 1216. Huang, R. C., Bonner, J., and Murray, K. (1964), J. Mol. Biol. 8, 54. Huang, R. C., and Huang, P. C. (1969), J . Mol. Biol. (in press). Izawa, M., Allfrey, V. G., and Mirsky, A. E. (1963), Proc. Natl. Acad. Sci. U . S . 49, 544. Johns, E. W. (1964), Biochem. J . 92, 55. Johns, E. W. (1967), Biochem. J. 104, 78. Kischer, C. W., Gurley, L. R., and Shepherd, G. R. (1966), Nature 212, 304. Lee, M. F., Walker, I. O., and Peacocke, A. R. (1963), Biochim. Biophys. Acta 72, 310. Lindsay, D. T’. (1964), Science 144,420. Lowry, 0. H., Rosebrough, N. J., Farr, A. L., and Randall, R. J. (1951), J . Biol. Chem. 193, 265. Marmur, J. (1961), J . Mol. Biol. 3, 208. Marushige, K., and Bonner, J. (1966), J . Mol. Biol. 15, 160. Marushige, K., and Ozaki, H. (1967), Decelop. Biol.
Id, 474. Murray, K. (1964), in The Nucleohistones, Bonner, J., andT’so, P. 0.P., Ed., San Francisco, Calif., HoldenDay, pp 21-35. Neelin, J. M. (1964), in The Nucleohistones, Bonner, J., andT’so, P. 0. P., Ed., San Francisco, Calif., HoldenDay, pp 66-71. Neelin, J. M., Callahan, P. X., Lamb, D. C., and Murray, K. (1964), Can. J. Biochem. Physiol. 42,1743. Neidle, A., and Waelsch, H. (1964), Science 145, 1059. Ohlenbusch, H. H., Olivera, B. M., Tuan, D., and Davidson, N. (1967), J . Mol. Bid. 25, 299. Olivera, B. M. (1966), Ph.D. Thesis, California Institute of Technology, Pasadena, Calif. Paul, J., and Gilmour, R. S. (1966a), J. Mol. Biol. 16, 242. Paul, J., and Gilmour, R. S. (1966b), Nature 210, 992. Phillips, D. M. P., and Johns, E. W. (1965), Biochem. J . 94, 127. Swift, H. (1964), in The Nucleohistones, Bonner, J., andT’so, P. 0. P., Ed., San Francisco, Calif., HoldenDay. Zubay, G., and Doty, P. (1959), J . Mol. Biol. 1, 1.
Determination of the Equilibrium Constants of Associating Protein Systems. 111. Evaluation of the Weight Fraction of Monomer from the Weight-Average Partition Coefficient (Application to Bovine Liver Glutamate Dehydrogenase)* P. W. Chun, S. J. Kim,t C . A. Stanley, and G. K . Ackers
ABSTRACT: Procedures for evaluation of weight fraction monomer from experimental weight-average partition coefficients have been derived for several important cases of association. Since the partition coefficients are sensitive to molecular asymmetry, the molecular sieve data can be combined with molecular weight data to provide inferences regarding the mode of aggrega-
I
n recent years the experimental study of associating protein systems has depended largely upon development of useful theories for interpretation of data that can be obtained by a variety of physical techniques such as osmotic pressure, light scattering, molecular sieve chromatography, and sedimentation equilibrium (Steiner, 1952; Squire and Li, 1961 ; Rao and Kegeles,
* From the Department of Biochemistry, College of Medicine, University of Florida, Gainesville, Florida 32601, and Department of Biochemistry, University of Virginia, Charlottesville, Virginia 22901. Receiwd December 30, 1968. This work was supported by National Institutes of Health Research Grants N I H F R 05362-06,07 and GM-14493. t Predoctoral fellow, Department of Mechanical Science, University of Florida, Gainesville, Fla. 32601.
tion for an associating system. These procedures have been applied to bovine liver L-glutamate dehydrogenase. For this enzyme the correlation of molecular weight and partition coefficient data indicated a linear aggregation of subunits. The subunit association was of the “indefinite” type in which a single equilibrium constant pertains to subunit addition for all species.
