Fig. 1: Challenges and solutions for the detection of molecular convergence.

a, False convergence is caused by tree topology errors. b, The overview of CSUBST. This programme processes substitution probabilities to derive observed (O_C^N and O_C^S) and expected (E_C^N and E_C^S) numbers of non-synonymous and synonymous convergence and evaluate their rates (dN_C and dS_C) in branch combinations in a phylogenetic tree. c, Generation of simulated datasets for performance evaluation in different evolutionary scenarios. The numbers of observed non-synonymous and synonymous convergence are indicated above trees (O_C^N and O_C^S, respectively; mean ± standard deviation). d, The estimated rates of protein convergence in different scenarios. Each box plot corresponds to the results of 1,000 simulations. Dashed lines indicate the theoretical expectation (= 1.0) except for the ratio of convergence and divergence events (C/D) (refs. ^13,17) for which no theoretical expectation is available. Values greater than the 95th percentile in the Neutral scenario are defined as true and false positives in Convergent and other scenarios, respectively, and are indicated at the top of the plot in pink (true) and blue (false). Box plot elements are defined as follows: centre line, median; box limits, upper and lower quartiles; whiskers, 1.5 × interquartile range. e, Performance of convergence metrics in empirical datasets. Known examples of protein convergences and HGTs are analysed with C/D, dN_C, dS_C and ω_C. Median values (bars) are overlaid on individual data points that correspond to gene trees. In trees where convergence occurred in more than two lineages, the median of all focal branch pairs is reported. The branch pairs sister to the focal branches are shown as a control¹⁰, except in cases where there is no substitution at all or the sister branches are phylogenetically not independent. Divergence time is according to timetree.org⁹⁵. The comparison with the background levels for each dataset is shown in Supplementary Fig. 5. The characteristics of the datasets are summarized in Supplementary Table 3. MYA, million years ago. Image credits for panel e: Cenchrus echinatus, Chutima Chaimratana/Shutterstock; Tetraopes tetrophthalmus, Paul Reeves Photography/Shutterstock; Myotis myotis, Agami Photo Agency/Shutterstock; Colobus guereza, Radek Karko/Shutterstock; Alloteropsis semialata Alexey Yakovlev under a Creative Commons license CC BY 4.0; Cuscuta europaea, ChWeiss/Shutterstock; alll other images except for Nepenthes cf. alata and Cephalotus follicularis from freepik.com.