Polytomies, the power of phylogenetic inference, and the stochastic nature of molecular evolution: a comment on Walsh et al. (1999).

Walsh et al. (1999) recently suggested an innovative application of power analysis (Cohen 1977) to explore the nature of polytomies (multifurcating rather than bifurcating relationships) in phylogenetic inference. The frequent recovery of polytomies in phylogenetic analyses has prompted substantial interest in the underlying biological nature of these inferred polytomies. Many authors assume that polytomies simply reflect the inability to resolve bifurcating relationships ("soft" polytomies) and suggest that additional data or improved analyses will allow the recovery of the underlying relationships (e.g., Maddison 1989; DeSalle et al. 1994). In contrast, other researchers consider polytomous relationships to be valid phylogenetic hypotheses reflecting multiple simultaneous speciation events (a "hard" polytomy; Hoelzer and Melnick 1994a,b). Examining the differences between these alternative viewpoints is extremely difficult, because soft polytomies do not exhibit clear differences from hard polytomies in phylogenetic analyses. Specifically, Walsh et al. (1999) sought to determine whether internal branch lengths which were not significantly different from zero simply reflect an insufficient sample size (amount of sequence data). The example developed by Walsh et al. (1999) involved the polytomous relationships among auklets (Charadriiformes: Alcidae) inferred using mitochondrial DNA sequences. Since this inferred polytomy might reflect a soft polytomy (defined by Walsh et al. [1999] as speciation during successive glacial/interglacial periods during the late Pliocene and early Pleistocene) or a hard polytomy (defined as multiple speciations during the same period of climatic oscillation), Walsh et al. (1999) used power analysis to determine whether the number of base pairs (sites) of sequence data obtained was sufficiently large to detect substitutions along the internal branches if the speciation was not simultaneous (a soft polytomy). We found this an innovative application of classical power analysis and an excellent alternative to Monte Carlo simulation (e.g., Saitou and Nei 1986; Hillis et al. 1994; Huelsenbeck et al. 1996). We believe that an extremely important aspect of the work by Walsh et al. (1999) is their clear statement of the alternative and null hypotheses concerning the differentiation between hard and soft polytomies. As they point out, "resolution of the biological reality of polytomies is complicated by the fact that a polytomy represents the null hypothesis for phylogenetic reconstruction-all taxa are equally relatedand therefore cannot be proven." By restating the problem in the context of the power necessary to detect short internal branches, Walsh et al. (1999) place the hypothesis of a hard polytomy in a much more testable context. However, we were surprised to find that power analysis suggests only 215 to 1237 base pairs of mitochondrial sequence data are required for resolution of a soft polytomy in the auklets, as defined by Walsh et al. (1999). To further explore the study of polytomies in a statistical context, we examined the implications of the inherent variance exhibited by the nucleotide substitution process. It is clear that molecular evolution is a stochastic process with the number, type, and position of fixed sequence differences being determined by the processes of mutation, selection, and genetic drift. The stochastic nature of molecular evolution may have a profound impact upon the number of differences observed during a specific time period and it is possible that many fewer (or many more) mutations will be fixed during a particular period of time than one expects based upon the total number of substitutions in the phylogenetic tree. Nucleotide substitution can be modeled as a Poisson process (e.g., Wilson et al. 1987), with the probability (P) of N nucleotide substitutions occurring during a given period of time given by: