Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.