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Phylogenomic studies based on multi-locus sequence data sets are usually characterized by partial taxon coverage, in which sequences for some loci are missing for some taxa. The impact of missing data has been widely studied in phylogenetics, but it has proven difficult to distinguish effects due to error in tree reconstruction from effects due to missing data per se. We approach this problem using a explicitly phylogenomic criterion of success, decisiveness, which refers to whether the pattern of taxon coverage allows for uniquely defining a single tree for all taxa.
We establish theoretical bounds on the impact of missing data on decisiveness. Results are derived for two contexts: a fixed taxon coverage pattern, such as that observed from an already assembled data set, and a randomly generated pattern derived from a process of sampling new data, such as might be observed in an ongoing comparative genomics sequencing project. Lower bounds on how many loci are needed for decisiveness are derived for the former case, and both lower and upper bounds for the latter. When data are not decisive for all trees, we estimate the probability of decisiveness and the chances that a given edge in the tree will be distinguishable. Theoretical results are illustrated using several empirical examples constructed by mining sequence databases, genomic libraries such as ESTs and BACs, and complete genome sequences.
Partial taxon coverage among loci can limit phylogenomic inference by making it impossible to distinguish among multiple alternative trees. However, even though lack of decisiveness is typical of many sparse phylogenomic data sets, it is often still possible to distinguish a large fraction of edges in the tree.
Ready access to vast sequence databases and new data emerging from large-scale sequencing efforts is transforming efforts to infer the phylogenetic history of species. Efforts to build ever larger subtrees of the "tree of life" are pushing the field to grapple with highly species-rich data sets (plants: 2538 taxa [1]; legumes: 2228 taxa [2]; Asterales: 13,533 taxa [3]; eukaryotes: 73,060 taxa [4]). At the same time, some recent studies exploiting complete genome sequences or genomic libraries include 100s - 1000s of loci [5-11]. The former tend to be data sets with many taxa and few loci; the latter few taxa and many loci. However, the distinction is blurring as larger data sets of both kinds have been constructed using semi-automated assembly pipelines [3,10] and high performance phylogenetic inference packages (e.g. [12]).
Although such vast data sets have effectively solved certain phylogenetic problems, such as the broad outline of yeast phylogeny [5], in other taxa they have uncovered a surprisingly high level of incongruence and uncertainty (reviewed in [13]). Case studies in Drosophila [14], rice [1,11,15], house mouse [16], and primates [17] document processes generating incongruence among gene trees in different parts of the genome that contribute nearly as much to the signal as the true species history does. Phylogenetic methods are struggling to keep pace with inferential complications arising from these processes, such as the potential for statistical inconsistency when sequences are concatenated [18,19]. Indeed, a minor paradigm shift appears to be occurring toward retaining the individuality of signals associated with small genomic regions ("gene trees") rather than simply concatenating all sequence data into an overall analysis [20-24]. This approach comprises so-called "supertree" methods [25] in the broadest sense of that term. In phylogenomic contexts this includes methods that explicitly try to build species trees while reconciling gene trees by modeling processes at the interface between phylogenetics and population genetics, such as lineage sorting [24,26,27], recombination [28] and hybridization [29,30].
In addition to biological processes which evidently confound phylogenomic inference are complexities that stem from the origin of the data themselves. One of the hallmarks of typical phylogenomic data sets is partial taxon coverage among loci–by which we mean not merely missing nucleotides within an alignment but loci for which an entire sequence is missing for some taxa (Sanderson et al. [31]; also referred to as "taxon occupancy" by Hejnol et al. [10]). Missing data arise from causes including biases in sequence databases, the technicalities of sequencing strategies, and real evolutionary processes of sequence divergence or gene loss. Whenever sequences from genomic libraries such as ESTs or BAC-ends provide only partial coverage of a genome, the mismatch between regions covered in different species will generate missing data. Whole genome shotgun sequencing using technologies that assemble at low coverage values will likewise entail missing data, especially at broader phylogenetic scales. Even finished whole genomes can fail to have homologs in a collection of species for various reasons ranging from accelerated sequence divergence to gene loss [32]. It is not unusual for finished genomes to have numerous genes with no detectable homologs in any other taxon (e.g., the maize genome has 2000 unique genes not found in close relatives: [33]).
The consequences of missing data for phylogenetic inference were first studied in the context of integrating fossil taxa, in which there are typically large subsets of missing morphological data, with more complete extant taxa [34-37]. More recently the impact of missing data on accuracy of inferred trees has been examined in the more general context of molecular and phylogenomic data sets [11,38-41] where missing data arises for a variety of reasons. Collectively this body of work treats missing data at the scale of individual cells in a taxon by character data matrix or sequence alignment up to entirely missing sequences for some taxa in a supermatrix or supertree setting, which is the phylogenomic context we consider here. Conclusions about the negative impact of missing data on accuracy have ranged from optimistic [37,42,43] to somewhat more cautious [38,41]; it is possibly fair to say that many empirical studies have argued that the impact of missing data tends to be unbiased and can be ameliorated by sufficient overall quantity of data (e.g., [10]), and most simulation work agrees (see exceptions: [35,41]). However, there has been frequent speculation about the distribution of missing data in the matrix (e.g. its "evenness", [38]: page 546; [10]: SOM). General conclusions have remained elusive both because of lack of characterization of this distribution and the difficulty of separating the usual problems of phylogenetic inference–error and bias caused by the data–from the simple absence of data.
In the phylogenomic context of multiple loci, a natural question is when can phylogenetic information from different loci be combined to produce a good tree with all taxa present, even when taxa are missing for some loci? Ideally this question should be abstracted away from the details of tree reconstruction methods. We derived some formal results recently for a relatively simple notion of combinability - "decisiveness" ([44]; note, Goloboff [45] used this term with a very different definition). A pattern of taxon coverage is decisive for some tree on all the taxa in a multi-locus data set if that tree is uniquely defined by combining the (correctly inferred and perfectly concordant) trees from the separate loci. Our previous paper [44] focused on a strong sense of decisiveness: decisiveness for all trees. In this paper we also consider weaker notions of decisiveness, derive further results on decisiveness in the important context of sampling from genome-scale data, and examine the utility of the concept in real phylogenomic data sets. The theory will permit us to make relatively precise statements about matters such as the amount and distribution of missing entries, at least within bounds. The scale of phylogenomic data sets and the sampling properties of genomic sequencing protocols is likely to increase the relevance of our mathematical theory.
This section outlines several mathematical results, so we begin with an intuitive description of its main point. We assume a phylogenomic data set consists of k loci (however defined) and n taxa. Loci with data present for fewer than n taxa have "missing data". Missing data across the entire data set can thus be characterized by the pattern of "taxon coverage", the set of sets of taxa present for each locus or tree (Figure 1). The basic idea of the paper is that we would like taxon coverage to be "decisive" for the true tree of all n taxa. This means that, taken together, all the subtrees of the true tree corresponding to the partial taxon sets at the different loci should be sufficient to uniquely define the true tree. Surprisingly, this is not true for many patterns of partial taxon coverage. This goal is independent of the sequence data, i.e., whether the loci agree on one tree or contain sufficient data to infer any tree. Setting those issues aside lets us isolate the effects of taxon coverage itself. Without sufficient coverage, even the best data will fail to recover a tree even if the underlying data themselves are of impeccable quality.
Figure 1. A simple taxon coverage pattern. The pattern is shown in both set notation and the more conventional notation as a matrix of taxa by loci. The taxon coverage density is 0.70.
Questions about taxon coverage arise in at least two contexts. In the first, taxon coverage is given a priori. This might happen when a data matrix has already been assembled from GenBank, or in an examination of a data set from the literature. In this case, the true tree is unknown, and the key question is whether the data set is decisive for all trees, some trees, or no trees. In the second context, taxon coverage can be modeled as the outcome of a sampling process. This might result from a sequencing strategy, for example, that induces a certain quantity of missing data, but for which those data are more or less randomly distributed. Here the pertinent question is how many loci must be sampled to guarantee decisiveness for the unknown true tree.
We begin by recalling some basic definitions from phylogenetic theory. Following Semple and Steel [46], given a set X of taxa, a phylogenetic X-tree is a (unrooted) tree T in which the leaves, or terminal taxa, of T consist of the label set X (taxon names) and all the remaining vertices (nodes) of T are unlabelled and have at least three incident edges. T is binary (completely bifurcating) if every internal vertex has exactly three incident edges. For a phylogenetic tree T and a subset Y of X, let T|Y denote the induced phylogenetic tree on leaf set Y (the tree obtained from the minimal subtree connecting Y by suppressing any vertices of degree two). We will let n = |X| throughout. A quartet tree is a binary phylogenetic tree on four leaves. For such a tree with leaves a, b, c, d, we write ab|cd if the internal edge of the tree separates the pair a, b from c, d.
