Convergence of one-dimensional diffusion processes to a jump process related to population genetics

A conjecture on the convergence of diffusion models in population genetics to a simple Markov chain model is proved. The notion of bi-generalized diffusion processes and their limit theorems are used systematically to prove the conjecture. Three limits; strong selection-weak mutation limit, moderate selection-weak mutation limit, weak selection-weak mutation limit are considered for typical diffusion models in population genetics.Supported in part by Research Grant from the Ministry of Education, Science and Culture of JapanSupported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144     

Isozyme genotype-environment relationships in natural populations of the harvester ant,Pogonomyrmex barbatus,from Texas

Three different allelic isozyme systems (two esterases, ESH and ESR, and a malic dehydrogenase, MDH) were analyzed in population samples of a species of ant, Pogonomyrmex barbatus, from Texas. Allelic frequencies were determined for several collection localities, and a number of significant differences were found. Principal component analysis was used to compare the patterns of variability of the allelic frequencies with environmental factors. Significant correlation was particularly evident with respect to weather and the pattern of variability in both esterases, and it is therefore suspected that natural selection is important in determining the allele frequency patterns. Observed and expected genotypic proportions were found in good agreement, generally, but in some localities homozygotes appeared in significantly greater numbers than expected. Heterotic selective maintenance was thus not indicated. Correlation found between patterns of variability in the enzyme systems themselves was consistent with the hypothesis that all three enzyme systems were affected by the environmental factors.Supported in part by USPHS Research Grants No. GM 15769, GM 11609, and GM 11546 and Texas A & I University Faculty Research Grant No. 449-N-68. The field work and laboratory analyses in this investigation were done while the senior author was at the University of Texas, and the data were analyzed at North Carolina State University. The computing was supported by Grant FR-00011 of the National Institutes of Health. Paper number 2784 of the Journal Series of the North Carolina Agricultural Experiment Station, Raleigh, North Carolina.     

The balance between pleiotropic mutation and selection,when alleles have discrete effects

The theory of pleiotropic mutation and selection is investigated and developed for a large population of asexual organisms. Members of the population are subject to stabilising selection on Omega phenotypic characters, which each independently affect fitness. Pleiotropy is incorporated into the model by allowing each mutation to simultaneously affect all characters. To expose differences with continuous-allele models, the characters are taken to originate from discrete-effect alleles and thus have discrete genotypic effects. Each character can take the values nxDelta where n=0,+/-1,+/-2, em leader, and the splitting in character effects, Delta, is a parameter of the model. When the distribution of mutant effects is normally distributed around the parental value, and Delta is large, a "stepwise" model of mutation arises, where only adjacent trait effects are accessible from a single mutation. The present work is primarily concerned with the opposite limit, where Delta is small and many different trait effects are accessible from a single mutation.In contrast to what has been established for continuous-effect models, discrete-effect models do not yield a singular equilibrium distribution of genotypic effects for any value of Omega. Instead, for different values of Omega, the equilibrium frequencies of trait values have very different dependencies on Delta. For Omega=1 and 2, decreasing Delta broadens the width of the frequency distribution and hence increases the equilibrium level of polymorphism. For all sufficiently large values of Omega, however, decreasing Delta decreases the width of the frequency distribution and the equilibrium level of polymorphism. The connection with continuous trait models follows when the limit Delta-->0 is considered, and a singular probability density of trait values is obtained for all sufficiently large Omega.     

Mutation-Selection Balance under Genomic Imprinting at an Autosomal Locus

I model the effect of genomic imprinting on the equilibrium allele frequencies at an autosomal diallelic locus subject to viability selection and mutation. The population size is assumed to be very large; male and female mutation rates may be unequal. Different models examine cases of the inactivation of one gene (with both complete and partial penetrance) and of differential expression of genes according to the parent of origin. In the simplest cases the frequency of the deleterious allele is approximately twice that of a dominant nonimprinting mutant, but considerably less than that of a recessive nonimprinting mutant. Under imprinting, selection and unequal mutation rates interact: other things being equal, male-biased mutation leads to lower mutant frequencies under maternal imprinting and higher frequencies under paternal imprinting. I also model cases where just one allele is imprintable (and the other not). These models allow us to predict the frequency of a failure to imprint in a normally imprinting system, as well as the frequency of imprinting at a standard nonimprinting locus.     

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.     

