Wakeley J. Coalescent Theory: An Introduction. Greenwood Village: Roberts & Company Publishers; 2009.Abstract
Not the abstract, but a note. This book is available from Macmillan. The title above links directly to their web site.
Book Chapter
Wakeley J, Wilton PR. Coalescent and models of identity by descent. In: Encyclopedia of Evolutionary Biology. Vol 1. Oxford: Academic Press ; 2016 :287-292. (pdf)
Wakeley J. Natural selection and coalescent theory. In: Evolution since Darwin: The First 150 Years. Sunderland, MA: Sinauer and Associates ; 2010 :119-149. short draft (pdf) long draft (pdf)
Wakeley J. Metapopulations and coalescent theory. In: Ecology, Genetics, and Evolution of Metapopulations. Amsterdam, Boston etc. Elsevier Academic Press ; 2004 :175-198.
Wakeley J. Inferences about the structure and history of populations: coalescents and intraspecific phylogeography. In: The Evolution of Population Biology - Modern Synthesis. Cambridge: Cambridge University Press ; 2002 :193-215. (pdf)
Wakeley J, Hey J. Testing speciation models with DNA sequence data. In: Molecular approaches to ecology and evolution. Basel, Boston & Berlin: Birkhauser ; 1998 :157-175.
Journal Article
King L, Wakeley J, Carmi S. A non-zero variance of Tajima’s estimator for two sequences even for infinitely many unlinked loci. Theoretical Population Biology. In Press.Abstract
The population-scaled mutation rate, θ, is informative on the effective population size and is thus widely used in population genetics. We show that for two sequences and n unlinked loci, the variance of Tajima’s estimator (ˆθ), which is the average number of pairwise differences, does not vanish even as n → ∞. The non-zero variance of ˆθ results from a (weak) correlation between coalescence times even at unlinked loci, which, in turn, is due to the underlying fixed pedigree shared by gene genealogies at all loci. We derive the correlation coefficient under a diploid, discrete-time, Wright–Fisher model, and we also derive a simple, closed-form lower bound. We also obtain empirical estimates of the correlation of coalescence times under demographic models inspired by large-scale human genealogies. While the effect we describe is small (Var [ˆθ]/θ2 ≈ O(N−1e)), it is important to recognize this feature of statistical population genetics, which runs counter to commonly held notions about unlinked loci.
McAvoy A, Fraiman N, Hauert C, Wakeley J, Nowak MA. Public goods games in populations with fluctuating size. Theoretical Population Biology. 2018;121 :72-84.Abstract
Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here,we consider evolutionary game dynamics in an extended Wright–Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation–selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait’s long-term success.
Wilton PR, Baduel P, Landon MM, Wakeley J. Population structure and coalescence in pedigrees: Comparisons to the structured coalescent and a framework for inference. Theoretical Population Biology. 2017;115 :1-12.Abstract

Contrary to what is often assumed in population genetics, independently segregating loci do not have completely independent ancestries, since all loci are inherited through a single, shared population pedigree. Previous work has shown that the non-independence between gene genealogies of independently segregating loci created by the population pedigree is weak in panmictic populations, and predictions made from standard coalescent theory are accurate for populations that are at least moderately sized. Here, we investigate patterns of coalescence in pedigrees of structured populations. We find that the pedigree creates deviations away from the predictions of the structured coalescent that persist on a longer timescale than in the case of panmictic populations. Nevertheless, we find that the structured coalescent provides a reasonable approximation for the coalescent process in structured population pedigrees so long as migration events are moderately frequent and there are no migration events in the recent pedigree of the sample. When there are migration events in the recent sample pedigree, we find that distributions of coalescence in the sample can be modeled as a mixture of distributions from different initial sample configurations. We use this observation to motivate a maximum-likelihood approach for inferring migration rates and mutation rates jointly with features of the pedigree such as recent migrant ancestry and recent relatedness. Using simulation, we show that our inference framework accurately recovers long-term migration rates in the presence of recent migration events in the sample pedigree.

King L, Wakeley J. Empirical Bayes estimation of coalescence times from nucleotide sequence data. Genetics. 2016;204 :249-257.Abstract

We demonstrate the advantages of using information at many unlinked loci to better calibrate estimates of the time to the most recent common ancestor (TMRCA) at a given locus. To this end, we apply a simple empirical Bayes method to estimate the TMRCA. This method is both asymptotically optimal, in the sense that the estimator converges to the true value when the number of unlinked loci for which we have information is large, and has the advantage of not making any assumptions about demographic history. The algorithm works as follows: we first split the sample at each locus into inferred left and right clades to obtain many estimates of the TMRCA, which we can average to obtain an initial estimate of the TMRCA. We then use nucleotide sequence data from other unlinked loci to form an empirical distribution that we can use to improve this initial estimate.

