The Space of Equidistant Phylogenetic Cactuses.

An equidistant X-cactus is a type of rooted, arc-weighted, directed acyclic graph with leaf set X, that is used in biology to represent the evolutionary history of a set X of species. In this paper, we introduce and investigate the space of equidistant X-cactuses. This space contains, as a subset, the space of ultrametric trees on X that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning ranked rooted X-cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of X that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted X-trees in terms of partitions of X, which provides an alternative proof that the space of ultrametric trees on X is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant X-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.

Incorporating Hierarchical Characters into Phylogenetic Analysis.

Popular optimality criteria for phylogenetic trees focus on sequences of characters that are applicable to all the taxa. As studies grow in breadth, it can be the case that some characters are applicable for a portion of the taxa and inapplicable for others. Past work has explored the limitations of treating inapplicable characters as missing data, noting that this strategy may favor trees where internal nodes are assigned impossible states, where the arrangement of taxa within subclades is unduly influenced by variation in distant parts of the tree, and/or where taxa that otherwise share most primary characters are grouped distantly. Approaches that avoid the first two problems have recently been proposed. Here, we propose an alternative approach which avoids all three problems. We focus on data matrices that use reductive coding of traits, that is, explicitly incorporate the innate hierarchy induced by inapplicability, and as such our approach extend to hierarchical characters, in general. In the spirit of maximum parsimony, the proposed criterion seeks the phylogenetic tree with the minimal changes across any tree branch, but where changes are defined in terms of dissimilarity metrics that weigh the effects of inapplicable characters. The approach can accommodate binary, multistate, ordered, unordered, and polymorphic characters. We give a polynomial-time algorithm, inspired by Fitch's algorithm, to score trees under a family of dissimilarity metrics, and prove its correctness. We show that the resulting optimality criteria is computationally hard, by reduction to the NP-hardness of the maximum parsimony optimality criteria. We demonstrate our approach using synthetic and empirical data sets and compare the results with other recently proposed methods for choosing optimal phylogenetic trees when the data includes hierarchical characters. [Character optimization, dissimilarity metrics, hierarchical characters, inapplicable data, phylogenetic tree search.].

Most Parsimonious Likelihood Exhibits Multiple Optima for Compatible Characters.

Maximum likelihood estimators are a popular method for scoring phylogenetic trees to best explain the evolutionary histories of biomolecular sequences. In 1994, Steel showed that, given an incompatible set of binary characters and a fixed tree topology, there exist multiple sets of branch lengths that are optima of the maximum average likelihood estimator. Since parsimony techniques-another popular method of scoring evolutionary trees-tend to exhibit favorable behavior on data compatible with the tree, Steel asked if the same is true for likelihood estimators, or if multiple optima can occur for compatible sequences. We show that, despite exhibiting behavior similar to parsimony, multiple local optima can occur for compatible characters for the most parsimonious likelihood estimator. We caution that thorough understanding of likelihood criteria is necessary before they are used to analyze biological data.

A new family of dissimilarity metrics for discrete character matrices that include inapplicable characters and its importance for disparity studies.

The use of discrete character data for disparity analyses has become more popular, partially due to the recognition that character data describe variation at large taxonomic scales, as well as the increasing availability of both character matrices co-opted from phylogenetic analysis and software tools. As taxonomic scope increases, the need to describe variation leads to some characters that may describe traits not found across all the taxa. In such situations, it is common practice to treat inapplicable characters as missing data when calculating dissimilarity matrices for disparity studies. For commonly used dissimilarity metrics like Wills's GED and Gower's coefficient, this can lead to the reranking of pairwise dissimilarities, resulting in taxa that share more primary character states being assigned larger dissimilarity values than taxa that share fewer. We introduce a family of metrics that proportionally weight primary characters according to the secondary characters that describe them, effectively eliminating this problem, and compare their performance to common dissimilarity metrics and previously proposed weighting schemes. When applied to empirical datasets, we confirm that choice of dissimilarity metric frequently affects the rank order of pairwise distances, differentially influencing downstream macroevolutionary inferences.

Review Paper: The Shape of Phylogenetic Treespace.

Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and in running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths.

Structured inquiry-based learning: Drosophila GAL4 enhancer trap characterization in an undergraduate laboratory course.

We have developed and tested two linked but separable structured inquiry exercises using a set of Drosophila melanogaster GAL4 enhancer trap strains for an upper-level undergraduate laboratory methods course at Bucknell University. In the first, students learn to perform inverse PCR to identify the genomic location of the GAL4 insertion, using FlyBase to identify flanking sequences and the primary literature to synthesize current knowledge regarding the nearest gene. In the second, we cross each GAL4 strain to a UAS-CD8-GFP reporter strain, and students perform whole mount CNS dissection, immunohistochemistry, confocal imaging, and analysis of developmental expression patterns. We have found these exercises to be very effective in teaching the uses and limitations of PCR and antibody-based techniques as well as critical reading of the primary literature and scientific writing. Students appreciate the opportunity to apply what they learn by generating novel data of use to the wider research community.

