Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

The effective size of fluctuating populations

We consider a Wright-Fisher model whose population size is an autocorrelated stochastic process. Our interest is in the effects of autocorrelated fluctuations of the population size on the effective size. We define the inbreeding effective size and the variance effective size and show that these effective sizes are the same for this model. In the literature, it is said that the effective size is equal to the harmonic mean of population size when the size fluctuates. We will show, however, that the effective size is not the same as the harmonic mean of population size unless the fluctuations of population size are uncorrelated. The effective size is larger (resp. smaller) than the harmonic mean when the fluctuations of population size are positively (resp. negatively) autocorrelated. Further, we obtain some asymptotic expressions for effective size when the population size is very large and/or the autocorrelation of the fluctuation is very strong.     

Substitution rates at neutral genes depend on population size under fluctuating demography and overlapping generations

It is widely accepted that the rate of evolution (substitution rate) at neutral genes is unaffected by population size fluctuations. This result has implications for the analysis of genetic data in population genetics and phylogenetics, and provides, in particular, a justification for the concept of the molecular clock. Here, we show that the substitution rate at neutral genes does depend on population size fluctuations in the presence of overlapping generations. As both population size fluctuations and overlapping generations are expected to be the norm rather than the exception in natural populations, this observation may be relevant for understanding variation in substitution rates within and between lineages.     

A note on effective population size with overlapping generations

A simple derivation is given for a formula obtained previously for the effective size of random-mating populations with overlapping generations. The effective population size is the same as that for a population with discrete generations having the same variance of lifetime family size and the same number of individuals entering the population per generation.     

Effective size of a fluctuating age-structured population

Previous theories on the effective size of age-structured populations assumed a constant environment and, usually, a constant population size and age structure. We derive formulas for the variance effective size of populations subject to fluctuations in age structure and total population size produced by a combination of demographic and environmental stochasticity. Haploid and monoecious or dioecious diploid populations are analyzed. Recent results from stochastic demography are employed to derive a two-dimensional diffusion approximation for the joint dynamics of the total population size, N, and the frequency of a selectively neutral allele, p. The infinitesimal variance for p, multiplied by the generation time, yields an expression for the effective population size per generation. This depends on the current value of N, the generation time, demographic stochasticity, and genetic stochasticity due to Mendelian segregation, but is independent of environmental stochasticity. A formula for the effective population size over longer time intervals incorporates deterministic growth and environmental stochasticity to account for changes in N.     

The effective size of annual plant populations: the interaction of a seed bank with fluctuating population size in maintaining genetic variation

Many annual plant populations undergo dramatic fluctuations in size. Such fluctuations can result in the loss of genetic variability. Here I formalize the potential for a seed bank to buffer against such genetic loss. The average time to seed germination (T) defines the generation time of "annuals" with a seed bank, and assuming random seed germination, I show that, under otherwise ideal conditions, a population's effective size (Ne) equals NT, where N is the number of adult plants. This result supports the general principle that lengthening the prereproductive period increases Ne. When adult numbers vary, Ne at any time depends on N and on the numbers contributing to the seed bank in previous seasons. Averaging these effects over time gives Ne approximately Nh + (T - 1)Na, where Nh and Na are the harmonic and arithmetic means of the adult population. Thus if T > 1, Ne is determined primarily by Na. Simulations showed that until fluctuations in N are large (>25x) this relationship is accurate. I extended the theory to incorporate a selfing rate (S) and reproductive variance (I) through seed production (k), outcrossed pollen (m), and variation in selfing rate: Ne = NT(1 -S/2)/(1 + I) = NT/[1 + FIS)(1 + I)]. Reproductive variance (I) equals [Ik(1 + S)2 + IM(1 - S)2 + 2(1 - S2)Ikm = S2IS(1 + Ik)]/4, , where Ij is the standardized variance (Vj/j2) of factor j and Ikm is the standardized covariance between k and m. These results are applicable to other organisms with a similar life history, such as freshwater crustaceans with diapausing eggs (e.g., tadpole shrimp, clam shrimp, and fairy shrimp) and other semelparous species with discrete breeding seasons and a variable maturation time (e.g., Pacific salmon).     

Seed banks, salmon, and sleeping genes: effective population size in semelparous, age-structured species with fluctuating abundance

Previous studies reached contrasting conclusions regarding how fluctuations in abundance affect Ne in semelparous species with variable age at maturity: that Ne is determined by the arithmetic mean N among the T years within a generation (Ne approximately = T(N)t; monocarpic plants with seed banks) or the harmonic mean (Ne approximately T[symbol: see text]; Pacific salmon). I show that these conclusions arise from different model assumptions rather than inherent differences between the species. Sequentially applying standard, discrete-generation formulas for inbreeding Ne to a series of nominal generations accurately predicts the multigenerational rate of increase in inbreeding. Variability in mean realized reproductive success across years (kt) is the most important factor determining Ne and Ne/N. When abundance is driven by random variation in kt, Ne < or = T[symbol: see text] < T(N)t. With random variation in Nt and constant per capita seed production (C), variation in kt is low and Ne approximately T[symbol: see text]; however, if C varies among years, Ne can be closer to T[symbol: see text]. Because population regulation affects the genetic contribution of entire cohorts of monocarpic perennials, Ne for these species may be more closely approximated by T[symbol: see text] than by T(N)t. With density-dependent compensation, Cov(kt, Nt) < 0, and Ne is further reduced because relatively few breeders make a disproportionate contribution to the next generation.     

