Eigenanalysis for the Kolmogorov backward equation for the neutral multi-allelic model

In a previous paper [9], we have given an algorithm for obtaining the time dependent solution of all polynomial moments of gene frequencies in neutral models. The recurrence formula for the moment generating functions obtained in [9] gives information about the eigenvalues and the eigenfunctions in neutral models. In this article, we solve the eigenvalue problem for the Kolmogorov backward equation for the case of neutral alleles under mutation pressure.     

The ESS under spatial variation with applications to sex allocation

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.     

Correlation of heterozygosity and the number of alleles in different frequency classes

Analytic expressions for the expectations and variance of the number of alleles with gene frequencies in a specified class are derived in the entire population as well as in a random sample of genes drawn from the population. The correlation of this quantity with heterozygosity at the locus is also obtained. The derivations are given in details for a steady state population of finite size under the infinite allele model of selectively neutral alleles. The results are extended to include weak selection pressures and non-stationarity of the population. The relevance of the correlation of heterozygosity and the number of rare alleles in connection with the neutralist-selectionist controversy is also discussed.     

The deviation from linkage equilibrium with multiple loci varying in a stepping-stone cline

The balance between the creation of associations between alleles at different loci by immigration and the convergence to linkage equilibrium due to the recombination process is studied in a theoretical model. The geographical structure of the model is a stepping-stone chain of populations linking two genetically constant source populations. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The gene frequencies at each locus will then show a linear cline through the stepping-stone chain of populations. The deviation from linkage equilibrium through the stepping-stone cline is characterized by an equation for linear measures that provide the linkage disequilibrium measures for a given set of loci in terms of the gene frequencies and the linkage disequilibria in the source populations and in terms of the linkage disequilibrium measures through the cline for lower numbers of loci. Numerical examples of this iterative solution are given, and it is shown that the build-up of the higher order Bennett-disequilibria through the cline is considerably more pronounced than the build-up of two-locus disequilibria.     

Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.     

General solution of cable theory with both ends sealed

General solution of the cable theory with both ends sealed when injecting an arbitrary current at an arbitrary point of the cable is presented, which is a time-dependent transient solution. The solution is an infinite series, each term of which is the product of a cosine term including a position variable only and an exponential term including a time variable only. The general solution contains almost all solutions reported hitherto as particular cases and the mutual relations among the various solutions of quite different forms are clarified by this general solution. Moreover the shorter the cable becomes, the more rapidly this solution converges, therefore it is useful for an analysis of the short cable in the case where the relative deviation error may grow large. The truncation error can also be estimated as the solution is an infinite series of simple functions.     

Selection, subdivision and extinction and recolonization

In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for fitness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity N(e)s(e), which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

The decay of linkage disequilibrium under random union of gametes: how to calculate Bennett's principal components

How rapidly does an arbitrary pattern of statistical association among a set of loci decay under meiosis and random union of gametes? This problem is non-trivial, even in the case of an infinitely large population where selection and other forces are absent. J. H. Bennett (1954, Ann. Hum. Genet. 18, 311-317) found that, for an arbitrary number of loci with an arbitrary linkage map, it is possible to define measures of linkage disequilibrium that decay geometrically with time. He found a recursive method for deriving expressions for these variables in terms of "allelic moments" (the factorial moments about the origin of the "allelic indicators"), and expressions for the allelic moments in terms of his new variables. However, Bennett no where stated his recursive algorithm explicitly, nor did he give a general formula for his measures of linkage disequilibrium, for an arbitrary number of loci. Recursive definitions of Bennett's variables were obtained by Lyubich. However, the expressions generated by these recursions are not the same as those found by Bennett. (They do not express Bennett's variables as functions of the allelic moments.) Lyubich's derivations employ genetic algebras. Here, I present a method for obtaining explicit expressions for Bennett's variables in terms of the allelic moments. I show that the transformation from the allelic moments to Bennett's variables and the inverse transformation always have the form that Bennett claimed. (This transformation and its inverse have essentially the same form.) I present general recursions for calculating the coefficients in the forward transformation and the coefficients in the inverse transformation. My derivations involve combinatorial arguments and ordinary algebra only. The special case of unlinked loci is briefly discussed.     

Continuous selective models

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.     

