A Mechanism for Turing Pattern Formation with Active and Passive Transport

Abstract

We propose a novel mechanism for Turing pattern formation that provides a possible explanation for the regular spacing of synaptic puncta along the ventral cord of $C.~elegans$ during development. The model consists of two interacting chemical species, where one is passively diffusing and the other is actively trafficked by molecular motors. We identify the former as the kinase CaMKII and the latter as the glutamate receptor GLR-1. We focus on a one-dimensional model in which the motor-driven chemical switches between forward and backward moving states with identical speeds. We use linear stability analysis to derive conditions on the associated nonlinear interaction functions for which a Turing instability can occur. We find that the dimensionless quantity $\gamma={\alpha d}/{v^2}$ has to be sufficiently small for patterns to emerge, where $\alpha$ is the switching rate between motor states, $v$ is the motor speed, and $d$ is the passive diffusion coefficient. One consequence is that patterns emerge outs...