Wright-Fisher model - senecio7の日記

Waxman D & Overall ADJ 2020 Influence of dominance and drift on lethal mutations in human populations. Front Genet 11:267.

a lethal disease-causing allele needs to be described by a modified Wright-Fisher model, which deviates from the standard model, where selection coefficients are assumed small compared with 1

most of the mutations do not conform to what is expected from the balance between mutation, purifying selection and random genetic drift
the majority of the mutations observed were at frequencies that were much elevated over what was theoretically expected
current data reveal an ascertainment bias
the disease alleles were the ones identified simply by being more frequent by chance
we present a theoretical investigation of the sensitivity of the mutation-selection dynamic to small changes in fitness of the carrier genotype, in particular when slightly overdominant

standard population genetics theory generally incorporates random genetic drift via a Wright-Fisher model (and its diffusion approximation) that is derived under the assumption of weak selection

X' = Bin(2N, X + F(X)) / (2N) ... (6)
Wright-Fisher model for weak selection
Bin(n, p) denotes a binomial random number (not a distribution), and gives the random number of successes on n independent trials, each of which has probability p of success
we need to modify the above Wright-Fisher model so it incorporates strong selection

the conventional Wright-Fisher model is based on the strong assumption that all randomness arises solely in the non-selective thinning of the population to the census population size

the conventional Wright-Fisher model also assumes census and effective population sizes coincide
it is possible to incorporate the effective population size into the Wright-Fisher model (Zhao et al., 2016)

selection is treated as a deterministic process, amounting to the population being effectively infinite during the time that selection occurs within the lifecycle
for humans in modern post-industrial populations, the number of offspring produced is typically little more than that required to replace the population
the number of zygotes produced is similar in number to the number of adults (i.e., similar to the census size)
we have used an explicitly probabilistic treatment of selection
for most practical purposes there is a negligible difference between such a model, and the model where selection is treated as acting deterministically

selection cannot be directly approximated as acting at the level of alleles (which is possible when selection is weak)
the equation X' = X + F(X) applies to the frequency of the disease-causing allele in an infinite population (see Equations 4 and 5)
we cannot simply extend this equation to become an equation describing a finite population, by using the weak-selection result
the frequency of the lethal genotype obeys the stochastic equation
X' = Bin(N, 2X + 2F(X)) / (2N) ... (7)
Wright-Fisher model for a lethal genotype
F(x) is given in Equation (5)

analytical insight into the Wright-Fisher model, and the phenomena occurring in a finite population, can be gained using a diffusion approximation of this model (Kimura, 1955)

an important consideration with lethality, is the explicit need to treat the action of selection on genotypes, rather than on alleles
the three genotype frequencies add to unity so just two are independent
when there is also lethality of one homozygote, this allows elimination of one of the two independent genotype frequencies, with the substantial simplification that just a single frequency is required to describe the population

for lethal alleles with a low mutation rate, even very weak overdominance can result in highly inflated equilibrium frequencies