The onset of an "eye-phase" and its role during the DNA unzipping is studied when a force is applied to the interior of the chain. The directionality of the hydrogen bond introduced here shows oscillations in force-extension curve similar to a "saw-tooth" kind of oscillations seen in the protein unfolding experiments. The effects of intermediates (hairpins) and stacking energies on the melting profile have also been discussed.
A study of the micromechanical unzipping of DNA in the framework of the Peyrard-Bishop-Dauxois model is presented. We introduce a Monte Carlo technique that allows accurate determination of the dependence of the unzipping forces on unzipping speed and temperature. Our findings agree quantitatively with experimental results for homogeneous DNA, and for $λ$-phage DNA we reproduce the recently obtained experimental force-temperature phase diagram. Finally, we argue that there may be fundamental differences between {\em in vivo} and {\em in vitro} DNA unzipping.
Based on the result of an unzipping phase transition by a force in a model of DNA, we hypothesize that the DnaA-type proteins act as a pulling agent with a force slightly less than the critical force for unzipping. The dynamic (space-time) correlation of unzipping then drives the subsequent events of replication. In such a correlation driven scenario, there is no need of a replisome as a structural unit and this elusive replisome may not exist at all.
We derive and solve a Fokker-Planck equation for the stationary distribution of the free energy, in a model of unzipping of double-stranded DNA under external force. The autocorrelation function of the random DNA sequence can be a general form, including long range correlations. In the case of Orstein-Uhlenbeck noise, characterized by a finite correlation length, our result reduces to the exact result of Allahverdyan et al, with the average number of unzipped base pairs going as <X>~1/f^2 in the white noise limit, where f is the deviation from the critical force. In the case of long range corr…
Using the criterion that the mechanical unzipping transition in a bound homopolymer is triggered when the average force exerted by the single unbound strands on the first base pair in the bound section exceeds the force binding the pair together, the temperature dependence of the critical unzipping force is obtained. In the resulting phase diagram, the critical force decreases monotonically with increasing temperature, from a finite value at zero Kelvin to zero at the critical temperature.
We study the unzipping of double stranded DNA (dsDNA) by applying a pulling force at a fraction $s$ $(0 \le s \le 1)$ from the anchored end. From exact analytical and numerical results, the complete phase diagram is presented. The phase diagram shows a strong ensemble dependence for various values of $s$. In addition, we show the existence of an ``eye'' phase and a triple point.
We propose a simple nonlinear scaler displacement model to calculate the distribution of effect created by a shear stress on a double stranded DNA (dsDNA) molecule and the value of shear force $F_c$ which is required to separate the two strands of a molecule. It is shown that as long as the force pulls entire strand in the direction of its application the value of $F_c$ depends linearly on the length; the deviation from linear behaviour takes place when part of a strand moves in opposite direction under the influence of force acting on the other strand. The calculated values of $F_c$ as a func…
We investigate the voltage-driven transport of hybridized DNA through membrane channels. As membrane channels are typically too narrow to accommodate hybridized DNA, the dehybridization of the DNA is the critical rate limiting step in the transport process. Using a two-dimensional stochastic model, we show that the dehybridization process proceeds by two distinct mechanisms; thermal denaturation in the limit of low driving voltage, and direct stripping in the high to moderate voltage regime. Additionally, we investigate the effects of introducing non-homologous defects into the DNA strand.
The unzipping of a double stranded DNA whose ends are subjected to a time dependent periodic force with frequency $ω$ and amplitude $G$ is studied using Monte Carlo simulations. We obtain the dynamical order parameter, $Q$, defined as the time average extension between the end monomers of two strands of the DNA over a period, and its probability distributions $P(Q)$ at various force amplitudes and frequencies. We also study the time autocorrelations of extension and the dynamical order parameter for various chain lengths. The equilibrium force-distance isotherms were also obtained at various f…
We study theoretically the unzipping of a double stranded DNA from a condensed globule state by an external force. At constant force, we find that the double stranded DNA unzips an at critical force Fc and the number of unzipped monomers M goes as M~(Fc-F)^{-3}, for both the homogeneous and heterogeneous double stranded DNA sequence. This is different from the case of unzipping from an extended coil state in which the number of unzipped monomers M goes as M~(Fc-F)^{-chi}, where the exponent chi is either 1 or 2 depending on whether the double stranded DNA sequence is homogeneous or heterogeneo…
We study the features on the unzipping process of a modified version of the Peyrard-Bishop- Dauxois model. We show that the inclusion of a barrier in the on-site potential allows for the existence of stable domain wall solutions between open and closed regions of the DNA chain. We analyze the linear stability of such solutions and study their relevance in the dynamical behavior of DNA under ac forces.
