The evolution of the full repertoire of proteins encoded in a given genome is mostly driven by gene duplications, deletions, and sequence modifications of existing proteins. Indirect information about relative rates and other intrinsic parameters of these three basic processes is contained in the proteome-wide distribution of sequence identities of pairs of paralogous proteins.
We introduce a simple mathematical framework based on a stochastic birth-and-death model that allows one to extract some of this information and apply it to the set of all pairs of paralogous proteins in H. pylori, E. coli, S. cerevisiae, C. elegans, D. melanogaster, and H. sapiens. It was found that the histogram of sequence identities p generated by an all-to-all alignment of all protein sequences encoded in a genome is well fitted with a power-law form ~ p-γ with the value of the exponent γ around 4 for the majority of organisms used in this study. This implies that the intra-protein variability of substitution rates is best described by the Gamma-distribution with the exponent α ≈ 0.33. Different features of the shape of such histograms allow us to quantify the ratio between the genome-wide average deletion/duplication rates and the amino-acid substitution rate.
We separately measure the short-term ("raw") duplication and deletion rates 
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View MathML"> which include gene copies that will be removed soon after the duplication event and their dramatically reduced long-term counterparts rdup, rdel. High deletion rate among recently duplicated proteins is consistent with a scenario in which they didn't have enough time to significantly change their functional roles and thus are to a large degree disposable. Systematic trends of each of the four duplication/deletion rates with the total number of genes in the genome were analyzed. All but the deletion rate of recent duplicates View MathML"> were shown to systematically increase with Ngenes. Abnormally flat shapes of sequence identity histograms observed for yeast and human are consistent with lineages leading to these organisms undergoing one or more whole-genome duplications. This interpretation is corroborated by our analysis of the genome of Paramecium tetraurelia where the p-4 profile of the histogram is gradually restored by the successive removal of paralogs generated in its four known whole-genome duplication events.
