Allele frequencies in a multidimensional Wright-Fisher model with general mutation

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 general and not necessarily symmetric. An example is given of a population in which one allele has a high mutation rate, and the others have an equal, low mutation rate. The matrix Q, with elements given by the probability of sampling two alleles of particular types, is calculated exactly and approximately for this case. A formula is given for the distribution of the number of segregating sites occurring in two randomly sampled finite sequences of completely linked sites, with general mutation at a site and identical mutation structure between sites.     

A bivariate frailty model with a cure fraction for modeling familial correlations in diseases

We suggest a cure-mixture model to analyze bivariate time-to-event data, as motivated by the article of Chatterjee and Shih (2001, Biometrics 57, 779-786), but with a simpler estimation procedure and the correlated gamma-frailty model instead of the shared gamma-frailty model. This approach allows us to deal with left-truncated and right-censored lifetime data, and accounts for heterogeneity, as well as for an insusceptible (cure) fraction in the study population. We perform a simulation study to evaluate the properties of the estimates in the proposed model and apply it to breast cancer incidence data for 5857 Swedish female monozygotic and dizygotic twin pairs from the so-called old cohort of the Swedish Twin Registry. This model is used to estimate the size of the susceptible fraction and the correlation between the frailties of the twin partners. Possible extensions, advantages, and limitations of the proposed method are discussed.     

Reciprocal sign epistasis is a necessary condition for multi-peaked fitness landscapes

Having multiple peaks within fitness landscapes critically affects the course of evolution, but whether their presence imposes specific requirements at the level of genetic interactions remains unestablished. Here we show that to exhibit multiple fitness peaks, a biological system must contain reciprocal sign epistatic interactions, which are defined as genetic changes that are separately unfavorable but jointly advantageous. Using Morse theory, we argue that it is impossible to formulate a sufficient condition for multiple peaks in terms of local genetic interactions. These findings indicate that systems incapable of reciprocal sign epistasis will always possess a single fitness peak. However, reciprocal sign epistasis should be pervasive in nature as it is a logical consequence of specificity in molecular interactions. The results thus predict that specific molecular interactions may yield multiple fitness peaks, which can be tested experimentally.     

Effects of genetic drift on variance components under a general model of epistasis

We analyze the changes in the mean and variance components of a quantitative trait caused by changes in allele frequencies, concentrating on the effects of genetic drift. We use a general representation of epistasis and dominance that allows an arbitrary relation between genotype and phenotype for any number of diallelic loci. We assume initial and final Hardy-Weinberg and linkage equilibrium in our analyses of drift-induced changes. Random drift generates transient linkage disequilibria that cause correlations between allele frequency fluctuations at different loci. However, we show that these have negligible effects, at least for interactions among small numbers of loci. Our analyses are based on diffusion approximations that summarize the effects of drift in terms of F, the inbreeding coefficient, interpreted as the expected proportional decrease in heterozygosity at each locus. For haploids, the variance of the trait mean after a population bottleneck is var(delta(z)) = sigma(n)k=1 FkV(A(k)), where n is the number of loci contributing to the trait variance, V(A(1)) = V(A) is the additive genetic variance, and V(A(k)) is the kth-order additive epistatic variance. The expected additive genetic variance after the bottleneck, denoted (V*(A)), is closely related to var(delta(z)); (V*(A)) = (1 - F) sigma(n)k=1 kFk-1V(A(k)). Thus, epistasis inflates the expected additive variance above V(A)(1 - F), the expectation under additivity. For haploids (and diploids without dominance), the expected value of every variance component is inflated by the existence of higher order interactions (e.g., third-order epistasis inflates (V*(AA. This is not true in general with diploidy, because dominance alone can reduce (V*(A)) below V(A)(1 - F) (e.g., when dominant alleles are rare). Without dominance, diploidy produces simple expressions: var(delta(z)) = sigma(n)k=1 (2F)kV(A(k)) and (V(A)) = (1 - F) sigma(n)k=1 k(2F)k-1V(A(k)). With dominance (and even without epistasis), var(delta(z)) and (V*(A)) no longer depend solely on the variance components in the base population. For small F, the expected additive variance simplifies to (V*(A)) approximately equal to (1 - F)V(A) + 4FV(AA) + 2FV(D) + 2FC(AD), where C(AD) is a sum of two terms describing covariances between additive effects and dominance and additive X dominance interactions. Whether population bottlenecks lead to expected increases in additive variance depends primarily on the ratio of nonadditive to additive genetic variance in the base population, but dominance precludes simple predictions based solely on variance components. We illustrate these results using a model in which genotypic values are drawn at random, allowing extreme and erratic epistatic interactions. Although our analyses clarify the conditions under which drift is expected to increase V(A), we question the evolutionary importance of such increases.     

