(See Arrival of the frequent for context)

See also Wright-Fisher model

The Hamming distance (i.e. the number of differing letters, or mutations) $d$ is then distributed binomially:

$h(d) = \binom{L}{d} \mu^{d} (1-\mu)^{L-d}$

The expected number of individuals with genotype $p$ that arises at generation $t$ can be written as:

where $\Phi_d (g_i, s_i, d)$ is the probability that a $d$-fold mutation of genotype $g_i$ (selected for reproduction according to fitness $1+s_i$) generates an individual with phenotype $p$. It takes into account the genotype-phenotype map. $g_i$ is the genotype of the $i$th member of the population, with a total of $N$ members. See derivation of this below:

As the number is distributed binomially, the average number is $m_p = N(\text{probability for single offspring to get phenotype p})$. Then we define $\tilde{\Phi_p}(g_i, s_i) = (\text{the probability for the single offspring to get to phenotype p}$ $\text{given it inherits a mutated version of parent i})$. Furthermore, $(\text{probability for single offspring to get phenotype p})$ = $\sum_{i=1}^N (\text{probability of single offspring to get phenotype p through parent } i)$ = $\sum_{i=1}^N \tilde{\Phi_p}(g_i, s_i) \times (\text{probability to inherit from parent } i)$ = $\sum_{i=1}^N \tilde{\Phi_p}(g_i, s_i) \frac{(1+s_i)}{\sum_{j=1}^N (1+s_j)}$. Finally,

$m_p = N(\text{probability for single offspring to get phenotype p})$ = $\sum_{i=1}^N \tilde{\Phi_p}(g_i, s_i) \frac{N(1+s_i)}{\sum_{j=1}^N (1+s_j)}$ $\equiv \sum_{i=1}^N \Phi_p'(g_i, s_i)$

By fine-graining the transitions from $g_i$ to a phenotype-$p$ genotype into transitions with particular mutation numbers $d$, we can write $\Phi_p'(g_i, s_i) \equiv \sum_{d=1}^L \Phi_p (g_i, s_i, d)$, recovering Eq. 1

[#[manual links]] (try to upgrade TW to make this work)

The actual number of individuals with genotype $p$ will follow a binomial distribution (as explained for a simple case in Wright-Fisher model), with probability $m_p(t)/N$, and number of trials $N$. The probability of none of the offspring having phenotype $p$ is: $(1-m_p(t)/N)^N \approx e^{-m_p(t)}$, the approximation holds for large $N$, and may be seen as approximating the Binomial distribution by a Poisson distribution.

If we assume that $Ld \ll 1$, i.e. the average number of mutations per genotype is very small, then $h(d) \ll h(1)$ for all $d>1$, and $h(1) \approx L \mu$ ($h(0) \approx 1$ while $h(0) < 1$ of course).

With the above assumption that $Ld \ll 1$, $\Phi_p'(g_i, s_i) = \sum_{d=1}^L h(d) \Phi_p (g_i, s_i, d) \approx \Phi_p (g_i, s_i, 0) + \Phi_p (g_i, s_i, 1) L\mu$. Also, $\Phi_p (g_i, s_i, 0) = 0$, if $p \neq q$. Next, if we assume, $s_i = 0$, for all $i$ with $g_i$ mapping to phenotype $q$ (i.e. in space $\mathcal{N}_q$), and that it all starts within $\mathcal{N}_q$, we have

We can also define the averaged {expected number of offspring with phenotype $p$ at one generation, which inherited from genotype $g_i$ at the previous generation via a single mutation}, i.e the average of $\Phi_p(g_i, 0, 1)$, over all $g_i$ in $\mathcal{N}_q$. We will write abuse notation, and use the label $i$ in $g_i$ to label a genotype in $\mathcal{N}_q$, so that $i = 1, 2, ... N_q$. The average is then:

$\Phi_{pq} = \frac{1}{N_q}\sum_{i=1}^{N_q} \Phi_p(g_i, 0, 1)$

Furtheremore, we should note that, as $\Phi_p'(g_i, s_i) = \tilde{\Phi_p}(g_i, s_i) \frac{N(1+s_i)}{\sum_{j=1}^N (1+s_j)}$ (and a similar expression for the $d$ dependent quantities). When $s_i = 0$, we find $\Phi_p'(g_i, s_i) = \tilde{\Phi_p}(g_i, s_i)$, and also, for example, that $\Phi_p(g_i, 0, 1) = \tilde{\Phi_p}(g_i, 0, 1)$, where $\tilde{\Phi_p}(g_i, s_i, d) = (\text{the probability for the single offspring to get to phenotype p}$ $\text{given it inherits a mutated version of parent i, via a single-point mutation (d=1)})$. Thus $\Phi_{pq}$ is the average of this probability.

We also define the robustness of phenotype $q$, $\rho$ as equal to the average probability over all $\mathcal{N}_q$ of a neutral mutation (i.e. one from $\mathcal{N}_q$ to $\mathcal{N}_q$). Under the approximate assumptions above, $\Phi_{qq} \approx \rho$. If we assume also that the population is large enough (more precisely, we are in the Polymorphic limit (Wright-Fisher model)), we can use a mean field approximation: approximate $\Phi_p (g_i, 0, 1)$ by $\Phi_{pq}$. This approximate works best if the population is large enough that most of the neutral space $\mathcal{N}_q$ is populated (or in the author of the paper word's "1-mutant neighbourhood of the population is similar to that of the whole neutral space"). Using this in Eq.2: