# 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.

## Key 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.