On the dispersion index of a Markovian molecular clock

The number of nucleotide substitutions accumulated in a gene or in a lineag e is an important random variable in the study of molecular evolution. Of p articular interest is the ratio of the variance to the mean of that random variable, often known as the dispersion index. Because nucleotide substitut ion is most commonly modeled by a continuous-time four-state Markov chain, this paper provides a systematic method of computing the dispersion indices exhibited by a continuous-time four-state Markov chain. Using this method along with computer algebra and Monte Carlo simulation, this paper offers p artially proven conjectures that were supported by thorough computer experi ments. It is believed that the Tamura model, the equal-input model and the Takahata-Kimura model always exhibit dispersion indices less than 2. It is also believed that a general four-state model can be chosen to exhibit a di spersion index of any desired magnitude, although the chance of a randomly chosen such model exhibiting a dispersion index greater than 2 is as small as about 2%. Relevance of these findings to the neutral theory is discussed . (C) 2001 Published by Elsevier Science Inc.