Simulation results with stepwise mutation model and their interpretations

Summary Monte Carlo simulations are performed to compare the predictions based on the two presently used theoretical models for studying genetic variations in natural populations, the infinite allele model and the stepwise mutation model. Distribution of heterozygosity is noticed to be similar under these models until the product of population size and mutation rate is large. It is seen that electromorphs with high population frequency usually contain older alleles (at the codon level) than an electromorph of low population frequency. The interpretations of these results in explaining the allelic variations at electrophoretic level is also discussed.Research supported by U.S. Public Health Service General Research Support Grant 5 SO 5 RR 07148 from the University of Texas Health Science Center, Graduate School of Biomedical Sciences, Houston, Texas     

Accurate partition of individuals into full-sib families from genetic data without parental information.

Two Markov chain Monte Carlo algorithms are proposed that allow the partitioning of individuals into full-sib groups using single-locus genetic marker data when no parental information is available. These algorithms present a method of moving through the sibship configuration space and locating the configuration that maximizes an overall score on the basis of pairwise likelihood ratios of being full-sib or unrelated or maximizes the full joint likelihood of the proposed family structure. Using these methods, up to 757 out of 759 Atlantic salmon were correctly classified into 12 full-sib families of unequal size using four microsatellite markers. Large-scale simulations were performed to assess the sensitivity of the procedures to the number of loci and number of alleles per locus, the allelic distribution type, the distribution of families, and the independent knowledge of population allelic frequencies. The number of loci and the number of alleles per locus had the most impact on accuracy. Very good accuracy can be obtained with as few as four loci when they have at least eight alleles. Accuracy decreases when using allelic frequencies estimated in small target samples with skewed family distributions with the pairwise likelihood approach. We present an iterative approach that partly corrects that problem. The full likelihood approach is less sensitive to the precision of allelic frequencies estimates but did not perform as well with the large data set or when little information was available (e.g., four loci with four alleles).     

The effect of intragenic recombination on the number of alleles in a finite population

A two-site infinite allele model is constructed to study the effect of intragenic recombination on the number of neutral alleles and the distribution of their frequencies in a finite population. The results of theory and Monte Carlo simulation of the two-site model demonstrate that intragenic recombination significantly increases the mean and variance of the number of alleles when the rates of mutation and recombination are as large as the reciprocal of the population size. Data from natural populations indicate that this may be a significant process in generating variation and determining its distribution.     

Bayesian mapping of QTL in outbred F2 families allowing inference about whether F0 grandparents are homozygous or heterozygous at QTL

In this paper, we propose a new Bayesian method for QTL analysis in outbred F2 families based on Markov chain Monte Carlo (MCMC) estimation allowing inference about whether each of F0 founders (grandparents) is homozygous or heterozygous at QTL. This, in turn, allows us to select a model accurately explaining observations of phenotypes for F2 individuals. The proposed method performs the fitting a statistical model of the two possible QTL states in each F0 grandparent, that is, homozygous and heterozygous at QTL, and gives a posterior distribution for the QTL states in each F0 grandparent. We confine ourselves to the discrimination of two QTL states, homozygous or heterozygous, for each of the F0 grandparents without taking into consideration whether common alleles are shared by F0 grandparents. The statistical model includes allelic effects and dominance effects for each QTL. The number of parameters representing allelic effects and dominance effects is therefore changed depending on the QTL states. A Reversible Jump MCMC technique is used for transition between the models of different dimensions. The effectiveness of the proposed method was investigated using simulation experiments. It was practicable to estimate the QTL states of F0 grandparents as well as the number, the locations and the effects of QTL segregating in an outbred F2 family.     

An estimator for pairwise relatedness using molecular markers

I propose a new estimator for jointly estimating two-gene and four-gene coefficients of relatedness between individuals from an outbreeding population with data on codominant genetic markers and compare it, by Monte Carlo simulations, to previous ones in precision and accuracy for different distributions of population allele frequencies, numbers of alleles per locus, actual relationships, sample sizes, and proportions of relatives included in samples. In contrast to several previous estimators, the new estimator is well behaved and applies to any number of alleles per locus and any allele frequency distribution. The estimates for two- and four-gene coefficients of relatedness from the new estimator are unbiased irrespective of the sample size and have sampling variances decreasing consistently with an increasing number of alleles per locus to the minimum asymptotic values determined by the variation in identity-by-descent among loci per se, regardless of the actual relationship. The new estimator is also robust for small sample sizes and for unknown relatives being included in samples for estimating allele frequencies. Compared to previous estimators, the new one is generally advantageous, especially for highly polymorphic loci and/or small sample sizes.     

