Isotensional and isometric force-extension response of chains with bistable units and Ising interactions

The combination of bistability and cooperativity plays a crucial role in several biological and artificial microand nanosystems. In particular, the exhaustive understanding of the mechanical response of such systems under the effect of thermal fluctuations is essential to elucidate a rich variety of phenomena. Here a linear chain composed of elastic units, which are bistable (folded or unfolded) and coupled through an Ising-like interaction, is selected as a case study. We assess the macroscopic thermoelastic response of this chain in terms of its microscopic description. For small systems, far from the thermodynamic limit, this response depends on the applied isometric or isotensional boundary conditions, which correspond to the Helmholtz or Gibbs ensembles of the statistical mechanics, respectively. The theoretical analysis is conducted through the spin variables approach, based on a set of discrete quantities able to identify the folded or unfolded state of the chain units. Eventually, this technique yields closed-form expressions for the force-extension curves and the average number of unfolded units, as function of the applied fields. In addition, it allows to unveil a critical behavior of such systems, characterizing the operating regions with negative differential stiffness (spinoidal phase).

devices like atomic-force microscopes, laser optical tweez- 48 ers, magnetic tweezers, and microelectromechanical systems 49 * stefano.giordano@iemn.univ-lille1.fr [24][25][26][27][28][29] have been employed to investigate proteins [30][31][32], 50 RNA [33,34], and DNA [35][36][37][38][39][40]. 51 While the force-extension response of molecular chains 52 without bistability is considered to be well understood 53 [41][42][43][44][45], the real complexity of chains with bistable units 54 has only been revealed through the introduction of the above 55 force-spectroscopy techniques. In this context, the mechani- 56 cally induced folding and unfolding of the units of a chain, 57 governed by the conformational transition between two states, 58 has been detected in polypeptides, nucleic acids, and other 59 molecules. Notably, for relatively short bistable molecular 60 chains (small systems thermodynamics), the applied boundary 61 conditions play an important role in defining their overall 62 response [46][47][48][49]. 63 On the one side, isotensional experiments (conducted at 64 constant applied force by soft devices) correspond to the 65 Gibbs statistical ensemble and lead to a plateau-like force-66 extension curve with a threshold force characterizing the 67 synchronized unfolding of all chain units [37,[50][51][52][53][54][55]. On the 68 other side, isometric experiments (conducted at prescribed 69 displacement by hard devices) represent a realization of the 70 Helmholtz statistical ensemble, and the corresponding force-71 extension curve shows a sawtooth-like shape, proving that 72 the units unfold sequentially in reaction to the increasing 73 extension [30,32,[54][55][56][57][58][59][60][61]. In any case, the differences between 74 isotensional and isometric force-extension curves disappear 75 whenever the number of units is very large since, in the 76 thermodynamic limit, the Gibbs and Helmholtz ensembles 77 become statistically equivalent [62,63]. A different point of 78 view about two-state systems driven by hard or soft devices 79 has been introduced to model plasticity, hysteretic behaviors 80 and martensitic transformations in solids [64][65][66][67][68][69][70][71]. [77,78]. This method has been recently applied to different 119 two-state systems and molecular chains as well [79][80][81][82]. Both 120 Gibbs and Helmholtz ensembles can be considered by the 121 spin variables approach, allowing to draw direct comparisons 122 between isotensional and isometric conditions at thermody-123 namic equilibrium. While the Gibbs ensemble will be stud-124 ied by means of the classical transfer matrix method [83], 125 typically adopted for one-dimensional interacting models, the 126 Helmholtz ensemble presents major difficulties and will be 127 approached by exploiting the Laplace transform relationship 128 between the Gibbs and Helmholtz partition functions [84]. 129 It is important to remark that we are studying small systems 130 (with the inequivalence of the ensembles), and we need there-131 fore to determine the exact value of the partition functions and 132 not their approximations holding for a large number of units, 133 as usually done for systems attaining the thermodynamic 134 limit. We provide evidence that the cooperativity, measured by 135 the Ising interaction coefficient, strongly modifies the force-136 extension response of the chain and its configurational prop-137 erties. In particular, under isometric conditions, we thoroughly 138 analyze the hierarchy of force peaks as function of the in-139 teraction coefficient. To complement the equilibrium picture, 140 we further characterize the criticality of the spinoidal phase, 141 describing the regions with negative differential stiffness. 142 The structure of the paper is the following. In Sec. II we 143 define the system under investigation and its Hamiltonian 144 function. In Secs. III and IV we analyze the behavior of 145 the chain with Ising interactions under isotensional (Gibbs) 146 and isometric (Helmholtz) conditions, respectively. Since the 147 problem of the Helmholtz ensemble is solved here through a 148 semianalytic procedure, we propose in Secs. V, VI, and VII 149 additional explicit asymptotic results describing the behav-150 ior of the system under weak and strong Ising interactions 151 (ferromagnetic-like and antiferromagnetic-like). Finally, in 152 Sec. VIII we generalize our results in order to take account 153 of a finite extensibility of the chain units, and we illustrate its 154 effect on the critical behavior of the system. While the first end terminal α is able to tether the first unit to a given substrate, the second one β is able either to apply a force (Gibbs condition) or a position (Helmholtz condition) to the last unit. (b) Potential energy of a single unit of the chain (dashed black curve). The potential wells are approximated through two parabolic (i.e., quadratic) profiles (solid blues curves), identified by S i = −1 (folded state) and S i = +1 (unfolded state). 156 We take into consideration a chain of m two-state elements 157 [see Fig. 1(a)], each described by a bistable potential energy 158 with a stable folded state and a metastable unfolded state 159 [see Fig. 1(b)]. The two potential wells in Fig. 1 Fig. 1(b)] [81]. In this case, the discrete variables belong to the 169 phase space of the system and allow us to specify the explored 170 well for each unit. The introduction of the discrete or spin vari-171 ables also allows the direct implementation of an interaction 172 between adjacent elements of the chain, e.g., described by a 173 classical Ising Hamiltonian. The overall Hamiltonian of this 174 system can be therefore written as