1958; Gilbert and Jenkins, 1963; Jeffrey and Coates, 1966; Adams and Filmer, 1966; Van Holde and Rossetti, 1967; Albright and Williams, 1968; Chun and Fried, 1967; Adams and Lewis, 1968; Chun et al., 1968). Recently it has been demonstrated (Ackers and Thompson, 1965; Ackers, 1967a; Chiancone et al., 1968) that molecular sieve chromatography may be used for quantitative studies which give results analogous to those of previous transport boundary analysis (Gilbert, 1955; Gilbert and Jenkins, 1963). It is clear from these studies that in order to determine the mode of association in experimental systems by molecular sieve chromatography, it is necessary to carry out an evaluation of the weight fraction of monomer from the
EQUILIBRIUM CONSTANTS OF ASSOCIATING PROTEIN
1625
SYSTEMS.
111
BIOCHEMISTRY
molecular sieve partition data. Such an evaluation is necessary in order to calculate stoichiometries and equilibrium constants and to correlate molecular sieve data with results of other measurements. At the present time only a few methods of analysis have been successfully applied to quantitative evaluation of interaction parameters (Steiner, 1952; Adams and Williams, 1964; Ackers, 1968; Chun and Fried, 1967). A general method for evaluation of the weight fraction of associating monomeric units based on the weight-average partition coefficient has been reported (Ackers, 1968) based on molecular sieve chromatographic experiments with gels of different porosity. The advantages of multiple porosity gel experiments have been described previously (Ackers, 1968). However, for relatively simple systems it may be possible to evaluate the desired reaction parameters from single-porosity experiments (Ackers and Thompson, 1965; Chiancone et al., 1968) which are technically simpler to carry out. We have therefore explored these possibilities by explicit evaluation of several important types of interaction models. In this communication procedures for quantitative evaluation of the weight fraction of monomer from weight-average partition coefficients on a singleporosity column are developed. One procedure depends upon the empirical relationship between the weight-average partition coefficient and the weightaverage molecular weight. The calculations have been applied to experimental data on the bovine liver L-glutamate dehydrogenase system. A correlation of data obtained by two independent methods will be shown to be consistent with an indefinite linear association of this protein. Evaluation of the Weight Fraction Monomer from the Weight-Average Partition Coefficient as a Function of Concentration. The basic equation for the molecular sieve partition isotherm is given by Ackers and Thompson (1965) and Ackers (1967a).
w h e r e 5 is the weight fraction of species i and Pi is a constant. Experimentally, the partition isotherm is linear at low concentration and the first-order term i
Hence (3) becomes (4) z
This is a fundamental expression of the isotherm applicable to conditions of low concentration. It is also possible to derive an empirical relationship between the partition coefficients and the molecular weight of B. The constant i-mer expressed as et = - A In M , A is a slope of the experimental plot of ut us. In M , and B is the intercept (Andrews, 1962, 1964; Whitaker, 1963). A proper substitution of u, = u1 - A In i (Ackers and Steere, 1967) into eq 4 gives empirical eq 4a.
