Influence of Gene Flow and Breeding Tactics on Gene Diversity within Populations

Expressions describing the accumulation of gene correlations within and among lineages and individuals of a population are derived. The model permits different migration rates by males and females and accounts for various breeding tactics within lineages. The resultant equations enable calculation of the probabilistic quantities for the fixation indices, rates of loss of genetic variation, accumulation of inbreeding, and coefficients of relationship for the population at any generation. All fixation indices were found to attain asymptotic values rapidly despite the consistent loss of genetic variation and accumulation of inbreeding within the population. The time required to attain asymptotic values, however, was prolonged when gene flow among lineages was relatively low (less than 20%). The degree of genetic differentiation among breeding groups, inbreeding coefficients, and gene correlations within lineages were found to be primarily functions of breeding tactics within groups rather than gene flow among groups. Thus, the asymptotic value of S. Wright's island model is not appropriate for describing genetic differences among groups within populations. An alternative solution is provided that under limited conditions will reduce to the original island model. The evolution of polygynous breeding tactics appears to be more favorable for promoting intragroup gene correlations than modification of migration rates. Inbreeding and variance effective sizes are derived for populations that are structured by different migration and breeding tactics. Processes that reduce the inbreeding effective population size result in a concomitant increase in variance effective population size.     

Effective Population Sizes with Multiple Paternity

While the concept of effective population size is of obvious applicability to many questions in population genetics and conservation biology, its utility has suffered due to a lack of agreement among its various formulations. Often, mathematical formulations for effective sizes apply restrictive assumptions that limit their applicability. Herein, expressions for effective sizes of populations that account for mating tactics, biases in sex ratios, and differential dispersal rates (among other parameters) are developed. Of primary interest is the influence of multiple paternity on the maintenance of genetic variation in a population. In addition to the standard inbreeding and variance effective sizes, intragroup (coancestral) and intergroup effective sizes also are developed. Expressions for effective sizes are developed for the beginning of nonrandom gene exchanges (initial effective sizes), the transition of gene correlations (instantaneous effective sizes), and the steady-state (asymptotic effective size). Results indicate that systems of mating that incorporate more than one male mate per female increase all effective sizes above those expected from polygyny and monogamy. Instantaneous and asymptotic sizes can be expressed relative to the fixation indices. The parameters presented herein can be utilized in models of effective sizes for the study of evolutionary biology and conservation genetics.     

Effective Size and F-Statistics of Subdivided Populations. I. Monoecious Species with Partial Selfing

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Effective Size and F-Statistics of Subdivided Populations. II. Dioecious Species

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Methods to estimate effective population size using pedigree data: Examples in dog,sheep, cattle and horse

Background

Effective population sizes of 140 populations (including 60 dog breeds, 40 sheep breeds, 20 cattle breeds and 20 horse breeds) were computed using pedigree information and six different computation methods. Simple demographical information (number of breeding males and females), variance of progeny size, or evolution of identity by descent probabilities based on coancestry or inbreeding were used as well as identity by descent rate between two successive generations or individual identity by descent rate.

Results

Depending on breed and method, effective population sizes ranged from 15 to 133 056, computation method and interaction between computation method and species showing a significant effect on effective population size (P < 0.0001). On average, methods based on number of breeding males and females and variance of progeny size produced larger values (4425 and 356, respectively), than those based on identity by descent probabilities (average values between 93 and 203). Since breeding practices and genetic substructure within dog breeds increased inbreeding, methods taking into account the evolution of inbreeding produced lower effective population sizes than those taking into account evolution of coancestry. The correlation level between the simplest method (number of breeding males and females, requiring no genealogical information) and the most sophisticated one ranged from 0.44 to 0.60 according to species.

Conclusions

When choosing a method to compute effective population size, particular attention should be paid to the species and the specific genetic structure of the population studied.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

Local extinction and recolonization, species effective population size, and modern human origins

A primary objection from a population genetics perspective to a multiregional model of modern human origins is that the model posits a large census size, whereas genetic data suggest a small effective population size. The relationship between census size and effective size is complex, but arguments based on an island model of migration show that if the effective population size reflects the number of breeding individuals and the effects of population subdivision, then an effective population size of 10,000 is inconsistent with the census size of 500,000 to 1,000,000 that has been suggested by archeological evidence. However, these models have ignored the effects of population extinction and recolonization, which increase the expected variance among demes and reduce the inbreeding effective population size. Using models developed for population extinction and recolonization, we show that a large census size consistent with the multiregional model can be reconciled with an effective population size of 10,000, but genetic variation among demes must be high, reflecting low interdeme migration rates and a colonization process that involves a small number of colonists or kin-structured colonization. Ethnographic and archeological evidence is insufficient to determine whether such demographic conditions existed among Pleistocene human populations, and further work needs to be done. More realistic models that incorporate isolation by distance and heterogeneity in extinction rates and effective deme sizes also need to be developed. However, if true, a process of population extinction and recolonization has interesting implications for human demographic history.     

