Valentine's Ballet

For all the biology-lovers out there: if you thought maths were boring, think again! Here's a mathematical model of cellular traffic on a sticky, heart-shaped surface, which yields a beautiful collective dancing motion. Try what happens when you change the "love factor". Happy Valentine's Day!

Try It Yourself

If you want to play with the simulation yourself, here's some suggestions (an explanation of the model follows below):

• Remove the movement boost by setting either its strength or memory to zero. You'll see that cells (pink) stop moving if we don't give them the incentive to do so. They lose the white patches that indicate they are actively moving.
• Increase the memory of the movement boost. You'll see that cells actually change shape and become more white; the increase in boost memory results in larger active (=white) areas.
• Free the cells by decreasing the "love factor". By reducing the difference in motion boost between inside and outside the heart shape, we take away the incentive for cells to stay inside.
• Confine the cells further by increasing the love factor. The heart shape should become even more defined.

About the Model

This is a so-called Cellular Potts Model. It sounds fancy, but essentially it is just like a zoomed-in photograph where you can see the pixels. Each pixel belongs to either the "surface" (dark / light gray, see below) or to a "cell" moving on that surface (the moving pink patches).

Cells move because they can conquer pixels from other cells in a game based on a few simple rules.

A Game of Pixels

In the game of conquering and losing pixels, cells prefer the following things:

• Staying intact, i.e. pixels of a cell stick together. A single pixel can't just go its own way; it wants to conquer new ground but also stay close to the rest of the cell.
• Staying in shape. A cell can stretch or compress a little, but not too much: when the cell becomes too big, it won't conquer new pixels until it has first lost some old ones (and the other way around). Have a look at the simulation: cells may deform and twist, but they are always roughly the same size.

But that's not enough to make them do more than float in place. If you watch the simulation for a while, you'll see that these cells do much more than float: they move all over the place. How?

Gas Pedal

If cells are intact, and at the right size, they won't have much incentive to conquer new pixels and move.

...Unless we give it to them, that is.

The model you are looking at is called the Act-CPM (see this and this paper), which adds one more ingredient to the game of pixels: a gas pedal.

This gas pedal gives the cells the "boost" to conquer new pixels, and it does this especially in parts of the cell that have actively conquered new ground just before. These "active" areas are recognisable by their white-ish color in the simulation.

So whenever a new pixel is conquered by the cell, that is a sign that that part of the cell is active. The new pixel then gets the boost to help it conquer even more pixels. This is controlled by two things: the activity "memory" (how long does the new pixel keep its boost) and the activity "strength" (how strong is this boost compared to the other rules of the game).

That's it! The cell is now moving. There is just one more thing...

Paving the Way

How do we make the cells stick to the heart shape? For that, we slightly modify the Act-CPM so that the boost-strength depends on the location in space. If we make the boost large inside the heart area, but small outside of it, cells end up stuck inside the heart (in other words, they have a high 'love factor').

(This heart shape is actually itself the result of a bit of math; see this page for details.)

The Result: Teamwork!

Have you noticed how cells in the simulation tend to follow and swirl around each other in "streams"? There's actually no rule telling them to do that! We call this "collective motion", and it arises naturally because the cells are trying to move while also pushing against each other. This is just one example of what we often see in biology: complex patterns and behaviours can emerge even from relatively simple rules.