After the pioneering experimental work in [1], mechanical molecular experiments allowed increasingly sophisticated analyses of the energy landscape and mechanical stability at the single molecule scale. Stretching experiment of a single B-DNA molecule in [2], [3] revealed the presence of a force plateau that has been interpreted as a cooperative transition from a double-stranded (ds) configuration to a single-stranded (ss) one [4], [5], [6], [7], [8]. From a modeling point of view, a large effort has been devoted to the development of numerical simulations so to be compared to experiments. These studies have been based on molecular dynamics or coarse grained models (also using MonteCarlo methods) to comprehend the mechanics of DNA, its stretching mechanism, the emergence of overstretching, the effects of structural defects and the role played by different pulling protocols [9], [10], [11], [12], [13], [14], [15].

Historically, a large amount of work in the theoretical analysis of DNA has been based on the Peyrard-Bishop (PB) model [16], [17] that allows to obtain numerical and, under specific material assumptions, analytical results. Starting from this seminal model, a large number of results have been obtained [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. With the explicit aim of determining the influence of chain vs intra-strands bonds stiffnesses, temperature and discreteness effects for DNA and macromolecules with similar configurations (RNA, hairpins), here we use a PB-type model for the analytical description of the different effects. As in the case of the PB model, we neglect three-dimensional effects and possible inhomogeneities of the base-pairs sequences that may play an important role, but would hide the analytical clearness of the model. Indeed, the main innovation of the proposed approach is that, by substituting the classical Morse potential [17], describing breakable links, with a parabolic/constant energy (see Fig. 2(a)) and by using a spin-type approach (see [31]), it is possible to deduce, both in the case of purely mechanical model (zero-temperature, Fig. 1(a)) and thermomechanically driven (Fig. 1(b)) melting processes, a fully analytical description of these phenomena. In particular, we determine the crucial role of a main non-dimensional parameter ν, measuring the relative stiffness of intra- vs inter-chains bonds on the melting transition behavior. As we show, in accordance with known experimental observations and reproducing typical AFM experiments, in the case of assigned displacement we observe that the melting phenomenon is signaled by stress serrations with a nucleation stress higher than the propagation stress plateaux [32]. Similarly, we deduce the effect of the parameter ν and of temperature on the cooperativity of the transition as recently detailed in the overstretching transition of double stranded polypeptide chains [33] as well as in focal adhesion phenomena [34].

As we show, the behavior in the important non zero temperature case, converges to the mechanical limit when temperature decreases. Moreover, the comparison with previous weel known results in the literature in the field, either numerical or theoretical – based on specific assumptions on the stiffness of the chain (e.g. continuum limit in [17] and in [21] or [22] in the extreme discretization limit) – show the consistency of our results, that remarkably do not require such type of specific assumptions. Finally we show the consistency of the proposed model regarding the (zero force) denaturation temperature by comparing the results with those obtained by the transfer integral technique.

In summary, based on the recalled simplification of the model, we are able to fill all the gaps of the several analyses proposed in the last decades of the popular PB model, by fully characterizing its energy landscape in the zero temperature limit and by determining the behavior of the system independently on any assumption of the stiffness ratio ν between the inter and intra-chains links. Two main assumptions let us obtain these results. On one hand, we exploited the widely used single domain wall hypothesis (decomposing the chain into two complementary attached and detached part). We explicitly analyze such hypothesis that can be theoretically justified in the end-bases loading protocols for temperature sufficiently lower than the limit (zero-force) denaturation value. On the other hand, a second important considered simplification is that the analytic evaluation of the partition function is based on an assumption of multivalued configurational energy as explained in detail in the following. This assumption, firstly proposed in [35], has been numerically shown to be effective in [36].

Finally, we obtain a clear phase diagram in the force-temperature plane (Fig. 7(a)) and assigned displacement-temperature plane (Fig. 7(b)) determining a stiffness-dependent analytic expression for the critical temperature. By substituting the known material parameters of the intra and inter-chains bonds we then show the possibility of quantitatively predicting the experimental behavior of the system with a very accurate result for all the values of temperature ranging from the zero limit to the denaturation value. As we show, the model can be extended for large systems also to quantitatively reproduce the experimental thermomechanical response of DNA hairpins. Again, we find a very good agreement between the theoretical predictions and the experimental results.

Model –

Following [17], we consider two chains of n+1 shear springs representing intrachains bonds (see Fig. 2) and elasto-fragile extensional springs reproducing interchains base interactions. We assume symmetric displacements of the two strands, so that we can minimize the energy of a single chain connected to a fixed (symmetry) axis by breakable links representing inter-strands interactions. The simple, but fundamental, innovation with respect to classical analyses of the PB model is the assumption of a simplified form on the base interactions potential energy (suggested in the context of biological peeling [37]) asψe(ui)=kel2. To reproduce classical experiments on DNA molecules, we suppose that one side of the chain is fixed with u0=0, whereas we assume that a displacement d is assigned to the last mass (hard device condition, see Fig. 2). More general boundary conditions taking care of the influence of the loading system can be assigned as proposed in [36] by considering another energy term introducing the elasticity of the loading device or interacting molecule (such as RNA in the case of DNA transcription), but for the sake of simplicity we here neglect this additional term.

After introducing the rescaled displacements wi=ui/ud we can write the (non dimensional) total elastic energy asnϕ=Φkel=ν22∑i=0n(wi+1−wi)2+12∑i=1n[(1−χi)wi2+χi].In Eq. (7) we introduced the main non dimensional parameter of the modelν=ktkeudl,measuring the energy per unit (relative) displacement of the shear (inter-strand/covalent) bonds vs the energy per unit displacement for the breakable (intra-strands/non-covalent) bonds.

As remarked in the introduction, previous analytical results in the literature are based on specific assumptions such as continuum limit in [17] and in [21] or [22] in the extreme discretization limit (corresponding to ν2n→0 and ν2n→∞, respectively). Remarkably, here we do not need any of these assumptions so that both limits result as particular cases of our analysis.