# Cellular Potts model

In computational biology, cellular Potts model (CPM) is a computational model of the collective behavior of cellular structures. It allows modelling of many phenomena, such as cell migration, clustering, and growth taking adhesive forces, environment sensing as well as volume and surface-area constraints into account. The first CPM was proposed for the simulation of cell sorting by Graner and Glazier as a modification of a large-Q Potts model.[1] Although the model was developed to model biological cells it can also be used to model individual parts of a biological cell, or even regions of fluid.

## Model description

The CPM works on a rectangular Euclidean lattice where it represents each cell as a subset of lattice sites sharing the same cell ID (analogical to spin in Potts models in physics). In order to evolve the model Metropolis-style updates are performed, that is,

1. choose a lattice site and propose a new cell ID to be assigned to it, and
2. decide if to accept or reject this change based on an energy function called the Hamiltonian.

### The Hamiltonian

The Hamiltonian is a central component of every CPM. It is determined by the configuration of the cell lattice. A basic Hamiltonian proposed by Graner and Glazier included adhesion energies and volume constraints:

{\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 _{i}\left(v(\sigma _{i})-V(\sigma _{i})\right)^{2}.\\\end{aligned}}}

Where i, j are lattice sites, σi is the cell at site i, τ(σ) is the cell type of cell σ, J is the boundary 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, λ is a Lagrange multiplier determining the strength of the volume constraint.

The Hamiltonian can be modified to control cell behaviors such as chemotaxis, elongation and haptotaxis by using other sub-lattices containing information such as the concentrations of chemicals.

## Extensions

Over time, the CPM has evolved from a specific model to a general framework with many extensions and even related methods that are entirely or partially off-lattice.[citation needed]

### 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.[2] Open source software Bionetsolver can be used to integrate intracellular dynamics with CPM algorithm.[3]