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.     

The effect on neutral gene flow of selection at a linked locus

The effect on gene flow at a neutral locus of a selective cline at a linked locus is investigated. A diffusion approximation for a two-locus island model is derived in which only one locus is subject to selection. The moments of the stationary distribution are obtained and compared to the corresponding moments from a one-locus, neutral island model. This comparison yields an effective migration rate. The effective migration rate is always less than the actual migration rate, but this effect is seen to be small for weak selection and loose linkage in the case of adult migration. The importance of selection at linked loci to the question of genetic differentiation in a subdivided population is discussed.     

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.     

Diversification before the most recent glaciation in Balanus glandula

A deep genetic cline between southern populations of the barnacle Balanus glandula (from about Monterey Bay southward) and northern populations (from northern California through Alaska) has recently been described. If this pattern is due to historical isolation and genetic drift, we expect it to have formed recently and represent a transient, nonequilibrium state. However, this cline appears to have formed well before the last glacial maximum. Our assays of sequence diversity at a region of mitochondrial cytochrome oxidase I, combined with coalescent estimators of the time of separation for these two regions, suggest that a late Pleistocene event more than 100 thousand years ago may be responsible for the initial separation. This suggests that either strong oceanographic mechanisms or natural selection have maintained the cline, because there has clearly been adequate time for this cline or polymorphism to resolve itself by genetic drift and migration. However, reliance on only a single mitochondrial marker for which the substitution rate has been estimated still limits the resolution of our analysis.     

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.     

Inference from Clines Stabilized by Frequency-Dependent Selection

Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between square root of s/square root of 8 sigma and square root of s/square root of 12 delta, where sigma is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations ("linkage disequilibria"), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequencies and of gametic correlations from a hybrid zone.     

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.     

Change and stability in a steep morph‐frequency cline in the snail Cepaea nemoralis (L.) over 43 years

Populations of the polymorphic land snail Cepaea nemoralis (L.) from Deepdale, Derbyshire, UK, sampled in 1965–67, showed a pattern of area effects, with steep clines among groups of populations differing in shell colour and banding morph frequencies. In 2010, most of these populations were resampled. In particular, a continuous transect made in 1967 of 42 quadrats (18.34 × 18.34 m) across a steep cline in several morph frequencies was completely resampled. In the dale as a whole, yellow shells had increased in frequency. In the transect, the frequencies of banding morphs showed no significant changes, although colour morphs showed some changes. Pink shells had increased in frequency in a section in which scrub had developed, and brown shells had increased in frequency in the area in which they had originally been at the highest frequency. In each case, the selection coefficients were of the order of 4%. Yellow had increased elsewhere. Nevertheless, both in the dale as a whole and in the transect, the pattern of geographical change in morph frequencies had remained essentially the same. Estimates of migration based on previous studies of marked snails and on modelling of the effect of drift and migration suggest that, regardless of whether the cline is a product of differential selection or of the gradual merging of previously separate founding populations, it has been in existence for a long time, and that migration occurs over greater distances than estimated from direct observation on marked snails. Although we can demonstrate that selection has occurred, the origin and maintenance of the cline and others similar to it remain in doubt; the development and maintenance of polymorphism in this species may require consideration of several processes operating on different time scales. © 2012 The Linnean Society of London     

Bayesian estimation of genomic clines

We developed a Bayesian genomic cline model to study the genetic architecture of adaptive divergence and reproductive isolation between hybridizing lineages. This model quantifies locus‐specific patterns of introgression with two cline parameters that describe the probability of locus‐specific ancestry as a function of genome‐wide admixture. ‘Outlier’ loci with extreme patterns of introgression relative to most of the genome can be identified. These loci are potentially associated with adaptive divergence or reproductive isolation. We simulated genetic data for admixed populations that included neutral introgression, as well as loci that were subject to directional, epistatic or underdominant selection, and analysed these data using the Bayesian genomic cline model. Under many demographic conditions, underdominance or directional selection had detectable and predictable effects on cline parameters, and ‘outlier’ loci were greatly enriched for genetic regions affected by selection. We also analysed previously published genetic data from two transects through a hybrid zone between Mus domesticus and M. musculus. We found considerable variation in rates of introgression across the genome and particularly low rates of introgression for two X‐linked markers. There were similarities and differences in patterns of introgression between the two transects, which likely reflects a combination of stochastic variability because of genetic drift and geographic variation in the genetic architecture of reproductive isolation. By providing a robust framework to quantify and compare patterns of introgression among genetic regions and populations, the Bayesian genomic cline model will advance our understanding of the genetics of reproductive isolation and the speciation process.     

