Cellular Potts model
Encyclopedia
The cellular Potts model is a 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 domain
of pixels with the same cell id (formerly 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.
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.
The primary rule base has three components:
The CPM can also be thought of as an agent based method in which cell agents evolve, interact via behaviors such as adhesion
, signalling, volume and surface area control, chemotaxis
and 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.
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:
Where for cell σ, λvolume is the volume constraint, Vtarget 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(x,y)={1,x=y; 0,x≠y}.
Many extensions to the original CPM Hamiltonian control cell behaviors including chemotaxis
, elongation and haptotaxis
.
Lattice model (physics)
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
-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
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
simulations of coarsening in metallic grains and soap froths, it has now been used to simulate foam
Foam
-Definition:A foam is a substance that is formed by trapping gas in a liquid or solid in a divided form, i.e. by forming gas regions inside liquid regions, leading to different kinds of dispersed media...
, biological tissues, fluid flow and reaction-advection-diffusion-equations. In the CPM a generalized "cell" is a simply-connected domain
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of pixels with the same cell id (formerly spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
). A generalized cell may be a single soap bubble
Soap bubble
A soap bubble is a thin film of soapy water enclosing air, that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also...
, an entire biological cell, part of a biological cell, or even a region of fluid.
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
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
(CA). Although it also closely resembles certain Monte Carlo methods, such as the large-q Potts model
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
, many subtle differences separate the CPM from Potts models and standard spin-based Monte Carlo schemes.
The primary rule base has three components:
- rules for selecting putative lattice updates
- a HamiltonianHamiltonian (quantum mechanics)In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
or effective energy function that is used for calculating the probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
of accepting lattice updates. - additional rules not included in 1. or 2..
The CPM can also be thought of as an agent based method in which cell agents evolve, interact via behaviors such as adhesion
Cell adhesion
Cellular adhesion is the binding of a cell to a surface, extracellular matrix or another cell using cell adhesion molecules such as selectins, integrins, and cadherins. Correct cellular adhesion is essential in maintaining multicellular structure...
, signalling, volume and surface area control, chemotaxis
Chemotaxis
Chemotaxis is the phenomenon in which somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food by swimming towards the highest concentration of food molecules,...
and 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.
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:
Where for cell σ, λvolume is the volume constraint, Vtarget 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(x,y)={1,x=y; 0,x≠y}.
Many extensions to the original CPM Hamiltonian control cell behaviors including chemotaxis
Chemotaxis
Chemotaxis is the phenomenon in which somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food by swimming towards the highest concentration of food molecules,...
, elongation and haptotaxis
Haptotaxis
Haptotaxis is the directional motility or outgrowth of cells, e.g. in the case of axonal outgrowth, usually up a gradient of cellular adhesion sites or substrate-bound chemoattractants...
.
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