concepts
edited by WILLIAM M. SCOVELL Bowling Green State University Bowling Green, OH 43403
In Vivo Kinetics and the Reversible Michaelis-Menten Model W. Grady Smith Deoartment of Biochemistw and Molecular Bioloov. -,. Universitv of Arkansas for Medical Sciences 4301 W. Markham, Little &ck, AR 72205
The Michaelis-Menten model accounts for many of the characteristics of enzyme reactions ( I ) .Although it is obviously a n oversimplification-that does not include cooperativity, allosteric effects, multiple intermediate complexes, or multiple substrates or products-it is used extensively as an introduction to enzyme kinetics in biochemistry and molecular biology Because the textbook presentations usually emphasize the enzyme assay in vitro, in which the initial product concentration is zero, it is easy for students to get the impression that the model is less applicable in vivo than it actually is. Most textbooks introduce the kinetic structure of metabolic pathways in a qualitative manner, if they consider it at all. Application of quantitative relationships is usually undertaken only when considering regulatory steps. Although there are shortcomings in using the model for any one specific enzyme, values derived from the model are frequently applied successfully in modeling metabolic networks (2, 31, for example, the kinetic constants obtained from in vitro determinations based on the Michaelian behavior of individual enzvmes. The exvanding literature on the kinetics of metabolic reactions inkvo h i s not yet been extensively incorporated in biochemistry textbooks. The purpose of this article is to illustrate that this popular teachinp model for enzvme kinetics can be successfullv applied tometabolic reactions in vivo. In Vivo Derivation
The Michaelis-Menten model can successfully serve as a n introduction to in vivo reaction kinetics if the derivation includes factors that are compatible with in vivo conditions: the presence of product and the reverse reaction. Then the model provides understandable rationales for the low substrate concentrations and high enzyme concentrations (or "excess" catalytic capacity) that are observed in vivo for many reactions. It also helps to account for the differing kinetic parameters of isozymes. In this article we present some examples that illustrate the utility of the reversible model. We use this model to explain the following phenomena associated with enzyme reactions.
I t is hoped that this article will facilitate the transition from the study of in vitro enzyme kinetics to an understanding of metabolic kinetics in vivo. The reversible Michaelis-Menten model, formulated with a single central complex X is shown below.
where the k's are rate constants; E is the concentration of free enzyme (Emd - X); and S and P a r e the substrate and product concentrations, respectively. The net velocity can be expressed as the difference between the forward and reverse components of the productforming step, as shown below.
where u,,, is the net velocity for the forward reaction. (The value of unetis negative in the reverse direction, and positive in the forward direction.) A similar differential equation may be written for d[Xlldt. Using this equation and the material balance for the enzyme, Ebb1 = [El + [XI we get two equations in two unknowns (X and E) Using the common condition &I
>> JLa
and the steady state approxhation
we can be obtain solutions for the steady state concentrations of E and X in terms of rate constants, substrate and product concentrations, and Ebbl. Then we substitute these expressions into the above equation for u.., and use the following definitions. Ks=-kz + k3 kl
1. Flux rates are maximized more effectively by increasing
,,V (by increasing kcat or enzyme concentration)than by increasing substrate concentrations. 2 'Excess" C D I G ~ rapacity ~ ~ ~ C ~ U L 1" . n large amount uf mryme or large kcat mahlss reactions to fitnrtion rffcctively in both directions near equilibrium. 3. Bv c s n be desimed . varvine.. kinetic mrameters.. enzvmes ,