Genetic Divergence under Uniform Selection. II. Different Responses to Selection for Knockdown Resistance to Ethanol among DROSOPHILA MELANOGASTER Populations and Their Replicate Lines

We have tested the hypothesis that genetic differences among conspecific populations may result in diverse responses to selection, using natural populations of Drosophila melanogaster. Selection for ethanol tolerance in a tube measuring knockdown resistance was imposed on five West Coast populations. In 24 generations the selected lines increased their mean knockdown times, on average, by a factor of 2.7. An initially weak latitudinal cline was steepened by selection. The two southernmost populations showed the same increases in the selected character, but differed consistently in their correlated responses in characters related to ethanol tolerance. This result indicates that the populations responded to selection by different genetic changes. Selection decreased female body weight and increased resistance to acetone, suggesting components of the response unrelated to ethanol metabolism. The Adhs allele was favored by selection in all populations at the onset, but increased in frequency only in the selected lines of the southernmost population. There was a correlation between latitude and Adh frequency changes, suggesting that fitnesses of the Adh alleles were dependent on the genetic background. Genetic background also had a large effect on the loss of fitness due to selection. Genetic drift between replicate lines caused more variation in selection response than initial genetic differences between populations. This result demonstrates the importance of genetic drift in divergence among natural populations undergoing uniform selection, since the effective population sizes approached those of small natural populations. Drift caused greater divergence between selected replicates than control replicates. Implications of this result for the genetic model of selection response are discussed.     

The adaptive value of epigenetic mutation: Limited in large but high in small peripheral populations

The fate of populations during range expansions, invasions and environmental changes is largely influenced by their ability to adapt to peripheral habitats. Recent models demonstrate that stable epigenetic modifications of gene expression that occur more frequently than genetic mutations can both help and hinder adaptation in panmictic populations. However, these models do not consider interactions between epimutations and evolutionary forces in peripheral populations. Here, we use mainland–island mathematical models and simulations to explore how the faster rate of epigenetic mutation compared to genetic mutations interacts with migration, selection and genetic drift to affect adaptation in peripheral populations. Our model focuses on cases where epigenetic marks are stably inherited. In a large peripheral population, where the effect of genetic drift is negligible, our analyses suggest that epimutations with random fitness impacts that occur at rates as high as 10^–3 increase local adaptation when migration is strong enough to overwhelm divergent selection. When migration is weak relative to selection and epimutations with random fitness impacts decrease adaptation, we find epigenetic modifications must be highly adaptively biased to enhance adaptation. Finally, in small peripheral populations, where genetic drift is strong, epimutations contribute to adaptation under a wider range of evolutionary conditions. Overall, our results suggest that epimutations can change outcomes of adaptation in peripheral populations, which has implications for understanding conservation and range expansions and contractions, especially of small populations.     

Seed and pollen flow and cline discordance among genes with different modes of inheritance

The relationships between seed and pollen flow and cline discordance/concordance between cytoplasmic and nuclear genes, with the incorporation of the effects of natural selection, are formulated for one locus with two alleles, under assumptions of random mating, no drift and no mutation. Results show that under certain conditions, the relative roles of seed and pollen flow in shaping cline discordance/ concordance are very similar to their roles in influencing population differentiation for selectively neutral markers with different modes of inheritance. Where the disequilibria between cytoplasmic and nuclear genes are of the order similar to selection coefficient, cline discordance/concordance can be predicted from the relative values of the ratio of pollen to seed flow and the ratio of selection coefficients. Where the disequilibria attained by seed and pollen flow are significant, the integrated cytonuclear data are recommended for cline analysis. In both cases, the relative rates of selection coefficients between cytoplasmic and nuclear genes can be roughly estimated according to their characteristic length.     

