The Distribution of Gene Frequencies in Populations

Selection intensity against deleterious mutations in RNA secondary structures and rate of compensatory nucleotide substitutions

A two-locus model of reversible mutations with compensatory fitness interactions is presented; single mutations are assumed to be deleterious but neutral in appropriate combinations. The expectation of the time of compensatory nucleotide substitutions is calculated analytically for the case of tight linkage between sites. It is shown that selection increases the substitution time dramatically when selection intensity Ns > 1, where N is the diploid population size and s the selection coefficient. Computer simulations demonstrate that recombination increases the substitution time, but the effect of recombination is small when selection is weak. The amount of linkage disequilibrium generated in the process of compensatory substitution is also investigated. It is shown that significant linkage disequilibrium is expected to be rare in natural populations. The model is applied to the mRNA secondary structure of the bicoid 3' untranslated region of Drosophila. It is concluded that average selection intensity Ns against single deleterious mutations is not likely to be much larger than 1.

Compensating for our load of mutations: freezing the meltdown of small populations

We have investigated the reduction of fitness caused by the fixation of new deleterious mutations in small populations within the framework of Fisher's geometrical model of adaptation. In Fisher's model, a population evolves in an n-dimensional character space with an adaptive optimum at the origin. The model allows us to investigate compensatory mutations, which restore fitness losses incurred by other mutations, in a context-dependent manner. We have conducted a moment analysis of the model, supplemented by the numerical results of computer simulations. The mean reduction of fitness (i.e., expected load) scaled to one is approximately n/(n+2Ne), where Ne is the effective population size. The reciprocal relationship between the load and Ne implies that the fixation of deleterious mutations is unlikely to cause extinction when there is a broad scope for compensatory mutations, except in very small populations. Furthermore, the dependence of load on n implies that pleiotropy plays a large role in determining the extinction risk of small populations. Differences and similarities between our results and those of a previous study on the effects of Ne and n are explored. That the predictions of this model are qualitatively different from studies ignoring compensatory mutations implies that we must be cautious in predicting the evolutionary fate of small populations and that additional data on the nature of mutations is of critical importance.

Heterosis increases the effective migration rate

Individuals coming from the same subpopulation are more likely to share deleterious mutations at any given locus than hybrids formed between parents from different populations. Offspring of migrants therefore may experience heterosis and have higher fitness than resident individuals. This will, in turn, result in the immigrant alleles being present in higher frequencies than predicted from neutral expectations and thus a higher effective migration rate. In this paper we derive a formula to calculate the effective migration rate in the presence of heterosis. It is shown that the effect of heterosis on the migration rate can be substantial when fitness reduction within local populations is severe. The effect will be more pronounced in species with relatively short map lengths. Furthermore the heterosis effect will be highly variable throughout the genome, with the largest effect seen near selected genes and in regions of high gene density.

Species delimitation in systematics: inferring diagnostic differences between species

Species are fundamental units in studies of systematics, biodiversity and ecology, but their delimitation has been relatively neglected methodologically. Species are typically circumscribed based on the presence of fixed (intraspecifically invariant or non-overlapping) diagnostic morphological characters which distinguish them from other species. In this paper, we argue that determining whether diagnostic characters are truly fixed with certainty is generally impossible with finite sample sizes and we show that sample sizes of hundreds or thousands of individuals may be necessary to have a reasonable probability of detecting polymorphisms in diagnostic characters at frequencies approaching zero. Instead, we suggest that using a non-zero frequency cut-off may be a more realistic and practical criterion for character-based species delimitation (for example, allowing polymorphisms in the diagnostic characters at frequencies of 5% or less). Given this argument, we then present a simple statistical method to evaluate whether at least one of a set of apparently diagnostic characters is below the frequency cut-off. This method allows testing of the strength of the evidence for species distinctness and is readily applicable to empirical studies.