Quantifying the Diversification of Hepatitis C Virus (HCV) during Primary Infection: Estimates of the In Vivo Mutation Rate

Hepatitis C virus (HCV) is present in the host with multiple variants generated by its error prone RNA-dependent RNA polymerase. Little is known about the initial viral diversification and the viral life cycle processes that influence diversity. We studied the diversification of HCV during acute infection in 17 plasma donors, with frequent sampling early in infection. To analyze these data, we developed a new stochastic model of the HCV life cycle. We found that the accumulation of mutations is surprisingly slow: at 30 days, the viral population on average is still 46% identical to its transmitted viral genome. Fitting the model to the sequence data, we estimate the median in vivo viral mutation rate is 2.5×10⁻⁵ mutations per nucleotide per genome replication (range 1.6–6.2×10⁻⁵), about 5-fold lower than previous estimates. To confirm these results we analyzed the frequency of stop codons (N = 10) among all possible non-sense mutation targets (M = 898,335), and found a mutation rate of 2.8–3.2×10⁻⁵, consistent with the estimate from the dynamical model. The slow accumulation of mutations is consistent with slow turnover of infected cells and replication complexes within infected cells. This slow turnover is also inferred from the viral load kinetics. Our estimated mutation rate, which is similar to that of other RNA viruses (e.g., HIV and influenza), is also compatible with the accumulation of substitutions seen in HCV at the population level. Our model identifies the relevant processes (long-lived cells and slow turnover of replication complexes) and parameters involved in determining the rate of HCV diversification.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

Statistics for Microsatellite Variation Based on Coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate × population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci—effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation–drift equilibrium.     

Evolutionary and statistical properties of three genetic distances

Many genetic distances have been developed to summarize allele frequency differences between populations. I review the evolutionary and statistical properties of three popular genetic distances: DS, DA, and theta;, using computer simulation of two simple evolutionary histories: an isolation model of population divergence and an equilibrium migration model. The effect of effective population size, mutation rate, and mutation mechanism upon the parametric value between pairs of populations in these models explored, and the unique properties of each distance are described. The effect of these evolutionary parameters on study design is also investigated and similar results are found for each genetic distance in each model of evolution: large sample sizes are warranted when populations are relatively genetically similar; and loci with more alleles produce better estimates of genetic distance.     

Mutation Size Optimizes Speciation in an Evolutionary Model

The role of mutation rate in optimizing key features of evolutionary dynamics has recently been investigated in various computational models. Here, we address the related question of how maximum mutation size affects the formation of species in a simple computational evolutionary model. We find that the number of species is maximized for intermediate values of a mutation size parameter μ; the result is observed for evolving organisms on a randomly changing landscape as well as in a version of the model where negative feedback exists between the local population size and the fitness provided by the landscape. The same result is observed for various distributions of mutation values within the limits set by μ. When organisms with various values of μ compete against each other, those with intermediate μ values are found to survive. The surviving values of μ from these competition simulations, however, do not necessarily coincide with the values that maximize the number of species. These results suggest that various complex factors are involved in determining optimal mutation parameters for any population, and may also suggest approaches for building a computational bridge between the (micro) dynamics of mutations at the level of individual organisms and (macro) evolutionary dynamics at the species level.     

What do simple models reveal about the population dynamics of a cooperatively breeding species?

Research on cooperatively breeding species has shown that their population dynamics differ from those of conventional breeders. Populations of cooperators are structured into groups, and group‐level Allee effects are likely common. We assess the ability of phenomenological models, lacking explicit group structure, to describe population dynamics in cooperative meerkats Suricata suricatta, and we assess potential Allee effects at the population level. Using maximum likelihood model fitting and information theoretic model selection, applied to time series data from a wild meerkat population, we find simple models that incorporate rainfall and conventional density dependence to be the most parsimonious of the models considered. Detecting no population‐level Allee effect, we conclude that explicit consideration of population structure will be key to understanding the mechanisms behind population dynamics in cooperatively breeding species.     

Some stochastic features of bacterial constant growth apparatus

Stochastic models for bacterial constant growth apparatus such as the chemostat are posed and studied. Approximations are given for the mean and variance of the size of the bacterial population when the population is in steady state. Procedures for stimulating a chemostat are developed and the approximate moments are compared with simulated values. The distribution is derived for the waiting time until the occurrence of a population change-over to a faster growing strain. Research supported by National Institutes of Health Grant 5-R01-GM21214.     

Discovery of biclonal origin and a novel oncogene SLC12A5 in colon cancer by single-cell sequencing

Single-cell sequencing is a powerful tool for delineating clonal relationship and identifying key driver genes for personalized cancer management. Here we performed single-cell sequencing analysis of a case of colon cancer. Population genetics analyses identified two independent clones in tumor cell population. The major tumor clone harbored APC and TP53 mutations as early oncogenic events, whereas the minor clone contained preponderant CDC27 and PABPC1 mutations. The absence of APC and TP53 mutations in the minor clone supports that these two clones were derived from two cellular origins. Examination of somatic mutation allele frequency spectra of additional 21 whole-tissue exome-sequenced cases revealed the heterogeneity of clonal origins in colon cancer. Next, we identified a mutated gene SLC12A5 that showed a high frequency of mutation at the single-cell level but exhibited low prevalence at the population level. Functional characterization of mutant SLC12A5 revealed its potential oncogenic effect in colon cancer. Our study provides the first exome-wide evidence at single-cell level supporting that colon cancer could be of a biclonal origin, and suggests that low-prevalence mutations in a cohort may also play important protumorigenic roles at the individual level.     

