Abstract While stretching of most polymer chains leads to rather featureless force-extension diagrams, some, notably DNA, exhibit non-trivial behavior with a distinct plateau region. Here we propose a unified theory that connects force-extension characteristics of the polymer chain with the convexity properties of the extension energy profile of its individual monomer subunits. Namely, if the effective monomer deformation energy as a function of its extension has a non-convex (concave up) region, the stretched polymer chain separates into two phases: the weakly and strongly stretched monomers. Simplified planar and 3D polymer models are used to illustrate the basic principles of the proposed model. Specifically, we show rigorously that when the secondary structure of a polymer is mostly due to weak non-covalent interactions, the stretching is two-phase, and the force-stretching diagram has the characteristic plateau. We then use realistic coarse-grained models to confirm the main findings and make direct connection to the microscopic structure of the monomers. We demostrate in detail how the two-phase scenario is realized in the α-helix, and in DNA double helix. The predicted plateau parameters are consistent with single molecules experiments. Detailed analysis of DNA stretching demonstrates that breaking of Watson-Crick bonds is not necessary for the existence of the plateau, although some of the bonds do break as the double-helix extends at room temperature. The main strengths of the proposed theory are its generality and direct microscopic connection. I. INTRODUCTION When pulled by the ends, a flexible linear polymer first undergoes entropic elongation where the work done by the stretching force reduces the conformational entropy of the chain [1, 2] . In this well-understood [2] weak extension regime the polymer obeys Hooke's law and its elastic properties are "universal", in that they are insensitive to details of the chemical structure and interactions within its monomers. As the polymer chain is extended further, and its end-to-end distance becomes comparable to the chain contour length, the intrinsic elasticity due to deformation and interaction of individual monomers begins to dominate the extension response [3] . Since short-scale chemical structures of real polymers differ substantially, as do their observed responses to strong tension forces, one wonders if polymer stretching in this regime can still be described by a universal principle? The question is important. Biopolymers such as DNA are subjected to a range of mechanical manipulations within the cell, they may change their conformations and undergo unexpected structural transitions [4] [5] [6] . Knowledge of elastic properties of biopolymers is required to understand the structural dynamics of many important cellular processes [7] [8] [9] . These properties can now be measured quite accurately by modern experimental techniques such as atomic force microscopy and optical tweezers. I. INTRODUCTION For DNA [4, 5, [10] [11] [12] and polypeptides [13] [14] [15] these experiments have revealed several peculiar features. When extended, the (double-stranded) DNA molecule exhibits the following behavior: until the end-to-end distance reaches 0.9 of the contour length, the stretching process is well described by established phenomenological models [2, 4, 16] . But then, when the molecule is subjected to forces of 65 ÷ 120 pN (depending of experimental conditions), a sudden structural tran- * imccormick@example.com sition occurs, in which the chain stretches up to 70% beyond its canonical B-form contour length. The extension force remains almost constant in this regime, which is manifested by a characteristic plateau on the experimental force-extension curve. Similar single molecule stretching experiments have also been performed on polypeptide molecules [13] [14] [15] . It was found that simple [49] helical polypeptide structures such as synthetic alpha-helices [14] and myosin molecules [13] , exhibit a force-extension plateau similar to that seen in DNA stretching experiments. In contrast, these features are not observed in many "non-biological" polymers such as polyethylene. I. INTRODUCTION Various microscopic models were proposed to explain these observations on a case-by-case basis. For example, forceextension plateau observed in single DNA molecule experiments (sometimes called the over-stretching plateau) is often explained by gradual un-zipping (force-induced melting) of the double helix in which Watson-Crick (WC) hydrogen bonds between base-pairs break [10] [11] [12] [17] [18] [19] , An alternative explanation involves cooperative transition of the whole structure into a new form called S-form where WC bonds remain intact [5, 20] , but the helix unwinds to form a straight ladder. In the case of polypeptides, force-extension plateau is attributed to alpha-helix unwinding [13, 21, 22] . Phenomenological descriptions based on various assumptions about stable monomer states were also proposed [23] . Still, no universal, microscopically based mechanism exists that can explain why some polymers do and some do not exhibit a plateau in forceextension experiments. Here we propose such a mechanism and show how the stretching properties of the polymer depend on the balance between valent and non-valent interactions on the scale of individual monomers. FIG . 