@article{4262, abstract = {Natural populations are structured spatially into local populations and genetically into diverse ‘genetic backgrounds’ defined by different combinations of selected alleles. If selection maintains genetic backgrounds at constant frequency then neutral diversity is enhanced. By contrast, if background frequencies fluctuate then diversity is reduced. Provided that the population size of each background is large enough, these effects can be described by the structured coalescent process. Almost all the extant results based on the coalescent deal with a single selected locus. Yet we know that very large numbers of genes are under selection and that any substantial effects are likely to be due to the cumulative effects of many loci. Here, we set up a general framework for the extension of the coalescent to multilocus scenarios and we use it to study the simplest model, where strong balancing selection acting on a set of n loci maintains 2n backgrounds at constant frequencies and at linkage equilibrium. Analytical results show that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. However, simulation results reveal that the structured coalescent approach breaks down when the number of backgrounds approaches the population size, because of stochastic fluctuations in background frequencies. A new method is needed to extend the structured coalescent to cases with large numbers of backgrounds.}, author = {Barton, Nicholas H and Navarro, Arcadio}, issn = {0016-6723}, journal = {Genetical Research}, number = {2}, pages = {129 -- 139}, publisher = {Cambridge University Press}, title = {{Extending the coalescent to multilocus systems: the case of balancing selection}}, doi = {10.1017/S0016672301005493}, volume = {79}, year = {2002}, } @article{3623, abstract = {We present the theoretical background to a new method for measuring genetic variation for total fitness in Drosophila. The method allows heterozygous effects on total fitness of whole wild-type chromosomes to be measured under normal demography with overlapping generations. The wild-type chromosomes are competed against two balancer chromosomes (B1, B2, say), providing a standard genotype B1/B2 against which variation in the fitness effects of the wild-type chromosomes can be assessed. Fitness can be assessed in two ways: (i) at equilibrium of all three chromosomes under heterozygote advantage, and (ii) during displacement of one balancer by the other. Equilibrium with all three chromosomes present will be achieved only if the wild-type homozygote is not too fit, and if the fitnesses of the three heterozygotes are not too unequal. These conditions were not satisfied for any of a sample of 12 lethal-bearing chromosomes isolated from a random-bred laboratory population of Drosophila. At equilibrium, genotypic frequencies show low sensitivity to changes in genotypic fitness. Furthermore, where all four genotypes are viable and fertile, supplementary information from cages with only two chromosomes present and from direct measurements of pre-adult viability are required to estimate fitnesses from frequencies. The invasion method has the advantages of a greater sensitivity and of not requiring further data to estimate fitnesses if the wild-type homozygote is fertile. However, it requires that multiple samples be taken as the invasion progresses. In a discrete generation model, generation time influences fitness estimates from this method and is difficult to estimate accurately from the data. A full age-structured model can also be applied to the data from both types of experiment. For the invasion method, this gives fitness estimates close to those from the discrete generation model.}, author = {Barton, Nicholas H and Patridge, Linda}, issn = {0016-6723}, journal = {Genetical Research}, number = {3}, pages = {297 -- 314}, publisher = {Cambridge University Press}, title = {{Measuring fitness by means of balancer chromosomes}}, doi = {10.1017/S0016672399004346}, volume = {75}, year = {2000}, } @article{4271, abstract = {Within hybrid zones that are maintained by a balance between selection and dispersal, linkage disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium causes selection on each locus to act on all other loci, thereby steepening dines, and generating a barrier to gene flow. Diffusion models predict simple relations between the strength of linkage disequilibrium and the dispersal rate, σ, and between the barrier to gene flow, B, and the reduction in mean fitness, W̄. The aim of this paper is to test the accuracy of these predictions by comparison with an exact deterministic