Evolution of ultraconserved elements by indels

Ultraconserved elements are segments of DNA that match identically in different species. Finding 100% identity over long evolutionary times is unexpected, but pioneering research in the human-mouse pairwise alignment uncovered something even more puzzling: these elements are not as rare as previously suspected, and their sizes are distributed as a power-law, a feature that cannot be explained by standard models of genome evolution where genome conservation is expected to decay exponentially over time. 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 to be fully clarified. To address this question, we 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. We also show that the model can be further used to predict other important aspects of genome evolution, such as indel rates and conservation in functional classes.