1: Two distinct forms of the effective site deformation energy E(∆l) of an individual monomeric unit (site) that lead to qualitatively different stretching scenarios of the linear polymer chain of N sites. The site extension is ∆l. Curve 1: A non-convex energy function (between ∆la and ∆l b ) leads to non-uniform, two-phase stretching of the chain. Some sites extend weakly by ∆la, and some strongly by ∆l b . The chain extension follows the convex hull (red line 2) of E(∆l), with only the relative fraction of weakly extended sites changing as the chain extends. The average (per site) energy of non-uniform extension ENU (∆L) is less than that of the corresponding uniform E(∆L) extension. Curve 3: A convex function E(∆l) leads to uniform extension of all the sites. II. RESULTS & DISCUSSION A. The general mechanism of polymer stretching under tension Consider a linear polymer chain of N ≫ 1 identical interacting sites (monomeric units). The effective site deformation energy E(∆l) can be defined as follows. Consider a configuration of the chain in which each site is stretched by the same amount ∆l. Then E(∆l) is simply the total deformation energy of the chain divided by N . Here we show that the shape of the effective site deformation energy E(∆l) determines force-induced stretching behavior of the chain in the general experimental scenario when the force is applied to the chain's ends and no restrictions are imposed on deformations of individual sites. A. The general mechanism of polymer stretching under tension In what follows we will use the following convention. Extension of a single monomeric site or, equivalently, that of each site of a uniformly stretched chain is denoted by ∆l. In general, including the case of non-uniform deformation, extension of site i is denoted by ∆l i . We use ∆L for the mean per site deformation ∆L = N −1 N i=1 ∆l i . If the function E(∆l) is convex down, line 3 in Fig. 1 , the most favorable structure of the chain with fixed total deformation i ∆l i corresponds to each site i stretched by the same amount ∆l i ≡ ∆l, see appendix A for details. The simplest example of such a polymer model is a chain of beads connected by harmonic springs. The total deformation energy of the chain in the case of convex E(∆l) is N E(∆l). Non-uniform stretching is energetically unfavorable in this scenario because any putative decrease in the total energy from under-stretching of a group of sites would be offset by a larger increase in the energy of the remaining sites that would have to over-stretch to keep the total deformation of the chain constant. In contrast, if the effective site extension energy is non-convex over some interval, (curve 1, Fig. 1 ), two-phase stretching becomes more favorable energetically: one part of chain consists of pN sites (0 < p < 1) stretched "strongly" by ∆l b , and another part consists of the rest (1 − p)N sites stretched "weakly" by ∆l a < ∆l b . Qualitatively, this is because the decrease of the chain energy (relative to the uniform stretching scenario) resulting from under-stretching of a group of sites is larger than the gain from over-stretching of the remaining sites. A detailed quantitative analysis is presented in the appendix A. Briefly, the mean deformation per site in this case is ∆L = p∆l b + (1 − p)∆l a , and the total energy of the chain in this non-uniform (NU) case equals N E N U (∆L) = N ((1 − p)E(∆l a ) + pE(∆l b )) (contribution from phase boundary can be neglected for long chains, N ≫ 1, typically used in experiment [19] ). Since the function E(∆l) is non-convex, N E N U (∆L) is less than N E(∆L) = N E((1 − p)∆l a + p∆l b ) -the total energy of the chain in the uniform case with the same total deformation, Fig. 1 . In the non-uniform regime, the chain extension is achieved via change in the relative fraction p of the strongly stretched sites, not by extension of individual sites. As p increases from 0 to 1, the mean deformation ∆L = (1 − p)∆l a + p∆l b depends on p linearly and ranges from ∆l a to ∆l b . The average per site chain energy E N U (∆L) also depends on p linearly, and ranges form E(∆l a ) to E(∆l b ) -that is the stretching process is described by a straight line connecting points ∆l a and ∆l b -see Fig. 1 , red line. The tension force dE N U /d∆L thus remains constant, and the characteristic plateau in the force-extension diagram appears. A. The general mechanism of polymer stretching under tension Real polymer chains may appear more complex, but the chain structure is always stabilized by interactions of two types: "strong" valent and "weak" non-valent. The former describes bond, angle and torsion deformations. The latter corresponds to "soft" non-valent interactions. These interactions include various combinations of electrostatic and van der Waals interactions, hydrogen bonds in polypeptide alpha-helix and stacking interactions between neighboring base pairs in DNA. The main feature of realistic non-valent interactions potentials W (r) is the existence of inflection points. If non-valent interactions contribute significantly to the extension energy E(∆l), the function E(∆l) may also have an inflection point and hence a non-convex region as in Fig. 1 , leading to a plateau in the force-extension diagram. A. The general mechanism of polymer stretching under tension We thus propose a general mechanism for the observed twophase stretching of liner polymers based on convexity properties of the effective potential energy of the monomeric units of the polymer. As we will demonstrate, the mechanism is able to explain the existence of force-extension plateaux for very different types of polymers. B. Stretching of a 2D zigzag molecular chain We begin by exemplifying the proposed mechanism of nonhomogeneous two-phase stretching on a 2D zigzag chain, FIG . 