Uniqueness and multiplicity of clines in an environmental pocket.

The number of clines (i.e., polymorphic equilibria) maintained by a step-environment in a unidimensional pocket at a single diallelic locus is investigated. The monoecious population is locally panmictic; its density is uniform. Migration and viability selection are both weak; the former is homogeneous and symmetric; the latter is directional and usually specified by a unimodal function f(p) of the gene frequency p. If the ratio of the selection intensity to the migration rate exceeds a critical value, at least one cline exists. The general theorems on equilibrium structure determine it in detail for many classes of f(p), including the cubic for frequency-independent selection. Numerical examples demonstrate that for suitable f(p), many equilibria can occur simultaneously.

Clines with partial panmixia across a geographical barrier.

In geographically structured populations, partial global panmixia can be regarded as the limiting case of long-distance migration. In the presence of a geographical barrier, an exact, discrete model for the evolution of the gene frequencies at a multiallelic locus under viability selection, local adult migration, and partial panmixia is formulated. For slow evolution, from this model a spatially unidimensional continuous approximation (a system of integro-partial differential equations with discontinuities at the barrier) is derived. For (i) the step-environment, (ii) homogeneous, isotropic migration on the entire line, and (iii) two alleles without dominance, an explicit solution for the unique polymorphic equilibrium is found. In most natural limiting cases, asymptotic expressions are obtained for the gene frequencies on either side of the barrier.

Clines with partial panmixia across a geographical barrier in an environmental pocket.

In a geographically structured population, partial global panmixia can be regarded as the limiting case of long-distance migration. On the entire line with homogeneous, isotropic migration, an environmental pocket is bounded by a geographical barrier, which need not be symmetric. For slow evolution, a continuous approximation of the exact, discrete model for the gene frequency p(x) at a diallelic locus at equilibrium, where x denotes position and the barrier is at x=±a, is formulated and investigated. This model incorporates viability selection, local adult migration, adult partial panmixia, and the barrier. The gene frequency and its derivatives are discontinuous at the barrier unless the latter is symmetric, in which case only p(x) is discontinuous. A cline exists only if the scaled rate of partial panmixia ß<1; several qualitative results also are proved. Formulas that determine p(x) in a step-environment when dominance is absent are derived. The maximal gene frequency in the cline satisfies p(0)<1-ß. A cline exists if and only if 0≤ß<1 and the radius a of the pocket exceeds the minimal radius a(∗), for which a simple, explicit formula is deduced. Given numerical solutions for p(0) and p(a±), an explicit formula is proved for p(x) in |x|>a; whereas in (-a,a), an elliptic integral for x must be numerically inverted. The minimal radius a(∗∗) for maintenance of a cline in an isotropic, bidimensional pocket is also examined.

Dying on the way: the influence of migrational mortality on neutral models of spatial variation.

Migrational mortality is introduced into the classical Malécot model for migration, mutation, and random genetic drift. To assess the influence of mortality, its effect on the backward migration rates and on the probabilities of identity in allelic state are studied. Perhaps surprisingly, some of the former may increase, but as is intuitive, their sum always decreases. As expected, in the island model, mortality does not change the migration pattern, but it decreases the migration rate. Furthermore, it decreases the expected heterozygosity, but increases the genetic diversity and differentiation. The circular habitat and the unbounded, linear stepping-stone model also illustrate the general results. Arbitrary migration is also analyzed. If migration is sufficiently weak, then mortality diminishes every migration rate; it decreases the expected heterozygosity and the genetic similarity between demes. In the strong-migration limit, mortality may raise or lower the probability of identity in state. Perhaps unexpectedly, under mild and reasonable biological assumptions, mortality does not alter the diffusion limit of the probabilities of identity.

Dying on the way: the influence of migrational mortality on clines.

Migrational mortality is introduced into the classical single-locus model for migration and selection. Genotype-independent migration follows selection, which may be soft or hard. For soft selection, the effect of mortality on the backward migration matrix is the same as in the Malécot model; for hard selection, some neutral results still hold, but some do not. For two diallelic demes, mortality can increase or decrease the stringency of the condition for protecting an allele from loss. In the discrete-space, continuous-time limit, mortality increases the diagonal elements of the migration rate matrix and decreases its off-diagonal elements. Were it not for the same result in the Malécot model, it would be surprising that mortality does not alter the general diffusion limit for multiple alleles, arbitrary multidimensional migration, and arbitrary selection.

Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat.