Population structure in the barn swallow,Hirundo rustica: a comparison between neutral DNA markers and quantitative traits

Local adaptation to variable environments can generate clinal variation in morphological traits. Alternatively, similar patterns of clinal variation may be generated simply as a result of genetic drift/migration balance. Teasing apart these different processes is a continuing focus in evolutionary ecology. We compare genetic differentiation at molecular loci and quantitative traits to analyse the effect of these different processes in a morphological latitudinal cline of the barn swallow, Hirundo rustica, breeding across Europe. The results obtained show no structuring at neutral microsatellite loci, which contrasts with positive structuring at five quantitative morphometric traits. This supports the hypothesis that the observed morphometric cline in barn swallows is the result of selection acting in a spatially heterogeneous environment. © 2010 The Linnean Society of London, Biological Journal of the Linnean Society, 2010, 99 , 306–314.     

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ₀ (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ₀ is an increasing function of the panmictic rate and to deduce the effect on λ₀ of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ₀; bounds on λ₀ are established; and λ₀ is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.     

Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.     

Tension versus ecological zones in a two-locus system

Previous theories show that tension and ecological zones are indistinguishable in terms of gene frequency clines. Here I analytically show that these two types of zones can be distinguished in terms of genetic statistics other than gene frequency. A two-locus cline model is examined with the assumptions of random mating, weak selection, no drift, no mutation, and multiplicative viabilities. The genetic statistics for distinguishing the two types of zones are the deviations of one- or two-locus genotypic frequencies from Hardy-Weinberg equilibrium (HWE) or from random association of gametes (RAG), and the deviations of additive and dominance variances from the values at HWE. These deviations have a discontinuous distribution in space and different extents of interruptions in the ecological zone with a sharp boundary, but exhibit a continuous distribution in the tension zone. Linkage disequilibrium enhances the difference between the deviations from HWE and from RAG for any two-locus genotypic frequency.     

Alternative forms for genomic clines

Understanding factors regulating hybrid fitness and gene exchange is a major research challenge for evolutionary biology. Genomic cline analysis has been used to evaluate alternative patterns of introgression, but only two models have been used widely and the approach has generally lacked a hypothesis testing framework for distinguishing effects of selection and drift. I propose two alternative cline models, implement multivariate outlier detection to identify markers associated with hybrid fitness, and simulate hybrid zone dynamics to evaluate the signatures of different modes of selection. Analysis of simulated data shows that previous approaches are prone to false positives (multinomial regression) or relatively insensitive to outlier loci affected by selection (Barton's concordance). The new, theory‐based logit‐logistic cline model is generally best at detecting loci affecting hybrid fitness. Although some generalizations can be made about different modes of selection, there is no one‐to‐one correspondence between pattern and process. These new methods will enhance our ability to extract important information about the genetics of reproductive isolation and hybrid fitness. However, much remains to be done to relate statistical patterns to particular evolutionary processes. The methods described here are implemented in a freely available package “HIest” for the R statistical software (CRAN; http://cran.r-project.org/ ).     

A comparison of nuclear and mitochondrial cline shapes in a hybrid zone in the Sceloporus grammicus complex (Squamata; Phrynosomatidae)

The F5 and FM2 chromosome races of the Sceloporus grammicus complex form a hybrid zone in the Mexican state of Hidalgo. Previous studies of this zone have assessed genetic structure by averaging estimates of shape and width across three diagnostic chromosome markers. This approach is likely to mask subtle differences in cline shape among loci (e.g. selected vs. neutral), and obscure any displacement of cline centres (if present). Here we use maximum likelihood methods to construct the best fitting individual clines for three chromosomal markers, and also add two new markers; the mitochondrial DNA (mtDNA) locus, and the nuclear ribosomal DNA (rDNA) repeat. For each locus, hybrid zone models were fitted by cline shape and width, and the position and number of segments describing the centre of the zone. Pairwise comparisons between all clines revealed concordance between chromosomes 2 and 6, but significant discordance in cline structure among all other paired combinations. The concordance of chromosomes 2 and 6 suggests that these clines are maintained by genome-wide forces. The discordance of the chromosome 1 cline suggests an influence of asymmetric introgression, while the mtDNA cline is probably influenced by selection and drift. The rDNA locus reveals a pattern best explained by either extreme asymmetric introgression or gene conversion. The structure of zone indicates that genome-wide processes and locus specific selective forces as well as drift, are operating to different degrees on different loci. The locus-by-locus approach used here permits a finer discrimination among possible mechanisms responsible for the maintenance of the individual clines.