Roth FP, Wakeley J. Taking exception to human eugenics. Genetics. 2016;204 :821-823. (pdf)
Wakeley J, King L, Wilton PR. Effects of the population pedigree on genetic signatures of historical demographic events. PNAS. 2016;113 (29) :7994-8001.Abstract

Genetic variation among loci in the genomes of diploid biparental organisms is the result of mutation and genetic transmission through the genealogy, or population pedigree, of the species. We explore the consequences of this for patterns of variation at unlinked loci for two kinds of demographic events: the occurrence of a very large family or a strong selective sweep that occurred in the recent past. The results indicate that only rather extreme versions of such events can be expected to structure population pedigrees in such a way that unlinked loci will show deviations from the standard predictions of population genetics, which average over population pedigrees. The results also suggest that large samples of individuals and loci increase the chance of picking up signatures of these events, and that very large families may have a unique signature in terms of sample distributions of mutant alleles.

Palamara PF, Francioli LC, Wilton PR, et al. Leveraging distant relatedness to quantify human mutation and gene-conversion rates. Am. J. Hum. Genet. 2015;97 (6) :775-789.Abstract

The rate at which human genomes mutate is a central biological parameter that has many implications for our ability to understand demographic and evolutionary phenomena. We present a method for inferring mutation and gene-conversion rates by using the number of sequence differences observed in identical-by-descent (IBD) segments together with a reconstructed model of recent population-size history. This approach is robust to, and can quantify, the presence of substantial genotyping error, as validated in coalescent simulations. We applied the method to 498 trio-phased sequenced Dutch individuals and inferred a point mutation rate of 1.66 x 10(-8) per base per generation and a rate of 1.26 x 10(-9) for <20 bp indels. By quantifying how estimates varied as a function of allele frequency, we inferred the probability that a site is involved in non-crossover gene conversion as 5.99 x 10(-6). We found that recombination does not have observable mutagenic effects after gene conversion is accounted for and that local gene-conversion rates reflect recombination rates. We detected a strong enrichment of recent deleterious variation among mismatching variants found within IBD regions and observed summary statistics of local sharing of IBD segments to closely match previously proposed metrics of background selection; however, we found no significant effects of selection on our mutation-rate estimates. We detected no evidence of strong variation of mutation rates in a number of genomic annotations obtained from several recent studies. Our analysis suggests that a mutation-rate estimate higher than that reported by recent pedigree-based studies should be adopted in the context of DNA-based demographic reconstruction.

Palacios JA, Wakeley J, Ramachandran S. Bayesian nonparametric inference of population size changes from sequential genealogies. Genetics. 2015;201 (1) :281-304.Abstract

Sophisticated inferential tools coupled with the coalescent model have recently emerged for estimating past population sizes from genomic data. Recent methods that model recombination require small sample sizes, make constraining assumptions about population size changes, and do not report measures of uncertainty for estimates. Here, we develop a Gaussian process-based Bayesian nonparametric method coupled with a sequentially Markov coalescent model that allows accurate inference of population sizes over time from a set of genealogies. In contrast to current methods, our approach considers a broad class of recombination events, including those that do not change local genealogies. We show that our method outperforms recent likelihood-based methods that rely on discretization of the parameter space. We illustrate the application of our method to multiple demographic histories, including population bottlenecks and exponential growth. In simulation, our Bayesian approach produces point estimates four times more accurate than maximum-likelihood estimation (based on the sum of absolute differences between the truth and the estimated values). Further, our method's credible intervals for population size as a function of time cover 90% of true values across multiple demographic scenarios, enabling formal hypothesis testing about population size differences over time. Using genealogies estimated with ARGweaver, we apply our method to European and Yoruban samples from the 1000 Genomes Project and confirm key known aspects of population size history over the past 150,000 years.

Carmi S, Wilton PR, Wakeley J, Pe'er I. A renewal theory approach to IBD sharing. Theoret. Pop. Biol. 2014;97 :35-48.Abstract

A long genomic segment inherited by a pair of individuals from a single, recent common ancestor is said to be identical-by-descent (IBD). Shared IBD segments have numerous applications in genetics, from demographic inference to phasing, imputation, pedigree reconstruction, and disease mapping. Here, we provide a theoretical analysis of IBD sharing under Markovian approximations of the coalescent with recombination. We describe a general framework for the IBD process along the chromosome under the Markovian models (SMC/SMC’), as well as introduce and justify a new model, which we term the renewal approximation, under which lengths of successive segments are independent. Then, considering the infinite-chromosome limit of the IBD process, we recover previous results (for SMC) and derive new results (for SMC’) for the mean number of shared segments longer than a cutoff and the fraction of the chromosome found in such segments. We then use renewal theory to derive an expression (in Laplace space) for the distribution of the number of shared segments and demonstrate implications for demographic inference. We also compute (again, in Laplace space) the distribution of the fraction of the chromosome in shared segments, from which we obtain explicit expressions for the first two moments. Finally, we generalize all results to populations with a variable effective size.

