The secant condition for instability in biochemical feedback control. II: Models with upper Hessenberg Jacobian matrices (Q1175542)

[For part I see the preceding review, Zbl 0751.92002.] An \(n\)-component biochemical system is considered whose Jacobian matrix \(J\) is of upper Hessenberg form, with principal subdiagonal elements \(b_ 1,b_ 2,\dots,b_{n-1}\) and upper right-hand corner element \(-f\). The open-loop Jacobian matrix \(J_ 0\) is formed from \(J\) by setting \(f=0\). It is shown that if the characteristic roots of \((-J_ 0)\) are real and non-negative then a necessary condition for instability at a critical point ( steady state) is: \[ (b_ 1b_ 2\dots b_{n-1}f/| -J_ 0|)\geq (\text{sec }\pi/n)^ 2. \] 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: \[ \beta_{N-1}X_{n+1}/\xi_{N-1}E_{0T}\geq (\text{sec }\pi/N)^ N, \] where \(N\) is the number of components, \(\beta_{N-1}\) the order of the enzyme-inhibitor reaction, \(\xi_{N-1}\) the order of reaction for the removal of the last metabolite, and \(X_{n+1}/E_{0T}\) the fraction of first enzyme blocked by inhibitor. The general theory that illustrates this condition is given in Section 2 of the paper. Practical details of the application of the theory are presented in the analysis of \textit{C. Walter's} model [Stability of controlled biological systems. J. Theor. Biol. 23, 23-38 (1969)] in Sections 3-6. In the biochemical model discussed in these sections, feedback from the end-product is indirect, and mediated by the formation of the enzyme-inhibitor complex. In Section 7, it is shown that if in addition to metabolites there are enzyme-substrate complexes and enzyme- inhibitor complexes as state variables, then it is possible to have instability from negative feedback in a metabolic sequence with fewer than three steps.