+
n
sw =
e1
- A C f , In i + O(C)
(44
1
The partition coefficient, e,, can be related to the molecular radius, a,, through the error function complement inverse as given by eq 4b. Here a0 and bo are calibration constants for a given gel (Ackers, 1967b) and are determined independently. Q,
= ao
+ boerfc-'u,
(4b)
This equation, which is a linear relationship, has been found to hold over a wide range of molecular sizes and a diversity of column bed material (Ackers, 1967b). The total concentration of macromolecular component is given by
C
=
Ci
+ K2Ci2 + K3Ci3 +
n * *
=
CK,Cit
(5)
1
Qiis the amount of solute, i, partitioned into gel phase per unit column length and cross-sectional area, Ci the solute concentration in the void spaces, and Vi is the gel internal volume fraction per unit column length. The molecular sieve coefficients, ci,characterize the interaction between the column gel phase and a given molecular species, i. Equation 1 is also written as (Brumbaugh and Ackers, 1968)
V p is the nonexcluded volume within the gel. 7 0 and Y, are the activity coefficients of solute in the void phase and gel solvent phase, respectively. Since these parameters are concentration dependent, a concentration dependence of ut results. Taking this concentration dependence into account, the weight-average partition coefficient, rw,is expressed (Ackers, 1967a) as
The concentration is expressed in grams per unit volume of solution and the weight fractions are given by n Ci - = f i and f; = 1 C i
Cirepresents the concentration of i-mers. The individual terms of eq 5 are to be regarded as single constituents in the description of the number of components of the system. nP1+ P,, n > 1 (A) Consider the case of an associating system of monomer-n-mer in chemical equilibrium. The weightaverage partition coefficient, cw,from eq 4 is given by (Ackers and Thompson, 1965). sw
n
1626 CHUN,
Cw
=
Cfdc,+ PZC,) i
KIM, S T A N L E Y , A N D A C K E R S
(3)
f1
= u n + f i ( c ~-
an)
(7)
The weight fraction of monomer, f i , is given by = (aN - un)/(ul - un) and consequently f n is
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given by fn = 1 - f i = (aw - a l ) / ( a n - ai). Once the weight fraction of monomer is known, the equilibrium constant of monomer-n-mer association is computed from eq 4 after substitution
It is also possible to evaluate fn and Kn by means of the empirical relationship described in eq 4a u1 f
n
=
- a, z
--
1-fl
(9)
It should be noted that this expression has exactly the same form as that obtained by eq 7 since both denominators A In n and an - u 1 are constants. The equilibrium constant obtained from eq 8 may be checked by the expression
Substituting Ci
=
un
K$li and fi = Ci/C into eq 12a
+ nul +
y'Jm f m 2
fmfn
?fn2
(13)
A proper substitution of eq 4 gives
The last two equations (eq 9 and 10) are based on the empirical relationship between molecular weight and partition coefficient. They may be useful as long as one is aware of the limitations involved in this empirical correlation. The variation of such a log model from the experimental curve in one such case, that of bovine liver L-glutamate dehydrogenase, will be shown.
- : =mf
, u,
-
6,
un
u1
-
+
f1
+
fm
+ f n = 1 into
0,
___
Jfl
=
Three species are involved in chemical equilibrium. This is a special case of the general analysis described previously for associating species in equilibrium (Ackers, 1967a). Method I. In order to analyze such a case in which three species are involved in chemical equilibrium by the partition isotherm experiments, one must consider the total concentration of macromolecular components as described in eq 5 and the derivatives of the concentration terms d C = d(C1
+ KmC1" + KnC1") (1 + mKmCY-l + nKnC;-')dC1 =
(11)
Substituting eq 11 into eq 4
(1
+ mKmC;"-' + nknC;-')dC1
(lla)
EQUILIBRIUM
A is a constant when m and n are given and a 1 and a
CONSTANTS OF ASSOCIATING
1627
P R O T E I N SYSTEMS. I l l
BIOCHEMISTRY
are evaluated at each concentration. q is readily obtainable from eq 12. Hence the weight fraction of monomer (eq 16) is solved by the quadratic equation fi =
- a,0 < f i < 1
-ai f z / c y i 2
(17)
Once f i is known, eq 14 and 15 are used to calculate and f m , respectively. The equilibrium constants, Km and K,, are computed from
Substituting the quantities Ci = KiCli and f;. = Ci/C into eq 19c cp=-
In m m + l
(fifm)
+
+ Inm +
. +In l ( fni f n )
T
f
m2
fn
From eq 4a and 6 one obtains (Inn)(l
Method 2. This method is based on eq 4a, from which the following empirical relationship may be derived. As the use of the basic empirical equation is risky, this method is less dependable than method 1. From eq 4a
Cm Cn -1nmf--Inn C c
In m
= KmC1"
+ KnCinI n n
(19)
Combining eq 11 and 19 and by integration
[ ' ( K m C l m In m (1
0 1 - a, n = fmIn- f1) - A m
and hence
The evaluation of the weight fraction quantities, f m and .fn, cannot be accomplished without first determining the weight fraction of monomer, f1. This quantity may be derived by substituting eq 19e and 19f into 19d.
