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 γ = αd/v2 has to be sufficiently small for patterns to emerge, where α 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 outside the parameter regime of fast switching where the model effectively reduces to a twocomponent reaction-diffusion system. Numerical simulations of the model using experimentally based parameter values generates patterns with a wavelength consistent with the synaptic spacing found in C. elegans. Finally, in the case of biased transport, we show that the system supports spatially periodic patterns in the presence of boundary forcing, analogous to flow distributed structures in reaction-diffusion-advection systems. Such forcing could represent the insertion of new motor-bound GLR-1 from the soma of ventral cord neurons.