Inferring Ancestral Recombination Graphs from Bacterial Genomic Data.

Homologous recombination is a central feature of bacterial evolution, yet it confounds traditional phylogenetic methods. While a number of methods specific to bacterial evolution have been developed, none of these permit joint inference of a bacterial recombination graph and associated parameters. In this article, we present a new method which addresses this shortcoming. Our method uses a novel Markov chain Monte Carlo algorithm to perform phylogenetic inference under the ClonalOrigin model. We demonstrate the utility of our method by applying it to ribosomal multilocus sequence typing data sequenced from pathogenic and nonpathogenic Escherichia coli serotype O157 and O26 isolates collected in rural New Zealand. The method is implemented as an open source BEAST 2 package, Bacter, which is available via the project web page at http://tgvaughan.github.io/bacter.

RECOMBINATION plays a crucial role in the molecular evolution of many bacteria, in spite of the clonal nature of bacterial reproduction. Indeed, for a large number of species surveyed in recent studies (Vos and Didelot 2009; Fearnhead ), homologous recombination was found to account for a similar or greater number of nucleotide changes than point mutation.

However, many traditional phylogenetic methods (Huelsenbeck and Ronquist 2001; Drummond ; Guindon and Gascuel 2003) do not account for recombination. This is regrettable for several reasons. First, ignoring recombination is known to bias phylogenetic analyses in various ways such as by overestimating the number of mutations along branches, artificially degrading the molecular clock hypothesis, and introducing apparent exponential population growth (Schierup and Hein 2000). Second, much of modern computational phylogenetics extends beyond the inference of phylogenetic relationships and instead focuses on the parametric and nonparametric inference of the dynamics governing the population from which the genetic data are sampled. In this context, the phylogeny is merely the glue that ties the data to the underlying population dynamics. Recombination events contain a strong phylogenetic signal, so incorporating recombination into the phylogenetic model can significantly improve analyses. For instance, Li and Durbin (2011) used a recombination-aware model to recover detailed ancestral population dynamics from pairs of human autosomes, a feat which would have been impossible without the additional signal provided by the recombination process.

The standard representation of the phylogenetic relationship between ancestral lineages when recombination is present is the ancestral recombination graph (ARG) (Griffiths 1981; Hudson 1983), a timed phylogenetic network describing the reticulated ancestry of a set of sampled taxa. Several inference methods based on the ARG concept have been developed, many of which (Wang and Rannala 2008; Bloomquist and Suchard 2010; Li and Durbin 2011) assume a symmetry between the contributions of genetic material from the parent individuals contributing to each recombination event, as is the expected result of the crossover resolution of the Holliday junction in eukaryotic recombination. This assumption, which is often anchored in the choice to base the inference on the coalescent with recombination (Wiuf and Hein 1999), is not generally appropriate for bacterial recombination, where there is usually a strong asymmetry between the quantity of genetic material contributed from each “parent.”

Alternatively, a series of methods introduced by Didelot and Falush (2007), Didelot , and Didelot and Wilson (2015) directly target bacterial recombination by employing models based on the coalescent with gene conversion (Hudson 1983; Wiuf 2000; Wiuf and Hein 2000). These models acknowledge that the asymmetry present in the bacterial context allows for the definition of a precisely defined clonal genealogy—the clonal frame (CF)—which represents not only the true reproductive genealogy of a given set of bacterial samples, but also the ancestry of the majority of their genetic material.

In the first article, Didelot and Falush (2007) presented a method for performing inference under a model of molecular evolution, which, in combination with a standard substitution model, includes effects similar to those resulting from gene conversion; instantaneous events that simultaneously produce character-state changes at multiple sites within a randomly positioned conversion tract. This model does not consider the origin of these changes: it dispenses entirely with the ARG and can be considered a rather peculiar substitution model applied to evolution of sequences down the CF. Despite this, it does allow the Markov chain Monte Carlo (MCMC) algorithm implemented in the associated ClonalFrame software package to jointly infer the bacterial CF, conversion rate, and tract-length parameters; neatly avoiding the branch-length bias described by Schierup and Hein (2000). Didelot and Wilson (2015) introduced a maximum likelihood method for performing inference under the same model, making it possible to infer CFs from whole bacterial genomes as opposed to the short sequences that the earlier Bayesian method could handle.