+ KnC1"In n) x
+ mKmC;"-l + nK,C;-')dCl
lc(v)
(19a)
fu1
- a,\
Cd C
+
is the analogous numerical integration of
n + l
1 C
-CMi =
o
Mnapp
Mwapp
described by Adams (1965a,b) for nonideal solutions. Adams and Williams (1964) developed and evaluated the nonideality term BM1 from this expression (see Appendix).
-[-+ -(-
In n 1 m m + l
C'p =
S,
+
K K
("
1628 CHUN,
KIM,
+
[KmClmIn m KnC1" Inn Km2C?"-'m In m K , 2C12n-1 n I n n
In
p+n--l
1
+
(n In m
+ m lnn)]dC1 =
:I:
STANLEY, A N D
a1 =
+
In )KmC1"KnCIn
ACKERS
(19~)
+
+~
n In m m In n m+n
-
) ] f :
n + l
In order to simplify eq 19g, the following relationships are set up ( = - Q1
01
m+1
In m In n In mn (In n/m)2 2
de-.!.!-
nlnm mlnn m+n
- Qw
A n I n m + mlnn m+n
CY2 =
2 In m In n (In n/m)2
a3 =
In mn 2(ln n / m ) 2
+
Inmlnnlnmn 2(1n n/m>2
m+n
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m+n In n 1 m m + l
-(-
-
+
'->I
n + l
Hence eq 19f can be simplified as (0
= a1
- a21:
+
a31:
- (a4
+ LY5l)fl +
a6f12
tion. The formulation of the weight-average partition coefficient as related to an indefinite association will be described. A new function, Gafu function for describing the convergence of the series as applied to weight fractions is discussed in this section. The total concentration, C, can be expressed as (Adams, 1967)
(20)
Analytical solution of eq 20 is as follows: (a) a l , ad, as, and a0 are solved by inserting the numerical values; (b) 1: = { ( c ) and letting Y = (a1 - a21: a312 - P>/aS, 271 = (a4 a 5 ) / a 6 , resulting in a further simplification of eq 20, and (c)
a 2 , a3,
+
+
f12
- 2Ylj-1 + Y
=
0
(21)
{ ( C , and cp quantities are computed at each known concentration (as obtained from the weight-average partition coefficient as a function of concentration). Then eq 21 is solved by the quadratic equation
f1 = 71
The weight-average partition coefficient, 5, from eq 4 as shown previously, becomes
f d?--=T
when the limits of 0 < f i < 1 are satisfied. From eq 19e and 19f, fm and f n can be evaluated by substitution off1
Equations 25 and 25a can be solved simultaneously for C1 and k , knowing kC1 = 1 - dcl/c
- z/Cl/c);-' . c-1
a,
= X + u i ( l
=
xiaifI(1 -
Similarly, using the empirical relationship of eq 4a 5, = u1
-A
x f; In i
=
u1 - A X -Ci In i
1
u1 - uw -fn
=
A In n / m
In m
C,
+-
In rn (71 f In n / m
= a1
iC
=
- A c i In i[fl(l - z/3)i-1] (26a) 1
Hence u1 A
--
The equilibrium constants of this system are computed from fm = KmClm/C, f n = KnCin/C, and f l = C1/C. Hence
dX);-'(26)
a,
- Z(f1) = x i l n i[fl(l
-
(27)
2
From eq 27 C i In i[fl(l - d\/fili-l] i
the convergence of the series can be verified by the ratio test lim When four or more species are present in chemical equilibrium, derivation and evaluation of the weight fraction based on the weight-average partition coefficient using method 1 becomes too involved to be practical as a research tool. This is not to say, however, that it cannot be accomplished, but the calculations are quite tedious. We propose the combination of the weight-average partition coefficient and molecular weight data to obtain weight fractions in such a case, as described in part IV of this communication. Indefinite Association. The theoretical considerations to be applied to indefinite association are similar to the procedure outlined by A d a m (1967). The quantity kC < 1, where k is the intrinsic equilibrium constant denoting quantities consistent with the Adams' nota-
i--tm
(n
+ 1) In (1 + 1
) -~4>"1 n In nil - d51n-'
where the limits of 0 < f l < 1 are satisfied. This new function (eq 27) designated as the Cafu function has the built-in error factor of an empirical derivation previously described. The range of the error can be seen from Figure 2. Equation 26 is the most useful function to be derived by this method. The resultant weight fraction data obtained (Table 11) from partition data compared with values obtained from the weight molecular weight data (Table I) show a remarkable degree of correlation even though two independent methods were used to
1629
EQUILIBRIUM C O N S T A N T S OF ASSOCIATING PROTEIN SYSTEMS.
