Detecting Isolation by Distance Using Phylogenies of Genes

We introduce a method for analyzing phylogenies of genes sampled from a geographically structured population. A parsimony method can be used to compute s, the minimum number of migration events between pairs of populations sampled, and the value of s can be used to estimate the effective migration rate M, the value of Nm in an island model with local populations of size N and a migration rate m that would yield the same value of s. Extensive simulations show that there is a simple relationship between M and the geographic distance between pairs of samples in one- and two-dimensional models of isolation by distance. Both stepping-stone and lattice models were simulated. If two demes k steps apart are sampled, then, s, the average value of s, is a function only of k/(Nm) in a one-dimensional model and is a function only of k/(Nm)2 in a two-dimensional model. Furthermore, log(M) is approximately a linear function of log(k). In a one-dimensional model, the regression coefficient is approximately -1 and in a two-dimensional model the regression coefficient is approximately -0.5. Using data from several locations, the regression of log(M) on log(distance) may indicate whether there is isolation by distance in a population at equilibrium and may allow an estimate of the effective migration rate between adjacent sampling locations. Alternative methods for analyzing DNA sequence data from a geographically structured population are discussed. An application of our method to the data of R. L. Cann, M. Stoneking and A. C. Wilson on human mitochondrial DNA is presented.     

Consequences of population structure on genes under balancing selection

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u < m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.     

The effects of subdivision on the genetic divergence of populations and species

Abstract. An island model of migration is used to study the effects of subdivision within populations and species on sample genealogies and on between-population or between-species measures of genetic variation. The model assumes that the number of demes within each population or species is large. When populations (or species), connected either by gene flow or historical association, are themselves subdivided into demes, changes in the migration rate among demes alter both the structure of genealogies and the time scale of the coalescent process. The time scale of the coalescent is related to the effective size of the population, which depends on the migration rate among demes. When the migration rate among demes within populations is low, isolation (or speciation) events seem more recent and migration rates among populations seem higher because the effective size of each population is increased. This affects the probability of reciprocal monophyly of two samples, the chance that a gene tree of a sample matches the species tree, and relative likelihoods of different types of polymorphic sites. It can also have a profound effect on the estimation of divergence times.     

PHASE III OF WRIGHT'S SHIFTING BALANCE PROCESS AND THE VARIANCE AMONG DEMES IN MIGRATION RATE

Interdemic selection by the differential migration of individuals out from demes of high fitness and into demes of low fitness (Phase III) is one of the most controversial aspects of Wright's Shifting Balance Theory. I derive a relationship between Phase III migration and the interdemic selection differential, S, and show its potential effect on F_ST. The relationship reveals a diversifying effect of interdemic selection by Phase III migration on the genetic structure of a metapopulation. Using experimental metapopulations, I explored the effect of Phase III migration on F_ST by comparing the genetic variance among demes for two different patterns of migration: (1) island model migration and (2) Wright's Phase III migration. Although mean migration rates were the same, I found that the variance among demes in migration rate was significantly higher with Phase III than with island model migration. As a result, F_ST for the frequency of a neutral marker locus was higher with Phase III than it was with island model migration. By increasing F_ST, Phase III enhanced the genetic differentiation among demes for traits not subject to interdemic selection. This feature makes Wright's process different from individual selection which, by reducing effective population size, decreases the genetic variance within demes for all other traits. I discussed this finding in relation to the efficacy of Phase III and random migration for effecting peak shifts, and the contribution of genes with indirect effects to among‐deme variation.     

Effect of population structure on the amount of polymorphism and the fixation probability under overdominant selection

Under overdominant selection, mutants substantially contribute to increase the amount of polymorphism. It is also known that under neutrality as the migration rates among demes decrease in a subdivided population, the amount of polymorphism increases along with the increase of the effective population size, N(e). In this study, under overdominant selection the effect of population subdivision on the amount of polymorphism was investigated using the diffusion approximation and the low migration approximation. It was shown that if selection is medium or strong (e.g., N(T)s > 1, where N(T) is the population size and s is the selective advantage of heterozygotes), the nucleotide diversity, pi, decreases along with the decrease of Nm against the increase of N(e), where N is the size of demes and m is the migration rate per deme. In addition, the ratio of the nucleotide diversity to the evolutionary rate also decreases along with the decrease of Nm. In some cases the ratio becomes smaller than that expected under neutrality as Nm decreases.     