model of unlinked loci (r = 0.5). Disruptive selection acts on the proportion of alleles from the parental populations (p, q): W = exp[-S(4pq)(β)], such that the least fit genotype has fitness e(-S). Where β << 1, fitness is reduced for a wide range of intermediate genotypes; where β >> 1, fitness is only reduced for those genotypes close to p = 0.5. Even with strong epistasis, linkage disequilibria are close to σ2p'(i)p'(j)/r(ij), where p'(i), p'(j) are the gradients in allele frequency at loci i, j. The barrier to gene flow, which is reflected in the steepening of neutral dines, is given by B = ∫(-∞)(∞) (W̄(1/r̄)-1) dx, where r̄, the harmonic mean recombination rate between the neural and selected loci, is here 0.5. This is a close approximation for weak selection, but underestimates B for strong selection. The barrier is stronger for small β, because hybrid fitness is then reduced over a wider range of p. The widths of the selected dines are harder to predict: though simple approximations are accurate for β = 1, they become inaccurate for extreme β because, then, fitness changes sharply with p. Estimates of gene number, made from neutral dines on the assumption that selection acts against heterozygotes, are accurate for weak selection when β = 1; however, for strong selection, gene number is overestimated. For β > 1, gene number is systematically overestimated and, conversely, when β < 1, it is underestimated. }, author = {Barton, Nicholas H and Shpak, Max}, issn = {0016-6723}, journal = {Genetical Research}, number = {2}, pages = {179 -- 198}, publisher = {Cambridge University Press}, title = {{The effects of epistasis on the structure of hybrid zones}}, doi = {10.1017/S0016672399004334}, volume = {75}, year = {2000}, } @misc{4276, author = {Barton, Nicholas H}, booktitle = {Genetics Research}, issn = {0016-6723}, number = {3}, pages = {371 -- 373}, publisher = {Cambridge University Press}, title = {{Population genetics of multiple loci}}, doi = {10.1017/S0016672300239220}, volume = {75}, year = {2000}, } @article{3625, abstract = {This article outlines theoretical models of clines in additive polygenic traits, which are maintained by stabilizing selection towards a spatially varying optimum. Clines in the trait mean can be accurately predicted, given knowledge of the genetic variance. However, predicting the variance is difficult, because it depends on genetic details. Changes in genetic variance arise from changes in allele frequency, and in linkage disequilibria. Allele frequency changes dominate when selection is weak relative to recombination, and when there are a moderate number of loci. With a continuum of alleles, gene flow inflates the genetic variance in the same way as a source of mutations of small effect. The variance can be approximated by assuming a Gaussian distribution of allelic effects; with a sufficiently steep cline, this is accurate even when mutation and selection alone are better described by the 'House of Cards' approximation. With just two alleles at each locus, the phenotype changes in a similar way: the mean remains close to the optimum, while the variance changes more slowly, and over a wider region. However, there may be substantial cryptic divergence at the underlying loci. With strong selection and many loci, linkage disequilibria are the main cause of changes in genetic variance. Even for strong selection, the infinitesimal model can be closely approximated by assuming a Gaussian distribution of breeding values. Linkage disequilibria can generate a substantial increase in genetic variance, which is concentrated at sharp gradients in trait means.}, author = {Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {3}, pages = {223 -- 236}, publisher = {Cambridge University Press}, title = {{Clines in polygenic traits}}, doi = {10.1017/S001667239900422X}, volume = {74}, year = {1999}, } @article{3627, abstract = {When a favourable mutation sweeps to fixation, those genes initially linked to it increase in frequency; on average, this reduces diversity in the surrounding region of the genome. In the first analysis of this 'hitch-hiking' effect, Maynard-Smith and Haigh followed the increase of the neutral allele that chanced to be associated with the new mutation in the first generation, and assumed that the subsequent increase was deterministic. Later analyses, based on either coalescence arguments, or on diffusion equations for the mean and variance of allele frequency, have also made one or both of these assumptions. In the early generations, stochastic fluctuations in the frequency of the selected allele, and coalescence of