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| The '''cellular Potts model''' is a [[lattice model (physics)|lattice]]-based computational modeling method to simulate the collective behavior of cellular structures. Other names for the CPM are '''extended large-q Potts model''' and '''Glazier and Graner model'''. First developed by [[James Glazier]] and [[Francois Graner]] in 1992 as an extension of large-q [[Potts model]] simulations of coarsening in metallic grains and soap froths, it has now been used to simulate [[foam]], [[biological tissues]], fluid flow and [[reaction-advection-diffusion-equations]]. In the CPM a generalized "cell" is a [[simply-connected]] [[subset|domain]] of [[pixels]] with the same ''cell id'' (formerly ''[[spin (physics)|spin]]''). A generalized cell may be a single [[soap bubble]], an entire [[biological cell]], part of a biological cell, or even a region of fluid.
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| The CPM is evolved by updating the cell lattice one pixel at a time based on a set of probabilistic rules. In this sense, the CPM can be thought of as a generalized [[cellular automaton]] (''CA''). Although it also closely resembles certain [[Monte Carlo methods]], such as the large-q [[Potts model]], many subtle differences separate the CPM from Potts models and standard spin-based Monte Carlo schemes.
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| The primary rule base has three components:
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| # rules for selecting putative lattice updates
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| # a ''[[Hamiltonian (quantum mechanics)|Hamiltonian]]'' or ''effective energy'' function that is used for calculating the [[probability]] of accepting lattice updates.
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| # additional rules not included in 1. or 2..
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| The CPM can also be thought of as an ''[[agent based]]'' method in which cell agents evolve, interact via behaviors such as [[cell adhesion|adhesion]], [[cell signalling|signalling]], volume and surface area control, [[chemotaxis]] and [[cell proliferation|proliferation]]. 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|date=December 2010}}
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| 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:
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| <math>
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| \begin{align}
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| H = & \sum_{ i, j \text{ neighbors}} J\left(\tau(\sigma(i)), \tau(\sigma( j))\right) \left(1 - \delta(\sigma( i), \sigma( j))\right) \\
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| + & \sum_{ i} \lambda_\text{volume}[V(\sigma( i)) - V_\text{target}(\sigma( i))]^2\\
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| + & \sum_{ i} \lambda_\text{surface}[S(\sigma(i)) - S_\text{target}(\sigma( i))]^2 .\\
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| \end{align}
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| </math>
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| Where for cell σ, λ<sub>volume</sub> is the volume constraint, V<sub>target</sub> is the target volume, and for neighbouring 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<sub>(x,y)</sub>={1,x=y; 0,x≠y}.
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| Many extensions to the original CPM Hamiltonian control cell behaviors including [[chemotaxis]], elongation and [[haptotaxis]].
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| ==References==
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| *{{cite journal |first=François |last=Graner |first2=James A. |last2=Glazier |year=1992 |title=Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=69 |issue=13 |pages=2013–2016 |doi=10.1103/PhysRevLett.69.2013 }}
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| *{{cite journal |first=Nan |last=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 }}
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| ==External links==
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| *[http://biocomplexity.indiana.edu/jglazier/ James Glazier] (professional website)
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| *[[CompuCell3D]], a CPM simulation environment: [http://sourceforge.net/projects/compucell/ Sourceforge]
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| *[https://simtk.org/home/compucell3d/ SimTK]
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| *[http://www.nd.edu/~lcls/compucell/linux.htm Notre Dame development site]
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| *[http://panmental.de/ALifeXIflag Artificial Life model of multicellular morphogenesis with autonomously generated gradients for positional information using the Cellular Potts model]
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| {{DEFAULTSORT:Cellular Potts Model}}
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| [[Category:Statistical mechanics]]
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| [[Category:Lattice models]]
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| [[Category:Stochastic models]]
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She is recognized by the name of Myrtle Shryock. To do aerobics is a thing that I'm completely addicted to. Managing individuals is what I do and the salary has been truly fulfilling. Years in the past we moved to North Dakota.
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