Using Monte Carlo simulations, we study the hysteresis in unzipping of a double stranded DNA whose ends are subjected to a time dependent periodic force with frequency ($ω$) and amplitude ($G$). For the static force, i.e., $ω\to 0$, the DNA is in equilibrium with no hysteresis. On increasing $ω$, the area of the hysteresis loop initially increases and becomes maximum at frequency $ω^{*}(G)$, which depends on the force amplitude $G$. If the frequency is further increased, we find that for lower amplitudes the loop area decreases monotonically to zero, but for higher amplitudes it has an oscilla…
Using theory and simulations, we carried out a first systematic characterization of DNA unzipping via nanopore translocation. Starting from partially unzipped states, we found three dynamical regimes depending on the applied force, f: (i) heterogeneous DNA retraction and rezipping (f < 17pN), (ii) normal (17pN < f < 60pN) and (iii) anomalous (f > 60pN) drift-diffusive behavior. We show that the normal drift-diffusion regime can be effectively modelled as a one-dimensional stochastic process in a tilted periodic potential. We use the theory of stochastic processes to recover the potential from …
A simple model of DNA based on two interacting polymers has been used to study the unzipping of a double stranded DNA subjected to a periodic force. We propose a dynamical transition, where without changing the physiological condition, it is possible to bring DNA from the zipped/unzipped state to a new dynamic (hysteretic) state by varying the frequency of the applied force. Our studies reveal that the area of the hystersis loop grows with the same exponents as of the isotropic spin systems. These exponents are amenable to verification in the force spectroscopic experiments.
We report studies of the equilibrium and the dynamics of a general set of lattice models which capture the essence of the force-induced or mechanical DNA unzipping transition. Besides yielding the whole equilibrium phase diagram in the force vs temperature plane, which reveals the presence of an interesting re-entrant unzipping transition for low T, these models enable us to characterize the dynamics of the process starting from a non-equilibrium initial condition. The thermal melting of the DNA strands displays a model dependent time evolution. On the contrary, our results suggest that the dy…
Using Langevin dynamics simulations, we study the hysteresis in unzipping of longer double stranded DNA chains whose ends are subjected to a time dependent periodic force with frequency $ω$ and amplitude $G$ keeping the other end fixed. We find that the area of the hysteresis loop, $A_{loop}$, scales as $1/ω$ at higher frequencies, whereas it scales as $(G-G_c)^αω^β$ with exponents $α=1$ and $β=1.25$ in the low frequency regime. These values are same as the exponents obtained in Monte Carlo simulation studies of a directed self avoiding walk model of a homopolymer DNA [R. Kapri, Phys. Rev. E 9…
Using Monte Carlo simulations, we study the hysteresis in unzipping of a double stranded block copolymer DNA with $-A_n B_n-$ repeat units. Here $A$ and $B$ represent two different types of base pairs having two- and three-bonds, respectively, and $2n$ represents the number of such base pairs in a unit. The end of the DNA are subjected to a time dependent periodic force with frequency ($ω$) and amplitude ($g_0$) keeping the other end fixed. We find that the equilibrium force-temperature phase diagram for the static force is independent of the DNA sequence. For the periodic force case, the resu…
The two strands of the DNA double helix can be `unzipped' by application of 15 pN force. We analyze the dynamics of unzipping and rezipping, for the case where the molecule ends are separated and re-approached at constant velocity. For unzipping of 50 kilobase DNAs at less than about 1000 bases per second, thermal equilibrium-based theory applies. However, for higher unzipping velocities, rotational viscous drag creates a buildup of elastic torque to levels above kBT in the dsDNA region, causing the unzipping force to be well above or well below the equilibrium unzipping force during respectiv…
We studied the unzipping kinetics of single molecules of double-stranded DNA by pulling one of their two strands through a narrow protein pore. PCR analysis yielded the first direct proof of DNA unzipping in such a system. The time to unzip each molecule was inferred from the ionic current signature of DNA traversal. The distribution of times to unzip under various experimental conditions fit a simple kinetic model. Using this model, we estimated the enthalpy barriers to unzipping and the effective charge of a nucleotide in the pore, which was considerably smaller than previously assumed.
