Stepwise Mutation Model Research Articles

Population structure, stepwise mutations, heterozygote deficiency and their implications in DNA forensics

In a substructured population the overall heterozygote deficiency can be predicted from the number of subpopulations (s), their time of divergence (t), and the nature of the mutations. At present the true mutational mechanisms at the hypervariable DNA loci are not known. However, the two existing mutation models (the infinite allele model (IAM) and the stepwise mutation model (SMM)) provide some guides to predictions from which the possible effect of population substructuring may be evaluated, assuming that the subpopulations do not exchange any genes among them during evolution. The theory predicts that the loci with larger mutation rate, and consequently showing greater heterozygosity within subpopulations, should exhibit a smaller proportional heterozygote deficiency (GST) and, hence, the effects of population substructuring should be minimal at the hypervariable DNA loci (an order of magnitude smaller than that at the blood group and protein loci). Applications of this theory to data on six Variable Number of Tandem Repeat (VNTR) loci and five short tandem repeat (STR) loci in the major cosmopolitan populations of the USA show that while the VNTR loci often exhibit a large significant heterozygote deficiency, the STR loci do not show a similar tendency. This discordant finding may be ascribed to the limitations, coalescence and nondetectability of alleles associated with the restriction fragment length polymorphism (RFLP) analysis through which the VNTR loci are scored. Such limitations do not apply to the polymerase chain reaction (PCR) method, through which the STR loci are scored. The implications of these results are discussed in the context of the forensic use of DNA typing data.

An evaluation of genetic distances for use with microsatellite loci.

Mutations of alleles at microsatellite loci tend to result in alleles with repeat scores similar to those of the alleles from which they were derived. Therefore the difference in repeat score between alleles carries information about the amount of time that has passed since they shared a common ancestral allele. This information is ignored by genetic distances based on the infinite alleles model. Here we develop a genetic distance based on the stepwise mutation model that includes allelic repeat score. We adapt earlier treatments of the stepwise mutation model to show analytically that the expectation of this distance is a linear function of time. We then use computer simulations to evaluate the overall reliability of this distance and to compare it with allele sharing and Nei's distance. We find that no distance is uniformly superior for all purposes, but that for phylogenetic reconstruction of taxa that are sufficiently diverged, our new distance is preferable.

A measure of population subdivision based on microsatellite allele frequencies.

A new measure of the extent of population subdivision as inferred from allele frequencies at microsatellite loci is proposed and tested with computer simulations. This measure, called R(ST), is analogous to Wright's F(ST) in representing the proportion of variation between populations. It differs in taking explicit account of the mutation process at microsatellite loci, for which a generalized stepwise mutation model appears appropriate. Simulations of subdivided populations were carried out to test the performance of R(ST) and F(ST). It was found that, under the generalized stepwise mutation model, R(ST) provides relatively unbiased estimates of migration rates and times of population divergence while F(ST) tends to show too much population similarity, particularly when migration rates are low or divergence times are long [corrected].

Population genetics of trinucleotide repeat polymorphisms.

Trinucleotide repeats at five disease loci (DM, DRPLA, HD, SBMA and SCA1) were surveyed in phenotypically normal individuals from three continental populations. This is the first analysis to examine the population dynamics of these five disease-related trinucleotide repeats in the same individuals from worldwide populations. Roughly half of all alleles observed at each locus are shared between all continental groups. For three loci, disease prevalence in each population corresponds with the number of alleles in the upper tail of the allele-size distribution. The allele-size distributions of African, Asian and Caucasian groups show a high degree of variation, and gene diversity estimates for trinucleotide repeat loci exceed estimates derived from dinucleotide or tetranucleotide repeats. Analyses that compared infinite alleles and stepwise mutation models suggest that normal variation at trinucleotide loci is not generated by stepwise mutation alone. Trees constructed for subpopulations using trinucleotide repeat loci show accurate continental clustering. Interpopulation genetic distance estimates show remarkable similarity to distance estimates produced from tetranucleotide repeats or nuclear restriction site polymorphisms. This finding is especially noteworthy in light of the fact that trinucleotide repeat polymorphisms at these loci can cause disease, while restriction site and tetranucleotide polymorphisms appear to be selectively neutral. In contrast, genetic distance estimates from trinucleotide loci are poorly correlated with genetic distance estimates from mitochondrial sequence data.

