Macroscopic fluctuation-response theory and its use for gene regulatory networks

Gaussian macroscopic fluctuation theory underpins the understanding of noise in a broad class of nonequilibrium systems. We derive exact fluctuation-response relations linking the power spectral density of stationary fluctuations to the linear response of stable nonequilibrium steady states. Both of these can be determined experimentally and used to reconstruct the kernel of the linearized dynamics and the diffusion matrix, and thus any features of the Gaussian theory. We apply our theory to gene regulatory networks with negative feedback, and derive an explicit internal-external noise decomposition of the power spectral density for any networks, including cross-correlations.

💡 Research Summary

The authors develop a comprehensive macroscopic fluctuation‑response framework for systems whose stochastic dynamics can be approximated by a weak‑noise Gaussian process. Starting from an N‑dimensional stochastic field x(t) that concentrates around a deterministic trajectory X(t) in the limit of vanishing noise amplitude ε, they linearize the dynamics around a stable fixed point x* to obtain the Ornstein‑Uhlenbeck equation
dx/dt = K (x−x*) + √ε η(t),
where K is the Jacobian of the deterministic flow at the fixed point and η(t) is a zero‑mean Gaussian white noise with covariance matrix D.

In the stationary regime the power spectral density (PSD) of the fluctuations is known to be
Z(ω) = (K−iωI)⁻¹ D