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The gPS model: biophysics version
This subsection, as well as § 2.1.4, can be skipped if one is not focusing on the biophyisics set-up. The model we consider has been introduced in [47]. The two DNA strands, of lengths M and N ≥ 1 – the length of course corresponds to the number of bases – interact by forming some base pairs. We talk of N-strand, M-strand and of base i of the N− or M−strand with the obvious meaning. An allowed configuration of our system is a collection of base pairs ((i1, j1), (i2, j2), . . . , (in, jn)) ∈ N2n , with N = {1, 2, . . .} , (2.1) where n ∈ N ∶= {1, 2, . . . ,N} and 1. (i1, j1) = (1, 1); 2. ik < ik′ , as well as jk < jk′ , for 1 ≤ k < k′ ≤ n.
The first condition is simply saying that the first two bases form a pair and the second condition is imposed by the geometric constraint (see Figure 2.1). The weight of every configuration is assigned by the following rules:
1. Each base pair is energetically favored and carries an energy −Eb < 0.
2. A base which is not in a pair is either in a loop or in the free ends:
• It is in a loop if it is in Lk ∶= ((ik, ik+1) ∪ (jk, jk+1))∩N for some k ∈ {1, . . . ,n−1}: the loop Lk has length ℓk ∶= (ik+1 − ik) + (jk+1 − jk) − 2 and we associate to Lk an entropy factor B(ℓk) with B(ℓ) ∶= sℓℓ−c .
The gPS model: renewal process viewpoint
From a mathematical perspective we take a more general viewpoint and we introduce a two-dimensional renewal pinning model. A discrete two-dimensional renewal issued from the origin is a random walk τ = {τn}n=0,1,… = (τ (1), τ (2)) = {(τ (1) n , τ (2) n )}n=0,1,… where τ0 = (0, 0) and, for n ∈ N ∶= {1, 2, . . .}, τn is a sum of n independent identically distributed random variables taking values in N2. So if we set K(n,m) ∶= P(τ1 = (n,m)) then given {(in, jn)}n=0,1,2,…, with (i0, j0) = (0, 0), for every k ∈ N P(τn = (in, jn) for n = 1, 2, . . . , k) = kM n=1 K(in − in−1, jn − jn−1) , (2.5) and, by construction, such a probability is zero unless the i’s and j’s form strictly increasing sequences. We can then introduce for given N and M ∈ N a pinning model of length (N,M) by forcing, i.e. conditioning, τ to visit (N,M) and by penalizing (h ≤ 0) or rewarding (h ≥ 0) the number of renewalsup to (N,M). More formally, we introduce the probability measure PN,M,h by setting for every k ∈ N such that k ≤ min(N,M) =∶ N ∧M and for every gPS model in the renewal process representation, now renewal points of a two-dimensional discrete renewal process: these points correspond to the (six) base pairs of Figure 2.1, except that the first base pair is now (0, 0) and also all the other ones are translated down of (1, 1) with respect to Figure 2.1.
The renewal trajectory is drawn up to the renewal point (9, 12) and the trajectory up to this point corresponds to one of the possible trajectories of Zc 9,12,h. The free ends, of lengths 2 and 6, are then represented as straight lines that go till the boundary of the rectangle with opposite vertices (0, 0) and (11, 18). Zf 11,18,h is obtained by summing up with respect to the position of the last renewal point – (9, 12) in this example – the contribution of the constrained partition function times the contribution due to the two free ends.
Matching biophysics and renewal process viewpoints
The first remark to make on the biophysics model is that the dependence on s of ZM N (cf. (2.3)) is trivial: ZM N for a given value of s coincides with sN+M ̃ZM N , where ̃ZM N is ZM N with s = 1 and both Eb and El are decreased by 2 β log s. We can therefore set s = 1 without true loss of generality. We then remark that we can match exp(−βEb)ZM+1 N+1 and Zf N,M,h by an appropriate choice of h, K(⋅) and Kf (⋅): the fact that we consider the biophysics model with lengths augmented by 1 and renormalized by the factor exp(−βEb) is just because in parallel with one-dimensional renewal pinning works we have chosen to start the renewal from time (or renewal epoch) zero, but without giving an energetic reward to the base pair 0. Matching is then made by observing that:
1. The match for the free end terms Kf (⋅) and A(⋅) is easily made.
2. A minimal inter-arrival step, that is (1, 1), in the renewal model corresponds to a base pair and contributes K(2) exp(h). It would then be matched to exp(βEb) in the biophysics model.
3. All other inter-arrival steps (i, i′), i.e. i + i′ = 3, 4, . . ., which give a contribution K(i+i′) exp(h) correspond to loops with i+i′−2 unpaired bases and the contribution in the biophysics model is exp(β(Eb − El))B(i + i′ − 2).
Then of course c = 2 + α and ¯c = ¯α and, at last, the matching between the two models is done thanks to the ample freedom that we have in choosing K(⋅), except that we have required that Σn,mK(n+m) = 1. This probability constraint in reality just corresponds to a shift in the parameter h. In a nutshell, renewal models include the biophysics ones: from a qualitative view point the matching is immediate, form the quantitative one it requires some bookkeeping care. Explicit matchings are presented in Section 2.3.4.
Table of contents :
1 Introduction
1.1 Motivations physiques et modélisations
1.1.1 Polymères
1.1.2 Le modèle de Poland-Scheraga
1.1.3 Le modèle de Poland-Scheraga généralisé
1.1.4 Modèles liés
1.2 Modélisation mathématique
1.2.1 Le modèle d’accrochage homogène
1.2.2 Le modèle d’accrochage désordonné
1.3 Perspectives et questions ouvertes
2 The generalized Poland-Scheraga model: homogeneous case
2.1 Introduction and main results
2.1.1 General overview
2.1.2 The gPS model: biophysics version
2.1.3 The gPS model: renewal process viewpoint
2.1.4 Matching biophysics and renewal process viewpoints
2.1.5 The free energy and the localization transition
2.1.6 Transitions in the localized regime
2.1.7 Outline of the approach, sharp estimates and limit path properties .
2.1.8 Open issues and perspectives
2.1.9 Organization of the paper
2.2 Large Deviations and Local Limit Theorems for bivariate renewals
2.3 The constrained model
2.3.1 The free energy of the constrained model
2.3.2 Analysis of the free energy singularities in the localized regime
2.3.3 Free energy analysis of the delocalization transition
2.3.4 The bio-physics model and other examples
2.4 The free model
2.4.1 A preliminary result and free energy equivalence
2.4.2 Sharp estimates on Zf N,M,h
2.4.3 Path properties: proof of Theorem 2.7
2.A Slowly and regularly varying functions
3 The generalized Poland-Scheraga model: disordered case
3.1 Introduction of the model and results
3.1.1 The generalized Poland-Scheraga model
3.1.2 Relevance and irrelevance of disorder
3.1.3 On the results, perspectives and related work
3.2 Free Energy: existence and properties
3.3 Upper bound on the critical point shift
3.4 Lower bound on the critical point shift
3.4.1 Finite-volume fractional moment estimate
3.4.2 Proof of (3.15) in Theorem 3.5: the case α > 2
3.4.3 Proof of (3.15) in Theorem 3.5: the case α ∈ (1, 2]
3.A Bivariate renewal theory, important estimates
3.B Irrelevance via a replica-coupling trick
Bibliography
