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:

• Space is represented as a discrete grid of pixels (like those in a blurry image)
• Each pixel belongs to a cell (black) or to the background (gray); we call this its "identity". These identities can change over time as cells continuously try to "conquer" pixels from each other
• Such conquests are stochastic but not random: the system tries to minimize a global energy, defined by the Hamiltonian $H$ (an equation defined by the modeller). Attempted conquests are more likely to succeed when they are energetically favourable such that:
  $$\Delta H_\text{attempt} = H_\text{before} - H_\text{after} \lt 0 $$
• This global energy $H$ differs per model, but in general it defines the "physical laws" followed by the cell. A typical $H$ contains energy terms that reward:
  • Adhesion: Pixels belonging to the same cell try to stick together; essentially, we put an energetic penalty on every black pixel next to a gray pixel. As you can see above, this ensures that the black cell stays intact, and that black pixels are not just scattered all over the grid.
  • Maintaining size and shape: Cells have a target volume and/or perimeter. They can deviate a little from that value by stretching or compressing, but they more or less maintain their size and membrane. As you can see above, the cell fluctuates at its borders but roughly maintains its size and circumference.

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, 2021), 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:

• If a pixel is newly added to a cell at time $t = t^*$, we say that the cell has protruded to gain that pixel. It then gets a "protrusive activity":
  $$A(t^*) = \text{max}_\text{act}$$
  (with $\text{max}_\text{act}$ a model parameter). The colored pixels in the simulation above represent "active" pixels;
• Over time, this activity decreases again with a point per time step (until it hits zero):
  $$A(t+1) = \begin{cases} A(t) - 1 & A(t) \geq 1\\ 0 & A(t) = 0 \end{cases}$$
  This is visible in the color gradient of the pixels in the simulation above; pixels gradually change from red to green as their activity drops, until they become black when their activity is completely gone. Thus, the parameter max_act represents an activity memory.
• Meanwhile, the activity feeds back on cell behavior because we add a term to the global energy equation:
  $$\Delta H_\text{act} (p_s \rightarrow p_t) = -\frac{\lambda_\text{act}}{\text{max}_\text{act}} (\text{GM}_\text{act}(p_s) - \text{GM}_\text{act}(p_t))$$
  Here, we consider an attempt of pixel $p_s$ to copy its identity into pixel $p_t$, and we assign an energetic reward if $p_s$ is in a more active local area than $p_t$. (This local activity is represented by the geometric mean $\text{GM}_\text{act}(p)$ of the activities in the Moore neighborhood of pixel $p$). The parameter $\lambda_\text{act}$ controls the strength of the energetic reward (or cost).

Try It Yourself

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

Suggestions:

• Set max_act to zero; you should see that the cell becomes black and stops moving. A zero max_act means that pixels do not remember their protrusive activity, and thus there is no positive feedback so that $\Delta H_\text{act} = 0$
• Reset max_act to a non-zero value and now set λ_act to zero. Again, the cell stops moving. It has some colored (active) pixels at its border, but these activities do not result in an energetic benefit because $\Delta H_\text{act} = 0$.
• Set max_act = 20 and λ_act = 400. The cell should form small protrusions which can also decay after some time in "stop-and-go" motion. Protrusions don't extend far into the cell, but there can be multiple, competing protrusions. The cell isn't very consistent in its direction.
• Keep max_act = 20 the same and increase λ_act. The cell should become faster. Protrusions may also become somewhat broader and more stable (although still not very stable).
• Set max_act = 60 and λ_act = 100. The larger activity memory means that the protrusion extends further into the cell; the cell shape also becomes broader. Protrusion hardly ever die out, and the cell hardly ever turns.

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