We consider an n-component biochemical system whose Jacobian matrix J is of upper Hessenberg form, with principal subdiagonal elements b l, b2,. _ _, b,_ ~ 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 -Jo are real and non-negative then a necessary condition for instability at a critical point (steady state) is bib2 b"-lf >>-(sec 7z/n)".
I-Jol
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 /~N-1X.+~ --/> (sec 7r/N) N, G,-1 /~or where N is the number of components (metabolites, enzyme-substrate complexes, and enzyme-inhibitor complex),/3 N_ ~ the order of the enzyme-inhibitor reaction (with respect to the inhibitor), IN ~ the order of reaction for the removal of the last metabolite, and X, + ~/Eor 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.