How do enzyme activities control metabolite concentrations?

Abstract

A simple theorem is derived relating the extent to which enzymes in a metabolic pathway control the steady-state concentration of metabolites to the kinetic properties of those enzymes. The theorem gives insight into the mechanism by which the concentration of a second messenger is controlled by the enzymes that form and degrade it, and provides an alternative to the 'cross-over theorem'.The theoretical principles developed by Kacser and Burns [I] and Heinrich and Rapoport [2] have greatly facilitated the quantitative determination of the extent to which certain enzymes control the flux through a metabolic pathway [3, 41 (the 'control strength' [2] of those enzymes). Through the 'connectivity theorem [I] for the control of fluxes', these principles allow the designation of the enzymic properties (i.e. elasticity coefficients [I, 51) that are responsible for a certain distribution of the control on such a flux between the enzymes in the pathway [I, 3,5 -81. In the present paper, a new theorem (the 'connectivity theorem for control of metabolite concentrations') is derived relating the extent to which metabolite concentrations are controlled by the different pathway enzymes to those enzymic properties (elasticity coefficients). THEORY The theorem .6).Remark : We shall assume throughout that both C and i: can bc defined. Cases with zero reaction rates or concentrations are formally excluded. For these cases alternative theorems are easily formulated.symbol for the elasticity coefficient of the reaction catalyzed by enzyme Ei with respect to metabolite M,, defined by Eqn (3) as (cf. [I, 2, 51):where ci is the (net) reaction rate through enzyme Ei. In the following the metabolite-concentration control coefficient C p and the elasticity coefficient cSkI will occassionally be abbre- ' [6, 10, 111, 'sensitization' [12], 'coefficient' [9]). For a review see [5]. The concentration control coefficient of a metabolite M, with respect to an enzyme Ei (identical to 'element of control matrix' [2]), i.e. Cg, is a measure of the relative change in concentration of metabolite M, occurring if the concentration of enzyme Ei is increased, whilst the concentrations of the other enzymes in the system are kept constant. It is not the immediate change in the metabolite concentration that is meant here, but the change in concentration from its initial value to the value it attains after the system has relaxed towards a new steady state. The concentrations of other metabolites may have changed as well. By dividing the relative change in metabolite concentration by the relative change in enzyme concentration, a measure is obtained for the importance of the enzyme Ei in the determination of the metabolite concentration [M,]. However, unless the relative change in enzyme concentration is small, the ratio depends on the magnitude of that relative change. To overcornc this complication, the changes are taken to be infinitesimally small in the definitions of control coefficients (and of elasticity coefficients).As an illustration of the...

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