Novel Distances for Dollo Data

Novel Distances for Dollo Data

Michael Woodhams, Dorothy A. Steane, Rebecca C. Jones,
Dean Nicolle, Vincent Moulton, Barbara R. Holland
Abstract

We investigate distances on binary (presence/absence) data in the context of a Dollo process, where a trait can only arise once on a phylogenetic tree but may be lost many times. We introduce a novel distance, the Additive Dollo Distance (ADD), which is consistent for data generated under a Dollo model, and show that it has some useful theoretical properties including an intriguing link to the LogDet distance. Simulations of Dollo data are used to compare a number of binary distances including ADD, LogDet, Nei Li and some simple, but to our knowledge previously unstudied, variations on common binary distances. The simulations suggest that ADD outperforms other distances on Dollo data. Interestingly, we found that the LogDet distance performs poorly in the context of a Dollo process, which may have implications for its use in connection with conditioned genome reconstruction. We apply the ADD to two Diversity Arrays Technology (DArT) datasets, one that broadly covers Eucalyptus species and one that focuses on the Eucalyptus series Adnataria. We also reanalyse gene family presence/absence data on bacteria from the COG database and compare the results to previous phylogenies estimated using the conditioned genome reconstruction approach.

KEY WORDS: Additive Dollo Distance, Dollo process, LogDet/paralinear distances, Diversity Arrays Technology, Eucalyptus phylogeny, Adnataria phylogeny, gene content phylogeny, conditioning genomes.

1: School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart 7001, Australia
2: School of Plant Science and CRC for Forestry, University of Tasmania, Private Bag 12, Hobart 7001, Australia
3: Faculty of Science, Health, Education and Engineering, University of the Sunshine Coast Maroochydore DC, 4558
4: School of Plant Science, University of Tasmania, Private Bag 55, Hobart 7001, Australia
5: Currency Creek Arboretum, P.O. Box 808, Melrose Park, South Australia 5039, Australia
6: School of Computing Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom
*: Corresponding author. barbara.holland@utas.edu.au; Fax number (61) 3 6226 2410; Phone number (61) 3 6226 1990

Feb 2012

A fundamental idea in evolutionary biology is that when two species share a complex trait the most likely explanation of the similarity is that both species have inherited the trait from a common ancestor. However, the absence of a particular trait carries far less information, for instance wings and eyes are complex traits that have been lost many times independently in different parts of the evolutionary tree of life. As long ago as 1893 Dollo captured this idea in what is now known as Dollo’s Law which states that complex traits may be gained once somewhere in evolutionary history, and may be subsequently lost independently many times.

In this paper we will only consider Dollo models that generate binary data, recording the presence or absence of some trait. For example — does an organism have any genes in a particular gene family? Does it have some skeletal feature, such as the mammalian inner ear? When its DNA is digested by a mix of restriction enzymes, is a particular DNA fragment produced? In reality, determining if a trait is present or absent can be less clear cut; for example, paralogues can confound gene presence/absence decisions. In a stochastic Dollo model the gain and loss of such traits is treated as arising from a simple probability model. While few situations match Dollo’s Law exactly it provides a useful model in many evolutionary scenarios of interest. Dollo models have been used to understand gene families (Huson and Steel, 2004; Dagan and Martin, 2007) and complex morphological traits (Gould, 1970), and stochastic Dollo models have been used to study cognates in language evolution (Ryder and Nicholls, 2011; Nicholls and Gray, 2008).

In some of these scenarios such data can have another interesting property: only traits that are present in particular reference taxa are visible; or, in other words, the data are censored. This happens, for example, with array-based studies where a small set of taxa is used to create a set of traits (i.e. a set of DNA fragments that make up an array) to which other taxa can be compared. The idea of a reference taxon also has parallels in the gene-content setting where some authors have proposed “conditioned genome reconstruction” (Lake and Rivera, 2004), where one genome is selected as a reference and, for the remaining genomes, only gene families present in the reference genome are analyzed.

Data thought to follow Dollo’s Law traditionally have been analysed using a parsimony approach (Quesne, 1974; Farris, 1977). As is normally the case with parsimony approaches, branch length information is not taken into account. The use of stochastic Dollo models is relatively recent (Ryder and Nicholls, 2011; Nicholls and Gray, 2008) and, so far, they have only been implemented in a Bayesian framework. Bayesian methods are computationally intensive so there is a need for an approach that is both computationally efficient and statistically consistent.
This motivated us to develop a distance-based approach to Dollo data. Our initial motivation was to derive a distance suitable for phylogenetic analysis of Diversity Array Technology (DArT) data (Jaccoud et al., 2001) which, by its nature, is censored. On further consideration we realized that the same formula can be derived directly from the mathematics of the stochastic Dollo process, or as a limiting case of the LogDet distance.

In the following sections we begin by deriving the ADD in a general Dollo context and then show why it also applies to the censored Dollo model that arises for DArT data. We then describe an intriguing link to the popular LogDet distances. After introducing a few other binary distances we present a simulation study that compares the performance of the new ADD to other binary distances when applied to Dollo data under a range of censoring schemes. As an illustration of our approach we apply the new distance to three case studies, two involving DArT data for Eucalyptus species and one using gene content information. We conclude with a discussion in which we point out some potential future directions.

