Documentation for EpithelialDynamics1D.

This is a package for simulating epithelial dynamics in one dimension, supporting cell migration and cell proliferation, implementing the model in Baker et al. (2019). In this model, cells are represented as intervals between points, with the $i$th cell given by $(x_i, x_{i+1})$, $i=1,\ldots,n-1$. The node positions $x_i$ are governed by the differential equation

\[\eta\dfrac{\mathrm dx_i}{\mathrm dt} = F\left(x_i - x_{i-1}\right) - F\left(x_{i+1} - x_i\right), \quad i=2,\ldots,n-1,\]

assuming $x_1 < x_2 < \cdots < x_n$. If the left boundary is fixed, then $\mathrm dx_1/\mathrm dt = 0$, otherwise $\eta\mathrm dx_1/\mathrm dt = -F(x_2 - x_1)$. Similarly, if the right boundary is fixed then $\mathrm dx_n/\mathrm dt = 0$, otherwise $\eta\mathrm dx_n/\mathrm dt = F(x_n - x_{n-1})$. The parameter $\eta$ is called the damping constant or the viscosity coefficient and $F$ is the force law. In this model, cells are modelled as springs, whose forces are governed by this force law $F$.

The proliferation mechanism that we use assumes that at most one cell can divide at a time. Let $C_i(t)$ denote the event that the $i$th cell divides in the time interval $[t, t+\mathrm dt)$. We assume that $\Pr[C_i(t)] = G_i\,\mathrm dt$, where $G_i = G(|x_{i+1} - x_i|)$ for some proliferation law $G$. When this division occurs we place a new position at $(x_i + x_{i+1})/2$, adjusting the indices of the $x_i$ accordingly so that they remain sorted. We implement this mechanism by attempting a proliferation event every $\mathrm dt = \Delta t$ units of time, so that the probability of a proliferation event occuring at the time $t$ is $\sum_{i=1}^{n-1} G_i\Delta t$, and the probability that the $i$th cell proliferates, given that a proliferation event does occur, is $G_i/\sum_{j=1}^{n-1} G_j$. DiffEqCallbacks.jl is used to implement the callback that attempts this event periodically while simulating the problem.

Examples of how to simulate these systems are given in the sidebar.

    Modules = [EpithelialDynamics1D]