A rooted phylogenetic tree is a tree obtained from a phylogenetic tree by deleting some leaf and its incident edge and regarding the other endpoint of that edge as a root vertex. Rather than defining the notions that follow for both rooted and unrooted phylogenetic trees separately, we will define them just for unrooted trees; the analogous definitions for rooted trees follow immediately by regarding any rooted tree on a set of n taxa as an unrooted phylogenetic tree on n+1 taxa, the additional taxon being some fixed reference outgroup (real or hypothetical).
We now define some key concepts. Let Y be the (label) set of taxa with nonmissing data for some locus (Yi if locus i is specified).
• Whenever Y ⊆ X, we say that a binary phylogenetic X-tree T displays a binary phylogenetic Y-tree, T' if T|Y = T'. More generally, for Y ⊆ X, we say that a phylogenetic X-tree T displays a phylogenetic Y-tree, T' if T|Y either equals T' or is a resolution of that tree (i.e. all the splits of T' are contained in T|Y).
• We say that a collection of phylogenetic trees T1,...,Tk defines a phylogenetic X-tree T if X is the union of the leaf sets of the trees T1,...,Tk and there exists one, and only one phylogenetic X-tree that displays these trees, and this tree is T. This implies T that must be binary.
• An internal edge e of a binary tree T is said to be distinguished by some other tree if both trees display a quartet tree xy|wz but the tree T/e (the tree obtained from T by collapsing e) does not display this quartet (this means that x, y, w, z comes from each of the four subtrees of T incident with e). There is an analogous notion of 'distinguished' for rooted trees based on rooted triples rather than quartet trees. For example, in Figure 2A the internal edge below the clade C, D of the original tree (left) is not distinguished by either of the two subtrees (middle) shown; however in Figure 2B all three interior edges of the original tree (left) are distinguished by the subtrees (middle).
Figure 2. Taxon coverage and defining trees. A. For the original tree and partial taxon coverage shown on the left, the two induced subtrees (middle) do not define the original tree, because other trees (right, middle and lower) also display these subtrees. B. In this case, the middle two subtrees do define the original tree. Notice that the taxon coverage pattern is the same in both cases. This coverage is phylogenetically decisive for the original tree in B, but not in A. Thus, it follows that this taxon coverage is not phylogenetically decisive for all trees.
• For a collection S = {Y1, ..., Yk} of subsets of X, we say that S is the pattern of taxon coverage or just taxon coverage (Figure 1). Any taxon found in all k subsets is called a reference taxon. The taxon coverage density is .
• We say that S is decisive for a phylogenetic tree T provided T|Y1,...,T|Yk defines (Figure 2).
• We say that S is phylogenetically decisive for all trees if it satisfies the following property: If T and T' are binary phylogenetic X-trees, with T|Y = T'|Y for all Y ∈ S then T = T'. In other words, for any binary phylogenetic X-tree T, the collection of induced subtrees {T|Y:Y ∈ S} defines T.
These last two definitions are from [44], although we now adopt the convention of saying "decisive for all trees" instead of merely "decisive". Notice that a pattern of taxon coverage that is decisive for a given tree can involve far fewer loci than any pattern of taxon coverage that is decisive for all trees; for instance, for every binary phylogenetic tree T, there is a set S of just n - 3 quadruples (set of four taxa) for which S is decisive for T [47] but any set of quadruples that is decisive for all trees must have size that is at least cubic in n (this follows from Theorem 1, below).
Notice that there is a subtle but fundamental distinction between questions about whether an arbitrary collection of input trees, T1,...,Tk, defines a unique tree, T, on the one hand, and whether a pattern of taxon coverage, S = {Y1,...,Yk}, together with a tree, T, induces a set of subtrees, T|Y1,...,T|Yk, that uniquely defines T. Intuitively, in the first case the focus is on the subtrees, whereas in the second the focus is the pattern of taxon coverage. The term "define" refers to the first context; "decisive" to the second. What does it mean when taxon coverage is not decisive for all trees? Then there exists two or more distinct trees which cannot be differentiated from one another based only on the subtrees induced by the taxon coverage pattern, because these trees induce the same set of subtrees. Of course, these may or may not include the "true tree" for any given problem in phylogenetic inference, but surely it is risky to be unable to distinguish among possibilities in case the true tree is involved.
Context 1: Taxon coverage is fixed a priori
In this context, we assume S is given. Steel and Sanderson [44] established exact conditions for when a pattern of taxon coverage, S, is decisive for all (unrooted) trees. Briefly, for every four-element partition of the label set, there must be a quadruple of taxa in some set in S such that each element of the quadruple is found in a different element of the partition. In general testing this "four-way partition" condition is computationally expensive, as the number of four-way partitions is exponential in the size of the label set. Easier to compute are (i) a necessary condition, which is that for each triple (set of three taxa) there must be at least one taxon set in S containing that triple [48], and (ii) a sufficient condition, which requires that, assuming a reference taxon exists, every quadruple containing that reference taxon has at least one taxon set in S containing this quadruple. See Figure 3 for some simple examples. The necessary condition can be quickly used to determine when taxon coverage is not decisive for all trees, and the fraction of triples present may provide an indication of the impact of low taxon coverage. Below we give several empirical examples.
Figure 3. Taxon coverage patterns and decisiveness. A. Trivially decisive for all trees because one locus contains all taxa and thus by itself satisfies the required four-way partition property. B. Not decisive for all trees. For two loci only two quartets are induced by any tree, and although two are necessary to define a tree, these two do not suffice. We also know that all triples must be present as a necessary condition for decisiveness for all trees, but the triple {C, D, E} is not present in any column. C. Decisive for all trees because there is a reference taxon and every quadruple of taxa containing the reference pattern is seen among the loci, which satisfies a sufficient condition.
It is also possible to derive some lower bounds on the number of loci required for phylogenetic decisiveness for all trees. We begin with the simpler case of subtrees defining a tree. A necessary condition for an arbitrary collection of subtrees to define a tree is given by [49]: if T1,...,Tk define a phylogenetic tree with n leaves then:
where ι(Tj) refers to the number of internal edges of Tj. For example, at least n-3 quartet trees are needed to define a tree with n leaves. A similar necessary condition holds for decisiveness for a given tree. Since ι(Tj) is at most the number of leaves of Tj minus 3 it follows that if S = {Y1,...,Yk} is phylogenetically decisive for T then:
[50], where nj = |Yj |. As noted in this last paper, Inequality (2) is not tight in the following sense: certain nine-leaf binary trees cannot be defined by any two six-leaf subtrees, even though Inequality (2) is satisfied since (6-3) + (6-3)≥ (9-3).
A consequence of Inequality (2) is that if the sets Y1, ..., Yk each have size at most m then a lower bound on the number of loci required for decisiveness for a given tree is:
Similar bounds can be obtained for phylogenetic decisiveness for all trees.
Theorem 1 If a collection S = {Y1, ..., Yk} of subsets of X is phylogenetically decisive for all trees then:.
where nj = |Yj|. In particular, if each of the sets in S has size at most m then:
(See Appendix for proof). This provides a necessary condition, or lower bound, for phylogenetic decisiveness for all trees. Roughly speaking, if some fraction of taxa, f = m/n, is present for each locus, then we need at least (1/f)3 loci for decisiveness for all trees, while at least (1/f) loci are required for decisiveness for a given tree.
Partial decisiveness
If we knew T, it would be sufficient to use results on defining trees to describe the taxon coverage needed to define T from subtrees. In practice we do not, and to play it safe, we require sufficient taxon coverage for all T (decisiveness for all trees). This may be too strong a requirement. Perhaps it is sufficient for taxon coverage to be decisive for most trees, assuming then that the true tree is likely among this set of trees. Figure 4 shows an example in which taxon coverage is not decisive for all trees (easy to see because not all triples are present), but is decisive for 6 out of 15 possible rooted trees for 4 taxa. This example suggests it would be useful to enumerate or estimate the fraction of trees for which S is decisive.
Figure 4. Partial taxon coverage. Shown is an example in which taxon coverage is not decisive for all trees, but it is decisive for 6 out of 15 possible rooted trees for 4 taxa. This is illustrated for the equivalent 15 unrooted trees, by taking advantage of the "trick" that there is an outgroup taxon, OG, that is common to both trees (a "reference taxon"), and thus we can regard it as the root. Then we can use known results [46] to determine whether the subtrees induced by the taxon coverage define the tree (and thus enumerate for which trees the coverage pattern is decisive).
