Investigation of a Biochemical Model with Recycling in Case of Negative Cooperativity

The objective of this paper is to find new dynamic perspectives in a well-known two dimensional nonlinear system which is a modification of the phosphofructo kinase model by incorporating recycling of the product, p, into the substrate, s. Specifically, we investigate the affect of the negative cooperativity on the number of equilibria and their stability. Moreover, in the parameter space, we analyze analytically and numerically the number of periodic oscillations (solutions) and their stability using Lyapunov coefficients (in other words, quantities and focus values). Thus, we obtain that three different dynamical conditions (regimes) take place: (1) structurally unstable, (2) the existence of an unstable limit cycle with an external stable limit cycle, and (3) the existence of a stable limit cycle with an external unstable limit cycle. Moreover, for a zero rate of product synthesis (due to e.g., defective enzyme), we obtain that the modified system has a first integral.