We study the evolution of a population in a two-locus genotype space, in which the negative effects of two single mutations are overcompensated in a high fitness double mutant. We discuss how the interplay of finite population size, $N$, and sexual recombination at rate $r$ affects the escape times $t_\mathrm{esc}$ to the double mutant. For small populations demographic noise generates massive fluctuations in $t_\mathrm{esc}$. The mean escape time varies non-monotonically with $r$, and grows exponentially as $\ln t_{\mathrm{esc}} \sim N(r - r^\ast)^{3/2}$ beyond a critical value $r^\ast$.
The evolutionary effect of recombination depends crucially on the epistatic interactions between linked loci. A paradigmatic case where recombination is known to be strongly disadvantageous is a two-locus fitness landscape dis- playing reciprocal sign epistasis with two fitness peaks of unequal height. Focusing on the occurrence of bistability in the equilibrium solutions, we consider here the deterministic, haploid two-locus model with reversible mu- tations, selection and recombination. We find analytic formulae for the criti- cal recombination probability rc above which two stable stationar…
We study the evolutionary dynamics of a haploid population of infinite size recombining with a probability $r$ in a two locus model. Starting from a low fitness locus, the population is evolved under mutation, selection and recombination until a finite fraction of the population reaches the fittest locus. An analytical method is developed to calculate the fixation time $T$ to the fittest locus for various choices of epistasis. We find that (1) for negative epistasis, $T$ decreases slowly for small $r$ but decays fast at larger $r$ (2) for positive epistasis, $T$ increases linearly for small $r…
Known results on the moments of the distribution generated by the two-locus Wright-Fisher diffusion model and a duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical methods for computing moments by a Markov chain Monte Carlo and a method to compute closed-form expressions of the moments are presented. By using the duality argument properties of the ancestral recombination graph are studied in terms of the moments.
Methods to effectively detect multi-locus genetic association are becoming increasingly relevant in the genetic dissection of complex trait in humans. Current approaches typically consider a limited number of hypotheses, most of which are related to the effect of a single locus or of a relatively small number of neighboring loci on a chromosomal region. We have developed a novel method that is specifically designed to detect genetic association involving multiple disease-susceptibility loci, possibly on different chromosomes. Our approach relies on the efficient discovery of patterns comprisin…
A precise estimate of allele and haplotype polymorphism is of great interest in theoretical population genetics, but also has practical applications, such as bone marrow registries management. Allele polymorphism is driven mainly by point mutations, while haplotype polymorphism is also affected by recombination. Current estimates treat recombination as mutations in an infinite site model. We here show that even in the simple case of two loci in a haploid individual, for a finite population, most recombination events produce existing haplotypes, and as such are silent. Silent recombination cons…
We study existence and stability of stationary solutions of a system of semilinear parabolic partial differential equations that occurs in population genetics. It describes the evolution of gamete frequencies in a geographically structured population of migrating individuals in a bounded habitat. Fitness of individuals is determined additively by two recombining, diallelic genetic loci that are subject to spatially varying selection. Migration is modeled by diffusion. Of most interest are spatially non-constant stationary solutions, so-called clines. In a two-locus cline all four gametes are p…
We have analyzed the relations between the mutational pressure, recombination and selection pressure in the bit-string model with sexual reproduction. For specific sets of these parameters we have found three phase transitions with one phase where populations can survive. In this phase, recombination enhances the survival probability. Even if recombination is associated, to some extent, with additional mutations it could be advantageous to reproduction, indicating that the frequencies of recombinations and recombination-associated mutations can self-organize in Nature. Partitioning the diploid…
Motivation: Recombination rates vary considerably at the fine scale within mammalian genomes, with the majority of recombination occurring within hotspots of ~2 kb in width. We present a method for inferring the location of recombination hotspots from patterns of linkage disequilibrium within samples of population genetic data. Results: Using simulations, we show that our method has hotspot detection power of approximately 50-60%, but depending on the magnitude of the hotspot. The false positive rate is between 0.24 and 0.56 false positives per Mb for data typical of humans. Availability: http…