Bayesian mapping of quantitative trait loci under complicated mating designs

Quantitative trait loci (QTL) are easily studied in a biallelic system. Such a system requires the cross of two inbred lines presumably fixed for alternative alleles of the QTL. However, development of inbred lines can be time consuming and cost ineffective for species with long generation intervals and severe inbreeding depression. In addition, restriction of the investigation to a biallelic system can sometimes be misleading because many potentially important allelic interactions do not have a chance to express and thus fail to be detected. A complicated mating design involving multiple alleles mimics the actual breeding system. However, it is difficult to develop the statistical model and algorithm using the classical maximum-likelihood method. In this study, we investigate the application of a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm to QTL mapping under arbitrarily complicated mating designs. We develop the method under a mixed-model framework where the genetic values of founder alleles are treated as random and the nongenetic effects are treated as fixed. With the MCMC algorithm, we first draw the gene flows from the founders to the descendants for each QTL and then draw samples of the genetic parameters. Finally, we are able to simultaneously infer the posterior distribution of the number, the additive and dominance variances, and the chromosomal locations of all identified QTL.     

Estimation of gene frequency and heterozygosity from pooled samples

Pooling of DNA samples can significantly reduce the effort of population studies with DNA markers. I present a statistical model and numerical method for estimating gene frequency when pooled DNA is assayed for the presence/absence of alleles. Analytical and Monte‐Carlo methods examined estimation variance and bias, and hence optimal pool size, under a triangular allele frequency distribution. For gene frequency of rarer alleles, the optimal number of pooled individuals is approximately the inverse of the gene frequency. For heterozygosity, the optimal pool is approximately half the allele number; this results in pools containing, on average, 60% of possible alleles.     

On the Divergence of Alleles in Nested Subsamples from Finite Populations

Within-population variation at the DNA level will rarely be studied by sequencing of loci of randomly chosen individuals. Instead, individuals will usually be chosen for sequencing based on some knowledge of their genotype. Data collected in this way require new sampling theory. Motivated by these observations, we have examined the sampling properties of a finite population model with two mutation processes and with no selection or recombination. One mutation process generates new alleles according to an infinite-alleles model, and the other generates polymorphisms at sites according to an infinite-sites model. A sample of n genes is considered. The stationary distribution of the number of segregating sites in a subsample from one of the allelic classes in the sample conditional on the allelic configuration of the sample is studied. A recursive scheme is developed to compute the moments of this distribution, and it is shown that the distribution is functionally independent of the number of additional alleles in the sample and their respective frequencies in the sample. For the case in which the sample contains only two alleles, the distribution of the number of segregating sites in a subsample containing both alleles conditional on the sample frequencies of the alleles is studied. The results are applied to the analysis of DNA sequences of two alleles found at the Adh locus of Drosophila melanogaster. No significant departure from the neutral model is detected.     

Comparison of methods used for recovering the line origin of alleles in a cross between outbred lines

Here, we introduce the idea of probabilities of line origins for alleles in general pedigrees as found in crosses between outbred lines. We also present software for calculating these probabilities. The proposed algorithm is based on the linear regression method of Haley, Knott and Elsen (1994) combined with the Markov chain Monte Carlo (MCMC) method for estimating quantitative trait locus coefficients used as regressors. We compared the relative precision of our method and the original method as proposed by Haley et al. (1994). The scenarios studied varied in the allelic distribution of marker alleles in parental lines and in the frequency of missing marker genotypes. We found that the MCMC method achieves a higher accuracy in all scenarios considered. The benefits of using MCMC approximation are substantial if the frequency of missing marker data is high or the number of marker alleles is low and the allelic frequency distribution is similar in both parental lines.     

Lattice simulations of aggregation funnels for protein folding.

A computer model of protein aggregation competing with productive folding is proposed. Our model adapts techniques from lattice Monte Carlo studies of protein folding to the problem of aggregation. However, rather than starting with a single string of residues, we allow independently folding strings to undergo collisions and consider their interactions in different orientations. We first present some background into the nature and significance of protein aggregation and the use of lattice Monte Carlo simulations in understanding other aspects of protein folding. The results of a series of simulation experiments involving simple versions of the model illustrate the importance of considering aggregation in simulations of protein folding and provide some preliminary understanding of the characteristics of the model. Finally, we discuss the value of the model in general and of our particular design decisions and experiments. We conclude that computer simulation techniques developed to study protein folding can provide insights into protein aggregation, and that a better understanding of aggregation may in turn provide new insights into and constraints on the more general protein folding problem.     

Monte Carlo experiments and the defense of diffusion models in molecular population genetics

In the 1960s molecular population geneticists used Monte Carlo experiments to evaluate particular diffusion equation models. In this paper I examine the nature of this comparative evaluation and argue for three claims: first, Monte Carlo experiments are genuine experiments: second, Monte Carlo experiments can provide an important meansfor evaluating the adequacy of highly idealized theoretical models; and, third, the evaluation of the computational adequacy of a diffusion model with Monte Carlo experiments is significantlydifferent from the evaluation of the emperical adequacy of the same diffusion model.     