Methods

Deriving an Additive Distance for the Stochastic Dollo Process

The description of a stochastic Dollo process which we follow is that of Huson and Steel (2004), whose discussion is in the context of the gene content of a genome. This model can be described as a constant-birth, proportional-death Markov process. New markers are acquired (e.g. genes added to the genome) at rate , existing markers are independently deleted with intensity rate . is the set of markers present at time . We make an initial observation of the set of markers at start time . It is assumed that the system is at equilibrium at time , i.e. it has been evolving by the stochastic Dollo process for long enough to be independent of initial conditions. The genome then evolves for a further time and we observe the marker set . We then put

(1)

i.e. is the number of shared presences (markers present in the genome at both time points), and are the counts of markers present at one time point but not the other.

Under these circumstances, Huson and Steel (2004) prove the following facts:

  1. is Poisson distributed with mean .

  2. If is chosen according to this equilibrium Poisson distribution, then the process is a time-reversible Markov process.

  3. is Poisson distributed with mean .

  4. is binomially distributed with trials each with probability of success, where a “success” in this context is the loss of a marker.

From 4 we have expected value and we solve for :

(2)

and so we get a distance that is proportional to time given by

(3)

Moreover, is a statistically consistent estimator for (i.e. will converge on as we collect more data.) Note that by the time reversibility property, we could equally well use . To make use of all available data (both and ) we average these two distances to give

(4)

and call this the Additive Dollo Distance (ADD).

Dollo models with censored data

As mentioned in the introduction, some datasets of interest have an additional property whereby only markers which are present in the reference taxon or taxa can be detected. This is referred to either as “censored data” or as an “ascertainment bias”. We want to extend the ADD distance to cases of both single and multiple reference taxa. As we will see below, the first case does not require any change in the formula.

We introduce our approach for censored data that arise in the context of DArT (Jaccoud et al., 2001). DArT uses particular restriction enzymes to create a “genomic representation” (DNA fragments typically 300-1000 bp in length) from one or more reference taxa. The restriction enzymes used consist of a ‘rare cutter’ (e.g. Pst I, that cuts at sequence CTGCAG) and a ‘frequent cutter’ (e.g. BstN I, that cuts at sequence TCGA.) Only fragments cut at both ends by the rare cutter become markers. The fragments are cloned and are then arrayed onto a glass microscope slide. Genomic representations are prepared for study samples using the same restriction enzymes. The study samples are screened (via DNA-DNA hybridisation) with the array. The presence or absence in the sample of the DArT markers on the slide is recorded to produce a binary dataset.

A DArT marker can be lost during evolution by mutations which disrupt the rare cutter target at either end of the marker, or introduce a new rare or common cutter target within the marker. Once lost, a marker can only be regained by reversing the mutation which caused loss, before another loss-causing mutation occurs. This is a rare event, so we can model DArT marker gain/loss as a Dollo process.

The data from a DArT analysis suffer from ascertainment bias: only markers which are present in the reference taxon or taxa (from which the array was prepared) can be detected. This can distort distances, depending on the proximity of the taxa to the reference. For example, the Hamming distance

(5)

will underestimate distances between taxa distant from the reference, as both taxa will have few markers in common with the reference, hence will be large. This illustrates a general theme: joint absences () do not carry the same meaning as joint presences () and should not be accorded equal weight in the distance formula.

The most appropriate way to infer phylogeny from DArT data is not obvious, and previous authors have taken a variety of approaches, often analyzing the same data multiple ways. The distance of Nei and Li (1979) is the most commonly used distance measure for DArT data (James et al., 2008; Wenzl et al., 2004; Xia et al., 2005). It is designed for restriction site data, which DArT does produce, but for fragments generated by a single restriction enzyme, rather than the two enzymes used by DArT. Furthermore, the Nei Li distance does not account for the ascertainment bias caused by observing only those markers that are present in the reference taxa. Other distances that have been used are LogDet (James et al., 2008) and DICE (Mace et al., 2008). Non distance-based methods include maximum parsimony (James et al., 2008; Steane et al., 2011) and Bayesian analysis (James et al., 2008; Steane et al., 2011). Sometimes a principal coordinate analysis is done in place of, or in addition to, a phylogenetic analysis (Jing et al., 2009; James et al., 2008; Yang et al., 2006).

Consider DArT data for taxa and generated from an array constructed with markers taken from taxon . The presence/absence data for and are summarized by the counts , , and (where is the number of markers present at but absent at , etc.). Consider the unrooted phylogenetic tree for , , and the reference taxon . We denote the unique internal node of this unrooted tree . As all markers in the dataset are present at , and the markers evolved by a Dollo process, any marker present at or must also be present at . In particular, a marker present at is present at , and the fraction of these markers which are lost between and is . In view of the arguments above,

(6)

where is the time (branch length) between and . From this, we find a distance proportional to time

(7)

By symmetry, we also have and so

(8)

and, once again, we arrive at the formula for . (Note that this formula will also work if one of or is the reference taxon .)

In this derivation, we assume the existence of a point in the tree such that any marker present at or is present at . Under a Dollo process, this assumption is valid if there is a single reference taxon, or if there are multiple reference taxa which are monophyletic with respect to and . This will not generally be the case, and indeed it is not the case for the Eucalyptus dataset (Steane et al., 2011) we will look at in one of the case studies.

