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Substitution rate matrices are used to correct multiple hits at the same sites, which requires the derivation of transition probabilities and evolutionary distances from substitution rate matrices. The derivation is essential in molecular phylogenetics and phylogenomics, and represents the only statistically sound way for developing scoring matrices used in sequence alignment and local string matching (e.g., BLAST and FASTA). Three different approaches are frequently used for deriving transition probabilities and evolutionary distances: 1) The probability reasoning, 2) Solving partial differential equations, and 3) Matrix exponential and logarithm. The first approach demands the least amount of mathematical skills but offers the best way for conceptual understanding, and can often generate nice mathematical expressions of transition probabilities and evolutionary distances. This review represents the most systematic and comprehensive numerical illustration of the first approach.

Deriving Transition Probabilities and Evolutionary Distances from Substitution Rate Matrix by Probability Reasoning