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1 c AT AT 1. DNA RNA EST

2 A B C D t 1 4 n A B C D h p q r s A B C D i j t 1 t 6 h 1 L(θ X h ) (1.1) L(θ X h )=f(x h θ)= i π ip ip(t 1)p iq(t 2) j p ij(t 5 + t 6)p jr(t 3)p js(t 4) π i A B C D i p ij(t) t i j i j Q h i,j L(θ X) n (1.2) L(θ X)= L(θ X) h=1 P Q

3 291 Q i j π j j AT Mycoplasma 75 Micrococcus 25 16S RNA AT Mooers and Holmes, 2000 AT AT Saccone et al., 1999 Chang and Campbell, 2000; Tarrio et al., 2001 AT π RNA Lockhart et al., 1994 Chang and Campbell, 2000 Stenico et al., rpl14 rpl16 rps3 rps11 rps12 rpob tufa Helicosporidium sp. Chromera velia Moore et al., 2008; Janouškovec et al.,

292 60 2 2012 2. 7 1 2 100 RAxML Ver.7.2.8 Stamatakis et al., 2008 GTR+Γ LG+Γ Helicosporidium sp.

4 RAxML Ver Stamatakis et al., 2008 GTR+Γ LG+Γ Helicosporidium sp. AT AAA AAT ATT TTT TTA TAA TAT ATA Ile Asn Lys Phe ρ 4 Euglena longa Helicosporidium sp. ρ = Ile Asn Lys Phe 2 Helicosporidium sp. =

5 293 AT Jermiin et al Ho and Jermiin π AT BppML BppSeqGen Dutheil and Boussau, 2008 AT rpl14 rpl16 rps3 rps11 rps12 rpob tufa Helicosporidium sp cdna NCBI MAFFT Katoh et al., ,279

6 cdna 3,837 1,279 cdna PAL2NAL Suyama et al., 2006 Helicosporidium sp. 2 Janouškovec et al cdna YN98 Yang and Nielsen, / YN DNA 4 3 =64 Standard Genetic Code TAG TAA TGA 3 YN98 61 YN98 YN98 i j 0 2 uπ j (2.1) q ij = uκπ j uωπ j uωκπ j π j j YN98 DNA π A,π T,π G,π C π j κ purine A, G pyrimidine C, T purine pyrimidine ω u 1 1 π j,κ,ω i j q ij = j:j i qij Q t n (2.2) P (t n)=e tnq 1 4 A B C D A B C D h p q r s A B C D i j (2.3) L(θ X h )=f(x h θ)= 61 i=1 π ip ip(t 1)p iq(t 2) 61 j=1 p ij(t 5 + t 6)p jr(t 3)p js(t 4) 1.2

7 295 YN98 m t m i j 0 2 (2.4) q (m) ij = u (m) π (m) j u (m) κ (m) π (m) j u (m) ω (m) π (m) j u (m) ω (m) κ (m) π (m) j YN98 π (m) j,κ (m),ω (m) / / t m Q (m) (2.5) P (t m)=e tmq(m) Boussau and Gouy, 2006 t x,y π xp xy(t) π yp yx(t) root 1 t 1 t 6 YN98 Q (1) Q (6) R X A B C D h p q r s (2.6) L(θ X h )=f(x h θ)= π x x=1 i=1 61 j=1 p (5) xi (t5)p(1) ip (t1)p(2) iq (t2) p (6) xj (t6)p(3) jr (t3)p(4) js (t4) Boussau and Gouy, A B / YN98 YN98 7 M 1 M B 3 B t 1 t 3 Q (1) t 4 t 8 Q (2) 3 M Γ Yang, 1994 YN98 + Γ Γ α / YN98 BppML Dutheil and Boussau, A B / YN98 + Γ

8 A B t M R root π j, κ, ω, α) BppSeqGen Dutheil and Boussau, ,279 3,837 M 1 M 7 R 3 B / 1 3, GTR + Γ

9 297 Tavaré, 1986 LG + Γ Le and Gascuel, 2008 RAxML Ver Stamatakis et al., / AT YN98 π j,κ,ω AT YN AT 1 Helicosporidium sp. 2 3 AT 1 YN98 AT YN AT AT AT AT AT 100 t 2 AT AT A AT B Helicosporidium

10 AT

11 A B sp. Pseudenoclonium akinetum AT

12 Helicosporidium sp. 3 B M 4 M 7 i π i A, π i T,π i G,π i C π i A,π i T Helicosporidium sp. AT AT AT Helicosporidium sp. 4 B AT A T AAA TTT Ile Asn Lys Phe Helicosporidium sp. 4 B 3 AT 3 Carlini et al., 2001 Helicosporidium sp. AT Hashimoto et al., AT Helicosporidium sp. 1 2 AT 1 4 B AT 2 1 Helicosporidium sp. 2 3 AT AT 3 AT 7

13 AT Helicosporidium sp BppML nhphylobayes Blanquart and Lartillot, 2008 Blanquart, S. and Lartillot, N A site- and time-heterogeneous model of amino-acid replacement, Molecular Biology and Evolution, 25, Boussau, B. and Gouy, M Efficient likelihood computations with nonreversible models of evolution, Systematic Biology, 55, Carlini, D. B., Chen, Y. and Stephan, W The relationship between third-codon position nucleotide content, codon bias, mrna secondary structure and gene expression in the Drosophilid

