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 .

Model description

The CPM works on a rectangular Euclidean lattice where it represents each cell as a subset of the lattice sites with the same cell ID' (analogical to spin in Potts models in physics). In general, a "cell" does not have to represent an entire biological cell but also a part of a biological cell, or even a region of fluid.

The CPM evolves by updating the cell lattice one site at a time based on a set of probabilistic rules which have to:

1. select the next putative update of some site, and
2. decide if to accept or reject the update.

This decision is made based on a an energy function called the Hamiltonian.

The Hamiltonian

The central component of the CPM is the definition of the Hamiltonian. The Hamiltonian is determined by the configuration of the cell lattice and perhaps other sub-lattices containing information such as the concentrations of chemicals. The original CPM Hamiltonian included adhesion energies, and volume and surface area constraints. We present a simple example for illustration:

${\begin{aligned}H=&\sum _{i,j{\text{ neighbors}}}J\left(\tau (\sigma (i)),\tau (\sigma (j))\right)\left(1-\delta (\sigma (i),\sigma (j))\right)\\+&\sum _{i}\lambda _{\text{volume}}[V(\sigma (i))-V_{\text{target}}(\sigma (i))]^{2}\\+&\sum _{i}\lambda _{\text{surface}}[S(\sigma (i))-S_{\text{target}}(\sigma (i))]^{2}.\\\end{aligned}}$

Where for cell σ, λ_volume is the volume constraint, V_target is the target volume, and for neighboring lattice sites i and j, J is the boundary coefficient between two cells (σ,σ') of given types τ(σ),τ(σ'), and the boundary energy coefficients are symmetric: J[τ(σ),τ(σ')]=J[τ(σ'),τ(σ)], and the Kronecker delta is δ_(x,y)={1,x=y; 0,x≠y}.

The Hamiltonian can also control cell behaviors including chemotaxis, elongation and haptotaxis.

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.

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. Open source software Bionetsolver can be used to integrate intracellular dynamics with CPM algorithm.

References

• Graner, François; Glazier, James A. (1992). "Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model". Phys. Rev. Lett. 69 (13): 2013–2016. Bibcode:1992PhRvL..69.2013G. doi:10.1103/PhysRevLett.69.2013.
• Chen, Nan; Glazier, James A.; Izaguirre, Jesus A.; Alber, Mark S. (2007). "A parallel implementation of the Cellular Potts Model for simulation of cell-based morphogenesis". Computer Physics Communications. 176 (11–12): 670–681. Bibcode:2007CoPhC.176..670C. doi:10.1016/j.cpc.2007.03.007.

External links

• James Glazier (professional website)
• CompuCell3D, a CPM simulation environment: Sourceforge
• SimTK
• Notre Dame development site
• Artificial Life model of multicellular morphogenesis with autonomously generated gradients for positional information using the Cellular Potts model
• Stochastic cellular automata