Distribution of Base Pair Alternations in a Periodic DNA Chain: Application of Polya Counting to a Physical System

TL;DR

This paper develops a mathematical framework using Polya counting to analyze the distribution of base pair alternations in periodic DNA, providing exact calculations validated by simulations and exploring how chain length and base ratios influence heterogeneity.

Contribution

It extends Polya's Enumeration Theorem to partitioned necklaces, enabling direct calculation of base pair alternation distributions in circular DNA models.

Findings

Distribution functions match Monte Carlo simulations.

Distribution characteristics depend on chain length and base ratios.

Gaussians effectively fit the distribution data.

Abstract

In modeling DNA chains, the number of alternations between Adenine-Thymine (AT) and Guanine-Cytosine (GC) base pairs can be considered as a measure of the heterogeneity of the chain, which in turn could affect its dynamics. A probability distribution function of the number of these alternations is derived for circular or periodic DNA. Since there are several symmetries to account for in the periodic chain, necklace counting methods are used. In particular, Polya's Enumeration Theorem is extended for the case of a group action that preserves partitioned necklaces. This, along with the treatment of generating functions as formal power series, allows for the direct calculation of the number of possible necklaces with a given number of AT base pairs, GC base pairs and alternations. The theoretically obtained probability distribution functions of the number of alternations are accurately reproduced by Monte Carlo simulations and fitted by Gaussians. The effect of the number of base pairs on the characteristics of these distributions is also discussed, as well as the effect of the ratios of the numbers of AT and GC base pairs.