2: Schematic of a 2D zigzag polymer chain. ǫ φ U (φ) and V (ρ) are the valent angle bending and bond stretching potentials; W (r) is the non-valent interaction between next-nearest neighbors. The longitudinal step of site n is ln, the transverse step is hn. Fig. 2 . While arguably among the simplest polymer geometries, it is often found in real polymers: for example, polyethylene (PE) molecule has a stable plane conformation of transzigzag. The 2D zigzag form is also common in hydrogenbonded chains · · ·X-H· · ·X-H· · ·X-H· · · of halides, where X=F, Cl, Br, I. B. Stretching of a 2D zigzag molecular chain We consider a dimensionless model of 2D zigzag chain, see section IV and appendix B for details. In the limiting case of zero angle bending potential ǫ φ = 0, Fig. 2 , only nonvalent interactions between next nearest neighbors determine elastic response of the chain. Numerical simulations of the zigzag with N ≫ 1 units, see section IV, show that in this case the chain stretching is accompanied by monotonous increase of the bond angle; once the mean extension reaches ∆L = 0.18, the zigzag straightens out completely into a line (the zigzag angle is φ = 180 B. Stretching of a 2D zigzag molecular chain • ), then valent bonds begin to stretch. The corresponding effective site deformation energy E(∆l) of a single monomer site is shown in Fig. 3 (a) . The function E(∆l) is not convex; its convex hull is given by a tangent line at points ∆l a = 0.04 and ∆l b = 0.21. According to our general mechanism, a fraction of the zigzag sites is expected to be in the weakly extended state with the longitudinal step l n = l 0 + ∆l a , while the rest will be in the strongly extended state with the step l n = l 0 + ∆l b . The tension force F = dE N U /d∆L remains constant between ∆l a and ∆l b , and the force-extension dependence F (∆L) has the typical plateau, Fig. 3 (b) . The zigzag effective site extension energy E(∆l) remains non-convex until the strength of the angle potential reaches a critical value (ǫ φ = 0.015). At this point, energetic benefit from non-uniform stretching relative to uniform stretching vanishes. As the angle bending potential becomes even stiffer, its relative contribution to chain stretching overwhelms that of the weak non-valent interactions that give rise to the non-convex behavior seen in Fig. 3 (a) . The effective site extension energy function becomes convex, Fig. 3 (c) , and the stretching behavior of the polymer is essentially that of a harmonic spring -single phase, uniform. These scenarios are further illustrated in the Appendixes for a zigzag chain of N = 400 "atoms". B. Stretching of a 2D zigzag molecular chain Thus, two-phase stretching and force-extension plateaux can be expected to occur in molecular chains in which secondary structure is supported by weak non-valent interactions. On the other hand, if the secondary structure is due mainly to angle deformation, then the stretching will be uniform. Such a scenario is typical for polyethylene (PE) trans-zigzag [24] . C. Stretching of the α-helix Consider a 3D molecular chain corresponding to an ideal [1] α-helix, Fig. 4 (a) . C. Stretching of the α-helix Here, the softest valent potential is the torsional potential; we vary its relative contribution ǫ θ to the total energy while keeping the other parameters fixed, see section IV and appendix C. Without the torsional rigidity (ǫ θ = 0), the helix is stabilized only by hydrogen bonds, connecting site i with sites (i + 3) and (i − 3) [1, 25, 26] . The effective site energy E(∆l) is shown in Fig. 5 . Upon stretching, the helix's angular step (deformation) monotonously increases and reaches its maximum value 180 C. Stretching of the α-helix • when ∆l = 0.16 (plane zigzag). The function E(∆l) is not convex, Fig. 5 (a) . Its convex hull is described by a tangent at points ∆l a = 0.06, ∆l b = 0.17. According to our general mechanism, a fraction of the helix is in the weakly extended state with the longitudinal step l 0 + ∆l a , while the rest is in the strongly extended (plane zigzag) state with the longitudinal step l 0 + ∆l b . As long as ∆l a < ∆L < ∆l b , the tension in the helix F = dE N U /d∆L remains constant, leading to the characteristic plateau in the force-extension diagram. If the torsional rigidity is increased, the non-convex shape of E(∆l) is preserved until ǫ θ = 0.0015 is reached. Once ǫ θ > 0.0015, the function E(∆l) becomes convex -see Fig. 5 . In this regime, only uniform stretching of the helix is possible. These scenarios are explicitly verified by numerical simulations for a helix consisting of N = 400 sites, Fig. 6 . When the torsional rigidity is zero ( ǫ θ = 0 ), and non-valent interactions dominate the elastic response, the stretching is two-phase. The distribution of longitudinal extension ∆l n along the chain completely matches the expectation based on the shape of E(∆L) function, that is for ∆L ≤ 0.07 the chain is stretched uniformly, while for 0.07 ≤ ∆L ≤ 0.165 non-uniform stretching is observed. The terminal regions of the chain are in a strongly stretched state, while the central region is stretched weakly [ Fig. 6 (a) ]. The transition boundary between the states is clearly seen in Fig. 4 (b). As the helix is extended further, the weakly stretched central region shrinks and vanishes when ∆L = 0.17. Beyond that point the helix is stretched uniformly. The domain of the non-uniform stretching regime decreases for higher torsional rigidity ǫ θ = 0.0015 Thus, if the torsional rigidity is small enough, stretching of the α-helix proceeds via