In spatially structured populations, global panmixia can be viewed as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection with complete dominance in an unbounded unidimensional habitat is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and symmetric; the latter is frequency independent. The spatial factor g̃(x) in the selection term, where x denotes position, is a single step at the origin: g̃(x)=-α<0 if x<0, and g̃(x)=1 if x>0. If α=1, there exists a globally asymptotically stable cline. For α<1, such a cline exists if and only if the scaled panmictic rate ß is less than the critical value ß(∗∗)=2α/(1-α). For α>1, a unique, asymptotically stable cline exists if and only if ß is less than the critical value ß(∗); then a smaller, unique, unstable equilibrium also exists whenever ß<ß(∗). Two coupled, nonlinear polynomial equations uniquely determine ß(∗). Explicit solutions are derived for each of the above equilibria. If ß>0 and a cline exists, some polymorphism is maintained even at x=±∞. Both the preceding result and the existence of an unstable equilibrium when α>1 and 0<ß<ß(∗) differ qualitatively from the classical case (ß=0).

Clines with partial panmixia in an environmental pocket.

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 (i.e., asymptotically stable equilibria) maintained by migration and selection in an isotropic environmental pocket in n dimensions is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and isotropic; the latter is directional. If the scaled panmictic rate ß≥1, then the allele favored in the pocket is ultimately lost. For ß<1, a cline is maintained if and only if the scaled radius a of the pocket exceeds a critical value an. For a step-environment without dominance, simple, explicit formulas are derived for a1 and a3; an equation with a unique solution and simple, explicit approximations are deduced for a2. The ratio of the selection coefficients outside and inside the pocket is -α. As expected intuitively, the cline becomes more difficult to maintain; i.e., the critical radius an increases for n=1,2,3,… as α, ß, or n increases.

Single-nucleotide mutation rate increases close to insertions/deletions in eukaryotes.

Mutation hotspots are commonly observed in genomic sequences and certain human disease loci, but general mechanisms for their formation remain elusive. Here we investigate the distribution of single-nucleotide changes around insertions/deletions (indels) in six independent genome comparisons, including primates, rodents, fruitfly, rice and yeast. In each of these genomic comparisons, nucleotide divergence (D) is substantially elevated surrounding indels and decreases monotonically to near-background levels over several hundred bases. D is significantly correlated with both size and abundance of nearby indels. In comparisons of closely related species, derived nucleotide substitutions surrounding indels occur in significantly greater numbers in the lineage containing the indel than in the one containing the ancestral (non-indel) allele; the same holds within species for single-nucleotide mutations surrounding polymorphic indels. We propose that heterozygosity for an indel is mutagenic to surrounding sequences, and use yeast genome-wide polymorphism data to estimate the increase in mutation rate. The consistency of these patterns within and between species suggests that indel-associated substitution is a general mutational mechanism.

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 ß(0) of the scaled panmictic rate ß 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 ß ≥ 0. Otherwise, if ß ≥ ß(0), the allele with the greater average selection coefficient is ultimately fixed. If ß < ß(0), 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.

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 λ(0) (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ(0) is an increasing function of the panmictic rate and to deduce the effect on λ(0) 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 λ(0); bounds on λ(0) are established; and λ(0) 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.

The influence of partial panmixia on neutral models of spatial variation.

Partial panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into neutral models of geographical variation is investigated. The monoecious, diploid population is subdivided into randomly mating colonies that exchange gametes independently of genotype. The gametes fuse wholly at random, including self-fertilization. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus; every allele mutates to new alleles at the same rate. Introducing some panmixia intensifies sufficiently weak migration. A general formula is derived for the migration effective population number, N(e), and N(e) is evaluated explicitly in a number of models with nonconservative migration. Usually, N(e) increases as the panmictic rate, b, increases; in particular, this result holds for two demes, and generically if the underlying migration is either sufficiently weak or panmixia is sufficiently strong. However, in an analytic model, there exists an open set of parameters for which N(e) decreases as b increases. Migration is conservative in the island and circular-habitat models, which are studied in detail. In the former, including some panmixia simply alters the underlying migration rate, increasing (decreasing) it if it is less (greater) than the panmictic value. For the circular habitat, the probability of identity in allelic state at equilibrium is calculated in a nonlocal, continuous-space, continuous-time approximation. In both models, by an efficient, general method, the expected homozygosity, effective number of alleles, and differentiation of gene frequencies are evaluated and discussed; their monotonicity properties with respect to all the parameters are determined; and in the model of infinitely many sites, the mean coalescence times and nucleotide diversities are studied similarly. For the probability of identity at equilibrium in the unbounded stepping-stone model in arbitrarily many dimensions, introducing some panmixia merely replaces the mutation rate by a larger parameter. If the average probability of identity is initially zero, as for identity by descent, then the same conclusion holds for all time.

Evolution under the multilocus Levene model without epistasis.

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, w(rho), depends only on the vector of gene frequencies, rho; it is nondecreasing, and the single-generation change is zero only on the set, Lambda, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of w(rho). If the equilibrium points rho of rho(t) (where t denotes time in generations) are isolated, as is generic, then rho(t) converges as t-->infinity to some rho. Generically, rho(t) converges to a local maximum of w(rho). Write the vector of gametic frequencies, p, as (rho,d)(T), where d represents the vector of linkage disequilibria. If rho is a local maximum of w(rho), then the equilibrium point (rho,0)(T) is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)-->0 globally as t-->infinity. In the second case, w(rho) has a unique maximum rho and (rho,0)(T) is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Lambda, (i) the number of demes, Gamma, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Gamma-1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.

