Homozygosity in a population of variable size and mutation rate

A formula is obtained for the probability that two genes at a single locus, sampled at random from a population at time t, are of particular types. The model assumed is a diffusion approximation to a neutral Wright-Fisher model in which mutation is not necessarily symmetric and the population size is a function of time. It is shown that for symmetric mutation in a population undergoing a step-function type bottleneck, homozygosity increases with decreasing population size. A formula is given for the distribution of the number of segregating sites occurring in two randomly sampled sequences of completely linked sites, with general mutation at a site and identical mutation structure between sites.We give similar results for a population of fixed size but for which the mutation rate is a function of time, and not necessarily symmetric. We confirm the intuitively clear effect that increasing the mutation rate decreases homozygosity.     

Genetic identity between populations when mutation rates vary within and across loci

A formula is derived for the probability that two genes taken at random from the same locus in two populations isolated at time t ago are of the same allelic type. The model assumed is a neutral one where there are possibly different mutation rates between different alleles. Inequalities are derived for this probability. A particular result is that for a fixed overall mutation rate, the probability is least for the infinite alleles model. Inequalities and approximations are found for Nei's genetic identity at one locus when mutation rates vary, and also for the identity across loci when the overall mutation rates per locus vary. Genetic identity at the molecular level is considered and a probability generating function found for the number of segregating sites between two randomly chosen gametes from two divergent populations, under various models.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

The frequency of shifts between alternative equilibria

We derive a formula giving the frequency with which random drift shifts a population between alternative equilibria. This formula is valid when such shifts are rare (Ns much greater than 1), and applies over a wide range of mutation rates. When the number of mutations entering the population is low (4 N mu much less than 1), the rate of stochastic shifts reduces to the product of the mutation rate and the probability of fixation of a single mutation. However, when many mutations enter the population in each generation (4 N mu much greater than 1), the rate is higher than would be expected if mutations were established independently, and converges to that given by a gaussian approximation. We apply recent results on bistable systems to extend this formula to the general multidimensional case. This gives an explicit expression for the frequency of stochastic shifts, which depends only on the equilibrium probability distribution near the saddle point separating the alternative stable states. The plausibility of theories of speciation through random drift are discussed in the light of these results.     

The effect of recurrent mutation on the frequency spectrum of a segregating site and the age of an allele

The sample frequency spectrum of a segregating site is the probability distribution of a sample of alleles from a genetic locus, conditional on observing the sample to be polymorphic. This distribution is widely used in population genetic inferences, including statistical tests of neutrality in which a skew in the observed frequency spectrum across independent sites is taken as a signature of departure from neutral evolution. Theoretical aspects of the frequency spectrum have been well studied and several interesting results are available, but they are usually under the assumption that a site has undergone at most one mutation event in the history of the sample. Here, we extend previous theoretical results by allowing for at most two mutation events per site, under a general finite allele model in which the mutation rate is independent of current allelic state but the transition matrix is otherwise completely arbitrary. Our results apply to both nested and nonnested mutations. Only the former has been addressed previously, whereas here we show it is the latter that is more likely to be observed except for very small sample sizes. Further, for any mutation transition matrix, we obtain the joint sample frequency spectrum of the two mutant alleles at a triallelic site, and derive a closed-form formula for the expected age of the younger of the two mutations given their frequencies in the population. Several large-scale resequencing projects for various species are presently under way and the resulting data will include some triallelic polymorphisms. The theoretical results described in this paper should prove useful in population genomic analyses of such data.     

Spatial Measures of Genetic Heterogeneity During Carcinogenesis

In this work we explore the temporal dynamics of spatial heterogeneity during the process of tumorigenesis from healthy tissue. We utilize a spatial stochastic model of mutation accumulation and clonal expansion in a structured tissue to describe this process. Under a two-step tumorigenesis model, we first derive estimates of a non-spatial measure of diversity: Simpson’s Index, which is the probability that two individuals sampled at random from the population are identical, in the premalignant population. We next analyze two new measures of spatial population heterogeneity. In particular we study the typical length scale of genetic heterogeneity during the carcinogenesis process and estimate the extent of a surrounding premalignant clone given a clinical observation of a premalignant point biopsy. This evolutionary framework contributes to a growing literature focused on developing a better understanding of the spatial population dynamics of cancer initiation and progression.     

