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/v 2 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.

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.

Intrinsic noise arising from the stochastic opening and closing of voltage-gated ion channels has been shown experimentally and mathematically to have important effects on a neuron's function. Study of classical neuron models with stochastic ion channels is becoming increasingly important, especially in understanding a cell's ability to produce subthreshold oscillations and to respond to weak periodic stimuli. While it is known that stochastic models can produce oscillations (quasicycles) in parameter regimes where the corresponding deterministic model has only a stable fixed point, little analytical work has been done to explore these connections within the context of channel noise. Using a stochastic hybrid Morris-Lecar (ML) model, we combine a system-size expansion in K(+) and a quasi-steady-state (QSS) approximation in persistent Na(+) in order to derive an effective Langevin equation that preserves the low-dimensional (planar) structure of the underlying deterministic ML model. (The QSS analysis exploits the fact that persistent Na(+) channels are fast.) By calculating the corresponding power spectrum, we determine analytically how noise significantly extends the parameter regime in which subthreshold oscillations occur.

Viruses and other cellular cargo that lack locomotion must rely on diffusion and cellular transport systems to navigate through a biological cell. Indeed, advances in single particle tracking have revealed that viral motion alternates between (a) diffusion in the cytoplasm and (b) active transport along microtubules. This intermittency makes quantitative analysis of trajectories difficult. Therefore, the purpose of this paper is to construct mathematical methods to approximate intermittent dynamics by effective stochastic differential equations. The coarse-graining method that we develop is more accurate than existing techniques and applicable to a wide range of intermittent transport models. In particular, we apply our method to two- and three-dimensional cell geometries (disk, sphere, and cylinder) and demonstrate its accuracy. In addition to these specific applications, we also explain our method in full generality for use on future intermittent models.

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