A multi-locus continuous-time selection model

Summary A continuous time selection model is formulated for a diploid monoecious population with multiple alleles at each of an arbitrary number of loci, incorporating differential fertility and mortality as well as arbitrary mating and age structure. The model is simplified in the case of age-independence and for the case of a stable age distribution. The age-independent model is examined in detail for the special case of multiple alleles at each of two loci. This model is analyzed under the assumptions of random mating and additive fertilities, with close attention given to the behavior of the system with respect to Hardy-Weinberg proportions and linkage equilibrium.M. M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Mathematical analysis of a model describing evolution of an asexual population in a changing environment

We investigate a mathematical model for an asexual population with non-overlapping (discrete) generations, that exists in a changing environment. Sexual populations are also briefly discussed at the end of the paper. It is assumed that selection occurs on the value of a single polygenic trait, which is controlled by a finite number of loci with discrete-effect alleles. The environmental change results in a moving fitness optimum, causing the trait to be subject to a combination of stabilising and directional selection.This model is different from that investigated by Waxman and Peck [Genetics 153 (1999) 1041] where overlapping generations and continuous effect alleles were considered. In this paper, we consider non-overlapping generations and discrete effect alleles. However in [Genetics 153 (1999) 1041] and the present work, there is the same pattern of environmental change, namely a constant rate of change of the optimum.From [Genetics 153 (1999) 1041], no rigorous theoretical conclusion can be drawn about the form of the solutions as t grows large. Numerical work carried out in [Genetics 153 (1999) 1041] suggests that the solution is a lagged travelling wave solution, but no mathematical proof exists for the continuous model. Only partial results, regarding existence of travelling wave solutions and perturbed solutions, have been established (see [Nonlin. Anal. 53 (2003) 683; An integral equation describing an asexual population in a changing environment, Preprint]). For the discrete case of this paper, under the assumption that the ratio between the unit of genotypic value and the speed of environment change is a rational number, we are able to give rigorous proof of the following conclusion: the population follows the environmental change with a small lag behind, moreover, the lag is represented using a calculable quantity.     

A Model Allowing Continuous Variation in Electrophoretic Mobility of Neutral Alleles

The infinite-sites model with no recombination is extended to include mutations that affect electrophoretic mobility. The model allows the effect of a single-site mutation to have a continuous effect on mobility. Formulae are obtained for the variance of electrophoretic mobility of alleles after an arbitrary lenght of time. A special case of the general model is the case of stepwise production of neutral alleles with an arbitrary number of steps.     

Random genetic drift and selection in a triallelic locus: a continuous diffusion model.

The Kolmogorov forward diffusion equation is used to examine the evolution of three alleles at one locus under viability selection and random genetic drift. Separation of variables and Chebyschev approximations are employed to solve this equation for long times. As an example, one artificial viability set is examined in detail; its general implications for the evolution at a triallelic locus are discussed.     

Biochemical and genetic properties of oligomeric structures: a general approach

The oligomers constituted by association of different subunits can exist under multiple forms. In the case of the genetically variable proteins, such a multiplicity leads to numerous questions (i) on the enumerations: what is the number of active forms when a given subunit can make the oligomer inactive, or when the subunits are encoded by s alleles; (ii) on the subunit effects on biochemical properties: how to estimate these effects, are they equal, are there interactions between subunits, etc. Theoretical methods for the study of such oligomeric structures are developed, which mainly rely on linear model techniques. Peculiar properties examined are Vmax and Km, but also the quantities of the various oligomers, which depend on their association law. This approach is extended to the oligomers composed of different sets of subunits, as are for example some enzymes. These aspects are discussed from numerous bibliographic examples, with special reference to molecular interactions (protein complementation or molecular heterosis). Otherwise the genetic application of this theoretical approach is presented: it is possible to consider a genotype as an oligomer of alleles, and thus to study their effects and their interactions, in the one-locus case as well as in the several-loci case. The relevance of this generalization is discussed in connection with two other concepts, the "sequence space" used in molecular evolution and the regression of the genotypic values on the number of alleles used in quantitative genetics.     

