Using jackknife to assess the quality of gene order phylogenies
 Jian Shi^{1},
 Yiwei Zhang^{1},
 Haiwei Luo^{2} and
 Jijun Tang^{1}Email author
DOI: 10.1186/1471210511168
© Shi et al; licensee BioMed Central Ltd. 2010
Received: 21 October 2009
Accepted: 6 April 2010
Published: 6 April 2010
Abstract
Background
In recent years, gene order data has attracted increasing attention from both biologists and computer scientists as a new type of data for phylogenetic analysis. If gene orders are viewed as one character with a large number of states, traditional bootstrap procedures cannot be applied. Researchers began to use a jackknife resampling method to assess the quality of gene order phylogenies.
Results
In this paper, we design and conduct a set of experiments to validate the performance of this jackknife procedure and provide discussions on how to conduct it properly. Our results show that jackknife is very useful to determine the confidence level of a phylogeny obtained from gene orders and a jackknife rate of 40% should be used. However, although a branch with support value of 85% can be trusted, low support branches require careful investigation before being discarded.
Conclusions
Our experiments show that jackknife is indeed necessary and useful for gene order data, yet some caution should be taken when the results are interpreted.
Background
Phylogenetic reconstruction is the process to determine the evolutionary histories among organisms. While biologists primarily use DNA or protein sequences to study phylogenies, higherlevel rearrangement events such as inversions and transpositions are proving to be useful in elucidating evolutionary relationships. As a result, researchers have used the rearrangement of gene orders to infer highquality phylogenies [1–4].
Given a set of DNA sequences, we can use procedures such as bootstrap to assign confidence values to edges (branches) in phylogenetic trees [5]. Edges with high confidence values (> 75  80%) are generally considered acceptable. However, such procedures are impossible for gene order data since essentially gene orders can be viewed as one character with a very large number of states [6].
Several papers presented a jackknife procedure to overcome the problem [1–3]. However, there are many questions to be answered regarding the performance of jackknife. For example, we need to know how many genes should be removed and how many replicates are needed. We even do not know if jackknife on gene order data will converge. We also need to know above what threshold of confidence values can we claim an edge correct.
In this paper, we conduct a set of experiments to tackle these questions. The remainder of this paper is organized as follows: We first review gene order data and genome rearrangements, along with general bootstrap and jackknife procedures. We then provide details of our experiments. In the Result section, we determine good rates of jackknife, the number of replicates required, and the accuracy of confidence values.
Gene orders and rearrangements
The inversion distance between two genomes is the minimum number of inversions needed to transform one into the other. Hannenhalli and Pevzner [7] developed a theory for signed permutations and provided a polynomialtime algorithm to compute the edit distance (and the corresponding minimum edit sequence) between two signed permutations under inversions. However, the minimum distance may significantly underestimate the true number of events that have occurred. Several true inversion distance estimators have been proposed and among them, the EDE correction [8] is the most used.
There are several widely used methods to reconstruct phylogenies from gene order data, including distancebased methods (neighborjoining [9] and FastME [10]), Bayesian (Badger [11]) and direct optimization methods (GRAPPA [12] and MGR [13]). Using corrected inversion distances, Wang et al. showed that highquality phylogenies can be obtained using distancebased methods such as Neighborjoining and FastME [14]. On the other hand, although Badger, GRAPPA and MGR are more accurate, these methods are computationally very demanding and may not be able to analyze datasets when genomes are distant.
Several other methods have been proposed. For example, MPBE [15] transforms adjacency pairs from the signed permutation into sequencelike strings, while the method proposed by Adam et al. [16] used common intervals (subsets of clusters contiguous in both genomes) to represent gene orders as binary strings. In MPBE, each gene ordering is translated into a binary sequence, where each site from the binary sequence corresponds to a pair of genes. For the pair (g_{ i }, g_{ j }), the sequence has a 1 at the corresponding site if g_{ i }is immediately followed by g_{ j }in the gene ordering and a 0 otherwise. These transformed strings are then inputs to the ordinary sequence parsimony software (e.g. PAUP* 4.0 [17]) to obtain a phylogeny. For a complete review, please see [18].
Bootstrap and jackknife
Bootstrap is commonly used to assess the quality of sequencebased phylogenies. The bootstrap procedure generally starts with creating new alignments by randomly picking alignment columns from the original input alignment and reconstruct a tree independently on each new alignment. A consensus tree is then constructed to summarize the results of all tree replicates. The confidence value for an edge in the consensus tree is defined to be the number of replicates in which it appears. If the confidence value for a given edge is 75% or higher, the topology at that branch is generally considered correct.
Although the above bootstrap procedure can be applied to methods such as MPBE where each character of the converted string is treated independently. However, it is not possible to perform this procedure in GRAPPA, MGR and most other methods (except e.g. [15, 16]), since for these methods, gene order data can be viewed as one character with 2^{ n }n! possible states for genomes with n genes [6].
There are several other ways to apply disturbance to gene order data and assess the robustness of the data. For example, one can randomly remove a genome from the dataset or randomly perform a number of events on the gene orders. However, even with 1000 genomes, removal of just one may not introduce enough disturbance. On the other hand, there are many parameters to consider in the latter approach: we need to determine what kind of events to be included, which evolutionary model to use and how to apply the events, how many events to apply, and if we should apply the same amount of events on each genome. Since we still do not have a good evolutionary model for genome rearrangements, it will be difficult to develop an assessment method based on this approach.
Several researchers (including our group) began to use a procedure called jackknife to overcome the problem [1–3].
However, to our knowledge, no detailed study on the performance of this method has been conducted.
In general, the jackknife procedure is performed using the following steps:

Generating k new sets of genomes by deleting some genes. Orders of the remaining genes are preserved with respect to their orders in the original genomes.

