Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond

TL;DR

This paper reviews the mathematical modeling of stochastic effects in biological systems, focusing on pattern formation and cell polarity, using the linear noise approximation and beyond to analyze noise-induced phenomena.

Contribution

It introduces a formalism for stochastic modeling in biology and applies it to analyze noise-driven pattern formation and cell polarity phenomena.

Findings

Stochastic amplification of Turing instabilities leads to observable patterns.

Analytic progress is made in understanding spontaneous cell polarity.

The linear noise approximation effectively explains certain noise-induced biological phenomena.

Abstract

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically-inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.