Stochastic Turing pattern formation in a model with active and passive transport

TL;DR

This paper explores how stochastic effects influence Turing pattern formation in a reaction-diffusion-advection model relevant to synaptogenesis, revealing that noise can extend the conditions under which patterns spontaneously emerge.

Contribution

It introduces a stochastic RDA model with a linear noise approximation, extending the concept of Turing instability to include stochastic power spectrum analysis.

Findings

Noise extends the parameter range for pattern formation.

A stochastic Turing instability can be identified via power spectrum peaks.

The model links molecular transport dynamics to pattern emergence in neural development.

Abstract

We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction diffusion advection (RDA) equation, which was previously introduced to model synaptogenesis in \textit{C. elegans}. The model describes the interactions between a passively diffusing molecular species and an advecting species that switches between anterograde and retrograde motor-driven transport (bidirectional transport). Within the context of synaptogenesis, the diffusing molecules can be identified with the protein kinase CaMKII and the advecting molecules as glutamate receptors. The stochastic dynamics evolves according to an RDA master equation, in which advection and diffusion are both modeled as hopping reactions along a one-dimensional array of chemical compartments. Carrying out a linear noise approximation of the RDA master equation leads to an effective Langevin equation, whose power spectrum provides a means of extending the definition of a Turing instability to stochastic systems, namely, in terms of the existence of a peak in the power spectrum at a non-zero spatial frequency. We thus show how noise can significantly extend the range over which spontaneous patterns occur, which is consistent with previous studies of RD systems.