A remark on the number of steady states in a multiple futile cycle

TL;DR

This paper analytically investigates the maximum number of positive steady states in multisite phosphorylation cycles, revealing bounds based on parameter ranges and cycle size, with implications for cellular signaling dynamics.

Contribution

It provides new analytical bounds on the number of steady states in multisite phosphorylation cycles, connecting parameter regimes to system behavior.

Findings

At least n+1 (if n even) or n (if n odd) steady states exist for some parameters.

Maximum of 2n-1 steady states across all parameters.

Near standard Michaelis-Menten conditions, at most n+1 steady states.

Abstract

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation-dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n+1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n-1 steady states (in particular, this implies that for n=2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n+1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.