3 Neutral Evolution in One- and Two-Locus Systems
Nature selection, neural models
genetic drift: random fluctuations in allele frequencies that necessarily result from sampling finite numbers of gametes in each generation. Sample size, time scale.
3.1 The Wright-Fisher Model
Single finite populations of constant size within which mating is random; other Variant models: separate sexes; family size; generation overlap
Its roots trace to Fisher (1922) and Wright (1931).
Assume:
- a diploid population with a fixed number (N) of monoecious (hermaphroditic, 雌雄同体) adults
- random mating (self-fertilization included)
- discrete generations
Consider a locus with two alleles B and b, without selection advantage:
the probability (\(P_{ij}\)) that i copies of allele B in generation t and j copies of allele B in generation t+1 follows a binomial distribution:
\[P_{ij} = \begin{pmatrix}2N \\j \end{pmatrix}(i/2N)^j[1-(i/2N)]^{2N-j}\]
Letting \(\boldsymbol{P} \in \boldsymbol{R}^{(2N+1)*(2N+1)}\) that
\[\boldsymbol{x}(t+1)=\boldsymbol{x}(t)\boldsymbol{P}\] , where \(\boldsymbol{x}(t) \in \boldsymbol{R}^{1*(2N+1)}\), the probabilities that the allele is present in i=0,1,…,2N copies in generation t
\[x(t) = x(0)P^t\]
Markov chain
Absorbing states: lost or fixed
transition-matrix; diffusion approximation
Moran model