# Oscillations and bistability in a model of ERK regulation

## TL;DR

This paper analyzes how oscillations and bistability arise in a model of ERK signaling, showing oscillations are robust while bistability depends on specific reversible reactions, with implications for understanding cellular signaling dynamics.

## Contribution

The study provides a detailed mathematical analysis of ERK network dynamics, revealing conditions for oscillations and bistability, and how these properties depend on reaction reversibility.

## Key findings

- Oscillations persist even with simplified, irreversible reactions.
- Bistability is lost when intermediates are removed or reactions are made irreversible.
- Reversible catalytic reactions are crucial for bistability.

## Abstract

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust -- this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network -- namely, the fully processive dual-site network -- which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.