DNA unzipping, the separation of its double helix into single strands, is crucial in modulating a host of genetic processes. Although the large-scale separation of double-stranded DNA has been studied with a variety of theoretical and experimental techniques, the minute details of the very first steps of unzipping are still unclear. Here, we use atomistic molecular dynamics (MD) simulations, coarse-grained simulations and a statistical-mechanical model to study the initiation of DNA unzipping by an external force. The calculation of the potential of mean force profiles for the initial separati…
We study theoretically the mechanical failure of a simple model of double stranded DNA under an applied shear. Starting from a more microscopic Hamiltonian that describes a sheared DNA, we arrive at a nonlinear generalization of a ladder model of shear unzipping proposed earlier by deGennes [deGennes P. G. C. R. Acad. Sci., Ser. IV; Phys., Astrophys. 2001, 1505]. Using this model and a combination of analytical and numerical methods, we study the DNA "unzipping" transition when the shearing force exceeds a critical threshold at zero temperature. We also explore the effects of sequence heteroge…
The mechanical separation of the double helical DNA structure induced by forces pulling apart the two DNA strands (``unzipping'') has been the subject of recent experiments. Analytical results are obtained within various models of interacting pairs of directed walks in the (1,1,...,1) direction on the hypercubic lattice, and the phase diagram in the force-temperature plane is studied for a variety of cases. The scaling behaviour is determined at both the unzipping and the melting transition. We confirm the existence of a cold denaturation transition recently observed in numerical simulations: …
The possibility of detecting mutations in a DNA from force measurements (as a first step towards sequence analysis) is discussed theoretically based on exact calculations. The force signal is associated with the domain wall separating the zipped from the unzipped regions. We propose a comparison method (``differential force microscope'') to detect mutations. Two lattice models are treated as specific examples.
We considered a dsDNA polymer in which distribution of bases are random at the base pair level but ordered at a length of 18 base pairs and calculated its force elongation behaviour in the constant extension ensemble. The unzipping force $F(y)$ vs. extension $y$ is found to have a series of maxima and minima. By changing base pairs at selected places in the molecule we calculated the change in $F(y)$ curve and found that the change in the value of force is of the order of few pN and the range of the effect depending on the temperature, can spread over several base pairs. We have also discussed…
We discuss the thermodynamic behaviour near the force induced unzipping transition of a double stranded DNA in two different ensembles. The Y-fork is identified as the coexisting phases in the fixed distance ensemble. From finite size scaling of thermodynamic quantities like the extensibility, the length of the unzipped segment of a Y-fork, the phase diagram can be recovered. We suggest that such procedures could be used to obtain the thermodynamic phase diagram from experiments on finite length DNA.
We study a class of micromanipulation experiments, exemplified by the pulling apart of the two strands of double-stranded DNA (dsDNA). When the pulling force is increased to a critical value, an ``unzipping'' transition occurs. For random DNA sequences with short-ranged correlations, we obtain exact results for the number of monomers liberated and the specific heat, including the critical behavior at the transition. Related systems include a random heteropolymer pulled away from an adsorbing surface and a vortex line in a type II superconductor tilted away from a fragmented columnar defect.
A double stranded DNA molecule when pulled with a force acting on one end of the molecule can become either partially or completely unzipped depending on the magnitude of the force F. For a random DNA sequence, the number M of unzipped base pairs goes as M~(F-Fc)^(-2) and diverges at the critical force Fc with an exponent χ=2. We find that when excluded volume effect is taken into account for the unzipped part of the DNA, the exponent χ=2 is not changed but the critical force Fc is changed. The force versus temperature phase diagram depends on only two parameters in the model, the persistence …
The force induced unzipping transition of a double stranded DNA is considered from a purely thermodynamic point of view. This analysis provides us with a set of relations that can be used to test microscopic theories and experiments. The thermodynamic approach is based on the hypothesis of impenetrability of the force in the zipped state. The melting and the unzipping transitions are considered in the same framework and compared with the existing statistical model results. The analysis is then extended to a possible continuous unzipping transition.
