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1.
A coalescent dual process for a multi-type Moran model with genic selection is derived using a generator approach. This leads to an expansion of the transition functions in the Moran model and the Wright–Fisher diffusion process limit in terms of the transition functions for the coalescent dual. A graphical representation of the Moran model (in the spirit of Harris) identifies the dual as a strong dual process following typed lines backwards in time. An application is made to the harmonic measure problem of finding the joint probability distribution of the time to the first loss of an allele from the population and the distribution of the surviving alleles at the time of loss. Our dual process mirrors the Ancestral Selection Graph of [Krone, S. M., Neuhauser, C., 1997. Ancestral processes with selection. Theoret. Popul. Biol. 51, 210–237; Neuhauser, C., Krone, S. M., 1997. The genealogy of samples in models with selection. Genetics 145, 519–534], which allows one to reconstruct the genealogy of a random sample from a population subject to genic selection. In our setting, we follow [Stephens, M., Donnelly, P., 2002. Ancestral inference in population genetics models with selection. Aust. N. Z. J. Stat. 45, 395–430] in assuming that the types of individuals in the sample are known. There are also close links to [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38–54]. However, our methods and applications are quite different. This work can also be thought of as extending a dual process construction in a Wright–Fisher diffusion in [Barbour, A.D., Ethier, S.N., Griffiths, R.C., 2000. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10, 123–162]. The application to the harmonic measure problem extends a construction provided in the setting of a neutral diffusion process model in [Ethier, S.N., Griffiths, R.C., 1991. Harmonic measure for random genetic drift. In: Pinsky, M.A. (Ed.), Diffusion Processes and Related Problems in Analysis, vol. 1. In: Progress in Probability Series, vol. 22, Birkhäuser, Boston, pp. 73–81].  相似文献   

2.
A computational study is made of the conditional probability distribution for the allelic type of the most recent common ancestor in genealogies of samples of n genes drawn from a population under selection, given the initial sample configuration. Comparisons with the corresponding unconditional cases are presented. Such unconditional distributions differ from samples drawn from the unique stationary distribution of population allelic frequencies, known as Wright's formula, and are quantified. Biallelic haploid and diploid models are considered. A simplified structure for the ancestral selection graph of S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210-237) is enhanced further, reducing the effective branching rate in the graph. This improves efficiency of such a nonneutral analogue of the coalescent for use with computational likelihood-inference techniques.  相似文献   

3.
Algorithms for generating genealogies with selection conditional on the sample configuration of n genes in one-locus, two-allele haploid and diploid models are presented. Enhanced integro-recursions using the ancestral selection graph, introduced by S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210-237), which is the non-neutral analogue of the coalescent, enables accessible simulation of the embedded genealogy. A Monte Carlo simulation scheme based on that of R. C. Griffiths and S. Tavaré (1996, Math. Comput. Modelling 23, 141-158), is adopted to consider the estimation of ancestral times under selection. Simulations show that selection alters the expected depth of the conditional ancestral trees, depending on a mutation-selection balance. As a consequence, branch lengths are shown to be an ineffective criterion for detecting the presence of selection. Several examples are given which quantify the effects of selection on the conditional expected time to the most recent common ancestor.  相似文献   

4.

We reconsider the deterministic haploid mutation–selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation’s solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

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5.
We consider non-neutral models for unlinked loci, where the fitness of a chromosome or individual is not multiplicative across loci. Such models are suitable for many complex diseases, where there are gene-interactions. We derive a genealogical process for such models, called the complex selection graph (CSG). This coalescent-type process is related to the ancestral selection graph, and is derived from the ancestral influence graph by considering the limit as the recombination rate between loci gets large. We analyse the CSG both theoretically and via simulation. The main results are that the gene-interactions do not produce linkage disequilibrium, but do produce dependencies in allele frequencies between loci. For small selection rates, the distributions of the genealogy and the allele frequencies at a single locus are well-approximated by their distributions under a single locus model, where the fitness of each allele is the average of the true fitnesses of that allele with respect to the distribution of alleles at other loci.  相似文献   

6.
Lessard S  Kermany AR 《Genetics》2012,190(2):691-707
We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.  相似文献   

7.
We study the equilibriumbehaviour of a deterministic four-statemutation-selection model as a model for the evolution of a population of nucleotide sequences in sequence space. The mutation model is the Kimura 3ST mutation scheme, and the selection scheme is assumed to be invariant under permutation of sites. Considering the evolution process both forward and backward in time, we use the ancestral distribution as the stationary state of the backward process to derive an expression for the mutational loss (as the difference between ancestral and population mean fitness), and we prove a maximum principle that determines the population mean fitness in mutation-selection balance.  相似文献   

8.
A population of constant size is subjected to mutation, such that each mutant is of a new allelic type. For the particular population model studied in this paper, the age of an allele, whose present frequency is known, is a random variable with distribution independent of the frequencies of other alleles. As a consequence of reversibility of the population process, the age of an allele, from the past to the present, has the same distribution as its time to extinction, from the present into the future. This verifies, and re-interprets, certain diffusion approximations found by Kimura and Ohta [Genetics 75, 199–212 (1973)] and Maruyama [Genet. Res. Cambridge 23, 137–143 (1974)].  相似文献   

9.
The Genealogy of Samples in Models with Selection   总被引:1,自引:0,他引:1  
C. Neuhauser  S. M. Krone 《Genetics》1997,145(2):519-534
We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman''s well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman''s coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models, DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.  相似文献   

10.

In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform.

