Fluctuations in Gene Regulatory Networks as Gaussian Colored Noise

The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. Firstly, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulae for the variances and covariance of molecular concentrations for a regulatory network near a stable equilibrium explicitly. Two examples, the gene auto-regulatory network and the toggle switch, are presented in details. In general, it is found that the finite correlation time of noise reduces the fluctuations and enhances the correlation between the fluctuations of the molecular components.

💡 Research Summary

The paper extends the stochastic analysis of gene regulatory networks by incorporating Gaussian colored noise, which possesses a finite correlation time, rather than the conventional white‑noise assumption. The authors begin by formulating the dynamics of a regulatory network near a stable steady state as a set of Langevin equations. To model colored noise, they adopt an Ornstein‑Uhlenbeck process for each stochastic term, characterized by a covariance function ⟨η_i(t)η_j(t′)⟩ = D_{ij} exp(−|t−t′|/τ), where τ denotes the noise correlation time. By augmenting the original state vector with the noise variables, the non‑Markovian system is transformed into an enlarged Markovian system. The solution of the resulting linear stochastic differential equation is expressed as an Itô integral:

x(t) = Φ(t)x(0) + ∫₀ᵗ Φ(t−s) B dW(s),

where Φ(t)=e^{At} is the matrix exponential of the deterministic Jacobian A, B is the noise‑coupling matrix, and W(s) is a standard Wiener process.

Two lemmas concerning Itô integrals are then proved. Lemma 1 states that the variance of a single Itô integral equals the time integral of the squared integrand, i.e., Var