Microsatellite DNA variation within and among European cattle breeds

Microsatellite markers offer great potential for genetic comparisons within and between populations. We report the analysis of 12 microsatellite loci in six breeds of European cattle. This yielded a wide spectrum of variability with observed heterozygosities ranging from 0.00 to 0.91. Deviations from Hardy-Weinberg equilibrium were noted for some locus-population combinations, particularly at a microsatellite located within the prolactin gene. Also, significant linkage disequilibrium was detected between two microsatellite loci located within the bovine major histocompatibility complex, and this association was maintained across breeds, providing evidence for marker stability during short-term evolution. The mode of mutation was investigated by comparing the observed data with that expected under the infinite alleles model of neutral mutation, and six of the microsatellite loci were found to deviate significantly, suggesting that a stepwise mutation model may be more appropriate. One indication of marker utility is that, when genetic distance estimates were computed, the resultant dendrogram showed concordance with known breed histories.

A new result on the support of the fleming-viot process, proved by non-standard construction

In Fleming and Viot [4] a measure-valued process was constructed as a weak limit of the Ohta-Kimura stepwise mutation model, under scalings of time and space. Dawson and Hochberg [3] obtained further results, including the fact that for fixed times, almost surely, the support of the random measure has Hausdorff dimension not greater than two. This paper uses a non standard construction of the Fleming-Viot process to establish this result for all times simultaneously

VNTR allele frequency distributions under the stepwise mutation model: a computer simulation approach.

Variable numbers of tandem repeats (VNTRs) are a class of highly informative and widely dispersed genetic markers. Despite their wide application in biological science, little is known about their mutational mechanisms or population dynamics. The objective of this work was to investigate four summary measures of VNTR allele frequency distributions: number of alleles, number of modes, range in allele size and heterozygosity, using computer simulations of the one-step stepwise mutation model (SMM). We estimated these measures and their probability distributions for a wide range of mutation rates and compared the simulation results with predictions from analytical formulations of the one-step SMM. The average heterozygosity from the simulations agreed with the analytical expectation under the SMM. The average number of alleles, however, was larger in the simulations than the analytical expectation of the SMM. We then compared our simulation expectations with actual data reported in the literature. We used the sample size and observed heterozygosity to determine the expected value, 5th and 95th percentiles for the other three summary measures, allelic size range, number of modes and number of alleles. The loci analyzed were classified into three groups based on the size of the repeat unit: microsatellites (1-2 base pair (bp) repeat unit), short tandem repeats [(STR) 3-5 bp repeat unit], and minisatellites (15-70 bp repeat unit). In general, STR loci were most similar to the simulation results under the SMM for the three summary measures (number of alleles, number of modes and range in allele size), followed by the microsatellite loci and then by the minisatellite loci, which showed deviations in the direction of the infinite allele model (IAM). Based on these differences, we hypothesize that these three classes of loci are subject to different mutational forces.

Fleming–Viot Processes in Population Genetics

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include:1. Introduction.2. Some measure-valued Markov chains.2.1. A diploid model. 2.2. The Wright–Fisher model.2.3. A Moran model.3. The Fleming–Viot process: characterization.4. Convergence.5. Ergodicity.6. An infinite particle system.7. Bounded mutation operators.8. Reversibility.9. Examples.9.1. Continuous-state stepwise-mutation model.9.2. Infinitely-many-neutral-alleles model.9.3. Infinitely-many-neutral-alleles model with ages.9.4. Two-locus model with recombination.9.5. n-locus model with gene conversion.9.6. Infinitely-many-sites model without recombination.9.7. Infinitely-many-neutral-alleles model with allelic genealogies.