Determining when taxon coverage is decisive for only some trees involves an important open question. When exactly does a set of subtrees define a tree? The bound stated earlier does not specify how to tell if some set of n-3 or more quartet trees defines a tree. The answer is known for rooted [47], but not for unrooted trees. A collection of rooted trees defines a tree if two conditions are met: each subtree is displayed by the tree, and each internal edge of the tree is distinguished by an internal edge of at least one of the input trees ([47]: see definition above). In contrast to the rooted setting, distinguishing all the edges of an unrooted tree is only a necessary (but not sufficient) condition for a collection of unrooted trees to define that tree. To proceed, we therefore restrict attention to the rooted case, which is equivalent to assuming there exists in the data set a reference taxon which can provisionally be used to root the input trees (in an artificial but useful sense of "rooting").
For rooted trees, the fraction of trees, ∏D, that are decisive for a given pattern of taxon coverage could be determined by checking these conditions for all possible trees. For small sets of taxa complete enumeration is feasible. For larger data sets, the true fraction of trees for which the taxon coverage is decisive can be estimated from the fraction in a large random sample of trees (under a uniform/equiprobable distribution of trees).
However, the condition of distinguishability of edges suggests a potentially more sensitive measure of "partial" decisiveness: the probability of an edge being distinguished on a tree for a given pattern of taxon coverage, ∏d. This is the fraction of edges on the tree that are distinguished by subtrees induced by the taxon coverage pattern, averaged across all trees, or a random sample of trees. We examine these measures of partial decisiveness in empirical examples below.
Context 2: Taxon coverage is the product of a random sampling process
In this context the taxon coverage is not fixed but instead is determined by a random process of sampling. This might be a reasonable model for data assembled from partial genomic coverage, for example. The question we want to address is how many loci must be sampled to guarantee with some high probability that the taxon coverage will be decisive for some unknown tree T. In this context we need not require that it be decisive for all trees, because the results we derive will hold for any tree T and the true tree will be one of these. This is a subtle difference compared to the first context of a fixed taxon coverage pattern, where it made sense to require decisiveness simultaneously for most if not all trees. One way to understand this is to realize that because the sampling scheme is generating random patterns of taxon coverage, there is a chance that they are decisive for any particular true tree, and the probability calculations will be derived on this assumption. [If there were reason to think that two or more trees are "correct", and we want to require the random taxon coverage will define all of them, we might insist on decisiveness for all trees in this context too–see Discussion for possible examples.]
Let T be an unknown binary phylogenetic X-tree, and suppose we independently sample k random subsets S1,...,Sk of X according to the following process: For each i, taxon x is included in Si with probability p(x), independently of other taxa. We will first consider two cases:
• Uniform coverage (UC): p(x) = p for all taxa x in X;
• Uniform (+1 complete) coverage(UC+1): p(y) = 1 for one taxon y in X, and p(x) = p for all other taxa x in X. This is the case in which a reference taxon is present. Later we will discuss possible extensions of this model to allow variable taxon coverage probabilities across a tree.
Recall that taxon coverage pattern leads to a set of induced subtrees for each locus, T|S1,...,T|Sk. Previously we assumed there was no conflict among these induced trees–they could be reconstructed accurately, but here we can obtain some results even if these induced trees are not fully resolved. We use a simple random model for lack of resolution in the induced trees. It does not explicitly model the possible sources of this loss of resolution, such as multiple hits in sequence data, but it provides a tractable starting point. Assume for some fixed probability q that each edge in T|Si (for each i) is retained with probability q or collapsed with probability 1-q, and this is done independently across the edges of T|Si, and across the different induced trees. Let denote the resulting set of partially-resolved induced subtrees of T. Let Ek(T) be the event that defines T.
We are interested in computing bounds on the probability P(Ek(T)) of this event that depend just on p, q and the total number of taxa n =|X|. It is useful to distinguish two further cases:
• Rooted: T (and thereby the induced subtrees) are rooted trees;
• Unrooted: T (and thereby the induced subtrees) are unrooted trees.
Let R(n) denote the set of rooted binary trees on n leaf taxa, and U(n) the set of unrooted binary trees on n leaf taxa. Note that by rooting a tree on an arbitrary taxon we can view U(n+1) as equivalent to R(n). Moreover, in what follows the UC condition applied to R(n) is essentially equivalent to the UC+1 condition applied to U(n+1).
Theorem 2 (i). For any rooted tree T in R(n), under UC (or an unrooted tree T in U(n+1) under UC+1) we have: P(Ek(T))≥ 1-(n-2) · (1-p3q)k. Consequently, for ε > 0, if .
Moreover, for any unrooted tree T in U(n+1) under UC, P(Ek(T)) ≥ 1 - ε holds if the term p3 in the above inequality is replaced by p4.
(ii). Conversely, for any ε > 0, if then there exist (exponentially many) choices of T ∈ R(n) under UC (and trees in U(n+1) under UC or UC+1) for which P(Ek(T)) < 1 - ε (See Appendix for proof).
Together these conditions give (respectively) upper and lower bounds on the number of loci needed for decisiveness relative to a single (unknown) tree. The precise number depends on the shape of the tree.
When q = 1 the event Ek(T) is equivalent to the event that S = {S1,...,Sk}is phylogenetically decisive for T, but for q < 1 the event is a stronger condition. Also, Theorem 2 readily extends to allow the probabilities of taxon coverage to vary across the tree, albeit at the expense of more complicated formulae. However, there is one simple case where the same formulae apply: Part (i) holds as stated if, in the Uniform Coverage conditions, the equality p(x) = p for all taxa x is weakened to p(x) ≥ p for all x (similarly, part (ii) holds if we impose an inequality in the reverse direction).
We can also consider models that explicitly reflect how taxon coverage is often much more thorough within a clade than across the entire tree, by considering the following clade-specific condition:
• Variable coverage (VC): p(x) ≥ p for all taxa x in clade C, and p(x) ≥ p' for any taxon x not in C (where p' <p).
Suppose that we have a rooted tree T in R(n), and for each of its n-2 nontrivial clades, we independently generate at least r taxon sets, where each taxon is included or not in each taxon set indepedently with a probability p(x) that satisfies the VC condition for that clade. Repeat this independently for all the n-2 clades of T (thus the resulting total number k of taxon sets must be at least r(n-2)). Then the proof of Theorem 2(i) easily modifies to show that P(Ek(T)) ≥ 1- ε provided that .
Empirical examples
We explore these theoretical results in several data sets derived from a diversity of phylogenomic data. We assess whether necessary and sufficient conditions for decisiveness for all trees are satisfied using our theoretical results. We also estimate the partial decisiveness quantities ∏D and ∏d using simulations, because no analytical results are known. We estimated these quantities for each data set using 1000 replicate simulated random trees generated using PAUP*'s equiprobable option [51]. Since the two measures of partial decisiveness can only be estimated on rooted trees, we estimate these quantities in a "reference subset" of the original data set. A reference subset is found by first identifying the taxon with the most loci present in the taxon coverage pattern, pruning all loci not present in this taxon from the entire matrix, and then removing any taxa that have been rendered data-less by this operation. This generates a data set with a reference taxon in the sense defined earlier, which can then be used to artificially "root" the analysis. Finally, we assess the bounds on data set size implied under random sampling of loci. In those analyses we assume the unrooted case without a reference taxon and require an error probability no greater than 5%.
Data mining examples
Driskell et al. [6] mined GenBank to build a sparse data set of 251 loci and 69 green plant taxa with a coverage density of 15% (Table 1). A necessary condition for decisiveness for all trees is the presence of all triples: since only 84% are present, the taxon coverage cannot be decisive for all trees. Indeed for the reference subset the coverage is only decisive for 2% of trees, so the probability of decisiveness for the true tree is quite low. However, on average 95% of internal edges across all trees can be distinguished by the taxon coverage. Thus, the coverage is likely inadequate to build the complete binary tree, but there is reason to be optimistic about most of it, at least from the perspective of missing data.
Table 1. Taxon coverage and decisiveness parameters for five data sets.
We also examined three data sets extracted from the PhyLoTA database [52], a tool for phylogenomic data mining of GenBank. These were Pleurodira (side-necked turtles: see Figure 5), Cactaceae (the cacti), and Amorpheae (a clade of legumes), all based on data from GenBank release 168 (Table 1). None of the three were phylogenetically decisive for all trees, because none exhibited all possible triples of taxa. The probability of the taxon coverage pattern being decisive for a random tree for their reference subsets was quite small, ranging from 0 to 1.7%. On the positive side, the fraction of edges that can be distinguished on average for a random tree is quite high, ranging from 61% to 95%.
Figure 5. An empirical taxon coverage pattern. This example was extracted from the PhyLoTA database (therein called the "data availability matrix"), for sequenced loci for the vertebrate clade Pleurodira (side-necked turtles). Only some of the taxa are shown.
However, the structure of the largest matrix, for Cactaceae is sobering. If we were to continue sampling loci with the same coverage density (14%) in the hopes of constructing a data set with high probability of being decisive for the true tree (Theorem 2, with perfect resolution), 2932 loci would be minimally necessary, and 23,891 would suffice. Of course, these results presuppose something like a uniform distribution of sampling effort across taxa, which is more likely to be found in genomic data sets than database mining studies. Nonetheless, these bounds are suggestive.