History of plant population genetics

This review of plant population genetics focuses on the genetic foundations of the processes that have led to documentable improvements in cultivated plants since the earliest domestications took place perhaps 13,000 years ago. Nearly all human civilizations have depended heavily on inbreeding plants (particularly wheat, barley, soybeans and other inbreeding legumes), as well as outbreeding vegetatively propagated species (white potatoes, yams) as their dietary standbys. The principal exception is maize (corn), an annual seed-produced outbreeder in nature. It is noteworthy that maize joined wheat, rice, and barley as a truly major crop worldwide only after its conversion to self-pollination combined with hybridization between favorably interacting inbred lines increased yield of maize several-fold in the twentieth century.

Sewall Wright's equation Deltaq=(q(1-q) partial differentialw/ partial differentialq)/2w

An equation of Sewall Wright's expresses the change in the frequency of an allele under selection at a multiallelic locus as a function of the gradient of the mean fitness "surface" in the direction in which the relative proportions of the other alleles do not change. An attempt to derive this equation using conventional vector calculus shows that this description leads to a different equation and that the purported gradient in Wright's equation is not a gradient of the mean fitness surface except in the diallelic case, where the two equations are the same. It is further shown that if Fisher's angular transformation is applied to the diallelic case the genic variance is exactly equal to one-eighth of the square of the gradient of the mean fitness with respect to the transformed gene frequency.

Mutation-selection balance at a modifier-of-imprinting locus

We propose a pair of population genetic models for a modifier-of-imprinting locus for which different genotypes imprint different proportions of an imprintable target locus in their gametes. The two models examine the situations in which imprinting is advantageous, and we discuss three cases for which the modifier is respectively partially dominant, dominant, or recessive. The models predict the stable equilibrium frequencies of the mutant modifier and functionally diploid individuals in a large population in terms of up to four parameters: the mutation rate at the modifier locus, nu; the selection coefficient against the disadvantageous phenotype, sigma; the proportion of unimprinted eggs produced by homozygotes for the mutant modifier, theta, and, in the partially dominant models, the dominance parameter, kappa. The equilibrium frequency of the mutant phenotypes is shown to be approximately twice that of standard Mendelian models: 2 nu/sigma or 4nu/sigma when the modifier is recessive or dominant, respectively. Mathematical equivalences between these and nonimprinting models are noted.

The Fundamental Theorem of Natural Selection in Ewens' sense (case of fertility selection)

We show that the Fundamental Theorem of Natural Selection in Ewens' sense is valid in the case of fertility selection: the additive genetic variance in fertility divided by the mean fertility is exactly equal to the partial change in the mean fertility from the current generation to the next. This partial change is the increase in the mean additive value caused by frequency changes from on generation to the next. This partial change is the increase in the mean additive value caused by frequency changes from one generation to the next but keeping unchanged the additive values. The only hypothesis on mating is that it does not affect the allelic frequencies in the sense that these are the same before and after mating in the parental generation, which occurs for a wide range of mating patterns going from random mating to several regular systems of inbreeding and cases of assortative mating. The fertility of couples is determined by the genes at an arbitrary number of loci, and the additive (average) allelic allelic effects are defined by a linear system of equations, which is used to extend Ewens' optimality principle to the case of fertility selection.

Failure of imprinting at Igf-2: two models of mutation-selection balance

The failure of maternal imprinting at the insulin-like growth factor II (Igf-2) locus predisposes individuals to several clinical conditions, including Wilms tumor. Having two functional Igf-2 genes, therefore, is selectively disadvantageous, and the condition is probably maintained in human populations by recurrent mutation. We propose two models that predict the expected frequency of functionally diploid individuals in a large population, in terms of a mutation rate, mu, and the selection coefficient against functionally diploid individuals, s. In the first model a mutant Igf-2 allele that cannot be imprinted arises from the standard, imprintable allele at a rate mu. Our second model hypothesizes a second modifier locus at which a recessive allele arises at rate mu. Mothers who are homozygous for this recessive modifier allele fail to imprint their eggs. Both models predict the expected frequency of affecteds to be 2 mu/s(1 + mu), approximately twice that predicted by the standard one-locus model of a recessive allele in mutation-selection balance. This frequency suggests that < or = 25% of the cases of Wilms tumor are due to the failure to imprint the maternal Igf-2 gene.