Towards improving searches for optimal phylogenies.

Finding the optimal evolutionary history for a set of taxa is a challenging computational problem, even when restricting possible solutions to be "tree-like" and focusing on the maximum-parsimony optimality criterion. This has led to much work on using heuristic tree searches to find approximate solutions. We present an approach for finding exact optimal solutions that employs and complements the current heuristic methods for finding optimal trees. Given a set of taxa and a set of aligned sequences of characters, there may be subsets of characters that are compatible, and for each such subset there is an associated (possibly partially resolved) phylogeny with edges corresponding to each character state change. These perfect phylogenies serve as anchor trees for our constrained search space. We show that, for sequences with compatible sites, the parsimony score of any tree [Formula: see text] is at least the parsimony score of the anchor trees plus the number of inferred changes between [Formula: see text] and the anchor trees. As the maximum-parsimony optimality score is additive, the sum of the lower bounds on compatible character partitions provides a lower bound on the complete alignment of characters. This yields a region in the space of trees within which the best tree is guaranteed to be found; limiting the search for the optimal tree to this region can significantly reduce the number of trees that must be examined in a search of the space of trees. We analyze this method empirically using four different biological data sets as well as surveying 400 data sets from the TreeBASE repository, demonstrating the effectiveness of our technique in reducing the number of steps in exact heuristic searches for trees under the maximum-parsimony optimality criterion.

Characterizing Local Optima for Maximum Parsimony.

Finding the best phylogenetic tree under the maximum parsimony optimality criterion is computationally difficult. We quantify the occurrence of such optima for well-behaved sets of data. When nearest neighbor interchange operations are used, multiple local optima can occur even for "perfect" sequence data, which results in hill-climbing searches that never reach a global optimum. In contrast, we show that when neighbors are defined via the subtree prune and regraft metric, there is a single local optimum for perfect sequence data, and thus, every such search finds a global optimum quickly. We further characterize conditions for which sequences simulated under the Cavender-Farris-Neyman and Jukes-Cantor models of evolution yield well-behaved search spaces.

On Determining if Tree-based Networks Contain Fixed Trees.

We address an open question of Francis and Steel about phylogenetic networks and trees. They give a polynomial time algorithm to decide if a phylogenetic network, N, is tree-based and pose the problem: given a fixed tree T and network N, is N based on T? We show that it is [Formula: see text]-hard to decide, by reduction from 3-Dimensional Matching (3DM) and further that the problem is fixed-parameter tractable.

US consumer attitudes toward sodium in baby and toddler foods.

Dietary data from a nationally representative survey indicate about 80% of US toddlers aged 1-3 years consume too much dietary sodium, which can influence their preference for salty foods in later life. Information on consumer attitudes can inform strategies to reduce sodium in baby and toddler foods. Data were obtained from a 2012 online survey sent to a sample of 11636 US adults aged ≥18 years enrolled in a national probability-based consumer panel; 6378 completed the survey and had non-missing responses to the question of interest, "It is important for baby and toddler foods to be low in sodium." Prevalence of agreement was estimated. Logistic regression was used to describe associations of respondent characteristics with agreement. The majority of respondents were non-Hispanic white and had a household income ≥$60,000. About 7 in 10 (68%, 95% CI: 66%-70%) respondents agreed it is important for baby or toddler foods to be low in sodium. More than 6 of 10 respondents in most subgroups agreed. Among parents with a child currently aged <2 years (N = 390), 82% agreed (95% CI: 77%-87%); the highest agreement included parents who thought sodium was very harmful to their own health (92%, 95% CI: 85%-99%) or who were watching/reducing their own sodium intake (95%, 95% CI: 90%-100%). After adjusting for sex, age, race-ethnicity, agreement was most strongly associated with being a parent of a child <2 years, thinking sodium was harmful, and watching/reducing sodium intake (adjusted odds ratios ≥ 2.5, 95% CI's ≠1.0). The majority of respondents including most parents agreed it is important for baby and toddler foods to be low in sodium, suggesting wide consumer support for strategies to lower sodium in these foods.

Maximum Covering Subtrees for Phylogenetic Networks.

Tree-based phylogenetic networks, which may be roughly defined as leaf-labeled networks built by adding arcs only between the original tree edges, have elegant properties for modeling evolutionary histories. We answer an open question of Francis, Semple, and Steel about the complexity of determining how far a phylogenetic network is from being tree-based, including non-binary phylogenetic networks. We show that finding a phylogenetic tree covering the maximum number of nodes in a phylogenetic network can be computed in polynomial time via an encoding into a minimum-cost flow problem.

