Correction: Bayesian Genome Assembly and Assessment by Markov Chain Monte Carlo Sampling

due to Figure 4 and Figure 5 being ordered incorrectly. Please download the PDF again to view the corrected article. The originally published, uncorrected article and the republished, corrected article are provided here for reference. Copyright: ß 2014 The PLOS ONE Staff. 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.


Introduction
Most current methods for de novo genome assembly generate point estimates of the genome sequence without explicit statistical information about confidence in this particular estimate, or its support relative to other alternative assemblies supported by the same sequence data [1]. Recently, several probabilistic approaches have been proposed to quantify assembly certainty and address these limitations. Computing Genome Assembly Likelihood (CGAL) approximates the likelihood of an assembly given the sequence reads and a generative model learned from the data [2]; Log Average Probability (LAP) approximates the likelihood under a similar model of sequence read generation [3]; and Assembly Likelihood Evaluation (ALE) uses an empirical Bayesian approach to estimate the posterior probability of a particular assembly (or components of that assembly) [4]. Historically, the idea of applying statistical criteria to the fragment assembly problem was first proposed by Myers [5], who formulated the problem as a maximum-likelihood search.
Here we present an alternative Bayesian approach to approximating posterior probabilities, using Markov chain Monte Carlo (MCMC) sampling [6] to generate an entire posterior distribution of assembly hypotheses. An MCMC framework for sequence assembly addresses many of the general problems of accommodating assembly uncertainty, and provides a probabilistic understanding of the support for an assembly as a whole, as well as for particular features of the assembly that the investigator has special interest in. For example, it may be that each individual assembly hypothesis has low overall confidence, but that some feature, such as the order of a set of genes or existence of a specific regulatory element, is consistent across all the assembly hypotheses and has high confidence.

Design and Implementation
To demonstrate the feasibility of the MCMC approach, we have implemented a prototype assembler called Genome Assembly by Bayesian Inference (GABI, https://bitbucket.org/ mhowison/gabi). Because our approach is computationally expensive, our current tests are limited to the small but wellstudied genome of the WX174 phage (which has NCBI Reference Sequencec NC_001422.1). Our input data consist of 2,000 readpairs drawn from Illumina WX174 sample data (http://www. illumina.com/systems/miseq/scientific_data.ilmn). Aavailable as File S1, this subset of reads has maximum length 251 bp, an estimated mean insert size of 357 bp, and approximately 936 coverage of the WX174 genome.
Conveniently, our approach has technical parallels and similar conceptual advantages as the advances made over the past 15 years in the application of Bayesian approaches to phylogenetic analysis [7][8][9]. As in phylogenetics and other fields, designing an MCMC sampling strategy that achieves good mixing and convergence poses a variety of challenges and opportunities. For sequence assembly, the challenge is to design a mechanism for proposing new assemblies, a means of calculating the likelihood of a proposed assembly under an explicit model of read generation, and the specification of a prior probability distribution that describes some prior beliefs about the assembly before the sequence data were collected. To implement an MCMC sequence assembler, we made the following choices:

Proposal mechanism
We first build an assembly graph starting from a de Bruijn graph of the reads. Then we remove all tips (since the WX174 genome is circular) and merge all unambiguous paths into single nodes that are annotated by the sequence of merged k-mers. The resulting unresolved assembly graph (no longer de Bruijn) is a directed graph that consists only of bubbles and is a minimal representation of the variants that can be inferred from the sequenced data. Concatenating the sequences across the nodes in a particular path through this graph gives a possible assembly sequence.
A particular assembly hypothesis is represented as a boolean vector, whose values indicate which of the enumerated edges in the assembly graph are in an active path ( Figure 1). We start with a random vector and propose new assemblies by toggling a value at random in the current vector. However, when toggling, we enforce the constraint that an active path cannot take multiple branches through the same bubble, since this would not spell a contiguous sequence. Instead, turning on an edge in an alternate branch turns on the entire alternate branch and turns off the existing path through the bubble ( Figure 2). In this way, we can propose alternate paths through a complex bubble in one move of the sampler, without having to split the path and wait for the alternate branch to randomly activate. Furthermore, we can repeat this perturbation process a specified number of times for each new sample. Using several perturbations per sample causes the proposal mechanism to take larger steps through the combinatorial space of possible assemblies.
In our prototype implementation, we only consider paths that visit an edge at most once. In the general case, however, the proposal mechanism should be extended to accommodate paths that revisit edges, in order to resolve repeat regions of a genome.