Finally, in this category we turn to several studies with very large taxon sets, for which we can only apply some of the simple results in Theorem 1. Goloboff et al. [4] assembled a supermatrix of 13 loci and 70,060 taxa. The locus with largest taxon coverage included ~20,000 taxa. By Theorem 1 the number of loci would have had to exceed 43 for phylogenetic decisiveness for all trees. Since most of the loci were sampled for far fewer than this number of taxa, a more accurate lower bound is probably higher.
This analysis, with coverage density of 8%, may be characteristic of taxon rich phylogenetic analyses published recently in which overall density is low, but a few loci have high coverage densities. For example, in [2] (2228 taxa of Leguminosae) and [3] (4954 taxa of Asterales) the locus with best coverage, nuclear ribosomal ITS, actually covered more than 73% and 84% of the label sets respectively, such that by Theorem 1 the lower bound on the number of loci is small, less than 3. However, in both cases, the ITS sequences were aligned by relatively "heroic" methods, and a more conservative procedure might have kept many separate taxonomically overlapping ITS alignments instead of one large one. We reported results from such an analysis (so-called "sparse" analysis in [2]) that ultimately included 1794 taxa in 72 loci. By Theorem 1 the lower bound for decisiveness for all trees in this case, assuming n/m = 1794/286, where 286 is the largest cluster (trnL), would have been 247 loci, far more than we had.
Partial genome sequences from shallow libraries
Hejnol et al. [10] constructed a large, sparse phylogenomic matrix of 94 taxa and 1487 loci derived mainly from ESTs. This is an expanded analysis of the widely cited phylogenomic study of 70 metazoan taxa [8]. Mean taxon coverage density is 18% across all loci, with the maximum density of 83% in one locus and a long tail of very low coverage loci. The given taxon coverage is not decisive for all trees, because it only includes 96% of possible triples (Table 1). Homo sapiens is present in 1351 of the loci. If we restrict the data to this reference subset, we can "root" the analysis with Homo as a reference taxon and test for a sufficient condition for decisiveness for all trees (a reference taxon plus all triples present: see above). For these loci there are still 94 taxa and 96% of possible triples are present. This fraction might have increased to 1 if the number of taxa in the new matrix had been less. This data set stands out by having a relatively high probability of decisiveness for random trees (34%), and a 99% chance of distinguishing a random edge in a tree. Under random taxon coverage (perfect resolution), by Theorem 2 the lower and upper bounds on loci are 1095 and 7148, which spans the actual data set.
Partial genome sequences from deep genomic libraries
Some genomic libraries are "deep", providing 10s to 100s of thousands of sequences, but still only partially cover a genome. Cranston et al. [11] analyzed data from BAC-end sequences from 10 species of Oryza including the complete genome of O. sativa and nine deep libraries averaging ~90,000 sequences per species. Collectively, these covered 44% of the CDS (coding regions) in O. sativa, but because of the random nature of library construction, of course, the pattern of overlap varied from species to species. The data were further filtered to include only single copy exonic regions with homology to O. sativa and having taxon coverage of at least four taxa, a total of 9481 exonic regions.
A posteriori the pattern of taxon coverage is trivially decisive for all trees because there are 6 loci (out of 9481) that have complete coverage–which automatically satisfies the four-way condition described earlier. An additional 60 loci are only missing one taxon. However, on average the taxon coverage among these exons averaged 48.3%. By Theorem 2 this value under random sampling and full resolution implies lower and upper bounds of 35 and 91 loci for decisiveness. However, if we consider partial resolution and if only one internal branch is present on average out of the 8 internal branches, the bounds increase to 292 and 742. Among a subset of loci most relevant for 6 of the species (the AA genome group) fully one quarter produced completely unresolved trees (the mean length of aligned exons was only 350 nt), so this is clearly an issue.
Complete genome sequences
Phylogenomic data sets constructed from complete genome sequences should suffer least from taxon coverage problems. We used Hahn et al.'s [32] analysis of gene families constructed from the 12 Drosophila complete genome sequences then available. Data were downloaded from Hahn's FRB Database http://www.indiana.edu/~hahnlab/fly/DfamDB/drosophila_frb.html webcite and parsed to quantify taxon coverage. Loci for which there was at most a single copy of a gene present were retained. Of these 13,677 loci, 6979 had some taxa missing. The average taxon coverage was 66.3%. Note that this number is somewhat inflated because all loci that had only one taxon sampled (singletons) were excluded (Hahn et al. suggested many of these represent annotation errors). Clearly with 6698 loci having complete taxon coverage, there is no problem with phylogenetic decisiveness for all trees. Moreover, at this rate of taxon coverage on average, by Theorem 2, the number of randomly sampled loci needed for decisiveness for the true tree is bounded by 12 - 25 loci for 12 taxa, and even if resolution among trees were dismal (say one branch only out of 9 internal branches) we would have bounds of 127 and 246 loci. Thus if one were setting out to build a phylogenomic matrix with these data, this simple analysis suggests there are plenty of loci available. Interestingly the fractional taxon coverage is so high that even were one able to do this across 1000 species of Drosophila, the number of loci is still bounded by 26 - 47.
Results and Discussion
Impact of missing data on multi-locus phylogenetic inference
Assessing the effect of missing data on phylogenetic inference has received substantial attention from phylogeneticists over many years (reviewed in [53]), but unambiguous conclusions have been elusive. For some amounts and patterns of missing data, simulation and empirical studies support the conclusion that its impact can be reduced to tolerable levels [37,38]; for other cases, significant problems remain [39-41]. Properties of the data themselves, such as rates of evolution, stationarity, etc., can have a strong impact on the accuracy of inference, irrespective of the amount or pattern of missing data, so attempts to understand the problem at a comprehensive level taking all impacts into account have had poor predictive success.
Our results relate the taxon coverage–the amount and pattern of missing data–to the ability of data to define an unambiguous phylogenetic tree for all the taxa in the input, by separating the impact of missing data from tree reconstruction errors per se. To simplify the problem, we have defined a notion of phylogenetic decisiveness [44], which is separate from any notion of correctness of the individual gene trees based on individual loci.
We infer limits (bounds) on decisiveness. In both the context of a fixed pattern of taxon coverage and a random pattern generated by sampling, there are lower bounds to how many loci must be included for a given fraction of missing data. For the sampling case it is also possible to identify upper bounds so that it is clear when sufficient data are present. For a fixed (i.e., observed) taxon coverage pattern, it is possible to characterize whether it is decisive for some or all possible trees, but computing the latter is nontrivial.
Our results may help clarify why some arguments in the literature appear at odds with each other. On the one hand, Wiens [37] has suggested that it is not the fraction of missing data that is most important, but the overall amount of character data present. The most direct support from this in our results is seen in the random sampling context. According to Theorem 2 (i), for any fraction of missing data (1 - p), there will be a number of loci that is sufficient to guarantee decisiveness. Another suggestive bit of evidence is seen in Table 1: Cactaceae and Metazoa have about the same fraction of missing data overall, but because Metazoa have so many more loci, the fraction of triples seen in that coverage pattern is much closer to the minimum level necessary for decisiveness (96% vs. 22%). A caveat is that the number of loci required can be quite large if the fraction of missing data is large or the number of taxa large (when p << 1 and or q << 1, the lower bound on number of loci k scales as 1/(p3q) (expanding the expression Theorem 2 part (ii) in a Taylor series approximation)).
On the other hand, Hartmann et al. [39] and Lemmon et al. [41] discuss cases in which missing data are more problematic. Both studies simulated missing data at the sequence level occurring in blocks, which in our context we can think of as "loci". Both attributed errors in phylogenetic inference in part to these patterns of taxon coverage, often in combination with other factors. Our results on phylogenetic decisiveness for a fixed pattern of taxon coverage make it clear that some patterns of taxon coverage can be pathologically bad. Once we cannot count on random taxon coverage to ultimately sample all 4-way partitions, it is easy to imagine patterns in which certain important 4-way partitions are never seen and thus ambiguity remains a real possibility even as the number of loci gets arbitrarily large. The obvious case is just to suppose that we sample many loci but all of them either have {A, B, C, D} or {A, B, C, F} taxon coverage. This will never be decisive according to the 4-way partition condition. Some of Hartmann et al.'s [39] EST-like patterns of missing data are reminiscent of this case because of blocks of missing data at the ends in many alignments.
