Molecular Evolution Modeled as a Fractal Renewal Point Process in Agreement with the Dispersion of Substitutions in Mammalian Genes David R. Bickel, 1 Bruce J. West 2 1 Houston Health Science Center, University of Texas, 1100 Holcombe No. 4.430, Houston, TX 77030, USA 2 Center for Nonlinear Science, University of North Texas, P.O. Box 5368, Denton, TX 76203-5368, USA Received: 12 January 1998 / Accepted: 19 May 1998 Abstract. A fractal renewal point process (FRPP) is used to model molecular evolution in agreement with the relationship between the variance and the mean numbers of nonsynonymous and synonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike certain other episodic models, it also accounts for the increase in the index of dispersion with the mean number of substitutions in Ohta’s (1995) data. We find that this correlation is significant for nonsynonymous substitutions at the 1% level and for synonymous substi- tutions at the 10% level, even after removing lineage effects and when using Bulmer’s (1989) unbiased esti- mator of the index of dispersion. This model is simpler than most other overdispersed models of evolution in the sense that it is fully specified by a single interevent prob- ability distribution. Interpretations in terms of chaotic dynamics and in terms of chance and selection are dis- cussed. Key words: Overdispersion — Episodic evolution — Fractal renewal process — Nonlinear — Chaos — Neu- tral theory Introduction Based on the study hemoglobin chains, Zuckerkandl and Pauling (1962) suggested that the number of amino acid substitutions between homologous chains in different species is roughly proportional to the time since the di- vergence of those organisms from a common ancestor. In comparing the number of cytochrome c amino acid dif- ferences between different organisms, Margoliash (1963) noted that the number of differences between members of a taxon and organisms outside that taxon is constant. For example, the number of differences between any of the mammal sequences and the bird sequence is about equal. He concluded that the number of substitutions between two organisms is a function of the time that elapsed since the two species diverged from a common ancestor. Zuckerkandl and Pauling (1965) coined the term ‘‘molecular clock’’ to designate constant rates of molecular evolution. Progress in nucleic acid sequencing led to the concept of a DNA clock in addition to the protein clock since synonymous and nonsynonymous substitutions seem to occur at different rates. Although the substitution rates in genes appear to be approximately constant over long periods of time, they are not precisely constant. The neutral theory claims that the rate of evolution is stochastically constant, much as the rate of radioactive decay is stochastically constant; see Easteal et al. (1995). However, Gillespie (1991, p. 119) has demonstrated that the differences in the number of substitutions calculated from molecular data are more pronounced than those expected from a simple Poisson process, even when lineage effects are removed. He uses the index of dispersion, I(t), the ratio of the variance in the number of nucleotide substitutions to the mean num- ber of substitutions in time t, as a measure of how much variation a process has. This quantity was first intro- duced by Fano (1947) to quantify dispersion in ioniza- Correspondence to: D.R. Bickel; e-mail: BickelDR@aol.com J Mol Evol (1998) 47:551–556 © Springer-Verlag New York Inc. 1998