From understanding genetic drift to a smart-restart parameter-less compact genetic algorithm

In our former paper, we have investigated the relation among the mean convergence time, the population size, and the chromosome length of genetic algorithms (GAs). Our analyses of GAs make use of the Markov chain formalism based on the Wright-Fisher model, which is a typical and well-known model in population genetics. The Wright-Fisher model is characterized by 1-locus, 2-alleles, fixed population size, and discrete generation. For these simple characters, it is easy to evaluate the behavior of genetic process. We have also given the mean convergence time under genetic drift. Genetic drift can be well described in the Wright-Fisher model, and we have determined the stationary states of the corresponding Markov chain model and the mean convergence time to reach one of these stationary states. The island model is also well-known model in population genetics, and it is similar to one of the most typical model of parallel GAs, which require parallel computer for high performance computing. We have also derived the most effective migration rate for the island model parallel GAs with some restrictions. The obtained most effective migration rate is rather small value, i.e. one immigrant per generation, however the behaviors of the island model parallel GAs at that migration rate are not revealed yet clearly. In this paper, we discuss the mean convergence time for the island model parallel GAs from both of exact solution and numerical simulation. As expected from the Wright-Fisher model’s analysis, the mean convergence time of the island model parallel GAs is proportional to population size, and the coefficient is larger with smaller migration rate. Since to keep the diversity in population is important for effective performance of GAs, the convergence in population gives a bad influence for GAs. On the other hand, mutation and crossover operation prevent converging in GAs population. Because of the small migration rate makes converging force weak, it must be effective for GAs. This means that the island model parallel GAs is more efficient not only to use large population size with parallel computers, but also to keep the diversity in population, than usual GAs.

Genetic linkage and molecular evolution

Populations of Human Immunodeficiency Virus type 1 (HIV-1) undergo a surprisingly large amount of genetic drift in infected patients despite very large population sizes, which are predicted to be mostly deterministic. Several models have been proposed to explain this phenomenon, but all of them implicitly assume that the process of virus replication itself does not contribute to genetic drift. We developed an assay to measure the amount of genetic drift for HIV populations replicating in cell culture. The assay relies on creation of HIV populations of known size and measurements of variation in frequency of a neutral allele. Using this assay, we show that HIV undergoes approximately ten times more genetic drift than would be expected from its population size, which we defined as the number of infected cells in the culture. We showed that a large portion of the increase in genetic drift is due to non-synchronous infection of target cells. When infections are synchronized, genetic drift for the virus is only 3-fold higher than expected from its population size. Thus, the stochastic nature of biological processes involved in viral replication contributes to increased genetic drift in HIV populations. We propose that appreciation of these effects will allow better understanding of the evolutionary forces acting on HIV in infected patients.

Global phylogeography of the dolphinfish ( Coryphaena hippurus): The influence of large effective population size and recent dispersal on the divergence of a marine pelagic cosmopolitan species

Population genetics: Peak shifts in large populations

I used a stochastic simulation model to examine the loss of heterozygosity and allelic diversity in a bottlenecked population and a sink population subject to various migration rates from source populations of different sizes. In the bottlenecked population, the initial founder size influenced the level of allelic diversity more than heterozygosity, although the subsequent patterns of loss of the 2 measures were similar. The rescue effect, whereby migration from a source population offsets genetic drift in a sink population, was shown to have a detrimental counterpart termed the imperil effect, which may lead to the erosion of genetic variation. This is manifested when the source population has less genetic variation than the sink population. When genetic data about a source population are lacking or scant, to avoid the imperil effect, migration rates should be <1 migrant per generation. J. WILDL. MANAGE. 54(4):676-682 A suggested goal of captive breeding and genetic management is to maintain 90% of the heterozygosity of the source (wild) population over a period of 200 years (Soule et al. 1986). This goal becomes complicated when economic and spatial constraints dictate that population sizes of managed species remain small, making the populations susceptible to genetic drift. In addition to heterozygosity, which has been correlated with individual fitness (Beardmore 1983, Allendorf and Leary 1986), another populationlevel measure of genetic variance, allelic diversity, is crucial to the long-term adaptability of a population (Frankel and Soule 1981) and should also be considered for genetic management. When a large population is reduced in size to N individuals, the average heterozygosity per locus is expected to decrease by 1/(2N) (Nei et al. 1975), and the number of polymorphic alleles at a particular locus is expected to decrease by 2 (1 pj)2N (Denniston 1978), where p, is the -1ee frequency, and k is the number of alleles allele frequency, and k is the number of alleles at a locus. Both measures will continue to decline from generation to generation until a drift-mutation equilibrium is attained (Lacy 1987). If the population rebounds from the bottleneck, heterozygosity may show little or no reduction, but rare alleles have a high probability of being lost (Allendorf 1986, Fuerst and Maruyama 1986). Management techniques designed to maintain allelic diversity and heterozygosity are conflicting. Allelic diversity is promoted through the subdivision of populations (Chesser 1983), whereas heterozygosity is maintained through a large effective population size (N,) (Simberloff 1988). Limited migration among subpopulations or 1-way migration from an infinitely large population decreases loss of heterozygosity caused by genetic drift (Wright 1931, Lacy 1987). This rescue effect (Brown and KodricBrown 1977) provided by an impulse of new genetic material could reduce the extinction probability of an insular population by preventing the loss of genetic variation. Migration This content downloaded from 157.55.39.243 on Wed, 05 Oct 2016 04:47:51 UTC All use subject to http://about.jstor.org/terms J. Wildl. Manage. 54(4):1990 RESCUE EFFECT * Weishampel 677 rates of <1 (Foose et al. 1986, Off. Technol. Assessment 1987), 1 (Avery 1978, Allendorf 1983), 1-2 (Franklin 1980), and 1-5 migrants per generation (Frankel and Soul6 1981) have been proposed. In my study, I used a stochastic simulation model to examine how bottleneck severity, immigration rate, and source and sink population sizes affect heterozygosity and allelic diversity. This work was inspired by E. F. Connor, J. J. Murray, and H. H. Shugart. I wish to thank them, J. A. Yeakley, and 3 anonymous reviewers for their input. This research was supported in part by an NSF grant presented under Interagency Agreement No. BSR-8718168 with the Department of Energy and by the National Aeronautics and Space Administration Earth Sciences Division (UPN 677-80-06-05) to W. E.

