evolution - Why does the number of mutations per individuals follow a Poisson distribution?

I was reading this review. On page 11, left column, first paragraph, one can read:

[..] there is a Poisson distribution of the equilibrium number of mutations per individual, if fitness effects are multiplicative.

with no further explanations. In other words, let the variable $X$ be the number of (deleterious) mutations found in one individual (irrespectively to whether the mutation occurred in the parents gonads or is older back in the lineage). The variable $X$ is Poisson distributed.

Why is this statement true? I suppose that the statement also assumes that all mutations have the same effect on fitness, is that true?

Answer

A Poisson process follows these postulates:

1. $\lim\limits_{h\to0+}\frac{P(N_h=1)}h=\lambda$

   i.e. the probability of occurrence one event in a very small interval of time is equal to the macroscopic rate or intensity ($\lambda\,$).



2. $P(N_h\geqslant2)=o(h)$

   i.e. the probability of occurrence of more than one events in an infinitesimal interval is essentially zero.



3. Events are independent.

If you consider a single individual (for simplicity assume a single cell), then the DNA will undergo mutations at some fixed rate (which we assume to be uniform for all loci). Now each mutation event is independent of the previous event and in a very small interval of time the chance of two or more mutations is negligible. Considering all these facts and assumptions, it can be said that mutation in a single cell would behave like a Poisson process.

From the Poisson postulates you can derive the expression for the Poisson distribution which describes the probability of $k$ number of events in a given time interval, $t$. Hence, the number of mutations in an individual for a fixed time window ($t\,$) follows a Poisson distribution.

$$P(N=k)=\frac{(\lambda t)^k e^{-\lambda t}}{k!}$$

You can find the derivation of Poisson distribution from the postulates from many sources. I referred to this book:

Hogg, Robert V., and Allen T. Craig. Introduction to mathematical statistics. New York: Macmillan, 1978.

EDIT

The effect of deleterious mutations, in the mentioned section of the linked paper talks about Muller's ratchet that describes the accumulation of deleterious mutations and its effect on the population (i.e. extinction). Like any mutation event, accumulation of deleterious mutations will also follow Poisson distribution. Muller's ratchet just says that beyond a tolerance limit, deleterious mutations will cause extinction of asexually reproducing organisms. Perhaps if each deleterious mutation had a strong effect on the fitness, then sampling from the population may lead to non-Poissonian estimates.