A graphical approach to multi-locus match probability computation: revisiting the product rule

The genealogical relationships of individuals in a finite population can create statistical non-independence of alleles at unlinked loci. In this paper, we introduce a flexible graphical method for computing the probabilities that two individuals in a finite, randomly mating population have the same haplotype or genotype at several loci. This method allows us to generalize the analysis of Laurie and Weir [2003. Dependency effects in multi-locus match probabilities. Theor. Popul. Biol. 63, 207-219] to cases with more loci and other models of mating. We show that monogamy increases the probabilities of genotypic matches at unlinked loci and that the effect of monogamy increases with the number L of loci. We conjecture a sharp upper bound on the effect of monogamy for a given L.     

The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population

We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population. From this we obtain new formulae(formally exact in the limit of extremely weak migration) for the mean fixation time from an arbitrary initial condition for Wright's island model, presenting two cases as examples. For more general models of population subdivision, formulae are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose position is known. These formulae contain parameters that typically have to be obtained numerically, a procedure we follow for two contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends strongly on the nature of subdivision. In particular, we demonstrate conditions under which the fixation time remains finite even in the limit of an infinite number of demes. In many cases-except this last where fixation in a finite time is seen--the time to fixation is shown to be in precise agreement with predictions from formulae for the asymptotic effective population size.     

Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Evolution of coadaptation in a subdivided population

The interplay between population subdivision and epistasis is investigated by studying the fixation probability of a coadapted haplotype in a subdivided population. Analytical and simulation models are developed to study the evolutionary fate of two conditionally neutral mutations that interact epistatically to enhance fitness. We find that the fixation probability of a coadapted haplotype shows a marked increase when the population is genetically subdivided and subpopulations are loosely connected by migration. Moderate migration and isolation allow the propagation of the mutant alleles across subpopulations, while at the same time preserving the favorable allelic combination established within each subpopulation. Together they create the condition most favorable for the ultimate fixation of the coadapted haplotype. On the basis of the analytical and simulation results, we discuss the fundamental role of population subdivision and restricted gene flow in promoting the evolution of functionally integrated systems, with some implications for the shifting-balance theory of evolution.     

A model for the length of tracts of identity by descent in finite random mating populations

Linkage disequilibrium (LD) reflects coinheritance of an ancestral segment by chromosomes in a population. To begin to understand the effects of population history on the extent of LD, we model the length of a tract of identity-by-descent (IBD) between two chromosomes in a finite, random mating population. The variance of an IBD tract is large: a model described by (Genet. Res. Cambridge 35 (1980) 131) underestimates this variance. Using Fisher's concept of junctions, we predict the mean length of an IBD tract, given the age of the population and the population sizes over time. We derive results also for subdivided populations, given times of subdivision events and sizes of the resulting subpopulations. The model demonstrates that population growth and subdivision strongly affect the expected length of an IBD tract in small populations. These effects are less dramatic in large populations.     

Inference from gene trees in a subdivided population

This paper studies gene trees in subdivided populations which are constructed as perfect phylogenies from the pattern of mutations in a sample of DNA sequences and presents a new recursion for the probability distribution of such gene trees. The underlying evolutionary model is the coalescent process in a subdivided population. The infinitely-many-sites model of mutation is assumed. Ancestral inference questions that are discussed are maximum likelihood estimation of migration and mutation rates; detection of population growth by likelihood techniques; determining the distribution of the time to the most recent common ancestor of a sample of sequences; determining the distribution of the age of the mutations on the gene tree; determining in which subpopulation the most recent common ancestor of all the sequences was; determining subpopulation ancestors, where they were, and times to them; and determining in which subpopulations mutations occurred. A computational technique of Griffiths and Tavaré used is a computer intensive Markov chain simulation, which simulates gene trees conditional on their topology implied by the mutation pattern in the sample of DNA sequences. The software GENETREE, which implements these ancestral inference techniques, is available.     

Variation after a selective sweep in a subdivided population

The effect of genetic hitchhiking on neutral variation is analyzed in subdivided populations with differentiated demes. After fixation of a favorable mutation, the consequences on particular subpopulations can be radically different. In the subpopulation where the mutation first appeared by mutation, variation at linked neutral loci is expected to be reduced, as predicted by the classical theory. However, the effect in the other subpopulations, where the mutation is introduced by migration, can be the opposite. This effect depends on the level of genetic differentiation of the subpopulations, the selective advantage of the mutation, the recombination frequency, and the population size, as stated by analytical derivations and computer simulations. The characteristic outcomes of the effect are three. First, the genomic region of reduced variation around the selected locus is smaller than that predicted in a panmictic population. Second, for more distant neutral loci, the amount of variation increases over the level they had before the hitchhiking event. Third, for these loci, the spectrum of gene frequencies is dominated by an excess of alleles at intermediate frequencies when compared with the neutral theory. At these loci, hitchhiking works like a system that takes variation from the between-subpopulation component and introduces it into the subpopulations. The mechanism can also operate in other systems in which the genetic variation is distributed in clusters with limited exchange of variation, such as chromosome arrangements or genomic regions closely linked to targets of balancing selection.     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

Persistence of an infectious disease in a subdivided population

The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspond to preferred mixing, proportionate mixing among subgroups, or mixing between social and nonsocial subgroups, is analyzed. It is shown that the threshold condition for the disease to persist is that either (i) the disease can persist within at least one group through intragroup contacts, or--if (i) does not hold--(ii) the intergroup transmission is sufficiently high. The among-group transmission is computed as an average where each subgroup's reproductive number is weighted according to its intragroup activity level squared and the total number of cases that one infectious individual will cause through intragroup contacts. The model thus allows for a study of the relative importance of communitywide disease transmission and of disease transmission within geographically or socially separate groups.     

