A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit

The genealogical structure of neutral populations in which reproductive success is highly-skewed has been the subject of many recent studies. Here we derive a coalescent dual process for a related class of continuous-time Moran models with viability selection. In these models, individuals can give birth to multiple offspring whose survival depends on both the parental genotype and the brood size. This extends the dual process construction for a multi-type Moran model with genic selection described in Etheridge and Griffiths (2009). We show that in the limit of infinite population size the non-neutral Moran models converge to a Markov jump process which we call the Λ-Fleming-Viot process with viability selection and we derive a coalescent dual for this process directly from the generator and as a limit from the Moran models. The dual is a branching-coalescing process similar to the Ancestral Selection Graph which follows the typed ancestry of genes backwards in time with real and virtual lineages. As an application, the transition functions of the non-neutral Moran and Λ-coalescent models are expressed as mixtures of the transition functions of the dual process.     

Exact coalescent for the Wright-Fisher model

The Kingman coalescent, which has become the foundation for a wide range of theoretical as well as empirical studies, was derived as an approximation of the Wright-Fisher (WF) model. The approximation heavily relies on the assumption that population size is large and sample size is much smaller than the population size. Whether the sample size is too large compared to the population size is rarely questioned in practice when applying statistical methods based on the Kingman coalescent. Since WF model is the most widely used population genetics model for reproduction, it is desirable to develop a coalescent framework for the WF model, which can be used whenever there are concerns about the accuracy of the Kingman coalescent as an approximation. This paper described the exact coalescent theory for the WF model and develops a simulation algorithm, which is then used, together with an analytical approach, to study the properties of the exact coalescent as well as its differences to the Kingman coalescent. We show that the Kingman coalescent differs from the exact coalescent by: (1) shorter waiting time between successive coalescent events; (2) different probability of observing a topological relationship among sequences in a sample; and (3) slightly smaller tree length in the genealogy of a large sample. On the other hand, there is little difference in the age of the most recent common ancestor (MRCA) of the sample. The exact coalescent makes up the longer waiting time between successive coalescent events by having multiple coalescence at the same time. The most significant difference among various summary statistics of a coalescent examined is the sum of lengths of external branches, which can be more than 10% larger for exact coalescent than that for the Kingman coalescent. As a whole, the Kingman coalescent is a remarkably accurate approximation to the exact coalescent for sample and population sizes falling considerably outside the region that was originally anticipated.     

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Integrating genealogy and epidemiology: the ancestral infection and selection graph as a model for reconstructing host virus histories

We model the genealogies of coupled haploid host-virus populations. Hosts reproduce and replace other hosts as in the Moran model. The virus can be transmitted between individuals of the same and succeeding generations. The epidemic model allows a selective advantage for susceptible over infected hosts. The coupled host-virus ancestry of a sample of hosts is embedded in a branching and coalescing structure that we call the Ancestral Infection and Selection Graph, a direct analogue to the Ancestral Selection Graph of Krone and Neuhauser [1997. Theoret. Population Biol. 51, 210-237]. We prove this and discuss various special cases. We show that the inter-host viral genealogy is a scaled coalescent. Using simulations, we compare the viral genealogy under this model to earlier published models and investigate the estimatability of the selection and infectious contact rates. We use simulations to compare the persistence of the disease with the time to the ultimate ancestor.     

The coalescent in two colonies with symmetric migration

Kingman's coalescent process is extended to two colonies with symmetric migration. The mean waiting time until a sample of genes taken from two colonies coalesces to a common ancestor is obtained. The final step in the waiting time before the process is absorbed at 1 is observed to have an intriguing behaviour. The distribution of this final waiting time converges to the known distribution of the corresponding waiting time in the case of a single population as the migration rate tends to zero. The mean, however, does not converge. The waiting time until a sample has two common ancestors is modeled as a function of the migration rate. Finally bounds for the expected waiting time for the two colonies to have j > 1 ancestors are derived.     

Structured coalescent processes from a modified Moran model with large offspring numbers

Structured coalescent processes are derived for the finite island model under a migration mechanism that conserves the subpopulation sizes. The underlying population model is a modified Moran model in which the reproducing individual can have very many offspring with some probability. Convergence to a structured coalescent process results when assuming that migration follows a coalescent timescale which can be much shorter than the usual Wright–Fisher timescale. Three different limit processes are possible depending on the coalescent timescale, two of which allow multiple mergers of ancestral lines. The expected time to most recent common ancestor, and the expected total size of the genealogy, of balanced and unbalanced samples can be very similar, even when migration is low, if the coalescent process allows multiple mergers. The expected total size increases almost linearly with sample size in some cases. The results have implications for inference about genetic population structure.     

