Numerical analysis of weak random drift in a cline

The equilibrium state of a diffusion model for weak random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric, and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a non-zero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending on the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. All the results are parameter free. Some asymptotic formulae are derived analytically.     

Clines with asymmetric migration

The consequences of asymmetric dispersion on the maintenance of an allele in a one-dimensional environmental pocket are examined. The diffusion model of migration and selection is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is further supposed that migration is homogeneous and independent of genotype, the population density is constant and uniform, and Hardy-Weinberg proportions obtain locally. If dispersion is preferentially out of an environmental pocket at the end of a very long habitat, the condition for maintaining the allele favored in the pocket becomes less stringent than for symmetric migration; dispersion preferentially into the pocket increases the severity of the condition for polymorphism. If an allele is harmful in large regions on both sides of an environmental pocket, the probability for polymorphism is decreased by asymmetric migration. The criterion for the existence of a cline is independent of the sense of the asymmetry; the cline itself is not. These phenomena are studied both analytically and numerically.—It is shown for symmetric migration and variable population density that the more densely populated parts of the habitat are more influential in determining gene frequency than the others. Thus, the higher the population density in an environmental pocket, the more easily an allele beneficial in the pocket will be maintained in the population.     

Genetic drift widens the expected cline but narrows the expected cline width

Random genetic drift shifts clines in space, alters their width, and distorts their shape. Such random fluctuations complicate inferences from cline width and position. Notably, the effect of genetic drift on the expected shape of the cline is opposite to the naive (but quite common) misinterpretation of classic results on the expected cline. While random drift on average broadens the overall cline in expected allele frequency, it narrows the width of any particular cline. The opposing effects arise because locally, drift drives alleles to fixation--but fluctuations in position widen the expected cline. The effect of genetic drift can be predicted from standardized variance in allele frequencies, averaged across the habitat: . A cline maintained by spatially varying selection (step change) is expected to be narrower by a factor of √1- relative to the cline in the absence of drift. The expected cline is broader by the inverse of this factor. In a tension zone maintained by underdominance, the expected cline width is narrower by about 1- relative to the width in the absence of drift. Individual clines can differ substantially from the expectation, and we give quantitative predictions for the variance in cline position and width. The predictions apply to clines in almost one-dimensional circumstances such as hybrid zones in rivers, deep valleys, or along a coast line and give a guide to what patterns to expect in two dimensions.     

Conditions for the Existence of Clines

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.     

Frequency distributions for chloroplast genes in Chlamydomonas zygote clones: Evidence for random drift

Mitochondria and chloroplasts of eucaryotic cells contain populations of DNA molecules. In certain cases, e.g., the chloroplasts of Chlamydomonas reinhardtii and the mitochondria of Saccharomyces cerevisiae, organelles contributed by the two parents are known to fuse in the zygote, creating a single population of DNA molecules. In a cross, this population will include molecules of both parental genotypes. There is reason to suspect that organelle DNA molecules in this population are selected randomly for replication and recombination. This would result in random changes in the frequency of a particular allele or genotype within the organelle gene pool of a single zygote and also within its clone of progeny cells. A given gene frequency would increase in some zygote clones and decrease in others, analogous to random drift of gene frequencies in small Mendelian populations. To test this, we have examined the distribution of chloroplast gene frequencies among the zygote clones produced in each of a number of crosses of Chlamydomonas. These distributions are typically U or L shaped as predicted by the random drift hypothesis. They include uniparental zygote clones, in which a chloroplast allele from one parent has been fixed (frequency 100%) and the alternative allele from the other parent has been lost (frequency 0%). Among the remaining (biparental) zygote clones, there is a linear distribution of allele frequencies, showing a great increase in variance over the input frequencies. In these experiments both biparental and uniparental zygotes show a bias favoring chloroplast alleles from the mt⁺ (maternal) parent, and there is no statistically significant mode at the allele frequency of 0.5 corresponding to the equal input of alleles from the maternal and paternal (mt^?) parents. The observed distributions support the hypothesis that both uniparental inheritance and the high variance of allele frequencies among zygote clones are due to random drift of allele frequencies, coupled with a directional force which favors fixation of the maternal allele. In addition, statistical analysis of the data shows a strong but incomplete tendency for linked chloroplast markers to be fixed or lost together in uniparental zygotes. Possible cellular and molecular mechanisms for these observations are discussed.     

