Infinite Sites Model Research Articles

An almost infinite sites model

Motivation:A main challenge in molecular evolution is to find computationally efficient mutation models with flexible assumptions that properly reflect genetic variation. The infinite sites model assumes that each mutation event occurs at a site never previously mutant, i.e. it does not allow recurrent mutations. This is reasonable for low mutation rates and makes statistical inference much more tractable. However, recurrent mutations are common enough to be observable from genetic variation data, even in species with low per-site mutation rates such as humans. The finite sites model on the other hand allows for recurrent mutations but is computationally unfeasible to work with in most cases. In this work, we bridge these two approaches by developing a novel molecular evolution model, the almost infinite sites model, that both admits recurrent mutations and is tractable. We provide a recursive characterisation of the likelihood of our proposed model under complete linkage and outline a parsimonious approximation scheme for computing it. Results:We show the usefulness of our model in simulated and human mitochondrial data. Our results show that the AISM, in combination with a constraint on the total number of mutation events, can recover accurate approximations to the maximum likelihood estimator of the mutation rate. Availability and implementation:An implementation of our model is freely available along with code for reproducing our computational experiments at https://github.com/Cronjaeger/almost-infinite-sites-recursions.

Polygenic dynamics underlying the response of quantitative traits to directional selection

We study the response of a quantitative trait to exponential directional selection in a finite haploid population, both at the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean Θ) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a supercritical Galton–Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation n, conditioned on non-extinction until n, is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution is related to the absolutely continuous part W+ of the random variable, typically denoted W, that characterizes the stochasticity accumulating during the mutant’s sweep. A suitable transformation yields the approximate dynamics of the mutant frequency distribution in a Wright–Fisher population of size N. Our expression provides a very accurate approximation except when mutant frequencies are close to 1. On this basis, we derive explicitly the (approximate) time dependence of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate approximations for all times, even for the quasi-stationary phase when the expected per-generation response and the trait variance have equilibrated. The latter refine classical results. In addition, we find that Θ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the initial allele-frequency dynamics are best described by sweep-like patterns at few loci or small allele-frequency shifts at many. The number of segregating sites is an appropriate indicator for these patterns. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested by comprehensive simulations in a Wright–Fisher framework. We argue that our results apply to more complex forms of directional selection.

Genomic Analysis of SARS-CoV-2 Alpha, Beta and Delta Variants of Concern Uncovers Signatures of Neutral and Non-Neutral Evolution.

Due to the emergence of new variants of the SARS-CoV-2 coronavirus, the question of how the viral genomes evolved, leading to the formation of highly infectious strains, becomes particularly important. Three major emergent strains, Alpha, Beta and Delta, characterized by a significant number of missense mutations, provide a natural test field. We accumulated and aligned 4.7 million SARS-CoV-2 genomes from the GISAID database and carried out a comprehensive set of analyses. This collection covers the period until the end of October 2021, i.e., the beginnings of the Omicron variant. First, we explored combinatorial complexity of the genomic variants emerging and their timing, indicating very strong, albeit hidden, selection forces. Our analyses show that the mutations that define variants of concern did not arise gradually but rather co-evolved rapidly, leading to the emergence of the full variant strain. To explore in more detail the evolutionary forces at work, we developed time trajectories of mutations at all 29,903 sites of the SARS-CoV-2 genome, week by week, and stratified them into trends related to (i) point substitutions, (ii) deletions and (iii) non-sequenceable regions. We focused on classifying the genetic forces active at different ranges of the mutational spectrum. We observed the agreement of the lowest-frequency mutation spectrum with the Griffiths-Tavaré theory, under the Infinite Sites Model and neutrality. If we widen the frequency range, we observe the site frequency spectra much more consistently with the Tung-Durrett model assuming clone competition and selection. The coefficients of the fitting model indicate the possibility of selection acting to promote gradual growth slowdown, as observed in the history of the variants of concern. These results add up to a model of genomic evolution, which partly fits into the classical drift barrier ideas. Certain observations, such as mutation "bands" persistent over the epidemic history, suggest contribution of genetic forces different from mutation, drift and selection, including recombination or other genome transformations. In addition, we show that a "toy" mathematical model can qualitatively reproduce how new variants (clones) stem from rare advantageous driver mutations, and then acquire neutral or disadvantageous passenger mutations which gradually reduce their fitness so they can be then outcompeted by new variants due to other driver mutations.