neutral lineages, can be accounted for correctly by following relationships between genes conditional on the number of copies of the favourable allele. This analysis shows that the hitch-hiking effect is increased because an allele that is destined to fix tends to increase more rapidly than exponentially. However, the identity generated by the selective sweep has the same form as in previous work, h[r/s] (2 Ns)(-2r/s), where h[r/s] tends to 1 with tight linkage. This analysis is extended to samples of many genes; then, genes may trace back to several families of lineages, each related through a common ancestor early in the selective sweep. Simulations show that the number and sizes of these families can (in principle) be used to make separate estimates of r/s and Ns.}, author = {Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {2}, pages = {123 -- 133}, publisher = {Cambridge University Press}, title = {{The effect of hitch-hiking on neutral genealogies}}, doi = {10.1017/S0016672398003462}, volume = {72}, year = {1998}, } @misc{4282, author = {Barton, Nicholas H}, booktitle = {Genetical Research}, issn = {0016-6723}, number = {1}, pages = {73 -- 73}, publisher = {Cambridge University Press}, title = {{Genetics and analysis of quantitative traits}}, doi = {10.1017/S0016672398219732}, volume = {72}, year = {1998}, } @misc{4290, author = {Barton, Nicholas H}, booktitle = {Genetical Research}, issn = {0016-6723}, number = {2}, pages = {178 -- 180}, publisher = {Cambridge University Press}, title = {{Natural hybridization and evolution}}, volume = {70}, year = {1997}, } @misc{4291, author = {Barton, Nicholas H}, booktitle = {Genetical Research}, issn = {0016-6723}, number = {2}, pages = {180 -- 181}, publisher = {Cambridge University Press}, title = {{The ecological detective: Confronting models with data}}, volume = {70}, year = {1997}, } @article{3635, abstract = {Experiments on Drosophila suggest that genetic recombination may result in lowered fitness of progeny (a 'recombination load'). This has been interpreted as evidence either for a direct effect of recombination on fitness, or for the maintenance of linkage disequilibria by epistatic selection. Here we show that such a recombination load is to be expected even if selection favours increased genetic recombination. This is because of the fact that, although a modifier may suffer an immediate loss of fitness if it increases recombination, it eventually becomes associated with a higher additive genetic variance in fitness, which allows a faster response to direction selection. This argument applies to mutation-selection balance with synergistic epistasis, directional selection on quantitative traits, and ectopic exchange among transposable elements. Further experiments are needed to determine whether the selection against recombination due to the immediate load is outweighed by the increased additive variance in fitness produced by recombination.}, author = {Charlesworth, Brian and Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {1}, pages = {27 -- 41}, publisher = {Cambridge University Press}, title = {{Recombination load associated with selection for increased recombination}}, doi = {10.1017/S0016672300033450}, volume = {67}, year = {1996}, } @article{3639, abstract = {A general representation of multilocus selection is extended to allow recombination to depend on genotype. The equations simplify if modifier alleles have small effects on recombination. The evolution of such modifiers only depends on how they alter recombination between the selected loci, and does not involve dominance in modifier effects. The net selection on modifiers can be found explicitly if epistasis is weak relative to recombination. This analysis shows that recombination can be favoured in two ways: because it impedes the response to epistasis which fluctuates in sign, or because it facilitates the response to directional selection. The first mechanism is implausible, because epistasis must change sign over periods of a few generations: faster or slower fluctuations favour reduced recombination. The second mechanism requires weak negative epistasis between favourable alleles, which may either be increasing, or held in check by mutation. The selection (si) on recombination modifiers depends on the reduction in additive variance of log (fitness) due to linkage disequilibria (υ1 < 0), and on non-additive variance in log (fitness) (V′2, V′3,.. epistasis between 2, 3.. loci). For unlinked loci and pairwise epistasis, si = − (υ1 + 4V2/3)δr, where δr is the average increase in recombination caused by the modifier. The approximations are checked against exact calculations for three loci, and against Charlesworth's analyses of mutation/selection balance (1990), and directional selection (1993). The analysis demonstrates a general relation between selection on recombination and observable components of fitness variation, which is open to experimental test.