A Linear Steady-State Treatment of Enzymatic Chains. General Properties, Control and Effector Strength

Abstract

A theoretical analysis of linear enzymatic chains is presented. By linear approximation simple analytical solutions can be obtained for the metabolite concentrations and the flux through the chain for steady-state conditions. The equations are greatly simplified if the common kinetic constants are expressed as functions of two parameters, i.e. the thermodynamic equilibrium constant and the "characteristic time". Three cardinal terms are proposed for the quantitative description of enzyme systems. The first two are the control strength and the control matrix ; these indicate the dependence of the flux and the metabolite concentrations, respectively, on the kinetic properties of a given enzyme. The third is the effector strength, which defines the dependence ofthe velocity of an enzyme on the concentration of an effector ; it expresses the importance of an effector. By linear approximation simple analytical expressions were derived for the control strength, the control matrix and the mass-action ratios. The effector strength was calculated for two cases: for a competitive inhibitor and for allosteric effectors according to the Monod (1965) model. The influence of an effector on the concentrations of the metabolites was considered.A mathematical analysis of an enzymatic chain is necessary if a comprehensive and quantitative description is intended. Verbal analysis even if sections of a chain are treated quantitatively do not permit the simultaneous consideration of diverse influences in the entire enzymatic chain. Therefore selections or weighing is involved which by necessity brings with it a certain arbitrariness.Prom a mathematical model one may expect not only a comprehensive presentation of the effects of a multitude of factors but also a quantitative appraisal of the regulatory features of an enzymatic chain and the identification of enzymes, effectors and metabolites of functional importance. Furthermore, mathematical descriptions make contradictions or insufficiencies of the data more easily apparent.Computer methods for the mathematical description of enzymatic chains were thoroughly explored by Garfinkel et al. in a number of papers ([l] and references cited therein) and also by other authors [2 -51. I n addition to mathematical problems there are several disadvantages of this approach. Firstly, from the computer output it appears difficult to differentiate between important and unimportant effects, enzymes, metabolites, etc. Secondly, it is difficult to see how some effects are brought about. Thirdly, such a computation is often impracticable for experimentalists. Fourthly, many ad hoc assumptions are even now necessary for the mechanisms of several of the constituent enzymes of a chain. The strong dependence of the model of an enzymatic chain on the detailed mechanisms of single enzymes is unfavorable.Analytical approaches, on the other hand, circumvent some of these difficulties, as they are usually connected with a reduction t o the essential parameters. They may show clearly the reasons for regulator...

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