Monoecious Population Research Articles

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population with random mating

Global convergence to equilibria is proved for the model in the title under the assumption that death rates (and fertilities) are constant and fertilities are additive and positive. The phase portraits are determined. There is at most one polymorphic equilibrium or there are a continuum of equilibria.

The evolution of multilocus systems under weak selection.

The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multi-allelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << cmin, where s and cmin denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t1 approximately (ln s)/ln(1 - cmin) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s-->0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s2). Then after a time t2 approximately 2t1, the linkage disequilibria are nearly constant, their rate of change being O(s2). Furthermore, with an error of O(s2), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s3), then the single-generation change in the mean fitness is delta W = W-1Vg+O(s3), where Vg designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s2) evolves at the rate delta Z = W-1Cg+O(s2), where Cg represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t2, the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.

Production and germination of seeds in Hydrilla verticillata

A monoecious population of Hydrilla verticillata (L.f.) Royle, occurring in the Wazirabad reservoir on the river Yamuna at New Delhi, produces seeds profusely during late winter. The seeds are light sensitive and germinate within a week at laboratory temperature (23–28°C). Seeds stored dry or in water in darkness for up to 1 year germinate readily in light. Thus, seeds offer a long-term strategy for survival of Hydrilla after prolonged dry periods in regions with a monsson climate.

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.

Rate of evolution of a quantitative character.

The rate of change of the mean of a quantitative character is evaluated exactly and also under the hypothesis of linear biparental regression. Generations are discrete and nonoverlapping; the monoecious population mates at random. The genotypic and environmental contributions to the character are additive and stochastically independent. The character is influenced by arbitrarily many multiallelic loci and has constant genotypic values; dominance, epistasis, and the linkage map are also arbitrary. The population is initially in linkage equilibrium, and there is no position effect. If the biparental regression is linear, then the regression coefficient is simply Vgam/V, and hence the single-generation change in the mean is deltaZ = (Vgam/V)S, where Vgam and V denote the gametic and total variances in the character and S designates the selection differential. The corresponding exact result is DeltaZ = (C/W) + (B/W2), where C, W and B represent the gametic covariance of the character and fitness, the mean fitness, and a correction term, respectively. If selection is weak, then DeltaZ approximately C/W. Furthermore, deltaZ = C/W if either there is no environmental contribution and the gametic effects are additive or the character is fitness itself. In the latter case, C is the gametic variance in fitness. Thus, even in linkage equilibrium, weakness of selection generally does not suffice to validate the linear-regression result. This conclusion holds even for additive loci.

Genetic variation in Ecballium elaterium (Cucurbitaceae): Breeding system and geographic distribution

AbstractDioecy, the separation of sexes, has arisen independently many times in the course of angiosperm evolution. Avoidance of inbreeding is clearly involved in the evolution of dioecy, and as a consequence we predict that dioecious populations should maintain higher levels of genetic variation than closely related nondioecious populations. We tested that prediction by comparing allozymic variation in two closely related taxa, the monoecious and dioecious subspecies of the Mediterranean cucurbit, Ecballium elaterium. Thirteen polymorphic loci were screened for seeds sampled from 10 monoecious and 13 dioecious populations spanning the geographic ranges of the subspecies in Spain. The dioecious subspecies showed strikingly greater allelic diversity and heterozygosity than the monoecious subspecies. A hierarchical F‐statistic analysis clearly demonstrated considerable genetic variation within populations for the dioecious populations, whereas for the monoecious populations almost all genetic variation resulted from differences among populations. The general pattern of homozygosity within monoecious populations suggests that they are highly inbred. In order to assess historical influences on current patterns of genetic variation, we conducted a genetic‐distance analysis. The observed relationship between genetic distance and geographic distance between populations supports the hypothesis that the subspecies' current allopatric distributions on the Iberian Peninsula are the result of separate waves of colonization from the north (monoecious) and south (dioecious).

