A proof of unlimited multistability for phosphorylation cycles

The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as $2\lfloor \frac{n}{2}\rfloor +1$ steady states (where ⌊x⌋ is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n = 1, 2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that $\lfloor \frac{n}{2}\rfloor +1$ out of $2\lfloor \frac{n}{2}\rfloor +1$ steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity $2\lfloor \frac{n}{2}\rfloor +1$. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.