The Parameters of the Menzerath-Altmann Law in Genomes

Abstract

The relationship between the size of the whole and the size of the parts in language and music is known to follow the Menzerath-Altmann law at many levels of description (morphemes, words, sentences, …). Qualitatively, the law states that the larger the whole, the smaller its parts, e.g. the longer a word (in syllables) the shorter its syllables (in letters or phonemes). This patterning has also been found in genomes: the longer a genome (in chromosomes), the shorter its chromosomes (in base pairs). However, it has been argued recently that mean chromosome length is trivially a pure power function of chromosome number with an exponent of −1. The functional dependency between mean chromosome size and chromosome number in groups of organisms from three different kingdoms is studied. The fit of a pure power function yields exponents between −1.6 and 0.1. It is shown that an exponent of −1 is unlikely for fungi, gymnosperm plants, insects, reptiles, ray-finned fishes and amphibians. Even when the exponent is very close to −1, adding an exponential component is able to yield a better fit with regard to a pure power-law in plants, mammals, ray-finned fishes and amphibians. The parameters of the Menzerath-Altmann law in genomes deviate significantly from a power law with a −1 exponent with the exception of birds and cartilaginous fishes.

Acknowledgements

This article is dedicated to G. Altmann and the late P. Menzerath. We thank G. Altmann for his clarifications and J. Perarnau for technical advice on the statistical analyses. This work was supported by the grant Iniciació i reincorporació a la recerca from the Universitat Politècnica de Catalunya and the grant BASMATI (TIN2011-27479-C04-03) from the Spanish Ministry of Science and Innovation (RFC and JB).