24 Theoretical Model for Determining Monomer-Polymer Reaction Stoichiometry
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from Equilibrium Gel Partition B R U C E F. C A M E R O N Papanicolaou Cancer Research Institute, 1155 Northwest 14th St., Miami, Fla. 33136, and Department of Medicine, Division of Hematology, University of Miami School of Medicine, POB 875—Biscayne Annex, Miami, Fla. 33152 A L A N D. A D L E R New England Institute, Grove St., Ridgefield, Conn.
06877
A theoretical analysis of an equilibrium gel partition system for determination of association reaction stoichiometry is described. Equations are generated for generalized mono mer-polymer equilibrium, including sequential equilibria. Numerical solutions of the generated polynomial equations were obtained for pM Ρ, p up to 6, and the slope of the graph of the ratio of monomer equivalents external to the gel phase to total monomer equivalents as a function of total initial concentration was shown to be a strong function of the stoichiometric order. A numerical analysis of slopes and values of this curve in the region of the inflection yielded a parameter independent of specific equilibrium constants and characteristic of the stoichiometric coefficient for simple generalized polymerization.
A relatively common feature of many problems involving molecular ^ weight determination of biopolymers is that of association-dissocia tion equilibrium. Subunit structure of enzyme proteins is well recognized ( I ) , and methods of dissociation of subunits to obtain monomer molec ular weight are widely utilized (2). A previous paper described the application of an equilibrium gel partition method to the analysis of macromolecular association in a monomer-dimer case (3). The experi mental parameters in a system utilizing the Sephadex series of gel filtrav
298 In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
24.
299
Equilibrium Gel Partition
C A M E R O N AND ADLER
tion materials were described and optimized, and the method was applied to the problem of the dissociation of the hemoglobin tetramer (3, 4). The present work is a theoretical extension of the mathematical model to generalized association stoichiometry, for like and unlike monomers, including sequential association with detectable intermediates for the system containing like interactants. Theory
Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch024
Case l a .
Multiple Association of Like Interactants.
F o r the reac
tion pM — P, where Ρ = M
p
the gel partition system is essentially K pM-P
a
pM -
Ρ
KM
M - M K? Ρ - Ρ
where the external phase is represented by a (volume = V « ) , and the internal phase is represented by β ( volume = ). The relevant equilib rium constants are defined i n terms of the number of moles of species M and Ρ and concentrations [P] and [M] as = [P«]/[MJ'
K
a
K κ
V Ρ
Β
+ α
t o
Μ
a
Μ + ρΡ
Α
0
Μ+ Α
β
ρΡ
α
and Mtot
Μ7
-, =
1
+
, 7 k x
where Χ represents equivalents of monomer. This still does not yield a constant independent of K ; however, recognizing that the original equaM
In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
308
P O L Y M E R M O L E C U L A R W E I G H T METHODS
Table I.
C u r v e Parameters for the Monomer—η-mer System
Ρ
Kp
Slope
Value
2 2 2 2 3 4 5 6
0.05 0.10 0.30 0.50 0.10 0.10 0.10 0.10
0.1683 0.1512 0.0960 0.0555 0.2362 0.2938 0.3366 0.3703
0.7035 0.6855 0.6271 0.5847 0.6673 0.6553 0.6461 0.6390
0
Slope = slope at the point of inflection, value = ordinate value at the point of inflec tion. Conditions: Va = Vç> = 1.0, K = 1 Χ 10 , K M = 0.90.
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α
5
A
Value of ξ as a Function of Κχ for the Monomer—Dimer Reaction
Table II.
α
b
K
K
M
0.90 0.90 0.90 0.90 α b
ξ ±
D
0.05 0.10 0.30 0.50
0.929 0.929 0.928 0.925
ξ has been defined operationally in the text. Conditions: V = 7 = 1.0, Ka = 1 Χ 10 . a
Table III.
&
5
Value of £ as a Function of Stoichiometry ξ ±
ρ
2 3 4 5 6 a
0.002
(p)
a
0.004
0.929 1.491 1.889 2.195 2.442
Conditions: Va = 7β = 1.0, Κα = 1 Χ ΙΟ , K M = 0.90, Kp = 0.10. 5
tion contains terms of the form (1 + k ) and (1 + kp), and that the abscissa is a logarithmic axis, division of this ratio by log [ ( 1 + & M ) / (1 + & P ) ] was attempted. The result is a number which is independent of the partition coefficients. A series of simulations with varying K and V indicated that this parameter, which is designated £, is independent of a l l parameters of the analysis except stoichiometric coefficient. The value of ξ for the various stoichiometries plotted i n Figure 2 is given i n Table III. It would be difficult i n practice to use the value of ξ to determine stoichiometry since it requires not only the slope but also the value at the inflection point, which may be very difficult if not impossible to M
a
a
In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
24.
C A M E R O N
A N D A D L E R
Equilibrium Gel Partition
309
recognize with sufficient accuracy in real error-prone data. Also, a knowl edge of the values of K M and K p is required. However, it has been shown that curve shape can give stoichiometric data i n and of itself; further numerical modeling is required to define practicable curve parameters for analysis of experimental data. For stoichiometry, some experimental uncertainty is tolerable since ξ is a discontinuous function, and to infer a stoichiometry one requires only that ξ be approximately equal to one of the calculated values for a given set of considered stoichiometries.
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Literature
Cited
1. Darnall, D. W., Klotz, I. M., Arch. Biochem. Biophys. (1972) 149, 1. 2. Shapiro, A. L . , Vinuela, E., Maizel, J. V., Biochem. Biophys. Res. Commun. (1967) 28, 815. 3. Cameron, B. F., Sklar, L . , Greenfield, V., Adler, A. D., Separ. Sci. (1971) 6, 217. 4. Kellett, G. L . , J. Mol. Biol. (1971) 59, 401. 5. Mostowski, Α., Stark, M., "Introduction to Higher Algebra," p. 285, Mac Millan, New York, 1964. 6. Adler, A. D., O'Malley, J. Α., Herr, A. J., Jr., J. Phys. Chem. (1967) 71, 2896. 7. Porath, J., Pure Appl. Chem. (1963) 6, 233. 8. Andrews, P., Biochem. J. (1964) 91, 222. 9. Guttman, C. M., DiMarzio, Ε. Α., Macromolecules (1970) 3, 681. January 17, 1972. Work supported in part by U . S. Public Health Grant AM-09 001. RECEIVED
In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
