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Notice à Reprendre pas de Clé UT au 12/12/2019; International audience; Detecting genomic regions under selection is an important objective of population genetics. Typical analyses for this goal are based on exploiting genetic diversity patterns in present time data but rapid advances in DNA sequencing have increased the availability of time series genomic data. A common approach to analyze such data is to model the temporal evolution of an allele frequency as a Markov chain. Based on this principle, several methods have been proposed to infer selection intensity. One of their differences lies in how they model the transition probabilities of the Markov chain. Using the Wright- Fisher model is a natural choice but its computational cost is prohibitive for large population sizes so approximations to this model based on parametric distributions have been proposed. Here, we compared the performance of some of these approximations with respect to their power to detect selection and their estimation of the selection coefficient. We developped a new generic Hidden Markov Model likelihood calculator and applied it on genetic time series simulated under various evolutionary scenarios. The Beta with spikes approximation, which combines discrete fixation probabilities with a continuous Beta distribution, was found to perform consistently better than the others. This distribution provides an almost perfect fit to the Wright-Fisher model in terms of selection inference, for a computational cost that does not increase with population size. We further evaluated this model for population sizes not accessible to the Wright-Fisher model and illustrated its performance on a dataset of two divergently selected chicken populations.

In this chapter we review the application of neutral theory to archaeological case studies. In particular, we focus on commonly used neutrality tests and discuss the interpretation of their results. We argue that one needs to be careful with interpreting the consistency (or inconsistency) of archaeological data with theoretical neutral expectations as evidence for the absence (or presence) of selective force in the population. While neutral theory does very well in replicating various population-level frequency patterns, many alternative processes of cultural transmission may do so, too, making the pattern-matching approach a fairly weak approach for inferring underlying transmission processes. We advocate the use of generative inference frameworks as one possibility to go beyond the hypothesis testing approach. These frameworks do not only allow for the identification of the most likely underlying learning process given the empirical data but for a description of the breath of processes that could have produced these data equally well. This brings us a step closer to understanding the limits to inferring underlying transmission processes from population-level frequency data and therefore the kinds of questions can be answered with which kinds of data.

The two-parameter Poisson--Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman's one-parameter Poisson--Dirichlet distribution and to certain Fleming--Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a $K$-allele Wright--Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $N\to\infty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson--Dirichlet diffusion as $K\to\infty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove \textit{a priori} that in the limit there is no "loss of mass", i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one., Comment: To appear in The Annals of Applied Probability

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.

We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright‒Fisher model, as well as a seed bank model with bounded age distribution considered in Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that, for age distributions with finite mean, the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Furthermore, we present a construction of the forward-in-time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced in Kaj, Krone and Lascoux (2001) as well as on a paper by Hammond and Sheffield (2013).

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to Pearson diffusions. Pearson diffusions have stationary distributions of Pearson type (i.e. normal, gamma, beta, Fisher- Snedecor, reciprocal gamma or Student stationary distribution). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn- scheme model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with nonindependent increments. According to the carefully chosen starting Markov chain, different limiting Pearson diffusions can be obtained. We present Ornstein-Uhlenbeck (OU) process, Cox- Ingersol- Ross (CIR) and Jacobi diffusion as the limiting Pearson diffusions.

Continuous time random walks have random waiting times between particle jumps. We define\ud the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions\ud (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme\ud model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but\ud diffusion processes with non-independent increments. The waiting times are selected from the domain of\ud attraction of a stable law.

We identify a new natural coalescent structure, which we call the seed-bank coalescent, that describes the gene genealogy of populations under the influence of a strong seed-bank effect, where "dormant forms" of individuals (such as seeds or spores) may jump a significant number of generations before joining the "active" population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright-Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size $N$. This extends earlier results of Kaj, Krone and Lascoux [J. Appl. Probab. 38 (2011) 285-300] who show that the genealogy of a Wright-Fisher model in the presence of a "weak" seed-bank effect is given by a suitably time-changed Kingman coalescent. The qualitatively new feature of the seed-bank coalescent is that ancestral lineages are independently blocked at a certain rate from taking part in coalescence events, thus strongly altering the predictions of classical coalescent models. In particular, the seed-bank coalescent "does not come down from infinity," and the time to the most recent common ancestor of a sample of size $n$ grows like $\log\log n$. This is in line with the empirical observation that seed-banks drastically increase genetic variability in a population and indicates how they may serve as a buffer against other evolutionary forces such as genetic drift and selection., Published at http://dx.doi.org/10.1214/15-AAP1106 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The joint and accurate inference of selection and demography from genetic data is considered a particularly challenging question in population genetics, since both process may lead to very similar patterns of genetic diversity. However, additional information for disentangling these effects may be obtained by observing changes in allele frequencies over multiple time points. Such data are common in experimental evolution studies, as well as in the comparison of ancient and contemporary samples. Leveraging this information, however, has been computationally challenging, particularly when considering multilocus data sets. To overcome these issues, we introduce a novel, discrete approximation for diffusion processes, termed mean transition time approximation, which preserves the long-term behavior of the underlying continuous diffusion process. We then derive this approximation for the particular case of inferring selection and demography from time series data under the classic Wright–Fisher model and demonstrate that our approximation is well suited to describe allele trajectories through time, even when only a few states are used. We then develop a Bayesian inference approach to jointly infer the population size and locus-specific selection coefficients with high accuracy and further extend this model to also infer the rates of sequencing errors and mutations. We finally apply our approach to recent experimental data on the evolution of drug resistance in influenza virus, identifying likely targets of selection and finding evidence for much larger viral population sizes than previously reported.

