A CPM of Cell Migration

How do cells move, and how do they deform along the way? This explorable contains a Cellular Potts Model (CPM) of cell migration. It first briefly explains how a relatively simple model allows for cell migration and realistic cell shapes, and ends with an interactive simulation to illustrate how the parameters work.

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

Modelling Cell Motion

The model of cell migration we will examine is a version of a Cellular Potts Model (CPM) (Graner and Glazier, 1992,Marée, 2007). You can find a more detailed description of the CPM in another tutorial, but we'll briefly revisit the basics here.

Cellular Potts Model

Here is an example of a very simple CPM model:

The basic mechanism is as follows:

These rules yield a cell with dynamic borders that can kind of float around, but there is no real "active" motion—for that, we'll need to add a new "rule" to the system.

Active Migration in the Act-CPM

As we have seen so far, cells in a basic CPM can move, but do not actively migrate like a real cell would. We here consider the Act-CPM (Niculescu, 2015, Wortel, 2020), an extension of the CPM that lets cells migrate actively:

Real cells migrate by manipulating their inner "cytoskeleton", which is made of so-called actin fibers. These actin fibers extend at the front of the cell and push against the cell membrane (like the wheels pushing against the caterpillar track of a tank, Elosegui-Artola and Roca-Cusachs, 2017). This force causes the membrane to "protrude" outward, and eventually allow the cell to drag itself forward. Importantly, the actin fiber extension process is subject to positive feedback: once a cell is polarized and is extending actin on one side, further extensions become more likely on that side. This lets the cell move and stabilizes its polarity, which then promotes further actin extension at the front.

On top of the basic CPM rules described above, we now add a positive feedback mechanism. Put simply: when a cell protrudes, it gains an active pixel, which is then more likely to protrude again. In more detail:

Try It Yourself

Below, you can explore the model and the effects of its two main parameters: λact and maxact.

Suggestions:

Obstacle Course

The nice thing about the CPM is that interactions between cells and their environment arise naturally, because pixels can only ever belong to one cell. For example, we can now explore what happens when the cell's internal protrusion dynamics start interacting with environmental obstacles:

Summary

A very simple encoding of actin-inspired dynamics in the CPM is sufficient to reproduce active cell migration and realistic cell shapes. Note that shape changes are not encoded in the model explicitly, but emerge spontaneously from the dynamics of local positive feedback (from the protrusive activity) and global negative feedback (from the area/membrane elasticity).

References

Elosegui-Artola and Roca-Cusachs. Amoebae as Mechanosensitive Tanks. Biophysical Journal, 2017.
Marée, Grieneisen, and Hogeweg. The Cellular Potts Model and Biophysical Properties of Cells, Tissues and Morphogenesis. Single-Cell-Based Models in Biology and Medicine. Mathematics and Biosciences in Interaction, 2007.
Niculescu et al. Crawling and Gliding: A Computational Model for Shape-Driven Cell Migration. PLoS Computational Biology, 2015.