Clonal interference, the competition between lineages arising from different beneficial mutations in an asexually reproducing population, is an important factor determining the tempo and mode of microbial adaptation. The standard theory of this phenomenon neglects the occurrence of multiple mutations as well as the correlation between loss by genetic drift and clonal competition, which is questionable in large populations. Working within the Wright-Fisher model with multiplicative fitness (no epistasis), we determine the rate of adaptation asymptotically for very large population sizes and show that the standard theory fails in this regime. Our study also explains the success of the standard theory in predicting the rate of adaptation for moderately large populations. Furthermore, we show that the nature of the substitution process changes qualitatively when multiple mutations are allowed for, because several mutations can be fixed in a single fixation event. As a consequence, the index of dispersion for counts of the fixation process displays a minimum as a function of population size, whereas the origination process of fixed mutations becomes completely regular for very large populations. We find that the number of mutations fixed in a single event is geometrically distributed as in the neutral case. These conclusions are based on extensive simulations combined with analytic results for the limit of infinite population size.

Genetic drift is the random change in allele frequencies by the chance success of some alleles relative to others. Genetic drift is more important in small populations, where chance plays stronger role. If the population size is small enough relative to the strength of selection, genetic drift can sometimes cause slightly deleterious alleles to rise in frequency or beneficial alleles to be lost from a population. Drift can lead to the fixation or loss of alleles, and therefore drift can contribute to the loss of genetic variation. As a consequence, genetic drift in small populations is a source of concern for the future evolutionary potential for some endangered species. Key Concepts: Alleles may increase or decrease in frequency by chance. The effects of chance on allele frequency change are most pronounced in small populations. Genetic drift tends to lead to fixation or loss of alleles over time, and therefore contributes to the loss of genetic variation. If the population size is small enough relative to the strength of selection, genetic drift can cause the fixation of deleterious alleles or loss of beneficial alleles. Genetic drift can cause genetic divergence between species or populations. Most genetic differences between species are probably due to genetic drift. Genetic drift is nonadaptive and nondirectional.

In discussions of the major features of evolution, Simpson (1953) applied population genetic models to the interpretation of the fossil record. Most population genetics theory concentrates on details of the genetic system, such as gene frequencies and recombination rates, which cannot be directly observed or inferred from measurements on polygenic characters. Analysis of phenotypic data, particularly fossil material, requires models which are framed as much as possible in phenotypic terms. Starting from a simple formula of quantitative genetics, the methods of population genetics are used here to make a theory of the evolution of the average phenotype in a population by natural selection and random genetic drift. By analogy with Wright's (1931) adaptive topography for genotypes, Simpson (1953) proposed the concept of adaptive zones for phenotypes. This is an intuitive method of visualizing the dynamics of phenotypic evolution in terms of the degree of adaptation of the various phenotypes in a population, it usually being thought that natural selection increases adaptation. Such qualitative ideas are used by most evolutionary biologists and the notion of adaptive zones is popular among paleontologists. In the present paper, the concept of adaptive zones is clarified by the construction of an adaptive topography for the average phenotype in a population. This shows that with constant fitnesses the average phenotype evolves toward the nearest adaptive zone in the phenotype space. But if fitnesses are frequency-dependent the average phenotype may evolve away from an adaptive zone. A method is developed for estimating the minimum selective mortality necessary to produce an observed rate of evolution. In examples of the evolution of tooth characters in Tertiary mammals, these minimum selective mortalities are found to be exceedingly small. In his paper on the measurement of rates of evolution, Haldane (1949) stated that slowness of the rate of change makes it clear that agencies other than natural selection cannot be neglected because they are extremely slow by laboratory standards or even undetectable during a human lifetime. He briefly discussed mutation pressure. Random genetic drift due to finite population size is another such agency. The relative importance of natural selection and random genetic drift has been debated since Wright (1931, 1932) proposed that evolution is a stochastic process. Fisher (1958), for example, believed that random genetic drift is insignificant in relation to natural selection. The debate continues today at a more biochemical level (Lewontin, 1974). In order to objectively evaluate the role of random genetic drift in macro-evolutionary events, it is necessary to use mathematical models to determine the rate of evolution which can occur by repeated samplings of genetic material in a finite population. This paper presents a statistical test for the hypothesis of evolution by random genetic drift, contingent on the effective population size. In examples from the fossil record, it is found that rates of evolution equal to or greater than those observed have a significant probability of occurring by random genetic drift