In a second article, Didelot present a different approximation to the coalescent with gene conversion which retains the ARG but assumes that the ARG has the form of a tree-based network (Zhang 2015), with the CF taking on the role of the base tree. While acknowledging that their model could be applied to jointly infer the CF and the conversions, the algorithm they present is limited to performing inference of the gene conversion ARG given a separately inferred CF. This choice permitted the application of their model to relatively large genomic data sets.

This model was also used recently by Ansari and Didelot (2014), who exploit the Markov property of the model with regard to the active conversions at each site along an aligned set of sequences to enable rapid simulation under the model. These simulations were used in an approximate Bayesian computation scheme (Beaumont ) to infer the homologous recombination rate, tract lengths, and scaled mutation rate from full genome data, as well as to assess the degree to which the recombination process favors DNA from donors closely related to the recipient. As with the earlier study, this method requires that the CF be separately inferred.

In this article we present a Bayesian method for jointly reconstructing the ARG, the homologous conversion events, the expected conversion rate and tract lengths, and the population history from genetic sequence data. Our approach assumes the ClonalOrigin model of Didelot , extended to allow for the piecewise-constant or piecewise-linear variations in population size. It relies upon a novel MCMC algorithm which uses a carefully designed set of proposal distributions to make traversing the vast state space of the model tractable for practical applications. Unlike earlier methods, our algorithm jointly infers the CF, meaning that the inference is a single-step process. This has a number of advantages such as improving the quality of the resulting uncertainty estimates when phylogenetic signal is poor, and allowing the CF itself to be inferred under a more realistic model of evolution under homologous gene conversion.

In addition to the inference method itself, we present a basic technique for summarizing the sampled ARG posterior. Our approach is an extension of the maximum clade credibility tree approach (as described by Heled and Bouckaert 2013) to summarizing phylogenetic tree posteriors in which a summary of the CF is annotated with well-supported conversion events.

We demonstrate that our method can accurately infer known parameters from simulated data and apply it to a set of Escherichia coli ribosomal multilocus sequence typing (rMLST) (Jolley ) sequences derived from isolates collected from in and around the Manawatu region in New Zealand. The method reveals details of previously unobserved gene flow between pathogenic and nonpathogenic populations belonging to the serotype O157.

A software implementation of our method is distributed as a publicly available BEAST 2 (Bouckaert ) package. This gives the sampler a substantial amount of flexibility, allowing it to be used in combination with complex substitution models and a wide variety of prior distributions. Details on how to obtain and use this package are given on the project website at http://tgvaughan.github.io/bacter.

Materials and Methods

The ClonalOrigin genealogical model

In contrast to eukaryotes where recombination primarily occurs during meiosis, bacteria generally undergo recombination due to mechanisms that are not directly related to the process of genome replication. These mechanisms generally only result in the transfer of small fragments of genetic material. A result of this is that every homologous recombination event in bacteria is comparable to a gene conversion event, regardless of the underlying molecular biology. A good model for the genealogy of bacterial genomes is therefore the coalescent with gene conversion: a straight-forward extension to the Kingman -coalescent (Kingman 1982a,b) in which (a) lineages may bifurcate as well as coalesce, and (b) lineages are associated with a subset of sites on each of the sampled genetic sequences to which they are ancestral. At each bifurcation event, a contiguous range of sites is chosen for “conversion” by selecting a starting site uniformly at random and a tract length from a geometric distribution. The ancestry of the converted sites follows one parental lineage, while that of the unconverted sites follows the other.

The ClonalOrigin model is a simplification of the coalescent with gene conversion in which lineages are labeled as either clonal or nonclonal, with nonclonal lineages assumed to be free from conversion events (i.e., they may not bifurcate) and pairs of these lineages forbidden from coalescing. As Didelot argue, this simplified process is a good approximation to the full model in the limit of small expected tract length (relative to genome length) and low recombination rate. It also possesses features that make it an attractive basis for practical inference methods. First among these is that, conditional on the CF, the conversion events are completely independent. In our context, this simplifies the process of computing the probability of a given ARG and proposing the modifications necessary when exploring ARG space using MCMC.