A genealogical interpretation of linkage disequilibrium

The degree of association between alleles at different loci, or linkage disequilibrium, is widely used to infer details of evolutionary processes. Here I explore how associations between alleles relate to properties of the underlying genealogy of sequences. Under the neutral, infinite-sites assumption I show that there is a direct correspondence between the covariance in coalescence times at different parts of the genome and the degree of linkage disequilibrium. These covariances can be calculated exactly under the standard neutral model and by Monte Carlo simulation under different demographic models. I show that the effects of population growth, population bottlenecks, and population structure on linkage disequilibrium can be described through their effects on the covariance in coalescence times.     

A comparison of rarefaction and bayesian methods for predicting the allelic richness of future samples on the basis of currently available samples

Rarefaction methods have been introduced into population genetics (from ecology) for predicting and comparing the allelic richness of future samples (or sometimes populations) on the basis of currently available samples, possibly of different sizes. Here, we focus our attention on one such problem: Predicting which population is most likely to yield the future sample having the highest allelic richness. (This problem can arise when we want to construct a core collection from a larger germplasm collection.) We use extensive simulations to compare the performance of the Monte Carlo rarefaction (repeated random subsampling) method with a simple Bayesian approach we have developed-which is based on the Ewens sampling distribution. We found that neither this Bayesian method nor the (Monte Carlo) rarefaction method performed uniformly better than the other. We also examine briefly some of the other motivations offered for these methods and try to make sense of them from a Bayesian point of view.     

Merging microsatellite data.

Genotype calling procedures vary from laboratory to laboratory for many microsatellite markers. Even within the same laboratory, application of different experimental protocols often leads to ambiguities. The impact of these ambiguities ranges from irksome to devastating. Resolving the ambiguities can increase effective sample size and preserve evidence in favor of disease-marker associations. Because different data sets may contain different numbers of alleles, merging is unfortunately not a simple process of matching alleles one to one. Merging data sets manually is difficult, time-consuming, and error-prone due to differences in genotyping hardware, binning methods, molecular weight standards, and curve fitting algorithms. Merging is particularly difficult if few or no samples occur in common, or if samples are drawn from ethnic groups with widely varying allele frequencies. It is dangerous to align alleles simply by adding a constant number of base pairs to the alleles of one of the data sets. To address these issues, we have developed a Bayesian model and a Markov chain Monte Carlo (MCMC) algorithm for sampling the posterior distribution under the model. Our computer program, MicroMerge, implements the algorithm and almost always accurately and efficiently finds the most likely correct alignment. Common allele frequencies across laboratories in the same ethnic group are the single most important cue in the model. MicroMerge computes the allelic alignments with the greatest posterior probabilities under several merging options. It also reports when data sets cannot be confidently merged. These features are emphasized in our analysis of simulated and real data.     

Importance sampling for the infinite sites model

Importance sampling or Markov Chain Monte Carlo sampling is required for state-of-the-art statistical analysis of population genetics data. The applicability of these sampling-based inference techniques depends crucially on the proposal distribution. In this paper, we discuss importance sampling for the infinite sites model. The infinite sites assumption is attractive because it constraints the number of possible genealogies, thereby allowing for the analysis of larger data sets. We recall the Griffiths-Tavaré and Stephens-Donnelly proposals and emphasize the relation between the latter proposal and exact sampling from the infinite alleles model. We also introduce a new proposal that takes knowledge of the ancestral state into account. The new proposal is derived from a new result on exact sampling from a single site. The methods are illustrated on simulated data sets and the data considered in Griffiths and Tavaré (1994).     

Cautions on direct gene flow estimation in plant populations

Through simulations we have investigated the statistical properties of two of the main approaches for directly estimating pollen gene flow (m) in plant populations: genotypic exclusion and mating models. When the assumptions about accurately known background pollen pool allelic frequencies are met, both methods provide unbiased results with comparable variances across a range of true m values. However, when presumed allelic frequencies differ from actual ones, which is more likely in research practice, both estimators are biased. We demonstrate that the extent and direction of bias largely depend on the difference (measured as genetic distance) between the presumed and actual pollen pools, and on the degree of genetic differentiation between the local population and the actual background pollen sources. However, one feature of the mating model is its ability to estimate pollen gene flow simultaneously with background pollen pool allelic frequencies. We have found that this approach gives nearly unbiased pollen gene flow estimates, and is practical because it eliminates the necessity of providing independent estimates of background pollen pool allelic frequencies. Violations of the mating model assumptions of random mating within local population affect the precision of the estimates only to a limited degree.     

