A stochastic spectral analysis of transcriptional regulatory cascades

TL;DR

This paper introduces a spectral method for analyzing the joint probability distributions in gene regulatory networks, enabling efficient modeling of noise and information flow in biological cascades.

Contribution

The spectral method provides a novel, efficient way to compute joint distributions in gene networks, facilitating analysis and optimization of regulatory cascades.

Findings

Strong regulations convert unimodal inputs to bimodal outputs

Multimodal inputs are not more informative than bimodal ones

Up-regulation chains outperform down-regulation chains

Abstract

The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here we introduce the spectral method for computation of the joint probability distribution over all species in a biological network. The spectral method exploits the natural eigenfunctions of the master equation of birth-death processes to solve for the joint distribution of modules within the network, which then inform each other and facilitate calculation of the entire joint distribution. We illustrate the method on a ubiquitous case in nature: linear regulatory cascades. The efficiency of the method makes possible numerical optimization of the input and regulatory parameters, revealing design properties of, e.g., the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of up-regulations outperforms a chain of down-regulations. We anticipate that this numerical approach may be useful for modeling noise in a variety of small network topologies in biology.