Cellular Potts Models (CPMs) are spatial models. In this explorable, we combine a simple CPM with stochastic cell divisions to show how spatial interactions can modify an otherwise exponential growth curve.

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

Suppose we start with $N(0)$ cells, which each have a chance $p_\text{div}$ to divide during each time interval of $\Delta t = 1$. This leads to the following differential equation: $$\frac{dN}{dt} = p_\text{div} \cdot N(t)$$ For this equation, the standard solution is exponential growth, where the expected number of cells at any time $t$ is given by: $$N(t) = N(0) \cdot e^{p_\text{div}t}$$ Note that these equations do not consider any spatial variables. What happens if we put this model in space?

The main problem with the exponential growth model is that $N(t)$ keeps growing indefinitely as $t$ increases. When we consider the growth of cells in space (say, on a petri dish), this is obviously not realistic: at some point, the finite space will be filled and growth will stop.

In this explorable we'll see what happens when we build these growth dynamics into
a simple CPM model, which *does* consider space and therefore prohibits
unlimited growth.

We'll start with a basic CPM where each cell:

- tries to stay together (adhesion)
- tries to maintain its size (volume constraint)
- tries to maintain its perimeter (perimeter constraint)

For model details, see this earlier tutorial; for now, we'll just assume that we have a CPM model of interacting cells in space.

The CPM describes how cells interact in space, but by default, it does not allow for
processes like cell proliferation. To include (stochastic) division in our model, we
perform the following *between* the time steps of the CPM model (i.e., between
Monte Carlo Steps, MCS):

- Loop over all cells currently present in the model
- With probability $p_\text{div}$, attempt a division event:
- If some conditions of our choice are met (see below), then
- Draw a line (perpendicular to the cell's longest axis) that divides the cell into two equal parts, and
- Assign the pixels on one side of this line to a new cell.

Thus, after a division event, we have two cells instead of one. They will be too small (only half their original size), but will rapidly grow because of the volume constraint.

The question is now: how do we encode information about space availability in this model? There are multiple ways to accomplish this. In the simulation below, we have chosen a relatively simple method: we set a condition that cells can only divide if they have a minimum size of: $$V_\text{min} = f_\text{min} \cdot V_\text{target}$$ with $V_\text{target}$ the target volume and $f_\text{min}$ a fraction we can choose.

As we'll see below, this interaction between the CPM volume and the division automatically puts an upper bound on the number of cells: when the grid is full, newly divided cells will not be able to grow into their target size; eventually, all cells on the grid will be smaller than $V_\text{min}$ and division will stop. This simple rule is of course not the only way to solve the infinite growth problem. Other possibilities would include:

- Letting $p_\text{div}$ depend on the fraction of the cell's interface that is in contact with "free" background space, rather than other cells
- Letting cells compete for resources by letting a nutrient diffuse during the simulation, which is consumed by the cells. The local nutrient concentration then determines the division rate $p_\text{div}$
- ...etc.

However, for simplicity, we'll stick with the simple rule for now. Below, we will explore how this changes the behaviour from the original exponential growth.

An interactive version of a CPM with cell division is shown below. Start/stop the simulation via the play/pause button, and reset it any time using "reset". Use the sliders to adjust the proliferation rate $p_\text{div}$ and the minimum volume fraction $f_\text{min}$. Cells that are currently too small to divide are shown in gray. Below the simulation, the graph shows $N(t)$ of the simulation ("CPM") compared to the theoretical "exponential" model.

Suggestions:

- Click start and watch the cells divide for a while. You should see that as time
progresses, new cells appear faster: this is because
*each cell*has a fixed chance to divide, so the more cells there already are, the bigger the chance of a division event. (In the differential equation, we see that $dN/dt$ increases linearly with $N(t)$). - Click reset and watch again for a while. Repeat this a number of times. Note that
in some cases, the CPM simulation may grow a little faster than the theoretical model,
whereas in others, it is somewhat slower. This is normal: we are looking at a stochastic
model, and the theoretical predicted value is only the
*average*you would get from running it many times. - Now watch the simulation for longer, until the entire grid is filled. You should see that once the grid is full, the exponential model keeps growing whereas the spatial CPM model stabilizes.
- Set $f_\text{min} = 0$ and $p_\text{div}$ to its maximum value. You should see that cells keep dividing even when the grid is full, and eventually lose their nice shape; this happens because they no longer "sense" their lack of space.

The "basic" dynamics of the CPM can be extended with other dynamic processes, like cell division. By linking these to the spatial characteristics of the cell, we can allow spatial interactions to take place in the model. This can also change the dynamics of the process we are simulating: for example, in the case of cell division, linking it to cell volume will ensure that growth ceases once cells run out of space.