The potential problem with the ADD for data generated from multiple reference taxa is illustrated by figure 1. If we consider only the characters which are present at (the leftmost 8 columns at each node), we get the correct answer: , , , (taking logarithms as base 2 for convenience.) (There is a 1/8 chance of a marker at surviving to , so the path length between and is 3.) However, using the full set of data (which includes as an additional reference taxon), now we have , , and . Not all taxon pairs will suffer this bias, for example in figure 1 if we added a taxon attaching at point , the distance would be unbiased.

If we know which markers are derived from which reference taxa, then we can partition the data by marker reference taxon, calculate an ADD distance matrix for each partition, and then form an average distance matrix from the partition distance matrices. Under the plausible assumption that the sampling variances in the partition distances are inversely proportional to the number of markers in that partition, the minimum variance estimator is obtained by a weighted average, with weights proportional to the square root of the number of markers in the partition. We call this weighted average of partitioned distances the Partitioned Additive Dollo Distance (PADD).

The derivation of the PADD assumes that the markers chosen from each reference are an independent and unbiased sample from the markers present in that reference taxon. In practice, this assumption may fail for two reasons. When constructing the DArT array, we attempt to eliminate redundancy - so a marker selected for the array from one reference taxon precludes the same marker being selected from a second reference taxon in which it may also be present. (Some redundancy is kept deliberately as an internal control, but omitted from the final alignments.) Also, markers are chosen which show useful levels of polymorphism. A marker present in all taxa is not useful, nor is one which is present only in the reference taxon from which it was derived.

Links to Conditioned Genome Reconstruction: The Additive Dollo Distance as a Limiting Case of LogDet

Another context in which Dollo models may be appropriate is gene-family presence/absence data. Such data are increasingly available as more and more genomes are studied. The COG database (Tatusov et al., 2003) sorts genes from 50 bacteria, 13 archaea and 3 eukaryotes into nearly 5000 gene families. The gene family presence/absence data have been used for phylogenetic inference by several authors (Lake and Rivera, 2004; Spencer et al., 2007; Cotton and McInerney, 2008; Sangaralingam et al., 2010), but Dollo models have not, to our knowledge, been applied. A problem noted by previous analysts of these data is that the taxa vary greatly in the number of gene families they contain. Thus, an unsophisticated analysis would be biased towards grouping together taxa that have small genomes. This is somewhat similar to the problem of inferring a phylogeny in the presence of base-frequency biases, which is an issue the LogDet distance (Lockhart et al., 1994; Lake, 1994) was designed to overcome. When Lake and Rivera (2004) sought to apply the LogDet distance to data from the COG database, they realised they needed to know the number of shared absences (). This in turn required the definition of a “universal” set of gene families. They achieved this by selecting a ‘conditioning genome’, and taking the set of gene families present in that genome as being the “universal” set.

In using the LogDet distance, Lake and Rivera (2004) explicitly assumed that gene family presence/absence is a Markov process where a gene family can disappear from a lineage and later reappear. Frequent horizontal gene transfer (HGT) was used to justify this assumption. That LogDet treats shared presences and absences ( and ) symmetrically, when physically they have very different meaning, seems to us to be a potential weakness in their method that has not been commented on previously.

Lake and Rivera’s assumption (that any gene family can be gained by any taxon at any time) can be thought of as one extreme. The opposite extreme is to discount the possibility of HGT completely, and adopt a Dollo model. Consider the standard (non-Dollo) multiple site two state Markov model: we have sites evolving independently between two states (‘present’ and ‘absent’) by a continuous time Markov process. The rate for absentpresent transitions is and for presentabsent transitions is . As before, we sample this process at two different times and get counts , and of shared presences, and of presences at one time point but not the other. In addition, we get , the number of shared absences. In this two state case, the LogDet distance formula is

(9)

If we now allow to vary, we can construct a Markov process so that in the limit it becomes a Dollo process. We require the rate of creation of new markers (sites in the ‘present’ state) to be , so we set . (The loss rate is independent of .) As (and ), the distributions of , and will remain finite and converge on those for the Stochastic Dollo process described above. As , and are finite, we must have . Letting in equation 9

(10)

showing that in this limit, is proportional to .

Comparison of binary distances

Various distances have been defined in the literature for presence/absence data. We have picked a number of these to compare to ADD including:

(Choi et al., 2010; Lockhart et al., 1994; Lake, 1994; Nei and Li, 1979).

Huson and Steel (2004) derived a maximum likelihood distance for gene presence/absence data under a Dollo process

(11)

where in our notation , and is the expected number of genes per genome. We do not know , but if we estimate it by the mean number of genes/markers at the two taxa then equation 11 simplifies to

(12)

This is a simple transformation of the DICE distance, being , so we name it the Log DICE distance. We can intuitively justify this transformation, arguing that logarithms correct for multiple events (e.g. gain, loss, mutation) on the same marker. We can also perform a similar transformation on the Jaccard distance to create the Log Jaccard distance . So in summary, in addition to the standard distances above, we introduce previously unstudied distances:

as well as the composite ADD distance method , defined above.

The Triangle Inequality and Additivity

Two important properties of distance functions (i.e. bivariate, non-negative functions with if and only if , and for all ), are the triangle inequality and additivity. The triangle inequality states that must hold for all (in which case is known as a ‘metric’). Additivity states that if was the last common ancestor of and , and the sequences evolved independently, then should hold (on average.) The desire for additivity accounts for the presence of the logarithm function in many phylogenetic distances.