Corridors for migration between large subdivided populations, and the structured coalescent

We study the ancestral genetic process for samples from two large, subdivided populations that are connected by migration to, from, and within a small set of subpopulations, or demes. We consider convergence to an ancestral limit process as the numbers of demes in the two large, subdivided populations tend to infinity. We show that the ancestral limit process for a sample includes a recent instantaneous adjustment to the sample size and structure followed by a more ancient process that is identical to the usual structured coalescent, but with different scaled parameters. This justifies the application of a modified structured coalescent to some hierarchically structured populations.     

A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal

The analysis of kin selection in subdivided populations has been hampered by the lack of well‐defined measures of genealogical relatedness in the presence of localized dispersal. Furthermore, the usual arguments underlying the definition of game‐theoretical measures of inclusive fitness are not exact under localized dispersal. We define such measures to give the first‐order effects of selection on the probability of fixation of an allele. The derived measures of kin selection and relatedness are valid in finite populations and under localized dispersal. For the infinite island model, the resulting measure of kin selection is equivalent to a previously used measure. In other cases its definition is based on definitions of relatedness which are different from the usual ones. To illustrate the approach, we reanalyse a model with localized dispersal. We consider sex ratio evolution under sex‐specific dispersal behaviour, and the results confirm the earlier conclusion that the sex ratio is biased towards the sex with the dispersal rate closer to the optimal dispersal rate in the absence of sex‐specific dispersal behaviour.     

Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Selection, mutation and sexual reproduction in an infinite haploid population with a genome of finite length

We present a model which describes mutation, selection and sexual reproduction in an infinite haploid population with a finite genome. Each generation is described using an approximation which assures a certain persistent form of the distribution of the number of deleterious elements. The steady state exists and is determined. In addition, we conclude that the introduction of sexual reproduction increases the mean fitness in the equilibrium.     

The evolution of social communication systems in a subdivided population

Evolution of social communication systems is modeled with a quantitative genetic model. The mathematical model describes the coevolutionary process of a social signal (a social character) and responsiveness (a social preference) to the signal. The responsiveness is postulated to influence fitness of senders of the signal. Considerations are extended to subdivided population structure by combining the social selection model with a group selection model. The numerical results derived from the models indicate that the evolutionary rate of social communication systems depends largely on genetic correlation between the signal and the responsiveness. Group selection can reinforce the evolutionary rate and relax its dependence on genetic correlation. The origin of genetic correlation is discussed in relation to group selection.     

Spatial population structure in the banner-tailed kangaroo rat,Dipodomys spectabilis

I attempted to characterize spatial units of local dynamics and dispersal in banner-tailed kangaroo rats (Dipodomys spectabilis), to determine if spatial structure influenced population dynamics in the way predicted by current metapopulation models. D. spectabilis exhibited a hierarchical spatial structure. Local populations that appeared as discrete entities on a scale of kilometers were subdivided into clusters of mounds on a scale of meters. This structure, however, cannot be characyerized in terms of the discrete habitat patches envisioned by the metapopulation models. Occupied areas were statistically distinguishable from the surrounding matrix, but this difference was only quantitative. There were no discrete boundaries between occupied areas and the matrix. Habitat within occupied areas was heterogeneous, and occupied areas in different locations were statistically distinguishable from each other. High heterogeneity within occupied areas, and high contrast among them, make it difficult to define what is a suitable habitat patch for D. spectabilis. On a smaller spatial scale, there was significant aggregation of resident mounds within occupied areas. These aggregations, however, do not correspond to discrete habitat patches. Rather, they appear to result from an interaction between fine-scale habitat heterogeneity and limited dispersal due to natal philopatry and low adult vagility. These complications make it difficult to identify habitat patches independent of the species' distribution. For species like D. spectabilis that are patchily distributed but do not occupy discrete habitat patches, a patch occupancy approach does not seem appropriate for describing spatial structure. Hierarchical spatial structure underscores the need for a framework that incorporates multiple scales of spatial structure, rather than one that pre-imposes a single spatial scale as being important for population dynamics. A framework that (i) considers patchiness as a combination of both habitat heterogeneity, and life-history and behavioral characteristics, and (ii) incorporates hierarchical spatial structure, appears to be the most suitable for conceptualizing spatial dynamics of behaviorally complex vertebrates such as D. spectabilis.     

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.