Change in US Adult Consumer Knowledge, Attitudes, and Behaviors Related to Sodium Intake and Reduction: SummerStyles 2012 and 2015.

PURPOSE: To describe changes in consumer knowledge, attitudes, and behaviors related to sodium reduction from 2012 to 2015. DESIGN: A cross-sectional analysis using 2 online, national research panel surveys. SETTING: United States. PARTICIPANTS: A total of 7796 adults (18+ years). MEASURES: Sodium-related knowledge, attitudes, and behaviors. ANALYSIS: Data were weighted to match the US population survey proportions using 9 factors. Wald χ2 tests were used to examine differences by survey year and hypertensive status. RESULTS: Despite the lack of temporal changes observed in respondent characteristics (mean age: 46 years, 67% were non-Hispanic white, and 26% reported hypertension), some changes were found in the prevalence of sodium-related knowledge, attitudes, and behaviors. The percentage of respondents who recognized processed foods as the major source of sodium increased from 54% in 2012 to 57% in 2015 ( P = .04), as did the percentage of respondents who buy or choose low/reduced sodium foods, from 33% in 2012 to 37% in 2015 ( P = .016). In contrast, the percentage of self-reported receipt of health professional advice among persons with hypertension decreased from 59% in 2012 to 45% in 2015 ( P < .0001). Other sodium-related knowledge, attitudes, and behaviors did not change significantly during 2012 to 2015. CONCLUSION: In recent years, some positive changes were observed in sodium-related knowledge and behaviors; however, the decrease in reported health professional advice to reduce sodium among respondents with hypertension is a concern.

Knowledge, Attitudes, and Behaviors Related to Sodium Intake and Reduction Among Adult Consumers in the United States.

PURPOSE: To describe the prevalence and determinants of sodium-related knowledge, attitudes, and behaviors among U.S. adults Design. A cross-sectional survey was used. SETTING: The study was set in the United States in 2012. SUBJECTS: Participants were 6122 U.S. adults. MEASURES: Sodium-related knowledge, attitudes, and behaviors were measured. ANALYSIS: Chi-squared tests were used to determine differences in sodium-related knowledge, attitude, and behaviors by respondent characteristics; multiple logistic regression was used to examine associations between selected respondent characteristics and health professional advice, reported action, or knowledge, attitudes, and behaviors (adjusted for all other respondent characteristics). RESULTS: About three-fourths of respondents answered eating too much sodium is "somewhat" or "very" harmful to their health. Twenty-six percent reported receiving health professional advice, and 45% reported taking action to reduce their sodium intake. The prevalence of reported action was highest among adults receiving advice, those with hypertension, blacks, and those aged ≥65 years. Sixty-two percent who reported action agreed that most of their sodium comes from processed or restaurant foods. Of those reporting action, the most common tactics to reduce sodium intake were checking nutrition labels, using other spices than salt, and choosing low-sodium foods; requesting lower-sodium options when eating out was the least common tactic. CONCLUSION: Results suggest almost half of adults overall and the vast majority of those receiving health professional advice are taking some action to watch or reduce sodium intake. Although a substantial proportion report using recommended tactics to lower intake, many are not using the most effective tactics. In order to reach the general population, health communication messages could be simpler and focus on the most effective tactics to reduce sodium intake. Furthermore, health professionals can help reduce sodium intake by discussing the benefits of sodium reduction and tactics to do so, regardless of a hypertension diagnosis.

Changes in Consumer Attitudes toward Broad-Based and Environment-Specific Sodium Policies-SummerStyles 2012 and 2015.

We examined temporal changes in consumer attitudes toward broad-based actions and environment-specific policies to limit sodium in restaurants, manufactured foods, and school and workplace cafeterias from the 2012 and 2015 SummerStyle surveys. We used two online, national research panel surveys to conduct a cross-sectional analysis of 7845 U.S. adults. Measures included self-reported agreement with broad-based actions and environment-specific policies to limit sodium in restaurants, manufactured foods, school cafeterias, workplace cafeterias, and quick-serve restaurants. Wald Chi-square tests were used to examine the difference between the two survey years and multivariate logistic regression was used to obtain odds ratios. Agreement with broad-based actions to limit sodium in restaurants (45.9% agreed in 2015) and manufactured foods (56.5% agreed in 2015) did not change between 2012 and 2015. From 2012 to 2015, there was a significant increase in respondents that supported environment-specific policies to lower sodium in school cafeterias (80.0% to 84.9%; p < 0.0001), workplace cafeterias (71.2% to 76.6%; p < 0.0001), and quick-serve restaurants (70.8% to 76.7%; p < 0.0001). Results suggest substantial agreement and support for actions to limit sodium in commercially-processed and prepared foods since 2012, with most consumers ready for actions to lower sodium in foods served in schools, workplaces, and quick-serve restaurants.

