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.