II. THE SYSTEM
introduced to deal with the isotensional conditions. Here f is 214 the force applied to the last unit, identified by its position r m . 215 We suppose that quantities r i ∈ R 3 and S i ∈ {−1, +1} ∀i ∈ 216 {1 . . . m} belong to the phase space of the system. Moreover, 217 to fix the ideas, we always consider r 0 = 0. The statistical 218 mechanics of the system can be introduced by calculating the 219 partition function, as where P = R 3m . The integral I = P · · · d r 1 · · · d r m , shown 221 in the last two lines of Eq. (3), can be developed by means 222 of the change of variables To further simplify this integral, by exploiting the isotropy 225 of the system, we suppose that f = (0, 0, f ), and we 226 introduce the spherical coordinates for the vectors ξ i , 227 namely, ξ i = (ξ i cos ϕ i sin θ i , ξ i sin ϕ i sin θ i , ξ i cos θ i ). There-228 fore, we easily obtain ξ i = ξ i , f · ξ i = f ξ i cos θ i and 229 d ξ i = ξ 2 i sin θ i dξ i dϕ i dθ i , and we get for I the expression where jointed chain [84], with elements of fixed lengths, i.e., without 237 elasticity. It is equivalent to say that k(+1) = k(−1) → +∞.

238
This hypothesis will be removed in a successive section of 239 the paper, where we will study an extensible chain with Ising 240 interactions. If we use the property α π e −αx 2 → δ(x) for α → 241 ∞, then we simplify the result for I as where we omitted a noninfluential multiplicative constant and 243 = 0 (−1) corresponds to the length of the folded units. We 244 finally obtain Eventually, the partition function assumes the simpler form We have now to approach the problem of calculating the sums which is a property valid for real numbers c i > 0 ∀i. Accord-250 ingly, we have where we defined To further elaborate the partition function, we also consider 253 254 where χ is the ratio between unfolded and folded lengths, and 255 E is the energy jump between the wells (see Fig. 1). We can 256 adopt the technique of the transfer matrix [83], and then we 257 can directly write where we have with the parameters and the coefficients representing the Boltzmann factor calculated with E and the 262 normalized force. Since we are studying the thermodynamics 263 of small systems (small values of m), we need to calculate 264 the exact value of the partition function given in Eq. (12) 265 and not its approximation evaluated for a large value of m, 266 corresponding to the thermodynamic limit. To develop the 267 calculation of Z G , we can simply calculate the eigenvalues 268 of T: (19) where we also introduced x = e − λ K B T . We underline that if λ 1 270 corresponds to the sign "+" and λ 2 to the sign "−," then we 271 get λ 1 > λ 2 > 0. Now, we need to explicitly determine the 272 matrix power T m−1 . Hence, we use the matrix function theory 273 [88], and we get after straightforward calculations where I is the 2 × 2 identity matrix. The partition function 275 then assumes the form where α and β can be obtained through long but straightfor-277 ward calculations as Finally, the explicit exact form of the Gibbs partition function written as function of x, λ 1 , and λ 2 . This is the most important 281 result of this section and allows us to determine the mechan-282 ical and configurational macroscopic behavior of the whole 283 chain under isotensional conditions. As usual, we obtain the 284 force-extension response as where r represents the average value of the extension, 286 measured in the direction of the applied force. We also note 287 that the quantity S i +1 2 gives 0 for folded elements and 1 288 for unfolded elements. Therefore, we have that is the average number of unfolded elements. On the other 290 hand, the term m i=1 v(S i ) of the Hamiltonian in Eq. (2) can 291 be also written as can be 293 evaluated through the expression where S = (S 1 , . . . , S m ) and r = ( r 1 , . . . , r m ). Then Eq. (26) 295 can be simplified to give which is the final expression for the average value of unfolded 297 domains. It is useful to introduce here the Gibbs free energy 298 of the system G = −K B T log Z G . The above expected values 299 can be reformulated in terms of this thermodynamic function 300 as The knowledge of Z G or G allows therefore the determination 302 of both the average extension of the chain and the average 303 number of unfolded units as function of the applied force and 304 temperature.

