When many noisy genes optimize information flow

TL;DR

The paper asks how much information a single transcription factor can convey about its concentration when regulating many target genes under a fixed molecular budget. It analyzes a broadcasting regulatory network within an information-theoretic framework, incorporating Poisson mRNA noise and Berg-Purcell arrival fluctuations, and derives how the optimal input distribution and network parameters shape information transmission. Key findings show that information capacity is maximized by distributing resources across many targets, biasing the input toward low $c$, and allowing per-gene noise; the parameter landscape is 'sloppy', enabling near-optimal performance without fine-tuning, and the results are extended with a Boltzmann-like ensemble interpretation. This work provides a principled explanation for the coexistence of noisiness and near-optimal information processing in biology and suggests that diverse, ensemble-like parameter settings can underpin robust information transmission under resource constraints.

Abstract

It often is emphasized that gene expression is noisy. A seemingly contradictory view is that control mechanisms have been optimized to squeeze as much information as possible out of a limited number of molecules. Here we revisit these issues in a simple model where a single transcription factor (TF) controls a large number of target genes. We include only the physically required noise sources: random arrival of TFs at their targets and counting noise in the synthesis and degradation of mRNA. If the cell has a limited total number of mRNA molecules, then the capacity to transmit information about TF concentration is maximized when these resources are distributed across the largest possible number of target genes. To realize this capacity the distribution of TF concentrations must be biased toward smaller values. Thus, in some limits, information transmission is optimized when individual expression levels are noisy. In addition, the dependence of information transmission on the parameters of this multi-gene system has a "sloppy" spectrum, so that optimal performance can co-exist with substantial variability.

Figures (5)

• Figure 1: A schematic of the broadcasting model. One input species, a transcription factor at concentration $c$, independently regulates the expression levels $\{g_i\}$ of $M$ output genes ($i = 1,\, 2,\, \cdots ,\, M$).
• Figure 2: Schematic input/output relation and noise associated with one arrow in Fig \ref{['fig:broadcast']}. Mean expression level $\bar{g} (c)$ as a function of the transcription factor concentration $c$ (solid blue) $\pm$ one standard deviation $\sigma_g(c)$ (dashed red). Expression is normalized so the maximal mean value is $\bar{g} = 1$ and TF concentration is in units of $c_0$ from Eq (\ref{['c0']}). We mark the scale of half--maximal activation $K/c_0 = 0.5$ and the sensitivity $n=4$ from Eq (\ref{['hill']}). Orthogonal error bars show how noise in expression (black) is equivalent to an error in input concentration $\sigma_c(g)$ (red).
• Figure 3: The partition function as a function of the dimensionless maximal input concentration. (A) $\mathcal{Z}_{\rm opt}(C,M)/\sqrt{N_{\rm max}}$ for networks with different numbers of genes. (B) $\mathcal{Z}_{\rm opt}(C,M)/\sqrt{N_{\rm max}M}$ for each network, shown on a log-log scale. The curves collapse onto $\mathcal{Z_{\rm opt}}\propto \sqrt{M} C^{1/4}$.
• Figure 4: Tiling the concentration axis. Functions $F_i(c/c_0)$ from Eq (\ref{['Fix']}); parameters $\{K_i,n_i\}$ are set to their optimal values for $M=20$ and $C=50$ (thin colored curves). The sum (solid black line) determines the effective noise level $\sigma_c(c)$ from Eq (\ref{['sumF']}). For comparison, if the effective noise level is given by the Berg-Purcell limit in Eq (\ref{['BP']}), we expect the sum to be proportional to $1/c$ (dashed red line).
• Figure 5: Spectrum of the Hessian matrix. Eigenvalues $\{\lambda_\mu\}$ of $\hat{H}$ from Eq (\ref{['HD']}) in a model with $M=25$ genes and $c_{\rm max}/c_0 = 50$.