A number of studies have suggested using comparisons between DNA sequences of closely related bacterial isolates to estimate the relative rate of recombination to mutation for that bacterial species. We consider such an approach which uses single-locus variants: pairs of isolates whose DNA differ at a single gene locus. One way of deriving point estimates for the relative rate of recombination to mutation from such data is to use composite likelihood methods. We extend recent work in this area so as to be able to construct confidence intervals for our estimates, without needing to resort to co…
Stronger selection implies faster evolution---that is, the greater the force, the faster the change. This apparently self-evident proposition, however, is derived under the assumption that genetic variation within a population is primarily supplied by mutation (i.e.\ mutation-driven evolution). Here, we show that this proposition does not actually hold for recombination-driven evolution, i.e.\ evolution in which genetic variation is primarily created by recombination rather than mutation. By numerically investigating population genetics models of recombination, migration and selection, we demo…
Maintenance of sexual reproduction and genetic recombination imposes physiological costs when compared to parthenogenic reproduction, most prominently: for maintaining the corresponding (molecular) machinery, for finding a mating partner, and through the decreased fraction of females in a population, which decreases the reproductive capacity. Based on principles from information theory, we have previously developed a new population genetic model, and applying it in simulations, we have recently hypothesized that all species maintain the maximum genomic complexity that is required by their nich…
Two-locus sampling probabilities have played a central role in devising an efficient composite likelihood method for estimating fine-scale recombination rates. Due to mathematical and computational challenges, these sampling probabilities are typically computed under the unrealistic assumption of a constant population size, and simulation studies have shown that resulting recombination rate estimates can be severely biased in certain cases of historical population size changes. To alleviate this problem, we develop here new methods to compute the sampling probability for variable population si…
The identification of the H3K4 trimethylase, PRDM9, as the gene responsible for recombination hotspot localization has provided considerable insight into the mechanisms by which recombination is initiated in mammals. However, uniquely amongst mammals, canids appear to lack a functional version of PRDM9 and may therefore provide a model for understanding recombination that occurs in the absence of PRDM9, and thus how PRDM9 functions to shape the recombination landscape. We have constructed a fine-scale genetic map from patterns of linkage disequilibrium assessed using high-throughput sequence d…
Bacteria are known to exchange genetic information by horizontal gene transfer. Since the frequency of homologous recombination depends on the similarity of recombining segments, several studies examined whether this could lead to the emergence of subspecies. Most of them used simulations of fixed size Wright-Fisher populations, in which the genetic drift should be taken into account. Here, we use non-linear Markov chains to describe a bacterial population evolving under mutation and recombination processes. We consider a population as a space of genomes and a probability measure on it. Thus, …
The persistence of life requires populations to adapt at a rate commensurate with the dynamics of their environment. Successful populations that inhabit highly variable environments have evolved mechanisms to increase the likelihood of successful adaptation. We introduce a $64 \times 64$ matrix to quantify base-specific mutation potential, analyzing four different replicative systems, error-prone PCR, mouse antibodies, a nematode, and Drosophila. Mutational tendencies are correlated with the structural evolution of proteins. In systems under strong selective pressure, mutational biases are sho…
We study the role of recombination, as practiced by genetically-competent bacteria, in speeding up Darwinian evolution. This is done by adding a new process to a previously-studied Markov model of evolution on a smooth fitness landscape; this new process allows alleles to be exchanged with those in the surrounding medium. Our results, both numerical and analytic, indicate that for a wide range of intermediate population sizes, recombination dramatically speeds up the evolutionary advance.
Conventional population genetics considers the evolution of a limited number of genotypes corresponding to phenotypes with different fitness. As model phenotypes, in particular RNA secondary structure, have become computationally tractable, however, it has become apparent that the context dependent effect of mutations and the many-to-one nature inherent in these genotype-phenotype maps can have fundamental evolutionary consequences. It has previously been demonstrated that populations of genotypes evolving on the neutral networks corresponding to all genotypes with the same secondary structure…
We investigate a multi-locus evolutionary model which is based on the DNA shuffling protocol widely applied in \textit{in vitro} directed evolution. This model incorporates selection, recombination and point mutations. The simplicity of the model allows us to obtain a full analytical treatment of both its dynamical and equilibrium properties, for the case of an infinite population. We also briefly discuss finite population size corrections.