Not all of the distances defined above satisfy the triangle inequality – see Table 1 for counterexamples to the triangle inequality holding for some of the distances. In Table 2 we summarize which distances are additive and which obey the triangle inequality. The additive Dollo distance is additive by construction in the stochastic Dollo context, and it is a limiting case of the LogDet distance which is additive (Lake, 1994). Notably, the only two distances ( ) known to obey the triangle inequality are not additive, and the only two distances known to be additive (, ) violate the triangle inequality. Phylogeneticists place greater value on additivity than on obeying the triangle inequality, as demonstrated by the popularity of LogDet.

It is worth noting that for and to be additive, it is necessary that they go to infinity as the evolutionary distance between and goes to infinity. For the stochastic Dollo process, for infinitely separated and , so and go to infinity as required, but for a Markov process where for unrelated , and will tend to a finite limit. We can generalize the formulae to correct for this, using

where is the expected value of evaluated on uncorrelated/infinitely separated sequences (and a similar formula applies for ). For example, for a Markov process where states 0 and 1 are equally likely at equilibrium, we have for and for .

Simulating censored Dollo data

The general scheme for our simulations is to create a random tree, simulate a Stochastic Dollo process along it, select reference taxa and, finally, select which markers are used to produce an alignment (on the basis of which markers are present at the reference taxa).

We generate clock-like and non-clock-like trees. For the non-clock-like trees, we generate the tree topology by a Yule process (Yule, 1924), then branch lengths are set so that they are distributed uniformly between lengths 0.05 and 0.40. For clock-like trees, we generate a tree by a Yule process with mean branch length 0.1, and repeat this process until we obtain a tree whose shortest branch is no shorter than than 0.01. (As short branch lengths are hard to resolve no matter how good the phylogenetic method, keeping such branches reduces the contrast between “good” and “poor” methods, which would make our simulation results harder to interpret.) Our simulated data are based on both 9- and 15-taxon trees.

For a given simulation run, we specify the expected number of markers per genome, . As a Dollo process in equilibrium is time reversible (Huson and Steel, 2004), we start the process at an arbitrary taxon, with the number of markers at that taxon drawn from a Poisson distribution with mean . Then we propagate the set of markers through the tree. On each branch with length , existing markers are lost with probability each and the number of new markers created has a Poisson distribution with mean .

We have a number of different models for selecting the markers that will be included in the alignment to be analyzed, and (for the PADD method) how the markers are partitioned.

incl1

(One reference taxon, included.) One taxon is chosen as a reference. Only markers present at that taxon are selected. There is only one partition of the markers.

excl1

(One reference taxon, excluded.) As ‘incl1’, except we discard the reference taxon from the alignment.

incl2

(Two reference taxa, included.) Two taxa are chosen as references. All markers present at either reference taxon are selected. For partitioning, a marker which is present at both reference taxa is assigned randomly to the partition of one of them. Markers which are present at only one reference taxon go into that taxon’s partition.

excl2

(Two reference taxa, excluded.) As ‘incl2’ except we discard both of the reference taxa from the alignment.

all

(All taxa are references.) All markers are included in the alignment. Each marker is assigned randomly to the partition of one of the taxa at which it is present. (We have as many partitions as taxa.)

p2inc

(Two reference taxa, included, predetermined partitioning.) Two reference taxa are chosen. Each marker is assigned randomly to the partition of one of the references. Only markers which are present at their partition’s reference taxon are included in the alignment.

p2exc

(Two reference taxa, excluded, predetermined partitioning.) As ‘p2inc’, except the two reference taxa are discarded from the alignment.

p_all

(All taxa are references, predetermined partitioning.) Each marker is assigned randomly to the partition of one of the taxa. Only markers which are present at their partition’s reference taxon are included in the alignment.

For the methods which discard reference taxa (excl1, excl2, p2exc) we simulate extra taxa at the tree generation stage to account for the taxa which will be discarded.

In the ‘incl2’, ‘excl2’ and ‘all’ models, if a marker is present in any reference taxon, it is included in the analysis. This simulates the circumstance when all possible markers found in the reference taxa have been included on the DArT array. The ‘p2inc’, ‘p2exc’ and ‘p_all’ models simulate the situation where the number of possible markers is very much greater than the number we can put on the array, so we get an independent random sampling of markers from each reference.

The models do not all produce the same quantity of data. Compared to the expected number of markers present at each taxon, the expected number of markers analyzed is equal for the ‘incl1’, ‘p2inc’ and ‘p_all’ models, lower for ‘excl1’ and ‘p2exc’, higher for ‘incl2’, several times higher for ‘all’, and for ‘excl2’ it depends on the tree, but for our data is higher on average.

For each scenario (number of taxa, clock like or not, the eight data selection models) we generate and analyze 5000 random trees.

Once a distance matrix has been calculated, the best tree is found by minimum evolution, using the program FastME (Desper and Gascuel, 2002). In addition, we obtain the most Dollo parsimonious tree using the “dollop” program from Phylip (Felsenstein, 2005). To measure the accuracy of different methods we record the proportion of splits in the generating tree that were present in the tree inferred by FastME.

Trees in this paper were plotted using the Interactive Tree Of Life (Letunic and Bork, 2011), and interactive versions may be viewed online at http://itol.embl.de/shared/mdw. Nexus files containing the raw data and tree files are available on TreeBase at http://purl.org/phylo/treebase/phylows/study/TB2:S12439.