Graph-based algorithms for Laplace transformed coalescence time distributions.

Extracting information on the selective and demographic past of populations that is contained in samples of genome sequences requires a description of the distribution of the underlying genealogies. Using the Laplace transform, this distribution can be generated with a simple recursive procedure, regardless of model complexity. Assuming an infinite-sites mutation model, the probability of observing specific configurations of linked variants within small haplotype blocks can be recovered from the Laplace transform of the joint distribution of branch lengths. However, the repeated differentiation required to compute these probabilities has proven to be a serious computational bottleneck in earlier implementations. Here, I show that the state space diagram can be turned into a computational graph, allowing efficient evaluation of the Laplace transform by means of a graph traversal algorithm. This general algorithm can, for example, be applied to tabulate the likelihoods of mutational configurations in non-recombining blocks. This work provides a crucial speed up for existing composite likelihood approaches that rely on the joint distribution of branch lengths to fit isolation with migration models and estimate the parameters of selective sweeps. The associated software is available as an open-source Python library, agemo.

High fraction of silent recombination in a finite-population two-locus neutral birth-death-mutation model.

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 considerably reduces the total number of haplotypes expected from the infinite site model for populations that are not much larger than one over the mutation rate. Moreover, in contrast with mutations, the number of haplotypes does not grow linearly with the population size. We hence propose a more accurate estimate of the total number of haplotypes that takes into account silent recombination. We study large-scale human leukocyte antigen (HLA) haplotype frequencies from human populations to show that the current estimated recombination rate in the HLA region is underestimated.

Biallelic mutations in cancer genomes reveal local mutational determinants

The infinite sites model of molecular evolution posits that every position in the genome is mutated at most once1. By restricting the number of possible mutation histories, haplotypes and alleles, it forms a cornerstone of tumor phylogenetic analysis2 and is often implied when calling, phasing and interpreting variants3,4 or studying the mutational landscape as a whole5. Here we identify 18,295 biallelic mutations, where the same base is mutated independently on both parental copies, in 559 (21%) bulk sequencing samples from the Pan-Cancer Analysis of Whole Genomes study. Biallelic mutations reveal ultraviolet light damage hotspots at E26 transformation-specific (ETS) and nuclear factor of activated T cells (NFAT) binding sites, and hypermutable motifs in POLE-mutant and other cancers. We formulate recommendations for variant calling and provide frameworks to model and detect biallelic mutations. These results highlight the need for accurate models of mutation rates and tumor evolution, as well as their inference from sequencing data.

Theoretical and Practical Considerations when using Retroelement Insertions to Estimate Species Trees in the Anomaly Zone.

A potential shortcoming of concatenation methods for species tree estimation is their failure to account for incomplete lineage sorting. Coalescent methods address this problem but make various assumptions that, if violated, can result in worse performance than concatenation. Given the challenges of analyzing DNA sequences with both concatenation and coalescent methods, retroelement insertions (RIs) have emerged as powerful phylogenomic markers for species tree estimation. Here, we show that two recently proposed quartet-based methods, SDPquartets and ASTRAL_BP, are statistically consistent estimators of the unrooted species tree topology under the coalescent when RIs follow a neutral infinite-sites model of mutation and the expected number of new RIs per generation is constant across the species tree. The accuracy of these (and other) methods for inferring species trees from RIs has yet to be assessed on simulated data sets, where the true species tree topology is known. Therefore, we evaluated eight methods given RIs simulated from four model species trees, all of which have short branches and at least three of which are in the anomaly zone. In our simulation study, ASTRAL_BP and SDPquartets always recovered the correct species tree topology when given a sufficiently large number of RIs, as predicted. A distance-based method (ASTRID_BP) and Dollo parsimony also performed well in recovering the species tree topology. In contrast, unordered, polymorphism, and Camin-Sokal parsimony (as well as an approach based on MDC) typically fail to recover the correct species tree topology in anomaly zone situations with more than four ingroup taxa. Of the methods studied, only ASTRAL_BP automatically estimates internal branch lengths (in coalescent units) and support values (i.e., local posterior probabilities). We examined the accuracy of branch length estimation, finding that estimated lengths were accurate for short branches but upwardly biased otherwise. This led us to derive the maximum likelihood (branch length) estimate for when RIs are given as input instead of binary gene trees; this corrected formula produced accurate estimates of branch lengths in our simulation study provided that a sufficiently large number of RIs were given as input. Lastly, we evaluated the impact of data quantity on species tree estimation by repeating the above experiments with input sizes varying from 100 to 100,000 parsimony-informative RIs. We found that, when given just 1000 parsimony-informative RIs as input, ASTRAL_BP successfully reconstructed major clades (i.e., clades separated by branches $>0.3$ coalescent units) with high support and identified rapid radiations (i.e., shorter connected branches), although not their precise branching order. The local posterior probability was effective for controlling false positive branches in these scenarios. [Coalescence; incomplete lineage sorting; Laurasiatheria; Palaeognathae; parsimony; polymorphism parsimony; retroelement insertions; species trees; transposon.].

