Deterministic cascade coarsening in a Bistable Gene Toggle model

Abstract

We investigate deterministic coarsening dynamics in a spatially extended bistable gene toggle model with diffusive coupling. Unlike classical curvature-driven coarsening, where domain walls move continuously and annihilate gradually, the present system exhibits a qualitatively different mechanism. The domain walls remain pinned for long intervals and disappear abruptly through collective cascade events. The density of domain walls decays approximately as $\ddot{I}(t)\sim t^{-\hat{I}´}$, but the coarsening exhibits clear log-periodic oscillations superimposed on the power-law behavior. For all values of the promoter strength $\hat{I}\pm$ considered, the measured exponent satisfies $\hat{I}´<0.5$, indicating a systematic deviation from the classical Allen--Cahn prediction $\hat{I}´=1/2$ for curvature-driven coarsening. We show that log-periodic oscillations are not controlled by the density of domain walls, but by the \emph{domains that disappear} in each cascade. The average size of disappearing domains grows roughly linearly with cascade index, producing a constant geometric spacing of cascade times, consistent with discrete scale invariance.