The effective size of mixed sexually and asexually reproducing populations

Using the transition matrix of inbreeding and coancestry coefficients, the inbreeding (N(eI)), variance (N(eV)), and asymptotic (N(e lambda)) effective sizes of mixed sexual and asexual populations are formulated in terms of asexuality rate (delta), variance of asexual (C) and sexual (K) reproductive contributions of individuals, correlation between asexual and sexual contributions (rho(ck)), selfing rate (beta), and census population size (N). The trajectory of N(eI) toward N(e lambda) changes crucially depending on delta, N, and beta, whereas that of N(eV) is rather consistent. With increasing asexuality, N(e lambda) either increases or decreases depending on C, K, and rho(ck). The parameter space in which a partially asexual population has a larger N(e lambda) than a fully sexual population is delineated. This structure is destroyed when N(1 - delta) < 1 or delta > 1 - 1/N. With such a high asexuality, tremendously many generations are required for the asymptotic size N(e lambda) to be established, and N(e lambda) is extremely large with any value of C, K, and rho(ck) because the population is dominated eventually by individuals of the same genotype and the allelic diversity within the individuals decays quite slowly. In reality, the asymptotic state would occur only occasionally, and instantaneous rather than asymptotic effective sizes should be practical when predicting evolutionary dynamics of highly asexual populations.     

Inbreeding depression due to mildly deleterious mutations in finite populations: size does matter

We studied the effects of population size on the inbreeding depression and genetic load caused by deleterious mutations at a single locus. Analysis shows how the inbreeding depression decreases as population size becomes smaller and/or the rate of inbreeding increases. This pattern contrasts with that for the load, which increases as population size becomes smaller but decreases as inbreeding rate goes up. The depression and load both approach asymptotic limits when the population size becomes very large or very small. Numerical results show that the transition between the small and the large population regimes is quite rapid, and occurs largely over a range of population sizes that vary by a factor of 10. The effects of drift on inbreeding depression may bias some estimates of the genomic rate of deleterious mutation. These effects could also be important in the evolution of breeding systems in hermaphroditic organisms and in the conservation of endangered populations.     

The Inbreeding Effective Population Number and the Expected Homozygosity for an X-Linked Locus

Assuming random mating and discrete nonoverlapping generations, the inbreeding effective population number, (see PDF), is calculated for an X-linked locus. For large populations, the result agrees with the variance effective population number. As an application, the maintenance of genetic variability by the joint action of mutation and random drift is investigated. It is shown that, if every allele mutates at rate u to new types, then the probabilities of identity in state (and hence the expected homozygosity of females) converge to the approximate value (see PDF) at the approximate asymptotic rate (see PDF).     

Pedigree analysis on small laboratory populations of the butterfly Bicyclus anynana: The effects of selection on inbreeding and fitness

The effect of small population size and gene flow on the rate ofinbreeding and loss in fitness in Bicyclus anynana populationswas quantified by means of a pedigree analysis. Laboratorymetapopulations each consisted of four subpopulations with breeding sizeof N = 6 or N = 12 and migration rate of m = 0 or m= 0.33. Pedigrees were established by individually marking about35,000 butterflies. The increase in inbreeding coefficients(F-coefficients) over time was compared to that of simulated populationswith similar N and m. In the seventh generation, the level of inbreedingin larger subpopulations did not deviate significantly from the expectedvalues, but smaller subpopulations were less inbred than expected.Individuals in the small populations still showed considerableinbreeding depression, indicating that only a small proportion of therecessive deleterious alleles had been purged by selection. Two opposingprocesses potentially affected the rate of inbreeding and fitness: (1)Inbreeding depression increased the variance in family size and reducedthe effective population size. This will accelerate the rate ofinbreeding and is expected to selectively purge deleterious recessivealleles. (2) Variance in reproductive success of families was reducedbecause individuals which had a large number of siblings in thepopulation were more likely to mate with a full-sib than individualswith a smaller number of siblings. Subsequent inbreeding depressionreduced the number of viable offspring produced by these full-sibmatings. As a consequence, natural selection purged only some of thedeleterious alleles from the butterfly populations during sevengenerations with inbreeding. These findings emphasise the potentialproblems of using only small numbers of breeding individuals (N10) incaptive populations for conservation purposes.     

