cellular Potts model

File:Cpm layout.pdf

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:

1. choose a random lattice site {{math|i}}
2. choose a random neighboring lattice site {{math|j}} to copy its ID into {{math|i}}.
3. calculate the difference in energy ($\backslash Delta\; H$) between the original and the proposed new configuration.
4. accept or reject this copy event based on the change in energy $\backslash Delta\; H$, as follows:
5. : if the new energy is lower, always accept the copy;
6. : if the new energy is higher, accept the copy with probability $e^\{-\backslash Delta\; H\; /\; T\}$ (the Boltzmann temperature {{math|T}} determines the likelihood of energetically unfavorable fluctuations).

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 {{math|i}}, {{math|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.

Over time, the CPM has evolved from a specific model of cell sorting to a general framework with many extensions, some of which are partially or entirely off-lattice.{{cite book|last1=Balter|first1=Ariel|last2=Merks|first2=Roeland M.H.|last3=Popławski|first3=Nikodem J.|last4=Swat|first4=Maciej|last5=Glazier|first5=James A.|title=Single-Cell-Based Models in Biology and Medicine| chapter=The Glazier-Graner-Hogeweg model: extensions, future directions, and opportunities for further study|series=Mathematics and Biosciences in Interaction|year=2007|pages=151–167|doi=10.1007/978-3-7643-8123-3_7|isbn=978-3-7643-8101-1|chapter-url=https://link.springer.com/chapter/10.1007/978-3-7643-8123-3_7 }} Various cell behaviours, such as chemotaxis, elongation and haptotaxis can be incorporated by extending either the Hamiltonian, H, or the change in energy $\backslash Delta\; H$. Auxiliary sub-lattices may be used to include additional spatial information, such as the concentrations of chemicals.

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 {{math|j}} into site {{math|i}} when the chemokine concentration is higher at {{math|j}}. This is done by modifying the change in energy $\backslash Delta\; H$ with a term that is proportional to the difference in concentration at {{math|i}} and {{math|j}}:

\begin{align}

\Delta H'=\Delta H-\mu(C_i-C_j)\\

\end{align}

Where $\backslash mu$ is the strength of chemotactic movement, and $C\_i$ and $C\_j$ 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.

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.{{Cite journal|title = Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution|last1 = Szabó|first1 = A|date = 2013|journal = Frontiers in Oncology|doi = 10.3389/fonc.2013.00087|pmid = 23596570|last2 = Merks|first2 = RM|volume=3|page = 87|pmc = 3627127|doi-access = free}} Open source software Bionetsolver can be used to integrate intracellular dynamics with CPM algorithm.{{Cite journal|title = Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion|last1 = Andasari|first1 = Vivi|date = 2012|journal = PLOS ONE|doi = 10.1371/journal.pone.0033726|pmid = 22461894|last2 = Roper|first2 = Ryan T|last3 = Swat|first3 = Maciej H|last4 = Chaplain|first4 = MA|bibcode = 2012PLoSO...733726A|volume=7|issue = 3|pages=e33726|pmc=3312894|doi-access = free}}

The Cellular Potts Model (CPM)—also known as the Glazier-Graner-Hogeweg (GGH) model—is widely used in computational biology to simulate multicellular systems. It allows modeling of cell shape, movement, adhesion, growth, and interactions with the environment, making it highly suitable for studying tissue morphogenesis, cancer, cell sorting, and pattern formation. Below is a table summarizing key biological applications of CPM along with representative references.

The following frameworks allow users to apply the CPM to multicellular model construction.

{{Reflist}}

• {{cite journal |first1=Nan |last1=Chen |first2=James A. |last2=Glazier |first3=Jesus A. |last3=Izaguirre |first4=Mark S. |last4=Alber |year=2007 |title=A parallel implementation of the Cellular Potts Model for simulation of cell-based morphogenesis |journal=Computer Physics Communications |volume=176 |issue=11–12 |pages=670–681 |doi=10.1016/j.cpc.2007.03.007 |pmid=18084624 |bibcode = 2007CoPhC.176..670C |pmc=2139985 }}

• [http://biocomplexity.indiana.edu/jglazier/ James Glazier] (professional website)
• CompuCell3D, a CPM simulation environment: [http://sourceforge.net/projects/compucell/ Sourceforge]
• [https://simtk.org/home/compucell3d/ SimTK]
• [https://web.archive.org/web/20061209220108/http://www.nd.edu/~lcls/compucell/linux.htm Notre Dame development site]
• [http://panmental.de/ALifeXIflag Artificial Life model of multicellular morphogenesis with autonomously generated gradients for positional information using the Cellular Potts model]
• Stochastic cellular automata

{{DEFAULTSORT:Cellular Potts Model}}

Category:Statistical mechanics

Category:Lattice models

Category:Stochastic models