Genetic Drift in Clines Which Are Maintained by Migration and Natural Selection

Genetic drift will cause a migration-selection cline to wobble about its expected position. A rough linear approximation is developed, valid when local populations are large. This is used to calculate effects of genetic drift on clines in a stepping-stone model with abrupt and with gradual changes of selection coefficients at a single haploid locus. Among the quantities calculated are measures of slope, standardized variation of gene frequencies around their expected values, and correlation among neighboring populations with respect to deviations from the expected gene frequencies. These quantities appear to be primarily functions of Ns and Nm for a given pattern of selection. Computer simulation gives rough confirmation of these results. Standardized variances of gene frequencies and correlation of neighbors differ along the cline in the case of smooth changes in selection. In no case is pathological behavior of gene frequency deviations found near the boundaries of selective regions. Local behavior of gene frequences of nearby colonies is approximately predicted by a simple adaptation of the stepping-stone theory of Kimura and Weiss. Approximate measures of the lateral variation of the midpoint of a cline and the probability of non-monotonicity are also calculated and discussed.     

The effects of genetic drift during range expansion on geographical patterns of variation: a computer simulation of the colonization of Australia by Bufo marinus

A computer simulation is performed of allele frequency changes resulting from genetic drift at eleven loci during the probable course of colonization of Australia by populations of the Giant Toad, Bufo marinus. The history of twelve populations for which allele frequency data are available is modelled. Account is taken of the likely pattern of relationship among the populations, the effective size of the populations (as indicated by the observed variance of allele frequencies) and the probable isolation of the populations from each other following their separation. In all simulations, allele frequencies at some loci show a significant association with latitude while others do not. In six of ten simulations, there is a significant association between degree of variation at a locus and the presence of a latitudinal cline of allele frequencies. There are also indications of this kind of association in simulations of a uni-directional range expansion. These results demonstrate that such associations, which were also observed in the data from the actual populations, can result from genetic drift during a range expansion, and therefore cannot be taken as evidence of the action of natural selection.     

Clines with partial panmixia in an unbounded unidimensional habitat

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an unbounded unidimensional habitat is investigated. Migration and selection are both weak. The former is homogenous and isotropic; the latter has no dominance. The population density is uniform. A simple, explicit formula is derived for the maximum value β₀ of the scaled panmictic rate β for which a cline exists. The former depends only on the asymptotic values of the scaled selection coefficient. If the two alleles have the same average selection coefficient, there exists a unique, globally asymptotically stable cline for every β≥0. Otherwise, if β≥β₀, the allele with the greater average selection coefficient is ultimately fixed. If β<β₀, there exists a unique, globally asymptotically stable cline, and some polymorphism is retained even infinitely far from its center. The gene frequencies at infinity are determined by a continuous-time, two-deme migration-selection model. An explicit expression is deduced for the monotone cline in a step-environment. These results differ fundamentally from those for the classical cline without panmixia.     

Intracellular Population Genetics: Evidence for Random Drift of Mitochondrial Allele Frequencies in SACCHAROMYCES CEREVISIAE and SCHIZOSACCHAROMYCES POMBE

We report evidence for random drift of mitochondrial allele frequencies in zygote clones of Saccharomyces cerevisiae and Schizosaccharomyces pombe. Monofactorial and bifactorial crosses were done, using strains resistant or sensitive to erythromycin (alleles E^R, E^S), oligomycin (O^R, O^S), or diuron (D^R, D^S). The frequencies of resistant and sensitive cells (and thus the frequencies of the resistant and sensitive alleles) were determined for each of a number of clones of diploid cells arising from individual zygotes. Allele frequencies were extremely variable among these zygote clones; some clones were "uniparental," with mitochondrial alleles from only one parent present. These observations suggest random drift of the allele frequencies in the population of mitochondrial genes within an individual zygote and its diploid progeny. Drift would cease when all the cells in a clone become homoplasmic, due to segregation of the mitochondrial genomes during vegetative cell divisions. To test this, we delayed cell division (and hence segregation) for varying times by starving zygotes in order to give drift more time to operate. As predicted, delaying cell division resulted in an increase in the variance of allele frequencies among the zygote clones and an increase in the proportion of uniparental zygote clones. The changes in form of the allele frequency distributions resembled those seen during random drift in finite Mendelian populations. In bifactorial crosses, genotypes as well as individual alleles were fixed or lost in some zygote clones. However, the mean recombination frequency for a large number of clones did not increase when cell division was delayed. Several possible molecular mechanisms for intracellular random drift are discussed.     