Efficiently Summarizing Relationships in Large Samples: A General Duality Between Statistics of Genealogies and Genomes.

As a genetic mutation is passed down across generations, it distinguishes those genomes that have inherited it from those that have not, providing a glimpse of the genealogical tree relating the genomes to each other at that site. Statistical summaries of genetic variation therefore also describe the underlying genealogies. We use this correspondence to define a general framework that efficiently computes single-site population genetic statistics using the succinct tree sequence encoding of genealogies and genome sequence. The general approach accumulates sample weights within the genealogical tree at each position on the genome, which are then combined using a summary function; different statistics result from different choices of weight and function. Results can be reported in three ways: by site, which corresponds to statistics calculated as usual from genome sequence; by branch, which gives the expected value of the dual site statistic under the infinite sites model of mutation, and by node, which summarizes the contribution of each ancestor to these statistics. We use the framework to implement many currently defined statistics of genome sequence (making the statistics’ relationship to the underlying genealogical trees concrete and explicit), as well as the corresponding branch statistics of tree shape. We evaluate computational performance using simulated data, and show that calculating statistics from tree sequences using this general framework is several orders of magnitude more efficient than optimized matrix-based methods in terms of both run time and memory requirements. We also explore how well the duality between site and branch statistics holds in practice on trees inferred from the 1000 Genomes Project data set, and discuss ways in which deviations may encode interesting biological signals.

The complexity of genome rearrangement combinatorics under the infinite sites model

Rearrangements are discrete processes whereby discrete segments of DNA are deleted, replicated and inserted into novel positions. A sequence of such configurations, termed a rearrangement evolution, results in jumbled DNA arrangements, frequently observed in cancer genomes. We introduce a method that allows us to precisely count these different evolutions for a range of processes including breakage-fusion-bridge-cycles, tandem-duplications, inverted-duplications, reversals, transpositions and deletions, showing that the space of rearrangement evolution is super-exponential in size. These counts assume the infinite sites model of unique breakpoint usage.

Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent

We study tree lengths in $\Lambda $-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches, we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen–Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e., branches carrying $a$ individuals out of the sample). These results immediately transform to sampling formulas in the infinite sites model. In particular, we obtain the asymptotic site frequency spectrum of the Bolthausen–Sznitman coalescent. The proofs rely on a new technique to obtain laws of large numbers for certain functionals of decreasing Markov chains.

Accurate and efficient cell lineage tree inference from noisy single cell data: the maximum likelihood perfect phylogeny approach