Prediction of effects of genetic drift on variance components under a general model of epistasis

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.     

Equilibrium solutions of age-specific population dynamics of several species

Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L ¹ of the dynamics and for existence as well as for stability of equilibrium solutions are given.     

Human genetic diversity,a critical resource for man's future

The human gene pool displays exuberant genetic variation; this is normal for a sexual species. Even small isolated populations contain a large percentage of the total variability, emphasizing the basic genetic unity of our species. As modern man spread across the world from its African source, the genetic basis for man's unique mental acuity was retained everywhere. Nevertheless, some geographical genetic variation such as skin color, stature and physiognomy was established. These changes were biologically relatively insignificant. Most of the genetic load in the genome has been carried throughout the history of the species. There is little hope of purging all of these harmful genes; we must accept them and continue to treat their syndromes medically. All populations carry extensive genetic variation due to genes that encode variations in quantitative traits. Of greatest importance among these is ubiquitous polygenic variability in brain function and intelligence. Mental acuity is what sets us apart from the rest of the biological world. Throughout our history, genetic recombination among the many genes involved in brain function has occurred. This has provided a genetic basis for the action of natural selection that favors intelligence in meeting the demands of the environment. As environments change in the future, this type of genetic variability will continue to be a crucial resource.This article is based on a contribution at the Session on Genetic Load chaired by Dr. Henretta Trent Band and presented at a meeting of the International Society for the History, Philosophy and Social Studies of Biology, Northwestern University, Evanston, Illinois in July 1991. The author is indebted to Professor Antonio Brito daCunha of the University of São Paulo, Brazil for his encouragement and comments.     

The impact of high‐order epistasis in the within‐host fitness of a positive‐sense plant RNA virus

RNA viruses are the main source of emerging infectious diseases because of the evolutionary potential bestowed by their fast replication, large population sizes and high mutation and recombination rates. However, an equally important property, which is usually neglected, is the topography of the fitness landscape. How many fitness maxima exist and how well they are connected is especially interesting, as this determines the number of accessible evolutionary pathways. To address this question, we have reconstructed a region of the fitness landscape of tobacco etch potyvirus constituted by mutations observed during the experimental adaptation of the virus to the novel host Arabidopsis thaliana. Fitness was measured for many genotypes and showed the existence of multiple peaks and holes in the landscape. We found prevailing epistatic effects between mutations, with cases of reciprocal sign epistasis being common among pairs of mutations. We also found that high‐order epistasis was as important as pairwise epistasis in their contribution to fitness. Therefore, results suggest that the landscape was rugged due to the existence of holes caused by lethal genotypes, that a very limited number of potential neutral paths exist and that it contained a single adaptive peak.     

Nuclear–mitochondrial epistasis: a gene's eye view of genomic conflict

We use population genetic models to investigate the cooperative and conflicting synergistic fitness effects between genes from the nucleus and the mitochondrion. By varying fitness parameters, we examine the scope for conflict relative to cooperation among genomes and the utility of the “gene's eye view” analytical approach, which is based on the marginal average fitness of specific alleles. Because sexual conflict can maintain polymorphism of mitochondrial haplotypes, we can explore two types of evolutionary conflict (genomic and sexual) with one epistatic model. We find that the nuclear genetic architecture (autosomal, X‐linked, or Z‐linked) and the mating system change the regions of parameter space corresponding to the evolution by sexual and genomic conflict. For all models, regardless of conflict or cooperation, we find that population mean fitness increases monotonically as evolution proceeds. Moreover, we find that the process of gene frequency change with positive, synergistic fitnesses is self‐accelerating, as the success of an allele in one genome or in one sex increases the frequency of the interacting allele upon which its success depends. This results in runaway evolutionary dynamics caused by the positive intergenomic associations generated by selection. An inbreeding mating system tends to further accelerate these runaway dynamics because it maintains favorable host–symbiont or male–female gene combinations. In contrast, where conflict predominates, the success of an allele in one genome or in one sex diminishes the frequency of the corresponding allele in the other, resulting in considerably slower evolutionary dynamics. The rate of change of mean fitness is also much faster with positive, synergistic fitnesses and much slower where conflict is predominant. Consequently, selection rapidly fixes cooperative gene combinations, while leaving behind a slowing evolving residue of conflicting gene combinations at mutation–selection balance. We discuss how an emphasis on marginal fitness averages may obscure the interdependence of allelic fitness across genomes, making the evolutionary trajectories appear independent of one another when they are not.     