Adaptation to a steep environmental gradient and an associated barrier to gene exchange in Littorina saxatilis

Steep environmental gradients offer important opportunities to study the interaction between natural selection and gene flow. Allele frequency clines are expected to form at loci under selection, but unlinked neutral alleles may pass easily across these clines unless a generalized barrier evolves. Here we consider the distribution of forms of the intertidal gastropod Littorina saxatilis, analyzing shell shape and amplified fragment length polymorphism (AFLP) loci on two rocky shores in Britain. On the basis of previous work, the AFLP loci were divided into differentiated and undifferentiated groups. On both shores, we have shown a sharp cline in allele frequencies between the two morphs for differentiated AFLP loci. This is coincident with a habitat transition on the shore where the two habitats (cliff and boulder field) are immediately contiguous. The allele frequency clines coincide with a cline in shell morphology. In the middle of the cline, linkage disequilibrium for the differentiated loci rises in accordance with expectation. The clines are extremely narrow relative to dispersal, probably as a result of both strong selection and habitat choice. An increase in F(ST) for undifferentiated AFLPs between morphs, relative to within-morph comparisons, is consistent with there being a general barrier to gene flow across the contact zone. These features are consistent either with an episode of allopatric divergence followed by secondary contact or with primary, nonallopatric divergence. Further data will be needed to distinguish between these alternatives.     

DISPERSAL,GENE FLOW,AND ALLELIC DIVERSITY BETWEEN LOCAL POPULATIONS OF THOMOMYS BOTTAE POCKET GOPHERS IN THE COASTAL RANGES OF CALIFORNIA

We studied Thomomys bottae pocket gophers from 1979 to 1982 to determine if the amount of gene flow between local populations was sufficient to reduce allele frequency differences between them. Dispersal was quantified using three different trap techniques, and genetic changes in the population were monitored using protein variants. Additional allele frequency data were available for 1976 and 1977. We observed dispersal to be common in pre-reproductive juvenile females throughout the breeding season of their birth. Males on the other hand tended to disperse only from the end of the breeding season. Although dispersal was common, 63% of adults appeared to be recruited within 40 m of where they were born. Gene flow occurred into both established populations and into vacant habitat, but it was too low to reduce the differences in gene frequencies between the fields over seven years. We conclude that allelic diversity in T. bottae populations is a balance between random drift due to small effective population size and gene flow.     

Genetic Drift in a Cline

A model is developed of genetic drift in a cline maintained by spatially varying natural selection and local dispersal of individuals. The model is analyzed by an approximation scheme which is valid for weak selection and small migration rates. The results, which are based on numerical iterations of the approximate equations, are that the cline is less steep than predicted on the basis of the deterministic theory but that for weak selection the correlation between random fluctuations in neighboring colonies is approximately the same as in models of migration and drift in the absence of selection.     

Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Analysis of CCR5Delta32 geographic distribution and its correlation with some climatic and geographic factors

We studied the possible effects of climatic-geographic factors on the world distribution of the mutant allele for the chemokine receptor gene CCR5, which has a 32-bp deletion (CCR5Delta32) preventing cell invasion by the primary transmitting strain of HIV-1. New data on CCR5 polymorphisms in Russian, Ukrainian, and Moldavian populations are presented. All available data on CCR5Delta32 frequencies in the Old World (number of populations n = 77) were used for construction of a geographical gene map to analyze possible correlations between allele frequencies and eight climatic-geographic parameters. A strong positive correlation was found between the allele frequency and latitude (r = 0.72), a strong negative correlation with annual radiation balance (r = -0.66), and a weaker negative correlation with longitude (r = -0.34). Partial correlations were calculated excluding the influence of latitude. The negative correlation between the allele frequency and annual radiation balance decreased (r = -0.42), but remained large and significant. We propose that the existence of correlations between the cline of CCR5Delta32 frequencies and climatic-geographic parameters provides evidence for a possible effect of either natural environmental factors or large-scale population movements on the distribution of this allele.     

The Spread of an Advantageous Allele across a Barrier: The Effects of Random Drift and Selection against Heterozygotes

A local barrier to gene flow will delay the spread of an advantageous allele. Exact calculations for the deterministic case show that an allele that is favorable when rare is delayed very little even by a strong barrier: its spread is slowed by a time proportional to log ((B/σ) &2S)/S, where B is the barrier strength, σ the dispersal range, and fitnesses are 1:1 + S:1 + 2S. However, when there is selection against heterozygotes, such that the allele cannot increase from low frequency, a barrier can cause a much greater delay. If gene flow is reduced below a critical value, spread is entirely prevented. Stochastic simulations show that with additive selection, random drift slows down the spread of the allele, below the deterministic speed of σ &2S. The delay to the advance of an advantageous allele caused by a strong barrier can be substantially increased by random drift and increases with B/(2Sρσ(2)) in a one-dimensional habitat of density ρ. However, with selection against heterozygotes, drift can facilitate the spread and can free an allele that would otherwise be trapped indefinitely by a strong barrier. We discuss the implications of these results for the evolution of chromosome rearrangements.     

