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Model

ToDo

  • Meshes
  • Morphomovie
  • More details of: Finite volume methods, and time discretisation/Euler-Maruyama

Reaction-Diffusion

LimbNET simulates systems of Reaction-Diffusion equations. For any given species, \(u\), the Reaction-Diffusion dynamics are specified as (Stochastic) Partial Differential Equations (SPDEs) of the form

\[ \frac{\partial u}{\partial t} = f(u, \mathbf{x}, t) + D_u \nabla^2 u + \eta(u, t) \]

Where

  • \(u\) is a single species;
  • \(t\) is time;
  • \(\mathbf{x}\) is our spatial coordinate, consisting of two spatial dimensions \(x\) and \(y\) in the standard 2D morphomovie;
  • \(D_u\) is the diffusivity (diffusion coefficient) of an individual species \(u\);
  • and \(\eta(u, t)\) is a noise term for species \(u\), such that \(\eta(u, t) = \nu(u, t) W_t^u\), where \(\nu(u, t)\) is a function (which may be constant) that defines the amount of multiplicative noise (intrinsic noise) in the system and \(W_t^u\) represents a random Wiener process (a one-dimensional Brownian motion), for species \(u\).

More generally, since we tend to model multiple species \(u\) interacting simultaneously, we actually specify a system of (coupled) SPDEs of the form

\[ \frac{\partial \mathbf{u}}{\partial t} = f(\mathbf{u}, \mathbf{x}, t) + \mathbf{D} \nabla^2 \mathbf{u} + \mathbf{\eta}(\mathbf{u}, t) \]

Where

  • \(\mathbf{u}\) is a vector of all reactants/species;
  • \(t\) is time;
  • \(\mathbf{x}\) is our spatial coordinate;
  • \(\mathbf{D}\) is a diagonal matrix of all per-species diffusivities, \(D_u\);
  • and \(\mathbf{\eta}(\mathbf{u}, t)\) is a vector of all per-species noise terms \(\eta(u,t)\) as defined above.

Simulation of Reaction-Diffusion

The simulation currently proceeds according to an explicit fixed-time-step scheme. An adaptive time-step discretisation is in process of being implemented.

Simulation of the Reaction step

The Euler-Maruyama scheme is employed to integrate the reaction component of the RHS of the system of SPDEs per time step.

Simulation of the Diffusion step

The mesenchyme domain is discretised into triangular elements and diffusion is simulated per time step using a finite volume method. The ectoderm is likewise discretised into linear elements over which diffusion is simulated per time step using a finite volume method.