111
BIOCHEMISTRY
TABLE I:
The Weight Fraction of Monomer to i-Mer Tabulation of Bovine Liver L-Glutamate Dehydrogenase Based
on the Weight-Average Molecular Weight as a Function of Concentration (for Indefinite Association)."
fi i
C = 0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.4217 0.2957 0.1555 0.0727 0.0319 0.0137 0.0059 0.0019 0.0011
'c[ +
= Ci/C = - 1
k
C
=
1.0
0.2820 0.2645 0.1860 0.1162 0.0683 0.0383 0.021 1 0.0111 0.0060 0.0030 0.0013 0.0007
1 -(1 2kC
C = 2.0
C
0.1749 0.2035 0.1777 0.1377 0.1002 0.0700 0.0474 0.0315 0.0206 0.0132 0.0086 0.0054 0.0035 0.0021
The concentration, C, is expressed as X g/ml. M, pH 7.0, 0.2 M phosphate b ~ f f e r - l o - ~M EDTA at 25".
I
- dl + 4kC) =
3.0
0.1286 0.1650 0.1587 0.1357 0.1088 0.0837 0.0627 0.0459 0.0332 0.0237 0.0168 0.0115 0.0081 0.0056 L'S.
C
4.0
C = 5.0
0.1023 0.1391 0.1420 0.1287 0.1095 0.0893 0.0709 0.0551 0.0422 0.0319 0.0238 0.0177 0.0131 0.0095
0.0852 0.1206 0.1282 0.1210 0.1071 0.0910 0.0752 0.0608 0.0485 0.0381 0.0298 0.0230 0.0177 0.0135
=
C data were taken from Eisenberg and Tomkins (1968),
1.0
obtain the data. This supports our original contention that these two methods may be used to confirm the mode of association of a particular model system. The weight-average partition coefficient as a function of concentration is regenerated if the weight fractions are known. The weight fraction of monomer is determined from a plot such as shown in Figure 1. The curve in this particular figure represents values using bovine liver L-glutamate dehydrogenase. Then, the equilibrium constant, k, is computed from the following expression. 1 1 k = -(I = -(I - dfi) (28) c1 Cfl
G)
0.9
0.8
0.7
0.6
0.5
c.u
0.3
0.2
3.:
1630 CHUN,
Application to Bovine Liver L-Glutamate Dehydrogenase. In the extensive research on the allosteric enzyme glutamate dehydrogenase which was pioneered by Olson and Anfinson in 1952 and has since been continued by many other workers, the primary objective has been a comprehesive investigation relating the association of this enzyme to its biological activity. In a recent publication, Eisenberg and Tomkins (1968) have found by light-scattering measurements that the molecular weight of the enzymatically active monomer is 313,000 and the molecular weights of its constituent subunits were found to be 53,500. From the angular dependence of scattered intensities it was proposed by these authors that the subunits associate to form linear rodlike aggregates. In the present study we have evaluated linear us. compact aggregation models for the enzyme by combining molecular weight data which are independent of molecular shape with
KIM,
STANLEY,
AND ACKERS
FIGURE 1: The weight-average partition coefficient plotted against the weight fraction of monomer (bovine liver L-glutamate dehydrogenase). See text eq 26, where runs were made at 0.2 M sodium phosphate buffer-1 X M EDTA (pH 7.0) at 25".
molecular sieve partition coefficient data which are sensitive to molecular asymmetry. From the accurate molecular weight data of Eisenberg and Tompkins it was found that the only model from a large number (see Chun and Kim, 1969, part IV of this series) which provided a good fit was that of an indefinite association, From this analysis the weight fraction monomer f1 was calculated as a function of
VOL.