The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates

We analyze a decoupled Moran model with haploid population size N, a biallelic locus under mutation and drift with scaled forward and backward mutation rates θ₁=μ₁N and θ₀=μ₀N, and directional selection with scaled strength γ=sN. With small scaled mutation rates θ₀ and θ₁, which is appropriate for single nucleotide polymorphism data in highly recombining regions, we derive a simple approximate equilibrium distribution for polymorphic alleles with a constant of proportionality. We also put forth an even simpler model, where all mutations originate from monomorphic states. Using this model we derive the sojourn times, conditional on the ancestral and fixed allele, and under equilibrium the distributions of fixed and polymorphic alleles and fixation rates. Furthermore, we also derive the distribution of small samples in the diffusion limit and provide convenient recurrence relations for calculating this distribution. This enables us to give formulas analogous to the Ewens-Watterson estimator of θ for biased mutation rates and selection. We apply this theory to a polymorphism dataset of fourfold degenerate sites in Drosophila melanogaster.     

Perfect simulation from population genetic models with selection

We consider using the ancestral selection graph (ASG) to simulate samples from population genetic models with selection. Currently the use of the ASG to simulate samples is limited. This is because the computational requirement for simulating samples increases exponentially with the selection rate and also due to needing to simulate a sample of size one from the population at equilibrium. For the only case where the distribution of a sample of size one is known, that of parent-independent mutations, more efficient simulation algorithms exist. We will show that by applying the idea of coupling from the past to the ASG, samples can be simulated from a general K-allele model without knowledge of the distribution of a sample of size one. Furthermore, the computation involved in generating such samples appears to be less than that of simulating the ASG until its ultimate ancestor. In particular, in the case of genic selection with parent-independent mutations, the computational requirement increases only quadratically with the selection rate. The algorithm is demonstrated by simulating samples at a microsatellite locus.     

Time to the MRCA of a sample in a Wright-Fisher model with variable population size

Determining the expected distribution of the time to the most recent common ancestor of a sample of individuals may deliver important information about the genetic markers and evolution of the population. In this paper, we introduce a new recursive algorithm to calculate the distribution of the time to the most recent common ancestor of the sample from a population evolved by any conditional multinomial sampling model. The most important advantage of our method is that it can be applied to a sample of any size drawn from a population regardless of its size growth pattern. We also present a very efficient method to implement and store the genealogy tree of the population evolved by the Galton–Watson process. In the final section we present results applied to a simulated population with a single bottleneck event and to real populations of known size histories.     

A coalescent model of background selection with recombination,demography and variation in selection coefficients

There is increasing evidence that background selection, the effects of the elimination of recurring deleterious mutations by natural selection on variability at linked sites, may be a major factor shaping genome-wide patterns of genetic diversity. To accurately quantify the importance of background selection, it is vital to have computationally efficient models that include essential biological features. To this end, a structured coalescent procedure is used to construct a model of background selection that takes into account the effects of recombination, recent changes in population size and variation in selection coefficients against deleterious mutations across sites. Furthermore, this model allows a flexible organization of selected and neutral sites in the region concerned, and has the ability to generate sequence variability at both selected and neutral sites, allowing the correlation between these two types of sites to be studied. The accuracy of the model is verified by checking against the results of forward simulations. These simulations also reveal several patterns of diversity that are in qualitative agreement with observations reported in recent studies of DNA sequence polymorphisms. These results suggest that the model should be useful for data analysis.     

Biologically constrained action selection improves cognitive control in a model of the Stroop task

The Stroop task is a paradigmatic psychological task for investigating stimulus conflict and the effect this has on response selection. The model of Cohen et al. (Cohen et al. 1990 Psychol. Rev. 97, 332-361) has hitherto provided the best account of performance in the Stroop task, but there remains certain key data that it fails to match. We show that this failure is due to the mechanism used to perform final response selection-one based on the diffusion model of choice behaviour (Ratcliff 1978 Psychol. Rev. 85, 59-108). We adapt the model to use a selection mechanism which is based on the putative human locus of final response selection, the basal ganglia/thalamo-cortical complex (Redgrave et al. 1999 Neuroscience 89, 1009-1023). This improves the match to the core human data and, additionally, makes it possible for the model to accommodate, in a principled way, additional mechanisms of cognitive control that enable better fits to the data. This work prompts a critique of the diffusion model as a mechanism of response selection, and the features that any response mechanism must possess to provide adaptive action selection. We conclude that the consideration of biologically constrained solutions to the action selection problem is vital to the understanding and improvement of cognitive models of response selection.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.     