A population genetic study of six VNTR loci in three ethnically defined populations

To investigate the population genetic characteristics of VNTR polymorphisms in human populations, we have studied the allele frequency distribution of six VNTR loci (D1S57, RB1, D1S77, D1S61, α-globin 5′HVR, D1S76) in three well-defined populations (Kachari of Northeast India; Dogrib Indian of Canada; and New Guinea Highlander of Papua New Guinea). Even though the number of alleles sampled is limited, 48 to 92 alleles per locus per population, significant variation is noticed in the number of alleles per locus for all the populations. Using alternate summary measures, we have observed that genotype distributions at the six VNTR loci apparently conform to their respective Hardy-Weinberg predictions. Multilocus genotype profiles of the individuals in each of the three populations suggest that the VNTR alleles are independently segregating with the exception of the two linked loci D1S76 and D1S77. Lack of fit of all VNTR loci to one particular model of mutational change, either the Infinite Allele Model or the Stepwise Mutation Model, suggests more than one mechanism for production of new VNTR alleles. This study also indicates that increased heterozygosity at VNTR loci in comparison to protein and blood group loci may lead to more accurate estimates of genetic distance.

Interacting Fleming-Viot processes

We construct a class of interacting Ohta-Kimura stepwise-mutation models and study their macroscopic behavior, i.e., we prove their weak convergence when the population size n increases to infinity, the evolution is speeded up by a factor n 2, the increment effects of a change in each population's distribution of characteristics under study is scaled down by a factor n -1 2 and the level of interaction between populations is decreased by a factor n -1. The measure-valued limits—interacting Fleming-Viot processes—are characterized as unique solutions to certain martingale problems using the method of duality. Finally, we prove a scaling theorem for these processes.

The quantitative genetic consequences of pleiotropy under stabilizing and directional selection.

The independence of two phenotypic characters affected by both pleiotropic and nonpleiotropic mutations is investigated using a generalization of M. Slatkin's stepwise mutation model of 1987. The model is used to determine whether predictions of either the multivariate normal model introduced in 1980 by R. Lande or the house-of-cards model introduced in 1985 by M. Turelli can be regarded as typical of models that are intermediate between them. We found that, under stabilizing selection, the variance of one character at equilibrium may depend on the strength of stabilizing selection on the other character (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the types of mutations that can occur. Similarly, under directional selection, the genetic covariance between two characters may increase substantially (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the kinds of mutations that are assumed to occur. Hence, even for the simple model we consider, neither the house-of-cards nor the multivariate normal model can be used to make predictions, making it unlikely that either could be used to draw general conclusions about more complex and realistic models.

Mutation-selection balance and continuum-of-alleles models

A discrete-time model for evolution of type densities in a haploid population that is governed by mutation and selection is analysed. Important special cases are the classical one-locus multiallele model, models like the stepwise-mutation model, and models with a continuum of possible allelic effects on a quantitative trait. Using methods from functional analysis, It is proved that for very general mutation and selection regimes a uniquely determined positive equilibrium density exists that is globally stable. Moreover, an upper bound for the equilibrium variance that can be maintained by a balance between mutation and Gaussian stabilizing selection is derived.

Mutation-selection balance and continuum-of-alleles models

A discrete-time model for the evolution of type densities in a haploid population that is governed by mutation and selection is analysed. Important special cases are the classical one-locus multiallele model, models like the stepwise-mutation model, and models with a continuum of possible allelic effects on a quantitative trait. Using methods from functional analysis, it is proved that for very general mutation and selection regimes a uniquely determined positive equilibrium density exists that is globally stable. Moreover, an upper bound for the equilibrium variance that can be maintained by a balance between mutation and Gaussian stabilizing selection is derived.

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.

Heritable variation and heterozygosity under a balance between mutations and stabilizing selection.

