The CPM consists of a rectangular
Euclidean lattice, where each cell is a
subset of lattice sites sharing the same
cell ID (analogous to spin in Potts models in physics). Lattice sites that are not occupied by cells are the medium. The dynamics of the model are governed by an energy function: the
Hamiltonian which describes the energy of a particular configuration of cells in the lattice. In a basic CPM, this energy results from adhesion between cells and resistance of cells to volume changes. The algorithm for updating CPM
minimizes this energy. In order to evolve the model
Metropolis-style updates are performed, that is: • choose a random lattice site • choose a random neighboring lattice site to copy its ID into . • calculate the difference in energy (\Delta H ) between the original and the proposed new configuration. • accept or reject this copy event based on the change in energy \Delta H , as follows: • : if the new energy is lower, always accept the copy; • : if the new energy is higher, accept the copy with probability e^{-\Delta H / T} (the
Boltzmann temperature determines the likelihood of energetically unfavorable fluctuations).
The Hamiltonian The original model proposed by Graner and Glazier contains cells of two types, with different adhesion energies for cells of the same type and cells of a different type. Each cell type also has a different contact energy with the medium, and the cell volume is assumed to remain close to a target value. The Hamiltonian is formulated as: \begin{align} H = \sum_{i,j\text{ neighbors}}J\left(\tau(\sigma_i),\tau(\sigma_j)\right)\left(1-\delta(\sigma_i,\sigma_j)\right) + \lambda\sum_{\sigma_i}\left(v(\sigma_i)- V(\sigma_i)\right)^2,\\ \end{align} where , are lattice sites, σi is the cell at site i, τ(σ) is the cell type of cell σ, J is the coefficient determining the adhesion between two cells of types τ(σ),τ(σ'), δ is the
Kronecker delta, v(σ) is the volume of cell σ, V(σ) is the target volume, and λ is a
Lagrange multiplier determining the strength of the volume constraint. Cells with a lower J value for their membrane contact will stick together more strongly. Therefore, different patterns of cell sorting can be simulated by varying the J values. == Extensions ==