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term ( n 2 ) where n denotes the number of lineages present in the genealogy. These two standard constraints of Kingman’s coalescent, obtained in the limit of a large population size, approximate the exact ancestral process of Wright-Fisher or Moran models under appropriate parameterization. Calculation of coalescence event probabilities with higher accuracy quantifies the dependence of sample and population sizes that adhere to Kingman’s coalescent process. The convention that probabilities of leading order N − 2 are negligible provided n ≪ N is examined at key stages of the mathematical derivation. Empirically, expected genealogical parity of the single-pair restricted Wright-Fisher haploid model exceeds 99% where n ≤ 1 2 N 3 ; similarly, per expected interval where n ≤ 1 2 N / 6 . The fractional cubic root criterion is practicable, since although it corresponds to perfect parity and to an extent confounds identifiability it also accords with manageable conditional probabilities of multi-coalescence.

We study a generalization of the Wright--Fisher model in which some individuals adopt a behavior that is harmful to others without any direct advantage for themselves. This model is motivated by studies of spiteful behavior in nature, including several species of parasitoid hymenoptera in which sperm-depleted males continue to mate de- spite not being fertile. We first study a single reproductive season, then use it as a building block for a generalized Wright--Fisher model. In the large population limit, for male-skewed sex ratios, we rigorously derive the convergence of the renormalized process to a diffusion with a frequency-dependent selection and genetic drift. This allows a quantitative comparison of the indirect selective advantage with the direct one classically considered in the Wright--Fisher model. From the mathematical point of view, each season is modeled by a mix between samplings with and without replacement, and analyzed by a sort of "reverse numerical analysis", viewing a key recurrence relation as a discretization scheme for a PDE. The diffusion approximation is then obtained by classical methods.

Whales have 1000-fold more cells than humans and mice have 1000-fold fewer; however, cancer risk across species does not increase with the number of somatic cells and the lifespan of the organism. This observation is known as Peto's paradox. How much would evolution have to change the parameters of somatic evolution in order to equalize the cancer risk between species that differ by orders of magnitude in size? Analysis of previously published models of colorectal cancer suggests that a two- to three-fold decrease in the mutation rate or stem cell division rate is enough to reduce a whale's cancer risk to that of a human. Similarly, the addition of one to two required tumour-suppressor gene mutations would also be sufficient. We surveyed mammalian genomes and did not find a positive correlation of tumour-suppressor genes with increasing body mass and longevity. However, we found evidence of the amplification of TP53 in elephants, MAL in horses and FBXO31 in microbats, which might explain Peto's paradox in those species. Exploring parameters that evolution may have fine-tuned in large, long-lived organisms will help guide future experiments to reveal the underlying biology responsible for Peto's paradox and guide cancer prevention in humans.

