A proof of unlimited multistability for phosphorylation cycles

Abstract

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 n 2 + 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 n 2 + 1 out of 2 n 2 + 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 n 2 + 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.

A robust method for designing multistable systems by embedding bistable subsystems

“…When external and/or internal conditions change, the system may switch from one steady state to another either randomly by perturbations or in a desired way according to the control strategies. In recent years mathematical models with multistability have been developed for theoretical analysis and computer simulations, which shed light on the mechanisms that generate multistability and control the transition between steady states 16 – 19 .…”

Autophosphorylation and the Dynamics of the Activation of Lck

“…( 2014 ) and for the maximal number of stable steady states in Feliu et al. ( 2020 ). Much less is known in the case of autophosphorylation.…”

“…There has been a lot of work on models for cases where there is a clear distinction between substrates and enzymes. A standard example is the multiple futile cycle where bounds for the maximal number of steady states were obtained in Wang and Sontag (2008) and Flockerzi et al (2014) and for the maximal number of stable steady states in Feliu et al (2020). Much less is known in the case of autophosphorylation.…”

Critical parameters for singular perturbation reductions of chemical reaction networks

“…From a qualitative vantage point, one strategy to prove special features such as the existence of periodic solutions, or multistationarity, is to prove such features for a reduced system and show that they persist for the full system in some parameter range. For a recent example of this strategy, see [17]. Thus it is of general interest to identify parameter domains where a systematic reduction is possible.…”