The Potts model (first described by Renfrey Potts in his 1952 dissertation) is a model of interacting spins on an n-dimensional lattice. The strength of the Potts model is not so much that it models statistical mechanics or solid state systems well; rather the one-dimensional case is exactly solvable and that its mathematical formulation has been studied extensively by mathematicians and physicists.

Figure 1: A simple q = 2 spin state Ising model

The Potts model is an extension of the Ising model (see previous tutorial) in which there are more than two states; as you should already know, the Ising model has only a 2-spin structure – spun up (+) or spin down (–).  An example of a q = 2 spin state system is shown on the Figure 1. Notice how simple the model is.  Of course, by adding more and more lattice points, the Ising model can become quite complicated, as can be seen in the illustration on the Figure 2.

Figure 2: A larger q = 2 spin state Ising model

The Potts model, however, employs multiple (n > 2) spin states.  Therefore, it is particularly useful for modeling transition states or phase changes in dynamical systems.  An example of a q = 3 spin state model can be seen on Figure 3.  Note that the various states can be randomly mixed, so the evolution of the model is dependent upon interactions between each lattice point.  Most Potts models use the Metropolis Algorithm, which uses the change in energy over time to describe how the model evolves.  This can be given in a general way by

                        ∆H = E_new – E_old

which, in turn, yields a new energy configuration.  The goal here is that the interactions between each lattice point will cause a gradual decrease in cellular energy.  That’s why it is sometimes referred to as the Cellular Potts model. 

The complete Metropolis Algorithm has the following conditions:

        {1}  P(∆E) = 1,                ∆E < 0

        {2}  P(∆E) = e^–∆E/kt,         ∆E > 0.

 An example of how this operates is given below on Figure 4.  In this model, there are q = 5 spin states.  You will notice how much more complicated the image is . . . much of this due to the increased number of phase transitions that occur between each lattice point and the necessary increase in time required for the model to evolve. 

Figure 3: A simple q = 3 spin state Potts model

Even though this is an extremely complex example, most Potts model simulations are substantially simpler because we restrict the number of parameters which are examined during interactions.  For most biological models, the cellular Potts model examines three energy interactions.  These are cell-cell adhesion, cell size and shape changes, and chemical effects.

Figure 4: A complex q = 5 spin state Potts model

Adhesion:  This behavior E_Adhesion defines the net adhesion and/or repulsion between two cell membranes.  It is a product of the binding energy per unit area and the area of interaction between the two cells.  This effect ensures that only the surface sites between two cells contribute to adhesion energy.

Cell size:  Cells have four different characteristics.  These are

• target volume,
• volume elasticity,
• target surface area, and
• membrane elasticity. 

Cell volume and surface area change due to growth and division of cells in which E_Volume exacts an “energy penalty” for deviating away from the target volume and the target surface area, respectively.

Chemical effects.  Cells can move up or down gradients of both diffusible chemical signals (chemotaxis) and insoluble extracellular matrix ECM molecules (haptotaxis). The energy terms for both chemotaxis and haptotaxis are driven by a gradient defining the local concentration of a signaling molecule in the ECM.

Figure 5: A chemical pre-pattern for the Potts model	Figure 6: The completed Cellular Potts model simulation

However, given the proper initial parameters, a wide variety of biological behaviors can be readily modeled using this system.  One such example is the cell type differentiation and cell condensation in order to simulate chicken limb bud formation.  The image shown on the Figure 5 is that predicted using a chemical concentration pre-pattern (in which the E_chemical and gradient behavior is specified).  After running a cellular Potts model simulation, the result is shown on the Figure 6.

The Potts model is widely used in solid state physics and a number of other research fields.  The cellular Potts model, describing the cell in a discrete lattice, has proven itself to be a simple and flexible system framework for simulating cell-morphogenesis.