Relationship between DNA Polymorphism and Fixation Time

When there is no recombination among nucleotide sites in DNA sequences, DNA polymorphism and fixation of mutants at nucleotide sites are mutually related. Using the method of gene genealogy, the relationship between the DNA polymorphism and the fixation of mutant nucleotide was quantitatively investigated under the assumption that mutants are selectively neutral, that there is no recombination among nucleotide sites, and that the population is a random mating population with N diploid individuals. The results obtained indicate that the expected number of nucleotide differences between two DNA sequences randomly sampled from the population is 42% less when a mutant at a particular nucleotide site reaches fixation than at a random time, and that heterozygosity is also expected to be less when fixation takes place than at a random time, but the amount of reduction depends on the value of 4Nv in this case, where v is the mutation rate per DNA sequence per generation. The formula for obtaining the expected number of nucleotide differences between the two DNA sequences for a given fixation time is also derived, and indicates that, even when it takes a large number of generations for a mutant to reach fixation, this number is 33% less than at a random time. The computer simulation conducted suggests that the expected number of nucleotide differences between the two DNA sequences at the time when an advantageous mutant becomes fixed is essentially the same as that of neutral mutant if the fixation time is the same. The effect of recombination on the amount of DNA polymorphism was also investigated by using computer simulation.     

Hidden Genetic Variability within Electromorphs in Finite Populations

The amount of hidden genetic variability within electromorphs in finite populations is studied by using the infinite site model and stepwise mutation model simultaneously. A formula is developed for the bivariate probability generating function for the number of codon differences and the number of electromorph state differences between two randomly chosen cistrons. Using this formula, the distribution as well as the mean and variance of the number of codon differences between two identical or nonidentical electromorphs are studied. The distribution of the number of codon differences between two randomly chosen identical electromorphs is similar to the geometric distribution but more leptokurtic. Studies are also made on the number of codon differences between two electromorphs chosen at random one from each of two populations which have been separated for an arbitrary number of generations. It is shown that the amount of hidden genetic variability is very large if the product of effective population size and mutation rate is large.     

The frequency spectrum of a mutation,and its age,in a general diffusion model

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.     

Allelic frequencies given the sample's common ancestral type

A general formula is derived for finding the expected proportion of genes in a random sample of n genes which are of a particular allelic type given the type of the sample's common ancestor. Expressions of these expectations are obtained for the 2-allele, K-allele, infinitely many alleles, and stepwise mutation models. Numerical examples for the stepwise mutation model are also tabulated.     

On the size distribution of private microsatellite alleles

Private microsatellite alleles tend to be found in the tails rather than in the interior of the allele size distribution. To explain this phenomenon, we have investigated the size distribution of private alleles in a coalescent model of two populations, assuming the symmetric stepwise mutation model as the mode of microsatellite mutation. For the case in which four alleles are sampled, two from each population, we condition on the configuration in which three distinct allele sizes are present, one of which is common to both populations, one of which is private to one population, and the third of which is private to the other population. Conditional on this configuration, we calculate the probability that the two private alleles occupy the two tails of the size distribution. This probability, which increases as a function of mutation rate and divergence time between the two populations, is seen to be greater than the value that would be predicted if there was no relationship between privacy and location in the allele size distribution. In accordance with the prediction of the model, we find that in pairs of human populations, the frequency with which private microsatellite alleles occur in the tails of the allele size distribution increases as a function of genetic differentiation between populations.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

The coalescent in an island model of population subdivision with variation among demes

A simple genealogical structure is found for a general finite island model of population subdivision. The model allows for variation in the sizes of demes, in contributions to the migrant pool, and in the fraction of each deme that is replaced by migrants every generation. The ancestry of a sample of non-recombining DNA sequences has a simple structure when the sample size is much smaller than the total number of demes in the population. This allows an expression for the probability distribution of the number of segregating sites in the sample to be derived under the infinite-sites mutation model. It also yields easily computed estimators of the migration parameter for each deme in a multi-deme sample. The genealogical process is such that the lineages ancestral to the sample tend to accumulate in demes with low migration rates and/or which contribute disproportionately to the migrant pool. In addition, common ancestor or coalescent events tend to occur in demes of small size. This provides a framework for understanding the determinants of the effective size of the population, and leads to an expression for the probability that the root of a genealogy occurs in a particular geographic region, or among a particular set of demes.     

On the divergence of genes in multigene families

Statistical properties of the amount of divergence of genes in multigene families are studied. The model considered is an infinite-site neutral model with unbiased intrachromosomal conversion, unbiased interchromosomal conversion, and recombination. By considering the time back to the most recent common ancestor of two genes, both the probability of identity and the moments of S, the number of sites that differ between two sampled genes, are obtained. We find that if recombination rates are large or conversion is always interchromosomal, then the expectation of S is 4N mu n where N is the population size, mu is the rate of mutation per generation per gene and n is the number of genes in the gene family, as the conversion rates approach zero, the moments of divergence do not approach the moments of divergence with conversion rates equal to zero, and it is possible for a decrease in the rate of intrachromosomal conversion to result in a higher probability of identity, but a greater mean divergence of the two genes.     