We briefly reiterate the mathematical details of the model described in Didelot using terminology more appropriate for our purposes. We define the ClonalOrigin recombination graph where represents the CF and is a set of recombinant edges connecting pairs of points on The CF is assumed to be generated by an unstructured coalescent process governed by a time-dependent effective population size where measures time before the present. That is, the probability density of can be writtenHere is the set of internal (coalescent) nodes between edges of including the root node and are the ages of these nodes. The term represents the number of CF lineages extant at time

Conversion events appear at a constant rate on each lineage of and thus their number |R| is Poisson distributed with mean with being the sum of all branch lengths in Here is the per-site, per-unit-time rate of homologous gene conversion, is the expected conversion tract length and are the loci for which length sequence alignments are available. Each conversion is defined by where and identify points on at which the recombinant lineage attaches, with the age of less than that of The element indicates the locus to which the conversion applies, and and identify the start and end, respectively, of the range of sites affected by the conversion. The point is chosen uniformly over while is drawn from the conditional coalescent distribution,where and are the ages of points and respectively. The locus is chosen with probability the site is drawn from the distribution and the site is drawn from [In these equations is the indicator function.]

The full probability density for a ClonalOrigin ARG is then simply the product:where the |R|! accounts for independence with respect to label permutations of the recombination set Figure 1 illustrates the various elements of the ClonalOrigin model and associated notation.

Figure 1 

Schematic representation of a recombination graph for a single locus with CF and conversion The conversion attaches to at points and and affects sites through of the sites belonging to locus The horizontal bars represent ancestral sequences belonging to each lineage and colors are used to denote which samples each site is ancestral to, with white indicating sites ancestral to no samples. The graph on the right displays the associated CF lineages-through-time function together with the times used in computing These include the conversion attachment times and together with ages of coalescent nodes and i′.

Bayesian inference

Performing Bayesian inference under the ClonalOrigin model shares many similarities with the process of performing inference under the standard coalescent. The goal is to characterize the joint posterior density:where represents multiple sequence alignments for each locus in and represents one or more parameters of the chosen substitution model. The distributions on the right-hand side include the likelihood of the recombination graph; the probability density of the graph under the ClonalOrigin model discussed above; and the joint prior density of the model parameters.

To define the ARG likelihood, first consider that every ARG may be mapped onto a set of “local” trees describing the ancestry of contiguous ranges of completely linked sites in the alignment. The likelihood of is expressed in terms of local trees as the following productwhere is the portion of the alignment whose ancestry is described by local tree and is the standard phylogenetic tree likelihood (Felsenstein 2003).

Since it is possible for conversions to have no effect on there is no one-to-one correspondence between and This suggests that certain features of may be strictly nonidentifiable in terms of the likelihood function. As Bayesian inference deals directly with the posterior distribution, this nonidentifiability will not invalidate any analysis, provided that is proper. However, the existence of nonidentifiability has practical implications for the design of sampling algorithms, as we discuss in the following section.

MCMC

We use MCMC to sample from the joint posterior given in Equation 4. This algorithm explores the state space of (or some subspace thereof) using a random walk in which steps from to x′ are drawn from some proposal distribution and accepted with a probability that depends on the relative posterior densities at x′ and

In practice, is often expressed as a weighted sum of proposal densities (also known as proposals or moves) which individually proposes alterations to some small part of While there is considerable freedom in choosing a set of moves, their precise form can dramatically influence the convergence and efficiency of the sampling algorithm. Proposals should not generate new states that are too bold (accepted with very low frequency) nor too timid (accepted with very high frequency): both extremes tend to lead to chains with long autocorrelation periods. In this section we present an informal outline of the moves used in our algorithm. (Refer to the Appendix for a detailed description.)

For the subspace made up of the continuous model parameters and choosing appropriate proposals is relatively trivial as standard proposals for sampling from are sufficient. In our algorithm we use the univariate scaling operator described by Drummond , which can be made more or less bold simply by altering the size of the scaling operation.