In sexual population, recombination reshuffles genetic variation and produces novel combinations of existing alleles, while selection amplifies the fittest genotypes in the population. If recombination is more rapid than selection, populations consist of a diverse mixture of many genotypes, as is observed in many populations. In the opposite regime, which is realized for example in the facultatively sexual populations that outcross in only a fraction of reproductive cycles, selection can amplify individual genotypes into large clones. Such clones emerge when the fitness advantage of some of th…
Genomes evolve as modules. In prokaryotes (and some eukaryotes), genetic material can be transferred between species and integrated into the genome via homologous or illegitimate recombination. There is little reason to imagine that the units of transfer correspond to entire genes; however, such units have not been rigorously characterized. We examined fragmentary genetic transfers in single-copy gene families from 144 prokaryotic genomes and found that breakpoints are located significantly closer to the boundaries of genomic regions that encode annotated structural domains of proteins than ex…
Pathogenic bacteria present a large disease burden on human health. Control of these pathogens is hampered by rampant lateral gene transfer, whereby pathogenic strains may acquire genes conferring resistance to common antibiotics. Here we introduce tools from topological data analysis to characterize the frequency and scale of lateral gene transfer in bacteria, focusing on a set of pathogens of significant public health relevance. As a case study, we examine the spread of antibiotic resistance in Staphylococcus aureus. Finally, we consider the possible role of the human microbiome as a reservo…
We consider how transfer of genetic information between individuals influences the phase diagram and mean fitness of both the Eigen and the parallel, or Crow-Kimura, models of evolution. In the absence of genetic transfer, these physical models of evolution consider the replication and point mutation of the genomes of independent individuals in a large population. A phase transition occurs, such that below a critical mutation rate an identifiable quasispecies forms. We generalize these models of quasispecies evolution to include horizontal gene transfer. We show how transfer of genetic informa…
How self-incompatibility systems are maintained in plant populations is still a debated issue. Theoretical models predict that self-incompatibility systems break down according to the intensity of inbreeding depression and number of S-alleles. Other studies have explored the function of asexual reproduction in the maintenance of self-incompatibility. However, the population genetics of partially asexual, self-incompatible populations are poorly understood and previous studies have failed to consider all possible effects of asexual reproduction or could only speculate on those effects. In this …
Numerous traits under migration-selection balance are shown to exhibit complex patterns of genetic architecture with large variance in effect sizes. However, the conditions under which such genetic architectures are stable have yet to be investigated, because studying the influence of a large number of small allelic effects on the maintenance of spatial polymorphism is mathematically challenging, due to the high complexity of the systems that arise. In particular, in the most simple case of a haploid population in a two-patch environment, while it is known from population genetics that polymor…
For a genetic locus carrying a strongly beneficial allele which has just fixed in a large population, we study the ancestry at a linked neutral locus. During this ``selective sweep'' the linkage between the two loci is broken up by recombination and the ancestry at the neutral locus is modeled by a structured coalescent in a random background. For large selection coefficients $α$ and under an appropriate scaling of the recombination rate, we derive a sampling formula with an order of accuracy of $\mathcal{O}((\log α)^{-2})$ in probability. In particular we see that, with this order of accuracy…
The genetic composition of a naturally developing population is considered as due to mutation, selection, genetic drift and recombination. Selection is modeled as single-locus terms (additive fitness) and two-loci terms (pairwise epistatic fitness). The problem is posed to infer epistatic fitness from population-wide whole-genome data from a time series of a developing population. We generate such data in silico, and show that in the Quasi-Linkage Equilibrium (QLE) phase of Kimura, Neher and Shraiman, that pertains at high enough recombination rates and low enough mutation rates, epistatic fit…
Solving the recombination equation has been a long-standing challenge of \emph{deterministic} population genetics. We review recent progress obtained by introducing ancestral processes, as traditionally used in the context of \emph{stochastic} models of population genetics, into the deterministic setting. With the help of an ancestral partitioning process, which is obtained by letting population size tend to infinity (without rescaling parameters or time) in an ancestral recombination graph, we obtain the solution to the recombination equation in a transparent form.
