  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 as a coding region
in a genome 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 other definitions such as
over-represented k-mers which are much shorter in length. 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. When tokens were randomly sampled and
sample sizes reach to the maximum level, a deviation from the Heaps' law was
observed, but a quadratic regression in log-log type-token plot fits the data
perfectly. Investigation of type-token plot and its regression coefficients
could provide an alternative narrative of reusage and redundancy of protein
domains as well as creation of new protein domains from a linguistic
perspective.