Experimental studies of the thermal denaturation of DNA yield a strong indication that the transition is first order. This transition has been theoretically studied since the early sixties, mostly within an approach in which the microscopic configurations of a DNA molecule are given by an alternating sequence of non-interacting bound segments and denaturated loops. Studies of these models neglect the repulsive, self-avoiding, interaction between different loops and segments and have invariably yielded continuous denaturation transitions. In this study we exploit recent results on scaling prope…
We consider force-induced unzipping transition for a heterogeneous DNA model with a correlated base-sequence. Both finite-range and long-range correlated situations are considered. It is shown that finite-range correlations increase stability of DNA with respect to the external unzipping force. Due to long-range correlations the number of unzipped base-pairs displays two widely different scenarios depending on the details of the base-sequence: either there is no unzipping phase-transition at all, or the transition is realized via a sequence of jumps with magnitude comparable to the size of the…
When pulled along the axis, double-strand DNA undergoes a large conformational change and elongates roughly twice its initial contour length at a pulling force about 70 pN. The transition to this highly overstretched form of DNA is very cooperative. Applying force perpendicular to the DNA axis (unzipping), double-strand DNA can also be separated into two single-stranded DNA which is a fundamental process in DNA replication. We study the DNA overstretching and unzipping transition using fully atomistic molecular dynamics (MD) simulations and argue that the conformational changes of double stran…
We review statistical-mechanical theories of single-molecule micromanipulation experiments on nucleic acids. First, models for describing polymer elasticity are introduced. We then review how these models are used to interpret single-molecule force-extension experiments on single-stranded and double-stranded DNA. Depending on the force and the molecules used, both smooth elastic behaviors and abrupt structural transitions are observed. Third, we show how combining the elasticity of two single nucleic acid strands with a description of the base-pairing interactions between them explains much of…
The unzipping transition under the influence of external force of a dsDNA molecule has been studied using the Peyrard-Bishop Hamiltonian. The critical force $F_c(T)$ is found to depend on the potential parameters $k$, represents the stiffness of single strand of DNA and the potential depth $D$. We used constant extension ensemble to calculate the average force needed to stretch a base pair $y$ distance apart. A very large peak around $y = 1 {\rm Å}$ is found. The value of $F(y)$ needed to stretch a base pair located far away from the ends of a dsDNA molecule is found twice the value of the for…
We present an extensive theoretical investigation of the mechanical unzipping of double-stranded DNA under the influence of an applied force. In the limit of long polymers, there is a thermodynamic unzipping transition at a critical force value of order 10 pN, with different critical behavior for homopolymers and for random heteropolymers. We extend results on the disorder-averaged behavior of DNA's with random sequences to the more experimentally accessible problem of unzipping a single DNA molecule. As the applied force approaches the critical value, the double-stranded DNA unravels in a ser…
Under constant applied force, the separation of double-stranded DNA into two single strands is known to proceed through a series of pauses and jumps. Given experimental traces of constant-force unzipping, we present a method whereby the locations of pause points can be extracted in the form of a pause point spectrum. A simple theoretical model of DNA constant-force unzipping is demonstrated to produce good agreement with the experimental pause point spectrum of lambda phage DNA. The locations of peaks in the experimental and theoretical pause point spectra are found to be nearly coincident bel…
We present a theoretical study of single-stranded DNA under stretching. Within the proposed framework, the effects of basepairing on the mechanical response of the molecule can be studied in combination with an arbitrary underlying model of chain elasticity. In a generic case, we show that the stretching curve of ssDNA exhibits two distinct features: the second-order "unfolding" phase transition, and a sharp crossover, reminiscent of the first-order "unzipping" transition in dsDNA. We apply the theory to the particular cases of Worm-like Chain (WLC) and Freely-Joint Chain (FJC) models, and dis…
We study by using Monte Carlo simulations the hysteresis in unzipping and rezipping of a double stranded DNA (dsDNA) by pulling its strands in opposite directions in the fixed force ensemble. The force is increased, at a constant rate from an initial value $g_0$ to some maximum value $g_m$ that lies above the phase boundary and then decreased back again to $g_{0}$. We observed hysteresis during a complete cycle of unzipping and rezipping. We obtained probability distributions of work performed over a cycle of unzipping and rezipping for various pulling rates. The mean of the distribution is fo…
Three, interrelated biologically-relevant examples of biased random walks are presented: (1) A model for DNA melting, modelled as DNA unzipping, which provides a way to illustrate the role of the Boltzmann factor in a venue well-known to biology and pre-medical students; (2) the activity of helicase motor proteins in unzipping double-stranded DNA, for example, at the replication fork, which is an example of a Brownian ratchet; (3) force generation by actin polymerization, which is another Brownian ratchet, and for which the force and actin-concentration dependence of the velocity of actin poly…
We separate double stranded lambda phage DNA by applying a fixed force at a constant temperature ranging from 15C to 50C, and measure the minimum force required to separate the two strands, providing the first experimental determination of the phase boundary between single stranded DNA and double stranded DNA in the temperature- force plane. The measurements also offer information on the free energy of dsDNA at temperatures where dsDNA does not thermally denature in the absence of force. While parts of the phase diagram can be explained using existing models and free energy parameters, others …
We study the effect of random forces on a double stranded DNA in unzipping the two strands, analogous to the problem of an adsorbed polymer under a random force. The ground state develops bubbles of various lengths as the random force fluctuation is increased. The unzipping phase diagram is shown to be drastically different from the pure case.
