Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes

TL;DR

This paper provides a rigorous mathematical analysis of a generalized Poland-Scheraga DNA denaturation model, extending previous work to include non-complementary strands and analyzing phase transitions using bivariate renewal processes.

Contribution

It introduces a comprehensive mathematical framework for the generalized model, analyzing critical points and path properties with precision, extending the solvable structure of the original model.

Findings

Complete analysis of free energy singularities

Identification of localization-delocalization critical points

Precise estimates on the model's path properties

Abstract

The Poland-Scheraga model describes the denaturation transition of two complementary - in particular, equally long - strands of DNA, and it has enjoyed a remarkable success both for quantitative modeling purposes and at a more theoretical level. The solvable character of the homogeneous version of the model is one of features to which its success is due. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. We present a mathematical analysis of the homogeneous generalized Poland-Scheraga model. Our approach is based on the fact that such a model is a homogeneous pinning model based on a bivariate renewal process, much like the basic Poland-Scheraga model is a pinning model based on a univariate, i.e. standard, renewal. We present a complete analysis of the free energy singularities, which include the localization-delocalization critical point and (in general) other critical points that have been only partially captured in the physical literature. We obtain also precise estimates on the path properties of the model.