Cells in an organism share a common evolutionary history, called cell lineage tree. Cell lineage tree can be inferred from single cell genotypes at genomic variation sites. Cell lineage tree inference from noisy single cell data is a challenging computational problem. Most existing methods for cell lineage tree inference assume uniform uncertainty in genotypes. A key missing aspect is that real single cell data usually has non-uniform uncertainty in individual genotypes. Moreover, existing methods are often sampling based and can be very slow for large data. In this article, we propose a new method called ScisTree, which infers cell lineage tree and calls genotypes from noisy single cell genotype data. Different from most existing approaches, ScisTree works with genotype probabilities of individual genotypes (which can be computed by existing single cell genotype callers). ScisTree assumes the infinite sites model. Given uncertain genotypes with individualized probabilities, ScisTree implements a fast heuristic for inferring cell lineage tree and calling the genotypes that allow the so-called perfect phylogeny and maximize the likelihood of the genotypes. Through simulation, we show that ScisTree performs well on the accuracy of inferred trees, and is much more efficient than existing methods. The efficiency of ScisTree enables new applications including imputation of the so-called doublets. The program ScisTree is available for download at: https://github.com/yufengwudcs/ScisTree. Supplementary data are available at Bioinformatics online.

S-Cluster++: a fast program for solving the cluster containment problem for phylogenetic networks.

Comparative genomic studies indicate that extant genomes are more properly considered to be a fusion product of random mutations over generations (vertical evolution) and genomic material transfers between individuals of different lineages (reticulate transfer). This has motivated biologists to use phylogenetic networks and other general models to study genome evolution. Two fundamental algorithmic problems arising from verification of phylogenetic networks and from computing Robinson-Foulds distance in the space of phylogenetic networks are the tree and cluster containment problems. The former asks how to decide whether or not a phylogenetic tree is displayed in a phylogenetic network. The latter is to decide whether a subset of taxa appears as a cluster in some tree displayed in a phylogenetic network. The cluster containment problem (CCP) is also closely related to testing the infinite site model on a recombination network. Both the tree containment and CCP are NP-complete. Although the CCP was introduced a decade ago, there has been little progress in developing fast algorithms for it on arbitrary phylogenetic networks. In this work, we present a fast computer program for the CCP. This program is developed on the basis of a linear-time transformation from the small version of the CCP to the SAT problem. The program package is available for download on http://www.math.nus.edu.sg/∼matzlx/ccp.

Variance in estimated pairwise genetic distance under high versus low coverage sequencing: The contribution of linkage disequilibrium

The mean pairwise genetic distance among haplotypes is an estimator of the population mutation rate θ and a standard measure of variation in a population. With the advent of next-generation sequencing (NGS) methods, this and other population parameters can be estimated under different modes of sampling. One approach is to sequence individual genomes with high coverage, and to calculate genetic distance over all sample pairs. The second approach, typically used for microbial samples or for tumor cells, is sequencing a large number of pooled genomes with very low individual coverage. With low coverage, pairwise genetic distances are calculated across independently sampled sites rather than across individual genomes. In this study, we show that the variance in genetic distance estimates is reduced with low coverage sampling if the mean pairwise linkage disequilibrium weighted by allele frequencies is positive. Practically, this means that if on average the most frequent alleles over pairs of loci are in positive linkage disequilibrium, low coverage sequencing results in improved estimates of θ, assuming similar per-site read depths. We show that this result holds under the expected distribution of allele frequencies and linkage disequilibria for an infinite sites model at mutation–drift equilibrium. From simulations, we find that the conditions for reduced variance only fail to hold in cases where variant alleles are few and at very low frequency. These results are applied to haplotype frequencies from a lung cancer tumor to compute the weighted linkage disequilibria and the expected error in estimated genetic distance using high versus low coverage.

Exact Likelihood Calculation under the Infinite Sites Model

A key parameter in population genetics is the scaled mutation rate θ = 4 N μ , where N is the effective haploid population size and μ is the mutation rate per haplotype per generation. While exact likelihood inference is notoriously difficult in population genetics, we propose a novel approach to compute a first order accurate likelihood of θ that is based on dynamic programming under the infinite sites model without recombination. The parameter θ may be either constant, i.e., time-independent, or time-dependent, which allows for changes of demography and deviations from neutral equilibrium. For time-independent θ, the performance is compared to the approach in Griffiths and Tavaré’s work “Simulating Probability Distributions in the Coalescent” (Theor. Popul. Biol. 1994, 46, 131–159) that is based on importance sampling and implemented in the “genetree” program. Roughly, the proposed method is computationally fast when n × θ < 100 , where n is the sample size. For time-dependent θ ( t ) , we analyze a simple demographic model with a single change in θ ( t ) . In this case, the ancestral and current θ need to be estimated, as well as the time of change. To our knowledge, this is the first accurate computation of a likelihood in the infinite sites model with non-equilibrium demography.

Coalescent: an open-science framework for importance sampling in coalescent theory.

Background. In coalescent theory, computer programs often use importance sampling to calculate likelihoods and other statistical quantities. An importance sampling scheme can exploit human intuition to improve statistical efficiency of computations, but unfortunately, in the absence of general computer frameworks on importance sampling, researchers often struggle to translate new sampling schemes computationally or benchmark against different schemes, in a manner that is reliable and maintainable. Moreover, most studies use computer programs lacking a convenient user interface or the flexibility to meet the current demands of open science. In particular, current computer frameworks can only evaluate the efficiency of a single importance sampling scheme or compare the efficiencies of different schemes in an ad hoc manner.Results. We have designed a general framework (http://coalescent.sourceforge.net; language: Java; License: GPLv3) for importance sampling that computes likelihoods under the standard neutral coalescent model of a single, well-mixed population of constant size over time following infinite sites model of mutation. The framework models the necessary core concepts, comes integrated with several data sets of varying size, implements the standard competing proposals, and integrates tightly with our previous framework for calculating exact probabilities. For a given dataset, it computes the likelihood and provides the maximum likelihood estimate of the mutation parameter. Well-known benchmarks in the coalescent literature validate the accuracy of the framework. The framework provides an intuitive user interface with minimal clutter. For performance, the framework switches automatically to modern multicore hardware, if available. It runs on three major platforms (Windows, Mac and Linux). Extensive tests and coverage make the framework reliable and maintainable.Conclusions. In coalescent theory, many studies of computational efficiency consider only effective sample size. Here, we evaluate proposals in the coalescent literature, to discover that the order of efficiency among the three importance sampling schemes changes when one considers running time as well as effective sample size. We also describe a computational technique called “just-in-time delegation” available to improve the trade-off between running time and precision by constructing improved importance sampling schemes from existing ones. Thus, our systems approach is a potential solution to the “28 programs problem” highlighted by Felsenstein, because it provides the flexibility to include or exclude various features of similar coalescent models or importance sampling schemes.

A Framework Including Recombination for Analyzing the Dynamics of Within-Host HIV Genetic Diversity

This paper presents a novel population genetic model and a computationally and statistically tractable framework for analyzing within-host HIV diversity based on serial samples of HIV DNA sequences. This model considers within-host HIV evolution during the chronic phase of infection and assumes that the HIV population is homogeneous at the beginning, corresponding to the time of seroconversion, and evolves according to the Wright-Fisher reproduction model with recombination and variable mutation rate across nucleotide sites. In addition, the population size and generation time vary over time as piecewise constant functions of time. Under this model I approximate the genealogical and mutational processes for serial samples of DNA sequences by a continuous coalescent-recombination process and an inhomogeneous Poisson process, respectively. Based on these derivations, an efficient algorithm is described for generating polymorphisms in serial samples of DNA sequences under the model including various substitution models. Extensions of the algorithm are also described for other demographic scenarios that can be more suitable for analyzing the dynamics of genetic diversity of other pathogens in vitro and in vivo. For the case of the infinite-sites model, I derive analytical formulas for the expected number of polymorphic sites in sample of DNA sequences, and apply the developed simulation and analytical methods to explore the fit of the model to HIV genetic diversity based on serial samples of HIV DNA sequences from 9 HIV-infected individuals. The results particularly show that the estimates of the ratio of recombination rate over mutation rate can vary over time between very high and low values, which can be considered as a consequence of the impact of selection forces.