Pennings PS, Kryazhimskiy S, Wakeley J. Loss and recovery of genetic diversity in adapting populations of HIV. PLoS Genet. 2014;10(1): e1004000.Abstract

The evolution of drug resistance in HIV occurs by the fixation of specific, well-known, drug-resistance mutations, but the underlying population genetic processes are not well understood. By analyzing within-patient longitudinal sequence data, we make four observations that shed a light on the underlying processes and allow us to infer the short-term effective population size of the viral population in a patient. Our first observation is that the evolution of drug resistance usually occurs by the fixation of one drug-resistance mutation at a time, as opposed to several changes simultaneously. Second, we find that these fixation events are accompanied by a reduction in genetic diversity in the region surrounding the fixed drug resistance mutation, due to the hitchhiking effect. Third, we observe that the fixation of drug-resistance mutations involves both hard and soft selective sweeps. In a hard sweep, a resistance mutation arises in a single viral particle and drives all linked mutations with it when it spreads in the viral population, which dramatically reduces genetic diversity. On the other hand, in a soft sweep, a resistance mutation occurs multiple times on different genetic backgrounds, and the reduction of diversity is weak. Using the frequency of occurrence of hard and soft sweeps we estimate the effective population size of HIV to be 1:5|105 (95% confidence interval ½0:8|105,4:8|105). This number is much lower than the actual number of infected cells, but much larger than previous population size estimates based on synonymous diversity. We propose several explanations for the observed discrepancies. Finally, our fourth observation is that genetic diversity at non-synonymous sites recovers to its pre-fixation value within 18 months, whereas diversity at synonymous sites remains depressed after this time period. These results improve our understanding of HIV evolution and have potential implications for treatment

Citation: Pennings PS, Kryazhimskiy S, Wakeley J (2014) Loss and Recovery of Genetic Diversity in Adapting Populations of HIV. PLoS Genet 10(1): e1004000.

Editor: Christophe Fraser, Imperial College London, United Kingdom
Received April 19, 2013; Accepted October 19, 2013; Published January 23, 2014

Copyright: 2014 Pennings et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: SK was supported by a Career Award at Scientific Interface from the Burroughs Wellcome Fund ( PSP was supported by a
long-term postdoctoral fellowship of the Human Frontier Science Program ( The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.

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Wakeley J. Coalescent theory has many new branches. Theoret. Pop. Biol. 2013;87 :1-4. (pdf)
Arnold B, Bomblies K, Wakeley J. Extending coalescent theory to autotetraploids. Genetics. 2012;192 (1) :195-204.Abstract

We develop coalescent models for autotetraploid species with tetrasomic inheritance. We show that the ancestral genetic process in a large population without recombination may be approximated using Kingman’s standard coalescent, with a coalescent effective population size 4N. Numerical results suggest that this approximation is accurate for population sizes on the order of hundreds of individuals. Therefore, existing coalescent simulation programs can be adapted to study population history in autotetraploids simply by interpreting the timescale in units of 4N generations. We also consider the possibility of double reduction, a phenomenon unique to polysomic inheritance, and show that its effects on gene genealogies are similar to partial self-fertilization.

Wakeley J, King L, Low BS, Ramachandran S. Gene genealogies within a fixed pedigree, and the robustness of Kingman's coalescent. Genetics. 2012;190 (4) :1433-1445.Abstract

We address a conceptual flaw in the backward-time approach to population genetics called coalescent theory as it is applied to diploid biparental organisms. Specifically, the way random models of reproduction are used in coalescent theory is not justified. Instead, the population pedigree for diploid organisms--that is, the set of all family relationships among members of the population--although unknown, should be treated as a fixed parameter, not as a random quantity. Gene genealogical models should describe the outcome of the percolation of genetic lineages through the population pedigree according to Mendelian inheritance. Using simulated pedigrees, some of which are based on family data from 19th century Sweden, we show that in many cases the (conceptually wrong) standard coalescent model is difficult to reject statistically and in this sense may provide a surprisingly accurate description of gene genealogies on a fixed pedigree. We study the differences between the fixed-pedigree coalescent and the standard coalescent by analysis and simulations. Differences are apparent in recent past, within ≈ <log(2)(N) generations, but then disappear as genetic lineages are traced into the more distant past.

Maruvka YE, Shnerb NM, Bar-Yam Y, Wakeley J. Recovering population parameters from a single gene genealogy: an unbiased estimator of the growth rate. Mol. Biol. Evol. 2011;28 (5) :1617-1631.Abstract
We show that the number of lineagesancestral to a sample, as a function of time back into the past, which we call the number
of lineages as a function of time (NLFT), is a nearly deterministic property of large-sample gene genealogies. We obtain analytic
expressionsfor the NLFT for both constant-sizedand exponentiallygrowing populations.The low level of stochastic variation
associated with the NLFT of a large sample suggests using the NLFT to make estimates of population parameters. Based on
this, we develop a new computational method of inferring the size and growth rate of a population from a large sample of
DNA sequences at a single locus. We apply our method first to a sample of 1,212 mitochondrial DNA (mtDNA) sequences
from China, confirming a pattern of recent population growth previously identified using other techniques, but with much
smaller confidence intervals for past population sizes due to
the low variation of the NLFT. We further analyze a set of 63
mtDNA sequences from blue whales (BWs), concluding that the population grew in the past. This calls for reevaluation of
previous studies that were based on the assumption that the BW population was fixed.