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:

${\displaystyle {\begin{aligned}H=\sum _{i,j{\text{ neighbors}}}J\left(\tau (\sigma _{i}),\tau (\sigma _{j})\right)\left(1-\delta (\sigma _{i},\sigma _{j})\right)+\lambda \sum _{\sigma _{i}}\left(v(\sigma _{i})-V(\sigma _{i})\right)^{2},\\\end{aligned}}}$

where i, j are lattice sites, σ_i 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.

Chemotaxis

In CPM, cells can be made to move in the direction of higher chemokine concentration, by increasing the probability of copying the ID of site j into site i when the chemokine concentration is higher at j. This is done by modifying the change in energy ${\displaystyle \Delta H}$ with a term that is proportional to the difference in concentration at i and j:^[2]

${\displaystyle {\begin{aligned}\Delta H'=\Delta H-\mu (C_{i}-C_{j})\\\end{aligned}}}$

Where ${\displaystyle \mu }$ is the strength of chemotactic movement, and ${\displaystyle C_{i}}$ and ${\displaystyle C_{j}}$ are the concentration of the chemokine at site i and j, respectively. The chemokine gradient is typically implemented on a separate lattice of the same dimensions as the cell lattice.

Multiscale and hybrid modeling using CPM

Core GGH (or CPM) algorithm which defines the evolution of the cellular level structures can easily be integrated with intracellular signaling dynamics, reaction diffusion dynamics and rule based model to account for the processes which happen at lower (or higher) time scale.^[4] Open source software Bionetsolver can be used to integrate intracellular dynamics with CPM algorithm.^[5]