Likelihood
Because of the constraints described above, the active paths in every proposed assembly can be output as a FASTA file of concatenated node sequences. We run LAP directly on this FASTA file to estimate the proposed assembly's likelihood.

Priors
An uninformative (e.g., flat) prior distribution could be used if nothing is known about the genome to be assembled. Since WX174 is well known, and it has previously been established that the genome is 5,386 bp and consists of one circular chromosome, we construct a prior probability distribution as the product of two gamma distributions, one for the sum of contig lengths (centered at 5,386) and the other for the number of contigs in the assembly (centered at 1). In other cases, external information about repeat structure or gene synteny could also be incorporated as priors.

Results
To evaluate the mixing and convergence of our MCMC sampler, we ran three independent chains and compared their traces and split frequencies, as is common practice for other applications of MCMC [10]. Our results show that our design achieve d good mixing when using a simple Metropolis sampler [11] with three perturbations per assembly proposal (Figure 3a). The cumulative frequencies of the individual nodes and edges were mostly stabilized after 4,000 accepted samples (Figure 3b). Those with weak support were likely assembled from reads with sequencing errors. Other nodes and edges were more ambiguous, with a cumulative frequency hovering at an intermediate value or varying across samples. Ten nodes achieved a frequency of 100% (darkest red nodes in Figure 4a) and these include the four nodes representating the longest unambiguous sequences in the assembly graph (each longer than 400 bp), for a total of 2,428 bp of sequence.
Overall, the standard deviation in edge frequencies between the chains decrease d with additional sampling, indicating that the independent chains converged to the same posterior distribution (Figure 3c). The split frequencies among the chains were mostly correlated after the last sample ( Figure 3d). We also tested the sampler with flat priors, with no likelihood calculations (priors only), and with different choices of the shape parameter for the gamma distributions ( Figure 5). Our acceptance rate was 22.4%, which is close to the heuristic of 25% that can be considered optimal for general Metropolis sampling [12]. Aggregated across GABI provides multiple ways for summarizing the results of an MCMC analysis and is built on top of BioLite, a light-weight bioinformatics framework with rich diagnostics and reporting capabilities [13]. The assembly graph can be annotated with the approximated posterior probabilities (Figure 4a), or a consensus assembly can be extracted, for instance a majority-rule consensus that shows all edges occurring with frequency w50% (Figure 4b).
The report also includes a FASTA file for the majority-rule consensus, annotated with the posterior probabilities of its components, and an interactive animation of the sampling for each MCMC chain, created with the D3js data visualization toolkit [14]. A sample report for the MCMC chains analyzed here is archived in the Brown Digital Repository [15].
GABI includes a tool to assign posterior probabilities to features of external assemblies that correspond to features in its own assembly graph. This provides an explicit and unified statistical framework for comparison of assemblies produced by multiple methods and software tools. Here, we compare NCBI Reference Sequence NC_001422.1 and de novo point-estimate assemblies generated by the String Graph Assembler [16] and Velvet [17]. We require exact matches to identify corresponding features, and this conservative strategy means that there is not an exact correspondence between the NCBI reference sequence and the GABI graph (Figure 4e). For this simple assembly problem, there is nearly universal agreement among the assemblies: both the majority rule consensus (Figure 4b) and NCBI reference sequence (Figure 4e) are proper subsets of the two de novo point-estimate assemblies (Figure 4c-d). The one notable difference is that the SGA assembly contains two contigs that choose alternate paths in one of the bubbles, and these alternate paths have similar posterior probabilities (green highlight in Figure 4c ).

