# The covarion model for the evolution of proteins: Parameter estimates and comparison with holmquist, cantor, and Jukes' stochastic model

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## Summary

W. Fitch used a mathematical model to estimate the covarion size (the number of codons which are variable at a given time) and the turnover rate of covarions in the evolution of cytochrome c. We improve and correct the mathematical derivations and statistical estimation procedures in Fitch's model, altered to account more fully for the redundancy in the genetic code. We also consider a closely related model, which assumes the covarion fixing the last minimum mutation distance increasing (MDI) substitution has the same probability of losing variability as the other covarions. The average number of covarions is estimated to be at most five. Roughly 35 to 65% of the covarions are predicted to lose variability after each MDI substitution; this is smaller than Fitch's estimate, but the estimate is quite sensitive to changes in the data, which are a phylogenetic tree derived by Fitch and Margoliash. Both covarion models predict that there are about 0.9 to 2.0 total substitutions per variable codon in cytochrome c during “short” periods of evolutionary time (at most 10 MDI substitutions). This is less than the prediction from Holmquist, Cantor, and Jukes' stochastic model, which emphasizes variability over the entire time of divergence, rather than variability at a given time as in the covarion model, but this difference is predicted by the differing model assumptions.

Both the covarion and interactive models provide clear descriptions for hypotheses of a stochastic evolutionary process operating within deterministic selective constraints. Both depend on only two parameters, one measuring selective constraints, and the other, the rate of a stochastic process. Both factors are important, so it is unlikely that one could describe the process with fewer parameters. Since both models provide similar estimates for the rate of substitution for closely related pairs of species, it is plausible that both describe the same process, but from different viewpoints. Extensive tests on the protein data using an improved covarion model are necessary to determine whether these models are in fact compatible.

## Key words

Molecular evolution Evolutionary rates Cytochrome c Codon variability Mathematical modeling## Preview

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