305
An application of Eqs. (28) and (29) where r m = r is fixed. The phase space is therefore composed 335 of r i ∀i = 1, . . . , m − 1 and S i ∀i = 1, . . . , m. Hence, the 336 partition function can be written as where Q = R 3(m−1) . It is not difficult to realize that the have now the combination of two forms of interaction among 346 the units, the first being implicitly encoded in the isometric 347 condition and the second explicitly implemented through the 348 Ising scheme. An useful technique to cope with this difficulty 349 is the following. By comparing Eqs. (3) and (31), we deduce 350 that the two partition functions Z G and Z H are related through 351 a three-dimensional bilateral Laplace transform, as where, as usual, we neglect the noninfluential multiplicative 353 constants in the partition functions. Moreover, by considering 354 the spherical symmetry of the problem, we easily obtain the 355 inverse relationship where Z G (iη) is the analytic continuation of the partition 357 function Z G (f ) for the Gibbs ensemble, given in Eq. (24). 358 The integral in Eq. (33) can be simplified by the change of 359 variable y = η K B T , leading to where, as before, we neglected the noninfluential multiplica-361 tive constant. Coherently with our assumptions, the variables 362 p and q assume the form Accordingly, the eigenvalues of the transfer matrix become wherep = py,q = qy, andλ 1,2 = λ 1,2 y. Hence, the analytic 365 continuation of the Gibbs partition function becomes where, importantly,λ 1 andλ 2 depend on y only through sin y 367 and sin χy. In particular, when χ is an integer (or also a 368 rational number), Z G is composed of a periodic function of 369 y divided by y m . So we have where P (y) = P (y + L y ) for a given L y , and we have If we consider integer values of χ , P (y) is periodic with a 372 period of L y = 2π , and it can be developed in Fourier series, where The Here we used the Euler formula e iry = cos ry + i sin ry , 382 and we observed that the integral with cos ry is zero since where the last integral is well defined since the path ex-391 cludes the singularity at the origin from the integration. We know that an application of the residue theorem delivers [81] Therefore, where 1(x) represents the Heaviside step function, defined as 395 where we used the property stating that C −h = (C h ) * , which 397 is valid for the Fourier coefficients of a real periodic function. 398 The result obtained in Eq. (48) is exact for χ ∈ N, but it 399 is based on the numerical computation of the coefficients 400 C k (semianalytic procedure). The limitation introduced by 401 considering integer values for χ does not restrict the physical 402 interpretation of the results. Moreover, this procedure can be 403 easily generalized in order to consider arbitrary rational values 404 for χ [of course, the function P (y) remains periodic with χ ∈ 405 Z]. Furthermore, in next sections, we also discuss additional 406 asymptotic results, which are not based on restrictions over 407 the values of the parameter χ .