Balancing selection in Pattern Recognition Receptor signalling pathways is associated with gene function and pleiotropy in a wild rodent

Pathogen‐mediated balancing selection is commonly considered to play an important role in the maintenance of genetic diversity, in particular in immune genes. However, the factors that may influence which immune genes are the targets of such selection are largely unknown. To address this, here we focus on Pattern Recognition Receptor (PRR) signalling pathways, which play a key role in innate immunity. We used whole‐genome resequencing data from a population of bank voles (Myodes glareolus) to test for associations between balancing selection, pleiotropy and gene function in a set of 123 PRR signalling pathway genes. To investigate the effect of gene function, we compared genes encoding (a) receptors for microbial ligands versus downstream signalling proteins, and (b) receptors recognizing components of microbial cell walls, flagella and capsids versus receptors recognizing features of microbial nucleic acids. Analyses based on the nucleotide diversity of full coding sequences showed that balancing selection primarily targeted receptor genes with a low degree of pleiotropy. Moreover, genes encoding receptors recognizing components of microbial cell walls etc. were more important targets of balancing selection than receptors recognizing nucleic acids. Tests for localized signatures of balancing selection in coding and noncoding sequences showed that such signatures were mostly located in introns, and more evenly distributed among different functional categories of PRR pathway genes. The finding that signatures of balancing selection in full coding sequences primarily occur in receptor genes, in particular those encoding receptors for components of microbial cell walls etc., is consistent with the idea that coevolution between hosts and pathogens is an important cause of balancing selection on immune genes.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population with random mating

Global convergence to equilibria is proved for the model in the title under the assumption that death rates (and fertilities) are constant and fertilities are additive and positive. The phase portraits are determined. There is at most one polymorphic equilibrium or there are a continuum of equilibria.     

A bayesian random-effects markov model for tumor progression in women with a family history of breast cancer

SUMMARY: It makes intuitive sense to model the natural history of breast cancer, tumor progression from preclinical screen-detectable phase (PCDP) to clinical disease, as a multistate process, but the multilevel structure of the available data, which generally comes from cluster (family)-based service screening programs, makes the required parameter estimation intractable because there is a correlation between screening rounds in the same individual, and between subjects within clusters (families). There is also residual heterogeneity after adjusting for covariates. We therefore proposed a Bayesian hierarchical multistate Markov model with fixed and random effects and applied it to data from a high-risk group (women with a family history of breast cancer) participating in a family-based screening program for breast cancer. A total of 4867 women attended (representing 4464 families) and by the end of 2002, a total of 130 breast cancer cases were identified. Parameter estimation was carried out using Markov chain Monte Carlo (MCMC) simulation and the Bayesian software package WinBUGS. Models with and without random effects were considered. Our preferred model included a random-effect term for the transition rate from preclinical to clinical disease (sigma(2)(2f)), which was estimated to be 0.50 (95% credible interval = 0.22-1.49). Failure to account for this random effect was shown to lead to bias. The incorporation of covariates into multistate models with random effect not only reduced residual heterogeneity but also improved the convergence of stationary distribution. Our proposed Bayesian hierarchical multistate model is a valuable tool for estimating the rate of transitions between disease states in the natural history of breast cancer (and possibly other conditions). Unlike existing models, it can cope with the correlation structure of multilevel screening data, covariates, and residual (unexplained) variation.     

A novel Arabidopsis–oomycete pathosystem: differential interactions with Phytophthora capsici reveal a role for camalexin,indole glucosinolates and salicylic acid in defence

Phytophthora capsici causes devastating diseases on a broad range of plant species. To better understand the interaction with its host plants, knowledge obtained from a model pathosystem can be instrumental. Here, we describe the interaction between P. capsici and Arabidopsis and the exploitation of this novel pathosystem to assign metabolic pathways involved in defence against P. capsici. Inoculation assays on Arabidopsis accessions with different P. capsici isolates revealed interaction specificity among accession‐isolate combinations. In a compatible interaction, appressorium‐mediated penetration was followed by the formation of invasive hyphae, haustoria and sporangia in leaves and roots. In contrast, in an incompatible interaction, P. capsici infection elicited callose deposition, accumulation of active oxygen species and cell death, resulting in early pathogen encasement in leaves. Moreover, Arabidopsis mutants with defects in salicylic acid signalling, camalexin or indole glucosinolates biosynthesis pathways displayed severely compromised resistance to P. capsici. It is anticipated that this model pathosystem will facilitate the genetic dissection of complex traits responsible for resistance against P. capsici.