A branching-process model for the evolution of transposable elements incorporating selection

We have formulated a very general mathematical model to analyze the evolution of transposable genetic elements in prokaryotic populations. Transposable genetic elements are DNA sequences able to replicate and insert copies of themselves at new locations in the genome. This work characterizes the equilibrium distribution of copy number under the influence of copy number-dependent selection, transposition and deletion. Our principal results concern the equilibrium distribution of copy number in response to various selective regimes. For particular transposition patterns (e.g. unregulated transposition or copy number-dependent transposition), equilibrium distributions are calculated numerically for a variety of specific selection patterns. Selection is quantified through specification of the expected number of offspring for individuals of each type, which is generally a non-increasing function of copy number, in accord with the usual evolutionary speculations.     

A model for the involvement of Okazaki fragments maturation in the expansion of short tandem repeats

Mutations were accumulated with a wide variety in the p53 pseudogene of various wild mouse species and subspecies captured at different localities, as extensively observed in the exon 4 – exon 5 region. The rate of mutation accumulation in the mouse p53 pseudogene was estimated to be 1.4–2.1×10⁻⁸ mutations/bp/year, which is 20–30 times faster than that of the functional p53 and makes the dating possible for the time range of 10⁶ years or more. From comparison of the mutation spectrum, the origin of laboratory mice was identified to one of two M. m. domesticus groups.     

A branching process model for the evolution of transposable elements

A discrete-time multitype branching process model is presented for the evolution of transposable elements in haploid populations. An individual is classified as type i if it possesses i copies of the TE, i0. The general model incorporates copy-dependent selection and transposition, and recursion relations are derived for the distribution of the number of individuals of the various types. The asymptotic relative proportions of individuals of the different types is studied in the neutral case. The behavior of this equilibrium distribution is examined for various patterns of regulated transposition and deletion.     

EvoluZion: a computer simulator for teaching genetic and evolutionary concepts

EvoluZion is a forward-in-time genetic simulator developed in Java and designed to perform real time simulations on the evolutionary history of virtual organisms. These model organisms harbour a set of 13 genes that codify an equal number of phenotypic features. These genes change randomly during replication, and mutant genes can have null, positive or negative effects on the organisms’ fitness, allowing to model effects of both selection pressures and drift on gene evolution. There are two versions of this program: version 1.6.x_haploid; focused on macroevolutionary events and depicting prokaryote-like organisms, and version 2.3.x_diploid that simulate diploid, sexually reproducing organisms, and it is more adequate to teach micro-evolution as well as key genetic concepts such as Mendel’s laws, epistasis, genetic linkage, genetic mapping among others. Different data sets can be collected periodically during running in order to perform further analyses. In addition, the complete genealogy of extant as well as extinct organisms can be recorded. EvoluZion is well suited for teaching evolutionary biology concepts to students of all levels in a pedagogic way. This is mainly due to three main program features: (i) its intuitive and simple graphical interface (ii) a visualisation similar to videogames (iii) flexible integration of a wide range of biological phenomena into a single simulation.     

Detecting the genomic signal of polygenic adaptation and the role of epistasis in evolution

Over the last decade, the genomic revolution has offered the possibility to generate tremendous amounts of data that contain valuable information on the genetic basis of phenotypic traits, such as those linked to human diseases or those that allow for species to adapt to a changing environment. Most ecologically relevant traits are controlled by a large number of genes with small individual effects on trait variation, but that are connected with one another through complex developmental, metabolic and biochemical networks. As a result, it has recently been suggested that most adaptation events in natural populations are reached via correlated changes at multiple genes at a time, for which the name polygenic adaptation has been coined. The current challenge is to develop methods to extract the relevant information from genomic data to detect the signature of polygenic evolutionary change. The symposium entitled “Detecting the Genomic Signal of Polygenic Adaptation and the Role of Epistasis in Evolution” held in 2017 at the University of Zürich aimed at reviewing our current state of knowledge. In this review, we use the talks of the invited speakers to summarize some of the most recent developments in this field.     