Conservation effort and assessment of population size in fluctuating environments

We consider optimal conservation strategies for endangered populations. We assume that the survival of the population is affected by unpredictable environmental fluctuation and can be improved by conservation effort. Furthermore, the exact value of the initial population size is assumed to be unknown. The conservation strategy involves two aspects: investment of assessment effort, to improve the estimate of the initial population size and investment of conservation effort. Both types of effort imply economic costs. The optimal management strategy is assumed to minimize the weighted sum of extinction probability and the economic cost of the conservation and the assessment effort. (1) We first analyse the optimal conservation effort when the current population size is known accurately. (2) Next, we consider the situation in which there is limited information (i.e. a cue) on population size. (3) We subsequently discuss the cases where the cue accuracy can be improved by assessment of the population. We study the optimal level of the assessment effort and discuss its dependence on various parameters.     

Joint estimation of migration rate and effective population size using the island model

Using the island model of population demography, I report that the demographic parameters migration rate and effective population size can be jointly estimated with equilibrium probabilities of identity in state calculated using a sample of genotypes collected at a single point in time from a single generation. The method, which uses moment-type estimators, applies to dioecious populations in which females and males have identical demography and monoecious populations with no selfing and requires that offspring genotypes are sampled following reproduction and prior to migration. I illustrate the estimation procedure using the infinite-island model with no mutation and the finite-island model with three kinds of mutation models. In the infinite-island model with no mutation, the estimators can be expressed as simple functions of estimates of the F-statistic parameters F(IT) and F(ST). In the finite-island model with mutation among k alleles, mutation rate, migration rate, and effective population size can be simultaneously estimated. The estimates of migration rate and effective population size are somewhat robust to violations in assumptions that may arise in empirical applications such as different kinds of mutation models and deviations from temporal equilibrium.     

Estimating effective population size or mutation rate with microsatellites

Microsatellites are short tandem repeats that are widely dispersed among eukaryotic genomes. Many of them are highly polymorphic; they have been used widely in genetic studies. Statistical properties of all measures of genetic variation at microsatellites critically depend upon the composite parameter theta = 4Nmicro, where N is the effective population size and micro is mutation rate per locus per generation. Since mutation leads to expansion or contraction of a repeat number in a stepwise fashion, the stepwise mutation model has been widely used to study the dynamics of these loci. We developed an estimator of theta, theta; (F), on the basis of sample homozygosity under the single-step stepwise mutation model. The estimator is unbiased and is much more efficient than the variance-based estimator under the single-step stepwise mutation model. It also has smaller bias and mean square error (MSE) than the variance-based estimator when the mutation follows the multistep generalized stepwise mutation model. Compared with the maximum-likelihood estimator theta; (L) by, theta; (F) has less bias and smaller MSE in general. theta; (L) has a slight advantage when theta is small, but in such a situation the bias in theta; (L) may be more of a concern.     

Genome size is negatively correlated with effective population size in ray-finned fish

A recent theory suggesting that genome size and complexity can increase as a passive consequence of small effective population size has generated much controversy. In this article, we demonstrate that freshwater fish species, which have smaller effective population sizes than marine fish species, have larger genomes. We show that genome size is negatively correlated with genetic variability, independent of phylogeny, body size and generation time. Genome duplication is also observed predominantly in freshwater fish. These results suggest that the raw materials of complexity originate under conditions of reduced selection efficiency.     

The effective population size of an age-structured population with a sex-linked locus

Let a population have the same age distribution and age-specific sex ratios at times 0, 1, 2,..., and let M, F, and L, respectively, be the numbers of males and females in the youngest age group and the generation interval. It can then be shown that if there is a sex-linked locus the fixation probabilities of a neutral allele are respectively 1/3LM or 1/3LF if the allele first appears in one newborn male or in one newborn female. The effective population size can then be derived. It is the same as for a population with discrete generations having the same means, variances, and covariances of male and female progeny during a lifetime and the same number of individuals entering the population per generation.     

Changing effective population size and the McDonald-Kreitman test

Artifactual evidence of adaptive amino acid substitution can be generated within a McDonald-Kreitman test if some amino acid mutations are slightly deleterious and there has been an increase in effective population size. Here I investigate the conditions under which this occurs. I show that fairly small increases in effective population size can generate artifactual evidence of positive selection if there is no selection upon synonymous codon use. This problem is exacerbated by the removal of low-frequency polymorphisms. However, selection on synonymous codon use restricts the conditions under which artifactual evidence of adaptive evolution is produced.     

Pacific salmon and the coalescent effective population size

Pacific salmon include several species that are both commercially important and endangered. Understanding the causes of loss in genetic variation is essential for designing better conservation strategies. Here we use a coalescent approach to analyze a model of the complex life history of salmon, and derive the coalescent effective population (CES). With the aid of Kronecker products and a convergence theorem for Markov chains with two time scales, we derive a simple formula for the CES and thereby establish its existence. Our results may be used to address important questions regarding salmon biology, in particular about the loss of genetic variation. To illustrate the utility of our approach, we consider the effects of fluctuations in population size over time. Our analysis enables the application of several tools of coalescent theory to the case of salmon.     

Genetic drift and estimation of effective population size

The statistical properties of the standardized variance of gene frequency changes (a quantity equivalent to Wright's inbreeding coefficient) in a random mating population are studied, and new formulae for estimating the effective population size are developed. The accuracy of the formulae depends on the ratio of sample size to effective size, the number of generations involved (t), and the number of loci or alleles used. It is shown that the standardized variance approximately follows the chi(2) distribution unless t is very large, and the confidence interval of the estimate of effective size can be obtained by using this property. Application of the formulae to data from an isolated population of Dacus oleae has shown that the effective size of this population is about one tenth of the minimum census size, though there was a possibility that the procedure of sampling genes was improper.