Evolution of alphaviruses in the eastern equine encephalomyelitis complex

Evolution of viruses in the eastern equine encephalomyelitis (EEE) complex was studied by analyzing RNA sequences and oligonucleotide fingerprints from isolates representing the North and South American antigenic varieties. By using homologous sequences of Venezuelan equine encephalomyelitis virus as an outgroup, phylogenetic trees revealed three main EEE virus monophyletic groups. A North American variety group included all isolates from North America and the Caribbean. One South American variety group included isolates from the Amazon basin in Brazil and Peru, while the other included strains from Argentina, Guyana, Ecuador, Panama, Trinidad, and Venezuela. No evidence of heterologous recombination was obtained when three separate regions of the EEE virus genome were analyzed independently. Estimates of the overall rate of EEE virus evolution (nucleotide substitution) were 1.6 x 10(-4) substitution per nucleotide per year for the North American group and 4.3 x 10(-4) for the Argentina-Panama South American group. Evolutionary rate estimates for the North American group increased over 10-fold (from about 2 x 10(-5) to 4 x 10(-4)) concurrent with divergence of two monophyletic groups during the early 1970s. The North and South American antigenic varieties diverged roughly 1,000 years ago, while the two main South American groups diverged about 450 years ago. Analysis of multiple strains isolated from an upstate New York transmission focus during the same years suggested that, in certain locations, EEE virus may be relatively isolated for short time periods.

Evolutionary consequences of DNA mismatch inhibited repair opportunity

Double strand breaks (DSBs) are often repaired via homologous recombination. Recombinational repair processes are expected to be influenced by nucleotide heterozygosity through mismatch detection systems. Unrepaired DSBs have severe biological consequences and are often lethal. We show that natural selection due to inhibition of recombinational repair associated with polymorphisms could influence their molecular evolution. The main conclusions from this analysis are that, for increasing population size, mismatch detection leads to a limit on average heterozygosity of otherwise selectively neutral polymorphism, an excess of rare variants, and a slowing down of the rate of neutral molecular evolution. The first two results suggest that mismatch detection may account for the surprisingly narrow range of observed average heterozygosities, given the great variation in population size between species.

Recent development of the neutral theory viewed from the Wrightian tradition of theoretical population genetics

In contrast to the Darwinian theory of evolution by natural selection, the neutral theory emphasizes the great importance of random genetic drift (due to finite population size) and mutation pressure as the main causes of molecular evolution. In this paper, after a brief review of the neutral theory, recent data strongly supporting the neutral theory are presented. Also discussed are such topics as compensatory neutral evolution and an approach to a unified understanding of molecular and phenotypic evolution. It is concluded that random genetic drift acting on selectively neutral mutants must have played some very important role in organic evolution, including the origin of life and macroevolution.

Genetic differentiation of quantitative characters between populations or species. II: Optimal selection in infinite populations

Using a new and more general genetic model called the discrete-allelic state model and assuming discrete-time process, the evolutionary changes of genetic variation of quantitative characters, controlled by a few loci, within and between populations during the process of genetic differentiation of populations or species, are studied under the effects of mutation and centripetal selection in infinitely large populations. While in a finite population and ignoring selection, the rate of change of additive genetic variance depends on mutation and effective population size, traits under optimal selection in infinitely large populations go through the dynamics of a rather complicated form depending on the relative intensities of selection and mutation. When a population, which has reached steady-state by mutation-selection balance, splits into two, in one of which the same optimum genotype holds but in the other the optimum shifts a few standard deviations away from the original optimum, the corresponding daughter population starts differentiating from its sister population by favouring certain class of mutant alleles and discarding others which were originally favoured. During this process of turn over of genes, both the intra- and inter-populational variances undergo a complicated change, and the ratio of the former to the latter is a non-linear function of time of divergence. This pattern is qualitatively very different from the case when selection is absent. The intra-population distribution of genotypic values, during this transition, is shown to deviate considerably from normality. The presence of linkage seems to retard the accumulation of intra-populational genetic variance. The implications of these results are discussed in comparison with the earlier findings of evolutionary models of quantitative traits.

