From nullomers to abundant motifs: Fractals, CpG Bias, and Chargaff's rules in genomic sequences

Nullomers - sequences entirely absent from a given genome - exhibit unexpected fractal structures when visualized using Chaos Game Representation (CGR). Unlike nullomers, rare sequences are defined as those that occur infrequently within the genome. Both nullomers and rare sequences conform to Generalised Chargaff's Second Parity rule at rates that far exceed random expectations. Beginning with nullomer analysis in Homo sapiens, we identified similar fractal patterns among rare sequences with low genomic frequency. We observe a continuous transition in various organisational properties, such as fractal geometry, CpG content, and Hamming distance between consensus sequences as a function of sequence frequency. In addition, our results reveal a fine-grained interpretation of Chargaff's rule: sequences exhibit frequency-dependent distributional characteristics. Rare sequences, in particular, display distinctive structural order that differentiates them from more abundant sequences, offering new insights into the underlying architecture of the genome as well as its informational structure. Moreover, these architectural and structural distinctions reinforce the perspective that information and meaning are encoded and managed independently within the genomic language.