FIG. 1. Various models of rate variation across sites and lineages

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FIG. 1. Various models of rate variation across sites and lineages FIG. 1. Various models of rate variation across sites and lineages. The thickness of the lines represents different rates: the thicker the line, the higher the rate is at the site for the lineage. (a) equal rates; (b) rates-across-sites; (c) covarion; (d) bivariate rates—under this model the two subtrees are allowed to have different rates and the rates vary across sites independently between the two subtrees; (e) branch length mixture; (f) lineage-specific variation in proportions of variable sites. (c)–(f) are heterotachy models. The solid and dashed lines in (c) represent that the lineage can be switched “ON” or “OFF” for substitution, respectively. This figure is partially adapted from Wu and Susko (2009). From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

FIG. 4. Subselecting taxa for the mixed branch length model FIG. 4. Subselecting taxa for the mixed branch length model. Here a pair of eight-taxon trees is shown in which one taxon has a different edge length than the other three taxa in the same clade. In another simulation setting, a 16-taxon tree is subselected so that 1 taxon has different edge length than the other 7 taxa in the same clade. The length of the central edge connecting the two groups is set to 1.0. The short and long edges within the two groups have a length of 0.1 and 0.7, respectively. From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

FIG. 3. One pair of the simulating trees for a mixed branch length model. Here a tree of four taxa per group is shown. Other simulating trees have 8, 12, and 16 taxa per group. The length of the central edge connecting the two groups is set to 1.0. The short and long edges in the two groups have a length of 0.1 and 0.7, respectively. From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

FIG. 2. One of the four simulating trees with two “star” subtrees separated by an internal edge. Here a tree of 15 taxa per group is shown. Other simulating trees have 5, 10, and 20 taxa per group. b is the length of the central edge connecting the two subtrees, which is set to 1.0; a is the length of the edges within the two subtrees, which is set to 0.1, 0.3, 0.5, and 0.7, respectively, for different simulations. From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

FIG. 5. The type I error rates of the three test statistics for data simulated under the RAS model (lower panel of lines) and the power of the tests for data simulated under a general covarion model (upper panel of lines). (A) w test; (B) w′ test; (C) the correlation test. The simulating tree (see fig. 2) has 5, 10, 15, and 20 taxa per clade and a terminal edge length a = 0.1, 0.3, 0.5, and 0.7, respectively. The power of correlation tests is always 100% for different sizes of data, merging all lines into a single line (blue line in C). (B) Also shows some overlapping of the lines due to the same power for different size of the data. From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

FIG. 6. Power of the w tests for data simulated under the bivariate rate model. The gamma shape parameter α = 1.0 was used in the simulations. The simulating tree (see fig. 2) has 5, 10, 15, and 20 taxa per clade and a terminal edge length a = 0.1, 0.3, 0.5, and 0.7, respectively. Because the power for the 10 and 15 taxa per clade trees are all same (100%) for a = 0.1 and a = 0.3, the black line is overlapped with the red line and becomes invisible. From: Fast Statistical Tests for Detecting Heterotachy in Protein Evolution Mol Biol Evol. 2011;28(8):2305-2315. doi:10.1093/molbev/msr050 Mol Biol Evol | © The Author 2011. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com