two-phase scenario with a typical plateau region where the tension remains constant. The scenario is confirmed by all-atom molecular dynamics simulations [22] and experiments [13, 14] . The root cause of the non-uniform stretching in this case is the typical form of the hydrogen bond potential (4), which has an inflection point. In contrast, polymer helices that have no hydrogen bonds, such as Polytetrafluoroethylene (PTFE) helix, are expected to stretch uniformly, without force-extension plateaux. D. Stretching of the DNA double helix The coarse-grained model [27, 28] used here to simulate stretching of the DNA double helix (dsDNA) is semiatomistic: each nucleotide consists of six united atom particle -three for the sugar-phosphate backbone and three for the nucleobase, see section IV and appendix D. D. Stretching of the DNA double helix The corresponding effective site (base pair) energy E(l) as a function of relative site extension is shown in Fig. 8 (a). To facilitate direct comparison with experiment, the energy is taken to depend on the relative extension l/l 0 instead of absolute deviation ∆l from equilibrium base-pair length l 0 ; l 0 = 3.352A calculated within our model agrees with the experimental value for B-DNA. The function E(l/l 0 ) is nonconvex between points l a /l 0 = 1.12, l b /l 0 = 1.84, its convex hull is shown by red line 2 in Fig. 8 (a) . Thus, when the mean relative site extension L/l 0 of a stretched dsDNA fragment is between the above two values, a part of the double helix is in the weakly extended state with the longitudinal step l a , while The proposed non-uniform stretching mechanism, so far explored without taking into account thermal fluctuations, holds at room temperature: only the range of the dsDNA overstretching plateau increases slightly, Fig. 8 . Critically, the room temperature value of the tension at the plateau coincides with the value obtained from the analysis of the minimum energy (ground) states described above. As the model chain stretches at room temperature, thermal fluctuations cause the WC hydrogen bonds to break, as expected from experiment, Fig. 9 In the plateau regime, the DNA double-helix consists of two fractions: a slightly stretched helix with hydrogen bonds intact, and a strongly stretched helix with some hydrogen bonds broken (Fig. 18 ). This exact behavior is observed in all-atom simulations [29, 30] . An important question arises whether the breaking of the hydrogen bonds between complementary bases is necessary [10, 11, 17, 18] for the observed over-stretching plateau, or is the unzipping of the helix simply the consequence of the helix stretching? Our analysis clearly shows that WC bond breaking is not necessary for the appearance of the over-stretching plateau. First, the two-phase stretching behavior of dsDNA and the force-extension plateau [ Fig. 8 (b) ] with virtually the same characteristics exist in the absence of thermal fluctuations when hydrogen bonds are only weakened in the strongly stretched regime, but not yet broken. Second, in a computational experiment in which the bond strength is artificially doubled to prevent breaking of WC bonds at room temperature, we find virtually the same plateau. This explains the somewhat puzzling result of a recent single-molecule experiment in which torsionally relaxed DNA exhibited the same over-stretching plateau when its unzipping was inhibited [31] . In the case of the double-stranded DNA, it is mainly the basestacking deformations that give the effective stretching energy its non-convex shape that is ultimately responsible for the on- set of the two-phase stretching with the characteristic plateau, Fig. 17 . Mathematically, base stacking is described by a combination of power law functions -Coulomb and LennardJones potentials -that give it the non-convex shape. (a). The plateau in dsDNA force-extension diagram was observed previously in single molecule stretching experiments [4, 5] ; the plateau was found in the range of (relative) extensions 1.1 < L/l 0 < 1.7. This agrees well with our result 1.12 < L/l 0 < 1.84, Fig. 8 (b) . The value of the plateau transition force within our model is 0.2 eV/A = 320 pN, which is somewhat higher than experimental estimates. A typical value of the force is often reported to be about 65 ÷ 70 pN [4, 5, 19] ; however one should keep in mind that the experimental value was obtained in the type of experiment when the DNA strands are pulled by the same type (3 ′ or 5 ′ ) end, while the other two ends remain free. In contrast, the simulations reported here correspond to uniform pulling by all four ends. Experimentally, larger values of the plateau tension were reported under the uniform pulling scenario: 105÷120pN [19, 32] . Although this value is still less than half of the 320 pN predicted by our model, we note that the above experiment used a specific, non-homogeneous sequence. Many DNA properties are strongly sequence-dependent: for example, pure poly(dG-dC) and poly(dA-dT) DNA sequences yield tension values at the plateau that differ by a factor of two [32] . Thus, only semiquantitative agreement of our homogeneous (poly-A) model with the above experiments may be expected. Reported differences between earlier estimates based on all-atom room temperature Molecular Dynamics simulations and experimental values are also of the same order [5, 20] ; therefore numerical agreement we have obtained with experiment can be considered as reasonable. III. CONCLUSION While all polymers behave very similar to each other under weak tension where their elastic properties are entropic