Approximating subtree distances between phylogenies.

We give a 5-approximation algorithm to the rooted Subtree-Prune-and-Regraft (rSPR) distance between two phylogenies, which was recently shown to be NP-complete. This paper presents the first approximation result for this important tree distance. The algorithm follows a standard format for tree distances. The novel ideas are in the analysis. In the analysis, the cost of the algorithm uses a "cascading" scheme that accounts for possible wrong moves. This accounting is missing from previous analysis of tree distance approximation algorithms. Further, we show how all algorithms of this type can be implemented in linear time and give experimental results.

Do Lower Calorie or Lower Fat Foods Have More Sodium Than Their Regular Counterparts?

The objective of this study was to compare the sodium content of a regular food and its lower calorie/fat counterpart. Four food categories, among the top 20 contributing the most sodium to the US diet, met the criteria of having the most matches between regular foods and their lower calorie/fat counterparts. A protocol was used to search websites to create a list of "matches", a regular and comparable lower calorie/fat food(s) under each brand. Nutrient information was recorded and analyzed for matches. In total, 283 matches were identified across four food categories: savory snacks (N = 44), cheese (N = 105), salad dressings (N = 90), and soups (N = 44). As expected, foods modified from their regular versions had significantly reduced average fat (total fat and saturated fat) and caloric profiles. Mean sodium content among modified salad dressings and cheeses was on average 8%-12% higher, while sodium content did not change with modification of savory snacks. Modified soups had significantly lower mean sodium content than their regular versions (28%-38%). Consumers trying to maintain a healthy diet should consider that sodium content may vary in foods modified to be lower in calories/fat.

Accuracy and Usefulness of Select Methods for Assessing Complete Collection of 24-Hour Urine: A Systematic Review.

Twenty-four-hour urine collection is the recommended method for estimating sodium intake. To investigate the strengths and limitations of methods used to assess completion of 24-hour urine collection, the authors systematically reviewed the literature on the accuracy and usefulness of methods vs para-aminobenzoic acid (PABA) recovery (referent). The percentage of incomplete collections, based on PABA, was 6% to 47% (n=8 studies). The sensitivity and specificity for identifying incomplete collection using creatinine criteria (n=4 studies) was 6% to 63% and 57% to 99.7%, respectively. The most sensitive method for removing incomplete collections was a creatinine index <0.7. In pooled analysis (≥2 studies), mean urine creatinine excretion and volume were higher among participants with complete collection (P<.05); whereas, self-reported collection time did not differ by completion status. Compared with participants with incomplete collection, mean 24-hour sodium excretion was 19.6 mmol higher (n=1781 specimens, 5 studies) in patients with complete collection. Sodium excretion may be underestimated by inclusion of incomplete 24-hour urine collections. None of the current approaches reliably assess completion of 24-hour urine collection.

Reconstructing reticulate evolution in species-theory and practice.

We present new methods for reconstructing reticulate evolution of species due to events such as horizontal transfer or hybrid speciation; both methods are based upon extensions of Wayne Maddison's approach in his seminal 1997 paper. Our first method is a polynomial time algorithm for constructing phylogenetic networks from two gene trees contained inside the network. We allow the network to have an arbitrary number of reticulations, but we limit the reticulation in the network so that the cycles in the network are node-disjoint ("galled"). Our second method is a polynomial time algorithm for constructing networks with one reticulation, where we allow for errors in the estimated gene trees. Using simulations, we demonstrate improved performance of this method over both NeighborNet and Maddison's method.

Hamiltonian walks of phylogenetic treespaces.

We answer Bryant's combinatorial challenge on minimal walks of phylogenetic treespace under the nearest-neighbor interchange (NNI) metric. We show that the shortest path through the NNI-treespace of n-leaf trees is Hamiltonian for all n. That is, there is a minimal path that visits all binary trees exactly once, under NNI moves.

Walks on SPR neighborhoods.

A nearest-neighbor-interchange (NNI)-walk is a sequence of unrooted phylogenetic trees, T1, T2, . . . , T(k) where each consecutive pair of trees differs by a single NNI move. We give tight bounds on the length of the shortest NNI-walks that visit all trees in a subtree-prune-and-regraft (SPR) neighborhood of a given tree. For any unrooted, binary tree, T, on n leaves, the shortest walk takes Θ(n²) additional steps more than the number of trees in the SPR neighborhood. This answers Bryant's Second Combinatorial Challenge from the Phylogenetics Challenges List, the Isaac Newton Institute, 2011, and the Penny Ante Problem List, 2009.