Discussion
A challenge to scaling MCMC assembly is that the combinatorial complexity of larger assembly graphs could become prohibitive for full de novo Bayesian assembly of large genomes. There are several ways to address this, such as applying more sophisticated sampling methods like Metropolis-coupled MCMC [18] or bridging states [19] that can explore larger combinatorial spaces more efficiently; constraining the assembly graph to focus on particular assembly hypotheses instead of attempting full de novo assembly; partitioning a larger genome based on an existing draft assembly and performing MCMC assembly concurrently on subgraphs of less complexity; or pruning the assembly graph using additional data from restriction site mappings [20]. Another promising application is transcriptome assembly, since the assembly graphs for individual transcripts should be less complex and can be sampled independently.
Our prototype assembler cannot yet consider repeats, as each edge can only be visited once. Repeats are common in many genomes, and addressing them is critical if this approach is to be widely adopted. Fortunately, a straight-forward extension of our graph representation from a vector space of booleans to a vector space of nonÂ -negative integers can represent repeats, where the integer value for an edge denotes how many times it is visited. The proposal mechanism can be modified to enforce constraints on the in-and out-degree of the nodes in the path, to ensure that a proposed path can be unwound into a valid, contiguous sequence. Because paired-end alignments are used to calculate the likelihood, a path with the correct number of repeats will score a higher likelihood. We expect this to work well because two maximumlikelihood assembly approaches have already used the same principle to demonstrate high accuracy in resolving repeat regions [21,22].
Like existing assemblers, GABI can be used to assemble a point estimate of a genome, but unlike most other assemblers, the resulting assembly will have been chosen according to explicit statistical criteria (posterior probability) and will have associated information on the confidence in various aspects of its sequence and structure. In addition, MCMC assembly provides new opportunities for investigators who are interested not only in the certainty of a particular inference (e.g. an ALE [4] estimate of posterior probability for a given assembly), but in the many alternative hypotheses that are also supported by the data. The MCMC approach addresses many of the general problems of summarizing assembly uncertainty and will allow assembly uncertainty to be propagated to downstream analyses [1].

Methods
The results presented were generated with GABI version 0.2.0 and are recomputable using included scripts (https://bitbucket. org/mhowison/gabi/src/master/phix-test). Here, we provide brief comments on the technical details.

Compute resources
All tests were run at the Center for Computation and Visualization, Brown University, on IBM iDataPlex nodes, each equipped with 8-core, dual-socket Intel Xeon E5540 (2.53 Ghz) processors and 24 GB of memory. CPU-hours were calculated as total walltime across all nodes, multiplied by 8 CPUs per node.

SGA and Velvet assemblies
To assemble the WX174 subset with SGA, we followed the example provided for an assembly of E. coli from similar MiSeq reads (https://github.com/jts/sga/blob/master/src/examples/ sga-ecoli-miseq.sh). To assemble with Velvet, we used the included VelvetOptimiser tool to sweep coverage and cutoff parameters for a k-mer size of 99. The estimated mean and standard deviation of the insert size reported earlier were obtained from VelvetOpti-miser's output. This procedure is available in the script https:// bitbucket.org/mhowison/gabi/src/master/phix-test/01-assemble. sh.

De Bruijn graph reduction
First, we recursively and completely trimmed all tips (edges with an incidence of one), because the WX174 genome is circular (for a linear genome, a softer tip trimming threshold would be more appropriate). Next, we merged all simple paths through the graph, similar to the process of reducing an overlap graph to a string graph [23]. We annotated the new node with the accumulated overlap of the merged nodes' k-mers, which accumulates one additional nucleotide for each merged node. Finally, we split the graph into weakly connected components and choose the largest one. A path of edges through the graph spells a contig by concatenating the annotations on the nodes in the path.

Odds ratio
To determine whether to accept a proposed assembly hypothesis, we calculate the odds ratio [11] of the posterior probabilities of the old assembly, P(HDD), and the new perturbed assembly, P(H Ã DD), as:

R~P
If this ratio is greater than 1, the new assembly is automatically accepted. If it is less than 1, it is accepted with probability R. The process is then repeated until there is a stationary distribution of assemblies in the sample, which occurs when the frequency of assembly attributes does not change with additional sampling.