Implications in the context of gene tree discordance
Evidence is mounting that high levels of gene tree discordance is driven in part by real biological processes of lineage sorting, hybridization and recombination [14,16,17,28]. Our analyses have by design set aside the latter issue, and by necessity the former, but incongruence is obviously real. Intuitively, since the lower bounds on the number of loci needed to infer a single tree decisively under random taxon coverage were derived assuming just one tree (see Appendix), we expect it would require many more loci to simultaneously be able to recover more than one tree. Interestingly in the Cranston et al. [11] analysis of rice phylogenomics, 118 out of 236 possible rooted 5-taxon trees were found among the collection of gene trees (some undoubtedly due to error rather than lineage sorting). If lineage sorting is this rampant in a data set, we would need to know bounds not on decisiveness for one tree under random sampling but decisiveness for all trees. In an a posteriori analysis of a given pattern of coverage (as with rice), we can check decisiveness for all trees after the fact, and gene tree discordance provides a strong motivation for requiring this property.
Implications for next-gen sequencing strategies
Next generation sequence technologies are currently characterized by assembly of a large number of small reads. In phylogenomic analyses in which de novo assembly (without a reference genome) with limited coverage may be unavoidable, partial taxon coverage in the final data set may be the rule rather than the exception [54]. For example, the 1KP project [55] aims to use a short-read high-throughput approach to sequence transcriptomes of 1000 green plant species across 1 billion years of evolutionary history. A combination of partial transcriptome coverage in the final assemblies and lack of homology across such deep divergences will certainly limit the overall taxon coverage densities. Given estimates of this fraction derived from pilot studies it should be possible to use Theorem 2 to predict how much of the transcriptome is needed to anticipate a decisive analysis.
Does conflict hurt or help?
A particular consequence of Inequality (1) is that a binary phylogenetic tree T for a large number of species cannot be defined by a small number of poorly resolved trees. For example, if T has 1000 leaves, then we require a total of 997 or more internal edges amongst any set of trees that defines T. Now suppose we use a supertree method like MRP (Matrix Representation with Parsimony, see review in [25]) on these input trees. An interesting question is whether Inequality (1) might still hold if we replace the condition that the trees "T1,...,Tk define T" by "T1,...,Tk have a unique MRP tree T".
This seems plausible since the two notions are equivalent when the input trees T1,...,Tk are compatible, and it might be supposed that conflict-free data is in some sense 'optimal'. However, it turns out that such a modification to Inequality (1) fails. For example, there is an incompatible set of six trees for 12 taxa, each with just one internal edge, that has a unique MRP tree T on those 12 taxa (results not shown); by contrast if we wished to additionally require the input trees to be compatible, we would require at least nine trees (each having one internal edge), not six. In other words, allowing the trees to conflict can in some cases actually 'help' reduce the amount of data required to determine a unique supertree by a method such as MRP.
Relation to phylogenetic "groves"
Decisiveness for all trees is a strong condition. It is sometimes possible to glean information about a tree that includes all taxa even when the taxon coverage pattern is not decisive for any such tree. A weaker sense of decisiveness is embodied in the notion of a "grove" of phylogenetic trees ([56]; reviewed in a less mathematical exposition in [57]). Figure 6 illustrates a taxon coverage pattern that is not decisive for any tree. However, for some tree(s) it exhibits the property that the collection of trees that display the subtrees induced by the taxon coverage pattern agree on parts of the tree. In the example it is only the position of one taxon that is variable among the trees consistent with the induced subtrees. The induced subtrees in this case are said to form a grove, because the trees discovered have information in them that is not contained in any of the induced subtrees alone.
Figure 6. Decisiveness and phylogenetic groves. Pattern of taxon coverage shown is not decisive for any tree but is a phylogenetic grove sensu Ané et al. [56]. There are five binary trees like the tree in the lower right, which differ only in the placement of taxon C (two of these are indicated by alternative placements of dashed lines). Although taxon coverage is not decisive for any of the five trees, there is no ambiguity about the relationships of the five trees if taxon C is ignored. Example modified from [56,57].
Future work
Several problems are suggested by this work and its application to empirical studies. First, many real taxon coverage patterns are sparse enough that they are not decisive for all trees. It would be useful to know what data should be obtained to convert an indecisive to a decisive data set, or to convert one that is decisive with low probability to one that is decisive with high probability. In both cases there are natural optimality problems that could be stated that would try to minimize the cost in additional sequencing. Second, along the lines of phylogenetic groves, when a taxon coverage pattern is not decisive for the entire label set, it might well be decisive for a large subset of it. What is the size of the largest label set for which a given pattern of taxon coverage is decisive? In some sense this conveys the "effective tree size" which a taxon coverage pattern can address.
We have characterized the impact of partial taxon coverage in multilocus phylogenomic data sets in the context of both fixed and random patterns of coverage. For fixed coverage patterns we have determined lower bounds on the number of loci needed for a coverage pattern to be phylogenetically decisive for all trees–that is to allow reconstruction of a unique tree on all taxa irrespective of what the tree is. Most real data sets having reasonably large numbers of taxa are not decisive for all trees, but in these cases we developed methods for assessing the fraction of trees for which they are decisive and the fraction of edges in those trees that can be distinguished. In general, the first measure is still low, but the second can be quite high. For random taxon coverage patterns, such as might be seen in data sets derived from EST or BAC libraries, upper and lower bounds on the number of loci needed for decisive tree inference were obtained. Some real data sets required relatively few loci for decisiveness, particularly those with few taxa and modest levels of missing data, whereas other data sets required tens of thousands of loci when the number of taxa and fraction of missing data was high. Clearly, the pattern of taxon coverage is a factor in large scale phylogenetic inference, regardless of the details of the sequence data themselves or tree reconstruction method.
Authors' contributions
MJS, MMM, and MS conceived the initial problems, which were then formalized as theorems and proofs by MS. MJS and MMM designed and analyzed the empirical studies and formulated implementations of partial decisiveness measures. MMM wrote code to support the latter. MJS and MS wrote the manuscript, with editing by MMM. All authors read and approved the final manuscript.
Appendix: Proofs
Proof of Theorem 1. If S is phylogenetically decisive for all trees then:
In other words, all three-taxon subsets of must be present as a subset of some element Y of S (this is Lemma 6.2.1 of [48]). Thus:
from which the results stated now follow.
Proof of Theorem 2: Part (i). First suppose that T is in R(n). For each internal edge e of T let Ae be the event that e is distinguished by at least one of the trees (for the definition of 'distinguished' see [46]). Let Be,ibe the event that edge e is distinguished by T|Si. By Theorem 3 of [47]T|S1,...,T|Sk defines T precisely if each internal edge of T is distinguished by at least one of these induced trees. Consequently:
where Eint(T) is the set of internal edges of T, and
By the Bonferroni inequality:
where overline denotes the complementary event. Now, by (5), we have: and for fixed e, the events are independent events of equal probability; consequently:
Now, if n1, n2, n3 denotes the number of leaves (in X) in the three subtrees of T that are incident with e but which do not contain the root of T then:
where the inequality arises because n1, n2, n3 ≥ 1. Applying this inequality to (7) we obtain: and so, by (6):
since a tree in R(n) has precisely n - 2 internal edges, as required. The resulting bound involving k now follows by routine algebra.
Turning to the case of an unrooted tree T ∈ U(n + 1), if condition UC+1 holds then we can directly apply the result for rooted trees under UC. However if the weaker condition UC holds for T ∈ U(n + 1)then we use the result (from [47]) that T is defined by any set of quartet trees that distinguishes every edge of T provided that each quartet contains a fixed leaf. We can then repeat the above argument but at the penalty of increasing the exponent on p from 3 to 4.
Proof of Part (ii): Let m = ⌊n/3⌋, and let T1 be any tree in R(m). Identify each leaf of T1 with the root of a separate 3-leaf binary tree to obtain a tree in T2 in R(3m). Consider the set E' of the m newly-created internal edges of this tree. If n > 3m then form an arbitrary rooted tree from the remaining n - 3m leaves, adjoin its root, and the root of T2 to a new root to obtain a tree T ∈ R(n) (if n = 3m, we simply take T = T1). Now each edge e ∈ E' has just one leaf in the three subtrees that are incident with the edge and that do not contain the root of T. Thus, for each of these m internal edges we have:
and so, by (5), and the independence of Be, j for different j, we have:
Now, although for a general tree the events Ae are not independent, for this particular tree, and for this set E' of m internal edges, these events are independent. Consequently, by (4),
from which part (ii) follows upon applying the inequality: This completes the proof.
This research was supported by a US NSF grant to MJS and MMM. MS thanks the Allan Wilson Centre for Molecular Ecology and Evolution, and the NZ Royal Society (James Cook Fellowship). We thank Tandy Warnow for discussion.
