Pattern‐oriented modelling of population genetic structure

Although several statistical approaches can be used to describe patterns of genetic variation and infer stochastic differentiation, selective responses, or interruptions of gene flow due to physical or environmental barriers, it is worthwhile to note that similar processes, controlled by several parameters in theoretical models, frequently give rise to similar patterns. Here, we develop a Pattern‐Oriented Modelling (POM) approach that allows us to determine how a complex set of parameters potentially driving empirical genetic differentiation among populations generate alternative scenarios that can be fitted to observed data. We generated 10 000 random combinations of parameters related to population size, gene flow and response to gradients (both driven by dispersal and selection) in a spatially explicit model, and analysed simulated patterns with F_ST statistics and mean correlograms using Moran's I spatial autocorrelation coefficients. These statistics were compared with observed patterns for a tree species endemic to the Brazilian Cerrado. For a best match with observed F_ST (equal to 0.182), the important parameters driving simulated scenario are mainly related to population structure, including low population size with closed populations (low N_m), strong distance decay of gene flow, in addition to a strong effect of the initial variance of allele frequencies. These scenarios present a low autocorrelation of allele frequencies. Best matching of correlograms, on the other hand, appears in simulations with a large population size, high N_m and low population differentiation and F_ST (as well as more gene flow). Thus, targeting the two statistics (correlograms and F_ST) shows that best matches with empirical data with two distinct sets of parameters in the simulations, because observed patterns involve both a relatively high F_ST and significant autocorrelation. This conflict can be resolved by assuming that initial variance in allele frequencies can be interpreted as reflecting deep‐time historical variation and evolutionary dynamics of allele frequencies, creating a relatively high level of population differentiation, whereas current patterns in gene flow creates spatial autocorrelation. This make sense in terms of the previous knowledge on population differentiation in D. alata, especially if patterns are explained by a combination of isolation‐by‐distance and allelic surfing due to range expansion after the last glacial maximum. This reveals the potential for more complex applications of POM in population genetics. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1152–1161.     

DYNAMICS OF SEXUAL SELECTION IN DIPLOID POPULATIONS

We describe results for a diploid, two-locus model for the evolution of a female mating preference directed at an attractive male trait that is subject to viability and/or fertility selection. Using computer simulation, we studied a large, random sample of parameter values, assuming additivity of alleles at the preference locus and partial dominance at the trait locus. Simulation results were classifiable into nine types of parameter sets, each differing in equilibria, evolutionary trajectories, and rates of evolution. For many parameters, evolutionary trajectories converged on curves within the allelic frequency plane and subsequently evolved along the curves toward fixation. Neutrally stable curves of equilibria did not occur in Fisherian models that assume only viability and sexual selection unless there is complete dominance at the trait locus. The Fisherian models also exhibited oscillation of allelic frequencies and unique polymorphic equilibria. “Sexy son” models in which attractive males had reduced fertility were much less likely to lead to increase in traits and preferences than were the Fisherian models. However, if less fertile males had increased viability, trait polymorphisms and fixation of rare “sexy” alleles occurred. In general, the behavior of the diploid model was much more complex than that of analogous haploid or polygenic models.     

James F. Crow and the stochastic theory of population genetics

Research in population genetics theory has two main strands. The first is deterministic theory, where random changes in allelic frequencies are ignored and attention focuses on the evolutionary effects of selection and mutation. The second strand, stochastic theory, takes account of these random changes and thus is more complete than deterministic theory. This essay is one in the series of Perspectives and Reviews honoring James F. Crow on the occasion of his 95th birthday. It concerns his contributions to, and involvement with, the stochastic theory of evolutionary population genetics.     