The average number of distinct sites visited by a one-dimensional random walker and its application to isotope exchange in polypeptides

The average number of distinct sites visited by a random walker moving with arbitrary transition probability on a one-dimensional lattice is calculated. Asymptotic forms of this quantity for both asymmetric and symmetric random walks are determined, and an exact solution for the latter case is also given for any number of steps. The average number of sites visited is then analyzed for intermediate numbers of steps by introducing an exponent. This approach is applied to explain the results of isotope exchange experiments in polypeptides, and applications of asymmetric random walks to other biological problems are briefly discussed.     

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.     

Genetic variability due to geographical inhomogeneity

The frequency of one of two alleles is studied as a function of position and time in a one, two, or three dimensional region. A nonlinear diffusion equation is employed. Each allele is assumed to have a selective advantage in some part of the region. An asymptotic solution is constructed for the case when the selection coefficient is large compared to the diffusion coefficient, i.e. when selection acts more rapidly than diffusion. Then as time increases, the solution tends to a cline, i.e. an equilibrium distribution in which both alleles are present everywhere, each predominating where it has the advantage. In a narrow region around the boundary where the selective advantage switches from one allele to the other, both alleles are present with comparable frequencies. Along a line normal to this boundary, the frequency varies as in a one dimensional habitat with a simple variation in selective advantage. The asymptotic solution is compared with the numerical solution for a special two dimensional case, and the agreement is found to be good.Research supported by the National Science Foundation.     

Preferential mating in symmetric multilocus systems: stability conditions of the central equilibrium

Several multilocus models that incorporate both preferential mating and viability selection are studied. Specifically, a class of symmetric heterozygosity models are considered that assign individuals to phenotypic classes according to which loci are in heterozygous state regardless of the actual allelic content. Otherwise, an arbitrary number of loci, number of alleles per locus, and arbitrary recombination scheme, viability parameters and preferential mating pattern based on phenotypes are allowed. The conditions for the stability of a central polymorphism are indicated and interpreted. The effects of viability and preference selection may be summarized in a single quantity for each phenotypic class, a generalized fitness. Preferential assortative mating alone can produce stability for a central polymorphism as in the case of viability selection when sexual attractiveness or general fitness increases with higher levels of heterozygosity. The situation is more complex with sexual selection.     

Exclusion probabilities obtainable by biochemical polymorphisms in dogs

General formulae are given to calculate the exclusion probabilities in false paternity and parentage cases by means of gene loci with an arbitrary number of alleles whereas in paternity cases an arbitrary number of offspring per litter is considered additionally.
By aid of these formulae and on the basis of the allele frequencies of four blood protein and enzyme systems the probabilities of excluding incorrect paternity and parentage are calculated in seven German dog breeds. The results are tabulated and discussed.
It can be shown that the exclusion probability in false paternity cases increases distinctly with an increasing number of offspring per litter and its maximum is nearly attained if 5 offspring are examined. Therefore it is of value to consider entire litters in paternity controls in dogs.     

Difficulties in parentage analysis: the probability that an offspring and parent have the same heterozygous genotype.

Parentage studies often estimate the number of parents contributing to half-sib progeny arrays by counting the number of alleles attributed to unshared parents. This approach is compromised when an offspring has the same heterozygous genotype as the shared parent, for then the contribution of the unshared parent cannot be unambiguously deduced. To determine how often such cases occur, formulae for co-dominant markers with n alleles are derived here for Ph, the probability that a given heterozygous parent has an offspring with the same heterozygous genotype, and Pa, the probability that a randomly chosen offspring has the same heterozygous genotype as the shared parent. These formulae have been derived assuming Mendelian segregation with either (1) an arbitrary mating system, (2) random mating or (3) mixed mating. The maximum value of Pa under random mating is 0.25 and occurs with any two alleles each at a frequency of 0.5. The behaviour with partial selfing (where reproduction is by selfing with probability s, and random mating otherwise) is more complex. For n < or = 3 alleles, the maximum value of Pa occurs with any two alleles each at a frequency of 0.5 if s < 0.25, and with three equally frequent alleles otherwise. Numerically, the maximum value of Pa for n > or = 4 alleles occurs with n* < or = n alleles at equal frequencies, where the maximizing number of alleles n* is an increasing function of the selfing rate. Analytically, the maximum occurs with all n alleles present and equally frequent if s > or = 2/3. In addition, the potential applicability of these formulae for evolutionary studies is briefly discussed.