Coalescent: an open-source and scalable framework for exact calculations in coalescent theory

BackgroundCurrently, there is no open-source, cross-platform and scalable framework for coalescent analysis in population genetics. There is no scalable GUI based user application either. Such a framework and application would not only drive the creation of more complex and realistic models but also make them truly accessible.ResultsAs a first attempt, we built a framework and user application for the domain of exact calculations in coalescent analysis. The framework provides an API with the concepts of model, data, statistic, phylogeny, gene tree and recursion. Infinite-alleles and infinite-sites models are considered. It defines pluggable computations such as counting and listing all the ancestral configurations and genealogies and computing the exact probability of data. It can visualize a gene tree, trace and visualize the internals of the recursion algorithm for further improvement and attach dynamically a number of output processors. The user application defines jobs in a plug-in like manner so that they can be activated, deactivated, installed or uninstalled on demand. Multiple jobs can be run and their inputs edited. Job inputs are persisted across restarts and running jobs can be cancelled where applicable.ConclusionsCoalescent theory plays an increasingly important role in analysing molecular population genetic data. Models involved are mathematically difficult and computationally challenging. An open-source, scalable framework that lets users immediately take advantage of the progress made by others will enable exploration of yet more difficult and realistic models. As models become more complex and mathematically less tractable, the need for an integrated computational approach is obvious. Object oriented designs, though has upfront costs, are practical now and can provide such an integrated approach.

Genealogies of regular exchangeable coalescents with applications to sampling

Cet article considère un modèle de généalogie qui correspond à un processus de coalescence échangeable régulier (appelé aussi un $\varXi$-coalescent) démarré d’une configuration à taille finie et grande, et subissant des mutations neutres. Des expressions asymptotiques pour le nombre de lignées actives ont été obtenues par l’auteur dans un travail précédent. Des résultats analogues pour le nombre de lignées actives et la longueur totale des lignées sont dérivés par les mêmes techniques martingales. Ils sont donnés en terme de la convergence en probabilité, pendant que des extensions à la convergence au sens des moments et la convergence presque sûre sont examinées. Ces résultats ont des conséquences directes sur la théorie d’échantillonnage dans le cadre de $\varXi$-coalescence. En particulier, les $\varXi$-coalescents réguliers qui descendent de l’infini (c.-à-d. qui ont des généalogies localement finies) ont des nombres de familles égaux au sens asymptotique sous le modèle d’allèles infinies et le modèle de site infinis. Dans des cas particuliers, on peut ainsi dériver des formules asymptotiques quantitatives pour le nombre de familles contenant un nombre fixe d’individus.

Genetic Estimates of Population Age in the Water Flea, Daphnia magna

Genetic datasets can be used to date evolutionary events, even on recent time scales if sufficient data are available. We used statistics calculated from multilocus microsatellite datasets to estimate population ages in data generated through coalescent simulations and in samples from populations of known age in a metapopulation of Daphnia magna in Finland. Our simulation results show that age estimates improve with additional loci and define a time frame over which these statistics are most useful. On the most recent time scales, assumptions regarding the model of mutation (infinite sites vs. stepwise mutation) have little influence on estimated ages. In older populations, size homoplasy among microsatellite alleles results in a downwards bias for estimates based on the infinite sites model (ISM). In the Finnish D. magna metapopulation, our genetically derived estimated ages were biased upwards. Potential sources of this bias include the underlying model of mutation, gene flow, founder size, and the possibility of persistent source populations in the system. Our simulated data show that genetic age estimation is possible, even for very young populations, but our empirical data highlight the importance of factors such as migration when these statistics are applied in natural populations.

A General Method for Calculating Likelihoods Under the Coalescent Process

Analysis of genomic data requires an efficient way to calculate likelihoods across very large numbers of loci. We describe a general method for finding the distribution of genealogies: we allow migration between demes, splitting of demes [as in the isolation-with-migration (IM) model], and recombination between linked loci. These processes are described by a set of linear recursions for the generating function of branch lengths. Under the infinite-sites model, the probability of any configuration of mutations can be found by differentiating this generating function. Such calculations are feasible for small numbers of sampled genomes: as an example, we show how the generating function can be derived explicitly for three genes under the two-deme IM model. This derivation is done automatically, using Mathematica. Given data from a large number of unlinked and nonrecombining blocks of sequence, these results can be used to find maximum-likelihood estimates of model parameters by tabulating the probabilities of all relevant mutational configurations and then multiplying across loci. The feasibility of the method is demonstrated by applying it to simulated data and to a data set previously analyzed by Wang and Hey (2010) consisting of 26,141 loci sampled from Drosophila simulans and D. melanogaster. Our results suggest that such likelihood calculations are scalable to genomic data as long as the numbers of sampled individuals and mutations per sequence block are small.