The effects of linkage and density-dependent regulation on gene flow

The rate of gene flow across a hybrid zone may be reduced by the presence of a physical barrier, by a reduction of population density caused by reduced fitness of hybrids (the "hybrid sink" effect), and by linkage. If the reduction in hybrid fitness is not extreme, the strength of the barrier to gene flow caused by these effects is B = w(rho *+/rho *0)2(W*+/W*0)(2/R + 1/r). Here, w is the width of the cline; rho * is the carrying capacity; W* is the mean fitness of the population, excluding effects of density; R is the strength of density-dependent regulation; and r is the harmonic mean recombination rate between the locus whose flow is being calculated, and loci under selection. +, 0 denote populations outside the hybrid zone, and at its centre, respectively. This relation is illustrated using data from hybrid zones in Bombina and Podisma, and its implications for interpretation of data from nature are discussed.

Marital mores as a mechanism for the maintenance of ethnic variations of lethal gene frequencies

A computer simulation model was developed to study the effects of various marital mores on the incidence of lethal autosomal recessive genes in populations that are subdivided into small isolates. The problem was studied in isolates where initial generation size was 30, 40, and 50 individuals. In each of these, the mean fertility rate was varied from 2.3 to 2.7 surviving (to adulthood) children per couple whose marriage had been contracted in accordance with the prevailing convention: marriage between first cousins and siblings prohibited; marriage between siblings prohibited; marriage allowed between any individuals; marriage prohibited between siblings but encouraged between cousins; and marriage encouraged between siblings. In all cases, the mean gene frequency in generation 20 was lower than that predicted by the deterministic model with random mating in an unsubdivided population of infinite size, due to gene loss through random drift (to zero) in many of the isolates. The mores that encouraged consanguineous marriages had the lowest final lethal-gene frequencies. Random mating produced intermediate values, and the restrictive mores, the highest final frequencies. The deterministic model (assuming infinite population size and random mating) predictions of the final gene frequency were exceeded only if there was reproductive compensation. It is concluded that restrictive marital mores significantly reduce the selective pressures on lethal recessive genes in small isolates, but that this is counteracted by the increased rate of gene loss through random drift.

Surnames in Sardinia. III. The spatial distribution of surnames for testing neutrality of genes

1. A study on heterogeneity of males' surnames over time and space in the island of Sardinia was carried out using data from consanguineous marriages (1800-1970) and telephone directories (1978). 2. Variation of frequency of surnames over time is barely significant and 10 times lower than that over space, which is very highly significant. 3. For sufficiently frequent surnames the estimate of the Wahlund variance, calculated from chi 2 for heterogeneity in space, is independent of the frequency of a surname: this supplies evidence of neutrality in line with that obtained from the frequency distribution of surnames. 4. The Wahlund variance, W (also called FST), decreases regularly as the average size of the area considered (number of individuals per area, N) increases. The estimate of the parameter beta in the relation W = KN beta could be of interest for the study of population structure. 5. A correction factor of 1/4 must be made on the surnames' variance in consideration of their haploid unisexual transmission. 6. It is suggested that surnames could provide a baseline for estimating the value of Wahlund variance under random genetic drift and hence evaluating whether a gene behaves as selectively neutral. 7. The distribution of the Wahlund variances obtained from two sets of gene frequency data as compared with that obtained in comparable conditions for surnames in the same areas seems to show that most genes behave as neutral, with the exception of a few, with high W values, which presumably have been under different selection pressures in the area examined.

