Freeman Dyson’s Model of a Cell

Life and Death in Random Processes

May 11, 2026

In this post I’ll describe Freeman Dyson’s toy model of a cell in his book Origins of Life.^[1] It’s a probabilistic model that gives us interesting behaviours. It has auto-catalysis, which means that a reaction increases the likelihood of the same reaction occuring. And it has two equilibrium points that can be interpreted as life and death.

It’s also a good example of how to reason about random systems and how they can tend to stable states. You may need to see this in a web-browser to see the simulations run.

The model is of a cell evolving in discrete time steps. The cell is a constrained volume of fluid, with a membrane that is meant to be the cell wall. Through diffusion, molecules called monomers can go in or out. These monomers are meant to be nutrients.

Within a cell there are N sites. Each monomer can attach to one of the N sites, or the site can be empty. The model evolves in discrete time steps. At every time step, a site can remain empty. If it has a monomer attached to it, the monomer can stay there, or it can unbind. The sites are meant to represent placeholders for monomers binding into polymers, but I’ll get into the biochem in a different post.

There are n+1 species of monomers that can bind to the sites. n of these are considered ‘inactive’, and one type is ‘active’. If a site has an active monomer, then the likelihood that other sites absorb an active monomer increases. A monomer being active is meant to represent a monomer being at the ‘right place at the right time’ that lets other active monomers bind to a cite. This is meant to capture ‘auto-catalysis’ — the binding of an active monomer makes it more likely that other active monomers will bind.

So each site has three possible states — empty, inactive monomer, or active monomer. Per unit time, the probabilities are modelled as follows:

If a site is filled by either an active or inactive monomer, then it has a probability pq of being emptied, or ‘desorbed’, where q is a constant.

On the other hand, if a site is empty,

the probability of an inactive monomer being absorbed is np, where n is the number of types of inactive monomers.
the probability of an active monomer being absorbed is ψ(x) · p, where x is the fraction of sites already occupied by active monomers.
- ψ(x) is a monotonically increasing function of the fraction of sites that are already active.
- This means that the more active sites there are, the more likely it is that active monomers will be absorbed.

This function psi is meant to encode the increased chance that an active monomer will be absorbed if there are active ones present already. This is shown in the video below.

For each site, the transition between each state at each time step is given by the following chain:

So if we let a simulation run, let’s see what happens to the fraction of empty, inactive, and active sites. If we start with a large number of inactive sites, the number of active sites doesn’t really seem to take off — and so the cell remains filled with inactive sites. In this case the cell is just filled with mostly ‘inactive’ monomers, and remains ‘dead’.

This is shown in the video below, where the blue monomers are inactive, and the green monomers are active.

On the other hand, if we initialise the system with a larger amount of active monomers, it seems to stay that way.

Now, if we increase the number of monomers and the time of the simulation rapidly, we see that the cell converges to one of two states. The green trajectory shows the convergence to an alive state in which there is a constant fraction of active monomers. And the blue one shows convergence to a dead state where most states are inactive monomers. This is pretty cool!

So we have got an apparently random, auto-catalytic system and have found that it has converged to two stable states. How do we know which initial conditions lead to which state, and what fraction there is of active monomers in each state? I will discuss this in my next post.