The evolutionary dynamics of HIV during the chronic phase of infection is driven by the host immune response and by selective pressures exerted through drug treatment. To understand and model the evolution of HIV quantitatively, the parameters governing genetic diversification and the strength of selection need to be known. While mutation rates can be measured in single replication cycles, the relevant effective recombination rate depends on the probability of coinfection of a cell with more than one virus and can only be inferred from population data. However, most population genetic estimato…
Recombination plays a fundamental role in meiosis, ensuring the proper segregation of chromosomes and contributing to genetic diversity by generating novel combinations of alleles. Using data derived from directUtoUconsumer genetic testing, we investigated patterns of recombination in over 4,200 families. Our analysis revealed a number of sex differences in the distribution of recombination. We find the fraction of male events occurring within hotspots to be 4.6% higher than for females. We confirm that the recombination rate increases with maternal age, while hotspot usage decreases, with no …
Recently, much attention has been given to understanding recombination events along a chromosome in a variety of field. For instance, many population genetics problems are limited by the inaccuracy of inferred evolutionary histories of chromosomes sampled randomly from a population. This evolutionary history differs among genomic locations as an artifact of recombination events along a chromosome. Thus, much recent attention has been focused on identifying these recombination points. However, many proposed methods either make simplifying, but unrealistic, assumptions about recombination along …
We study the stationary state of a population evolving under the action of random genetic drift, selection and recombination in which both deleterious and reverse beneficial mutations can occur. We find that the equilibrium fraction of deleterious mutations decreases as the population size is increased. We calculate exactly the steady state frequency in a nonrecombining population when population size is infinite and for a neutral finite population, and obtain bounds on the fraction of deleterious mutations. We also find that for small and very large populations, the number of deleterious muta…
A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evolutionary history. In this paper, we develop theoretical foundations for this approach. First, we present a novel formulation of the underlying inference problem. Specifically, we introduce and study the novelty profile, a simple, stable statistic of an evolutionary history which not only counts recombination events but also quantifies how recombination creat…
For population genetics models with recombination, obtaining an exact, analytic sampling distribution has remained a challenging open problem for several decades. Recently, a new perspective based on asymptotic series has been introduced to make progress on this problem. Specifically, closed-form expressions have been derived for the first few terms in an asymptotic expansion of the two-locus sampling distribution when the recombination rate $ρ$ is moderate to large. In this paper, a new computational technique is developed for finding the asymptotic expansion to an arbitrary order. Computatio…
We consider a population evolving due to mutation, selection and recombination, where selection includes single-locus terms (additive fitness) and two-loci terms (pairwise epistatic fitness). We further consider the problem of inferring fitness in the evolutionary dynamics from one or several snap-shots of the distribution of genotypes in the population. In the recent literature this has been done by applying the Quasi-Linkage Equilibrium (QLE) regime first obtained by Kimura in the limit of high recombination. Here we show that the approach also works in the interesting regime where the effec…
This work presents a population genetic model of evolution, which includes haploid selection, mutation, recombination, and drift. The mutation-selection equilibrium can be expressed exactly in closed form for arbitrary fitness functions without resorting to diffusion approximations. Tractability is achieved by generating new offspring using n-parent rather than 2-parent recombination. While this enforces linkage equilibrium among offspring, it allows analysis of the whole population under linkage disequilibrium. We derive a general and exact relationship between fitness fluctuations and respon…
Large populations may contain numerous simultaneously segregating polymorphisms subject to natural selection. Since selection acts on individuals whose fitness depends on many loci, different loci affect each other's dynamics. This leads to stochastic fluctuations of allele frequencies above and beyond genetic drift - an effect known as genetic draft. Since recombination disrupts associations between alleles, draft is strong when recombination is rare. Here, we study a facultatively outcrossing population in a regime where the frequency of out-crossing and recombination, r, is small compared t…
The rate of recombination affects the mode of molecular evolution. In high-recombining sequence, the targets of selection are individual genetic loci; under low recombination, selection collectively acts on large, genetically linked genomic segments. Selection under linkage can induce clonal interference, a specific mode of evolution by competition of genetic clades within a population. This mode is well known in asexually evolving microbes, but has not been traced systematically in an obligate sexual organism. Here we show that the Drosophila genome is partitioned into two modes of evolution:…
Biochemical and regulatory interactions central to biological networks are expected to cause extensive genetic interactions or epistasis affecting the heritability of complex traits and the distribution of genotypes in populations. However, the inference of epistasis from the observed phenotype-genotype correlation is impeded by statistical difficulties, while the theoretical understanding of the effects of epistasis remains limited, in turn limiting our ability to interpret data. Of particular interest is the biologically relevant situation of numerous interacting genetic loci with small indi…