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Terrific Triplet Transportation! 3 Thumbs up! By Emnickalex, USA
January 1, 2004
We bought a previously owned Peg Perego Triplette stroller from another MOM on a triplet support site when I was around 15 weeks pregnant. My trio are now almost 2 yrs old and this stroller is still going strong. When trying to decide which triple stroller to buy, we found that the Peg Perego consistently had very good reviews. We have found ourselves, that this stroller is indestructible and additionally it has very good resale value, which is great, since money is always tight when you have multiples! Even though my children are almost 2, they still have plenty of room in the stroller seats. The seats recline if they want to nap, and the sun canopies are each removable, as are the seat covers (which is very important when you have 3 hungry toddlers devouring french fries, ice cream, etc from within this stroller). The stroller is rather long and heavy but I feel that is to be expected when you are dealing with a stroller built for 3. Once you get hang of it, it is not difficult to push or manuever. The triplette stroller does not fold up but instead has to be taken apart to fit in the back of the minivan. I don't know that it would fit in the standard car trunk although I am guessing that most families with multiples are in need of a minivan anyways. Once you have taken it apart and put it back together a few times, it is easy to do and can be done rather quickly. I have found that it is almost impossible to push this stroller while carrying a cup. And unfortunately the stroller does not have a cup holder. However we were able to buy a cup holder for around $5 which just snaps on to the stroller handle and that has worked very well for us. My only other complaint is that the stroller's storage basket is rather shallow. My large diaper bag does not fit there. If you are only using the basket for small shopping bags or for jackets, it is just fine. Overall I think this stroller is a great value. The frame is strong and the wheels are large and heavy. I have used it on grass, gravel, concrete, as well as at the mall, with no problems. I would highly recommend it!
LIght-Weight enough for moms and roomie enough for babies. By Savvymommy
September 15, 2006
After weeks of research and debate, I finally decided on the Pliko Lite in Rose over the Maclaren Quest. I was originally looking into buying a light-weight umbrella stroller for my 17 months old daughter to hand down the bugaboo to my newborn. And after using the Bugaboo Frog for over a year, I was worried that not many strollers can fit my numerous needs. I really liked the Maclaren Quest for its light-weight, compact fold, and good reviews from its users but I was a little hesistant to spend that much money on a stroller that I can only use for quick trips. I was looking for a stroller that can do it all..be light-weight with an umbrella fold yet still have an accessible basket, comfortable and roomie seat for baby, and a good sized canopy. I live in So. California where I'm constantly in and out of the car, but once I'm out, I stroll with the baby for longer than an hour and didn't want to lose all the nice features that a full or medium-sized stroller has for an umbrella stroller. Fortunately I came across the Pliko lite during my research and it seemed to fit my requirements the best. The basket is roomie (only accessible from the sides or the front but I can live with that), the seat is plush with extra paddings on the sides so my daughter can easily lean her head to the side when taking a nap, the seat reclines far enough so she can be comfortable sleeping. Websites say that this stroller has a 2 point recline system which is a little misleading because it actually has three positions. One upright, one about 1/3 down and one 2/3 down, which makes it good enough for an older baby. The canopy comes down pretty far but it is missing a viewing window which I would like. But once again, my daughter is old enough to sit well on her own so I don't feel too bad about not checking up on her constantly through the canopy. It also comes with a decent cup holder but it is too small to hold large drinks. It can probably fit a small water bottle or sippy cups. But once again, I can't expect a full sized cup holder on an umbrella stroller. The fold is really easy and most importantly, the stroller stands on its own when folder which is a huge plus for me. It is not TOO light-weight but not too heavy either. I'd say about 16 lbs. So basically if you want a really light-weight umbrella stroller for quick trips, Maclarens are probably better. But if you're williing to deal with a little more weight for the comfort of your baby, I'd definitely recommend the Pliko Lite. My daughter loves this stroller and looks so comfortable in it that I don't think she'll miss her old Bugaboo. I'm glad I made this purchase and would definitely recommend it as an every day stroller for malls, outdoor areas, amusement parks, etc.
Great stroller! By S. Larrea
December 11, 2006
I have bought several strollers over the years especially as my needs have changed. I love lightweight umbrella type strollers as they fit the needs of my lifestyle, but I always miss features like cupholders, a snack tray, etc. This stroller has the both of best worlds. I read all the complaints about this stroller. I have to admit I was a little hesitant about purchasing it. My only advice is to take your time putting it together--make sure you put the wheels and cupholder on right. I have not had any problems with steering, cup holder popping off, etc. Great stroller!
Great Stroller By Mom, Pocatello, ID
January 1, 2004
I purchased this stroller as a travel stroller to fly with and to leave in my SUV. We just returned from a trip to Disneyland with our 4 year old and 8 month old. The stroller was wonderfull from the minute we used it. My Graco Snugride carseat fits in the stroller very securely. The stroller pushes very nicely with my 8 month old in the front and my 40 pound 4 year old on the back standing. We navigated through the crowds of Disneyland with no problems. My 4-year old dressed up like Ariel mermaid for dinner one night and we had her stand on the back of the stroller to get to dinner in California Adventure and people commented that she looked like she was riding on a "Chariot." She enjoyed riding both in front and on the back. The great thing about this stroller is, even with my 40 pound daugher riding,either in front or stand in on the back, it still pushed and turned very nicely. We own 4 strollers total. This one is by far my favorite for travel and to keep in the car. It is very compact compared to my Graco and very stable compared to my Combi travel stroller. I still use my Inglesina Magnum for walks from home because of the big air filled wheels.Over all the Pliko 3 is a wonderful stoller with being lightwieght, compact, stable, a smooth ride and both of my children can ride without compromising anything. Definetly worth the money. I wished I would have purchased this before I bought the Graco and Combi. It combines the smooth ride and stability of the Graco with the compactness of the Combi. Overall the "complete stroller."
Awesome Stroller, UNBEATABLE QUALITY! CADILLAC OF ALL TRIPLET STROLLERS! By Anonymous
January 1, 2004
For safety, quality, and comfort, you can't beat it! Although it is more expensive than most brands, it is truly worth it--you definitely get what you paid for. The wheel suspension allow a smooth and comfortable ride. My favorite feature is the fact that you can face the seats toward you; this helps you to have more control over strangers trying to reach in and touch the babies, and you can keep a constant eye on the little ones.
We LOVE this stroller! By Shari, OK
January 1, 2004
This is the ultimate side-by-side double stroller in my opinion! I have the single Aria and love it so much that I knew I wanted the double (now that we are expecting our second child) as soon as it came out! This stroller weighs only 14lbs and pushes like a DREAM! It folds SO compactly that I could easily fit TWO of them in the trunk of my Nissan Altima (I have the Twin and the single both in their right now - with plenty of room to spare). The basket underneath is HUGE - I can't imagine not being able to fit everything I would need in there PLUS tons of shopping bags ;) The stroller has two independent canopies which is great if one child wants to be in the sun and the other doesn't (which happens more than you think just because they want to be difficult!). The front bumper bars in the Aria Twin can either be removed completely or swiveled open (again, independently) so the child can climb in and out on their own (which my 2-year old LOVES to do - you know how independent they are!). The 5-point harness is great because it keeps them from climbing out of the stroller but still allows them to move freely as it's not attached to the back of the seat. The Aria Twin pushes SO easily, even with just one child in it - you would think having weight on one side would throw it off balance but it doesn't at all! I bought the Aurora pattern because I ordered it from a website that had a coupon code so I could get a discount (it was the only website I could find that carried this stroller that had a coupon code) and this was the only pattern they had. I wasn't too fond of the colors based on the pictures but it is MUCH nicer in person! I'm sooooo glad I got the one I did! I can't think of anything I DON'T like about this stroller - I highly recommend it to anyone with twins or two children of different ages, like me =)


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Book your room at Ramada Rock Hill
Please enter your travel dates.
Get an overview of this hotelHotel in Rock Hill with free breakfast and bar/lounge
Situated in Rock Hill, this hotel is 1 mi (1.6 km) from Giordana Velodrome and within 3 mi (5 km) of Cherry Park and Piedmont Medical Center Trailhead. Rock Hill Galleria and Glencairn Gardens are also within 6 mi (10 km).
Hotel Features
A bar/lounge, a seasonal outdoor pool, and meeting rooms are available at this hotel. Free continental breakfast and free WiFi in public areas are also provided. Additionally, laundry facilities, barbecue grills, and a picnic area are onsite.
Room Amenities
All 124 rooms offer free WiFi, cable TV, and hair dryers.Information missing or incorrect? Tell us!
Hotel Amenities
Hotel Amenities
A bar/lounge is on site where guests can unwind with a drink. A complimentary breakfast is offered each morning. High-speed wireless Internet access is complimentary.
Event facilities measuring 2500 square feet (232 square meters) include meeting rooms. This Rock Hill hotel also offers a seasonal outdoor pool, barbecue grills, and a picnic area. The property may provide extended parking privileges to guests after check-out (surcharge).
