Selection Response in Finite Populations

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (σ(AW0)(2)) and mutational (σ(M)(2)) variance. The cumulative response to selection until generation t(CR(t)) can be approximated as & where N(e) is the effective population size, σ(AW &)(2) = N(e)σ(M)(2) is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R(0) is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. β is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R(0) is the major determinant of the short term selection response, but σ(M)(2), N(e) and β are also important for the long term. A selection method of high accuracy using family information gives a small N(e) and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.     

Demographic and genetic factors shaping contemporary metapopulation effective size and its empirical estimation in salmonid fish

The preservation of biodiversity requires an understanding of the maintenance of its components, including genetic diversity. Effective population size determines the amount of genetic variance maintained in populations, but its estimation can be complex, especially when populations are interconnected in a metapopulation. Theory predicts that the effective size of a metapopulation (meta-N(e)) can be decreased or increased by population subdivision, but little empirical work has evaluated these predictions. Here, we use neutral genetic markers and simulations to estimate the effective size of a putative metapopulation in Atlantic salmon (Salmo salar). For a weakly structured set of rivers, we find that meta-N(e) is similar to the sum of local deme sizes, whereas higher genetic differentiation among demes dramatically reduces meta-N(e) estimates. Interdemic demographic processes, such as asymmetrical gene flow, may explain this pattern. However, simulations also suggest that unrecognized population subdivision can also introduce downward bias into empirical estimation, emphasizing the importance of identifying the proper scale of distinct demographic and genetic processes. Under natural patterns of connectivity, evolutionary potential may generally be maintained at higher levels than the local population, with implications for conservation given ongoing species declines and habitat fragmentation.     

Evolutionarily stable mutation rate in a periodically changing environment

Evolution of mutation rate controlled by a neutral modifier is studied for a locus with two alleles under temporally fluctuating selection pressure. A general formula is derived to calculate the evolutionarily stable mutation rate μ(ess) in an infinitely large haploid population, and following results are obtained. (I) For any fluctuation, periodic or random: (1) if the recombination rate r per generation between the modifier and the main locus is 0, μ(ess) is the same as the optimal mutation rate μ(op) which maximizes the long-term geometric average of population fitness; and (2) for any r, if the strength s of selection per generation is very large, μ(ess) is equal to the reciprocal of the average number T of generations (duration time) during which one allele is persistently favored than the other. (II) For a periodic fluctuation in the limit of small s and r, μ(ess)T is a function of sT and rT with properties: (1) for a given sT, μ(ess)T decreases with increasing rT; (2) for sT </= 1, μ(ess)T is almost independent of sT, and depends on rT as μ(ess)T & 1.6 for rT << 1 and μ(ess)T & 6/rT for rT >> 1; and (3) for sT >/= 1, and for a given rT, μ(ess)T decreases with increasing sT to a certain minimum less than 1, and then increases to 1 asymptotically in the limit of large sT. (III) For a fluctuation consisting of multiple Fourier components (i.e., sine wave components), the component with the longest period is the most effective in determining μ(ess) (low pass filter effect). (IV) When the cost c of preventing mutation is positive, the modifier is nonneutral, and μ(ess) becomes larger than in the case of neutral modifier under the same selection pressure acting at the main locus. The value of c which makes μ(ess) equal to μ(op) of the neutral modifier case is calculated. It is argued that this value gives a critical cost such that, so long as the actual cost exceeds this value, the evolution rate at the main locus must be smaller than its mutation rate μ(ess).     

How Often Do Duplicated Genes Evolve New Functions?

A recently duplicated gene can either fix a null allele (becoming a pseudogene) or fix an (advantageous) allele giving a slightly different function, starting it on the road to evolving a new function. Here we examine the relative probabilities of these two events under a simple model. Null alleles are assumed to be neutral; linkage effects are ignored, as are unequal crossing over and gene conversion. These assumptions likely make our results underestimates for the probability that an advantageous allele is fixed first. When new advantageous mutations are additive with selection coefficient s and the ratio of advantageous to null mutations is ρ, the probability an advantageous allele is fixed first is ([1 - e(-S)]/[ρS] + 1)(-1), where S = 4N(e)s with N(e) the effective population size. The probability that a duplicate locus becomes a pseudogene, as opposed to evolving a new gene function, is high unless ρS & 1. However, even if advantageous mutations are very rare relative to null mutations, for sufficiently large populations ρS & 1 and new gene function, rather than pseudogene formation, is the expected fate of most duplicated genes.     

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.     

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.     

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ_∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if , where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.     

A Phylogenetic Estimator of Effective Population Size or Mutation Rate

A new estimator of the essential parameter θ = 4N(e)μ from DNA polymorphism data is developed under the neutral Wright-Fisher model without recombination and population subdivision, where N(e) is the effective population size and μ is the mutation rate per locus per generation. The new estimator has a variance only slightly larger than the minimum variance of all possible unbiased estimators of the parameter and is substantially smaller than that of any existing estimator. The high efficiency of the new estimator is achieved by making full use of phylogenetic information in a sample of DNA sequences from a population. An example of estimating θ by the new method is presented using the mitochondrial sequences from an American Indian population.     