Distribution of Nucleotide Differences between Two Randomly Chosen Cistrons in a Finite Population

Watterson''s (1975) formula for the steady-state distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population has been extended to transient states. The rate for the mean of this distribution to approach its equilibrium value is 1/2 N and independent of mutation rate, but that for the variance is dependent on mutation rate, where N denotes the effective population size. Numerical computations show that if the heterozygosity (i.e., the probability that two cistrons are different) is low, say of the order of 0.1 or less, the probability that two cistrons differ at two or more nucleotide sites is less than 10 percent of the heterozygosity, whereas this probability may be as high as 50 percent of the heterozygosity if the heterozygosity is 0.5. A simple estimate for the mean number (d) of site differences between cistrons is d = h/(1 - h) where h is the heterozygosity. At equilibrium, the probability that two cistrons differ by more than one site is equal to h², the square of heterozygosity.     

The expected number of heterozygous sites in a subdivided population.

A simple, exact formula is derived for the expected number of heterozygous sites per individual at equilibrium in a subdivided population. The model of infinitely many neutral sites is posited; the linkage map is arbitrary. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). With suitable weighting, the expected number of heterozygous sites is 4Neu, where Ne denotes the migration effective population number and u designates the total mutation rate per gene (or DNA sequence). For diploid migration, this formula is a good approximation if Ne >> 1.     

Investigation of Heteroplasmy in the Human Mitochondrial DNA Control Region: A Synthesis of Observations from More Than 5000 Global Population Samples

Instances of point and length heteroplasmy in the mitochondrial DNA control region were compiled and analyzed from over 5,000 global human population samples. These data represent observations from a large and broad population sample, representing nearly 20 global populations. As expected, length heteroplasmy was frequently observed in the HVI, HVII and HVIII C-stretches. Length heteroplasmy was also observed in the AC dinucleotide repeat region, as well as other locations. Point heteroplasmy was detected in approximately 6% of all samples, and while the vast majority of heteroplasmic samples comprised two molecules differing at a single position, samples exhibiting two and three mixed positions were also observed in this data set. In general, the sites at which heteroplasmy was most commonly observed correlated with reported control region mutational hotspots. However, for some sites, observations of heteroplasmy did not mirror established mutation rate data, suggesting the action of other mechanisms, both selective and neutral. Interestingly, these data indicate that the frequency of heteroplasmy differs between particular populations, perhaps reflecting variable mutation rates among different mtDNA lineages and/or artifacts of particular population groups. The results presented here contribute to our general understanding of mitochondrial DNA control region heteroplasmy and provide additional empirical information on the mechanisms contributing to mtDNA control region mutation and evolution. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Hypermutable Non-Synonymous Sites Are under Stronger Negative Selection

Mutation rate varies greatly between nucleotide sites of the human genome and depends both on the global genomic location and the local sequence context of a site. In particular, CpG context elevates the mutation rate by an order of magnitude. Mutations also vary widely in their effect on the molecular function, phenotype, and fitness. Independence of the probability of occurrence of a new mutation''s effect has been a fundamental premise in genetics. However, highly mutable contexts may be preserved by negative selection at important sites but destroyed by mutation at sites under no selection. Thus, there may be a positive correlation between the rate of mutations at a nucleotide site and the magnitude of their effect on fitness. We studied the impact of CpG context on the rate of human–chimpanzee divergence and on intrahuman nucleotide diversity at non-synonymous coding sites. We compared nucleotides that occupy identical positions within codons of identical amino acids and only differ by being within versus outside CpG context. Nucleotides within CpG context are under a stronger negative selection, as revealed by their lower, proportionally to the mutation rate, rate of evolution and nucleotide diversity. In particular, the probability of fixation of a non-synonymous transition at a CpG site is two times lower than at a CpG site. Thus, sites with different mutation rates are not necessarily selectively equivalent. This suggests that the mutation rate may complement sequence conservation as a characteristic predictive of functional importance of nucleotide sites.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.     

Reconstructing the prior probabilities of allelic phylogenies

In general when a phylogeny is reconstructed from DNA or protein sequence data, it makes use only of the probabilities of obtaining some phylogeny given a collection of data. It is also possible to determine the prior probabilities of different phylogenies. This information can be of use in analyzing the biological causes for the observed divergence of sampled taxa. Unusually "rare" topologies for a given data set may be indicative of different biological forces acting. A recursive algorithm is presented that calculates the prior probabilities of a phylogeny for different allelic samples and for different phylogenies. This method is a straightforward extension of Ewens' sample distribution. The probability of obtaining each possible sample according to Ewens' distribution is further subdivided into each of the possible phylogenetic topologies. These probabilities depend not only on the identity of the alleles and on 4N(mu) (four times the effective population size times the neutral mutation rate) but also on the phylogenetic relationships among the alleles. Illustrations of the algorithm are given to demonstrate how different phylogenies are favored under different conditions.