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/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.

Catalytic Membrane Reactor Model as a Laboratory for Pattern Emergence in Reaction‐diffusion‐advection Media

“…3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 subsets, mass conservation, non-local interactions, higher codimension bifurcations and applications thereof. [87,[92][93][94][95][96][97][98][99][100][101][102][103][104][105][106][107][108] 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56…”

Multi-spike solutions of a hybrid reaction–transport model

“…Here ρ 1 , ρ 2 represent the strength of interactions, and μ 1 , μ 2 are degradation rates. Equations (2.1) are supplemented by reflecting boundary conditions at the ends x = ± L , normal∂Afalse(x,tfalse)normal∂x|x=±L=0,1emvH+false(±L,tfalse)=vH−false(±L,tfalse). (Note that, in the specific application to synaptogenesis in C. elegans , A represented CaMKII and H ± represented GLR-1 [24,25]. )…”

“…We thus obtain the condition D A ≪ D . In the particular application to synaptogenesis in C. elegans the following order of magnitude parameter values were used [24,25]: μ 1 , μ 2 , ρ 1 ∼ 1/s, ρ 2 ∼ 1/( μ M · s), α ∼ 0.1 s −1 , v ∼ 1 μ m s −1; and D A ∼ 0.01 μ m 2 s −1 . In addition, the typical spacing of synaptic puncta is 3 or 4 per 10 μ m. It is clear that, for this application, the effective diffusivity D ≈ 1 μ m 2 s −1 , and thus the inequality D A ≪ D is satisfied.…”

“…Based on experimental studies of synaptogenesis in Caenorhabditis elegans , we recently introduced a hybrid reaction–transport model for intracellular pattern formation, which involves the interaction between a passively diffusing component and an advecting component that switches between anterograde and retrograde motor-driven transport (bidirectional transport) [24,25]. These components were identified as the protein kinase CaMKII and the glutamate receptor GLR-1, respectively.…”

“…The structure of the paper is as follows. The hybrid reaction–transport model of [24,25] is introduced in §2; this is a three-component model consisting of a passively diffusing activator concentration A and a pair of actively transported inhibitor concentrations H ± , which correspond to left-moving and right-moving velocity states, respectively. In steady state, the full model is shown to be equivalent to the classical two-component GM model for ( A , H ), where H = H + + H − .…”