Emergence of geometric order in proliferating metazoan epithelia

by Ankit B. Patel, Radhika Nagpal

Introduction

We use a simple Markov Chain Model to show why many different multicellular organisms all have the same epithelial cell topology. In other words, if you were to count how many neighbors each cell has, you would find the same distribution in all of these organisms! The organisms we examined include the fruitfly (Drosophila melanogaster), the frog (Xenopus), and the Hydra (Hydra). Since these are each on different brnaches of multicellular tree of life, we conclude that this distribution of cells is universal. That is, we expect it show up in the epidermis of all multicellular animals.

Watching real epithelial cells divide

Let's go further in detail. Suppose we have an epithelium of cells, i.e. a 2D sheet of cells that "stick" to each other. Here's a picture of an epithelium:

Assumption I: Cells stick to each other

The flourescent green outlines (GFP staining) marks the boundaries of cells. The cellular adhesion between adjacent cells is crucial for our model, and so our work does not apply to cells that don't adhere to each other. The big rounded cell is a cell that is about to divide. After this "rounding up", the cell will divide like this:

If this is hard to see, here's a more schematic diagram:

Assumption 2: Cells do not rearrange with each other

See how the neighbors of the dividing cell do not rearrange? The fact that cells do not rearrange is another crucial assumption in the model that, in this case, is validified by experiment. In reality, some cells do rearrange but they number less than 5% of the total number of cells.

Moving on, we next make a mathematical model of this division process. We model real cells as polygons with vertices a.k.a junctions. Each side of a polygonal cell , determined by two junctions, corresponds to exactly one of its neighbors (see epithelium figure to verify that this is a good assumption).

Assumption 3: Cells choose cleavage planes by random junction distribution + NO triangular cells allowed

Now here's where we add some assumptions that we cannot see from the pictures above: We assume that when cells divide, the junctions are randomly distributed to each of the daughter cells, in such a way that no daughter cell is allowed to have only 3 sides. In other words, each daughter cell must have at least 4 sides. If you look at the epithelium again, you can verify that there are absolutely no 3-sided cells observed. As of yet, we do not know why this happens, but this is how we incorporate this crucial piece of empirical information into our model. Thus, we have defined our cleavage plane model, i.e. the algorithm for how we choose to divide a polygonal cell into two daughter cells. Schematically, a cell division looks like

where the choice of the plane that split the 6-cell into two 5-cells was chosen according to our cleavage plane model above.

Cells lose sides by division and gain sides by neighboring divisions

Note that the neighboring cells of the dividing cell gained sides! Thus the daughter 5-cells have on average less sides then the parent 6-cell, but the neighboring cells gained sides so that the average is still 6! We shall see later that balance between losing and gaining sides is underlying reason why epithelia converge to a universal equiliubrium distribution. But we are getting ahead of ourselves.

Calculating transition probabilities from the cleavage plane model

Back to the cleavage plane model. The specific rules we've chosen allow us to calculate the probability that an i-sided cell becomes j-sided after a single division. These probabilities are computed in the following tables:

Okay I lied. Note that the table entries aren't probabilities; they are actually likelihoods or relative odds. Also, there are two tables. The left table represents a single cell division and the right table represents a single cell division + any extra sides you gain from your neighbors' dividing. As mentioned earlier, daughter cells lose sides (on average) after a parent cell division, but can gain sides from neighboring cells' division. To read these tables appropriately, here's an example: if you are a 7-sided cell, there is a 3 in 8 chance that you will be 6-sided after you and your neighbors all divide (look at row 7 column 6 in right table).

A surprise: the transitions probability table is Pascal's Triangle

The left matrix is called the Pascal matrix since, amazingly, it contains the rows of Pascal's Triangle!! Of course, this is no surprise to mathematicians since we are randomly distributing junctions from a parent cell to two daughters, a process analogous to flipping coins and counting the number of heads and tails.The right matrix, which we call the overall transition matrix T , is just a rightward shift of the left matrix, and takes into account the neighbor effects we mentioned above. It turns out that neighboring divisions contribute 1 extra side (on average), leading to the 1-column shift.

The Markov Chain Model can be visualized as a network

After careful computation of the probabilities, we build a Markov Chain model that can be visualized as follows:

This colored network diagram just encodes all the transitions probabilities of the overall transition matrix T. You can see the arrow from the 7-cell to the 6-cell labeled by its probability of 3/8.

The balance between losing and gaining sides leads to an equilibrium distribution: Comparison to real epithelia yields excellent match

Given our model, we can calculate the equiliubrium distribution of sides, p*. The distribution p* turns out to be the principal eigenvector of the overall transition matrix T, and can thus be easily computed in MATLAB. Comparing p* to empirical data from the fruitfly, the frog, and Hydra, we see an excellent agreement:

The mathematical model is in yellow, and the three emprical distributions are in other colors, as indicated. The excellent fit confirms our model, though it is possible that other models yield the same distribution. To date, however, we have yet to find any other models that fit this distribution.