The secant condition for instability in biochemical feedback control—I. The role of cooperativity and saturability

Abstract

This paper concerns oscillations arising in a sequence of biochemical reactions

whose end-product inhibits the rate of the first reaction. Except for the end-product inhibition, each reaction rate is assumed to be unaffected by S _i other than its reactant. It is shown that with a constant concentration of S₀ a necessary condition for instability at a critical point (steady state) is

$$\frac{{\beta _1 \beta _2 ...\beta _{n - 1} \phi }}{{\alpha _1 \alpha _2 ...\alpha _n }} \geqslant (\sec \pi /n)^n $$

whereβ _i is the order of reaction (with respect to the concentrationS _i of S _i ) of the formation of S _i+1 from S _i ;α _i is the order of reaction (with respect toS _i ) of the combined processes of metabolic conversion and all other processes removing S _i ; and ϕ is the order of the negative feedback, i.e.—ϕ is the order of reaction (with respect toS _n ) of the conversion of S₀ to S₁. The order of a chemical reaction with respect toS _i is defined as ∂ logV/∂ logS _i , whereV is the rate of reaction. So defined, order of reaction may be concentration-dependent; and in the above inequality any concentration-dependent reaction orders must be evaluated at the critical point. The above inequality, with strict inequality, is a sufficient condition for instability in the special case where all the diagonal elements of the Jacobian matrix are equal at the critical point. With a saturable final elimination process (α _n >1), the above condition may be met even if ϕ≤1, i.e. feedback is not cooperative. Saturable elimination of an intermediate metabolite has a similar effect, but this effect is weak unless there is a relatively large loss of that metabolite from the reaction sequence. A 3-component model is constructed which has saturable final elimination and no feedback cooperativity and shows stable oscillations.