Synaptic pattern formation during cellular recognition

The plan is to

1. Describe the variables present via their domains and images
2. Describe the equations, ideally

Constants

$$\{\begin{array}{rl}\Omega & \text{Set of Molecules available}\\ k_{B}T& \text{Thermal energy at temperature T}\\ D_j& \text{Diffusion coefficient for the jth membrane protein in the appropriate membrane}\\ {\lambda }_j& \text{Curvature of binding energy well for jth complex}\\ j& \text{I think this is an index for membrane complexes?}\end{array}$$

Comparing the variables/constants with the equations above, it seems like the membrane complexes $j$ might be in the set $Ai$, $Bi$, $TM$, $T$. It is unclear what these mean.

Variables

$$\{\begin{array}{rll}F:& x,y,t\mapsto \mathbb{R}& \text{Force}\\ C_T:& x,y,t\mapsto \mathbb{R}& \text{TCR Concentration}\\ C_M:& x,y,t\mapsto \mathbb{R}& \text{MHC-peptide Concentration}\\ C_{Ai}:& x,y,t\mapsto \mathbb{R},i\in \Omega & \text{Concentration of the ith adhesian molecule}\\ C_{Bi}:& x,y,t\mapsto \mathbb{R},i\in \Omega & \text{Concentration of the ith complementary ligand}\\ C_{TM}:& x,y,t\mapsto \mathbb{R}& \text{Concentration of the TCR-MHC peptide complex}\\ C_i:& x,y,t\mapsto \mathbb{R},i\in \Omega & \text{Concentration of the complex formed between Ai and Bi}\\ C_{Tt}:& x,y,t\mapsto \mathbb{R}& \text{Concentration of triggered TCRs}\\ z:& x,y,t\mapsto \mathbb{R}& \text{Local intermembrane separation}\\ k_{on}:& \mathbb{R}\mapsto \mathbb{R}& \text{On rate for TCT/MHC peptide binding}\\ k_{off}:& \mathbb{R}\mapsto \mathbb{R}& \text{Off rate for TCT/MHC peptide binding}\\ k_i:& \mathbb{R}\mapsto \mathbb{R},i\in \Omega & \text{On rate for the complexation of Ai and Bi}\\ k_{-i}:& \mathbb{R}\mapsto \mathbb{R},i\in \Omega & \text{Off rate for the complexation of Ai and Bi}\\ z:& x,y,z\mapsto \mathbb{R}& \text{Local intermembrane separation}\\ z_j:& ???\mapsto ???& \text{Natural length of jth protein complex}\\ t:& t\mapsto \mathbb{R}& \text{Time (identity function when seen as a variable)}\\ \gamma :& ???\mapsto ???& \text{Interfacial tension of cell membrane}\\ \kappa :& ???\mapsto ???& \text{Bending rigidity of cell membrane}\\ \zeta :& ???\mapsto ???& \text{Thermal noise}\\ M:& ???\mapsto ???& \text{Phenomenological constant for membrane response to free energy changes}\end{array}$$

Equations

$$\{\begin{array}{rll}F& ={\lambda }_T{\int }_{{\mathbb{R}}^2}{C}_{TM}\cdot [z-z_{TM}{]}^{2}dxdy& \\ & +{\sum }_i\frac{{\lambda }_i}{2}{\int }_{{\mathbb{R}}^2}{C}_i\cdot [z-z_i{]}^{2}dxdy& \\ & +\frac{1}{2}{\int }_{{\mathbb{R}}^2}[\gamma (\nabla z{)}^2+\kappa ({\nabla }^{2}z{)}^2]dxdy& \\ \frac{\partial }{\partial t}{C}_T& =D_T{\nabla }^2{C}_T-k_{on}(z)C_T{C}_M+k_{off}(1-P)C_{TM}+{\zeta }_T& \text{What is P ? Shouldn’t we have koff(z) instead?}\\ \frac{\partial }{\partial t}{C}_M& =D_M{\nabla }^2{C}_M-k_{on}(z)C_T{C}_M+k_{off}{C}_{TM}+{\zeta }_M& \text{Shouldn’t koff depend on z ?}\\ \frac{\partial }{\partial t}{C}_{Ai}& =D_{Ai}{\nabla }^2{C}_{Ai}-k_i(z)C_{Ai}{C}_{Bi}+k_{-i}{C}_i+{\zeta }_{Ai}& \text{What is ζAi ? Shouldn’t k−i depend on z ?}\\ \frac{\partial }{\partial t}{C}_{Bi}& =D_{Bi}{\nabla }^2{C}_{Bi}-k_i(z)C_{Ai}{C}_{Bi}+k_{-i}{C}_i+{\zeta }_{Bi}& \text{What is ζBi ? Shouldn’t k−i depend on z ?}\\ \frac{\partial }{\partial t}{C}_{TM}& =D_{TM}\left[{\nabla }^2{C}_{TM}+\frac{1}{k_{B}T}\nabla \cdot C_{TM}\nabla \frac{\partial }{\partial C_{TM}}F\right]+k_{on}(z)C_T{C}_M-k_{off}{C}_{TM}+{\zeta }_{TM}& \text{How is ∂CTM∂F calculated?}\\ \frac{\partial }{\partial t}{C}_i& =D_i\left[{\nabla }^2{C}_i+\frac{1}{k_{B}T}\nabla \cdot C_i\nabla \frac{\partial }{\partial C_i}F\right]+k_i(z)C_{Ai}{C}_{Bi}-k_{-i}{C}_i+{\zeta }_i& \\ \frac{\partial }{\partial t}{C}_{Tt}& =D_T{\nabla }^2{C}_{Tt}+P{k}_{off}{C}_{TM}-k_t{C}_{Tt}+{\zeta }_{Tt}& \text{What is P ? What is ζTt ?}\\ \frac{\partial }{\partial t}z& =-M\frac{\partial }{\partial z}F+\zeta & \end{array}$$

I assume that:

• All the “unknown” ${\zeta }_X$ are independent noise functions
• $P\in [0,1]$ is a constant