Theory of the effects of population structure and sampling on patterns of linkage disequilibrium applied to genomic data from humans

We develop predictions for the correlation of heterozygosity and for linkage disequilibrium between two loci using a simple model of population structure that includes migration among local populations, or demes. We compare the results for a sample of size two from the same deme (a single-deme sample) to those for a sample of size two from two different demes (a scattered sample). The correlation in heterozygosity for a scattered sample is surprisingly insensitive to both the migration rate and the number of demes. In contrast, the correlation in heterozygosity for a single-deme sample is sensitive to both, and the effect of an increase in the number of demes is qualitatively similar to that of a decrease in the migration rate: both increase the correlation in heterozygosity. These same conclusions hold for a commonly used measure of linkage disequilibrium (r(2)). We compare the predictions of the theory to genomic data from humans and show that subdivision might account for a substantial portion of the genetic associations observed within the human genome, even though migration rates among local populations of humans are relatively large. Because correlations due to subdivision rather than to physical linkage can be large even in a single-deme sample, then if long-term migration has been important in shaping patterns of human polymorphism, the common practice of disease mapping using linkage disequilibrium in "isolated" local populations may be subject to error.     

Methods to Measure the Impact of Home,Social, and Sexual Neighborhoods of Urban Gay,Bisexual, and Other Men Who Have Sex with Men

Men who have sex with men (MSM) accounted for 61% of new HIV diagnoses in the United States in 2010. Recent analyses indicate that socio-structural factors are important correlates of HIV infection. NYCM2M was a cross-sectional study designed to identify neighborhood-level characteristics within the urban environment that influence sexual risk behaviors, substance use and depression among MSM living in New York City. The sample was recruited using a modified venue-based time-space sampling methodology and through select websites and mobile applications. This paper describes novel methodological approaches used to improve the quality of data collected for analysis of the impact of neighborhoods on MSM health. Previous research has focused predominately on residential neighborhoods and used pre-determined administrative boundaries (e.g., census tracts) that often do not reflect authentic and meaningful neighborhoods. This study included the definition and assessment of multiple neighborhoods of influence including where men live (home neighborhood), socialize (social neighborhood) and have sex (sexual neighborhood). Furthermore, making use of technological advances in mapping, we collected geo-points of reference for each type of neighborhood and identified and constructed self-identified neighborhood boundary definitions. Finally, this study collected both perceived neighborhood characteristics and objective neighborhood conditions to create a comprehensive, flexible and rich neighborhood-level set of covariates. This research revealed that men perceived their home, social and sexual neighborhoods in different ways. Few men (15%) had the same home, social and sexual neighborhoods; for 31%, none of the neighborhoods was the same. Of the three types of neighborhoods, the number of unique social neighborhoods was the lowest; the size of sexual neighborhoods was the smallest. The resultant dataset offers the opportunity to conduct analyses that will yield context-specific and nuanced understandings of the relations among neighborhood space, and the well-being and health of urban MSM.     

The average number of sites separating DNA sequences drawn from a subdivided population

The "infinite sites" model in the absence of recombination is examined in a subdivided population in which there is arbitrary migration among demes. It is shown that, if the migration matrix is symmetric and irreducible, the average number of sites that differ in two alleles chosen from the same deme depends only on an effective size of the whole population and not on either the elements of the migration matrix or the size of each deme separately. If there are n demes all of size N, the average number of sites that differ in two alleles chosen from the same deme is 4nN mu, where mu is the average mutation rate per site. This is the same value as for two alleles drawn from a panmictic population of size nN. The average number of sites that differ in alleles drawn from the same and from different demes can provide some information about the degree of population subdivision, as is illustrated by using the data of Kreitman and Aquadé (1986, Proc. Nat. Acad. Sci. U.S.A., 83, 3562) on Drosophila melanogaster.     

