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

arXiv:2605.16049v1 Announce Type: cross
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.