Abstract
The frequency distributions of DNA k-mers are shaped by fundamental biological processes and offer a window into genome structure and evolution. Inspired by analogies to natural language, prior studies have attempted to model genomic k-mer usage using Zipf's law, a rank-frequency law originally formulated for words in human language. However, the extent to which this law accurately captures the distribution of k-mers across diverse species remains unclear. Here, we systematically analyze k-mer frequency spectra across more than 225,000 genome assemblies spanning all three domains of life and viruses. We demonstrate that Zipf's law consistently underperforms in modeling k-mer distributions. In contrast, we propose the truncated power law and Zipf-Mandelbrot distributions, which provide substantially improved fits across taxonomic groups. We show that genome size and GC content influence model performance, with larger and GC-content imbalanced genomes yielding better fits. Additionally, we perform an extensive analysis on vocabulary expansion and exhaustion across the same organisms using Heaps' law. We apply our modeling framework to evaluate simulated genomes generated by k-let preserving shuffling and deep generative language models. Our results reveal substantial differences between organismal genomes and their synthetic or shuffled counterparts, offering a novel approach to benchmark the biological plausibility of artificial genomes. Collectively, this work establishes new standards for modeling genomic k-mer distributions and provides insights relevant to synthetic biology, and evolutionary sequence analysis.