Abstract
We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in [35, 7], via a 2-dimensional renewal approach, with a loop exponent 2+α (α > 0): it was found to undergo a localization/delocalization phase transition of order ν = min(1, α) −1 , together with – in general – other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for d-dimensional disordered systems (here d = 2). First, we prove that when α d/2), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order ν = α −1. On the other hand, when α > 1, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.
Highlights
Introduction of the model and resultsThe analysis of the DNA denaturation phenomenon, i.e. the unbinding at high temperature of two strands of DNA, has lead to the proposal of a very elementary model, the Poland–Scheraga (PS) model [PS70], that turns out to be relevant at a conceptual and qualitative level [Fis84, Gia07], and at a quantitative level [BBB+99, BD98]
The quantitative analysis is based on finite length chains with a given sequence of pairs, but in order to analyse general properties of inhomogeneous chains bio-physicists focused on the cases in which the base sequence is the realization of a sequence of random variables, that is often referred to as disorder in statistical mechanics
In the generalized Poland–Scheraga model (gPS) model, other phase transitions exist, beyond the denaturation transition. Another relevant remark is that PS and gPS models contain a parameter that, in a mathematical or theoretical physics perspective, can be chosen arbitrarily and on which depends the critical behavior
Summary
The analysis of the DNA denaturation phenomenon, i.e. the unbinding at high temperature of two strands of DNA, has lead to the proposal of a very elementary model, the Poland–Scheraga (PS) model [PS70], that turns out to be relevant at a conceptual and qualitative level [Fis, Gia07], and at a quantitative level [BBB+99, BD98]. In the gPS model, other phase transitions exist, beyond the denaturation transition Another relevant remark is that PS and gPS models contain a parameter (the loop exponent) that, in a mathematical or theoretical physics perspective, can be chosen arbitrarily and on which depends the critical behavior. Our aim is to analyze the disordered gPS model and to understand the effect of disorder on the denaturation transition for this generalized, 2–dimensional, model
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