408
It is important to remark that our semianalytic procedure, 409 leading to Eq. (48) and based on the numerical implemen-410 tation of Eq. (42), is very efficient for the determination of 411 the Helmholtz partition function. Indeed, the direct numerical 412 calculation of the original integral in Eq. (34), grounded on the 413 knowledge of the Gibbs partition function given in Eq. (38), 414 is a really hard, if not impossible, task since the integrand 415 function is decreasing (as 1/y m−1 ) and oscillating for any r 416 in the whole interval between 0 and mχ . Since we need the 417 quantity log Z H (r ) to analyze the system behavior, all the 418 oscillations of the integrand function (also for large values 419 of |y|) play an important role in defining the result. For this 420 reason, our procedure leads to very accurate results, being 421 based on the analytic determination of the integral over 422 and on the numerical evaluation of the integrals over (0, 2π ) 423 defined in Eq. (42), which are much more stable than the one 424 defined in Eq. (34).
and the average value of unfolded domains with the relation in Filamin A) [76].

460
A form of criticality can be noticed for the Helmholtz 461 response of the bistable Ising chain. To do this, in the 462 force-extension curves shown in Fig. 4(a), we can identify 463 the spinoidal regions, characterized by a negative slope or, 464 equivalently, by a negative differential stiffness. It means 465 that, for each force peak observed in Fig. 4(a), we have a 466 spinoidal interval with ∂f/∂r < 0. It is interesting to study 467 the evolution of these spinoidal regions in terms of the tem-468 perature. In general, we can say that the system is or not 469 in a spinoidal phase depending on values of r and T . We 470 can therefore determine a sort of phase diagram, as shown 471 in Fig. 5, where the end points of each spinoidal interval 472 (on the extension axis) are shown versus the temperature. 473 While the left end-point corresponds to the maximum of the 474 force peak, the right end point corresponds to the following 475 minimum. These curves have been represented for different 476 values of the interaction coefficient λ to explore the effects 477 of the Ising scheme on this critical behavior. Importantly, we 478 can observe that each spinoidal interval disappears for a given 479 temperature, which is a critical temperature for the system. 480 Hence, for a given chain composed of m units, there are m 481 different critical temperatures, one for each unfolding process. 482 We remark that, for a system without Ising interactions, the 483 critical temperature is larger for the last unfolded units. This 484 contrast among critical temperatures is further amplified for 485 antiferromagnetic-like systems. On the other hand, a given in-486 tensity of ferromagnetic-like interactions is able to equilibrate 487 the critical temperatures among the unfolding processes (see, 488 e.g., the curves in  [2]. We remark that the observation 499 of a negative differential stiffness for subcritical temperatures 500 and of a positive differential stiffness for supercritical temper-501 atures can be interpreted by stating that the system behaves 502 as a metamaterial [79,80]. In a following section, we will also 503 explore the effect of the intrinsic stiffness of the units on this 504 critical behavior. 507 We investigate in more detail the particular case with 508 weak Ising interaction, i.e., |λ| K B T , by considering both 509 ferromagnetic-like and antiferromagnetic-like interactions.

510
Under this condition, we will introduce an asymptotic devel- The first-order approximation in Eq. ( In this section, we consider μ = 0 to simplify the following 520 calculations. We have to determine 521 Z H (r ) = −i Z G iy K B T y r e i ry dy (54) where we used the approximation of Z G given in Eq. (51). We 522 firstly calculate the quantity I 1 , as Since where − = m − k − 2p + χk − 2χq + r . Similarly, we 527 calculate the integral I 2 given by and, by using again the integral in Eq. (46), we eventually 529 obtain 530 Finally, the partition function reads which is valid for strong ferromagnetic-like Ising interactions. 561 An application of this expression is shown in Fig. 7. In partic-562 ular, we compare the approximated result in Eq. (64) (yellow 563 or light gray curve) with the exact response obtained from 564 Eq. (48) for λ = 0.5, 1, 1.5, . . . , 7K B T (red or gray curves) 565 and with the response without Ising interactions (black dashed 566 curve). In Fig. 7 one can find the force-extension curves, 567 the average number of unfolded units and the Helmholtz 568 free energy. It is interesting to discuss the evolution of the 569 overall behavior of the system with an increasing interaction 570 coefficient. Indeed, as λ is increased, the units are progres-571 sively favored to be in the same state, and therefore there 572 is an increasing average number of units which unfolds at 573 r = m . It means that the number of unfolding processes at 574 r = m is a growing function of the Ising coefficient λ, going 575 from 1 with λ = 0 to m with λ approaching infinity. This 576 can be seen in Fig. 7(b), where this process is represented 577 by the series of red curves (or gray) with increasing λ, 578 and it ends with the yellow (or light gray) curve obtained 579 through Eq. (64). The latter means that all units unfold at 580 the same time at r = m when λ → ∞, and this behavior is 581 perfectly caught by the asymptotic development. Accordingly, 582 the peaks in the force-extension curve are strongly modified 583 by increasing λ: while the first peak becomes more and more 584 pronounced, the others are progressively reduced, as shown 585 in Fig. 7(a). As a matter of fact, the first peak corresponds to 586 the simultaneous unfolding of the units when λ is very large. 587 Hence, in the limiting case of λ → ∞, the force-extension 588 curve is composed of only one peak (yellow or light gray 589  Fig. 7(c). Here we can see the evolution of the typical 594 cusps with the increasing Ising coefficient. As an example, 595 the collapse of all the force peaks into a single unfolding 596 event explains the tandem repeats behavior in red cell spectrin, 597 where two units unfold simultaneously because of a strong 598 cooperativity [75]. 600 We discuss here the development of the theory under strong Ising antiferromagnetic-like interactions. As before, we can 601 develop the analytic continuation of the Gibbs partition function in a power series of λ → −∞, i.e., for x = e − λ K B T → +∞. The 602 result can be eventually obtained as

