Turing mechanism for homeostatic control of synaptic density during C. elegans growth

We propose a mechanism for the homeostatic control of synapses along the ventral cord of Caenorhabditis elegans during development, based on a form of Turing pattern formation on a growing domain. C. elegans is an important animal model for understanding cellular mechanisms underlying learning and memory. Our mathematical model consists of two interacting chemical species, where one is passively diffusing and the other is actively trafficked by molecular motors, which switch between forward and backward moving states (bidirectional transport). This differs significantly from the standard mechanism for Turing pattern formation based on the interaction between fast and slow diffusing species. We derive evolution equations for the chemical concentrations on a slowly growing one-dimensional domain, and use numerical simulations to demonstrate the insertion of new concentration peaks as the length increases. Taking the passive component to be the protein kinase CaMKII and the active component to be the glutamate receptor GLR-1, we interpret the concentration peaks as sites of new synapses along the length of C. elegans, and thus show how the density of synaptic sites can be maintained.