Despite the practically unlimited number of possible protein sequences, the number of basic shapes in which proteins fold seems not only to be finite, but also to be relatively small, with probably no more than 10,000 folds in existence. Moreover, the distribution of proteins among these folds is highly non-homogeneous -- some folds and superfamilies are extremely abundant, but most are rare. Protein folds and families encoded in diverse genomes show similar size distributions with notable mathematical properties, which also extend to the number of connections between domains in multidomain proteins. All these distributions follow asymptotic power laws, such as have been identified in a wide variety of biological and physical systems, and which are typically associated with scale-free networks. These findings suggest that genome evolution is driven by extremely general mechanisms based on the preferential attachment principle.

The evolutionary rates of protein-coding genes in an organism span, approximately, 3 orders of magnitude and show a universal, approximately log-normal distribution in a broad variety of species from prokaryotes to mammals. This universal distribution implies a steady-state process, with identical distributions of evolutionary rates among genes that are gained and genes that are lost. A mathematical model of such process is developed under the single assumption of the constancy of the distributions of the propensities for gene loss (PGL). This model predicts that genes of different ages, that is, genes with homologs detectable at different phylogenetic depths, substantially differ in those variables that correlate with PGL. We computationally partition protein-coding genes from humans, flies, and Aspergillus fungus into age classes, and show that genes of different ages retain the universal log-normal distribution of evolutionary rates, with a shift toward higher rates in ''younger'' classes but also with a substantial overlap. The only exception involves human primate-specific genes that show a heavy tail of rapidly evolving genes, probably owing to gene annotation artifacts. As predicted, the gene age classes differ in characteristics correlated with PGL. Compared with ''young'' genes (e.g., mammal-specific human ones), ''old'' genes (e.g., eukaryotespecific), on average, are longer, are expressed at a higher level, possess a higher intron density, evolve slower on the short time scale, and are subject to stronger purifying selection. Thus, genome evolution fits a simple model with approximately uniform rates of gene gain and loss, without major bursts of genomic innovation.gene age ͉ gene expression ͉ genome evolution ͉ intron density

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.