Range-Limited Heaps' Law for Functional DNA Words in the Human Genome

Heaps' or Herdan's law is a linguistic law describing the relationship between the vocabulary/dictionary size (type) and word counts (token) to be a power-law function. Its existence in genomes with certain definition of DNA words is unclear partly because the dictionary size in genome could be much smaller than that in a human language. We define a DNA word in a genome as a DNA coding region that codes for a protein domain. Using human chromosomes and chromosome arms as individual samples, we establish the existence of Heaps' law in the human genome within limited range. Our definition of words in a genomic or proteomic context is different from that in large language models for DNA or protein sequences where words are usually short. Although an approximate power-law distribution of protein domain sizes due to gene duplication and the related Zipf's law is well known, their translation to the Heaps' law in DNA words is not automatic. Several other animal genomes are shown herein also to exhibit range-limited Heaps' law with our definition of DNA words, though with various exponents, partially depending on their level of complexity. Investigation of Heaps' law and its exponent value could provide an alternative narrative of reusage and redundancy of protein domains as well as creation of new protein domains from a linguistic perspective.