Mathematische Modelle in der Populationsgenetik

In mathematical population genetics, the influence of selection and mutation on the evolution of a population is modelled. Because all populations and particularly the samples used for their analysis are finite, the stochastic nature of these models plays an important role. Relevant genetic models include the Wright–Fisher model and the coalescence model for the genealogy of samples, as well as the infinite alleles model and the infinite sites model for the mutation processes superimposed upon these genealogies.     

Studying models of balancing selection using phase-type theory

Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyze models with BLS acting on a biallelic locus: an equilibrium model with long-term BLS, a model with long-term BLS and recent changes in population size, and a model of recent BLS. Using phase-type theory, a mathematical tool for analyzing continuous time Markov chains with an absorbing state, we examine how BLS affects polymorphism patterns in linked neutral regions, as summarized by nucleotide diversity, the expected number of segregating sites, the site frequency spectrum, and the level of linkage disequilibrium (LD). Long-term BLS affects polymorphism patterns in a relatively small genomic neighborhood, and such selection targets are easier to detect when the equilibrium frequencies of the selected variants are close to 50%, or when there has been a population size reduction. For a new mutation subject to BLS, its initial increase in frequency in the population causes linked neutral regions to have reduced diversity, an excess of both high and low frequency derived variants, and elevated LD with the selected locus. These patterns are similar to those produced by selective sweeps, but the effects of recent BLS are weaker. Nonetheless, compared to selective sweeps, nonequilibrium polymorphism and LD patterns persist for a much longer period under recent BLS, which may increase the chance of detecting such selection targets. An R package for analyzing these models, among others (e.g., isolation with migration), is available.     

The evolution of isochores: evidence from SNP frequency distributions

The large-scale systematic variation in nucleotide composition along mammalian and avian genomes has been a focus of the debate between neutralist and selectionist views of molecular evolution. Here we test whether the compositional variation is due to mutation bias using two new tests, which do not assume compositional equilibrium. In the first test we assume a standard population genetics model, but in the second we make no assumptions about the underlying population genetics. We apply the tests to single-nucleotide polymorphism data from noncoding regions of the human genome. Both models of neutral mutation bias fit the frequency distributions of SNPs segregating in low- and medium-GC-content regions of the genome adequately, although both suggest compositional nonequilibrium. However, neither model fits the frequency distribution of SNPs from the high-GC-content regions. In contrast, a simple population genetics model that incorporates selection or biased gene conversion cannot be rejected. The results suggest that mutation biases are not solely responsible for the compositional biases found in noncoding regions.     

An overview of the TAMBEETLE model of Dendroctonus frontalis population dynamics

Population dynamics of the southern pine beetle Dendroctonus frontalis has been the subject of intensive research in the USA for more than a decade. TAMBEETLE, an acronym for the mechanistic model of population dynamics of D. frontalis , was developed to abstract contemporary knowledge on the insect. This model was developed as a joint venture between the Biosystems Research Group of Industrial Engineering and the Department of Entomology at Texas A&M University. The approach used to develop the model and the structure of component submodels is described. Research leading to the development of the model involved a series of field and laboratory studies aimed at various aspects of the population system of the insect. This research is reviewed. Important areas needing further research are identified. Studies dealing with factors involved in initiation of infestions, biome-level epidemiology, host susceptibility, stand modeling, and dynamics of within-tree mortality were emphasized as important topics for further research.     

Effect of conjugate size on the kinetics of cell-mediated cytotoxicity at the population level

We developed a model for the kinetics of target cell lysis by cytotoxic T-lymphocytes which accounts for most facts observed at the population level. In contrast to previous models, the following facts: conjugate frequency of cytotoxic T-lymphocytes bound to target cell, dependence of this frequency on the lymphocyte-to-target ratio (R), variation of R with time as target cells are destroyed, and population distributions of the different types of conjugates formed between lymphocytes and target cells, which are involved in the kinetics of these kinds of effector-target systems have been contemplated in the model. The relationship with effector-kinetic analogy models for the lytic process has been discussed. Predictions of the model have been explored and compared with experimental observations about target cell lysis reported in the literature.     

Combining mathematical models and statistical methods to understand and predict the dynamics of antibiotic-sensitive mutants in a population of resistant bacteria during experimental evolution

Temporarily discontinuing the use of antibiotics has been proposed as a means to eliminate resistant bacteria by allowing sensitive clones to sweep through the population. In this study, we monitored a tetracycline-sensitive subpopulation that emerged during experimental evolution of E. coli K12 MG1655 carrying the multiresistance plasmid pB10 in the absence of antibiotics. The fraction of tetracycline-sensitive mutants increased slowly over 500 generations from 0.1 to 7%, and loss of resistance could be attributed to a recombination event that caused deletion of the tet operon. To help understand the population dynamics of these mutants, three mathematical models were developed that took into consideration recurrent mutations, increased host fitness (selection), or a combination of both mechanisms (full model). The data were best explained by the full model, which estimated a high mutation frequency (lambda = 3.11 x 10(-5)) and a significant but small selection coefficient (sigma = 0.007). This study emphasized the combined use of experimental data, mathematical models, and statistical methods to better understand and predict the dynamics of evolving bacterial populations, more specifically the possible consequences of discontinuing the use of antibiotics.     

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.