Dynamics of cytonuclear disequilibria in finite populations and comparison with a two-locus nuclear system

We study the behavior of cytonuclear disequilibria in a finite monoecious population due to (1) random drift alone, (2) random drift and mutation, and (3) random drift and migration, using exact results on the RUZ (Random Union of Zygotes) model and diffusion approximations. We also show that the RUG (Random Union of Gametes) model is not suitable for a cytonuclear system. The study is also accompanied by a comparison with a two-locus nuclear system. We show that in a finite population of size N without mutation, the rate of decrease of the cytonuclear allelic disequilibrium is the same as that in the corresponding unlinked two-locus nuclear system. The principal rate of decrease of variance in allelic disequilibrium in a cytonuclear system is slightly faster than that in the corresponding nuclear system. However, the expected value of the variance in cytonuclear disequilibria is larger than that in a two-locus nuclear system for at least the first N generations. With mutation, the expected value of steady state variances of both systems are about the same; however, the normalized variance in linkage disequilibrium σ d 2 of the cytonuclear system is about twice as large as that for the corresponding nuclear system. For the migration process, two sets of steady state solutions are provided, one for the variables before migration and the other for the variables after migration. Diffusion approximations for both the principal rate of decay and steady state solutions in both systems are found to be satisfactory. A more accurate backward diffusion equation for a two-locus nuclear system is provided when the recombination fraction R is large.

Error bounds for the fundamental and secondary theorems of natural selection.

Error bounds are derived for the approximation delta Z approximately C/W, where delta Z, C, and W denote the rate of change of the mean of a character under selection, the genic (or additive genetic) covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character), and the mean fitness, respectively. Generations are discrete and nonoverlapping; the monoecious population mates at random. The character is determined by arbitrarily many multiallelic loci without epistasis; the linkage map is also arbitrary. The genotypic values of the character are constant. Rounds on the absolute error in the above approximation are deduced for an arbitrary character, and these are converted to bounds on the relative error when the character is fitness itself. In that case, C is the genic variance in fitness, and the relative error cannot exceed one half the greatest genotypic selection coefficient.

Gene conversion, linkage, and the evolution of repeated genes dispersed among multiple chromosomes.

The evolution of the probabilities of genetic identity within and between the loci of a multigene family dispersed among multiple chromosomes is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, but the same for every chromosome; the dependence of the probabilities of identity on the location on each chromosome is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. Under the assumption of loose linkage (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), explicit approximations are obtained for the equilibrium values of the probabilities of identity and of the linkage of disequilibria. The probabilities of identity are of order one [i.e., O(1)] and do not depend on location; the linkage disequilibria are of O(epsilon) and, within each chromosome, depend on location through the crossover rates. It is demonstrated also that the ultimate rate and pattern of convergence to equilibrium are close to that of a much simpler, location-independent model. If intrachromosomal conversion is absent, the above results hold even without the assumption of loose linkage. In all cases, the relative errors are of O(epsilon). Even if the conversion rate between genes on nonhomologous chromosomes is considerably less than between genes on the same chromosome or homologous chromosomes, the probabilities of identity between the former genes are still almost as high as those between the latter, and the rate of convergence is still not much less than with equal conversion rates. If the crossover rates are much less than 1/2, then most of the linkage disequilibrium is due to intrachromosomal conversion. If linkage is loose, the reduction of the linkage disequilibria to O(epsilon) requires only O(-ln epsilon) generations.

The variance of sample heterozygosity

The variance of sample heterozygosity, averaged over several loci, is studied in a variety of situations. The variance depends on the sampling implicit in the mating system as well as on that explicit in the loci scored and individuals sampled. There are also effects of allelic distributions over loci and of linkage or linkage disequilibrium between pairs of loci. Results are obtained for populations in drift and mutation balance, for infinite populations undergoing mixed self and random mating, and for finite monoecious populations with or without selfing. For unlinked loci in drift/mutation balance, variances appear to be lessened more by increasing the number of loci scored than by increasing the number of individuals sampled. For infinite populations under the mixed self and random mating system, however, the reverse is true. Methods for estimating the variance of sample heterozygosity are discussed, with attention being paid to unbalanced data where not all loci are scored in all individuals.

Rate of evolution of a character without epistasis

IT IS SHOWN THAT THE RATE OF CHANGE OF THE MEAN OF A CHARACTER UNDER SELECTION IS [FORMULA: see text] where C, W, DeltaZ, and B denote the genic (additive genetic) covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character), the mean fitness, the average of the changes in the genotypic values of the character over the next generation, and a correction term, respectively. Generations are discrete and nonoverlapping; the monoecious population mates at random. The character is determined by arbitrarily many multiallelic loci without epistasis; the linkage map is also arbitrary. If there is no dominance, then B = 0; if, in addition, the genotypic values of the character are constant, then Delta Z = C/ W. Even if dominance is present, for weak selection and slowly varying genotypic values, Delta Z approximately C/ W. If the character is fitness itself, then C is the genic variance in fitness.