- Free breakfast
- Total number of rooms - 124
- Number of floors - 2
- Conference space size (meters) - 232
- Bar/lounge
- Barbecue grill(s)
- Free WiFi
- Laundry facilities
- Meeting rooms
- Picnic area
- Safe-deposit box at front desk
- Conference space size (feet) - 2500
- Number of meeting rooms - 3
- Number of buildings/towers - 1
- Outdoor seasonal pool
- Television in lobby
Family Friendly Amenities
- Free Wi-Fi
- Rollaway/extra beds available
- Laundry facilities
Available in all rooms: Free WiFi
Available in some public areas: Free WiFi
Room Amenities
- Air conditioning
- Hair dryer
- Rollaway/extra beds available
- Cribs/infant beds available
- Cable TV service
- Free WiFi
- Blackout drapes/curtains
Where to Eat
Guests are offered a complimentary continental breakfast each morning.
To make requests for specific accessibility needs, complete the "Special or Accessibility Requests" section of "Trip Preferences" when you book your hotel.
- Accessible bathroom
- Accessible parking
Hotel Policies
Payment types
Children and extra beds
- Children are welcome.
- Rollaway/extra beds are available.
- Cribs (infant beds) are available.
You need to know
Extra-person charges may apply and vary depending on hotel policy.
Government-issued photo identification and a credit card or cash deposit are required at check-in for incidental charges.
Special requests are subject to availability upon check-in and may incur additional charges. Special requests cannot be guaranteed.


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Moving to Bahrain
Often referred to as "Middle East lite", Bahrain can be considered a starter course set out to introduce foreigners to the characteristics so commonplace in many Persian Gulf nations. The country is considerably more liberal than neighbours Saudi Arabia and Qatar, but is still rooted in an Arab belief system.
There is a sizeable expat community of more than 500,000 individuals, but not enough of a population to overshadow the local Bahrainis.Thus, expats relocating to this Gulf state will still have to contend with elements of culture shock and make some adjustments to their lifestyle.
Most expats moving to Bahrain do so for improved career opportunities. The earning potential for expats is high in Bahrain and expats report enjoying a higher standard to living than they had experienced in their home country.
As the financial centre of the Middle East, highly-skilled foreigners will find plenty of positions available in the banking and construction sectors. Furthermore, as Bahrain had made intense efforts to diversify its economy beyond the petroleum industry, jobs in many multinational firms exist as well.
Moving to Bahrain also means bigger houses for many. Properties of all shapes and sizes designed to suit all budgets and tastes eagerly await new tenants. Expats can choose between towering high-rises, stand-alone villas or even homes enclaved in a secure compound.
While expats with children won't be able to enrol their little ones in any of Bahrain's public schools, plenty of excellent private and international options exist; though spots disappear quickly so be sure to make registration a priority upon relocation.
The largest challenge for most new arrivals seems to be adjusting to the intense Middle Eastern heat, Bahrain temperatures can reach 45°C and the humidity more than 95 percent. However, as the "Kingdom of Two Seas" there's plenty of shallows, and air conditioning and pool facilities abound.
As far as Middle Eastern destinations are concerned, Bahrain is increasingly becoming an expat favourite, and a fantastic alternative location for those based in neighbouring countries looking to situate their families in a more liberal environment.


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Tokyoflash Equalizer BPM watch for dee-jays
Tokyoflash have released a watch with a special feature to appeal to budding DJs – it can measure BPMs (Beats Per Minute, idiot).
We think the theory goes that if you understand how many BPMs are in a rhythm then you can use that knowledge to mix tracks with similar BPMs, or do success "blends" between two songs. We think.
If you tap the watch's D button to the beat after a few moments, the watch automatically displays the BPMs on the LCD display.
Er, great. We guess. At least we know what to get Norman Cook for Christmas now, so that's a weight off.


text: games Comments 8 Pixel Puzzle Challenge 3: TV Shows! By Becky Ferreira Sep 24, 2012 Back 7 of 7 Next More More Slideshows We Celebrate Get Pop Cultured's Doctor Who Day With a Super Quiz 24 Perfect Songs for Book Lovers in This Week's Geeky Twitter! Clumsy Nazgul, TARDIS Manicures, & Nerd Weddings: Tales from Denver Comic Con! Back Next more > Back Next more > Kyle McCoyCredit: Tags: tv, games, quizzes, trivia, pixel puzzle challenge, kyle mccoy Melissa Marr's "Carnival of Souls" Cast a Spell On Us! Melissa Marr's "Carnival of Souls" Cast a Spell On Us! Memorize these Quotes if You Write Science Fiction Memorize these Quotes if You Write Science Fiction Related Posts like a stapler to the face 5 Reasons We LOVED Agent Carter check out this helmet hair Magneto Cat: Like Your Tungsten-Jawed Crush From X-Men Only With Furrier Motives hold my staff for a second I'm swooning Review: Witchy Women Put Wet-Shirted Boys Through the Ringer in Magical Seventh Son Write your own comment! Hold up there partner! Before you comment on a post there's one more step. You'll have to choose a nickname to be displayed on your comment. Give 'er a shot below! Write your own comment! OR Remember me OR Please log in first Email Password Remember me Log in Create an account | Lost password ...or log in using


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Because why wear thongs when you can wear comfier, more stylish footwear?
How to Look Great When You're Hosting
The holiday season means party-going—and where there's a party, there's a camera. Learn how to look your best in photos at every fete with these simple tips. From perfecting your smile to smoothing your strands, these snapshot secrets will ensure that you never call yourself un-photogenic again.
Stash a whitening pen in your purse to quickly erase surface stains (use twice a day for two weeks to lighten teeth).
Go Smile On The Go Teeth Whitening Pen, $22; GoSmile.com
How to Shrink Pores


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PLATTSBURGH - Have you ever wanted to know more about the family behind the "Battle of Lake Champlain" painting that hangs in the KeyBank Gallery of the Battle of Plattsburgh Association?
Dr. Joy Demarse will be speaking about the family, their poetry, and their lives at 7 p.m. Wednesday, Jan. 14, at the Battle of Plattsburgh Association, 31 Washington Road.
The event is free and open to the public.
For more information, call 566-1814, e-mail email@example.com or visit www.battleofplattsburgh.org.
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Texas on the Potomac
Washington news with a Texas accent
Posts filtered on Tag
Ted Cruz says he voted against storm aid because 'cynical politicians' larded bill with 'spending utterly unrelated to Sandy'
If anyone had a doubt that Ted Cruz would be a more conservative senator than Kay Bailey Hutchison, we got an example of the difference between the two Texas Republicans yesterday. Cruz voted against the bipartisan "emergency" aid package designed to help the victims of Superstorm Sandy. Last December, Hutchison voted for a similar package. […] [Read More]
Sen. Ted Cruz joins John Cornyn, says he would support partial government shutdown
While appearing on PBS Newshour this week, the newly sworn in Texas Sen. Ted Cruz said that he is ready to use the debt ceiling as a bargain tool. [Read More]
Rick Perry's war on Washington takes a day off
If you watched Rick Perry's 2012 presidential campaign, you know important anti-Washington rhetoric was to the Texas governor. Too much federal spending. Too much federal regulation. Too many unfunded federal mandates. Too much micromanagement from Washington. Too many Cabinet agencies. If you expected to hear an undercurrent of Washington-bashing today in Perry's remarks to the […] [Read More]
Ron Paul: Fiscal cliff deal was 'dishonest and unconstitutional'
Texas on the Potomac is pleased to share guest commentary from across the political spectrum. Today, we share Ron Paul's first "Texas Straight Talk" column since leaving Congress last Thursday. Last week the Senate and House demonstrated again why their approval ratings are so low. The 154 page "fiscal cliff" bill was made available to […] [Read More]
Rep. Ted Poe says lawmakers are addicts, should attend Spending Anonymous
"Congress is addicted to spending money. Maybe Congress should go to Spending Anonymous," he said on the floor of the House on Jan. 1. [Read More]
John Culberson says he's 'dumbfounded and deeply disappointed' by fiscal cliff deal
Texas on the Potomac welcomes guest commentary from across the political spectrum. Today, we offer a column by Houston Republican Rep. John Culberson explaining his strong opposition to the bipartisan "fiscal cliff" deal that was approved late last night by the House. In the dead of the night, with no committee hearings, no public input […] [Read More]
Texas House Republicans resoundingly say 'no' to bipartisan fiscal cliff deal
Texas Republican House members just said no to the bipartisan deal to pull the nation back from the "fiscal cliff." While both Texas senators — Republicans Kay Bailey Hutchison and John Cornyn — joined 87 other colleagues in an overwhelming bipartisan vote of 257 to 167 to make the Bush-era tax cuts permanent for about […] [Read More]
Hutchison, Cornyn explain votes for 'fiscal cliff' deal
Happy New Year, y'all! In case you weren't watching C-SPAN2 in the hours following your champagne toast, you might have missed the first nes story of 2013: The Senate mustered an 89-vote majority for the "fiscal cliff" deal negotiated by Vice President Joe Biden and Senate Republican leader Mitch McConnell. Texas Sens. Kay Bailey Hutchison, […] [Read More]
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Tennsco - Multimedia Cabinet For 5 x 8 Cards, 7-Drawer, 28" - Putty
by Tennsco | Item #: 100542 | Model #: TNNCF758PY |
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Non-members pay 10% surchargeJoin now!