We study the unzipping of a double stranded DNA (dsDNA) by applying an external force on a single strand while leaving the other strand free. We find that the dsDNA can be unzipped to two single strands if the external force exceeds a critical value. We obtain the phase diagram which is found to be different from the phase diagram of unzipping by pulling both the strands in opposite directions. In the presence of an attractive surface near DNA, the phase diagram gets modified drastically and shows richer surprises including a critical end point and a triple point.
We develop a simple model to study the effects of an applied force on the melting of a double stranded DNA (dsDNA). Using this model, we could study the stretching, unzipping, rupture and slippage like transition in a dsDNA. We show that in absence of an applied force, the melting temperature and the melting profile of dsDNA strongly depend on the constrained imposed on the ends of dsDNA. The nature of the phase boundary which separates the zipped and the open state for the shearing like transition is remarkably different than the DNA unzipping
We present a critique and extension of the mean-field approach to the mechanical pulling transition in bound polymer systems. Our model is motivated by the theoretically and experimentally important examples of adsorbed polymers and double-stranded DNA, and we focus on the case in which quenched disorder in the sequence of monomers is unimportant for the statistical mechanics. We show how including excluded volume interactions in the model affects the phase diagram for the critical pulling force, and we predict a re-entrancy phase at low temperatures which has not been previously discussed. We…
Motivated by the experiments on DNA under torsion, we consider the problem of pulling a polymer out of a potential well by a force applied to one of its ends. If the force is less than a critical value, then the process is activated and has an activation energy proportinal to the length of the chain. Above this critical value, the process is barrierless and will occur spontaneously. We use the Rouse model for the description of the dynamics of the peeling out and study the average behaviour of the chain, by replacing the random noise by its mean. The resultant mean-field equation is a nonlinea…
A double stranded DNA molecule under the stress of a pulling force acting on the strand terminals exhibits a partially denatured structure or can be completely unzipped depending the magnitude of the pulling force. A scaling argument for relationships amongst basic length scales is presented that takes into account the heterogeneity of the sequence. The result agrees with our numerical simulation data, which provides a critical test of the power laws in the unzipping transition region.
Replication and transcription are two important processes in living systems. To execute such processes, various proteins work far away from equilibrium in a staggered way. Motivated by this, aspects of hysteresis during unzipping of DNA under a periodic drive in non-equilibrium conditions are studied. A steady state phase diagram of a driven DNA is proposed which is experimentally verifiable. As a two state system, we also compare the results of DNA with that of an Ising magnet under an asymmetrical variation of magnetic field.
Using Monte Carlo simulations, we study the dynamic transitions in the unzipping of an adsorbed homogeneous polymer on a surface (or wall). We consider three different types of surfaces. One end of the polymer is always kept anchored, and other end monomer is subjected to a periodic force with frequency $ω$ and amplitude $g_0$. We observe that the force-distance isotherms show hysteresis loops in all the three cases. For all the three cases, it is found that the area of the hysteresis loop, $A_{loop}$, scales as $1/ω$ in the higher frequency regime, and as $g_0^α ω^β$ with exponents $α= 1$ and…
We show the existence of a critical endpoint in the phase diagram of unzipping of an adsorbed double-stranded (ds) polymer like DNA. The competition of base pairing, adsorption and stretching by an external force leads to the critical end point. From exact results, the location of the critical end point is determined and its classical nature established.