A GENETIC AND BEHAVIORAL ANALYSIS OF MATE CHOICE AND SONG NEIGHBORHOODS IN INDIGO BUNTINGS

Neighboring males of indigo buntings (Passerina cyanea) share songs in southern Michigan. We sampled polymorphic enzymes to compare the genetic variation between mates and the variation among contiguous song neighborhoods. Mate choice was independent of the genetic and morphometric similarity of female and male, and these measures were independent of each other. The incidence of extrapair copulations and fertilizations was independent of the song of cuckolding males. Breeding success of the mated pairs was independent of their genetic or morphological similarity. Males characterized by different song dialects did not differ in mean lifetime reproductive success. We found no significant genetic differences among the neighborhoods. Most birds that bred in one song neighborhood were born in another, and neighborhoods were not isolated demes. Bunting songs may provide no information to a female about genetic quality of males. The results are consistent with a neutral model of no mate choice for genes.     

EVOLUTION OF ALTRUISM IN KIN-STRUCTURED AND RANDOM SUBDIVIDED POPULATIONS

A. population structure favorable to the evolution of an altruistic trait is studied by Monte Carlo simulation. The model is based on a small-scale nonindustrial human society but seems generalizable to other highly social mammals. Three hierarchical levels are recognized: 1) the ecologically isolated local group (hamlet) which may be composed of kin and/or unrelated individuals; 2) the deme (settlement) comprising several such groups which interbreed; and 3) the set of demes (metapopulation) among which gene flow occurs. The first two levels of the model are based on D. S. Wilson's structured deme concept; the third allows for gene flow among demes in the metapopulation and for the structured diffusion of alleles across a wider area than might be included within the scope of a single deme. The simulation models genetic drift by a process of hamlet formation which may be random, or variously kin-structured. Hamlets may then become extinct based on a probability function of their gene frequencies. Individual selection within settlements is modeled deterministically, and gene flow among settlements is modeled as two-dimensional steppingstone migration of random or kin-structured groups. Results of the simulations show that, with realistic values for group sizes, moderate extinction rate, and high rates of migration (m > 27%), disadvantageous alleles (s = 10% and 25%) may increase markedly due to differential hamlet extinction over the course of 50 generations. The greater the degree of kin-structuring of founder groups, the higher the variance among hamlets and the faster the rate of increase of the allele for altruism. Nonetheless, even in some randomly founded groups, a clear increase in the altruism gene frequency occurred. It is also notable that kin-structured group selection by hamlet extinction may be effective when the initial frequency of altruism genes is very low (average of one per deme) and among a relatively small number of demes (25). Thus the process of group extinction in a hierarchically structured population allows rapid increase of an allele for altruism under plausible demographic conditions.     

Neutral additive genetic variance in a metapopulation

For neutral, additive quantitative characters, the amount of additive genetic variance within and among populations is predictable from Wright's FST, the effective population size and the mutational variance. The structure of quantitative genetic variance in a subdivided metapopulation can be predicted from results from coalescent theory, thereby allowing single-locus results to predict quantitative genetic processes. The expected total amount of additive genetic variance in a metapopulation of diploid individual is given by 2Ne sigma m2 (1 + FST), where FST is Wright's among-population fixation index, Ne is the eigenvalue effective size of the metapopulation, and sigma m2 is the mutational variance. The expected additive genetic variance within populations is given by 2Ne sigma e2(1-FST), and the variance among demes is given by 4FSTNe sigma m2. These results are general with respect to the types of population structure involved. Furthermore, the dimensionless measure of the quantitative genetic variance among populations, QST, is shown to be generally equal to FST for the neutral additive model. Thus, for all population structures, a value of QST greater than FST for neutral loci is evidence for spatially divergent evolution by natural selection.     