}, author = {Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {2}, pages = {123 -- 144}, publisher = {Cambridge University Press}, title = {{A general model for the evolution of recombination}}, doi = {10.1017/S0016672300033140}, volume = {65}, year = {1995}, } @article{3641, abstract = {The probability of fixation of a mutation with selective advantage s will be reduced by substitutions at other loci. The effect of a single substitution, with selective advantage S0016672300032857inline1, can be approximated as a sudden reduction in the frequency of the favourable allele, by a fraction w = 1 −(s/S)r/s (where r is the recombination rate). An expression for the effect of a given sequence of such catastrophes is derived. This also applies to the ecological prxoblem of finding the probability that a small population will survive, despite occasional disasters. It is shown that if substitutions occur at a rate Δ, and are scattered randomly over a genetic map of length R, then an allele is unlikely to be fixed if its advantage is less than a critical value, Scrit = (π2/6)(2ΔS/(Rlog(S/s))). This threshold depends primarily on the variance in fitness per unit map length dueto substitutions, var(W)/R = 2ΔS/R. With no recombination, the fixation probability can be calculated for a finite population. If Δ > s, it is of the same order as for a neutral allele ( ≈ Δ/(2N(Δ−s))), whilst if S0016672300032857inline2, fixation probability is much higher than for a neutral allele, but much lower than in the absence of hitch-hiking S0016672300032857inline3. These results suggest that hitch-hiking may substantially impede the accumulation of weakly favoured adaptations.}, author = {Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {3}, pages = {199 -- 208}, publisher = {Cambridge University Press}, title = {{The reduction in fixation probability caused by substitutions at linked loci}}, doi = {10.1017/S0016672300032857 }, volume = {64}, year = {1994}, } @article{3643, abstract = {We investigate the establishment and spread of new adaptive peaks within Wright's ‘shifting balance’. The third phase of the ‘shifting balance’ involves a kind of group selection, since demes in which a superior peak has been established contain more individuals, and so send out more migrants. We assume that population size, N, increases with mean fitness, , according to the exponential relation, . Here, k is a measure of the weakness of density-dependent regulation, and equals the inverse of the regression of log (fitness) on log(N). In the island model, we find that just as with soft selection (k = 0), two distinct types of behaviour exist: group selection makes no qualitative difference. With low numbers of migrants, demes fluctuate almost independently, and only one equilibrium exists. With large numbers of migrants, all the demes evolve towards the same adaptive peak, and so the whole population can move towards one or other of the peaks. Group selection can be understood in terms of an effective mean fitness function. Its main consequence is to increase the effect of selection relative to drift (Ns), and so increase the bias towards the fitter peak. However, this increased bias depends on the ratio between k and the deme size (k/N), and so is very small when density-dependence is reasonably strong.}, author = {Rouhani, Shahin and Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetical Research}, number = {2}, pages = {127 -- 136}, publisher = {Cambridge University Press}, title = {{Group selection and the 'shifting balance'}}, doi = {10.1017/S0016672300031232}, volume = {61}, year = {1993}, } @article{3644, abstract = {Wright proposed that there is a ' shifting balance' between selection within demes, random drift, and selection between demes at different 'adaptive peaks'. We investigate the establishment and spread of new adaptive peaks, considering a chromosome rearrangement, and a polygenic character under disruptive selection. When the number of migrants (Nm) is small, demes fluctuate independently, with a bias towards the fitter peak. When Nm is large, the whole population can move to one of two stable equilibria, and so can be trapped near the lower peak. These two regimes are separated by a sharp transition at a critical Nm of order 1. Just below this critical point, adaptation is most efficient, since the shifting balance greatly increases the proportion of demes that reach the global optimum. This is so even if one peak is only slightly fitter than the other (AWx \/N), and for both strong and weak selection (Ns <§ 1 or Ns > 1). Provided that Nm varies sufficiently gradually from place to place, the fitter peak can be established in regions where Nm « 1, and can then spread through the rest of the range. Our analysis confirms Wright's argument that if selection, migration and drift are of the same order, the ' shifting balance' leads to efficient evolution towards the global optimum.