Although the attempt of sociobiology to provide a unified account of human biological and cultural evolution foundered on the sheer plasticity of human behaviour, subsequent attempts to rescue the relevance of evolutionary biology for the study of cultural phenomena have proved fruitful. A—perhaps the—major driver of this success has been the development of dual inheritance models (Boyd and Richerson 1985) that explicitly acknowledge the operation of both genetic and non- genetic inheritance in human sociocultural evolution. The development of such models has come about through, on the one hand, a willingness to apply Darwinian “population thinking” to social and cultural phenomena and, on the other, the recognition that biological genetic evolution is but one specific example of a more general ‘algorithm’ applicable to several domain beyond biology (Hull, 1982; Dennett, 1995). Nevertheless, there remains considerable debate about whether cultural change is an evolutionary process operating on the same kinds of functional entities (replicators, interactors and lineages), or whether it is simply an analogous process. A common refrain in the debate about the status of cultural change as an evolutionary process, voiced by biologists and social scientists (Gould, 1987), is that if cultural change is evolutionary then it is Lamarckian rather that Darwinian. None of the protagonists believe that cultural evolution is literally Lamarckian in the sense that acquired cultural characteristics might somehow become encoded in genetic material. Rather, the question arises out of uncertainty surrounding what is genotypic and what is phenotypic in purely cultural evolution (e.g. Lake 1998). Indeed, in the case of the evolution of material culture, objects are often conceived as phenotypic expressions of genotypic ideas, but there may well be circumstances in which they actually function as cultural genotypes or even conflate both functions (Lake 1998). Although the genotype-phenotype distinction in material culture is philosophically interesting, the question we address in this paper is essentially pragmatic: does the uncertainty surrounding the physical permanence of material culture actually matter for the application of models derived from evolutionary biology? We seek to explore potential implications of this phenomena for the application of one particular model that has been widely adopted for the study of cultural evolution:the neutral allele theory (Kimura, 1983). Originally developed in population genetics, its flexible and broad mathematical basis can serve as a null model for a variety of applications, including circumstances where the frequency of cultural variants change as a function of innovation rate and unbiased copying processes. The latter implies that variants are replicated without any particular selective pressure, and random events associated with sampling errors can lead to the spread or loss of knowledge. Archaeological applications of this null hypothesis produced a variety of results. Some exhibit empirical patterns predicted by the neutral model (e.g. Bentley et al. 2004), while others suggest the effect of systematic social choices or biases (e.g. Shennan and Wilkinson 2001). Premo (2014) has however demonstrated that techniques developed to identify neutrality are not necessarily capable of identifying unbiased cultural transmission in samples collected from time-averaged archaeological assemblages. In this paper we argue that there is potentially another problem with using the standard biological neutral model to detect the emergence of systematic preferences in the evolution of material culture, i.e the possibility that the effective population of cultural models is the number of artefacts in circulation rather than the number of people producing them. In addition, considering the differential durability of material culture, it is also possible that this number includes artefacts created by previous generations of producers. The first issue was noted by Shennan and Wilkinson (2001). As we are aware there have been explorations of the concept of memory in language evolution (Bentley et al. 2011), but there are no formal studies concerning the consequences of the persistence of material culture on the application of the neutral model. In order to investigate this problem we develop a simulation model where standard unbiased cultural transmission has been modified to incorporate a ‘production’ and a ‘persistence’ bias. Rather than formalising knowledge transfer as an individual-to- individual process, we build a model where individuals update their ‘genotype’ by copying from objects produced by other agents. This slight change in the model introduces two new mechanisms: the expression of the phenotype might be affected by stochastic events (‘production bias’), and objects might persist in the physical world for a given amount of time, potentially outliving their creator/genotype (‘persistence bias’). We generate a series of artificial archaeological records to examine whether these two biases modify the result of tests commonly used to detect neutrality in cultural datasets (Slatkin’s Exact test and Ewens-Watterson homozygosity test). Preliminary results suggest that this is the case, with the frequency of cultural variants showing significantly greater diversity than expected according to the standard neutral model. Our result thus indicates that in the presence of production and persistence bias there is a higher chance of incorrectly rejecting the null hypothesis of random copying/neutral cultural transmission.

This is an Erratum for ECP volume 18 paper 72 (2013) . The proof of statement 2) in Theorem 1.1 may not hold true. Indeed it relies on Lemma 2.5 which claims the transience of a certain Markov chain. The proof of Lemma 2.5 is incomplete because the arguments presented. In this note, we establish this transience.

International audience; The article "Population genetics models with skewed fertilities: a forward and backward analysis" Stochastic Models, 27, 521-554 (2011) contains on top of page 536 a formula for the joint falling factorial moments of the offspring numbers which is wrong in that generality. It is clarified for which compound Poisson model this formula holds true.