A review on Monte Carlo simulation methods as they apply to mutation and selection as formulated in Wright-Fisher models of evolutionary genetics

A case has made for the use of Monte Carlo simulation methods when the incorporation of mutation and natural selection into Wright-Fisher gametic sampling models renders then intractable from the standpoint of classical mathematical analysis. The paper has been organized around five themes. Among these themes was that of scientific openness and a clear documentation of the mathematics underlying the software so that the results of any Monte Carlo simulation experiment may be duplicated by any interested investigator in a programming language of his choice. A second theme was the disclosure of the random number generator used in the experiments to provide critical insights as to whether the generated uniform random variables met the criterion of independence satisfactorily. A third theme was that of a review of recent literature in genetics on attempts to find signatures of evolutionary processes such as natural selection, among the millions of segments of DNA in the human genome, that may help guide the search for new drugs to treat diseases. A fourth theme involved formalization of Wright-Fisher processes in a simple form that expedited the writing of software to run Monte Carlo simulation experiments. Also included in this theme was the reporting of several illustrative Monte Carlo simulation experiments for the cases of two and three alleles at some autosomal locus, in which attempts were to made to apply the theory of Wright-Fisher models to gain some understanding as to how evolutionary signatures may have developed in the human genome and those of other diploid species. A fifth theme was centered on recommendations that more demographic factors, such as non-constant population size, be included in future attempts to develop computer models dealing with signatures of evolutionary process in genomes of various species. A brief review of literature on the incorporation of demographic factors into genetic evolutionary models was also included to expedite and stimulate further development on this theme.     

A random model approach to mapping quantitative trait loci for complex binary traits in outbred populations.

Mapping quantitative trait loci (QTL) for complex binary traits is more challenging than for normally distributed traits due to the nonlinear relationship between the observed phenotype and unobservable genetic effects, especially when the mapping population contains multiple outbred families. Because the number of alleles of a QTL depends on the number of founders in an outbred population, it is more appropriate to treat the effect of each allele as a random variable so that a single variance rather than individual allelic effects is estimated and tested. Such a method is called the random model approach. In this study, we develop the random model approach of QTL mapping for binary traits in outbred populations. An EM-algorithm with a Fisher-scoring algorithm embedded in each E-step is adopted here to estimate the genetic variances. A simple Monte Carlo integration technique is used here to calculate the likelihood-ratio test statistic. For the first time we show that QTL of complex binary traits in an outbred population can be scanned along a chromosome for their positions, estimated for their explained variances, and tested for their statistical significance. Application of the method is illustrated using a set of simulated data.     

Unequal allelic frequencies at the self‐incompatibility locus within local populations of Prunus avium L.: an effect of population structure?

In this paper, we investigated the genetic structure and distribution of allelic frequencies at the gametophytic self-incompatibility locus in three populations of Prunus avium L. In line with theoretical predictions under balancing selection, genetic structure at the self-incompatibility locus was almost three times lower than at seven unlinked microsatellites. Furthermore, we found that S-allele frequencies in wild cherry populations departed significantly from the expected isoplethic distribution towards which balancing selection is expected to drive allelic frequencies (i.e. identical frequency equal to the inverse of the number of alleles in the population). To assess whether this departure could be caused either by drift alone or by population structure, we used numerical simulations to compare our observations with allelic frequency distributions expected : (1) within a single deme from a subdivided population with various levels of differentiation; and (2) within a finite panmictic population with identical allelic diversity. We also investigated the effects of sample size and degree of population structure on tests of departure from isoplethic equilibrium. Overall, our results showed that the observed allele frequency distributions were consistent with a model of subdivided population with demes linked by moderate migration rate.     

Estimating effective paternity number in social insects and the effective number of alleles in a population

Estimating paternity and genetic relatedness is central to many empirical and theoretical studies of social insects. The two important measures of a queen's mating number are her actual number of mates and her effective number of mates. Estimating the effective number of mates is mathematically identical to the problem of estimating the effective number of alleles in population genetics, a common measure of genetic variability introduced by Kimura & Crow (1964). We derive a new bias-corrected estimator of effective number of types (mates or alleles) and compare this new method to previous methods for estimating true and effective numbers of types using Monte Carlo simulations. Our simulation results suggest that the examined estimators of the true number of types have very similar statistical properties, whereas the estimators of effective number of types have quite different statistical properties. Moreover, our new proposed estimator of effective number of types is approximately unbiased, and has considerably lower variance than the original estimator. Our new method will help researchers more accurately estimate intracolony genetic relatedness of social insects, which is an important measure in understanding their ecology and social behaviour. It should also be of use in population genetic studies in which the effective number of alleles is of interest.