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.     

Bottleneck Effect on Evolutionary Rate in the Nearly Neutral Mutation Model

Variances of evolutionary rates among lineages in some proteins are larger than those expected from simple Poisson processes. This phenomenon is called overdispersion of the molecular clock. If population size N is constant, the overdispersion is observed only in a limited range of 2Nσ under the nearly neutral mutation model, where σ represents the standard deviation of selection coefficients of new mutants. In this paper, we investigated effects of changing population size on the evolutionary rate by computer simulations assuming the nearly neutral mutation model. The size was changed cyclically between two numbers, N(1) and N(2) (N(1) > N(2)), in the simulations. The overdispersion is observed if 2N(2)σ is less than two and the state of reduced size (bottleneck state) continues for more than ~0.1/u generations, where u is the mutation rate. The overdispersion results mainly because the average fitnesses of only a portion of populations go down when the population size is reduced and only in these populations subsequent advantageous substitutions occur after the population size becomes large. Since the fitness reduction after the bottleneck is stochastic, acceleration of the evolutionary rate does not necessarily occur uniformly among loci. From these results, we argue that the nearly neutral mutation model is a candidate mechanism to explain the overdispersed molecular clock.     

Prognostic Value of Tumor Size in Patients with Remnant Gastric Cancer: Is the Seventh UICC Stage Sufficient for Predicting Prognosis?

Background

The 7th UICC N stage may be unsuitable for remnant gastric cancer (RGC) because the original disease and previous operation usually cause abnormal lymphatic drainage. However, the prognostic significance of the current TNM staging system in RGC has not been studied.

Methods

Prospective data from 153 RGC patients who underwent curative gastrectomy from Jan 1995 to Aug 2009 were reviewed. All patients were classified according to tumor size (<3 cm as N0;>3&≤5 cm as N1;>5&≤7 cm as N2; and>7 cm as N3). The overall survival was estimated using the Kaplan-Meier method, and hazard ratios (HRs) were calculated using the Cox proportional hazard model.

Results

Tumor sizes ranged from 1.0 to 15.0 cm (median 5.0 cm). Tumor size, depth of invasion and lymph node (LN) metastasis were significant prognostic factors based on both the univariate and multivariate analyses (P<0.05). In the survival analysis, the seventh edition UICC-TNM classification provided a detailed classification; however, some subgroups of the UICC-TNM classification did not have significantly different survival rates. The combination of the seventh edition T classification and the suggested N classification, with ideal relative risk (RR) results and P value, was distinctive for subgrouping the survival rates except for the IA versus IB and II A versus IIB. A modified staging system based on tumor size, predicted survival more accurately than the conventional TNM staging system.

Conclusions

In RGCs, tumor size is an independent prognostic factor and a modified TNM system based on tumor size accurately predicts survival.     

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.     

Demographic consequences of population subdivision on the long-furred woolly mouse opossum (Micoureus paraguayanus) from the Atlantic Forest

Habitat destruction and fragmentation severely affected the Atlantic Forest. Formerly contiguous populations may become subdivided into a larger number of smaller populations, threatening their long-term persistence. The computer package VORTEX was used to simulate the consequences of habitat fragmentation and population subdivision on Micoureus paraguayanus, an endemic arboreal marsupial of the Atlantic Forest. Scenarios simulated hypothetical populations of 100 and 2000 animals being partitioned into 1–10 populations, linked by varying rates of inter-patch dispersal, and also evaluated male-biased dispersal. Results demonstrated that a single population was more stable than an ensemble of populations of equal size, irrespective of dispersal rate. Small populations (10–20 individuals) exhibited high instability due to demographic stochasticity, and were characterized by high rates of extinction, smaller values for metapopulation growth and larger fluctuations in population size and growth rate. Dispersal effects on metapopulation persistence were related to the size of the populations and to the sexes that were capable of dispersing. Male-biased dispersal had no noticeable effects on metapopulation extinction dynamics, whereas scenarios modelling dispersal by both sexes positively affected metapopulation dynamics through higher growth rates, smaller fluctuations in growth rate, larger final metapopulation sizes and lower probabilities of extinction. The present study highlights the complex relationships between metapopulation size, population subdivision, habitat fragmentation, rate of inter-patch dispersal and sex-biased dispersal and indicates the importance of gaining a better understanding of dispersal and its interactions with correlations between disturbance events.     

Extinction times for closed epidemics: the effects of host spatial structure

Although of practical importance, the relationship between the duration of an epidemic and host spatial structure is poorly understood. Here we use a stochastic metapopulation model for the transmission of infection in a spatially structured host population. There are three qualitatively different regimes for the extinction time, which depend on patch population size, the within‐patch basic reproductive number and the strength of coupling between patches. In the first regime, the extinction time for the metapopulation (i.e. from all patches) is approximately equal to the extinction time for a single patch. In the second regime, the metapopulation extinction time is maximal but also highly variable. In the third regime, the extinction time for the metapopulation (T_E) is given by T_E = a + bn^1/2 where a is the local extinction time (i.e. from last patch), b is the transit time (i.e. the time taken for infection to spread from one patch to another) and n is the total number of patches.     