}, author = {Barton, Nicholas H and Rouhani, Shahin}, issn = {0016-6723}, journal = {Genetical Research}, number = {1}, pages = {57 -- 74}, publisher = {Cambridge University Press}, title = {{Adaptation and the 'shifting balance'}}, doi = {10.1017/S0016672300031098 }, volume = {61}, year = {1993}, } @misc{4302, author = {Barton, Nicholas H}, booktitle = {Genetical Research}, issn = {0016-6723}, number = {1}, pages = {77 -- 85}, publisher = {Cambridge University Press}, title = {{Review of "The causes of molecular evolution" by J.H. Gillespie}}, doi = {10.1017/S001667230003158X }, volume = {62}, year = {1993}, } @article{4303, abstract = {In a stably subdivided population with symmetric migration, the chance that a favoured allele will be fixed is independent of population structure. However, random extinction introduces an extra component of sampling drift, and reduces the probability of fixation. In this paper, the fixation probability is calculated using the diffusion approximation; comparison with exact solution of the discrete model shows this to be accurate. The key parameters are the rates of selection, migration and extinction, scaled relative to population size (S = 4Ns, M = 4Nm, Λ = 4Nλ); results apply to a haploid model, or to diploids with additive selection. If new colonies derive from many demes, the fixation probability cannot be reduced by more than half. However, if colonies are initially homogeneous, fixation probability can be much reduced. In the limit of low migration and extinction rates (M, Λ 1), it is 2s/{1 + (Λ/MS)(1 −exp(−S))}, whilst in the opposite limit (S 1), it is 4sM/{Λ(Λ + M)}. In the limit of weak selection (M, Λ 1), it is 4sM/{Λ(Λ + M)}. These factors are not the same as the reduction in effective population size (Ne/N), showing that the effects of population structure on selected alleles cannot be understood from the behaviour of neutral markers.}, author = {Barton, Nicholas H}, issn = {0016-6723}, journal = {Genetics Research}, number = {2}, pages = {149 -- 158}, publisher = {Cambridge University Press}, title = {{The probability of fixation of a favoured allele in a subdivided population}}, doi = {10.1017/S0016672300031748}, volume = {62}, year = {1993}, } @article{4314, abstract = {Polygenic variation can be maintained by a balance between mutation and stabilizing selection. When the alleles responsible for variation are rare, many classes of equilibria may be stable. The rate at which drift causes shifts between equilibria is investigated by integrating the gene frequency distribution W2N II (pq)4N mu-1. This integral can be found exactly, by numerical integration, or can be approximated by assuming that the full distribution of allele frequencies is approximately Gaussian. These methods are checked against simulations. Over a wide range of population sizes, drift will keep the population near an equilibrium which minimizes the genetic variance and the deviation from the selective optimum. Shifts between equilibria in this class occur at an appreciable rate if the product of population size and selection on each locus is small (Ns alpha 2 less than 10). The Gaussian approximation is accurate even when the underlying distribution is strongly skewed. Reproductive isolation evolves as populations shift to new combinations of alleles: however, this process is slow, approaching the neutral rate (approximately mu) in small populations.