Some Mathematical Models from Population Genetics

The main purpose of theoretical population genetics is to understand the complex patterns of genetic variation that we observe in the world around us. Its origins can be traced to the pioneering work of Fisher, Haldane and Wright. Their contributions were fundamental in establishing the Modern Evolutionary Synthesis, in which Darwin’s theory of evolution by natural selection was finally reconciled with Mendelian genetics. Darwin’s theory of evolution Darwin (1859) can be simply stated: “Heritable traits that increase reproductive success will become more common in a population”. Thus, in order for natural selection to act, there must be variation within a population and offspring must be similar to their parents. So to fully understand evolution we need a mechanism whereby variation is created and inherited. This is provided by Mendelian genetics Mendel (1866). Again the idea can be simply stated. Traits are determined by genes. Each gene occurs in finitely many different types that we call alleles and different alleles may produce different traits. Offspring are similar to their parents because they inherit genes from their parents. The difficulty is that Darwin had argued that evolution of complex, welladapted organisms depends on selection acting on a large number of slight variants in a trait and much ofMendel’s work deliberately focused on discontinuous changes in traits determined by a single gene.

Evolution is a random process. Random events enter in many ways, from errors in copying genetic material to small and large scale environmental changes, but the most basic source of randomness that we must understand is due to reproduction in a finite population leading to random genetic drift. The simplest model of random genetic drift was developed independently by Sewall Wright and R.A. Fisher and is known as the Wright–Fisher model.We consider a population in which every individual is equally likely to mate with every other and in which all individuals experience the same conditions. Such a population is called panmictic.We also suppose that the population is neutral (everyone has an equal chance of reproductive success). Most species are either haploid meaning that they have a single copy of each chromosome (for example, most bacteria), or diploid meaning that they have two copies of each chromosome (for example, humans). We suppose that the population is haploid, so that each individual has exactly one parent. Although in a diploid population individuals have two parents, each gene can be traced to a single parental gene in the previous generation and so it is customary in this setting to model the genes in a diploid population of size N as a haploid population of size 2N.¹ As we shall see in Sect. 5.6, this device fails once we are interested in tracing several genes at the same time.

In this chapter we are going to remind ourselves of some useful facts about one–dimensional diffusions. It is not an exhaustive study. Excellent references for this material are Karlin and Taylor (1981) and Knight (1981). We start in a fairly general setting.

In Remark 2.18 we introduced the notion of nucleotide diversity – the proportion of nucleotides that differ between two randomly chosen sequences. Its expected value is θ = 4Neμ (for a diploid population) where μ is the mutation probability per base pair per individual per generation and N _e is the effective population size. The mutation rate can be estimated directly (or from the divergence between species with a known divergence time) and this gives an estimate of N _e (Barton et al. (2007), p.426). This approach yields N _e ~ 106 for Drosophila melanogaster, far lower than the actual (census) population size or indeed than the population size is likely to have been in the past. Moreover, although genetic variation is certainly higher in more abundant organisms, the relationship is rather weak. For example there’s only about a factor of ten difference between Drosophila melanogaster and humans. Abundant species have much less genetic diversity than expected from the neutral theory, something else is going on.

Most models of spatially structured populations have the same basic format. The population is assumed to be subdivided into demes, which one can think of as ‘islands’ of population. The demes sit at the vertices of a graph and interaction between the subpopulations in different demes is through migration (or more accurately exchange) of individuals along the edges of the graph. The most elementary example is Wright’s island model. This is how he introduced it in (Wright (1943))