14 alcohol dehydrogenase genes, Adh and Adhr, Genetics, 159, Chang, B. S. W. and Campbell, D. L Bias in phylogenetic reconstruction of vertebrate rhodopsin sequences, Molecular Biology and Evolution, 17, Dutheil, J. and Boussau, B Non-homogeneous models of sequence evolution in the Bio++ suite of libraries and programs, BMC Evolutionary Biology, 8: 255. doi: / Hashimoto, T., Nakamura, Y., Nakamura, F., Shirakura, T., Adachi, J., Goto, N., Okamoto, K. and Hasegawa, M Protein phylogeny gives a robust estimation for early divergences of eukaryotes: phylogenetic place of a mitochondria-lacking protozoan, Giardia lamblia, Molecular Biology and Evolution, 12, Ho,S.Y.W.andJermiin,L.S Tracing the decay of the historical signal in biological sequence data, Systematic Biology, 53, Janouškovec, J., Horak, A., Obornik, M., Lukes, J. and Keeling, P. J A common red algal origin of the apicomplexan, dinoflagellate, and heterokont plastids, Proceedings of the National Academy of Science of the United States of America, 107, Jermiin,L.S.,Ho,S.Y.W.,Ababneh,F.,Robinson,J.andLarkum,A.W.D The biasing effect of compositional heterogeneity on phylogenetic estimates may be underestimated, Systematic Biology, 53, Katoh, K., Kuma, K., Toh, H. and Miyata, T MAFFT version 5 : Improvement in accuracy of multiple sequence alignment, Nucleic Acids Research, 33, Le, S. Q. and Gascuel, O An improved general amino acid replacement matrix, Molecular Biology and Evolution, 25, Lockhart,P.J.,Steel,M.A,Hendy,M.D.andPenny,D Recovering evolutionary trees under a more realistic model of sequence evolution, Molecular Biology and Evolution, 11, Mooers,A.andHolmes,E.C The evolution of base composition and phylogenetic inference, Trends in Ecology and Evolution, 15, Moore, R. B., Obornik, M., Janouškovec, J., Chrudimsky, T., Vancova, M., Green, D. H., Wright, S. W., Davies, N. W., Bolch, C. J., Heimann, K., Slapeta, J., Hoegh-Guldberg, O., Logsdon, J. M. and Carter, D. A A photosynthetic alveolate closely related to apicomplexan parasites, Nature, 451, Saccone, C., De Giorgi, C., Gissi, C., Pesole, G. and Reyes, A Evolutionary genomics in Metazoa: The mitochondrial DNA as a model system, Gene, 238, Stamatakis, A., Hoover, P. and Rougemont, J A rapid bootstrap algorithm for the RAxML web servers, Systematic Biology, 57 5, Stenico,M.,Lloyd,A.T.andSharp,P.M Codon usage in Caenorhabditis elegans: Delineation of translational selection and mutational biases, Nucleic Acids Research, 22, Suyama, M., Torrents, D. and Bork, P PAL2NAL: Robust conversion of protein sequence alignments into the corresponding codon alignments, Nucleic Acids Research, 34, Tarrio, R., Rodríguez-Trelles, F. and Ayala, F. J Shared nucleotide composition biases among species and their impact on phylogenetic reconstructions of the Drosophilidae, Molecular Biology and Evolution, 18, Tavaré, S Some probabilistic and statistical problems on the analysis of DNA sequences, Lecture on Mathematics in the Life Science, 17, Yang, Z Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: Approximate methods, Journal of Molecular Evolution, 39, Yang, Z. and Nielsen, R Synonymous and nonsynonymous rate variation in nuclear genes of mammals, Journal of Molecular Evolution, 46,

15 Proceedings of the Institute of Statistical Mathematics Vol. 60, No. 2, (2012) 303 Assessment of the Performance of Phylogenetic Inference Based on Simulated Protein-coding Sequences with Significant Compositional Heterogeneity Sohta A. Ishikawa and Tetsuo Hashimoto Graduate School of Life and Environmental Sciences, University of Tsukuba Phylogenetic analyses of molecular sequence data with commonly-used homogeneous substitution models assume the stationarity of nucleotide or amino-acid composition across tree, but real world data sometimes violate the assumption. This report assesses how significantly the violation of compositional stationarity affects the performance of homogeneous model-based phylogenetic inference by using simulated protein-coding sequences. In order to estimate parameters for sequence simulation, we prepared a realworld sequence data set of seven plastid genome-encoded protein genes with adenine plus thymine content (AT content) in all the 1st, 2nd, and 3rd codon positions extraordinarily biased between species, and subjected it to a maximum-likelihood analysis for a given model tree to estimate the parameters. The analysis was carried out assuming a non-homogeneous codon substitution model that can accommodate the heterogeneity of nucleotide composition in three codon positions across the tree. Using the parameters estimated and the model tree, we simulated protein-coding sequence data with compositional heterogeneity between species by the Monte-Carlo method. Finally, we tested the performance of homogeneous model-based phylogenetic analyses both at nucleotide and amino acid sequence levels for recovering the model ( correct ) tree. The results clearly demonstrated that both of the two analyses mostly failed to recover the correct tree but instead strongly favored artifactual trees attracted by the parallel compositional convergence between distantly-related species. This is de facto the first simulation study that assessed the appropriateness of applying homogeneous substitution models in phylogenetic analyses to protein-coding sequence data containing significant compositional heterogeneity. Key words: Phylogenetic inference, maximum-likelihood analysis, compositional heterogeneity, codon substitution model, simulation, model misspecification.

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