Chromosomal Inversion Symmetry: Generalized Chargaff Rules

The generalization of the second Chargaff rule to values of k larger than 1, states that the frequency of any k-mer on a single strand almost equals that of its inverse (reverse-complement). We demonstrate the validity of the generalized rule up to k=10 for all human chromosomes. Moreover, this holds for many species, both eukaryotes and prokaryotes, for ranges of k which may vary from 7 to 10 as chromosomal lengths vary from 2Mbp up to 200 Mbp. We demonstrate that the statistical distributions of inverted pairs of k-mers are very different from other natral pairings of k-mers, implying that inversion symmetry is a basic principle of chromosomal structure. We suggest that it came into being because genomic evolution employed many rearrangements which conisted of inversions of chromosomal sections; on length scales down to order 1-10Kbp. Model simulations substantiate this claim.Low-scale inversions during chromosomal evolution imply that IS may exist for short sections of human chromosomes. This is indeed the case: we find that chromosome sections of length 5Kbp satisfy IS for k=1 and k=2. The largest value of k for which IS holds,which we call the k-limit of IS, increases logarithmically as the section length increases. The logarithmic dependence of the k-limit on the length of the chromosome is a universal characteristic, observed throughout the tree of life.