Does spatial genetic structure increase with altitude? An answer from Picea abies in Tyrol, Austria

Harsh environment at high altitude may affect the mating system of plant species, especially those with wide ecological amplitude. Smaller effective neighbourhood size, less pollen and seed production, higher rate of inbreeding and a shift towards vegetative propagation may be involved. These changes can be reflected in spatial genetic structure (SGS). Populations of Norway spruce [Picea abies (L.) Karst.] were analysed along an altitudinal cline to verify whether SGS increases with altitude. Three putatively autochthonous populations in Tyrol (Austria) at 800, 1,200 and 1,600?m above sea level (asl) were studied. Six highly polymorphic DNA markers (expressed sequence tag?Cderived simple sequence repeats, EST-SSRs) were used to genotype a total of 450 contiguous trees (150 trees per population). Loiselle??s kinship coefficient was used to quantify SGS. Against expectation no significant SGS was found in any of the populations, indicating a random spatial pattern. Significant SGS was observed when all populations were treated as a single one conforming to an isolation-by-distance pattern. Nearly identical allelic frequencies were found resulting in very small population differentiation (F _ST?=?0.002). The fixation index decreased with diameter at breast height (a proxy for age) indicating natural selection against inbred trees. The results of this study indicate that seed and pollen dispersal mechanisms in Norway spruce are strongly counteracting spatial aggregation of similar genotypes even at high elevations.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

Genetic structure of populations of the threatened eastern massasauga rattlesnake, Sistrurus c. catenatus: evidence from microsatellite DNA markers

Throughout its distribution in North America, the threatened eastern massasauga rattlesnake ( Sistrurus c. catenatus ) persists in a series of habitat-isolated disjunct populations of varying size. Here, we use six microsatellite DNA loci to generate information on the degree of genetic differentiation between, and the levels of inbreeding within populations to understand how evolutionary processes operate in these populations and aid the development of conservation plans for this species. Samples were collected from 199 individuals from five populations in Ontario, New York and Ohio. Our results show that all sampled populations: (i) differ significantly in allele frequencies even though some populations are < 50 km apart, and may contain genetically distinct subpopulations < 2 km apart; (ii) have an average of 23% of alleles that are population specific; and (iii) have significant F _IS values (mean overall F _IS= 0.194) probably due to a combination of Wahlund effects resulting from fine-scale genetic differentiation within populations and the presence of null alleles. Our results imply that massasauga populations may be genetically structured on an extremely fine scale even within continuous populations, possibly due to limited dispersal. Additional information is needed to determine if dispersal and mating behaviour within populations can account for this structure and whether the observed differentiation is due to random processes such as drift or to local adaptation. From a conservation perspective, our results imply that these massasauga populations should be managed as demographically independent units and that each has high conservation value in terms of containing unique genetic variation.     

Genetic variability in bryophytes: does mating system really matter?

Abstract

Mating patterns do not alter allelic frequencies in natural populations. Therefore, if evolution is defined as changes in allelic frequencies in a population through time, then mating system is not an evolutionary force. There is no direct causal link between mating patterns and the familiar measures of genetic variability applied to allele frequency data from haploid gametophyte populations. In this paper we focus on how mating patterns may affect intralocus variability when acting together with evolutionary forces, like selection and drift. Moreover, we address how levels of inbreeding can have a profound influence on haplotypic variation. Measures for estimating the extent of recombination from haplotypic variation are presented. It seems that high levels of selfing are not necessarily associated with bisexuality in bryophytes.     

Cautions on direct gene flow estimation in plant populations

Through simulations we have investigated the statistical properties of two of the main approaches for directly estimating pollen gene flow (m) in plant populations: genotypic exclusion and mating models. When the assumptions about accurately known background pollen pool allelic frequencies are met, both methods provide unbiased results with comparable variances across a range of true m values. However, when presumed allelic frequencies differ from actual ones, which is more likely in research practice, both estimators are biased. We demonstrate that the extent and direction of bias largely depend on the difference (measured as genetic distance) between the presumed and actual pollen pools, and on the degree of genetic differentiation between the local population and the actual background pollen sources. However, one feature of the mating model is its ability to estimate pollen gene flow simultaneously with background pollen pool allelic frequencies. We have found that this approach gives nearly unbiased pollen gene flow estimates, and is practical because it eliminates the necessity of providing independent estimates of background pollen pool allelic frequencies. Violations of the mating model assumptions of random mating within local population affect the precision of the estimates only to a limited degree.     

Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns?1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.     

Clarification of genetic terms and their use in the management of captive populations

Some of the concepts, terms, and methods used in the genetic management of captive populations have not been defined precisely in the scientific literature and consequently have been misunderstood and misused. The definitions and interrelationships among gene diversity, effective population size, founder genome equivalents, inbreeding, allelic diversity, mean kinship, and kinship value are presented here. It is important to understand what populations and generations are used as the baselines against which losses of genetic variation are measured. Gene diversity and founder genome equivalents are defined relative to a source population from which founders of the captive population were randomly sampled. Inbreeding and allelic diversity are assessed relative to the founders. The potential gene diversity that would result from an equalization of frequencies of founder alleles retained in the population can never be achieved because, among other limitations, the random process of gene transmission will prevent equalization of allele frequencies even if animals are bred optimally. The gene diversity achievable with the population can be determined by iterative production of hypothetical offspring from the pairs with lowest mean kinship. The long-term objective for offspring production from each animal is also thereby generated. Mean kinships should be recalculated with each real or hypothetical birth and death, because offspring objectives based on current mean kinships might correlate poorly with the optimal long-term offspring objectives. © 1995 Wiley-Liss, Inc.     

Fixation probability with multiple alleles and projected average allelic effect on selection

The first-order effect of selection on the probability of fixation of an allele, with respect to an intensity of selection s>0 in a diploid population of fixed finite size N, undergoing discrete, non-overlapping generations, is shown to be given by the sum of the average effects of that allele on the coefficient of selection in the current generation and all future generations, given the population state in the current generation. This projected average allelic effect is a weighted sum of average allelic effects in allozygous and autozygous offspring in the initial generation, with weights given in terms of expected coalescence times, under neutrality, for the lineages of two or three gametes chosen at random in the same generation. This is shown in the framework of multiple alleles at one locus, with genotypic values determining either viability or fertility differences, and with either multinomial or exchangeable reproduction schemes. In the limit of weak selection in a large population such that Ns tends to zero, the initial average allelic effects in allozygous offspring and autozygous offspring have the same weight on the fixation probability only in the domain of application of the Kingman coalescent. With frequency-dependent selection in a linear-game-theoretic context with two phenotypes determined by additive gene action, the first-order effect on the fixation probability is a combination of two effects of frequency-independent selection, one in a haploid population, the other in a diploid population. In the domain of application of the Kingman coalescent as the population size goes to infinity and Ns to zero, the first effect is three times more important than the second effect. This explains the one-third law of evolutionary dynamics in this domain, and shows how this law can be extended beyond this domain.     

GENETIC DIVERSITY AND POPULATION STRUCTURE IN CAMELLIA JAPONICA L. (THEACEAE)

Camellia japonica is a widespread and morphologically diverse tree native to parts of Japan and adjacent islands. Starch gel electrophoresis was used to score allelic variation at 20 loci in seeds collected from 60 populations distributed throughout the species range. In comparison with other plant species, the level of genetic diversity within C. japonica populations is very high: 66.2% of loci were polymorphic on average per population, with a mean number of 2.16 alleles per locus; the mean observed and panmictic heterozygosities were 0.230 and 0.265, respectively. Genotypic proportions at most loci in most populations fit Hardy-Weinberg expectations. However, small heterozygote deficiencies were commonly observed (mean population fixation index = 0.129). It is suggested that the most likely cause of the observed deficiencies is population subdivision into genetically divergent subpopulations. The overall level of population differentiation is greater than is typically observed in out-breeders: The mean genetic distance and identity (Nei's D and I) between pairs of populations were 0.073 and 0.930, respectively, and Wright's Fst was 0.144. Differences among populations appeared to be manifested as variation in gene frequencies at many loci rather than variation in allelic composition per se. However, the patterns of variation were not random. Reciprocal clinal variation of gene frequencies was observed for allele pairs at six loci. In addition, principal components analysis revealed that populations tended to genetically cluster into four regions representing the geographic areas Kyushu, Shikoku, western Honshu, and eastern Honshu. There was a significant relationship between genetic and geographic distance (r = 0.61; P < 0.01). Analysis of variance on allozyme frequencies showed that there was approximately four times as much differentiation among populations within regions, as among regions. It is likely that the observed patterns of population relationships result from the balance between genetic drift in small subpopulations and gene flow between them.     