in nature and are virtually independent of the structure of the monomers, striking differences are observed in experiments when stronger forces are applied and short-scale details start to dominate. We show that these differences can be explained by a very general mechanism based on convexity of the (effective) deformation energy function of individual monomers. We demonstrate that when this energy is a convex function of the extension, the chain stretching is single-phase uniform, without a plateau in the force-extension diagram. The scenario is realized in polymers such as polyethylene whose structure is supported by strong covalent interactions. In contract, when the secondary structure of a polymer is mostly due to weak non-covalent interactions, the deformation function may become non-convex, leading to two-phase stretching: a part of the chain is stretched weakly, while the other is stretched strongly. In this regime, extension of the whole chain proceeds by increasing of the fraction of the strongly stretched sites, so the tension remains constant. The forcestretching diagram has the characteristic plateau seen in experiment. Examples include α-helix polypeptide, and DNA double helix, consistent with earlier observations based on all-atom molecular dynamics simulations and previous single molecules experiments. We illustrate the general mechanism by numerical simulations based on realistic coarse-grained models and atomistic potentials of several polymers, from planar "zig-zag", to the more complex helix and B-DNA. Numerical modeling is in complete agreement with the general mechanism, and in acceptable agreement with experiment. III. CONCLUSION The main strengths of the proposed theory are its complete generality and direct connection to microscopic structure of the monomers. Our framework applies to any polymer in the strong deformation regime where short-scale details dominate -to the best of our knowledge, no such universal description was available before. Although division of the polymer into two stretching phases was discussed earlier in the context of phenomenological models [21, 23] , the second equilibrium state was assumed to exist and to be known a priori. Within our framework, no assumptions of multiple stable equilibrium states [13, 21] or additional kinetic arguments are necessary [21] . IV. METHODS A. Computing the force-extension diagram The effective site deformation energy function E(∆l) (Fig. 1) is found by minimizing the total potential energy H of the chain under the constraint that each site is stretched by the same amount ∆l: A. Computing the force-extension diagram nition of E(∆l) includes contributions from both short-range interactions within each site and short-and long-range interactions between the sites. Next, we consider the same chain of N ≫ 1 effective sites, but without the ∆l i = ∆l constraint, that is with the possibility of non-uniform extension. The dependence of the mean chain energy E N U (∆L) upon its mean longitudinal extension ∆L = N −1 N i=1 ∆l i is found by minimizing the total energy of the chain H under the condition of fixed total deformation A. Computing the force-extension diagram The resulting function E N U (∆L) is the convex hull of the effective site deformation energy E(∆l) (see appendix A for details ). Unless otherwise specified, the extension force (ten-sion) is obtained as F = dE N U (∆L)/d∆L. No torsional constraints are imposed in any case. Unless otherwise stated, polymer chain is modeled as quasi-one-dimensional crystal. B. 2D plane zigzag The 2D zigzag chain, Fig. 2 , is specified by the distance between its neighboring sites ρ 0 and the zigzag angle φ 0 (equilibrium longitudinal step is l 0 = ρ 0 sin(φ 0 /2)). We consider dimensionless zigzag model, details of its parameters values are given in Refs. [33] [34] [35] and the appendix B. B. 2D plane zigzag The chain potential energy is given by: B. 2D plane zigzag where V (ρ n ) is the valent bond energy between neighboring atoms n and (n + 1) separated by distance ρ n . B. 2D plane zigzag with the bond rigidity K = 2. The ǫ φ U (φ n ) term is the deformation energy of the angle between atoms (n − 1), n and (n + 1), where ǫ φ ≥ 0 is the angle deformation stiffness. B. 2D plane zigzag The last term W (r n ) corresponds to weak non-valent interaction between atoms n and (n + 2) (next-nearest neighbors) separated by r n ; its form is typical for non-valent interactions; it may be used for description of hydrogen bonds and van der Waals interactions alike. B. 2D plane zigzag where ǫ hb ≥ 0 is the interaction energy, r 0 = 2h x = 1.633 is the equilibrium length, d = 0.5 is the diameter of the inner hard core. The balance between valent and non-valent interactions is varied by changing the angle stiffness ǫ φ , while keeping other interactions fixed. C. α-helix The model we describe here, Fig. 4 (a) , is similar to the ones [25, 26] used previously for analysis of ultrasonic soliton motion. The equilibrium atomic helix coordinates: C. α-helix with n = 0, ±1, ±2, ... being the atom number, R 0 -helical radius, ϕ 0 and l 0 -the angular and longitudinal helix period. The chain potential energy is C. α-helix The term V (ρ n ) gives the energy of interaction between neighbor sites n and (n + 1), where ρ n is the distance between them. The angle deformation energy is described by ǫ φ U (φ n ), where φ n is the angle between sites (n − 1), n, and (n + 1) (the vertex is on site n). The third term ǫ θ Z(θ n ) is the energy of the torsional deformation (rotation) around n-th bond: C. α-helix where ǫ θ ≥ 0 is torsional rigidity; different values of ǫ θ are considered while keeping other interactions fixed. The function W (r n ) is the energy of the n-th hydrogen bond connecting sited n and (n + 3).We consider dimensionless model of the helix, see Appendixes for specific parameters. The bond deformation energy is described by the potential (2) with the rigidity K = 10. Hydrogen bond energy is given by eq. (4), angle deformation energy is described by (3). D. DNA double helix The potential energy of the double helix consists of 4 terms: D. DNA double helix The first two terms describe deformation energy of complementary strands A and B, respectively, within the 12CG coarse-grained model [27] . These terms include essentially the same energetic contributions as in the case of the alphahelix: internal energy (bond stretching, angle bending and torsion twisting) plus non-valent interaction between the grains within the same strand. The last two terms are non-valent interactions: E st -between two neighboring base pairs, and E * hb -between two complementary bases (within the same base-pair), including hydrogen bonds. D. DNA double helix Within the framework of our coarse-grained model [27] , the nitrogen bases are treated most accurately, at all-atom level. WC hydrogen bonds and stacking interactions are modeled via Coulomb and van der Waals potentials taken from current all-atom AMBER [36] force-field widely used to model nucleic acids. To make the computations feasible, solvent effects are treated via the so-called implicit solvation model [37] at the generalized Born level often used in all-atom simulations of DNA [38] . Within the model, water is treated as a continuum with the (room-temperature) dielectric and hydrophobic properties of water; screening effect of salt ions is also taken into account. Hydrogen bonding with the solvent is present, albeit in an "average" sense. The balance between solute-solute and solute-solvent h-bond strength is controlled by adjusting E hb -the interactions between complementary bases taken from the all-atom AMBER explicit solvent forcefield [36] . Here we use E * hb = c 0 E hb , with c 0 = 0.4, which leads to quantitative agreement with experiment, Fig. 9 . Further details of the calculation can be found in Ref. [27] and SI. To avoid sequence-dependence issues that do not affect the basic physics, we consider homogeneous poly(A)-poly(T) sequence. D. DNA double helix The tension at T = 300K, Fig. 8 (b) , is obtained as F = d H /dL, where · denotes ensemble averaging over a Molecular Dynamics trajectory. The simulation employed a 500 base-pair poly(A)-poly(T) fragment in the 12CG coarsegrained representation, see Appendixes for details. The same trajectory was used to obtain results in Fig. 9 . Acknowledgments This research was supported by RFBR (grant no. 08-04-91118-a), CRDF (grant no. RUB2-2920-MO-07), and, in part, NIH GM076121 to A.V.O. The authors thank Erwin J.G. Peterman for a helpful discussion and providing experimental data points in Fig. 9 . Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site Consider a linear polymer chain of N ≫ 1 identical sites (monomeric units). Each site i is extended by ∆l i , with the corresponding deformation energy E(∆l i ). When pulled by both ends, the equilibrium energy of the entire chain can be found by solving the constrained (conditional) minimization problem: Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site where N ∆L is the total extension of the chain. Here ∆L denotes the mean per site extension of the chain. Using the method of Lagrange multipliers, the problem is converted to the unconditional minimization problem over (N+1) variables: Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site Differentiating with respect to ∆l i yields E ′ (∆l i ) − λ that should equal zero for each i. Derivative with respect to λ also equals zero; this condition yields the original constraint. Excluding λ from all of the equations, we obtain the following system of N equations: Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site When the function E(∆l) is convex, its derivative is a monotonically increasing function, and the condition A1 can be satisfied only when extensions of all the sites are equal to each other: ∆l a = . . . = ∆l n = ∆l = ∆L, that is when the chain is extended uniformly. In contrast, when the energy function E(∆l) is non-convex, it is possible for its derivative (tangent) to have the same value at two distinct points ∆l a and ∆l b , see Fig. 1 . In this case, some ∆l i (for simplicity, i = 1, . . . , pN ) are equal to ∆l b , while the rest of ∆l i (i = pN + 1, . . . , N ) are equal to ∆l a , ∆l a < ∆l b . If N ≫ 1 (technically, if N → ∞), one can always find such p, 0 < p < 1, that the constraint ∆L = p∆l b + (1 − p)∆l a is satisfied. The deformation energy in this case of non-uniform stretching is equal to pN E(∆l (∆l a , E(∆l a )), (∆l b , E(∆l b ) ). This linear function is the convex hull of E(∆l), see Fig 1 in the main text. By definition of non-convex function, pE(∆l b ) + (1 − p)E(∆l a ) < E(p∆l b + (1 − p)∆l a ) = E(∆L), which proves that the two-phase extension is energetically preferred relative to the uniform extension in this case. As the chain is stretched further, and ∆L increases, p increases accordingly so that ∆L = p∆l b + (1 − p)∆l a is satisfied. The extension of the chain via the change in the fraction of sites extended by ∆l b can continue until p = 1 at ∆L = ∆l b . Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site The above reasoning does not take into account phase boundary effects. However, these, as well as end effects, are negligible as long as the polymer chain is long, N gg1, which is always the case experimentally. Our numerical calculations give the same results supporting our general conclusions. Appendix A: Minimal energy conformation of the polymer chain is determined by convex properties of the effective deformation energy curve of individual monomer site In what follows we consider three microscopic polymer models in detail: 2D zigzag, α-helix, and double-stranded DNA. Since we are interested in the regime where polymer extension approaches its contour length, entropic contributions (key in weak stretching ) are neglected. Appendix B: 2D Zigzag Consider a dimensionless 2D model of zigzag chain shown in Fig. 2 . Such a chain can be considered as a quasi-onedimensional crystal with the elementary cell being two neighboring sites; that is each cell can be obtained from the previous one by translation along the x-axes. The potential energy of the zigzag chain is given by the Eq. (1). The equilibrium bond length is ρ 0 = 1, zigzag angle is φ 0 = arccos(−1/3) = 109.47 Appendix B: 2D Zigzag • (so that equilibrium longitudinal step is l 0 = 0.8165). The non-valent interaction is given by Appendix B: 2D Zigzag , where ǫ hb ≥ 0 is the interaction energy, r 0 = 2h x = 1.633 is the equilibrium length, d = 0.5 is the diameter of the inner hard core. For simplicity, but without loss of generality, we set the equilibrium length r 0 to correspond to the equilibrium value of the angle θ 0 . The non-valent interaction energy coefficient was fixed at ǫ hb = 0.0178 (so that W ′′ (r 0 ) = 72ǫ hb /(r 0 − d) 2 = 1). Let the x-axis be along the chain, and the y-axis in the perpendicular direction, Fig. 2 . The longitudinal (x-) step l and transverse (y-) step h for site n are shown in Fig. 2 . To find the uniform extension energy that is energy of a single zigzag site extended by ∆l > 0 along the x-axis, one has to solve the minimization problem over transverse step h Appendix B: 2D Zigzag with fixed value of the longitudinal step l 0 + ∆l. Note that the distance between next-nearest neighbors is r = 2(l 0 + ∆l), and the value of the angle φ is defined by the bond length ρ. Solving the minimization problem (B1) yields the energy (per one site) a uniformly stretched zigzag as a function of the relative longitudinal extension ∆l. Conjugate gradient is used to find the minimum. The angle φ, energy and tension force as functions of the extension are shown in Fig. 10, 11 for different values of the coefficient ǫ φ . This coefficient controls the relative contribution of a valent interaction (angle deformation energy) to the elastic response of the chain. Appendix B: 2D Zigzag To find the energy of the extended (generally nonuniformly) chain for each value of the mean (per site) extension ∆L ≥ 0, one has to solve the minimization problem: Appendix B: 2D Zigzag (B2) with the condition of fixed ends: Appendix B: 2D Zigzag Conjugate gradient is used to find the minimum. The initial condition is chosen to be that of a uniformly stretched chain. It should be mentioned that in what follows only the xcoordinate is fixed, all other coordinates are allowed to change freely. If {x n , y n } N n=1 is a solution of the minimization problem (B2), then the distribution of the longitudinal extension in the chain is given by the function ∆l n = x n+1 − x n − l 0 , and the distribution of the transverse step is given by the function h n = |y n+1 − y n |. Appendix B: 2D Zigzag One can see from Fig. 12 that when ǫ φ = 0 the distribution of the longitudinal step extension ∆l n and the transverse step h n along the chain is in agreement with the prediction that can be made based on convexity of the function E(∆l). That is when 0 ≤ ∆l ≤ 0.037 and ∆l ≥ 0.2056, uniform stretching takes place, while when 0.037 < ∆l < 0.2056 nonuniform one is observed. Here, the end sites of the zigzag are in the strongly extended state with extension ∆l n = 0.2056, while the central part is in weakly stretched state with extension ∆l n = 0.037. Gradual transition from one state to the other is observed along the domain boundary. As the chain extends, the size of the weakly stretched part in the center monotonously decreases and vanishes at ∆l = 0.2056. At even higher tension, the zigzag is stretched uniformly. The two-phase stretching scenario persists for as long as 0 ≤ ǫ φ ≤ 0.015. The range of extensions where the scenario is realized gradually shrinks as the contribution of the valent bond poten- Appendix C: Alpha-helix Consider a 3D molecular chain corresponding to an ideal [1] α-helix. The equilibrium atomic helix coordinates: Appendix C: Alpha-helix with n = 0, ±1, ±2, ... being the atom number, R 0 -helical radius, ϕ 0 and l 0 -the angular and longitudinal helix period. For the sake of simplicity we consider dimensionless model of the helix [see Fig. 4 (a) ], where the (dimensionless) helix radius is R 0 = 0.4919, angular step is ϕ 0 = 100 Appendix C: Alpha-helix • , longitudinal step is l 0 = 0.6572 [25] . Such a helix can be treated as a quasi-one-dimensional crystal; that is each site can be obtained from the previous one by the appropriate translation along longitudinal axes and rotation around the same axis. The chain potential energy is The term V (ρ n ) gives the energy of interaction between neighbor sites n and (n + 1), where ρ n is the distance between them. Bond rigidity is K = 10, the equilibrium bond length is ρ 0 = 1. The angle deformation energy is described by ǫ φ U (φ n ), where φ n is the angle between