# Maths

## Reaction-Diffusion

The system of PDEs that we simulate in each compartment is the two-dimensional reaction-diffusion equation:

where

\(c_s\) is the concentration of species \(s\) at position \((x, y)\) and time \(t\)

\(D_s\) is the diffusion constant for species \(s\)

\(R_s\) is the reaction term for species \(s\)

and we assume that

the diffusion constant \(D_s\) is a scalar that does not vary with position or time

the reaction term \(R_s\) is a function that can depend on the concentrations of other species in the model, but only locally, i.e. the concentrations at the same spatial coordinate.

Note

This is equivalent to simulating a 3-d system with no spatial variation in the z-direction. In our simulations we then assume that we are simulating a 2-d slice of such a 3-d system with unit length in the z-direction (i.e. the system has extent 1 in the length units of our model in the z-direction). This allows the user to use the usual 3-d units for concentration, etc.

## Compartment Reactions

Compartment reaction terms correspond to the \(R_s\) term in the reaction-diffusion equation, and describe the rate of change of species concentration with time, and are evaluated at every point inside the compartment

## Membrane reactions

Membrane reactions are reactions that occur on the membrane between two compartments, and describe the species amount that crosses the membrane per unit membrane area per unit time.

## Boundary Conditions

All boundaries have “zero-flux” Neumann boundary conditions, whether they are boundaries between two compartments or boundaries between a compartment and the outside (except for the flux caused by any membrane reactions).