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

The emergence and persistence of polymorphism within populations generally requires specific regimes of natural or sexual selection. Here, we develop a unified theoretical framework to explore how polymorphism at targeted loci can be generated and maintained by either disassortative mating choice or balancing selection due to, for example, heterozygote advantage. To this aim, we model the dynamics of alleles at a single locus A in a population of haploid individuals, where reproductive success depends on the combination of alleles carried by the parents at locus A. Our theoretical study of the model confirms that the conditions for the persistence of a given level of allelic polymorphism depend on the relative reproductive advantages among pairs of individuals. Interestingly, equilibria with unbalanced allelic frequencies were shown to emerge from successive introduction of mutants. We then investigate the role of the function linking allelic divergence to reproductive advantage on the evolutionary fate of alleles within the population. Our results highlight the significance of the shape of this function for both the number of alleles maintained and their level of genetic divergence. Large number of alleles are maintained with substantial replacement of alleles, when disassortative advantage slowly increases with allelic differentiation . In contrast, few highly differentiated alleles are predicted to be maintained when genetic differentiation has a strong effect on disassortative advantage. These opposite effects predicted by our model explain how disassortative mate choice may lead to various levels of allelic differentiation and polymorphism, and shed light on the effect of mate preferences on the persistence of balanced and unbalanced polymorphism in natural population.

     

Decoupling of differentiation between traits and their underlying genes in response to divergent selection

We dissected the relationship between genetic differentiation (Q(ST)) for a trait and its underlying genes (G(STq), differentiation for a quantitative locus) in an evolutionary context, with the aim of identifying the conditions in which these two measurements are decoupled. We used two parameters (θ(B) and θ(W)) scaling the contributions of inter- and intrapopulation allelic covariation between genes controlling the trait of interest. We monitored the changes in θ(B) and θ(W), Q(ST) and G(STq) over successive generations of divergent and stabilizing selection, in simulations for an outcrossing species with extensive gene flow. The dynamics of these parameters are characterized by two phases. Initially, during the earliest generations, differentiation of the trait increases very rapidly and the principal and immediate driver of Q(ST) is θ(B). During subsequent generations, G(STq) increases steadily and makes an equal contribution to Q(ST). These results show that selection first captures beneficial allelic associations at different loci at different populations, and then targets changes in allelic frequencies. The same patterns are observed when environmental change modifies divergent selection, as shown by the very rapid response of θ(B) to the changes of selection regimes. We compare our results with previous experimental findings and consider their relevance to the detection of molecular signatures of natural selection.     

Selection with two alleles and complete dominance

For a plant selection model with frequency-independent viabilities, fertilities and selfing rates, it is shown that apart from global fixation, for certain parameter combinations a protected polymorphism and facultative fixation (either allele may become fixed according to initial frequencies) may both occur. Facultative fixation requires different selling rates for the dominant and recessive type. Protection of the polymorphism requires resource allocation for male and female function. In this connection the problem of purely genetically caused population extinction is discussed.
For general frequency dependence and regular segregation, the chances for establishment of a completely recessive gene are compared to those of a completely dominant gene. It is proven that the process of establishment of the recessive gene, despite a fitness advantage, may be considerably endangered by drift effects if random mating prevails. The recessive gene may reach the same effectivity in establishment as a dominant gene, only if the recessive homozygote mates exclusively with its own type during the period of establishment.     