Predicting the annual effective size of livestock populations

Effective population size (Ne) is an important parameter determining the genetic structure of small populations. In natural populations, the number of adults (N) is usually known and Ne can be estimated on the basis of an assumed ratio Ne/N, usually found to be close to 0.5. In farm animal populations, apart from using pedigrees or genetic marker information, Ne can be estimated from the number N of breeding animals, and a value of 1 is commonly assumed for the ratio Ne/N. The purpose of this paper is to show the relation between effective population size and breeding herd size in livestock species. With overlapping generations, Ne can be predicted knowing the number of individuals entering the population per generation and the variance of family size, the latter being directly related to the survival pattern (or replacement policy) in the breeding herd. Assuming an ideal survivorship leading to a geometric age distribution, it can be shown that the number of breeding animals tends to overestimate effective size, particularly in early-maturing species. The ratio of annual effective size to the number of breeding animals is shown to be equal to [1 + (a- 1)(1 - s)]2/(1 - s2), where a is the age at first offspring and s is the survival rate (including culling) of the parents between successive births. This expression shows to what extent inbreeding may be determined by demography or culling policy independently of the actual herd size. In many situations a fast replacement or an early culling will increase annual effective size. Consequences for the management of small populations are discussed.     

Effective Size of Nonrandom Mating Populations

Nonrandom mating whereby parents are related is expected to cause a reduction in effective population size because their gene frequencies are correlated and this will increase the genetic drift. The published equation for the variance effective size, Ne, which includes the possibility of nonrandom mating, does not take into account such a correlation, however. Further, previous equations to predict effective sizes in populations with partial sib mating are shown to be different, but also incorrect. In this paper, a corrected form of these equations is derived and checked by stochastic simulation. For the case of stable census number, N, and equal progeny distributions for each sex, the equation is [formula: see text], where Sk2 is the variance of family size and alpha is the departure from Hardy-Weinberg proportions. For a Poisson distribution of family size (Sk2 = 2), it reduces to Ne = N/(1 + alpha), as when inbreeding is due to selfing. When nonrandom mating occurs because there is a specified system of partial inbreeding every generation, alpha can be substituted by Wright's FIS statistic, to give the effective size as a function of the proportion of inbred mates.     

Do estimates of contemporary effective population size tell us what we want to know?

Estimation of effective population size (N_e) from genetic marker data is a major focus for biodiversity conservation because it is essential to know at what rates inbreeding is increasing and additive genetic variation is lost. But are these the rates assessed when applying commonly used N_e estimation techniques? Here we use recently developed analytical tools and demonstrate that in the case of substructured populations the answer is no. This is because the following: Genetic change can be quantified in several ways reflecting different types of N_e such as inbreeding (N_eI), variance (N_eV), additive genetic variance (N_eAV), linkage disequilibrium equilibrium (N_eLD), eigenvalue (N_eE) and coalescence (N_eCo) effective size. They are all the same for an isolated population of constant size, but the realized values of these effective sizes can differ dramatically in populations under migration. Commonly applied N_e‐estimators target N_eV or N_eLD of individual subpopulations. While such estimates are safe proxies for the rates of inbreeding and loss of additive genetic variation under isolation, we show that they are poor indicators of these rates in populations affected by migration. In fact, both the local and global inbreeding (N_eI) and additive genetic variance (N_eAV) effective sizes are consistently underestimated in a subdivided population. This is serious because these are the effective sizes that are relevant to the widely accepted 50/500 rule for short and long term genetic conservation.  The bias can be infinitely large and is due to inappropriate parameters being estimated when applying theory for isolated populations to subdivided ones.     

Dynamics of inbreeding depression due to deleterious mutations in small populations: mutation parameters and inbreeding rate

A multilocus stochastic model is developed to simulate the dynamics of mutational load in small populations of various sizes. Old mutations sampled from a large ancestral population at mutation-selection balance and new mutations arising each generation are considered jointly, using biologically plausible lethal and deleterious mutation parameters. The results show that inbreeding depression and the number of lethal equivalents due to partially recessive mutations can be partly purged from the population by inbreeding, and that this purging mainly involves lethals or detrimentals of large effect. However, fitness decreases continuously with inbreeding, due to increased fixation and homozygosity of mildly deleterious mutants, resulting in extinctions of very small populations with low reproductive rates. No optimum inbreeding rate or population size exists for purging with respect to fitness (viability) changes, but there is an optimum inbreeding rate at a given final level of inbreeding for reducing inbreeding depression or the number of lethal equivalents. The interaction between selection against partially recessive mutations and genetic drift in small populations also influences the rate of decay of neutral variation. Weak selection against mutants relative to genetic drift results in apparent overdominance and thus an increase in effective size (Ne) at neutral loci, and strong selection relative to drift leads to a decrease in Ne due to the increased variance in family size. The simulation results and their implications are discussed in the context of biological conservation and tests for purging.     