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11.
Damgaard C 《Genetics》2000,154(2):813-821
The expected fixation probability of an advantageous allele was examined in a partially self-fertilizing hermaphroditic plant species using the diffusion approximation. The selective advantage of the advantageous allele was assumed to be increased viability, increased fecundity, or an increase in male fitness. The mode of selection, as well as the selfing rate, the population size, and the dominance of the advantageous allele, affect the fixation probability of the allele. In general it was found that increases in selfing rate decrease the fixation probability under male sexual selection, increase fixation probability under fecundity selection, and increase when recessive and decrease when dominant under viability selection. In some cases the highest fixation probability of advantageous alleles under fecundity or under male sexual selection occurred at an intermediary selfing rate. The expected mean fixation times of the advantageous allele were also examined using the diffusion approximation.  相似文献   

12.
We study a generalisation of Moran’s population-genetic model that incorporates density dependence. Rather than assuming fixed population size, we allow the number of individuals to vary stochastically with the same events that change allele number, according to a logistic growth process with density dependent mortality. We analyse the expected time to absorption and fixation in the ‘quasi-neutral’ case: both types have the same carrying capacity, achieved through a trade-off of birth and death rates. Such types would be competitively neutral in a classical, fixed-population Wright–Fisher model. Nonetheless, we find that absorption times are skewed compared to the Wright–Fisher model. The absorption time is longer than the Wright–Fisher prediction when the initial proportion of the type with higher birth rate is large, and shorter when it is small. By contrast, demographic stochasticity has no effect on the fixation or absorption times of truly neutral alleles in a large population. Our calculations provide the first analytic results on hitting times in a two-allele model, when the population size varies stochastically.  相似文献   

13.
We extend the one-locus two allele Moran model of fixation in a haploid population to the case where the total size of the population is not fixed. The model is defined as a two-dimensional birth-and-death process for allele number. Changes in allele number occur through density-independent death events and birth events whose per capita rate decreases linearly with the total population density. Uniquely for models of this type, the latter is determined by these same birth-and-death events. This provides a framework for investigating both the effects of fluctuation in total population number through demographic stochasticity, and deterministic density-dependent changes in mean density, on allele fixation. We analyze this model using a combination of asymptotic analytic approximations supported by numerics. We find that for advantageous mutants demographic stochasticity of the resident population does not affect the fixation probability, but that deterministic changes in total density do. In contrast, for deleterious mutants, the fixation probability increases with increasing resident population fluctuation size, but is relatively insensitive to initial density. These phenomena cannot be described by simply using a harmonic mean effective population size.  相似文献   

14.
Concepts of substitutional load in finite populations   总被引:2,自引:0,他引:2  
In defining the substitutional load in a finite population, one can use a definition relating to the total load suffered until substitution of a favoured allele occurs (this will normally involve a contribution to the load from a number of cases where the favoured allele is lost), or one can use a definition using the load only for a single substitution process in which successful substitution of the favoured allele does occur. It is not evident at the moment which definition will be the more useful and meaningful in discussing how load arguments might limit the rate of selectively controlled gene substitutions. In this paper relatively simple methods are used to find formulae for both loads (the formula for the first confirming a previous result of Kimura and Maruyama, 1969). Numerical comparisons of the two loads are given, together with the results of a Monte Carlo experiment confirming the theoretical value for the second load. The latter theoretical value is found by considering a conditional diffusion process, the condition being that the favoured allele is eventually fixed. This diffusion process can be used, amongst other things, to arrive at the formulae for the mean and variance of the conditional fixation time found by other methods by  and .  相似文献   

15.
16.
We determine fixation probabilities in a model of two competing types with density dependence. The model is defined as a two-dimensional birth-and-death process with density-independent death rates, and birth rates that are a linearly decreasing function of total population density. We treat the 'quasi-neutral case' where both types have the same equilibrium population densities. This condition results in birth rates that are proportional to death rates. This can be viewed as a life history trade-off. The deterministic dynamics possesses a stable manifold of mixtures of the two types. We show that the fixation probability is asymptotically equal to the fixation probability at the point where the deterministic flow intersects this manifold. The deterministic dynamics predicts an increase in the proportion of the type with higher birth rate in growing populations (and a decrease in shrinking populations). Growing (shrinking) populations therefore intersect the manifold at a higher (lower) than initial proportion of this type. On the center manifold, the fixation probability is a quadratic function of initial proportion, with a disadvantage to the type with higher birth rate. This disadvantage arises from the larger fluctuations in population density for this type. These results are asymptotically exact and have relevance for allele fixation, models of species abundance, and epidemiological models.  相似文献   

17.
Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka–Volterra-like dynamics and single locus genetics (as in Kimura’s in Proc Natl Acad Sci USA 54: 731–736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the ‘long’ time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M 1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as ‘canonical equation’.  相似文献   

18.
It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.  相似文献   

19.
The Coalescent Process in Models with Selection   总被引:23,自引:12,他引:11       下载免费PDF全文
N. L. Kaplan  T. Darden    R. R. Hudson 《Genetics》1988,120(3):819-829
Statistical properties of the process describing the genealogical history of a random sample of genes are obtained for a class of population genetics models with selection. For models with selection, in contrast to models without selection, the distribution of this process, the coalescent process, depends on the distribution of the frequencies of alleles in the ancestral generations. If the ancestral frequency process can be approximated by a diffusion, then the mean and the variance of the number of segregating sites due to selectively neutral mutations in random samples can be numerically calculated. The calculations are greatly simplified if the frequencies of the alleles are tightly regulated. If the mutation rates between alleles maintained by balancing selection are low, then the number of selectively neutral segregating sites in a random sample of genes is expected to substantially exceed the number predicted under a neutral model.  相似文献   

20.
Uecker H  Hermisson J 《Genetics》2011,188(4):915-930
A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.  相似文献   

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