Steady States in Dyson's Theory of the Cell
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Steady States in Dyson's Theory of the Cell<br>Using Markov chains to calculate equilibrium states in Dyson's cell model.<br>CasualPhysicsEnjoyer<br>Jun 19, 2026
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In my last post I described Freeman Dyson's toy model of a cell. Dyson made this model to capture qualitative aspects of autocatalysis, which is a process that involves products that speed up their own creation.<br>But I feel that his explanation of it in his origins of life book wasn't as clear as it could be. In some of the more mathematical parts, he tended to skip over some points which are disguised as easy, but are not straightforward. This is a shame because people who are rusty at math, or don't have mathematics in their formal training, are probably the most likely to find these kinds of models useful.<br>In this post, I wanted to fill those gaps with some interesting math on Markov chains, which I will explain in this post.<br>What is a reasonable toy model of a cell?
In the cell, we have N sites, where N is some ridiculously large number. These sites are meant to host monomers, which are simple chemical units. Each of these sites can be in one of three states. It can be empty, filled with an inactive monomer, or filled with an active monomer. For now, it doesn't matter too much what inactive versus active means. The word active is used because the more active monomers there are in the cell, the more likely it is that active monomers will spawn, and Dyson wanted a way to model this.
At any given time step, if a site is empty, then the probability that the site then gets an active monomer is ψ(x)p, where x is the fraction of sites that are already active, and ψ(x) is an increasing function. Dyson did it this so he could model autocatalysis, which means that the presence of something makes it more likely that this thing will be produced.<br>To be precise, x is just the fraction of all N sites that are currently active,<br>\(x = \frac{\text{number of active sites}}{N},\)
so x is a number between 0 and 1. The function ψ(x) is then any increasing function of that fraction, ψ: [0,1] → [0, ∞) with ψ'(x) > 0. The increasing part is the whole point: the more of the cell that is already active (larger x), the larger ψ(x), and so the more likely an empty site is to become active next. Dyson deliberately left the exact shape of ψ general — all that matters for the qualitative behaviour is that it goes up with x.<br>On the other hand, if a site is empty, there is also a chance that it gets an inactive monomer, where the probability of this is np, where n is the number of chemical species. Unlike the active monomers, having more inactive monomers doesn't increase the chances of inactive monomers occurring. The odds of an inactive monomer binding to a site stays fixed.<br>And if a site is filled by either an active or an inactive monomer, the probability of it becoming empty is qp. Dyson called this 'desorption'.<br>In all of the sentences above, the values p, q, and n are all constants. If we put this all together, we get a diagram that mathematicians use to reason about these kinds of systems.<br>This diagram has three nodes, each meant to represent a state, and each node has arrows showing the odds of transitioning to another node. In this diagram, it's worth noting that the probabilities leaving any node must add up to one. This is because if you're in a given state, you must go somewhere, so all the probabilities need to add up.<br>That means we can calculate the chance that an active monomer stays active, which would be 1 - pq, since the only way to leave the active state is to desorb. Similarly, the chance that an inactive monomer stays inactive is 1 - pq. We can also calculate the chance that an empty site stays empty: it's just one minus the chance that it becomes active minus the chance that it becomes inactive, which means that the chance an empty site stays empty is 1 - ψ(x)p - np.
Interestingly, in this model, Dyson did not assign a probability that an active monomer would transform into an inactive monomer, or vice versa. For a site to have an active monomer changing to an inactive one, it must become empty first. If we were to add this transition probability, it would lead to some interesting math to do with loops in the chain, that I will explain in a future post.<br>So we now have this weird diagram, but how can we actually use it? In my last post, I didn't go into detail about how to think about these systems rigorously, so let's ask some dumb questions.<br>Suppose we had just 100 sites, all empty in the beginning. What happens next? Let's let the simulation run. The first thing you notice is that it seemingly settles to a stable proportion of inactive, active, and empty states. It wiggles around a bit, but the proportion quickly settles. The speed at which it settles is also an interesting question which I will not be covering here. The chart on the right shows the proportion of...