For the ARG itself, assembling an appropriate set of moves is more difficult. Even determining exactly what constitutes a timid or bold move in space is hard to determine without detailed knowledge of the target density. Our general approach here is to design proposals that (a) only minimally affect the likelihood of where possible, and (b) draw any significant changes from the prior that the ClonalOrigin model places on The design of these proposals is assisted by our knowledge of the identifiability issue considered in the previous section: there is a many-to-one mapping from to the local tree set and the ARG likelihood depends only on Thus, ARG proposals that minimally effect the likelihood are those that propose a G′ that maps to the same or similar

The proposals for fall into two groups, the first of which deals exclusively with the set of conversions These include all three moves described by Didelot (we consider the conversion add/remove pair to be two halves of a single proposal), along with six additional simple moves aimed at quickly exploring the ARG state space conditional on Examples include a conversion merge/split proposal that merges pairs of conversions between the same pair of edges on the CF that affect nearby ranges of sites or splits single conversions into such pairs, a proposal which reversibly replaces a single conversion between two edges with a pair involving a third intermediate edge, and a proposal which adds or removes conversions that do not alter the topology of the CF.

Proposals in the second group propose joint updates to both the CF and the conversions Some of these moves are quite bold (and thus tend to be accepted rarely), but are very important for dealing with topological uncertainty in the CF. The general strategy for each move is to apply one of the tree proposals from Drummond to and to simultaneously modify the conversions in to ensure both compatibility with the C′ and to minimize the effect of the proposal on both the likelihood and the ARG prior. The changes to can for the most part be decomposed into primitive operations that involve selecting a subtree, deleting the edge attaching that subtree to the rest of the CF at time then reconnecting the subtree via a new edge e′ to a new point on at time Modification of is done using an approach (depicted in Figure 2) that consists of two distinct forms. The first form, the “collapse,” is applied whenever and involves finding conversions for which or are on the edge above the subtree and attach at times or greater than These attachment points are moved from their original position to contemporaneous points on the lineage ancestral to e′. The second form, the “expansion,” is applied when and is the inverse of the first: conversion attachments or at times are moved with some probability to contemporaneous positions on e′.

Figure 2 

Schematic representation of the collapse/expand strategy used by the MCMC algorithm to update conversions following the movement of a CF edge. (A) Illustrates a proposal to replace the thick black edge portion of the CF edge joined to with the thick gray edge portion joint to Since the collapse procedure is applied by moving affected conversion attachment points, highlighted with •, to contemporaneous points on the lineage ancestral to Any conversion with a new arrival point above the root is deleted from the new ARG. (B) Illustrates the reverse situation, where a CF edge attached at is reattached at Since the expand procedure is applied by moving any attachment points contemporaneous with a point on the newly extended portion of the CF edge to that point with some probability. Since becomes the new CF root, new conversions with arrival points on the new CF edge at times older than the previous CF root are drawn from the ClonalOrigin prior.

In concert, these proposals allow us to effectively explore the entire state space of

Summarizing the ARG posterior

Bayesian MCMC algorithms produce samples from posterior distributions rather than point estimates of inferred quantities. These approaches are superior because they give us the means to directly quantify the uncertainty inherent in the inference. For the very high dimensional state space that ARGs (even the ClonalOrigin model’s tree-based networks) occupy, actually visualizing this uncertainty and extracting an overall picture of the likely ancestral history of the sequence data are nontrivial.

A similar problem exists for Bayesian phylogenetic tree inference. Given the maturity of that field, it should not be surprising that a large number of solutions exist. The majority of these solutions involve the assembly of some kind of summary or consensus tree (see chapter 30 of Felsenstein 2003 for an overview, or Heled and Bouckaert 2013 for a recent discussion). While conceptually appealing, the replacement of a posterior distribution with a single tree can very easily lead to the appearance of signal where there is none, so care must be taken. At least one method exists that avoids this problem: the DensiTree software (Bouckaert 2010) simply draws all of the trees in a given set with some degree of transparency, making it possible to actually visualize the distribution directly.

Unfortunately, the approach taken by DensiTree cannot be easily applied to ARGs, since the recombinant edges introduce significant visual noise, making patterns difficult to discern. Nor can any of the standard summary methods be applied directly.

