We investigate noise-induced bimodal distributions in self-regulated gene networks with fast dimerization, where dimerized proteins enhance gene expression. Despite their fundamental role in gene regulation, analytical study of bimodal behaviour in such networks is challenging because the nonlinear interactions introduced by dimer formation render exact steady-state distributions infeasible. To address this, we reformulate the problem as a reduced self-regulated gene-expression model that approximates fast dimerization, in which the transition rate from the promoter-off to promoter-on state depends nonlinearly on protein levels. We introduce two diagnostic quantities: the promoter activity ratio , which quantifies promoter activation as a function of protein level, and the mode detection ratio , which identifies peaks of the steady-state protein distribution. Analysis of their recurrence relations reveals how promoter activity shapes the steady-state law, and how intrinsic stochasticity can generate multimodal protein distributions in self-regulated expression circuits. We further show that the corresponding mean-field ODE system admits a unique non-negative equilibrium when the protein synthesis-to-degradation ratio lies below an explicit threshold determined by the inactivation and dimer-induced activation rates. Hence, the bimodality we observe can arise purely from stochastic effects rather than deterministic bistability. Our approach provides a general framework for diagnosing noise-induced multimodality in gene networks with nonlinear promoter transitions, without relying on exact probability distributions, which are typically infeasible for nonlinear reaction rates, particularly in our case. Beyond its theoretical contribution, this work has conceptual relevance to sustainability: our mode-detection diagnostics and the distinction between deterministic multistability and noise-induced multimodality can inform assessments of resilience, early-warning indicators, and state persistence.
Hung et al. (Sun,) studied this question.