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is a solution of equation (1) such that, once R is given, the probability P(t) of substitution between two states can be computed for any t and the evolution of sequences through time is completely described as long as all GTR-assumptions are verified.
In the framework of ML phylogeny inference from multiple sequence alignments, Yang and Kumar (Yang and Kumar 1996) proposed to use a mean F# matrix (ie, the average of all F# matrices, each computed from the corresponding pairwise sequence comparison) for computing a single R for the whole tree. This procedure reduces, but does not eliminate, the risk of computing a complex R1. On the other hand, many phylogeny inference softwares implement optimization techniques that yield a single R for the whole tree. This approach removes the possibility of observing negative eigenvalues in P (because computation of log(P) is by-passed) but sacrifices the possibility of locally optimizing the transition rates (eg, for each pair of nodes) and thus constrains the hypothesis of homogeneity along the whole evolutionary tree (an assumption that can be unreasonable with some data sets). When locally computing a R matrix (eg, for a pair of sequences) using the procedure of Waddell and Steel (1997), the homogeneity assumption only holds for the corresponding portion of the tree.
The use of maximum likelihood (ML) algorithms in developing phylogenetic hypotheses requires a model of evolution. The frequently used General Time Reversible (GTR) family of nested models encompasses 64 models with different combinations of parameters for DNA site substitution. The models are listed here from the least complex to the most parameter rich. 781b155fdc