## P P

Time (arbitrary units)

FIGURE 4.28 Approach of enzyme-catalyzed reactions to thermodynamic equilibrium. Determination of an equilibrium constant (Keq = [P]n/[A]n) for a reaction by determining the time-invariant end-point (t = tN), by monitoring depletion of reactant A or accumulation of product P. In the lower right, the reaction is irreversible, such that [A]0 = x and [P]0 = 0 and [A] N = 0 and [P] N = x. See text for details.

4.11.1b. Prepare a Series of Reaction Samples with Various Concentrations of Substrates and Products, Add Enzyme, and Determine the Resulting Equilibrium Concentrations of Substrates and Products

In this approach, the experimenter seeks to determine the reaction equilibrium concentration starting with excess substrate in some samples and excess product in other samples. If the pH and other conditions remain unchanged during the ensuing reaction, the same equilibrium constant should be obtained for each run.

Determination of the equilibrium constant for choline acetyltransferase (Reaction: Acetyl-Coenzyme A + Choline # Acetyl-Choline + Coenzyme A) by Hersh (1982) is an excellent example of this approach. The equilibrium constant (Keq = [CoA][Acetyl-Choline]/ [Acetyl-CoA][Cho1ine]) had previously been determined to be 12.3 ± 0.6 by Pieklik and Guynn (1975), who utilized high concentrations of acetylcholine (100 mM) to avoid interference by a small contaminating level of acetylcholinesterase (Reaction: Acetyl-Choline + H2O # Acetate + Choline) in their choline acetyltransferase preparation. The choline acetyltransferase preparation used by Hersh (1982) was devoid of acetylcholinesterase. Because he intended to use much lower reactant concentrations in a series of isotope exchange studies, he decided to re-determine the reaction equilibrium constant. As shown in Table 4.19, Hersh chose a variety of initial substrate and/or product concentrations, added enzyme, and allowed sufficient time for the reaction to reach equilibrium. The experimental Keq value was found to be 13.3 ± 0.7, a value that was consistently higher than that obtained earlier.

4.11.1c. Prepare a Series of Reaction Samples with all Substrates and Products Present at Concentrations Corresponding Roughly to the Equilibrium Constant

After enzyme is added to mixtures having slightly different mass-action ratios, the investigator can plot these changes versus the initially chosen mass-action ratio (Fig. 4.29). If the ratio is lower than the equilibrium constant, there will be net conversion to product (that is, measuring A[P], possibly using labeled products and/or substrates). If the ratio is greater than the equilibrium constant, there will be net conversion to reactant. The mass action ratio that yields a zero deviation must exactly match the equilibrium constant.

A double-label isotopic assay method may also be employed to sense a reaction's equilibrium poise. For example, if [14C]-labeled substrate is mixed with [3H]-

labeled product in the presence of enzyme, when equilibration has been reache product must be equal.

bration has been reached, the 3H/14C ratio in substrate and

4.11.1d. Use Thermochemical Data for Partial Reactions to Estimate DGrxn

Although there are excellent databases of thermochemical data, differences in reaction conditions for the reference data and the actual experimental run can be problematic. Alberty (1997; 2000) has introduced the use of Legendre transformations to obtain equilibrium constants on pH, metal ion concentration, and other independent variables (see Section 3.11.4: The Alberty Treatment Redefines the Thermodynamics of Biochemical Systems). For ionic substrates and/or products, pH and free metal ion concentrations often influence the apparent equilibrium constant.

4.11.1e. Use Initial-Rate Kinetic Experiments to Obtain the Equilibrium Constant by Applying the Haldane Relation

As shown in Chapter 5, the Haldane Relationship specifies how the values of the Michaelis constants and maximal velocities for the forward and reverse reactions are constrained by the equilibrium constant. 