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.

Figure 1

Schematic figure showing distribution of synaptic punta along ventral cord of early and late stage C. elegans. New synapses are inserted during development to maintain the synaptic density.

Figure 2

Regulation of transport and delivery of GLR-1 to synapses by CaMKII. (a) Calcium influx through voltage-gated calcium channels activates CaMKII, which enhances the active transport and delivery of GLR-1 to synapses. (b) Under conditions of increased excitation, higher calcium levels results in constitutively active CaMKII which fails to localize at synapses, leading to the removal of GLR-1 from synapses.

Figure 3

Numerical simulations showing temporal evolution of the concentrations $C,R_{\pm }$ on a slowly growing domain. (a) $t=0\text{s}$; (b) $t=100\text{s}$; (c) $t=850\text{s}$; (d) $t=1300\text{s}$; (e) $t=1400\text{s}$; (f) $t=1500\text{s}$. Initial concentrations are taken to be random fluctuations about steady-state values. A spatially periodic pattern is becoming established with five potential synapse sites by $t=100$ s, and as the domain continues to grow, the synapse sites begin to spread out. At $t=1300$ s, we can see the beginnings of pattern reorganization between the synapse sites as CaMKII concentrations start to grow there. New peaks are inserted after the second, third, and fourth existing synapse sites by $t=1400$ s; the beginnings of a new CaMKII peak can also be observed between the first and second peak. Numerical simulations were performed using a combination of Crank-Nicolson and Lax-Wendroff schemes, with no flux boundary conditions. Full solutions were computed in the Lagrangian coordinate system, and then converted back into physical coordinates. We take space step $dx=0.05$ and time step $dt=0.025$. The 1D domain grows with growth parameters $r=0.001\mu \mathrm{m}/\mathrm{s}$ and ${\mathrm{\Lambda }}_0=10$. Other parameters are ${\rho }_1=1/\mathrm{s}$, ${\rho }_2=1/(\mu M\mathrm{s}$), ${\mu }_1=0.25/\mathrm{s}$, ${\mu }_2=1/\mathrm{s}$, $\alpha =0.11/\mathrm{s}$, $D=0.01\mu {\mathrm{m}}^2/\mathrm{s}$, $\gamma =0.02\mu M/\mathrm{s}$, and $v=1\mu \mathrm{m}/\mathrm{s}$.

Figure 4

Space-time plot showing the insertion of new potential synaptic sites as the domain representing a section of the ventral cord grows over the course of 2.5 h. The horizontal axis represents position along the C. elegans ventral cord and the vertical axis represents time in seconds. Colors represent local concentration of GLR-1 (a combined total of both leftward and rightward trafficking species) in $\mu \mathrm{M}$. Areas of high concentration represent potential synapse sites. Numerical simulations and parameter values are as in Fig. 3.