Gene conversion, linkage, and the evolution of multigene families.

The evolution of the probabilities of genetic identity within and between the loci of a multigene family is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, and the location dependence of the probabilities of identity is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. For interchromosomal conversion, the equilibrium probabilities of identity are within order epsilon [i.e., O(epsilon)] of those in a simple model that has no location dependence and, at equilibrium, no linkage disequilibrium. At equilibrium, the linkage disequilibria are of O(epsilon); they are evaluated explicitly with an error of O(epsilon 2); they may be negative if symmetric heteroduplexes occur. The ultimate rate and pattern of convergence to equilibrium are within O(epsilon 2) and O(epsilon), respectively, of that of the same simple model. If linkage is loose (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), the linkage disequilibria are reduced to O(epsilon) in a time of O(-ln epsilon). If intrachromosomal conversion is incorporated, the same results hold for loose linkage, except that, if the crossover rates are much less than 1/2, then the linkage disequilibria generally exceed those for pure interchromosomal conversion.

INBREEDING AND VARIANCE EFFECTIVE POPULATION NUMBERS.

In this paper, a correction and extension of earlier work, we derive expressions for the inbreeding effective number, NeI , and the variance effective number, NeV , with various models. Diploidy, random mating, and discrete generations are assumed and formulas for NeI are given for six situations: isogamous monoecious populations with self-fertilization permitted or excluded; monoecious populations, male and female gametes distinguished, with self-fertilization permitted or excluded; and separate sexes with or without male and female progeny distinguished. NeV is given for monoecious and separate-sexed populations. Most higher animals have separate sexes, and male and female progeny are distinguished. Letting the subscript t represent the generation in which the inbreeding effect is manifest, the inbreeding effective number is [Formula: see text] in which σsm,sf is the covariance of the number of male (m) and female (f) progeny of a parent of sex s (s = m or f), μsm and μsf are the mean numbers of sons and daughters of a parent of sex s, and Ns,t-2 is the number in the grandparental generation. For the variance effective number [Formula: see text] in which [Formula: see text] and σsm2 and σsf2 are the variances in the number of sons and daughters of a parent of sex s. Observations are made at the same age in each generation.

Permanency of response to selection for quantitative characters in finite populations.

To study the permanency of response to selection for a quantitative character in finite populations and the nature of the genetic effects that contribute to this response, we have used the covariance between ancestors and descendents within populations. Effects and variances are defined for an initial equilibrium random mating monoecious population that gives rise to replicate finite populations. After a prescribed history of restricted population size, the replicate populations are expanded, and the covariance between ancestors and descendents is quantified in terms of descent measures and genetic components in the initial population as a means of determining the additive variance within populations. Several dominance components including joint dominance effects of loci contribute to the additive variance, some of which can be negative. There is always a positive contribution of additive by additive variance to the additive variance within populations, which can be large. With the new definitions of components of genetic variance within populations, selection response is formulated in the same manner as for the initial random mating population, but the components have been modified considerably by the restricted population size.

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.

On the theory of partially inbreeding finite populations. I. Partial selfing.

Some stochastic theory is developed for monoecious populations of size N in which there are probabilities beta and 1 - beta of reproduction by selfing and by random mating. It is assumed that beta much greater than N-1. Expressions are derived for the inbreeding coefficient of one random individual and the coefficient of kinship of two random separate individuals at time t. The mean and between-lines variance of the fraction of copies of a locus that are identical in two random separate individuals in an equilibrium population are obtained under the assumption that there is an infinite number of possible alleles. It is found that the theory for random mating populations holds if the effective population number is Ne = N'/(1 + FIS), where FIS is the inbreeding coefficient at equilibrium when N is infinite and N' is the reciprocal of the probability that two gametes contributing to random separate adults come from the same parent. When there is a binomial distribution of successful gametes emanating from each adult, N' = N. An approximation to the probability that an allele A survives if it is originally present in one AA heterozygote is found to be 2(N'/N)(FISS1 + (1 - FIS)S2), where S1 and S2 are the selective advantages of AA and AA in comparison with AA. In the last section it is shown that if there is partial full sib mating and binomial offspring distributions Ne = N/(1 + 3FIS).