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4,
APRIL
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1969
3.c
L.O
FIGURE 2 : The weight-averagepartition
5.0
coefficients of bovine
liver L-glutamate dehydrogenase as a function of concentra-
tion generated from various calibration models. (A) Association model based on the relationship between ui and log Mi (log model). 7w = 0 1 - A Z f i I n i i
(A) Association based on a spherical model (sphere model) ui = e r f c [ ( a ~ i ” -~ao)/bo], where a1 = 54.5 A, ao =
9.1 A, b o = 157.5 A. (0) Indefinite linear association model from the weight fractions rw =
Acknowledgment The authors thank Dr. Robert Langdon for numerous valuable suggestions in the course of this work. All calculations were made by Wang Model 370 calculater with 320 electronic keyboard. Appendix
zuifi
54.5 A, obtained curve at 0.2 M sodium phosphate b~ffer-lo-~M EDTA (pH 7.0) at 25”. Column gel was composed of Sepharose-4B. where a0
calculated from the logarithmic calibration relationship (es 27). It can be seen that the linear indefinite association model provides a remarkably good fit to the experimental weight-average partition coefficients whereas the compact aggregation model deviates very substantially from the experimental curve. The extreme failure of the logarithmic model confirms our earlier suspicions regarding the difficulties of this method. A subsequent series of calculations were carried out to determine the best fit value of the equilibrium constant from the molecular sieve data based on the linear aggregation model. For this calculation a theoretical plot of f l as a function of CW was made according to eq 26 and is shown in Figure 1. The corresponding values of f l are obtained from this plot for each experimental Cw. For each 1 7 then, ~ the values of the equilibrium constants k were calculated from f i and C by means of eq 25. The closest correspondence in values was 1650 ml/g as compared with k = 1660 ml/g calculated from the molecular weight data. The distribution of components through 14-mer are shown in Table 11. It can be seen from this table that the weight fractions and partition coefficients of these components become increasingly minute. Thus it may be assumed that the contribution of aggregates higher than 14-mer to the weight-average partition coefficients is negligible. Therefore in the refinement of calculations described above, only aggregates up to 14-mer were included in the calculation of Cn..
u1 e r f c [ ( ( f / f ~ ) u ~-j ao)/b?], l/~ a1 = = 9.1 A, bo = 157.5 A. ( 0 ) Experimentally
C as shown in Table I. The resulting values of f l were then used to calculate curves of CW as a function of concentration by means of eq 26 using values of ut calculated according to two extreme molecular shape models. (a) Compact aggregation in this limiting case all species are assumed to be spherical and a, = u1i1I3. (b) Linear aggregation for this limiting case the aggregates were approximated as rods and the respective frictional ratios f / f o determined assuming a length to diameter ratio of i. Then a, = ( f / f ~ ) , a l i ” ~For . both models the value of a1 = 54.4 A was obtained by independent calibration of a column (Sepharose 4B, Pharmacia Fine Chemicals) on which experimental values of Cw were obtained for comparison with the theoretical curves. The results of this comparison are shown in Figure 2 along with the corresponding curve
Euuluation of the Weight Fraction of Monomer from Sedimentation Equilibrium or Light-Scattering Measurements. In the derivation given below the weight fractions of both the ideal case and nonideal case are considered, but it must be noted that this derivation can be extended to any discrete or indefinite case described by this communication. Previous assumptions (Adams, 1965a,b, 1967; Adams and Filmer, 1966) regarding the partial specific volume, the refractive index increment, and the natural logarithm of the activity coefficient will be used. In addition, it is assumed that a plot of M1/MwaPp GS. C has been constructed. 1. Ideal case, BM1 = 0 fl =
Cl/C
f2 =
Cz/C
f %=
__
=
K2Cl2/C
KaC1% C
1631
E Q U I L I B R I U M C O N S T A N T S O F A S S O C I A T I N G P R O T E I N S Y S T E M S . 111
BIOCHEMISTRY
TABLE 11: The Weight Fraction of Monomer to i-Mer Tabulation of Bovine Liver L-Glutamate Dehydrogenase Based on the Weight-Average Partition Coefficients, Utilizing Figure 1, eq 26."