Effect of group selection on the evolution of altruistic behavior

By using a Monte Carlo simulation, we studied the effect of group selection on the altruistic trait that is controlled by a single locus. The altruistic trait is disadvantageous to the bearer but advantageous to the others. Group selection is defined as the differential reproductive rate among demes caused by genotypic difference among demes. We found that the simulation reproduced many results of former studies. Additionally, when the mutation rate and the migration rate are small enough, we observed two new phenomena: (1) When the effect of the group selection is as large as that of the individual selection, the gene frequency is quite unstable. We found two local stable states, the A- and the S-state. When the metapopulation is in the A-state, altruists are nearly fixed. When in the S-state, on the contrary, altruists are almost lost. The metapopulation shifted quickly from one state to another. We call this phenomenon as the S-A transition. (2) When the mutation rate and migration rate are small enough we found an extremely strong mechanism to stop the non-altruists from expanding no matter how strong the individual selection coefficient is. This is caused by a phenomenon, which we call SA splitting, in which most demes are fixed either by altruists or non-altruists; thus, the relatedness of the metapopulation becomes nearly equal to one. We show SA splitting plays an important role in S-A transition. We define a parameter d to see the degree of SA splitting. We found that d is roughly proportional to mutation rate and deme size.     

Exact Inbreeding Coefficient and Effective Size of Finite Populations under Partial Sib Mating

An exact recurrence equation for inbreeding coefficient is derived for a partially sib-mated population of N individuals mated in N/2 pairs. From the equation, a formula for effective size (N(e)) taking second order terms of 1/Ninto consideration is derived. When the family sizes are Poisson or equally distributed, the formula reduces to N(e) = [(4 - 3β)N/(4 - 2β)] + 1 or N(e) = [(4 - 3β)N/(2 - 2β)] - 8/(4 - 3β), approximately. For the special case of sib-mating exclusion and Poisson distribution of family size, the formula simplifies to N(e) = N + 1, which differs from the previous results derived by many authors by a value of one. Stochastic simulations are run to check our results where disagreements with others are involved.     

Selection for High Adult Body Weight in Drosophila Populations with Different Structures

Selection for high adult body weight in Drosophila melanogaster was practiced for 18 generations in three selection lines. These lines were genetically similar and of equal size but different in population structure. One line represented a large mass-selected, random-mating population, while the other two lines simulated large populations that had been subdivided into partial isolates or demes. Mass selection and random mating occurred within each deme. These two subdivided lines were different only in the rate of effective migration among the demes (5% and 10%). Selection intensities of approximately 20% were applied to these populations. A fourth line served as a random mating control. Heritability of adult body weight in the base population was estimated to be 0.58± 0.22. The results indicate that significantly greater responses were achieved in the subdivided lines than in the large mass-selected line, in spite of the fact that larger selection differentials were applied to the latter. No significant differences in response were observed between the two subdivided lines. Wright (1930, 1931) postulated that selection would be most efficient in subdivided populations with limited interdeme migration. The present findings appear to support this theory.     

Spatial heterogeneity in the strength of selection against deleterious alleles and the mutation load

According to current estimates of genomic deleterious mutation rates (which are often of the order 0.1-1) the mutation load (defined as a reduction in the average fitness of a population due to the presence of deleterious alleles) may be important in many populations. In this paper, I use multilocus simulations to explore the effect of spatial heterogeneity in the strength of selection against deleterious alleles on the mutation load (for example, it has been suggested that stressful environments may increase the strength of selection). These simulations show contrasted results: in some situations, spatial heterogeneity may greatly reduce the mutation load, due to the fact that migrants coming from demes under stronger selection carry relatively few deleterious alleles, and benefit from a strong advantage within demes under weaker selection (where individuals carry many more deleterious alleles); in other situations, however, deleterious alleles accumulate within demes under stronger selection, due to migration pressure from demes under weaker selection, leading to fitness erosion within those demes. This second situation is more frequent when the productivity of the different demes is proportional to their mean fitness. The effect of spatial heterogeneity is greatly reduced, however, when the response to environmental differences is inconsistent across loci.     