The resolution of sexual antagonism by gene duplication

Disruptive selection between males and females can generate sexual antagonism, where alleles improving fitness in one sex reduce fitness in the other. This type of genetic conflict arises because males and females carry nearly identical sets of genes: opposing selection, followed by genetic mixing during reproduction, generates a population genetic "tug-of-war" that constrains adaptation in either sex. Recent verbal models suggest that gene duplication and sex-specific cooption of paralogs might resolve sexual antagonism and facilitate evolutionary divergence between the sexes. However, this intuitive proximal solution for sexual dimorphism potentially belies a complex interaction between mutation, genetic drift, and positive selection during duplicate fixation and sex-specific paralog differentiation. The interaction of these processes--within the explicit context of duplication and sexual antagonism--has yet to be formally described by population genetics theory. Here, we develop and analyze models of gene duplication and sex-specific differentiation between paralogs. We show that sexual antagonism can favor the fixation and maintenance of gene duplicates, eventually leading to the evolution of sexually dimorphic genetic architectures for male and female traits. The timescale for these evolutionary transitions is sensitive to a suite of genetic and demographic variables, including allelic dominance, recombination, sex linkage, and population size. Interestingly, we find that female-beneficial duplicates preferentially accumulate on the X chromosome, whereas male-beneficial duplicates are biased toward autosomes, independent of the dominance parameters of sexually antagonistic alleles. Although this result differs from previous models of sexual antagonism, it is consistent with several findings from the empirical genomics literature.     

Effect of epistasis and linkage on fixation probability in three-locus models: an ancestral recombination-selection graph approach

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination-Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained. The coefficients of the expansion are expressed in terms of the initial state of the population and the epistatic interactions among the selected loci. Under the assumption of weak selection at tightly linked loci, the sign of the leading term, which depends on the signs of epistasis and initial linkage disequilibrium, determines whether an increase in recombination rates increases the chance of ultimate fixation of the new mutant. If mutants are advantageous, this is the case when epistasis is positive or null and the initial linkage disequilibrium is negative, which is an expected state in a finite population under directional selection. Moreover, this is also the case for a neutral mutant modifier coding for higher recombination rates if the same conditions hold at the selected loci. Under the same conditions, deleterious mutants are disfavored for ultimate fixation and neutral modifiers for higher recombination rates still favored. The recombination rates between the modifier locus and the selected loci do not come into play in the leading terms of the approximation for the fixation probability, but they do in higher-order terms.     

Demographic and genetic structures of two partially isolated communities of Santa Catarina Island,southern Brazil

The objectives of this study were to analyze the population structure and genetic variability of two communities, Costa da Lagoa (CLG) and S?o Jo?o do Rio Vermelho (SJRV), located on Santa Catarina Island in southern Brazil. The two populations descend from Azores Archipelago immigrants (Portuguese), with a minor contribution of sub-Saharan Africans and Amerindians. To estimate the relative contribution of the different ethnic groups to the current gene pool of the two communities, values of admixture were obtained using the weighted least-squares method based on allelic frequencies of the loci ABO, RHD-RHCE, GPA-GPB (MNSs), HBB, HP, TF, CP, AK, and ACP1. The origins of the studied populations can be quantified as follows: for CLG, sub-Saharan Africans (A) = 17.3%, Iberian Europeans (P) = 75.0%, and Southern Amerindians (I) = 7.7%; for SJRV, A = 48.8%, P = 44.5%, and I = 6.7%. Because haplotype frequencies of the GPA-GPB loci in SJRV were unusual, possibly as a consequence of random genetic drift, the values of admixture were recalculated after exclusion of GPA-GPB, as follows: A = 28.0%; P = 53.3%, and I = 18.7%. The total diversity (HT) was estimated as 42.29%, of which 99.6% can be attributed to the intrapopulational variability (HS). The interpopulational genetic variation (or standard distance, DST) corresponds to 0.19%, while the gene differentiation coefficient is 0.28%, indicative of low genetic difference. These results led to the conclusion that random genetic drift may have had an important effect on the Costa da Lagoa community, while presently gene flow might be the predominant evolutionary factor potentially capable of changing allele frequencies in SJRV.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.