We show how single-molecule unzipping experiments can provide strong evidence that the zero-force melting transition of long molecules of natural dsDNA should be classified as a phase transition of the higher-order type (continuous). We study a model for a long molecule of dsDNA, and compute the equilibrium phase diagram for the experiment in which the molecule is unzipped under force. We consider a perfect-matching dsDNA model, in which the loops are volume-excluding chains with arbitrary loop exponent c. We include stacking interactions, hydrogen bonds, and main-chain entropy, including sequ…
We propose a phase coexistence based mechanism for activity of helicases, ubiquitous enzymes that unwind double stranded DNA. The helicase-DNA complex constitutes a fixed-stretch ensemble that entails a coexistence of domains of zipped and unzipped phases of DNA, separated by a domain wall. The motor action of the helicase leads to a change in the position of the fixed constraint thereby shifting the domain wall on dsDNA. We associate this off-equilibrium domain wall motion with the unzipping activity of helicase. We show that this proposal gives a clear and consistent explanation of the main …
The flexibility and the extension along the direction of the force are shown to be related to the bubble number fluctuation and the average number of bubbles respectively, when the strands of the DNA are subjected to a force along the same direction, here we call a stretching force. The force-temperature phase diagram shows the existence of a tricritical point (TCP), where the first-order force induced zipping transition becomes continuous. On the other hand, when the forces are being applied in opposite directions, here we call an unzipping force, the transition remains first-order,with the p…
The cations, in form of salt, present in the solution containing DNA play a crucial role in the opening of two strands of DNA. We use a simple non linear model and investigate the role of these cations on the mechanical unzipping of DNA. The Hamiltonian is modified to incoporate the solvent effect and the cations present in the solution. We calculate the melting temperature as well as the critical force that is required to unzip the DNA molecule as a function of salt concentration of the solution. The phase diagrams are found to be in close agreement with the experimental phase diagrams.
Experiments indicate that unbinding rates of proteins from DNA can depend on the concentration of proteins in nearby solution. Here we present a theory of multi-step replacement of DNA-bound proteins by solution-phase proteins. For four different kinetic scenarios we calculate the depen- dence of protein unbinding and replacement rates on solution protein concentration. We find (1) strong effects of progressive 'rezipping' of the solution-phase protein onto DNA sites liberated by 'unzipping' of the originally bound protein; (2) that a model in which solution-phase proteins bind non-specificall…
The rupture of double-stranded DNA under stress is a key process in biophysics and nanotechnology. In this article we consider the shear-induced rupture of short DNA duplexes, a system that has been given new importance by recently designed force sensors and nanotechnological devices. We argue that rupture must be understood as an activated process, where the duplex state is metastable and the strands will separate in a finite time that depends on the duplex length and the force applied. Thus, the critical shearing force required to rupture a duplex within a given experiment depends strongly o…
Single molecule studies, at constant force, of the separation of double-stranded DNA into two separated single strands may provide information relevant to the dynamics of DNA replication. At constant applied force, theory predicts that the unzipped length as a function of time is characterized by jumps during which the strands separate rapidly, followed by long pauses where the number of separated base pairs remains constant. Here, we report previously uncharacterized observations of this striking behavior carried out on a number of identical single molecules simultaneously. When several singl…
We study the effects of the sequence on the propagation of nonlinear excitations in simple models of DNA in which we incorporate actual DNA sequences obtained from human genome data. We show that kink propagation requires forces over a certain threshold, a phenomenon already found for aperiodic sequences [F. Dom\'ınguez-Adame {\em et al.}, Phys. Rev. E {\bf 52}, 2183 (1995)]. For forces below threshold, the final stop positions are highly dependent on the specific sequence. The results of our model are consistent with the stick-slip dynamics of the unzipping process observed in experiments. We…
By extending the classical Peyrard-Bishop model, we are able to obtain a fully analytical description for the mechanical resistance of DNA under stretching at variable values of temperature, number of base pairs and intrachains and interchains bonds stiffness. In order to compare elasticity and temperature effects, we first analyze the system in the zero temperature mechanical limit, important to describe several experimental effects including possible hysteresis. We then analyze temperature effects in the framework of equilibrium statistical mechanics. In particular, we obtain an analytical e…