Evolution under Fertility and Viability Selection

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.

Selection caused by self-fertilization I. Four measures of self-fertilization and their effects on fitness

The standard models of selfing in seed plants consider only the ovules, which are assumed to have a constant selfing rate. It has recently become clear, however, that hermaphrodite or monoecious populations frequently show sexual asymmetry (nonconstant pollen:ovule fertilities among individuals). Such asymmetry usually results in pollen selfing rates which differ from those for the ovules and are frequency-dependent even for constant ovule selfing rates. A recent study of selfing rates for all gametes of an individual is extended here to include four selfing rates (for ovules, pollen, all gametes, and zygotes), and simple mathematical relationships linking the four rates are obtained. Unlike earlier models of selfing, it is not assumed that the ovule selfing rate is constant, but instead that this rate, like all the others, is determined by the mobility of the pollen, which in turn is determined by the floral biology and ecology. It is found that all four selfing rates are usually frequency-dependent. The selfing rate for all gametes (the combined selfing rate) is usually intermediate between those for the ovules and pollen, and the zygotic rate is usually the smallest of the four. The exceptions to the above statements occur for relatively extreme situations, such as complete selfing for pollen or ovules, no selfing, or sexual symmetry. Three modes of selfing are considered: prior (PS), competing (CS), and delayed (DS) self-fertilization. It is shown that if there are at least two types with different selfing rates in the population, then the ranking of their selfing rates may depend upon the frequencies of the types (for the combined and the zygotic rates), may be frequency-independent (ovule rate), or may be dependent or independent, according to the mode of selfing (pollen rate). The effects of the various influences on the amount of selfing are by no means negligible. Thus a numerical study shows pollen selfing rates for one type which vary from 0.09 to 0.96, according to its frequency. Another numerical result shows a change in combined selfing rate from 0.13 to 0.86, depending solely on the mode of selfing. Results for Scots Pine show that an ovule selfing rate of 0.5 was accompanied by a combined rate of 0.143. The population selfing rate is not the same as the mean of individual selfing rates, and can only be obtained if female fitnesses as well as ovule selfing rates are known for each type. Previous models of selfing have failed to distinguish between the effects of increased selfing and increased pollen fertility, with the result that increased selfing always resulted in greater fitness. In the present models the two effects are distinguishable, and it is found that increased selfing may result in increased or decreased fitness, depending also on population density and on a form of pollen density. Thus the old dogma that in the absence of viability and fertility selection increased selfing always results in increased fitness is finally refuted, and the importance of the influence of ecological parameters on selfing and fitness is emphasized, since population density and pollen density influence the selfing rates.

Life history and taxonomy of two populations of ligulate Desmarestia (Phaeophyceae) from Chile

Two populations of broad-bladed (ligulate) Desmarestia from central and southern Chile were studied in laboratory cultures. They differ in sporophyte morphology and gametophyte characters (monoecism versus dioecism). The dioecious population is found to be related to D. firma from South Africa and D. munda from the northeast Pacific, while the taxonomic status of the broad-bladed monoecious population is left undetermined. In both populations sporophytes originate after fertilization as well as by apomixis. Chromosome numbers alternate between ca. 23 and ca. 46. Evidence is presented that spermatozoids of the dioecious Desmarestia can develop into sporophytes (androgenesis or ephebogenesis).

Intrachromosomal gene conversion, linkage, and the evolution of multigene families

The evolution of the probabilities of genetic identity within and between tandemly repeated loci of a multigene family is investigated analytically and numerically. Unbiased intrachromosomal gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Under the restriction that there is at most one crossover in the multigene family per individual per generation, the dependence on location of the probabilities of identity is treated exactly. In the “homogeneous” approximation to this “exact” model, end effects are disregarded; in the “exchangeable” approximation, to which all previous work was confined, all position dependence is neglected. Numerical results indicate that (i) the exchangeable and homogeneous models are both qualitatively correct, (ii) the exchangeable model is sometimes too inaccurate for quantitative conclusions, and (iii) the homogeneous model is always more accurate than the exchangeable one and is always sufficiently accurate for quantitative conclusions.