A demonstration Perl program, and instructions on its use, for calculating the ADD, log Jaccard and log DICE distances is included in the supplementary material.

Results

Simulated data

Table 3 shows the proportion of splits (i.e. edges) in the reconstructed trees which were incompatible with the true tree. (Additional tables for differing numbers of characters and number of taxa are provided in the supplementary material, tables 1–10.) The LogDet, Hamming and Jaccard distances consistently perform very poorly. ADD has the best overall performance, producing either the best results, or results that are not significantly different from the best results in six of the eight models; it also gives quite reasonable results in the remaining two cases. PADD has six near-best results, but fares worse on the remaining two. (The ‘all’ model violates the assumptions of PADD. Possibly ‘p_all’ performs poorly because there are so few markers in each partition.) Log Jaccard is not far behind the leading methods. DICE, Nei-Li and LogDice round out the middle of the field. Table 4 shows rather different results for the clock-like trees, with the best distances being Jaccard, DICE, Log Jaccard, PADD then ADD. We were somewhat surprised that the Jaccard distance did so well here given its poor performance on the non-clock-like trees. Variation between distance methods is smaller, as Yule trees have some very short branches, which are hard for any method to resolve. LogDet performs consistently poorly for both clock-like and non-clock-like trees.

Figure 2 plots the accuracy of branch length reconstruction against sequence length. The Hamming, LogDet, Jaccard and (to a lesser extent) DICE distances all show signs of ceasing to improve with increasing sequence length. This is expected when the method’s bias exceeds the sampling error. Only ADD improves at the optimal rate (lower dotted line.)

In figure 3, we investigate the possibility of bias in the distances due to tree shape. We divide the true trees according to how many cherries are in the unrooted tree. (A ‘cherry’ is an internal node directly connected to exactly two leaves.) The minimum number is 2 (a maximally unbalanced or ‘caterpillar’ tree). For 15 taxa, the maximum is 7. Two-cherry trees were too rare to get reliable statistics, so figure 3 shows results for 3 to 7 cherries. If a method is biased towards producing unbalanced trees, it will be more accurate when the true tree is unbalanced than when it is balanced. The non-logarithmic methods (Hamming, Jaccard, DICE) generally have high error rates on unbalanced trees, indicating a bias in favour of balanced trees. For the other methods, there is no obvious consistent bias. For example, Nei-Li, LogDice and ADD are slightly biased towards unbalanced trees for the non-clock like tree simulations and towards balanced trees for the clock like tree simulations. More plots demonstrating this lack of consistency are provided in the supplementary material, figures 1–6.

Applications to real data

Case Study 1 - Eucalyptus DArT data

We analysed the DArT Eucalyptus dataset of Steane et al. (2011). This includes 94 species of Eucalyptus from across the full taxonomic range (excluding Corymbia). The dataset comprised 7490 non-redundant DArT markers (newly acquired sequence data have allowed us to eliminate 864 markers as redundant, reducing the number of markers from the 8354 reported by Steane et al. 2011). This dataset was generated during the development phase of the Eucalyptus DArT array and only about 32% of the markers in this dataset are included on the final, publicly available Eucalyptus DArT array (Sansaloni et al., 2010).

Analysis of this data using the PADD distance yielded the tree shown in Figure 4. Branch support for this and subsequent trees are from 1000 nonparametric (resampling randomly with replacement) bootstraps. Rooting on E. curtisii (subgenus Acerosae) was based on previous studies (Drinnan and Ladiges, 1991; Steane et al., 2002). The topology of the PADD tree was highly concordant with the most recently published classification (Brooker, 2000) and previous molecular studies using ITS sequence data (Steane et al., 2002). The tree was also highly congruent with a cladistic analysis of the same data (Steane et al., 2011), but the PADD tree provided increased resolution at some key nodes. For example, sections Latoangulatae (SL), Exsertaria (SE) and Racemus (SR) in the PADD tree form a cluster that is distinct from all other sections, largely in agreement with results from ITS sequence data of Steane et al. (2002). The cladistic analysis of the DArT data (Steane et al., 2011) did not resolve the relationships between these sections and section Maidenaria. However, the position of section Racemus in relation to sections Latoangulatae and Maidenaria remains equivocal, with cladistic analysis of the DArT data and cladistic analysis of ITS sequence data placing it close to section Maidenaria. The bootstrap values on the branches of the PADD tree are generally high compared to those obtained by the cladistic analysis. Bootstrap values tended to be higher on internal nodes where there were longer branches (i.e., splits with more character support), although bootstrap values were generally >50% even when branches were very short. This contrasts with the cladistic analysis (Steane et al., 2011) where many branches throughout the cladogram had <50% bootstrap support.

Case Study 2 - Adnataria (Eucalyptus) DArT data

We screened 90 species of Eucalyptus from section Adnataria (Brooker, 2000) plus three outgroup taxa (E. cornuta, sect. Bisectae; E. torquata, sect. Dumaria; and E. staeri, sect. Longistylus) using the publicly available Eucalyptus DArT array (Sansaloni et al., 2010). Leaf samples were collected from Currency Creek Arboretum, South Australia; details of the samples will be given in a subsequent paper, but are available from the authors upon request. DNA was prepared as described previously (Steane et al., 2011). DArT analysis was conducted by DArT P/L (Canberra, Australia) using their standard protocol (Sansaloni et al., 2010).