VII. HELMHOLTZ RESPONSE UNDER STRONG ISING ANTIFERROMAGNETIC-LIKE INTERACTIONS
where 604 S = sin(y) y + χφ sin(χy) y , We will develop the asymptotic theory for both the cases with m odd and even. We first elaborate the Helmholtz partition function 605 for m odd: Here we use the relation and we get 608 Then straightforward calculations deliver where 610 Hence, by using Eq. (46), we get the final result which is valid for m odd. 612 We calculate now the same quantity for m even: Finally, by using again Eq. (46), we get value of |λ| (λ < 0), we observe that the first force peaks 622 tend to disappear, while the last ones become more and more 623 pronounced [ Fig. 8(a)]. This is coherent with the assumption 624 that, in an antiferromagnetic-like system, the favored states 625 are alternatively folded and unfolded. Accordingly, with an 626 increasing value of |λ| (λ < 0), we have an increasing number 627 of unfolded units in the initial configuration with r = 0. 628 Now the maximum values of p and q are m+1 2 and m−1 2 , 639 respectively. Hence, we have Equivalently, where mχ is the upper limit of r , attained when all elements 644 are unfolded. It means that the total length of the unit cannot 645 [which is the highest value between 646 Eqs. (81) and (82) 687 We consider now a chain of bistable units characterized by 688 a finite elastic constant. We start the analysis by considering 689 the Gibbs ensemble defined through the extended Hamiltonian 690 given in Eq. (2). Here, for the sake of simplicity, we suppose 691 that the folded and unfolded basins of the potential energy 692 shown in Fig. 1

AND EXTENSIBLE UNITS
which can be easily proved by calculating the exact solution can be obtained as where, with respect to Eq. (24) of Sec. III, we added an 703 exponential term, which is quadratic in the normalized force where P (y) is the periodic function defined in Eq. (40) where we used the Fourier development of the function P (y) 722 in order to perform the calculation. To complete the task, we 723 have to calculate a sequence of integral of the form where a ∈ R, b > 0, N ∈ N and the path is given in Fig. 3. 725 An application of the complex variable method allows us to 726 obtain the closed-form expression for this integral, as [82] 727 and they can be obtained recursively through the formula 732 [82,90] 733 Ref. [90].