Instead, we use a summary of the CF posterior as a starting point to produce summary ARGs, as described in Algorithm 1. In the algorithm, MCC refers to the maximal clade credibility tree (see, for instance, Heled and Bouckaert 2013), and the value of in step 3(c) imposes a threshold on the posterior support necessary for a conversion to appear in the summary. The relationship between the sampled conversions and the summary conversions is illustrated in Figure 3.

Figure 3 

This diagram illustrates the way that conversions are summarized by Algorithm 1. The solid tree on the left depicts the MCC summary of the CF, with each node labeled by its set of descendant leaves. The dashed edges represent distinct conversions that exist between a given pair of edges and in ARGs sampled from the posterior (with overlapping pairs of conversions present on single ARGs merged). The horizontal boxes on the right indicate the site regions affected by each conversion, with the graph above showing the fraction of sampled ARGs possessing conversions at each site. A summary conversion is recorded only when this fraction exceeds the threshold

Algorithm 1. Method used to summarize samples for from the marginal posterior for

1. Produce an MCC summary of and denote this

2. Label internal nodes in and every with their descendant leaf sets.

3. For each ordered triple where i,j are nodes in and is a locus in

(a) For each assemble the set of all conversions affecting locus with on the edge above and on the edge above

(b) Merge any in each with overlapping site ranges, averaging the attachment times, and collect all resulting merged conversions into the set

(c) Identify disjoint site ranges affected by at least conversions in and replace all contributing conversions with a single summary conversion with values for and averaged from the contributing conversions.

(d) Use the number of contributing conversions divided by as a proxy for the posterior support for the summary conversion.

Testing with simulated data demonstrates that the method is capable of recovering useful summaries. However, one significant drawback is that the algorithm only groups together sampled conversions that appear between identical (in the sense described in the algorithm) pairs of CF edges. This means that a single conversion with significant uncertainty in either of its attachment points or may appear as multiple conversions in the summary. As a result, we still consider the problem of how best to summarize the posterior distribution over ARGs a target for future research.

Data availability

The methods presented in this article are implemented in the open source BEAST 2 package, Bacter (http://tgvaughan.github.io/bacter). The BEAST 2 XML files necessary to reproduce both the simulated and real data analyses are provided as Supplemental Material, File S2.

Results

Implementation and validation

The methods described here are implemented as a BEAST 2 package. This allows the large number of substitution models, priors, and other phylogenetic inference methods already present in BEAST 2 to be used with the ClonalOrigin model.

Despite the reuse of an existing phylogenetic toolkit, the implementation is still complex. As such, the importance of validating the implementation cannot be overstated. Our validation procedure involved two distinct phases: sampling from the ARG prior and performing inference of known parameter values from simulated data.

Sampling from the ARG prior:

This first phase of the validation involves using the MCMC algorithm to generate samples from i.e., the prior distribution over ARG space implied by the ClonalOrigin model. Unlike the full posterior density, we can also sample from this distribution via direct simulation of ARGs. Statistical comparisons between these two distributions should yield perfect agreement. Assuming that errors in both the MCMC algorithm implementation and the ARG simulation algorithm are unlikely to produce identically erroneous results, this is a stringent test of all aspects of our implementation besides calculation of the ARG likelihood.

Figure 4 displays a comparison between the histograms for a number of summary statistics computed from ARGs with five (noncontemporaneous) leaves sampled using our implementation of each method. The MCMC chain was allowed to run for iterations with ARGs sampled every steps, while the simulation method was used to generate independent ARGs. The close agreement between the two sets of histograms is very strong evidence that our implementation of both algorithms is correct.

Figure 4 

Comparison between distributions of summary statistics computed from ARGs simulated directly under the model (gray lines) and ARGs sampled using the MCMC algorithm (black lines). These include (A) the age of the CF root node, (B) the number of recombinations, and the average length of the recombinant (C) edges and (D) tracts on each sampled ARG. Exact agreement for each summary suggests that both algorithms are correct.

