The secant condition for instability in biochemical feedback control—II. Models with upper Hessenberg Jacobian matrices

We consider an n-component biochemical system whose Jacobian matrix J is of upper Hessenberg form, with principal subdiagonal elements b1, b2, …, bn−1 and upper right-hand corner element −f. The open-loop Jacobian matrix J0 is formed from J by setting f=0. It is shown that if the characteristic roots of −J0 are real and non-negative then a necessary condition for instability at a critical point (steady state) is b 1 b 2 … b n − 1 f | − J 0 | ⩾ ( sec ⁡ π / n ) n . This condition is analyzed in terms of reaction orders. For a metabolic sequence with some reversible steps, no loss of intermediate metabolites, and competitive inhibition of the first enzyme by the last metabolite, the above necessary condition becomes β N − 1 ξ N − 1 X N + 1 E 0 T ⩾ ( sec ⁡ π / N ) N , where N is the number of components (metabolites, enzyme-substrate complexes, and enzyme-inhibitor complex), βN−1 the order of the enzyme-inhibitor reaction (with respect to the inhibitor), ζN−1 the order of reaction for the removal of the last metabolite, and Xn+1/E0T the fraction of first enzyme blocked by inhibitor. It is shown that, under certain assumptions, a critical point is always stable in a single two-step enzymatic process (formation of enzyme-substrate complex, followed by conversion to product, then loss of product) with slow negative feedback by competitive product inhibition. A model is constructed showing that stable oscillations can occur in a feedback system with only two metabolic steps and negative feedback by competitive inhibition with no cooperativity. The instability is due to a slow feedback reaction and saturable removal of the second metabolite.