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About this item
38,100 card capacity. For tab cards, mag cards, video cassettes and CDs. Durable steel construction ensures strength and stability. One lock secures all drawers. Center divider and two follower blocks per drawer.
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19 1/8"W x 28 1/2"D x 52"H
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The scribe for Sporting News befriended Ghawi after exchanging messages with her online, he told The Daily Beast.
Former Clinton scribe Mark Katz remembers his eloquence, deference—and classic Gridiron zingers.
He had beautiful handwriting, a very important talent for a scribe.
c.1200, "professional interpreter of the Jewish Law" (late 11c. as a surname), from Church Latin scriba "teacher of Jewish law," used in Vulgate to render Greek grammateus (corresponding to Hebrew sopher "writer, scholar"), special use of Latin scriba "keeper of accounts, secretary, writer," from past participle stem of scribere "to write;" see script (n.). Sense "one who writes, official or public writer" in English is from late 14c.
"to write," mid-15c., from Latin scribere "to write" (see script (n.)).
A text-formatting language by Brian Reid.


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First Drives
Auto Shows
Spy Shots
Car Tech
MUST SEE: Watch Senna Catch Prost In The Rain At Monaco, Announced By James Hunt: Video
The original Iceni is an open-top model best described as a roadster. It has since been joined by a fastback coupe called the Iceni Magna and a shooting brake called the Iceni Venturer, which is yet to be revealed. All three are currently available to order.
What makes the Iceni range of cars so interesting is the diesel powertrain common to each of them. The powertrain is a turbocharged 6.6-liter V-8 diesel engine, which is mounted up front and spins the rear wheels via a six-speed automatic transmission. The engine develops 395 horsepower and 700 pound-feet of torque, and will rocket the Iceni from 0-60 mph in just 3.7 seconds. Top speed is estimated to be over 190 mph.
DON'T MISS: Lamborghini Dealer Gives 7-Year-Old Ultimate Birthday Party: Video
Trident says the particular setup makes the Iceni the world's fastest and most fuel-efficient diesel sports car. The key is a patented method of utilizing torque multiplication to improve performance and efficiency. For example, the Iceni's engine runs at just 980 revs when the car travels at a constant speed 70 mph, such as cruising on a highway. In this way, it can return an average fuel economy of up to 57.4 mpg and cover a range of more than 2,000 miles on a single tank.


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-- Trenddr.tv reports that this week's WWE SmackDown scored a 84,846 in social media activity. This was a 39% increase up from last week. This is the highest social score in 14 weeks for SmackDown, going back to September.
The return of The Rock and a big World Title change bolstered SmackDown's ranking to rank #4 on cable TV on Friday night. SmackDown trailed behind two NBA games on ESPN and SpongeBob SquarePants on Nickelodeon.
-- Here is this week's edition of WWE SmackDown "Backstage Fallout":


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Jay DaVinci Artist/Painter
Jacqueline UreelModel Female Bozeman, Montana, US My MM URL: http://www.modelmayhem.com/jureel
Mayhem # 2297581
About MeMy Name is Jacqueline Ureel, I am 21 years old and I am a college student at Montana State University studying psychology. I have always been interested in acting/ modeling and I feel I can go far in this industry. I am hard-working and dedicated to my projects and always go into a situation with an open mind. I hope to get contacts out of this site and hopefully this will be the right stepping stone into making my dreams a reality.
Jay DaVinci Artist/Painter
Merrit Photography Photographer
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Cattle Zone Apparel Clothing Designer
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6th Guards Tank Brigade (United Kingdom)
|6th Guards Tank Brigade
|British Crown
|British Army
|Part of
|Guards Armoured Division 1941–1942
Independent 1942–1945
The 6th Guards Tank Brigade was formed in 1941 as the 6th Guards Armoured Brigade when Great Britain was under the threat of invasion and more armoured formations were required.
Permission was granted from King George VI and the Colonels of the Regiments involved and over the summer of 1941, the Infantry of the Guards converted into an armoured formation and so the Guards Armoured Division was formed which contained the 5th Guards Armoured Brigade and the 6th Guards Armoured Brigade.[1]
In 1942 all British armoured divisions were reorganised to have one armoured brigade and one motor brigade. The 6th Guards Armoured became an independent tank brigade being renamed as the 6th Guards Tank Brigade.[2] The brigade now equipped with the Churchill tank,[3] served in the North West Europe Campaign landing in Normandy on 20 July 1944.[4]
Correspondence in Winston Churchill's The Second World War (Volume V: Closing the Ring, Annex C) in April 1944 appears to indicate there was consideration of breaking the brigade up and making its personnel available as replacements for other army formations. Churchill was opposed to this, and nothing appears to have been done.[5]
The brigade went onto take part in Operation Bluecoat,[6] Operation Veritable finally ending the war at Lübeck on the Baltic Sea where they captured a U-boat.[7]
- 15 January 1943 – 3 August 1944 Brigadier GL Verney
- 3 August 1944 – 16 August 1944 Brigadier Sir Walter Bartellot, Bt
- 18 August 1944 – 2 February 1945 Brigadier WDC Greenacre
- 4th Motor Battalion Coldstream Guards
- 4th Armoured Battalion Grenadier Guards
- 3rd Armoured Battalion Scots Guards
- 2nd Armoured Battalion Welsh Guards
- ^ The Guards Divisions 1914 – 45 By Mike Chappell,p 28
- ^ "Guards Armoured".
- ^ 6th Guards Tank Brigade: The Story of Guardsmen in Churchill Tanks
- ^ "D-Day tanks".
- ^ Winston Churchill, The Second World War: Volume V: Closing the Ring, Reprint Society, London, 1954 (Hardback), Appendix C, notes of 4 April 1944, 4 April 1944, and 9 April 1944, pages 541-542.
- ^ British Armour in the Normandy Campaign 1944 By John Buckley,p 101
- ^ "brief history of the brigade".
- The Guards Divisions 1914–45 By Mike Chappell,Published by Osprey Publishing, 1995 .ISBN 1-85532-546-2
- British Armour in the Normandy Campaign 1944 By John Buckley,Published by Routledge, 2004 .ISBN 0-7146-5323-3
- 6th. Guards Tank Brigade: The Story of Guardsmen in Churchill Tanks by Patrick Forbes, Published by Sampson Low, 1946. OCLC 464061159
External links[edit]
- 6 Guards Armoured Brigade at Orders of Battle.com
- 6 Guards Tank Brigade at Orders of Battle.com
- 6 Guards Brigade at Orders of Battle.com


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Find a Doctor
Related Services
Cardiovascular Services
Heart Transplant
Saint Luke's Mid America Heart Institute's Heart Transplant Program:
- Ranks among the top 10 centers nationwide for number of adult heart transplants
- Is the only active heart transplant center in Kansas and the western two-thirds of Missouri
- Has one of the shortest waiting times for donor hearts in the country
- Is a Medicare-approved transplant facility
Top-ranked experience
Saint Luke's Hospital began its heart transplant program in 1985. Since that time, over 600 individuals have received a second chance at life through a transplant. While 80 percent of U.S. cardiac transplant programs perform 20 or fewer heart transplants a year, Saint Luke's performed 47 in 2012—the most ever in a single year at Saint Luke's. Our previous record was 43 in 2010, when Saint Luke's celebrated its 25th anniversary of heart transplants.
Shorter waiting times
Our program has one of the shortest waiting times for donor hearts in the country. Between 80 to 90 percent of our patients receive transplants within six months of joining the national waiting list, compared to 30 percent nationally.


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About Sandia Laboratory Federal Credit Union
SLFCU serves the high-tech and professional communities in Albuquerque, NM and Livermore, CA.
As a not-for-profit, member-owned financial cooperative that exists solely to serve its members, SLFCU can usually offer better rates on almost everything from car loans to credit cards. We don't focus on making profits and paying stockholders. Instead, we use earnings to lower loan rates, raise savings rates, and reduce fees.
Credit Unions all share the same philosophy: People helping people. We pride ourselves on serving members not only as a financial institution but also through financial education and community and charitable causes.
SLFCU helps you get the low-cost mortgages, consumer loans, and secure deposit accounts you need for financial success.
Board of Directors & Supervisory Committee
Learn about our governing body and our audit processes, and read our annual reports.
Community Involvement
SLFCU is proud to support our local communities. Learn about the ways we give back.