Reconstructing tree replicates from these new genomes.

Computing a consensus tree and corresponding confidence values on all internal edges.
A consensus tree can be obtained using majority rule, i.e. the consensus only contains edges that exist in more than half of the input trees. The extended majority rule method uses the majority rule result as a start and greedily adds edges that occur in less than half of the input trees, with the aim that a full binary tree can be obtained. In this paper, we use the CONSENSE program in PHYLIP [19]. We find that the extended majority rule consensus trees generally outperform those computed with majority rule.
Results
Determining jackknife rate
Indeed, jackknife has been used for sequence data before, although it is not as common as bootstrap. Felsenstein suggested for DNA sequences, that "one way to make the jackknife vary as much as the bootstrap would be to delete half of the characters, at random, in each replicate [5]." Farris later stated that 50% deletion is too severe [20] and suggested the rate of 1/e ≈ 37% should be used. The jackknife rate (how many genes should be deleted) is critical for gene order data as well: leaving too few genes out would not produce enough disturbances to the original data, while removing too many genes would make the data totally unrecognizable. The jackknife rate of 50% was adopted by the limited number of papers where jackknife were used [1–3]. However, no discussion was given on the choice of such rate.
Number of replicates required
In [1–3], the authors used 100 replicates to obtain the confidence values, following traditions in bootstrap. Pattengale et al. [22] discussed the number of replicates for DNA bootstrap and conducted a complete research about finding the correct number of bootstrap replicates. They found that this number indeed varies in a big range. To find out the requirement of replicates in gene order data, we conduct similar testing:

For a given dataset, generate k replicates using jackknife rate of 40%, starting from k = 50.

Randomly split the k replicates into two equal sized subsets s_{1} and s_{2}, each containing k/ 2 replicates.

Compute a consensus tree t_{1} from subset s_{1} and compare it with the consensus tree t_{2} obtained from s_{2}.

Stop if t_{1} and t_{2} are very close; otherwise, increase k by 50 and repeat the above procedures.
Threshold of confidence values
The confidence values of internal edges are perhaps the most valuable information obtained through the jackknife procedure. However, as in bootstrap, the meaning of these values is always up for interpretation. The most important question is to determine where to draw the threshold so that edges with confidence values higher than this threshold can be trusted, whereas edges with lower values can be discarded.
We design the following experiments to find out a good threshold value:

For each dataset, determine its converging point k and compute a consensus tree on these k replicates.

For a given threshold value M, contract all edges with confidence values below M.

Compare the true trees with the contracted trees to obtain FP and FN rates.

Repeat the above procedures for 60 ≤ M ≤ 95.
However, the FN rates are very high for these low thresholds, especially when the genomes are distant.
By comparing all values presented in Figures 4 to 6, we suggest the use of threshold value of 85%, which results in the best balance of FP and FN. Under the extreme case, using M = 85%, almost 50% true branches can be resolved with only 10% chance of error, and the expected FP rates are ≤ 3%.
Methods
In this paper, we concentrate our experiments on simulated datasets so that the quality of jackknife replicates can be assessed against the known true tree. In our simulations, we generate model tree topologies from the uniform distribution on binary trees, each with 20 leaves. On each tree, we evolve signed permutations of 100 and 1000 genes using various numbers of evolutionary rates: letting r denote the expected number of inversions along an edge of the true tree, we use values of r = 2, 4, 8, ⋯, 32 for 100 genes and r = 20, 40, 80, ⋯, 320 for 1000 genes. The actual number of inversions along each edge is sampled from a uniform distribution on the set . For each combination of parameter settings, we run 100 datasets and average the results.
We always use FastME to obtain phylogenies since it is very accurate with corrected inversion distances [14]. Other methods (GRAPPA and MGR) will take very long time for datasets with 20 genomes and large r values.
We assess topological accuracy via false negatives and false positives [21]. Let T be the true tree and let T' be the inferred tree. An edge e in T is "missing" in T' if T' does not contain an edge defining the same bipartition; such an edge is called a false negative (FN). The false negative rate is the number of false negative edges in T' with respect to T divided by the number of internal edges in T. The false positive (FP) rate is defined similarly, by swapping T and T'. The RobinsonFoulds (RF) rate is defined as the average of the FN and FP rates. An RF rate of more than 5% is generally considered too high [24].
Conclusions
We have conducted extensive experiments to validate the performance of jackknife on gene order phylogenies. These testings show that jackknife is very useful to determine the confidence level of a phylogeny, and a jackknife rate of 40% should be used. However, although a branch with support value of 85% can be trusted, low support branches should not be discarded without further investigation. The jackknife rate of 40% is very close to the suggested rate of 37% for sequence data [20], thus we need to conduct theoretical analysis on the foundation of jackknife on genome rearrangements. All our experiments are conducted with FastME, experiments using other methods should be conducted to further evaluate the performance of jackknife.
Declarations
Acknowledgements
The authors were supported by US National Institutes of Health (grant number 3R01GM07899103S1) and National Science Foundation (grant number OCI 0904179). All experiments were conducted on a 128core shared memory computer supported by US National Science Foundation grant (NSF grant number CNS 0708391).
Authors’ Affiliations
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