A diffusion approximation for selection and drift in a subdivided population

The population-genetic consequences of population structure are of great interest and have been studied extensively. An area of particular interest is the interaction among population structure, natural selection, and genetic drift. At first glance, different results in this area give very different impressions of the effect of population subdivision on effective population size (N(e)), suggesting that no single value of N(e) can completely characterize a structured population. Results presented here show that a population conforming to Wright's island model of subdivision with genic selection can be related to an idealized panmictic population (a Wright-Fisher population). This equivalent panmictic population has a larger size than the actual population; i.e., N(e) is larger than the actual population size, as expected from many results for this type of population structure. The selection coefficient in the equivalent panmictic population, referred to here as the effective selection coefficient (s(e)), is smaller than the actual selection coefficient (s). This explains how the fixation probability of a selected allele can be unaffected by population subdivision despite the fact that subdivision increases N(e), for the product N(e)s(e) is not altered by subdivision.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.     

Source-Sink Colonization as a Possible Strategy of Insects Living in Temporary Habitats

Continuous colonization and re-colonization is critical for survival of insect species living in temporary habitats. When insect populations in temporary habitats are depleted, some species may escape extinction by surviving in permanent, but less suitable habitats, in which long-term population survival can be maintained only by immigration from other populations. Such situation has been repeatedly described in nature, but conditions when and how this occurs and how important this phenomenon is for insect metapopulation survival are still poorly known, mainly because it is difficult to study experimentally. Therefore, we used a simulation model to investigate, how environmental stochasticity, growth rate and the incidence of dispersal affect the positive effect of permanent but poor (“sink”) habitats on the likelihood of metapopulation persistence in a network of high quality but temporary (“source”) habitats. This model revealed that permanent habitats substantially increase the probability of metapopulation persistence of insect species with poor dispersal ability if the availability of temporary habitats is spatio-temporally synchronized. Addition of permanent habitats to a system sometimes enabled metapopulation persistence even in cases in which the metapopulation would otherwise go extinct, especially for species with high growth rates. For insect species with low growth rates the probability of a metapopulation persistence strongly depended on the proportions of “source” to “source” and “sink” to “source” dispersal rates.     

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

For two genotypes that have the same mean number of offspring but differ in the variance in offspring number, naturalselection will favor the genotype with lower variance. In such cases, the average growth rate is not sufficient as a measure of fitness or as a predictor of fixation probability. However, the effect of variance in offspring number on the fixationprobability of mutant strategies has been calculated under several scenarios with the general conclusion that variance in offspring number reduces fitness in proportion to the inverse of the population size [Gillespie, J., Genetics 76:601–606, 1974; Proulx, S.R., Theor. Popul. Biol. 58:33–47, 2000]. This relationship becomes more complicated under a metapopulation scenario where the “effective” population size depends on migration rate, population structure, and lifecycle. It is shown that in a life cycle where reproduction and migration (the birth-migration-regulation life cycle, or BMR)occur prior to density regulation within every deme, the fitness of a strategy depends on migration rate. When migration rates are near zero, the fitness of the strategy is determined by the size of individual demes, so that the strategy favoredin small populations tends to be fixed. As migration rate increases and approaches panmixis between demes, the fitness ofa reproductive strategy approaches what its value would be in a single, panmictic deme with a population size correspondingtothe census size of the metapopulation. Interestingly, when the life cycle is characterized by having density regulation in each deme prior to migration (the BRM life cycle) the fixation probability of a strategy is independent of migration rate. These results are found to be qualitatively consistent with the individual-based simulation results in Shpak [Theor. Biosci.124:65–85, 2005]. An erratum to this article can be found at     

Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation

Approximate Bayesian computation (ABC) is useful for parameterizing complex models in population genetics. In this study, ABC was applied to simultaneously estimate parameter values for a model of metapopulation coalescence and test two alternatives to a strict metapopulation model in the well‐studied network of Daphnia magna populations in Finland. The models shared four free parameters: the subpopulation genetic diversity (θ_S), the rate of gene flow among patches (4Nm), the founding population size (N₀) and the metapopulation extinction rate (e) but differed in the distribution of extinction rates across habitat patches in the system. The three models had either a constant extinction rate in all populations (strict metapopulation), one population that was protected from local extinction (i.e. a persistent source), or habitat‐specific extinction rates drawn from a distribution with specified mean and variance. Our model selection analysis favoured the model including a persistent source population over the two alternative models. Of the closest 750 000 data sets in Euclidean space, 78% were simulated under the persistent source model (estimated posterior probability = 0.769). This fraction increased to more than 85% when only the closest 150 000 data sets were considered (estimated posterior probability = 0.774). Approximate Bayesian computation was then used to estimate parameter values that might produce the observed set of summary statistics. Our analysis provided posterior distributions for e that included the point estimate obtained from previous data from the Finnish D. magna metapopulation. Our results support the use of ABC and population genetic data for testing the strict metapopulation model and parameterizing complex models of demography.