A framework for estimating the fixation time of an advantageous allele in stepping-stone models

Determining how population subdivision increases the fixation time of an advantageous allele is an important problem in evolutionary genetics as this influences many processes. Here, I lay out a framework for calculating the fixation time of a positively selected allele in a subdivided population, as a function of the number of demes present, the migration rate between them and the manner in which they are connected. Using this framework, it becomes clear that a beneficial allele's fixation time is significantly reduced through migration continuously introducing copies of the allele into a newly colonized subpopulation, increasing its frequency within these demes. The effect that migration has on allele frequency needs to be explicitly taken into account to produce a realistic estimate of fixation time. This behaviour is most prominent when demes are arranged on a two-dimensional torus, in comparison with populations where demes are arranged in a circle. This is because each subpopulation is connected to several neighbours over a torus, so that there are multiple paths that an allele can take in order to fix. As a consequence, some demes experience a greater influx and efflux of migrants than others. Analytical results are found to be very accurate when compared to stochastic simulations, and are generally robust if there are a large number of demes, or if the allele is weakly selected for.     

Population Differentiation under the Charge State Model

The extent of divergence between partially isolated sub-populations for electrophoretically detectable alleles was formulated assuming the island model of migration and the charge state model of mutation. At equilibrium the ratio of the variance of charge between the means of k different islands to the average within-island variance of charge was shown to be approximately 4Nemk2/(k-1)2 where Ne is the effective size of each island population and m is the migration rate. This ratio was calculated from published data for eight polymorphic loci in six island populations of Drosophila willistoni. Under the assumption that all variants are selectively neutral, migration rates of greater than 10 adults per generation per island are required to explain the observed similarity of the allelic profiles in D. willistoni. Since the islands studied appear to be virtually completely isolated it was concluded either that the observed protein variants are adaptive and maintained in populations by some form of balancing selection or that the observed variants themselves are neutral but natural selection acts to restrict the appearance of more extreme variants in the charge carried.     

The loss of adaptive plasticity during long periods of environmental stasis

Adaptive plasticity allows populations to adjust rapidly to environmental change. If this is useful only rarely, plasticity may undergo mutational degradation and be lost from a population. We consider a population of constant size N undergoing loss of plasticity at functional mutation rate m and with selective advantage s associated with loss. Environmental change events occur at rate theta per generation, killing all individuals that lack plasticity. The expected time until loss of plasticity in a fluctuating environment is always at least tau, the expected time until loss of plasticity in a static environment. When mN > 1 and N theta > 1, we find that plasticity will be maintained for an average of at least 10(8) generations in a single population, provided tau > 18/theta. In a metapopulation, plasticity is retained under the more lenient condition tau > 1.3/theta, irrespective of mN, for a modest number of demes. We calculate both exact and approximate solutions for tau and find that it is linearly dependent only on the logarithm of N, and so, surprisingly, both the population size and the number of demes in the metapopulation make little difference to the retention of plasticity. Instead, tau is dominated by the term 1/(m+s/2).     

THE PROBABILITY OF FIXATION OF A NEW KARYOTYPE IN A CONTINUOUS POPULATION

We investigate the probability of fixation of a chromosome rearrangement in a subdivided population, concentrating on the limit where migration is so large relative to selection (m ? s) that the population can be thought of as being continuously distributed. We study two demes, and one- and two-dimensional populations. For two demes, the probability of fixation in the limit of high migration approximates that of a population with twice the size of a single deme: migration therefore greatly reduces the fixation probability. However, this behavior does not extend to a large array of demes. Then, the fixation probability depends primarily on neighborhood size (Nb), and may be appreciable even with strong selection and free gene flow (≈exp(-B ≈ Nb√s) in one dimension, ≈exp(-B ≈ Nb) in two dimensions). Our results are close to those for the more tractable case of a polygenic character under disruptive selection.     

Dynamics of Fst for the island model

F(st) is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that F(st)=1/1+2Nm in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population. Natural populations are not infinite and therefore they change over time due to genetic drift. In a finite population, F(st) becomes a random variable that evolves over time. In this work we ask, given an initial population state, what are the dynamics of the mean and variance of F(st) under the finite island model? In application both of these quantities are critical in the evaluation of F(st) data. We show that after a time of order N generations the mean of F(st) is slightly biased below 1/1+2Nm. Further we show that the variance of F(st) is of order 1/d where d is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.