AFLP genome scans suggest divergent selection on colour patterning in allopatric colour morphs of a cichlid fish

Genome scan-based tests for selection are directly applicable to natural populations to study the genetic and evolutionary mechanisms behind phenotypic differentiation. We conducted AFLP genome scans in three distinct geographic colour morphs of the cichlid fish Tropheus moorii to assess whether the extant, allopatric colour pattern differentiation can be explained by drift and to identify markers mapping to genomic regions possibly involved in colour patterning. The tested morphs occupy adjacent shore sections in southern Lake Tanganyika and are separated from each other by major habitat barriers. The genome scans revealed significant genetic structure between morphs, but a very low proportion of loci fixed for alternative AFLP alleles in different morphs. This high level of polymorphism within morphs suggested that colour pattern differentiation did not result exclusively from neutral processes. Outlier detection methods identified six loci with excess differentiation in the comparison between a bluish and a yellow-blotch morph and five different outlier loci in comparisons of each of these morphs with a red morph. As population expansions and the genetic structure of Tropheus make the outlier approach prone to false-positive signals of selection, we examined the correlation between outlier locus alleles and colour phenotypes in a genetic and phenotypic cline between two morphs. Distributions of allele frequencies at one outlier locus were indeed consistent with linkage to a colour locus. Despite the challenges posed by population structure and demography, our results encourage the cautious application of genome scans to studies of divergent selection in subdivided and recently expanded populations.     

Clines with Variable Migration

The consequences of a discontinuity in the migration rate and of a geographical barrier in the habitat are studied in a diffusion model of migration and selection. The treatment is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is supposed further that migration is independent of genotype, the population density remains constant and uniform, and Hardy-Weinberg proportions obtain locally. It is shown that a discontinuity in the migration rate leads to a jump in the slope of the gene frequency, but not in the gene frequency itself, while a localized geographical barrier has precisely the opposite effect. These features of the gene frequency behavior are quantitatively related to the migration rate. The influence of the above inhomogeneities in migration on the maintenance of an allele in an environmental pocket is examined. The extent to which the critical condition for polymorphism is made less stringent by decreased migration outside the pocket and by a geographical barrier between the pocket and the rest of the habitat is evaluated.     

Segregational Drift Constrains the Evolutionary Rate of Prokaryotic Plasmids

Plasmids are extrachromosomal genetic elements in prokaryotes that have been recognized as important drivers of microbial ecology and evolution. Plasmids are found in multiple copies inside their host cell where independent emergence of mutations may lead to intracellular genetic heterogeneity. The intracellular plasmid diversity is thus subject to changes upon cell division. However, the effect of plasmid segregation on plasmid evolution remains understudied. Here, we show that genetic drift during cell division—segregational drift—leads to the rapid extinction of novel plasmid alleles. We established a novel experimental approach to control plasmid allele frequency at the levels of a single cell and the whole population. Following the dynamics of plasmid alleles in an evolution experiment, we find that the mode of plasmid inheritance—random or clustered—is an important determinant of plasmid allele dynamics. Phylogenetic reconstruction of our model plasmid in clinical isolates furthermore reveals a slow evolutionary rate of plasmid-encoded genes in comparison to chromosomal genes. Our study provides empirical evidence that genetic drift in plasmid evolution occurs at multiple levels: the host cell and the population of hosts. Segregational drift has implications for the evolutionary rate heterogeneity of extrachromosomal genetic elements.     

Near Fixation of 374l Allele Frequencies of the Skin Pigmentation Gene SLC45A2 in Africa

The L374F polymorphism of the SLC45A2 gene, encoding the membrane-associated transporter protein that plays an important role in melanin synthesis, has been suggested to be associated with skin color in human populations. In this study, the detailed distribution of the 374f and 374l alleles has been investigated in 2,581 unrelated subjects from 36 North, East, West, and Central African populations. We found once more the highly significant (p < 0.001) correlation coefficient (r = 0.957) cline of 374f frequencies with degrees of latitude in European and North African populations. Almost all the African populations located below 16° of latitude are fixed for the 374l allele. Peul, Toucouleur, and Soninké populations have 374l allele frequencies of 0.06, 0.03, and 0.03, respectively.