Effects of population size and isolation on heterosis, mean fitness, and inbreeding depression in a perennial plant

? In small isolated populations, genetic drift is expected to increase chance fixation of partly recessive, mildly deleterious mutations, reducing mean fitness and inbreeding depression within populations and increasing heterosis in outcrosses between populations. ? We estimated relative effective sizes and migration among populations and compared mean fitness, heterosis, and inbreeding depression for eight large and eight small populations of a perennial plant on the basis of fitness of progeny produced by hand pollinations within and between populations. ? Migration was limited, and, consistent with expectations for drift, mean fitness was 68% lower in small populations; heterosis was significantly greater for small (mean?=?70%, SE?=?14) than for large populations (mean?=?7%, SE?=?27); and inbreeding depression was lower, although not significantly so, in small (mean?=?-0.29%, SE?=?28) than in large (mean?=?0.28%, SE?=?23) populations. ? Genetic drift promotes fixation of deleterious mutations in small populations, which could threaten their persistence. Limited migration will exacerbate drift, but data on migration and effective population sizes in natural populations are scarce. Theory incorporating realistic variation in population size and patterns of migration could better predict genetic threats to small population persistence.     

Within and between population variation in inbreeding depression in the locally threatened perennial Scabiosa columbaria

Inbreeding depression plays a central role within the conservation genetics paradigm. Until now inbreeding depression is incorporated into models of population viability as a mean value (e.g. number of lethal equivalents) for all traits in a population. In this study of the locally threatened perennial plant species Scabiosa columbaria we investigated both the mean and the variance among families of inbreeding depression in eight life history traits for five natural populations varying in size from 300 to more than 120,000 individuals. Significant inbreeding depression was found in all populations and all traits. The mean inbreeding depression value per trait was never correlated to population size. Within each population, highly significant variation in inbreeding depression between families (VIFLID) was found. Per trait, families with inbreeding depression next to families with outbreeding depression were often found within the same population. Inbreeding depression at the family level was in many cases not correlated among traits and independent of correlations between traits themselves. VIFLID was negatively correlated with population size: in two traits these correlations were significant. The results underline that inbreeding depression is a complex, highly dynamic phenomenon. Models of viability should incorporate inbreeding depression distributions, with a trait specific mean and variance. Moreover, models of metapopulation dynamics should incorporate genotype quality as factor in colonization success.     

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).     

Impact of density and environmental factors on population fluctuations in a migratory passerine

1. Populations of plants and animals typically fluctuate because of the combined effects of density-dependent and density-independent processes. The study of these processes is complicated by the fact that population sizes are typically not known exactly, because population counts are subject to sampling variance. Although the existence of sampling variance is broadly acknowledged, relatively few studies on time-series data have accounted for it, which can result in wrong inferences about population processes. 2. To increase our understanding of population dynamics, we analysed time series from six Central European populations of the migratory red-backed shrike Lanius collurio by simultaneously assessing the strength of density dependence, process and sampling variance. In addition, we evaluated hypotheses predicting effects of factors presumed to operate on the breeding grounds, at stopover sites in eastern Africa during fall and spring migration and in the wintering grounds in southern Africa. We used both simple and state-space formulations of the Gompertz equation to model population size. 3. Across populations and modelling approaches, we found consistent evidence for negative density-dependent population regulation. Further, process variance contributed substantially to variation in population size, while sampling variance did not. Environmental conditions in eastern and southern Africa appear to influence breeding population size, as rainfall in the Sahel during fall migration and in the south African wintering areas were positively related to population size in the following spring in four of six populations. In contrast, environmental conditions in the breeding grounds were not related to population size. 4. Our findings suggest negative density-dependent regulation of red-backed shrike breeding populations and are consistent with the long-standing hypothesis that conditions in the African staging and wintering areas influence population numbers of species breeding in Europe. 5. This study highlights the importance of jointly investigating density-dependent and density-independent processes to improve our understanding of factors influencing population fluctuations in space and time.     

INBREEDING AND VARIANCE EFFECTIVE SIZES FOR NONRANDOM MATING POPULATIONS

Following an inbreeding approach and assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, I have derived expressions for the inbreeding effective size, N_eI, for a finite diploid population with variable census sizes for three cases: monoecious populations with partial selfing; dioecious populations of equal numbers of males and females with partial sib mating; and unequal numbers of males and females with random mating. For the first two cases, recurrence equations for the inbreeding coefficient are also obtained, which allow inbreeding coefficients to be predicted exactly in both early and late generations. Following the variance of change in gene frequency approach, a general expression for variance effective size, N_eV, is obtained for a population with unequal numbers of male and female individuals, arbitrary family size distribution, and nonrandom mating. All the parameters involved are allowed to change over generations. For some special cases, the equation reduces to the simple expressions approximately as derived by previous authors. Comparisons are made between equations derived by the present study and those obtained by previous authors. Some of the published equations for N_eI and N_eV are shown to be incomplete or incorrect. Stochastic simulations are run to check the results where disagreements with others are involved.