Section Adnataria (subgenus Symphyomyrtus) includes 100–130 terminal taxa of which 90 were included in the DArT analysis. Because of the large amount of potential homoplasy in DArT data, it is preferable to include as much genetic variation as possible from within the study group, in order to minimise the risk of long-branch attraction and misleading results. Accordingly, the samples in the study represented eight of the nine series delineated by Brooker (2000) and represented the full geographic distribution of the section. (DArT data for E. dawsonii, the single species in series Dawsonianae, were not available). Of the 7680 markers on the DArT array, 3707 provided potentially phylogenetically informative data, of which 1230 were later found to be redundant and removed from the analysis, leaving 2477 markers to be analysed. The topology of the PADD tree shown in figure 5 is not entirely congruent with the established classification (Brooker, 2000), but it does have some interesting features. While at first glance it appears that - apart from series Aquilonares - most of the series within section Adnataria are polyphyletic, on closer inspection the series cluster into more-or-less discrete groups. Series Rhodoxylon and Siderophloiae form a distinct cluster (apart from the intrusion of E. rummeryi, series Buxeales) even though these series are in different subsections (Brooker, 2000). Most of series Buxeales and supraspecies Mollucanae form a discrete cluster that is “sister” to a cluster comprising series Heterophloiae, Melliodorae and Submelliodorae (and one species from series Buxeales), none of which are considered by Brooker (2000) to be particularly close. Close examination of the data reveals that there may be a biogeographic aspect to the clusters identified by the PADD algorithm. This is being explored in a separate paper (Steane et al., in prep.).

Notable features of figure 5 are the shortness of the internal branches and the poor bootstrap values. The internal edges of this tree have mean bootstrap support of 29% and ratio of internal to total edge lengths ("stemminess", Fiala and Sokal, 1985) of 0.052. Compared to the Adnataria phylogeny (figure 5), the Eucalyptus phylogeny has much higher bootstrap supports (mean 81%) and higher stemminess (0.138). The Adnataria dataset differs from Eucalyptus in that it has fewer markers (2477 compared to 7490) and its taxa are much more closely related to one another (i.e., they span a much narrower taxonomic range). To test whether the structural and statistical differences between the trees were simply a function of sample size (i.e., the number of markers), we generated 100 samples of 2477 markers from the Eucalyptus dataset (randomly chosen with replacement) and then bootstrapped each sample 1000 times. The resampled Eucalyptus data had mean bootstrap support of 60% and mean stemminess of 0.153 (std. deviation 0.003), demonstrating that the differences are not simply due to sample size. We conclude that the short internal branch lengths in the Adnataria phylogeny may be indicative of the aftermath of a period of adaptive radiation that produced many species over a short period of time, or of a loss of phylogenetic signal as a result of hybridization (or both).

If a marker is monomorphic (all 0 or all 1) within a dataset, the DArT data-processing cannot distinguish whether it is present or absent (fig. 4 of Jaccoud et al., 2001). Monomorphic data are automatically excluded from the final dataset. This is a potential problem with the Adnataria dataset, as it is much less genetically diverse than the full Eucalyptus genus from which the array was developed. We are not concerned by markers which are absent from all samples being omitted from the data, as they are ignored by the ADD formula, but there is a prospect that some markers which are present in all taxa have been omitted. To test the effect of such an omission, we added to the dataset 500 markers that were scored as present in all taxa, and reanalysed. The distances were uniformly smaller, there was very little change to the topology, bootstrap values or stemminess. We conclude that this potential for missing data does not materially affect our results. (The trees for this test are shown in the supplementary material, figures 7 and 8.)

Case study 3 - Gene Family Bacterial Phylogeny

In this case study we reanalyse the gene presence/absence data used by Spencer et al. (2007). A potential weakness of the conditioned genome reconstruction approach which has attracted attention recently is the influence of the choice of conditioning genome on the outcome. The most sophisticated attempt to address this, Spencer et al. (2007), used many conditioning genomes, produced a tree for each one, and finally constructed a consensus tree from these. The tree they derived for 40 of the bacterial genomes is shown in figure 6.

It has been noted previously (Lake and Rivera, 2004; Spencer et al., 2007) that parallel loss of genes required for free living causes parasitic bacteria from all bacterial phyla to falsely group together in this analysis.

As we have derived ADD as a special case of LogDet that is applicable to a Dollo process, we can analyze these COG data without the need for a conditioning genome. In effect, where Lake and Rivera (2004) use LogDet distances and a conditioning genome to find , we use LogDet distances assuming is infinite (equation 10.) The resulting phylogeny is shown in figure 7. It differs from the Spencer et al. (2007) phylogeny on only three edges, two of which have less than 50% bootstrap support in their tree.

Discussion

ADD is a simple, consistent distance suitable for use with binary data which evolve under a stochastic Dollo process. We have shown that ADD is also consistent for data which has been censored in such a way that only traits that are present in a single reference taxon are observable. Censoring by multiple reference taxa is more complex but in the case where each trait is known to derive from a particular reference taxon (e.g. in the DArT data presented above), ADD can be extended by a straight-forward partitioning scheme (PADD).

