Conditions for spatial instabilities and pattern formation from monomial steady state parameterizations

TL;DR

This paper derives algebraic conditions based on polynomial inequalities that predict the onset of spatial pattern formation in reaction networks with monomial steady state parameterizations, focusing on Turing-like instabilities.

Contribution

It introduces a new algebraic criterion for Turing instability in reaction networks using monomial parameterizations, linking domain size, rate constants, and diffusion coefficients.

Findings

Derived a sufficient condition for Turing-like instability based on polynomial inequalities.

Applied the condition to a phosphorylation network, identifying key parameters for instability.

Provided explicit algebraic criteria involving rate constants and diffusion coefficients.

Abstract

We study the onset of spatial instabilities in reaction networks where the spatially homogeneous system admits a steady state parameterization. We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains $\Omega$. We also present a specific condition on the domain size $|\Omega|$ required to trigger this instability. As a consequence of employing a monomial parameterization, these conditions take the form of algebraic polynomial inequalities involving only rate constants and diffusion coefficients. We apply these ideas to a network describing the sequential and distributive (de-)phosphorylation of a protein at two binding sites, ultimately deriving a condition involving only the four catalytic constants of the enzymes and the diffusion coefficients of the four enzyme-substrate complexes that guarantees a Turing-like instability.