Modeling the evolution of ultraconserved elements by indels

Ultraconserved elements are segments of DNA that are identical or nearly identical in distantly related species. Finding 100% identity over long evolutionary times is unexpected, but pioneering research in human-mouse pairwise alignment uncovered something even more puzzling: these elements are not as rare as previously suspected. Furthermore, their sizes are distributed as a power-law, a feature that cannot be explained by standard models of genome evolution where conservation is expected to decay exponentially. Despite the power-law behavior having been reported and investigated in a wide variety of biological and physical contexts, from cell-division to protein family evolution, why it appears in the size distribution of ultraconserved elements remains elusive. To address this question, I propose a model of DNA sequence evolution by mutations of arbitrary length based on a classical integro-differential equation that arises in various applications in biology. The model captures the ultraconserved size distribution observed in pairwise alignments between human and 40 other vertebrates, encompassing more than 400 million years of evolution, from chimpanzee to zebrafish. I also show that the model can be used to predict other important aspects of genome evolution, such as indel rates and conservation in functional classes.