Our simulation supports the expectation from theory that ADD and PADD should perform well on data generated under a stochastic Dollo model with various censoring schemes. By contrast, several other distances that are in common use for both DArT data and for analysis of gene-content data do not perform well in the simulations. This suggests that they should probably not be used for data which are thought to be generated under a Dollo model. We used DArT data and gene presence/absence data as illustrations of our new approach, but our distances can be applied to any data derived from complex traits that are unlikely to evolve more than once independently.

More specifically, our simulations indicate that LogDet performs very poorly on stochastic Dollo data. This is a concern for the use of LogDet distances in conditioned genome reconstruction (Lake and Rivera, 2004; Spencer et al., 2007). The use of LogDet on gene presence/absence data in deep bacterial phylogeny not only necessitates an extra layer of complication with conditioning genomes to find the number of shared absences, but also depends critically on horizontal gene transfer (HGT) being sufficiently common to justify the use of a Markov model. For the gene-content phylogeny based on the COG data, it is interesting that the tree found using ADD is very similar to the tree found by Spencer et al. (2007). Indeed the two approaches have very different underlying assumptions. ADD ignores the possibility of HGT, which certainly does occur for these data, and the augmented conditioning genome reconstruction method of Spencer et al. (2007) ignores the Dollo aspect of the data treating shared absences and shared presences equivalently. Both methods recover the (presumed artefactual) parasite clade (Sangaralingam et al., 2010). We could imagine an intermediate model for this type of data, where markers primarily evolve via a stochastic Dollo process, but are occasionally subject to horizontal gene transfer. In this situation, a suitable distance might be the LogDet distance, with augmented, but still finite.

ADD also appears to be an appropriate distance to use for DArT data, and we have applied it to two closely related datasets. The results obtained using the PADD algorithm for the Eucalyptus data are satisfyingly congruent with traditional morphology-based taxonomies (e.g. Brooker, 2000) and phylogenies based on ITS sequence data. In addition, the PADD analysis of DArT data appears to somewhat improve on the parsimony-based analysis as it provides more resolution, possibly because the parsimony-based approach does not take account of branch length information and can thus be more easily misled by homoplasy.

In pilot studies we demonstrated that DArT data had the potential to resolve relationships among closely related Eucalyptus species (Steane et al., 2011). This level of resolution has always been elusive to eucalypt systematists for various reasons including recent/incomplete speciation and the high incidence of inter-specific hybridisation (Byrne, 2008). To test the efficacy of the partitioned additive Dollo distance at this level of divergence we applied it to a set of DArT data for section Adnataria. The PADD-derived Adnataria phylogeny suggests that the series delimited by Brooker (2000) are not robust groups. However, the lack of bootstrap support and the short branch lengths do not provide us with confidence that PADD analysis of the DArT data has uncovered the “true tree”. Further analyses of this dataset (currently underway) may reveal patterns of variation in other traits (e.g., morphology, physiology, biogeography) that will inform us about the plausibility of this DArT-based phylogeny derived using the PADD algorithm.

With the increasing debate about the appropriateness of the Tree of Life metaphor for many domains of life, it would be timely to attempt to extend the basic Dollo model to incorporate borrowing of traits. Indeed, both the bacterial gene-content example and the Adnataria example are cases where Dollo models that include “borrowing” of traits would be very appropriate. Another avenue to pursue would be the development of consistent distances in the case of multi-state Dollo models (such as discussed by Alekseyenko et al., 2008). With the ever-increasing abundance of genomic data, finding good models for the evolution of complex traits is as appropriate now as it was in the time of Dollo.

Acknowledgments

This research was funded by BH’s ARC Future Fellowship FT100100031 and ARC grant DP0986491 to Brad Potts. This manuscript was completed while DS was a CRN (Collaborative Research Network) research fellow at the University of the Sunshine Coast. VM thanks the Royal Society and the New Zealand Marsden Fund for supporting his visit to New Zealand where work started on this project. Andrzej Kilian helped us understand details of the DArT data processing pipeline. Lars Jermin suggested considering the triangle inequality and additivity properties of our distances. We also acknowledge Mark Robertson for assistance in the field, Sascha Wise for technical laboratory assistance, René Vaillancourt and Brad Potts for academic input, and the Central Science Laboratory at the University of Tasmania for use of their facilites.

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Tables

Distance DICE LogJaccard LogDICE LogDet ADD
A:10 A:00111 A:001 A:00101 A:011
Alignment: B:01 B:11001 B:111 B:00011 B:101
C:11 C:11111 C:011 C:00001 C:111
1 log(5)=1.61 log(2)=0.69 log(6)=1.79 log(4)=1.39
2/3 2 log(5/3)=1.02 log(1.5)+log(1.25)=0.63 log(8/3)=0.98 2 log(1.5)=0.81
Table 1: Counterexamples to the Triangle Inequality
Distance Hamming Jaccard DICE LogJaccard LogDICE Nei-Li LogDet ADD
Triangle ineq. Yes Yes No No No Unknown No No
Additive No No No Unknown Unknown Unknown Yes Yes

