How do cells orient themselves into organised structures while a tissue grows? This explorable describes the very first Cellular Potts Model (CPM): the cell sorting model of François Graner and James Glazier. After a bit of history, the interactive model will illustrate how cells can sort themselves into ordered spatial patterns—a key ability for processes like embryogenesis and tissue development.

This explorable was published in: Wortel & Textor. eLife 2021;10:e61288. doi:10.7554/eLife.61288.

The first version of a CPM was developed in the early nineties by
François Graner and James Glazier (Graner and Glazier, 1992).
At the time, developmental biologists had found that when two different types of embryonic
cells were mixed, they would sort into large, homogeneous same-cell
patches—spontaneously! Experimental studies suggested that this sorting process
arose from differences in adhesion between the cell types.
To understand how this might work, Graner and Glazier wanted to model the
phenomenon and see if they could indeed reproduce sorting based on such
"differential adhesion" alone (in other words: they wanted to test the so-called
*differential adhesion hypothesis*).

Their model was an extension of the existing
*Potts model*,
originally developed by physicists to simulate magnets. The Potts model
simulates the interaction between so-called "spins" on a
crystalline lattice.

The idea is simple: we have a grid with a bunch of pixels, representing points
on the crystal lattice. These points all have a property called their *spin*,
which can be either "down" (0) or "up" (1). They then interact as follows:

- we assign an energetic
*penalty*to each pair of neighbor pixels with differing spin; - pixels continuously try to change (or "flip") their spin, but...
- they are more likely to succeed if this is energetically favourable (i.e. if the flip decreases the number of opposite-spin neighbors).

For more details on the model, see this tutorial; the model below is essentially the model from that tutorial with Adhesion only. For now though, let's just look at the result (with red and gray representing the "up" and "down" spins):

While the Potts model does not much look like a cell, Graner and Glazier
noticed that it does have an interesting property: the spins on the lattice
*automatically sort* into large patches of same-spin sites. And even the
mechanism somewhat resembled their differential adhesion hypothesis: the energetic
cost for having adjacent pixels with opposite spin could be interpreted as a
"contact energy" or "adhesion". Could they apply
a similar principle to the cell sorting question?

A problem with the Potts model is that pixels can only have one of two
states, since magnetic spins can only be "up" or "down". While you could
interpret this as "cell" or "empty background" instead, Graner and
Glazier needed to model *many* different cells for
their differential adhesion simulation—not just a single cell on an
empty background. Their model replaced the binary spin
property with a "cell identity", a number indicating
the cell to which a grid point belonged. (The trick here is that in contrast to
the spin, the cell identity number could be any integer $\geq$0 rather than
only 0 or 1—allowing the co-existence of multiple cells on the same grid).

A second problem is that the patches formed in the Potts model can be of any
size and shape, unlike cells. Graner and Glazier therefore extended the
energy equations so that "cells" would roughly keep the
same size (number of grid points), by assigning energetic penalties for any
deviations from this "target volume" (see this tutorial
for details; Graner and Glazier, 1992,
Glazier *et al*, 2007).

They then used this to model their differential adhesion hypothesis. They did this in a similar way as the original Potts model described earlier, but now:

- Instead of the binary "spin" property (0 or 1), pixels can now have a value $v$ in the range of 0-$n$, where $n$ is the number of cells
- Each cell tries to maintain its size as described above
- Each pixel also has a
*second*property $k$, describing the kind of "cell" it belongs to: the empty background (0) or one of the two self-sorting cell types 1 and 2 - Like with the spins, we assign a penalty $J$ to each pair of neighbor pixels
that do not belong to the same cell (i.e. have different $v$). But now the
size of this penalty depends on the
*kind*$k$ of the contacting pixels: we have $J_{\text{bg,kind 1}}$, $J_{\text{bg,kind 2}}$, $J_{\text{kind 1, kind 1}}$, $J_{\text{kind 2, kind 2}}$, $J_{\text{kind 1, kind 2}}$, and - To encode the "differential adhesion":
$$J_{\text{kind 1, kind 1}}, J_{\text{kind 2, kind 2}} \lt J_{\text{kind 1, kind 2}}$$(i.e., assign a larger penalty to cell contacts if the cells belong to different types)

The result was pretty magical:

Here, the black and gray cells start out randomly mixed, but automatically sort themselves into large, same-cell patches. This simulation supported the differential adhesion hypothesis: apparently, cells could find each other based on adhesion alone, because this model contained no other mechanism for cells to find each other.

Let's now explore the cell sorting behaviour in more detail, using an interactive model.

Below, you'll find Graner and Glaziers model of cells sorting through differential adhesion. You can adjust the different adhesion parameters using the sliders below.

Suggestions:

- Click play (►) and let the simulation run for a while. You should see that most of the cells on the outside of the "blob" are red; this is because the gray cells have a higher contact penalty with the background than the red cells do, so the gray cells prefer to "hide" behind the red ones.
- Set J
_{bg,gray}to 6 and J_{bg,red}to 12. What happens and why? - Now increase J
_{bg,red}further, to 30. You'll see that the gray cells on the outside stretch out; preventing the red cells from contacting the background becomes more important than preventing the gray cells from falling apart. - Set both J
_{bg,red}and J_{bg,gray}to 8, and increase J_{gray,red}to 12. You should see that cells start sorting. Once they have, you can click on "blender" to see them do it again (just to make sure it wasn't a fluke...) - What happens when you set J
_{gray,red}to its maximum value? - Now set both J
_{bg,red}and J_{bg,gray}to 8, J_{red,red}and J_{gray,gray}to 14, and J_{red,gray}to 6. What do you see?

The CPM was developed in the nineties as an extension of
an existing model of magnetism, in which dynamics arise from an energetic optimization
process. The model of Graner and Glazier considered only energies associated with
cell-cell *adhesion* and with cell *size*, and was designed to answer a
specific question: can cells of different types sort themselves in space based on
"differential adhesion" alone? Indeed, spatial patterning arose spontaneously in the
Graner-Glazier model, showing that differential adhesion was *sufficient* to
explain the cell sorting process.

This was the first "cellular" Potts model, or CPM. After Graner and Glazier
showed how the central energy equation of the original Potts model could be extended to
incorporate processes relevant to cells, others quickly followed with their own
variations to the cellular Potts model. In particular, Paulien Hogeweg developed
important extensions to generalize the CPM—which is why it is also
referred to as the Glazier-Graner-Hogeweg (GGH) model
(Glazier *et al*, 2007).

Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model.
Physical Review Letters,
1992.

Magnetization to Morphogenesis: A Brief History of the
Glazier-Graner-Hogeweg Model..
Single-Cell-Based Models in Biology and Medicine. Mathematics and Biosciences in Interaction,
2007.