Yuri M. Svirezhev

Multi-Locus Models

Let us look at the special features of the genetic content dynamics in a diploid population with respect to several autosomal loci. The state of the population will be presented in terms of frequencies of various types of gametes in the space Σ. The dimensionality of Σ rapidly grows with the number of loci. If distinguish gametes with respect to the genic content of the ith locus with n i alleles, then the dimensionality of Σ is n i−1. For two loci with n i and n j alleles it will be equal to n i n j−1, if gametes are distinguished with respect to l loci, the dimensionality of Σ will be already Πl i=1 n i−1, where n i is the number of alleles at the ith locus. Thus for the minimal amount of alleles at each polymorphic locus, which is two, the dimensionality of Σ exceeds 103 in case of ten loci. This means it is really hard to study multi-locus systems. Investigations of the system in the level of the whole genome come across practically unmanageable difficulties already due to the dimensionality of the problem.

Population Dynamics in Changing Environment

Up till now in all the models considered the environment where the population evolves, was assumed to be invariable. The environmental effects in our models were characterized only by the coefficients of the relative viability, therefore they were considered to be dependent neither on time (the environment is invariable in time), nor on gene frequencies. The latter implies that we do not treat the so-called ‘genotypic environment’, the pressure of selection being independent of the concentrations of particular genotypes in the population. Otherwise the population in evolution makes up its ‘genotypic environment’, and since the pressure of selection is a function of gene frequencies, the ‘genotypic environment’, in turn, determines further population dynamics. The arrival of this feedback can result in most unexpected effects, for instance a stable state may be substituted by a similar but unstable one.

Basic Equations of Population Genetics

We regard a population as a group of individuals that can cross and produce viable offspring. Individuals within population may differ with respect to genotype, age and sex. Let a population be made of N genotypic groups with numbers x i(t, T x) and y i(t, T y) where x i(t, T x) is the number of males of age T x with the genotype denoted by index {i} at time t, y i(t, T y) is the number of females of age T y with the genotype denoted by index {i} at time t. Naturally, T x, T y≥0. The distributions of x i(t, T x) and y i(t, T y) with respect to non-negative T are called age distributions of the corresponding genotypes, and the time variation of these variables is called the evolution of population.

Sex-Limited and Sex-Linked Characters. Models Taking Account of Sex Distinctions

Up till now we treated models of populations with both sexes being equivalent as regards heredity and selection. It was tacitly assumed that characters are controlled by genes pertaining to both sexes (autosomal genes) and equally manifested both in males and in females. However these conditions may not hold. They can be violated in two ways.

Populations with Deviations from Panmixia

Up till now we have examined populations, which mating system was panmixia. Though panmixia is wide-spread in natural populations, just as often various deviations from panmixia occur. For instance, they may be caused by different kinds of preference in mating, when individuals are more likely to form mating pairs by the principle of the greater kinship. It is well known the among birds that particular marriage ritual plays a great role in pair formation, and these behavior reactions are inherited controlled only by the genotype. In a word, everything the suits the saying ‘Gentlemen prefer blondes’ results in deviations from panmixia.

Systems of Linked Populations. Migration

Up till now we have been looking at the evolution of populations, occupying a fixed uniform area, within which some kind of panmixia was realized. However in real life it is more common, when populations of one and the same species occupy different areas, so that the spatial distribution of the species is represented by a patchy pattern. The areas themselves are not absolutely isolated from each other: there are always some flows of migrants between them, that alter both the sizes of separate populations and their genetic structure. Migration flows connect the originally isolated populations into a single system — the system of linked populations, which evolution can be essentially different from the evolution of an isolated population.

Properties of Single-Locus Models under Several Microevolutionary Pressures

The narrow-sense genetic drift model considered above is a kind of starting or reference point in a study of the effect a microevolutionary pressure has on the fate of a population. This role is similar to that of the random mating (panmixia) model among deterministic models of population genetics.

Diffusion Models of Population Genetics

Random influences of various factors of evolution, which may be highly important in population life, are not covered by deterministic analysis of population genetics models. Therefore, stochastic (probability) models arise quite naturally in population genetics, reflecting conditions which are not ‘exotic’.

Random Genetic Drift in Subdivided Populations

Natural populations are scarcely represented by a panmictic entity but rather by a collection of groups of individuals (subpopulations) which are linked one to another by migrations. Mating occurs mainly within a subpopulation, which is partly isolated from other groups. Isolation barriers of this kind arise in spreading of population over space (due to mere geographic barriers or isolation by distance).

Simplest Population Models

In this chapter we will show how even simple models, describing the most elementary populational genetic processes, bring us to conclusions, which are interesting due to their biological clearness. Simple models are still better since they are readily verified in experiments, clearly demonstrating their merits and drawbacks and showing to what extent mathematical models may be applied to the analysis of populational genetic processes. A complex model of great generality is hard to apply to concrete biological situations. The arising mathematical complications can totally shade the biological origin of the problem, shifting the basic accent of the study onto the lines of their overcoming.