SummaryA model of the balance between mutations and stabilizing selection affecting a quantitative character is developed and analysed. This model is essentially a discretized version of the continuum-of-alleles models analysed previously by Kimura, Lande, Turelli and others, and is formally similar to the stepwise mutation models used to interpret electrophoretic data. The complete model cannot be solved even for a haploid species, but there are useful approximations for most parameter values of interest. The ‘house-of-cards’ approximation can be used when selection is strong relative to mutation, and a normal approximation can be used when selection is relatively weak. For intermediate levels of selection a new ‘five-allele’ approximation provides accurate results over a wide range of parameter values. The house-of-cards and five-allele approximations applied to recessive alleles in a diploid population show that, for a given mutation rate, a somewhat larger genetic variance is maintained at equilibrium than in a comparable model of additive alleles. Under directional selection, the increase in genetic variance is largest for alleles of large effect and is much smaller for alleles of intermediate or small effect. At an equilibrium under stabilizing selection, homozygotes would tend to have a higher average fitness than heterozygotes when each mutation has a relatively large effect (the house-of-cards approximation), with the reverse if each mutation has a small effect (the normal approximation).

Models of evolution of reproductive isolation.

Mathematical models are presented for the evolution of postmating and premating reproductive isolation. In the case of postmating isolation it is assumed that hybrid sterility or inviability is caused by incompatibility of alleles at one or two loci, and evolution of reproductive isolation occurs by random fixation of different incompatibility alleles in different populations. Mutations are assumed to occur following either the stepwise mutation model or the infinite-allele model. Computer simulations by using Itô's stochastic differential equations have shown that in the model used the reproductive isolation mechanism evolves faster in small populations than in large populations when the mutation rate remains the same. In populations of a given size it evolves faster when the number of loci involved is large than when this is small. In general, however, evolution of isolation mechanisms is a very slow process, and it would take thousands to millions of generations if the mutation rate is of the order of 10(-5) per generation. Since gene substitution occurs as a stochastic process, the time required for the establishment of reproductive isolation has a large variance. Although the average time of evolution of isolation mechanisms is very long, substitution of incompatibility genes in a population occurs rather quickly once it starts. The intrapopulational fertility or viability is always very high. In the model of premating isolation it is assumed that mating preference or compatibility is determined by male- and female-limited characters, each of which is controlled by a single locus with multiple alleles, and mating occurs only when the male and female characters are compatible with each other. Computer simulations have shown that the dynamics of evolution of premating isolation mechanism is very similar to that of postmating isolation mechanism, and the mean and variance of the time required for establishment of premating isolation are very large. Theoretical predictions obtained from the present study about the speed of evolution of reproductive isolation are consistent with empirical data available from vertebrate organisms.

Wandering Random Measures in the Fleming-Viot Model

Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for describing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately rescaled, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.

DETECTING NATURAL SELECTION: THE STEPWISE MUTATION MODEL WITH HETEROSIS

SummaryThe problem of testing whether heterozygotes have a selective advantage is considered, assuming the stepwise mutation model for electrophoretic alleles. The population is observed for a number of generations, and the likelihoods of the changes in allele frequencies under the heterozygote advantage model and under the strict neutrality model are compared. It is shown by simulation that statistical tests based on the likelihoods have little power to detect selection if only a few generations of data are available.Many population genetic models, including the stepwise mutation model, have been formulated as diffusion processes in continuous time. The paper gives expressions for the likelihoods of multivariate diffusions observed over a time interval.

Composite stepwise mutation model under the neutral mutation hypothesis.

As an extension of the conventional ("Ohta-Kimura") stepwise mutation model, a new model is proposed. In this model, it is assumed that each charge state ("electromorph") is represented by K alleles and that a mutation changes an allele either by one step in the charge space or to one of the other members of the identical electromorph. It is shown that the net genetic variability within a population is similar to that predicted by the infinite-allele model ("Kimura-Crow" model) rather than to that predicted by the stepwise mutation model, and the K-dependence of genetic variability is rather weak when K greater than or equal to 2 and the effective population size is not much greater than the reciprocal of mutation rate. The results are compared with the recent observations at the xanthine dehydrogenase locus in Drosophila pseudoobscura.