}, author = {Barton, Nicholas H}, issn = {1469-5073}, journal = {Genetical Research}, number = {1}, pages = {59 -- 78}, publisher = {Cambridge University Press}, title = {{The divergence of a polygenic system under stabilising selection, mutation and drift}}, doi = {10.1017/S0016672300028378}, volume = {54}, year = {1989}, } @article{3660, abstract = {The maintenance of polygenic variability by a balance between mutation and stabilizing selection has been analysed using two approximations: the ‘Gaussian’ and the ‘house of cards’. These lead to qualitatively different relationships between the equilibrium genetic variance and the parameters describing selection and mutation. Here we generalize these approximations to describe the dynamics of genetic means and variances under arbitrary patterns of selection and mutation. We incorporate genetic drift into the same mathematical framework. The effects of frequency-independent selection and genetic drift can be determined from the gradient of log mean fitness and a covariance matrix that depends on genotype frequencies. These equations describe an ‘adaptive landscape’, with a natural metric of genetic distance set by the covariance matrix. From this representation we can change coordinates to derive equations describing the dynamics of an additive polygenic character in terms of the moments (means, variances, …) of allelic effects at individual loci. Only under certain simplifying conditions, such as those derived from the Gaussian and house-of-cards approximations, do these general recursions lead to tractable equations for the first few phenotypic moments. The alternative approximations differ in the constraints they impose on the distributions of allelic effects at individual loci. The Gaussian-based prediction that evolution of the phenotypic mean does not change the genetic variance is shown to be a consequence of the assumption that the allelic distributions are never skewed. We present both analytical and numerical results delimiting the parameter values consistent with our approximations.}, author = {Barton, Nicholas H and Turelli, Michael}, issn = {1469-5073}, journal = {Genetical Research}, number = {2}, pages = {157 -- 174}, publisher = {Cambridge University Press}, title = {{Adaptive landscapes, genetic distance, and the evolution of quantitative characters}}, doi = {10.1017/S0016672300026951}, volume = {49}, year = {1987}, } @article{4322, abstract = {A method is developed for calculating the probability of establishment of an allele which is favoured in some places, but not others, in a large subdivided population. This method is quite general, and could be used to calculate the chance that any system which is linear near an absorbing boundary will move away from that boundary. The results are applied to a population distributed along one dimension. Only mutants which arise within a distance σ/ √2s of the region in which they are favoured stand an appreciable chance of establishment. The net chance of establishment of mutations distributed randomly across the habitat will be decreased by gene flow if selection against them is sufficiently strong. However, if the mutations are only weakly deleterious outside some limited region, gene flow may increase the net chance of establishment.}, author = {Barton, Nicholas H}, issn = {1469-5073}, journal = {Genetical Research}, number = {1}, pages = {35 -- 40}, publisher = {Cambridge University Press}, title = {{The probability of establishment of an advantageous mutation in a subdivided population}}, doi = {10.1017/S0016672300023314}, volume = {50}, year = {1987}, } @article{4324, abstract = {The maintenance of polygenic variation through a balance between mutation and stabilizing selection can be approximated in two ways. In the ‘Gaussian’ approximation, a normal distribution of allelic effects is assumed at each locus. In the ‘House of Cards’ approximation, the effect of new mutations is assumed to be large compared with the spread of the existing distribution. These approximations were developed to describe models where alleles may have a continuous range of effects. However, previous analyses of models with only two alleles have predicted an equilibrium variance equal to that given by the ‘House of Cards’ approximation. These analyses of biallelic models have assumed that, at equilibrium, the population mean is at the optimum. Here, it is shown that many stable equilibria may coexist, each giving a slight deviation from the optimum. Though the variance is given by the ‘House of Cards’ approximation when the mean is at the optimum, it increases towards a value of the same order as that given by the ‘Gaussian’ approximation when the mean deviates from the optimum. Thus, the equilibrium variance cannot be predicted by any simple model, but depends on the previous history of the population.}, author = {Barton, Nicholas H}, issn = {1469-5073}, journal = {Genetical Research}, number = {3}, pages = {209 -- 216}, publisher = {Cambridge University Press}, title = {{The maintenance of polygenic variation through a balance between mutation and stabilising selection}}, doi = {10.1017/S0016672300023156}, volume = {47}, year = {1986}, }