We consider a population subdivided into two demes connected by migration in which selection acts in opposite direction. We explore the effects of recombination and migration on the maintenance of multilocus polymorphism, on local adaptation, and on differentiation by employing a deterministic model with genic selection on two linked diallelic loci (i.e., no dominance or epistasis). For the following cases, we characterize explicitly the possible equilibrium configurations: weak, strong, highly asymmetric, and super-symmetric migration, no or weak recombination, and independent or strongly rec…
Understanding patterns of selectively neutral genetic variation is essential in order to model deviations from neutrality, caused for example by different forms of selection. Best understood is neutral genetic variation at a single locus, but additional insights can be gained by investigating genetic variation at multiple loci. The corresponding patterns of variation reflect linkage disequilibrium and provide information about the underlying multi-locus gene genealogies. The statistical properties of two-locus genealogies have been intensively studied for populations of constant census size, a…
This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models are most appropriate if the dispersal distance is short compared to the scale at which the environment changes, or if the habitat is fragmented. The general goal of such models is to study the influence of population subdivision and gene flow among subpopulations on the amount and pattern of genetic variation maintained. Only deterministic models are treated. Because space is discrete, they are formulated in terms of systems o…
In sexual populations, selection operates neither on the whole genome, which is repeatedly taken apart and reassembled by recombination, nor on individual alleles that are tightly linked to the chromosomal neighborhood. The resulting interference between linked alleles reduces the efficiency of selection and distorts patterns of genetic diversity. Inference of evolutionary history from diversity shaped by linked selection requires an understanding of these patterns. Here, we present a simple but powerful scaling analysis identifying the unit of selection as the genomic "linkage block" with a c…
This contribution is concerned with mathematical models for the dynamics of the genetic composition of populations evolving under recombination. Recombination is the genetic mechanism by which two parent individuals create the mixed type of their offspring during sexual reproduction. The corresponding models are large, nonlinear dynamical systems (for the deterministic treatment that applies in the infinite-population limit), or interacting particle systems (for the stochastic treatment required for finite populations). We review recent progress on these difficult problems. In particular, we p…
The shape of allele-frequency clines maintained by migration-selection balance depends not only on the properties of migration and selection, but also on the dominance relations among alleles and on linkage to other loci under selection. We investigate a two-locus model in which two diallelic, recombining loci are subject to selection caused by an abrupt environmental change. The habitat is one-dimensional and unbounded, selection at each locus is modeled by step functions such that in one region one allele at each locus is advantageous and in the other deleterious. We admit an environmentally…
We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasi-species theory to modern biology are anal…
The dynamics of recombination in genetics leads to an interesting nonlinear differential equation, which has a natural generalization to a measure valued version. The latter can be solved explicitly under rather general circumstances. It admits a closed formula for the semigroup of nonlinear positive operators that emerges from the forward flow and is, in general, embedded in a multi-parameter semigroup.
Genetic contribution of advantaged ancestors in the biparental Moran model -- finite selection
Université Paris-Saclay, AgroParisTech, INRAE, UMR MIA Paris-Saclay（巴黎萨克雷大学、农业巴黎技术学院、法国国家农业食品环境研究院）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学、法国国家科学研究中心）
The key findings of classical population genetics are derived using a framework based on information theory using the entropies of the allele frequency distribution as a basis. The common results for drift, mutation, selection, and gene flow will be rewritten both in terms of information theoretic measurements and used to draw the classic conclusions for balance conditions and common features of one locus dynamics. Linkage disequilibrium will also be discussed including the relationship between mutual information and r^2 and a simple model of hitchhiking.
Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genea…
We discuss two different ways of chromosomes' and genomes' evolution. Purifying selection dominates in large panmictic populations, where Mendelian law of independent gene assortment is valid. If the populations are small, recombination processes are not effective enough to ensure an independent assortment of linked genes and larger clusters of genes could be inherited as the genetic units. There are whole clusters of genes which tend to complement in such conditions instead of single pairs of alleles like in the case of purifying selection. Computer simulations have shown that switching in-be…
Homologous recombination is an important operator in the evolution of biological organisms. However, there is still no clear, generally accepted understanding of why it exists and under what circumstances it is useful. In this paper we consider its utility in the context of an infinite population haploid model with selection and homologous recombination. We define utility in terms of two metrics - the increase in frequency of fit genotypes, and the increase in average population fitness, relative to those associated with selection only. Explicitly, we exhaustively explore the eight-dimensional…
