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Binary phylogenetic trees inferred from biological data are central to understanding the shared evolutionary history of organisms. Inferring the placement of latent nodes in a tree by any optimality criterion (e.g., maximum likelihood) is an NP-hard problem, propelling the development of myriad heuristic approaches. Yet, these heuristics often lack a systematic means of uniformly sampling random trees or effectively exploring a tree space that grows factorially, which are crucial to optimisation problems such as machine learning. Accordingly, we present Phylo2Vec, a new parsimonious representation of a phylogenetic tree. Phylo2Vec maps any binary tree with $n$ leaves to an integer vector of length $n$. We prove that Phylo2Vec is both well-defined and bijective to the space of phylogenetic trees. The advantages of Phylo2Vec are twofold: i) easy uniform sampling of binary trees and ii) systematic ability to traverse tree space in very large or small jumps. As a proof of concept, we use Phylo2Vec for maximum likelihood inference on five real-world datasets and show that a simple hill climbing-based optimisation efficiently traverses the vastness of tree space from a random to an optimal tree., Comment: 20 pages, 6 figures

Phylogenetics is now fundamental in life sciences, providing insights into the earliest branches of life and the origins and spread of epidemics. However, finding suitable phylogenies from the vast space of possible trees remains challenging. To address this problem, for the first time, we perform both tree exploration and inference in a continuous space where the computation of gradients is possible. This continuous relaxation allows for major leaps across tree space in both rooted and unrooted trees, and is less susceptible to convergence to local minima. Our approach outperforms the current best methods for inference on unrooted trees and, in simulation, accurately infers the tree and root in ultrametric cases. The approach is effective in cases of empirical data with negligible amounts of data, which we demonstrate on the phylogeny of jawed vertebrates. Indeed, only a few genes with an ultrametric signal were generally sufficient for resolving the major lineages of vertebrate. With cubic-time complexity and efficient optimisation via automatic differentiation, our method presents an effective way forwards for exploring the most difficult, data-deficient phylogenetic questions., Comment: 13 pages, 4 figures, 14 supplementary pages, 2 supplementary figures; typo in references corrected

Additional file 1: Fig. S1. Distribution of t-statistics for variables including across 1000 replicate regression analyses, each excluding a random set of 25% of species. The model evaluated the explanatory power of wing loading, body mass, median annual temperature and UV radiation on mitochondrial and nuclear molecular rates. Overlap between t values and 0 indicates non-significant relationships. Fig. S2. Distribution of t-statistics for variables including across 1000 replicate regression analyses, each excluding a random set of 25% of species. The model evaluated the explanatory power of hand-wing index, body mass, median annual temperature and UV radiation on mitochondrial and nuclear molecular rates. Overlap between t values and 0 indicates non-significantrelation ships. Fig. S3. Phylogenetic estimate of the family Furnariidae [36], showing the taxa included in regression models with mitochondrial (red) and nuclear (orange) data. Inclusion depended on availability of data on all of the variables describing environment, wing loading, hand-wing index, and molecular evolutionary rates. Table S1. Flight habit scores for 45 species in the Furnariidae. Scores are based on seasonal movement, foraging behaviour, and foraging stratum according to natural history data published. Higher score, more intensive is the flight habit. Table S2. PGLS regression coefficients from models testing whether the hand-wing index and mean values for environmental variables explain molecular rates. A Box-Cox transformation was used for all molecular rates variables. Table S3. PGLS regression coefficients from models testing whether the wing loading index and mean values for environmental variables explain molecular rates. A Box-Cox transformation was used for all molecular rates variables.

Figure S5. Relationships between phylogenetic stemminess and error in rate estimates using least-squares dating in LSD for 12 simulation treatments. Dashed lines indicate half an order of magnitude above or below the rates used for simulation. Light grey lines indicate lines of best fit for the estimates. (PDF 634 kb)

Figure S3. Examples of phylogenetic trees with different degrees of stemminess, as measured by the proportion of the overall tree length represented by internal branches (values in parentheses). (PDF 40 kb)

Figure S4. Relationships between phylogenetic stemminess and error in rate estimates using regression of root-to-tip distances in TempEst for 12 simulation treatments. Dashed lines indicate half an order of magnitude above or below the rates used for simulation. Light grey lines indicate lines of best fit for the estimates. (PDF 639 kb)

Table S1. One-sample Wilcoxon tests of errors in rate estimates obtained using TempEst, LSD, and BEAST. Significant results are indicated in bold font. Table S2. Mann-Whitney-Wilcoxon pairwise comparisons between standardized errors in rate estimates obtained using TempEst. Rows and columns correspond to the 12 simulation scenarios. Significant results are indicated in bold font. Table S3. Mann-Whitney-Wilcoxon pairwise comparisons between standardized errors in rate estimates obtained using LSD. Rows and columns correspond to the 12 simulation scenarios. Significant results are indicated in bold font. Table S4. Mann-Whitney-Wilcoxon pairwise comparisons between standardized errors in rate estimates obtained using BEAST. Rows and columns correspond to the 12 simulation scenarios. Significant results are indicated in bold font. Table S5. Marginal likelihoods of different combinations of clock models and tree priors for six mitogenomic data sets. (DOCX 36 kb)

