Dispersion Control in Stochastic Biomolecular Systems Without Peak Shifts

Intracellular biomolecular circuits often exhibit multimodal stationary distributions due to intrinsic noise, and the dispersion around each peak governs phenotypic robustness and adaptability. However, tuning dispersion by changing reaction parameters typically shifts peak positions or even alters modality. In this paper, we derive conditions that enable peak-shape control without peak shifts using the Chemical Fokker-Planck Equation. First, we formalize sharpness as a peak-local measure via probability ratios and show that the positions of peaks and valleys remain invariant as a control parameter varies. Second, we prove that sharpness varies monotonically as the control parameter increases, while the modality and the positions of extrema remain fixed. We validate these results with Monte Carlo simulations of two univariate networks: the burst-and-trickle gene expression system (unimodal) and the Schlögl system (bimodal), achieving dispersion tuning without peak shifts. Finally, we present preliminary multivariate evidence on the Genetic Toggle Switch, where the marginal distribution of one protein exhibits similar sharpness control. Our results provide structural design rules for engineering stochastic phenotypes while safeguarding the desired modal structure.