Maths

Reaction-Diffusion

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

\[\frac{\partial c_s}{\partial t} = D_s \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) c_s + R_s\]

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. They are evaluated on the membrane, and can be implemented as interface conditions between the PDEs in neighbouring compartments, or as additional reaction terms localised on these membranes.

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).