Polymorphism in multiallelic migration-selection models with dominance.

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.

The Rate of Change of a Character Correlated with Fitness.

An extended form of Fisher's Fundamental Theorem of Natural Selection gives the rate of change of the mean value, [Formula: see text], of a measured character. For a character determined by multiple alleles at two loci, this is [Formula: see text] where the Newtonian superior dot means the time derivative and the circle is the time derivative of the logarithm. Covg (m, γ) is the genic (additive genetic) covariance of the character and fitness. Specifically, it is the covariance of the average excess of an allele for fitness and its average effect on the character. [Formula: see text] is the average rate of change of the value of the character for individual genotypes, weighted by their frequencies. The value could be nonzero because of changing environments or change in the age distribution of the population. The third term on the right is the average over all pairs of alleles at both loci of the product of the dominance deviation and the rate of change of ln θ(n), where θ(n) is a measure of departure from random proportions. The last term is a similar expression for epistatic interactions. If selection is much weaker than recombination, after several generations, the last two terms are much smaller than the first. When the measured character is fitness, our result reduces to Kimura's generalization of Fisher's Fundamental Theorem of Natural Selection.

Evolution under multiallelic migration-selection models.

The loss of a specified allele and the convergence of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection are investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Sufficient conditions are established for global fixation and for global loss of a particular allele. When migration is either sufficiently weak or sufficiently strong relative to selection, the equilibria are described, convergence of the gene frequencies is demonstrated, and sufficient conditions for the increase of a suitably defined mean fitness are offered. If the selection pattern is the same in every colony and such that in a panmictic population there is a globally asymptotically stable, internal (i.e., completely polymorphic) equilibrium point, then under certain weak assumptions on migration, the gene frequencies in the subdivided population converge globally to that equilibrium point. Thus, in this case, the ultimate state of the population is unaffected by geographical structure.

Evolution at a multiallelic locus under migration and uniform selection.

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The drift and diffusion coefficients may depend on position, but the selection coefficients do not. It is established that if p is a uniform equilibrium point under pure selection, then p is a migration-selection equilibrium, and that generically the introduction of migration does not change the stability of p. It is also proved that if p is a uniform, globally asymptotically stable, internal equilibrium point under pure selection, then the gene frequencies converge to p when both migration and selection are present. Thus, in this case, after a sufficiently long time, there is no genetic indication of the spatial distribution of the population.

Multiallelic selection polymorphism.

The existence and stability of an internal (i.e., completely polymorphic) equilibrium for viability selection at a single multiallelic locus is investigated. Generations are discrete and nonoverlapping; the population is panmictic, monoecious, and diploid. Various necessary and sufficient conditions for the existence of an internal equilibrium are established and applied to the loss of alleles. Some necessary conditions for the existence of an asymptotically stable internal equilibrium are also established. All these conditions are simpler and yield general biological conclusions more easily than the classical necessary and sufficient conditions.

Evolution under the multiallelic Levene model.

The evolution of the multiallelic Levene model is investigated. New sufficient conditions for nonexistence of a completely polymorphic equilibrium and for global loss of an allele and information on which allele(s) will be lost are deduced from some new results on multidimensional recursion relations. In the absence of dominance, a more detailed analysis is presented. Sufficient conditions for global fixation of a particular allele are established. When the number of alleles equals the number of demes, necessary and sufficient conditions for the existence of an isolated, globally asymptotically stable, completely polymorphic equilibrium point are derived, and this equilibrium is explicitly determined. Three examples, one with arbitrarily many alleles and two with three alleles, illustrate the theory.

A stochastic model for a progressive chronic disease.

For many progressive chronic diseases, there exist useful prognostic indicators for the course of the disease and the survival of the patient. The evolution of such an indicator is modelled as a monotone transformation of a pure birth process with killing. Explicit formulas are derived for the probability distribution of this process at an arbitrary time, the distribution of the first-passage times, the joint distribution of the survival time and the maximum of the process, and the marginals of this joint distribution. In two examples, the general formulas are evaluated in closed form.

When and where was the most recent common ancestor?

The structured coalescent is investigated for single-locus, digenic samples in the diffusion limit of the unidimensional stepping-stone model for homogeneous, isotropic migration and random genetic drift. Let T denote the scaled time to the most recent common ancestor (MRCA) of the two genes, and let Z designate the scaled deviation of the position of the MRCA from the average position of the two genes. The joint probability density of T and Z is evaluated explicitly. Both the marginal and conditional distributions of T have infinite expectation, as does the marginal distribution of Z. Conditioned on T = tau, the distribution of Z is Gaussian with mean zero and variance 2tau. The main results are extended to anisotropic migration. The results establish the existence of and define in the diffusion limit a retrospective stochastic process for digenic samples in one spatial dimension.