Inference from simulated data:

A common way to determine the validity and usefulness of an inference algorithm is to assess its ability to recover known truths from simulated data. In contrast with sampling from the prior, inference from simulated data is sensitive to the implementation of the ARG likelihood. Here we use a well-calibrated (Dawid 1982) form of the test, which requires that known true values fall within the estimated 95% highest posterior density (HPD) interval 95% of the time.

The details of the validation procedure are as follows. First, 100 distinct 10-leaf ARGs were simulated under the ClonalOrigin model with parameters and These ARGs were then used to produce an equivalent number of two-locus alignments, with each locus containing sites. Finally, each simulated alignment was used as the basis for inference of the ARG using the MCMC algorithm described above, conditional on the known true parameters.

The circles in the graphs shown as Figure 5 display the fraction of the sampled marginal MCMC posteriors for the CF time to most recent common ancestor (tMRCA) and recombination event count which included the known true values as a function of the relative HPD interval width. The dashed lines indicate the fractions expected of a well-calibrated analysis. This close agreement therefore suggests that our analysis method is internally consistent in this regard, a result which strongly implies that our implementation is correct.

Figure 5 

Coverage fraction vs. HPD interval width for (A) the CF tMRCA and (B) the recombination event count posteriors inferred from simulated sequence data. The ○ represents the observed coverage fraction, while the dashed lines indicate the coverage fraction to be expected from a well-calibrated analysis.

Example: E. coli

We applied our new method to the analysis of sequence data collected from a set of 23 E. coli isolates. The isolates were derived from from both humans and cattle and include both Shiga toxin-producing E. coli (STEC) and non-STEC representatives of the O26 and O157 serotypes. The analysis focused on the 53 loci targeted by rMLST (Jolley ).

The analysis was performed under the assumption of a constant population, the size of which was given a log-normal prior The Hasegawa–Kishino–Yano substitution model (Hasegawa ) was used, with uniform priors placed on the relative site frequencies and a log-normal prior placed on the transition/transversion relative rate parameter We also infer the relative substitution rate with being the average substitution rate per site. For this we use an informative log-normal prior whose 95% HPD includes a previously published estimate of 1.024 (Didelot ). The expected tract length parameter was fixed at sites.

Six unique instances of the MCMC algorithm were run in parallel. Five of these were run for iterations while the sixth was run for iterations, the longest of these taking ∼1 week to run on a modern computer. Comparison of the posteriors sampled by each of these chains demonstrated that convergence had been achieved. Final results were obtained by removing the first 10% of samples from each chain to account for burn-in and then concatenating the results. Once complete, the effective sample size for every model parameter and summary ARG statistic recorded surpassed 200.

The final results of this analysis are presented as Figure 6. First, Figure 6A displays a summary ARG produced from the sampled ARG posterior using a conversion posterior cutoff threshold of 0.4. This summary shows that four conversion events have posterior support exceeding this threshold. Three of these depict gene conversion events that transfer nucleotides between lineages ancestral to samples with O157 serotype. More specifically, the conversions result in gene flow from lineages ancestral to pathogenic (+STEC) samples to lineages ancestral to nonpathogenic (−STEC) samples. The remaining conversion event is indicative of a recent introgression from the O26 serotype into −STEC O157.

Figure 6 

(A) Summary ARG produced by applying our method to sequences obtained from 23 E. coli isolates. Dashed edges represent summary conversions, with the numbers giving the estimated posterior support values. Conversions originating from the root edge of the CF have been omitted. (B) Posterior distributions over nucleotides transferred between lineages ancestral to +STEC and −STEC O157 samples. (C) Posterior and prior distributions for the relative recombination rate,

This overall pattern is also reflected in Figure 6B, which displays the posterior distributions for the total number of nucleotides transferred by conversion events between +/−STEC O157 ancestral lineages: the gene flow from +STEC to −STEC O157 is on average greater than that in the reverse direction. This asymmetry is, however, very slight—a fact that may be attributed to the presence of a large number of “background” conversions which individually lack the posterior support to be included in the summary, but which nevertheless contribute to the particular gene flow metric we have chosen.

Finally, Figure 6C displays the posterior distribution for the relative recombination rate parameter, giving a 95% HPD interval of [0.21, 1.44]. The log-normal prior density for the recombination rate is also shown and indicates that the data are informative for this parameter.