On the perpetuation of relic genes having an inviable homozygote

Reproductive compensation may be prezygotic or postzygotic. Prezygotic compensation is the replacement of inviable infants by one or more additional births. When the marital fertility rates are greater than necessary to replace the parents, however, many traditional societies ensured that any surviving progeny in excess of two (on average) did not marry, or married only if they replaced an older sib as heir to the family estate. We call this postzygotic compensation. We show that the coefficient of biologically effective compensation approximates unity when normal and genetic-disease-affected sibships were both sufficiently large to ensure that, on average, two or more sibs survived to the end of the reproductive period, irrespective of the level of prezygotic compensation. The impact of these types of compensation on the population genetics of Tay-Sachs disease and cystic fibrosis is discussed.

Demographic components of gene frequency change in free-ranging macaques on Cayo Santiago

Gene frequency profiles from January 1973 to January 1977 for three polymorphic loci were examined in Cayo Santiago rhesus social groups. The effects of demographic components (i.e., births, deaths, immigrations, emigrations, and group fission and fusion) on total change in gene frequencies are assessed. Allelic frequencies at the carbonic anhydrase II, 6-phosphogluconate dehydrogenase, and transferrin loci were analyzed in four social groups. In the two groups that underwent fission and fusion during the study period, the timing of these processes was related to the largest short-term changes in gene frequences. However, immigration and emigration had the greatest effect on total change in gene frequency in all groups during the study period. The relative importance of births and deaths in producing gene frequency change varied among the social groups. These results suggest that the relative importance of the demographic components of gene frequency change in primate populations is determined by behavioral patterns and ecological conditions specific to the population considered.

Evolution of a finite population under gene conversion

Evolution at a multiallelic locus under the joint action of gene conversion, mutation, selection, and random genetic drift is studied. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Under the assumption that all four evolutionary forces are weak, a diffusion approximation is established for the dynamics of the gene frequencies. For two alleles, the inclusion of gene conversion merely alters one of the two selection parameters of the thoroughly investigated diffusion process without conversion. Therefore, all results for this classical process, some of which are reviewed and extended here, are immediately applicable to the biologically more general problem. Small conversional disparities can dramatically affect the fixation probability (and hence the rate of gene substitution) and can greatly reduce the mean conditional fixation time of a new mutant. The mean absorption and fixation times are often sufficiently short to imply that biased gene conversion can be an important mechanism for the loss of genetic variability in and the genetic divergence of isolated populations.

Theories of social selection in human populations

Using Huntington disease, mental retardation, and schizophrenia, it has been shown that two individuals with identical genotypes or phenotypes have different fitnesses because of affected nuclear family members. Such fitness interaction seems to occur because of cultural and social reactions due to the presence of affected individuals, and the interaction has been termed "social selection." Without assuming any specific genetic control for the social behavior, we can study the effect of social behavior on the incidence of a genetic disease.

Social selection in human populations. I. Modification of the fitness of offspring by an affected parent

The concept of social selection for deleterious genes has been introduced by considering two alleles at one locus. A social selection model is constructed by assuming that the fitness of an individual is determined by his or her own as well as the parental phenotypes. It is shown that the equilibrium gene frequency depends on the loss of fitness of an individual due to the trait (gamma), due to affected parents (beta), and the probability that the heterozygote develops the trait (h). With mutational changes from the wild-type allele to the deleterious gene at a rate of alpha per generation, the equilibrium frequency of deleterious genes is approximately alpha/hs for 0 less than h less than or equal to 1 and square root alpha/s for h = 0, where s = gamma + beta(1 -- gamma)/2. Implications of the social selection model have been discussed for several diseases in man.