Figure S6. Relationships between phylogenetic stemminess and error in median posterior estimates using Bayesian inference in BEAST for 12 simulation treatments. Dashed lines indicate half an order of magnitude above or below the rates used for simulation. Light grey lines indicate lines of best fit for the estimates. (PDF 643 kb)

Figure S2. Pairwise comparisons of rate estimates from regression of root-to-tip distances in TempEst, least-squares dating in LSD, and Bayesian inference in BEAST. Comparisons are between BEAST and TempEst (top), BEAST and LSD (middle), and TempEst and LSD ottom). The closer the fit of the points to the solid lines, the greater the congruence between the estimates from the two methods being compared. Dashed lines indicate a line of best fit for the estimates. The two distinct clouds of points within each panel represent the estimates from the data simulated with low and high rates. Proportional difference and bias were calculated as in a previous study by Duchêne et al. [31]. Proportional difference is the difference in the estimates between two methods, divided by the first rate estimate. Bias is the proportion of data sets for which the estimate along the x-axis is greater than that along the y-axis. (PDF 199 kb)

Figure S1. Simulations of sequence evolution were performed across 12 different scenarios, representing different combinations of a mean rate, b rate variation among lineages, and c phylo-temporal clustering. The conditions of the simulation scenarios are based on those observed in time-structured mitogenomic data sets. One hundred replicates were performed for each of the 12 scenarios. d Data were analysed using three different methods. (PDF 171 kb)

Sexual selection is a powerful agent of evolution, driving microevolutionary changes in the genome and macroevolutionary rates of lineage diversification. The mechanisms by which sexual selection might influence macroevolution remain poorly understood. For example, sexual selection might drive positive selection for key adaptations that facilitate diversification. Furthermore, sexual selection might be a general driver of molecular evolutionary rate. We lay out some of the potential mechanisms that create a link between sexual selection and diversification, based on causal effects on other life-history traits such as body mass and the rate of molecular evolution. Birds are ideally suited for testing the importance of these relationships because of their diverse reproductive systems and the multiple evolutionary radiations that have produced their astounding modern diversity. We show that sexual selection (measured as the degree of polygyny) interacts with the rate of molecular evolution and with body mass to predict species richness at the genus level. A high degree of polygyny and rapid molecular evolution are positively associated with the net rate of diversification, with the two factors being especially important for explaining diversification in large-bodied taxa. Our findings further suggest that mutation rates underpin some of the macroevolutionary effects of sexual selection. We synthesize the existing theory on sexual selection as a force for diversity and propose avenues for exploring this association using genome data.

Sexual selection is a powerful agent of evolution, driving microevolutionary changes in the genome and macroevolutionary rates of lineage diversification. The mechanisms by which sexual selection might influence macroevolution remain poorly understood. For example, sexual selection might drive positive selection for key adaptations that facilitate diversification. Furthermore, sexual selection might be a general driver of molecular evolutionary rate. We lay out some of the potential mechanisms that create a link between sexual selection and diversification, based on causal effects on other life-history traits such as body mass and the rate of molecular evolution. Birds are ideally suited for testing the importance of these relationships because of their diverse reproductive systems and the multiple evolutionary radiations that have produced their astounding modern diversity. We show that sexual selection (measured as the degree of polygyny) interacts with the rate of molecular evolution and with body mass to predict species richness at the genus level. A high degree of polygyny and rapid molecular evolution are positively associated with the net rate of diversification, with the two factors being especially important for explaining diversification in large-bodied taxa. Our findings further suggest that mutation rates underpin some of the macroevolutionary effects of sexual selection. We synthesize the existing theory on sexual selection as a force for diversity and propose avenues for exploring this association using genome data.

Sexual selection is a powerful agent of evolution, driving microevolutionary changes in the genome and macroevolutionary rates of lineage diversification. The mechanisms by which sexual selection might influence macroevolution remain poorly understood. For example, sexual selection might drive positive selection for key adaptations that facilitate diversification. Furthermore, sexual selection might be a general driver of molecular evolutionary rate. We lay out some of the potential mechanisms that create a link between sexual selection and diversification, based on causal effects on other life-history traits such as body mass and the rate of molecular evolution. Birds are ideally suited for testing the importance of these relationships because of their diverse reproductive systems and the multiple evolutionary radiations that have produced their astounding modern diversity. We show that sexual selection (measured as the degree of polygyny) interacts with the rate of molecular evolution and with body mass to predict species richness at the genus level. A high degree of polygyny and rapid molecular evolution are positively associated with the net rate of diversification, with the two factors being especially important for explaining diversification in large-bodied taxa. Our findings further suggest that mutation rates underpin some of the macroevolutionary effects of sexual selection. We synthesize the existing theory on sexual selection as a force for diversity and propose avenues for exploring this association using genome data.