A modified Wright–Fisher model that incorporates Ne: A variant of the standard model with increased biological realism and reduced computational complexity

• The Wright–Fisher model does not incorporate the effective population size, Ne.
• We present a method that directly incorporates Ne into the Wright–Fisher model.
• This leads to reduced computational complexity, with matrices of size determined by Ne.
• The method may be applied where it is impossible to numerically treat diffusion results.
• We thus extend the realism and tractability of a key model of genetics and evolution.

The Wright–Fisher model is an important model in evolutionary biology and population genetics. It has been applied in numerous analyses of finite populations with discrete generations. It is recognised that real populations can behave, in some key aspects, as though their size that is not the census size, N, but rather a smaller size, namely the effective population size, Ne. However, in the Wright–Fisher model, there is no distinction between the effective and census population sizes. Equivalently, we can say that in this model, Ne coincides with N. The Wright–Fisher model therefore lacks an important aspect of biological realism. Here, we present a method that allows Ne to be directly incorporated into the Wright–Fisher model. The modified model involves matrices whose size is determined by Ne. Thus apart from increased biological realism, the modified model also has reduced computational complexity, particularly so when Ne⪡NNe⪡N. For complex problems, it may be hard or impossible to numerically analyse the most commonly-used approximation of the Wright–Fisher model that incorporates Ne, namely the diffusion approximation. An alternative approach is simulation. However, the simulations need to be sufficiently detailed that they yield an effective size that is different to the census size. Simulations may also be time consuming and have attendant statistical errors. The method presented in this work may then be the only alternative to simulations, when Ne differs from N. We illustrate the straightforward application of the method to some problems involving allele fixation and the determination of the equilibrium site frequency spectrum. We then apply the method to the problem of fixation when three alleles are segregating in a population. This latter problem is significantly more complex than a two allele problem and since the diffusion equation cannot be numerically solved, the only other way Ne can be incorporated into the analysis is by simulation. We have achieved good accuracy in all cases considered. In summary, the present work extends the realism and tractability of an important model of evolutionary biology and population genetics.