Studies of population structure often focus on the effects of population size and migration rates on genetic variation. Few studies, however, have investigated the relationship between these two factors. The purpose of this paper is to determine the extent to which migration (and gene flow) is density-dependent (that is, affected by population size) for populations in historical Massachusetts. Data from 4,859 marriage records were analyzed from four populations in north-central Massachusetts during the time period 1741 to 1849. These data were placed into 29 samples defined in terms of population and time cohort. Within each cohort the overall exogamy rate was computed along with three estimates of gene flow based on marital migration: local migration (k), long-distance migration (m), and effective migration rate (me). Three samples show unusually low rates that reflect the history of settlement. Regression analyses were used with the remaining samples, and they show nonlinear density-dependent migration that is unrelated to temporal trends. Migration is highest in samples with small population sizes (less than 800) and large population sizes (greater than 1,600). Migration is lowest in medium-sized populations. Two processes are suggested to explain this curvilinear relationship of migration and population size. In small populations, the lack of suitable potential mates and/or availability of settled land leads to an increase in migration into the population. As population size increases, this migration decreases. After populations reach a certain size, migration increases again, most likely reflecting the economic pull of larger populations. These patterns could act to enhance, or counter, genetic drift, depending on the direction of density dependence.

Conservation and the genetics of populations

How population size influences quantitative genetic variation and differentiation among natural, fragmented populations remains unresolved. Small, isolated populations might occupy poor quality habitats and lose genetic variation more rapidly due to genetic drift than large populations. Genetic drift might furthermore overcome selection as population size decreases. Collectively, this might result in directional changes in additive genetic variation (VA ) and trait differentiation (QST ) from small to large population size. Alternatively, small populations might exhibit larger variation in VA and QST if habitat fragmentation increases variability in habitat types. We explored these alternatives by investigating VA and QST using nine fragmented populations of brook trout varying 50-fold in census size N (179-8416) and 10-fold in effective number of breeders, Nb (18-135). Across 15 traits, no evidence was found for consistent differences in VA and QST with population size and almost no evidence for increased variability of VA or QST estimates at small population size. This suggests that (i) small populations of some species may retain adaptive potential according to commonly adopted quantitative genetic measures and (ii) populations of varying sizes experience a variety of environmental conditions in nature, however extremely large studies are likely required before any firm conclusions can be made.

A modified Wright–Fisher model that incorporates Ne: A variant of the standard model with increased biological realism and reduced computational complexity

Selection pressures on proteins are usually measured by comparing homologous nucleotide sequences (Zuckerkandl and Pauling 1965). Recently we introduced a novel method, termed volatility, to estimate selection pressures on proteins on the basis of their synonymous codon usage (Plotkin and Dushoff 2003; Plotkin et al. 2004). Here we provide a theoretical foundation for this approach. Under the Fisher-Wright model, we derive the expected frequencies of synonymous codons as a function of the strength of selection on amino acids, the mutation rate, and the effective population size. We analyze the conditions under which we can expect to draw inferences from biased codon usage, and we estimate the time scales required to establish and maintain such a signal. We find that synonymous codon usage can reliably distinguish between negative selection and neutrality only for organisms, such as some microbes, that experience large effective population sizes or periods of elevated mutation rates. The power of volatility to detect positive selection is also modest--requiring approximately 100 selected sites--but it depends less strongly on population size. We show that phenomena such as transient hyper-mutators can improve the power of volatility to detect selection, even when the neutral site heterozygosity is low. We also discuss several confounding factors, neglected by the Fisher-Wright model, that may limit the applicability of volatility in practice.

Estimation-of-distribution algorithms (EDAs) are optimization algorithms that learn a distribution from which good solutions can be sampled easily. A key parameter of most EDAs is the sample size (population size). Too small values lead to the undesired effect of genetic drift, while larger values slow down the process. Building on a quantitative analysis of how the population size leads to genetic drift, we design a smart-restart mechanism for EDAs. By stopping runs when the risk for genetic drift is high, it automatically runs the EDA in good parameter regimes. Via a mathematical runtime analysis, we prove a general performance guarantee for this smart-restart scheme. For many situations where the optimal parameter values are known, this shows that the restart scheme automatically finds these optimal values, leading to the asymptotically optimal performance. We also conduct an extensive experimental analysis. On four classic benchmarks, the smart-restart scheme leads to a performance close to the one obtainable with optimal parameter values. We also conduct experiments with PBIL (cross-entropy algorithm) on the max-cut problem and the bipartition problem. Again, the smart-restart mechanism finds much better values for the population size than those suggested in the literature, leading to a much better performance.