We consider diffusion processes x_{t} on the unit interval. Doob-transformation techniques consist of a selection of x_{t}-paths procedure. The law of the transformed process is the one of a branching diffusion system of particles, each diffusing like a new process tilde{x}_{t}, superposing an additional drift to the one of x_{t}. Killing and/or branching of tilde{x}_{t}-particles occur at some space-dependent rate lambda. For this transformed process, so in the class of branching diffusions, the question arises as to whether the particle system is sub-critical, critical or super-critical. In the first two cases, extinction occurs with probability one. We apply this circle of ideas to diffusion processes arising in population genetics. In this setup, the process x_{t} is a Wright-Fisher (WF) diffusion, either neutral or with mutations. We study a particular Doob transform which is based on the exponential function in the usual fitness parameter sigma. We have in mind that this is an alternative way to introduce selection or fitness in both WF-like diffusions, leading to branching diffusion models ideas. For this Doob-transform model of fitness, the usual selection drift sigma x(1-x) should be superposed to the one of x_{t} to form tilde{x}_{t} which is the process that can branch, binarily. In the first neutral case, there is a trade-off between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally extinct in finite time with exponential tails. In the second case with mutations, there is a trade-off between killing events removing some particles from the system and reflection at the boundaries where the particles survive. This branching diffusion process also gets eventually globally extinct but in very long finite time with power-law tails. Our approach relies on the spectral expansion of the transition probability kernels of both x_{t} and tilde{x}_{t}., Comment: To appear in: Intern. Math. Forum

The behaviour of ion channels within cardiac and neuronal cells is intrinsically stochastic in nature. When the number of channels is small this stochastic noise is large and can have an impact on the dynamics of the system which is potentially an issue when modelling small neurons and drug block in cardiac cells. While exact methods correctly capture the stochastic dynamics of a system they are computationally expensive, restricting their inclusion into tissue level models and so approximations to exact methods are often used instead. The other issue in modelling ion channel dynamics is that the transition rates are voltage dependent, adding a level of complexity as the channel dynamics are coupled to the membrane potential. By assuming that such transition rates are constant over each time step, it is possible to derive a stochastic differential equation (SDE), in the same manner as for biochemical reaction networks, that describes the stochastic dynamics of ion channels. While such a model is more computationally efficient than exact methods we show that there are analytical problems with the resulting SDE as well as issues in using current numerical schemes to solve such an equation. We therefore make two contributions: develop a different model to describe the stochastic ion channel dynamics that analytically behaves in the correct manner and also discuss numerical methods that preserve the analytical properties of the model.

We derive the joint density of market weights, at fixed times and suitable stopping times, of the volatility-stabilized market models introduced by Fernholz and Karatzas in [Ann. Finan. 1 (2005) 149-177]. The argument rests on computing the exit density of a collection of independent Bessel-square processes of possibly different dimensions from the unit simplex. We show that the law of the market weights is the same as that of the multi-allele Wright-Fisher diffusion model, well known in population genetics. Thus, as a side result, we furnish a novel proof of the transition density function of the Wright-Fisher model which was originally derived by Griffiths by bi-orthogonal series expansion., Published in at http://dx.doi.org/10.1214/10-AAP725 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; This article is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

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We point out that a proper use of the Hoeffding--ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet--Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the $n$th orthogonal space of multiple integrals can be represented as the $L^2$ limit of $U$-statistics with degenerate kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations., Published in at http://dx.doi.org/10.3150/07-BEJ5169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

(15pp); International audience; The P\'{o}lya process is an urn scheme arising in the context of the contagion spreading. It exhibits unstable persistence effects. The Friedman urn process is dual to the P\'{o}lya one with antipersistent stabilizing effects. It appears in a safety campaign problem. A P\'{o}lya-Friedman urn process is investigated with a tuning persistence parameter extrapolating the latter two extreme processes. The study includes the diffusion approximations of both the P\'{o}lya-Friedman proportion process and the population gap random walk. The structure of the former is a generalized Wright-Fisher diffusion appearing in Population Genetics. The correlation structure of the latter presents an anomalous character at a critical value of the persistence parameter.

The probability distribution of haplotype frequencies in a population, and the way it is influenced by genetical forces such as recombination, selection, random drift … is a question of fundamental interest in population genetics. For large populations, the distribution of haplotype frequencies for two linked loci under the classical Wright–Fisher model is almost impossible to compute because of numerical reasons. However the Wright–Fisher process can in such cases be approximated by a diffusion process and the transition density can then be deduced from the Kolmogorov equations. As no exact solution has been found for these equations, we developed a numerical method based on finite differences to solve them. It applies to transient states and models including selection or mutations. We show by several tests that this method is accurate for computing the conditional joint density of haplotype frequencies given that no haplotype has been lost. We also prove that it is far less time consuming than other methods such as Monte Carlo simulations.

One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented for classes of haploid and two-sex population models. For example dim U = N+ 3 for the classical haploid Wright-Fisher model with fixed population size N.