Discussion

Dealing appropriately with recombination in a phylogenetic setting is a difficult task for a number of reasons. First, the progressive bifurcation of lineages with increasing age steadily decrease the signal for these features in a given data set. Furthermore, the possibility of these bifurcations drastically increases the size of the state space occupied by the genealogy. Indeed, even for a small number of aligned sequences, the upper bound of the number of coalescent events influencing the evolution of those sequences is potentially huge: the total number of nucleotide sites in the alignment. Considering that the superexponential rate at which the number of binary trees grows as a function of sample size already presents complexity problems for computational phylogenetics, it is no surprise that models that explicitly consider recombination are not as widely used in genealogical inference.

Despite these challenges, Didelot and coauthors have shown repeatedly that traditional coalescent-based phylogenetic inference methods can be applied to such models, by applying carefully chosen simplifications to the coalescent with gene conversion which reduce the state space while maintaining sufficient realism in the important context of bacterial evolution. In our article we have sought to continue in this tradition, and have demonstrated that one can indeed perform full joint inference of tree-based ARGs using a carefully constructed MCMC algorithm. Also, in our effort to narrow the technological gap between inference using the ClonalOrigin model and Bayesian inference performed using common nonrecombination-aware models, we have introduced a means of summarizing sampled tree-based ARG posteriors that is reminiscent of the methods often employed to summarize sampled tree posteriors.

Our joint approach has several advantages over the earlier method described by Didelot . That method involves separately inferring a point estimate of the CF under the model described by Didelot and Falush (2007) and conditioning inference of the rest of the ARG on this point estimate. First, as it does not rely on a point estimate of the CF, the joint approach more accurately characterizes the posterior for the ARG (and associated model parameters) and should yield more accurate estimates of statistical uncertainty when the statistical signal for the CF is weak. Properly representing this uncertainty is extremely important, as it is used to assess the strength of biological conclusions drawn from the inference.

Second, our joint estimation algorithm allows the CF, the recombinant edges, and the parameters to be inferred under a single self-consistent model (the ClonalOrigin model); a model which is a good approximation to a well-known mathematical model for bacterial evolution in the presence of homologous gene conversion (Hudson 1983; Wiuf 2000; Wiuf and Hein 2000). In contrast, the earlier method of Didelot relies on a distinctly different model (the ClonalFrame model) of sequence evolution that does not allow for topological differences in marginal trees. It is therefore unsurprising that the joint method recovers the truth more often than the earlier approach (see File S1, and Figures S1 and S2 in particular, for details).

We must emphasize, however, that despite making significant headway we do not consider either the ClonalOrigin inference problem nor the problem of summarizing posterior distributions over tree-based networks to be in any way “solved.” In the case of the inference problem, computational challenges relating to the way the algorithm scales with increasing frequency of recombination remain. This problem is tied directly to the large amount of computation required to calculate the ARG likelihood (Equation 5). The tree likelihood calculation is often the most computationally expensive calculation even in standard phylogenetic analyses, and recombination only multiplies this expense. It may be the case that improving this situation will require replacing the mathematically exact likelihood evaluation under a given substitution model with a carefully chosen approximation, but the feasibility and usefulness of this approach has yet to be fully investigated.

The problem of summarizing posterior distributions over tree-based networks would seem to be a fruitful line of future research. The algorithm presented here does seem to perform relatively well from an empirical standpoint, and to our knowledge is the first of its kind. However, it does have drawbacks relating to its propensity to misclassify conversions for which topological uncertainty exists (i.e., uncertainty in the CF edge to which one or both of its end-points attach) as multiple distinct conversions with a proportionally smaller posterior support. Solving this problem would seem to be nontrivial, as it requires the algorithm to identify a conversion in one sampled ARG with a conversion in a second ARG even when those conversions join distinct pairs of edges on the CF. However, we feel that tackling these and other related problems is a worthwhile endeavor, and one which should encourage mainstream adoption of recombination-aware Bayesian phylogenetic inference methods.

Supplementary Material

Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.193425/-/DC1.

Click here for additional data file.

Click here for additional data file.

Click here for additional data file.

Click here for additional data file.

Click here for additional data file.