The effect of social selection on population dynamics of rare deleterious genes

Genetically abnormal or phenotypically affected individuals may influence the reproductive behaviour of other members in the family, regardless of their phenotype. This social selection is modelled by considering that whenever the normal individual has at least one affected member in the nuclear family the person has a reduced chance in mating success or in fertility compared to those from the normal nuclear family. The usual individual selection has also been incorporated using a single locus with two alleles. The gene frequency change is divided into two parts: the change due to individual selection and that due to social selection. It has been shown that a low mutation rate, a small population size and strong selection at both stages lower the equilibrium frequency of deleterious genes. The possibility of extending the single locus model for common diseases is also discussed.

Ethnic variation of genetic disease: roles of drift for recessive lethal genes

Using Wright's distribution of gene frequencies for a recessive lethal gene, a method is given to analyze the probability that any particular gene frequency is greater than a given threshold gene frequency. The method is introduced to analyze the plausibility of drift for explaining observed data.

Elevated frequency of Tay-Sachs disease among Ashkenazic Jews unlikely by genetic drift alone

Using the steady-state distribution of recessive lethal gene the probability of finding the elevated frequency of Tay-Sachs (TSD) gene among Ashkenazic Jews is computed. For various estimated values of mutation rate and population size, this probability is found to be statistically significant. This probabiltiy, in fact, becomes even smaller if a steady influx of foreign genes into the Ashkenazic Jewish populations is considered. It is suggested that heterozygote advantage together with random genetic drift should be considered as the most probable mechansim for the elevation of TSD gene frequency among the Ashkenazic Jews.

The genetic variability of third chromosomes in a local population of Drosophila melanogaster

Five hundred and two third chromosomes were extracted from a large cage population of Drosophila melanogaster initiated two months after collection of the progenitors near Raleigh, North Carolina in 1970.---Salivary gland chromosomes of 489 chromosome lines were examined and 54 chromosomes were found to carry inversions. The inversions were classified into three polymorphic types [In (3L)P, In (3R)P, and In (3R)C] and two unique types. The polymorphic inversions were found in frequencies of 0.012, 0.88, and 0.010, respectively.--Viabilities of homozygotes and heterozygotes were examined. Chromosomes with lethals occurred with a frequency of 0.495: 0.537 in the group of inversion-carrying chromosomes and 0.490 in the group of inversion-free chromosomes. The average homozygote viability computed on the basis of an average heterozygote viability of 1.0000 was 0.3235 if lethal lines were included and 0.6290 if they were excluded. The detrimental load to lethal load ratio (D:L ratio) was 0.70 (=0.4636-0.6650). The average viability of lethal heterozygotes was significantly larger than that of lethal-free heterozygotes. It appears, however, that lethal genes in heterozygotes have deleterious effects on fitness as a whole.--The average degree of dominance for viability polygenes was estimated to be about 0.3-0.4 in lethal-free individuals and nearly zero in lethal heterozygotes. Overdominance or some form of balancing selection was suggested at some loci. The difference between the values obtained for average degree of dominance due to genetic backgrounds and superior vibaility of lethal heterozygotes (but not fitness as a whole) suggest that some epistasis or coadaptation occurs.--The results described above are similar to those obtained for the second chromosomes.

The genetic variability of polygenic characters under optimizing selection, mutation and drift

The effect of optimizing selection, mutation and drift on a metric character determined by a large number of loci with equal effects without dominance was investigated theoretically. Conditions for a stable equilibrium under selection and mutation, in the absence of drift, have been obtained, and hence the amount of genetic variability which can be maintained by mutation has been determined. An approximate expression for the average amount of genetic variability to be expected in the presence of drift in a population of finite size has also been obtained and evaluated.

The effect of genetic linkage on the mean fitness of a population

If many gene loci are kept in a segregating state by natural selection, the equilibrium frequencies of the genotypes in the population may be a function of the amount of recombination among the genes. It is shown that if the equilibrium vector of gametic frequencies is a continuous function of the set of recombination frequencies among genes, then the mean fitness of the population at equilibrium is a maximum in the absence of recombination. Thus, in general, restriction of recombination increases fitness.