742
An application of the Gibbs and Helmholtz partition func-743 tions, stated in Eqs. (84) and (86), respectively, is presented 744 in Fig. 10, where we show the force-extension curves for two 745 values of the constant κ and for three values of the coefficient 746 λ. First, we note that the constant slope of the final part of the 747 force-extension curves represents the finite effective stiffness 748 of the chain, after the unfolding processes. Moreover, it is 749 interesting to remark that the softer systems exhibit a sensibly 750 reduced force peaks in the Helmholtz response. This point can 751 be also noticed by drawing a comparison between Fig. 4(a), 752 obtained for κ → ∞, and Fig. 10, corresponding to finite 753 values of κ. A similar phenomenon can be also observed in the 754 phase diagram showing the critical behavior of the spinoidal 755 response of the system. Indeed, we plotted in Fig. 11 four 756 phase diagrams corresponding to four different values of the 757 elastic constant. We observe that the critical temperature of the 758 unfolding processes is an increasing function of κ, similarly 759 to the previously discussed force peaks of the Helmholtz 760 response. Besides, as already seen in Fig. 5 concerning the 761 case with κ → ∞, antiferromagnetic-like interactions am-762 plify the dissimilarity among the critical temperatures, while 763 ferromagnetic-like interactions reduce this contrast, eventu-764 ally producing a more uniform response of the unfolding 765 processes. 767 We investigated the properties of a chain of two-state 768 units coupled through an Ising interaction scheme, providing 769 a paradigmatic description of the effects of bistability and 770 cooperativity in biological and artificial micro-and nanosys-771 tems. Accordingly, we studied our model by means of the 772 statistical mechanics of small systems, i.e., far from the 773 thermodynamic limit. It means that, for a limited number 774 m of units of the chain, the Gibbs and Helmholtz statistical 775 ensembles are not equivalent, and we ultimately obtain two 776 different isotensional and isometric responses, well recog-777 nized, e.g., in force-spectroscopy experiments. Some of the 778 most interesting findings of this paper concern the influence of 779 the cooperativity, measured by the Ising coefficient λ, on the 780 mechanical behavior and on the configurational features of the 781 system. In particular, we analyzed the force-extension curve 782 under isotensional conditions, obtaining a sharper or smoother 783 transition depending on λ, and under isometric conditions, 784 getting a variable hierarchy of force peaks as function of 785 the cooperativity. Also, the unfolding processes of the units 786 have been characterized by plotting the number of unfolded 787 units versus the mechanical quantity (f or r) inducing the 788 chain stretching. This point allows the interpretation of the 789 unfolding processes as synchronized or simultaneous under 790 isotensional conditions and as nonsynchronized or sequential 791 under isometric conditions. This result underlines the conve-792 nience of the spin variables to investigate the configurational 793 properties of the system.

794
From the methodological point of view, we underline that 795 the spin variables approach is useful to elaborate semianalytic 796 or closed-form expressions for the relevant observables. More 797 specifically, to solve the problem within the Gibbs ensemble, 798 we coupled this spin variables approach with the classic 799 transfer matrix technique to take account of the interactions.  To give a complete picture of the equilibrium behavior of 811 the system, we also investigated a form of criticality exhibited 812 by the system. In particular, our analysis highlights the critical 813 behavior of the spinoidal regions, characterizing the part of 814 the isometric response showing a negative differential elastic 815 stiffness. We prove that each unfolding process exhibits a 816 critical temperature defined by stating that we measure a 817 negative differential stiffness for subcritical temperatures and 818 a positive differential stiffness for supercritical temperatures.

819
This behavior is influenced by the cooperativity, which has the 820 capability to make the critical temperatures of the unfolding 821 processes more uniform. We can therefore state that a positive 822 cooperativity increases the resistance to fluctuations, 823 making the spinoidal intervals equally stable to temperature 824 variations.

825
While being a paradigmatic model for the understanding of 826 several phenomena, our chain with Ising interactions should 827 be improved to better represent more realistic situations. One 828 drawback concerns the uniformity of all parameters defining 829 the properties of the units. Indeed, in order to correctly 830 model the actual mechanical behavior of heterogeneous struc-831 tures, such as proteins, we would have the possibility to 832 freely choose these parameters for each unit. Nevertheless, 833 this heterogeneity consists in a form of quenched disorder, 834 which is much more complicated to be taken into account 835 by classical statistical mechanics methods. However, it should 836 be important to introduce this point since it could allow to 837 determine the full unfolding pathway, which depends on the 838 system microstructure. As an example, this is directly related 839 to the biological function of a protein. Another improvement 840 concerns the dynamics of the unfolding processes, which 841 should be studied in the context of the out-of-equilibrium 842 statistical mechanics. It is worth noting here that the spin 843 variable approach can be used for decoupling two kinds of 844 characteristic times: (i) the purely mechanical times induced 845 by the stiffness of each basin of the potential energy and 846 (ii) the times induced by the transition rates between the 847 basins, which depend on the energy barrier as classically 848 described by the Kramers theory. This approach should permit 849 to consider out-of-equilibrium unfolding processes, typically 850 induced in isometric force-extension experiments conducted 851 at fixed pulling velocity of the tethered chain.

852
ACKNOWLEDGMENT 853 We acknowledge the region "Hauts de France" for financial 854 support under project MEPOFIB.