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

TL;DR

The Gaussian macroscopic fluctuation theory is applied to gene regulatory networks with negative feedback, and an explicit internal-external noise decomposition of the power spectral density for any networks, including cross-correlations is derived.

Abstract

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.

Figures (3)

• Figure 1: Linking the stationary fluctuations and responses far-from-equilibrium.
• Figure 2: (a): Sketch of a simple transcription-translation process from gene to mRNAs to proteins with negative feedback control modeled by, $w_{+1}=k_m/[1+(\mathcal{C}_2/c_0)^H]$. (b): fixed points $c^*_1$ and $c^*_2$; (c): $\Psi(c^*_2)$ for the negative feedback; (d): mRNA scaled PSD $\varphi_1$; (e): protein scaled PSD $\varphi_2$; (f): mRNA-proteins PSD covariance $Z_{12}(0)$. Arrows denotes the direction of increasing parameter $H=0.1, 0.5, 1,2$. For calculations we used: $\tau_1=1$, $\tau_2 = 5$, $c_0 = 1$, $k_m = 1$.
• Figure 3: The parameter $\mathcal{R}$ defined in \ref{['eq:mathcal-r']} as a function of $c_B$. The arrow denotes direction of the increasing $\Omega=50,100,200,1000$. Parameters: $c_A=k_{\pm 1}=k_{\pm 2}=1$, $n_c=2\Omega$.