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Divergent Evolution of a Structural Proteome: Phenomenological Models

C. Brian Roland ^* and Eugene I. Shakhnovich 

^* Chemical Physics Program, and Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts

Correspondence: Address reprint requests to E. I. Shakhnovich, Dept. of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138. Tel.: 617-495-4130; Fax: 617-384-9228; E-mail: eugene{at}belok.harvard.edu.

We develop models of the divergent evolution of genomes; the elementary object of sequence dynamics is the protein structural domain. To identify patterns of organization that reflect mechanisms of evolution, we consider the individual genomes of many procaryote species, studying the arrangement of protein structural domains in the space of all polypeptide structures. We view the network of structural similarities as a graph, called the organismal Protein Domain Universe Graph (oPDUG); vertices represent types of structural domains and edges represent strong structural similarity. As observed before, each oPDUG is a highly nonrandom graph, as evidenced in the vertex degree distribution, which resembles a Pareto law (which has a power-law asymptotic). To explain this and other peculiar properties of the oPDUGs, we construct an evolving-graph model for the long-timescale evolutionary dynamics of oPDUGs, containing only divergent mechanisms of domain discovery. The model generates degree distributions (resembling Pareto laws) and clustering-coefficient distributions that are characteristic of the oPDUGs. In the infinite-graph limit, we analytically compute the exponent for specific biological parameters, as well as the complete phase diagram of the model, finding two distinct regimes of domain innovation dynamics. Thus, divergent evolutionary dynamics quantitatively explains the nonrandom organization of oPDUGs.