Capacity and Expressiveness of Genomic Tandem Duplication

The majority of the human genome consists of repeated sequences. An important type of repeated sequences common in the human genome are tandem repeats, where identical copies appear next to each other. For example, in the sequence $AGTC\underline{TGTG}C$, $TGTG$ is a tandem repeat, that may be generated from $AGTCTGC$ by a tandem duplication of length $2$. In this work, we investigate the possibility of generating a large number of sequences from a \textit{seed}, i.e.\ a small initial string, by tandem duplications of bounded length. We study the capacity of such a system, a notion that quantifies the system's generating power. Our results include \textit{exact capacity} values for certain tandem duplication string systems. In addition, motivated by the role of DNA sequences in expressing proteins via RNA and the genetic code, we define the notion of the \textit{expressiveness} of a tandem duplication system as the capability of expressing arbitrary substrings. We then \textit{completely} characterize the expressiveness of tandem duplication systems for general alphabet sizes and duplication lengths. In particular, based on a celebrated result by Axel Thue from 1906, presenting a construction for ternary square-free sequences, we show that for alphabets of size 4 or larger, bounded tandem duplication systems, regardless of the seed and the bound on duplication length, are not fully expressive, i.e. they cannot generate all strings even as substrings of other strings. Note that the alphabet of size 4 is of particular interest as it pertains to the genomic alphabet. Building on this result, we also show that these systems do not have full capacity. In general, our results illustrate that duplication lengths play a more significant role than the seed in generating a large number of sequences for these systems.