# What is the basis for modelling the random assortment of molecules using the hypergeometric distribution instead of the binomial distribution?

## Background

I wrote a simulation to investigate a system in which molecules proliferate in cells.

There are cells. They contain two kinds of molecules: enzymes and parasites. The setting is early life, so we assume that during cell division the cells don't have any mechanism to regulate the assortment of their molecules between the daughter cells, so the molecules get randomly to daughter-cell-1 or daughter-cell-2.

So in our model assortment happens as if we flipped a fair coin in the case of each molecule. If it's heads, the molecule gets to daughter-cell-1, if it's tails, it gets to daughter-cell-2. So the distribution of the number of molecules between, say, daughter-cell-1s is binomial.

The implementation of this model is like the following:

parent cell:

• number of enzymes: $$x_0$$
• number of parasites: $$y_0$$

daughter-cell-1:

• number of enzymes: $$x_1 \sim B(n=x_0, p=0.5)$$
• number of parasites: $$y_1 \sim B(n=y_0, p=0.5)$$

daughter-cell-2:

• number of enzymes: $$x_2 = x_0-x_1$$
• number of parasites: $$y_2 = y_0-y_1$$

### Side question

Is drawing random numbers separately for enzymes and parasites consistent with our idea of flipping a fair coin in the case of each molecule of a parent cell?

Or should we do it by drawing a random number for the total number of molecules of daughter-cell-1 and after that, determining somehow how many of this should be enzymes and how many should be parasites? This latter solution seems to be quite complicated though. I don't really have a clue how I would do it.

## Main question

There are at least two published scientific articles, investigating very similar systems, in which they draw random numbers from hypergeometric distributions instead of binomial distributions.

"A protocell splits into two (by hypergeometric sampling, i.e., without replacement) ..."
"... fission by reassorting ribozymes according to the hypergeometric distribution, ..."

From the viewpoint of the individual molecules, how should one imagine this? I don't understand at all what's going on here. Is there an instructive idea analogous to the coin flipping idea? In the case of cell division with random assortment, I don't think there is sampling in the sense of drawing balls with or without replacement from an urn in which there are $$b$$ blue and $$r$$ red balls.

• How many molecules are you talking about? In the limit it becomes all Gaussian distributed anyways. Dec 11 '20 at 8:16
• @SextusEmpiricus The maximum total number of molecules in a cell is a parameter of the model: split size ($s$). Cell division takes place when the total number of molecules reaches $s$. I have examined $s$ in the range of 20 and 1000 so far. Dec 11 '20 at 12:24
• the reason that I was asking is because 'molecules' sounds like 'a lot' and makes me think of the thermodynamic limit with extremely many molecules (which should distribute fifty-fifty without much variation). Dec 11 '20 at 12:33
• @SextusEmpiricus Ah, I see. Well, with a lot of molecules per cell the outcome is that enzymes (≈ useful information) go extinct. It actually seems that the variation between cells generated by the random assortment of molecules during cell division is one of the keys to survival. Dec 11 '20 at 12:55

The hypergeometric distribution occurs when the total number of molecules $$x_1+y_1$$ is fixed. With your mechanism this is not the case $$x_1+y_1 \sim B(n=x_0+y_0, p=0.5)$$.

If the total number is fixed then the situation would be like drawing $$k$$ molecules from the pool of parasites and enzymes without repetition.

### Edit

Is drawing random numbers separately for enzymes and parasites consistent with our idea of flipping a fair coin in the case of each molecule of a parent cell?

Or should we do it by drawing a random number for the total number of molecules of daughter-cell-1 and after that, determining somehow how many of this should be enzymes and how many should be parasites? This latter solution seems to be quite complicated though. I don't really have a clue how I would do it.

This can be equivalent. Say you have two parasites and two enymes then the joint distribution of $$x_1$$ and $$y_1$$ could be:

$$\begin{array}{cc|cc} & &&x_1 \\ &&0&1&2\\ \hline &0 &\frac{1}{16}&\frac{1}{8}&\frac{1}{16} \\ y_1 &1&\frac{1}{8}&\frac{1}{4}&\frac{1}{8} \\ &2 &\frac{1}{16}&\frac{1}{8}&\frac{1}{16} \end{array}$$

It might be possible to first determine $$x_1+y_1$$ and then $$x_1,y_1$$ based on the distribution of $$x_1,y_1$$ conditional on $$x_1+y_1$$. The distribution of $$x_1+y_1$$ is binomial (if the probabilities to transfer to the cell are the same, in your case you have $$p=0.5$$ so this is ok). The distribution of $$x_1$$ (and $$y_1$$), conditional on $$x_1 + y_1$$ is hypergeometric.

• I may have figured out how to resolve the discrepancy between my implementation and the mentioned published implementations. I have posted a question on Mathematics SE. (It seemed more fitting there because I'm interested in suggestions regarding proof-writing also.) Jan 14 at 0:30