An evolutionary perspective on pathogenic mtDNA mutations: haplogroup associations of clinical disorders

More than 75 human diseases have been associated with mitochondrial dysfunction, and many of these are directly caused by overtly pathogenic mutations in the mitochondrial genome (mtDNA). In addition, there have been a number of reports that posit a different, subtler role for mtDNA substitutions in the disease process. As we review here, mtDNA evolution has resulted in the distribution of sequences into continent-specific haplogroups, which are defined by a relatively small number of polymorphisms. Thus, mtDNA sequences can be assigned to European, African, or Asian/Native American haplogroups. There are numerous reports that various diseases are haplogroup-associated, and it has been suggested that some of these haplogroup-associated polymorphisms act as risk factors in these disorders. It has also been suggested that there are haplogroup-associations for aging. As we note here, however, such associations have usually been observed only in single studies and it is difficult to draw broad conclusions on the basis of the available evidence. At a minimum, we suggest that, a haplogroup-group association must be detected in multiple subpopulations or in a large, carefully controlled population survey.     

IS LIFE IMPOSSIBLE? INFORMATION,SEX, AND THE ORIGIN OF COMPLEX ORGANISMS

The earliest organisms are thought to have had high mutation rates. It has been asserted that these high mutation rates would have severely limited the information content of early genomes. This has led to a well‐known “paradox” because, in contemporary organisms, the mechanisms that suppress mutations are quite complex and a substantial amount of information is required to construct these mechanisms. The paradox arises because it is not clear how efficient error‐suppressing mechanisms could have evolved, and thus allowed the evolution of complex organisms, at a time when mutation rates were too high to permit the maintenance of very substantial amounts of information within genomes. Here, we use concepts from the formal theory of information to calculate the amount of genomic information that can be maintained. We identify conditions under which much higher levels of genomic information can be maintained than previously considered possible among origin‐of‐life researchers. In particular, we find that the highest levels of information are maintained when many genotypes produce identical phenotypes, and when reproduction occasionally involves recombination between multiple parental genomes. There is a good reason to believe that these conditions are relevant for very early organisms, and thus the results presented may provide a solution to a long‐standing logical problem associated with the early evolution of life.     

Perspectives for the use of structural information and chemical genetics to develop inhibitors of Janus kinases

Gain-of-function mutations in the genes encoding Janus kinases have been discovered in various haematologic diseases. Jaks are composed of a FERM domain, an SH2 domain, a pseudokinase domain and a kinase domain, and a complex interplay of the Jak domains is involved in regulation of catalytic activity and association to cytokine receptors. Most activating mutations are found in the pseudokinase domain. Here we present recently discovered mutations in the context of our structural models of the respective domains. We describe two structural hotspots in the pseudokinase domain of Jak2 that seem to be associated either to myeloproliferation or to lymphoblastic leukaemia, pointing at the involvement of distinct signalling complexes in these disease settings. The different domains of Jaks are discussed as potential drug targets. We present currently available inhibitors targeting Jaks and indicate structural differences in the kinase domains of the different Jaks that may be exploited in the development of specific inhibitors. Moreover, we discuss recent chemical genetic approaches which can be applied to Jaks to better understand the role of these kinases in their biological settings and as drug targets.     

Homoplasy and mutation model at microsatellite loci and their consequences for population genetics analysis

Homoplasy has recently attracted the attention of population geneticists, as a consequence of the popularity of highly variable stepwise mutating markers such as microsatellites. Microsatellite alleles generally refer to DNA fragments of different size (electromorphs). Electromorphs are identical in state (i.e. have identical size), but are not necessarily identical by descent due to convergent mutation(s). Homoplasy occurring at microsatellites is thus referred to as size homoplasy. Using new analytical developments and computer simulations, we first evaluate the effect of the mutation rate, the mutation model, the effective population size and the time of divergence between populations on size homoplasy at the within and between population levels. We then review the few experimental studies that used various molecular techniques to detect size homoplasious events at some microsatellite loci. The relationship between this molecularly accessible size homoplasy size and the actual amount of size homoplasy is not trivial, the former being considerably influenced by the molecular structure of microsatellite core sequences. In a third section, we show that homoplasy at microsatellite electromorphs does not represent a significant problem for many types of population genetics analyses realized by molecular ecologists, the large amount of variability at microsatellite loci often compensating for their homoplasious evolution. The situations where size homoplasy may be more problematic involve high mutation rates and large population sizes together with strong allele size constraints.