a: Hamming 1950; b: Lipkus 1999; c: Lake 1994

Table 2: Summary of the mathematical properties of the distances tested in this paper.
Method incl1 excl1 incl2 excl2 all p2inc p2exc p_all
LogDet n/a 0.0624(15) 0.3472(227) 0.1313(64) 0.0824(273) 0.2616(121) 0.1242(42) 0.0950(63)
Hamming 0.1204(51) 0.1047(35) 0.1008(59) 0.0857(37) 0.0370(121) 0.0966(36) 0.0931(28) 0.0457(26)
Jaccard 0.0649(22) 0.0699(19) 0.0508(25) 0.0578(21) 0.0665(220) 0.0600(17) 0.0679(16) 0.0512(30)
DICE 0.0435(10) 0.0464(7) 0.0246(7) 0.0288(5) 0.0381(125) 0.0347(4) 0.0402(3) 0.0231(9)
Nei-Li 0.0549(16) 0.0541(11) 0.0438(20) 0.0405(11) 0.0011(2) 0.0398(7) 0.0446(6) 0.0168(4)
LogDICE 0.0497(14) 0.0495(9) 0.0363(15) 0.0337(8) 0.0006 0.0362(5) 0.0398(3) 0.0140(2)
LogJaccard 0.0406(9) 0.0416(5) 0.0230(6) 0.0222(1) 0.0017(4) 0.0293(2) 0.0329(0) 0.0117
ADD 0.0241 0.0308 0.0303(11) 0.0269(4) 0.0006 0.0273(1) 0.0326 0.0139(2)
PADD 0.0241 0.0308 0.0146 0.0206 0.0082(25) 0.0260 0.0354(1) 0.0341(17)
DolloP 0.0396 0.0363 0.0236 0.0228 0.0004 0.0245 0.0296 0.0074
Table 3: The proportion of incorrect splits in minimum evolution trees derived from the various distances. Columns show the results for each of the various models of data censoring. Numbers in parentheses indicate how many standard deviations worse this result is than the best result for this model. Within each column, for the distance-based methods, the best (lowest) value is shown in bold, along with any which are not significantly worse than the best value. Distance-based results are colour coded, from green (best in column) to red (at least 20 standard deviations worse than the best.) Also included for comparison are the Dollo parsimony results. These data are derived from simulations using 15 taxa and a mean of 500 markers per taxon, with 5000 random trials.
Method incl1 excl1 incl2 excl2 all p2inc p2exc p_all
LogDet n/a 0.0677(19) 0.2309(139) 0.1663(88) 0.0223(24) 0.1769(80) 0.1326(50) 0.0811(31)
Hamming 0.0432(8) 0.0401(6) 0.0355(12) 0.0327(8) 0.0044(1) 0.0365(6) 0.0385(4) 0.0253(1)
Jaccard 0.0272 0.0287 0.0162 0.0189 0.0039 0.0248 0.0295 0.0235
DICE 0.0325(3) 0.0336(2) 0.0209(3) 0.0240(3) 0.0042(0) 0.0285(2) 0.0337(2) 0.0259(1)
Nei-Li 0.0408(7) 0.0420(7) 0.0399(15) 0.0369(11) 0.0069(4) 0.0365(6) 0.0408(5) 0.0314(4)
LogDICE 0.0393(6) 0.0407(6) 0.0364(13) 0.0347(9) 0.0062(3) 0.0353(6) 0.0395(5) 0.0308(4)
LogJaccard 0.0340(3) 0.0357(3) 0.0254(6) 0.0275(5) 0.0049(1) 0.0302(3) 0.0352(3) 0.0269(2)
ADD 0.0336(3) 0.0364(4) 0.0366(13) 0.0346(9) 0.0062(3) 0.0336(5) 0.0390(5) 0.0310(4)
PADD 0.0336(3) 0.0364(4) 0.0237(5) 0.0284(6) 0.0108(9) 0.0238(4) 0.0396(5) 0.0364(7)
DolloP 0.0411 0.0404 0.0288 0.0311 0.0061 0.0332 0.0374 0.0270
Table 4: As table 3, but for Yule trees. Unexpectedly, the Jaccard distance performs very well in these simulations. Only LogDet performs poorly.

Figures

Figure 1: Multiple reference taxa. The binary strings indicate the presence or absence of DArT markers. There is a 0.5 probability of marker loss along any edge. Only markers present at reference taxon R or S are shown on the diagram. Each character (the 1s and 0s for a given marker) occurs a number of times proportional to its probability.
Figure 2: Root mean square (RMS) error in branch length estimation plotted as a function of the number of characters for the various distance methods. If the method is biased, the RMS error cannot fall below the error caused by the bias, and the line is approximately horizontal (e.g. Hamming distance.) The grey “1/sqrt(n)” line illustrates the expected slope of an optimal method, with variance inversely proportional to sequence length.
Figure 3: The effect of tree shape (measured by the number of cherries in the unrooted true tree) on split error rate. In each panel, unbalanced trees are on the left, balanced trees on the right. (Points are for 3 to 7 cherries.) Downward sloping lines indicate a bias towards constructing balanced trees, and upward sloping lines a bias for unbalanced trees. The vertical axis is linear and originates at zero, but is of different scale in each panel.
Figure 4: Eucalyptus phylogeny derived from DArT data analyzed with PADD distance and minimum evolution tree building.
Figure 5: Phylogeny for Eucalyptus section Adnataria, derived from DArT data using PADD distance and minimum evolution tree building. (Species names anonymized for arXiv submission only.)
Figure 6: Gene family presence/absence phylogeny with bootstrap support from Spencer et al. (2007, figure 9). (Edges are not drawn to scale.)
Figure 7: Phylogeny of bacteria by COG gene family presence/absence data, using the additive Dollo distance. Edges are to scale.
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