Fluctuation theory in space and time: white noise in reaction-diffusion models of morphogenesis

TL;DR

This paper develops a coarse-grained approach to estimate finite fluctuations in reaction-diffusion models of morphogenesis, addressing the infinite variance issue in traditional stochastic equations and enabling practical predictions of fluctuation magnitudes.

Contribution

It introduces a novel coarse-graining method to quantify finite fluctuations in reaction-diffusion systems, avoiding ad hoc microscopic correlations.

Findings

Fluctuations are finite when estimated at a relevant mesoscopic scale.

The approach successfully models positional information encoding in biophysical systems.

Numerical methods are provided for practical fluctuation prediction.

Abstract

The precision of reaction-diffusion models for mesoscopic physical systems is limited by fluctuations. To account for this uncertainty, Van Kampen derived a stochastic Langevin-like reaction-diffusion equation that incorporates spatio-temporal white noise. The resulting solutions, however, have infinite standard deviation. Ad hoc modifications that address this issue by introducing microscopic correlations are inconvenient in many physical contexts of wide interest. We instead estimate the magnitude of fluctuations by coarse-graining solutions of the Van Kampen equation at a relevant mesoscopic scale. The ensuing theory yields fluctuations of finite magnitude. Our approach is demonstrated for a specific biophysical model--the encoding of positional information. We discuss the properties of the fluctuations and the role played by the macroscopic parameters of the underlying reaction-diffusion model. The analysis and numerical methods developed here can be applied in physical problems to predict the magnitude of fluctuations. This general approach can also be extended to other classes of dynamical systems that are described by partial differential equations.