ci, = Ci.,fl(l
- X1fi)t-l
i
i
C = 0.5
1 2 3 4 5 6 7 8 9
0.4100 0.2949 0.1591 0.0763 0.0343 0.0148 0.0062 0.0025 0.0010
C
~-
=
1.0
0.2800 0.2637 0.1862 0.1169 0.0688 0.0389 0.0214 0.0115 0.0061 0.0032 0.0016 0.0008
10
11 12 13 14
C
=
2.0
0.1800 0.2073 0.1790 0.1374 0.0989 0.0683 0.0459 0.0302 0.0196 0.0125 0.0079 0.0050 0.0031 0.0019
C
=
3.0
0.1350 0.1708 0.1621 0.1367 0.1081 0.0820 0.0605 0.0438 0.0312 0.0219 0.01 52 0.0105 0.0072 0.0049
C = 4.0
C = 5.0
0.1120 0.1490 0.1487 0.1319 0.1097 0.0876 0.0680 0.0517 0.0387 0.0286 0.0209 0.0152 0.0110 0.0078
0.0980 0.1346 0.1387 0.1271 0.1091 0.0900 0.0721 0.0566 0.0437 0.0334 0.0252 0.0189 0.0141 0.0104
a The concentration, C , is expressed as X10-3 g/rnl. Runs were made at 0.2 M sodium phosphate buffer-10-3 EDTA (pH 7.0) at 25". Column gel was composed of Sepharose-4B.
2. Nonideal case, BM1 # 0
3. Indefinite association 1
where fl is evaluated from part 2.
References
1632
Ackers, G. K. (1967a), J . Bid. Chem. 242, 3026. Ackers, G. K. (1967b), J . Biol. Chem. 242, 3237. Ackers, G . K. (1968), J . Bid. Chem. 243, 2056. Ackers, G. K., and Steere, R. L. (1967), Methods Virol. 2, 325. Ackers, G. K., and Thompson, T. E. (1965), Proc. Natl. Acad. Sci U.S. 53, 342. Adams, E. T., Jr. (1965a), Biochemistry 4, 1646. Adarns, E. T., Jr. (1965b), Biochemistry 4, 1655. Adams, E. T., Jr. (1967), Biochemistry 5, 2971. Adarns, E. T., Jr., and Filrner, D. L. (1966), Biochemistry 5, 2971.
C H U N , KIM,
STANLEY,
AND ACKERS
M
Adams, E. T., Jr., and Lewis, M. S. (1968), Biochemistry 7, 1044. Adarns, E. T., Jr., and Williams, J. W. (1964), J . Am. Chem. Soc. 86, 3454. Albright, D. A., and Williams, J. W. (1968), Biochemistry 7, 67. Andrews, P. (1962), Nafure 196, 36. Andrews, P. (1964), Biochem. J . 91, 222. Brumbaugh, E. E., and Ackers, G. K. (1968), J . Biol. Chem. 243, 6317. Chiancone, E., Gilbert, L. M., Gilbert, G. A., and Kellet, G. L. (1968), J . Biol. Chem. 243, 1212. Chun, P. W., and Fried, M. (1967), Biochemistry 6, 3094. Chun, P. W., Fried, M., and Yee, K. S. (1968), J . Theoret. Biof. 19, 147. Chun, P. W., and Kim, S. J. (1969), Biochemistry 8, 1633 (this issue, following paper). Eisenberg, H . , and Tomkins, G. (1968), J . Mol. Biof. 31, 37. Gilbert, G. A. (1955), Discussions Faraday SOC.20, 24. Gilbert, G. A., and Jenkins, R. C. L. (1963), in Ultracentrifugal Analysis in Theory and Experiment, Williams, J. W., Ed., New York, N. Y., Academic, pp 59, 73. Jeffrey, P. D., and Coates, J. H. (1966), Biochemistry 5 , 489. Olson, J. A., and Anfinson, C . B. (1952), J . Biol. Chem. 197, 67. Rao, M. S . N., and Kegeles, G. (1958), J. Am. Chem. SOC.80, 5724. Squire, P. G., and Li, C. H. (1961), J . Am. Chem. SOC.83, 3521. Steiner, R. F. (1952), Arch Biochem. Biophys. 39, 333. Van Holde, K . E., and Rossetti, G. P. (1967), Biochemistry 6, 2189. Whitaker, J. R. (1963), Anal. Chem. 35, 1950.
