Potential for regulatory genetic networks of gene expression near a stable point

A description for regulatory genetic network based on generalized potential energy is constructed. The potential energy is derived from the steady state solution of linearized Fokker-Plank equation, and the result is shown to be equivalent to the system of coupled oscillators. The correspondence between the quantities from the mechanical picture and the steady-state fluctuations is established. Explicit calculation is given for auto-regulatory networks in which, the force constant associated with the degree of protein is very weak. Negative feedback not only suppresses the fluctuations but also increases the steepness of the potential. The results for the fluctuations agree completely with those obtained from linear noise Fokker-Planck equation.

💡 Research Summary

The paper presents a theoretical framework that describes gene‑regulatory networks near a stable steady‑state in terms of a generalized potential energy landscape. Starting from the stochastic description of transcription and translation, the authors derive a Fokker‑Planck equation for the probability density of molecular concentrations. By linearizing the dynamics around the deterministic fixed point, the drift term becomes a linear function characterized by a Jacobian matrix A, while the intrinsic biochemical noise is captured by a diffusion matrix D. The steady‑state solution of this linearized Fokker‑Planck equation is a multivariate Gaussian distribution whose covariance matrix Σ satisfies the Lyapunov equation AΣ + ΣAᵀ = 2D.

Recognizing that any Gaussian can be written as ρ_ss(x) ∝ exp