Self-organization and dynamics of an open futile cycle

Abstract

The dynamic behaviour of an open futile cycle composed of two enzymes has been investigated in the vicinity of a steady-state. A necessary condition required for damped or sustained oscillations of the system is that enzyme E 2, which controls recycling of the substrate S 2, be inhibited by an excess of this substrate. In order for the system to be neutrally stable and therefore to exhibit sustained oscillations, it is not necessary for antagonist enzyme E 1 to be activated by its product S 2. If it is enzyme E 1 which is inhibited by an excess of its substrate S 1, the system has a saddle point. Other conditions for stability or instability of the system have been determined. If the enzyme E 1, which is not inhibited by the substrate, exhibits a slow conformational transition of the mnemonical type, this transition dramatically alters the stability behavior of the system. If the mnemonical enzyme E 1 were exhibiting a positive kinetic co-operativity, decreasing the rate of the conformational transition of the mnemonical enzyme will increase the stability of the whole system and will tend to damp the oscillations in the vicinity of the steady-state. If conversely the mnemonical enzyme E 1 were exhibiting a negative kinetic co-operativity, decreasing the rate of the enzyme conformational transition will decrease the stability of the system and will tend to create or amplify oscillations of the system taken as a whole. If these results may be extended to more complex metabolic cycles, involving more than two enzymes, it may be tentatively considered that positive co-operativity associated with slow transition has emerged in the course of evolution in order to limit temporal instabilities of metabolic cycles. Alternatively one may speculate that the “biological function” of negative co-operativity is to create or amplify these temporal instabilities.