sites (n − 1), n, and (n + 1) (the vertex is on site n). Equilibrium angle is φ 0 = arccos(−1/3) = 109.47 The chain potential energy is • , coefficient ǫ φ = 1. The specific form of these terms is described in "Methods" section. The third term ǫ θ Z(θ n ) is the energy of the torsional deformation (rotation) around n-th bond. The chain potential energy is where ǫ θ ≥ 0 is the torsional rigidity; different values of ǫ θ are considered while keeping other interactions fixed. The equilibrium torsion angle is θ 0 = arccos(0.2395). The function W (r n ) is the energy of the n-th hydrogen bond connecting sited n and (n + 3). It is given by the formula The chain potential energy is The equilibrium hydrogen bond length is r 0 = 2.0322. Other parameters of the hydrogen bonding potential are: d = 0.7(inner core diameter), ǫ hb = 0.0246, so that rigidity of non-valent interactions W ′′ (r 0 ) = 72ǫ hb /(r 0 − d) 2 = 1. Equilibrium values of angles and distances correspond to specified values of helix radius R 0 , angle step ϕ 0 and longitudinal step l 0 of the helix in ground state. The chain potential energy is To determine the dependence of the effective helix site energy E (per one step) upon the relative uniform extension ∆l, we solve the following conditional minimization problem: The chain potential energy is with the fixed value of longitudinal step l 0 + ∆l. Here R is the helix radius, and ϕ is its angular step, see section "Methods". Conjugate gradient is used to find the minimum. The chain potential energy is In general, to find variable extensions ∆l n of each helix site in the case of non-uniform stretching, we solve: The chain potential energy is with the fixed ends condition: The chain potential energy is Here, as before, N ∆L is the total extension of the chain of N sites. Conjugate gradient is used to find the minimum. The initial condition is chosen to be that of the uniformly stretched helix. The chain potential energy is Appendix D: dsDNA Model details The 12CG model of the DNA double-helix used in this work is shown in Figs. 15 and 16 . To provide additional information we switch to atomic units. Model details The total potential energy of the system has the following form: Model details The terms in the brackets describe the deformation energy of both strands. In the main part, a short-hand notation was used to avoid unnecessary details. For example, the energy E A of strand "A", which appears in the main part, equals the bracketed terms above in which only the untied atoms from strand "A" are retained. The terms E v , E a , E t correspond to valent bond, angle and torsion deformation energy respectively. These potentials have a common form: bond deformation energy is calculated as Model details with the rigidity coefficients K αβ and equilibrium values R αβ being different for different type grains. Angle deformation energy has the form Model details where values of the coefficient ǫ a and the equilibrium angle differ for different angle types. Torsion deformation energy is described by the potential Model details where values of the coefficient ǫ t and the equilibrium torsion angle θ 0 differ for different torsions. Rotational axis of the torsional potential are shown in Fig. 16 base is a rather rigid chemical structure modeled here by rigid harmonic potentials which keep four points near one plain: grain C1 and the three points on each nucleobase. Model details The next two terms E el , E vdW describe electrostatic and van der Waals interactions between backbone grains. Solvent is treated implicitly, via the Generalized Born (GB) model [39, 40] . The methodology has been used to model free DNA in solution [41, 42] , binding between proteins and nucleic acids [43] [44] [45] , conformational changes such as the A → B transition [38] , as well as for exploring dynamics of long DNA fragments [46] . The GB model approximates solvation energy of two interacting charges by the following formula originally proposed by Still et al. [39] Model details where ǫ out is the dielectric constant of water, r ij is the distance between atoms i and j, q i is the partial charge of atom i, R i is the so-called effective Born radius of atom i, and Model details The empirical function f is designed to interpolate between the limits of large r ij ≫ R i R j where the Coulomb law applies, and the opposite limit where the two atomic spheres fuse into one, restoring the famous Born formula for solvation energy of a single ion. The effective Born radius of an atom represents its degree of burial within the low dielectric interior of the molecule: the further away is the atom from the solvent, the larger is its effective radius. In our model, we assume constant effective Born radii which we calculate once from the first principles [40] for the B-form DNA. The screening effects of monovalent salt are introduced approximately, at the Debye-Huckel level by substitution Model details The 0.73 pre-factor was found empirically to give the best agreement with the numerical Poisson-Boltzmann (PB) treatment [47] . Here κ is the Debye-Huckel screening parame- Model details Implementation details in the context of the 12CG DNA coarse-grained model can be found in Ref. [27] . Model details The last two terms E * hb and E st in Eq. (D1) describe interactions between nitrogen bases (including stacking and hydrogen bonds). Since nitrogen base is a rather rigid structure, we can calculate coordinates of all the original atoms (corresponding to the all-atom representation) from positions of the three united atoms. This allows us to utilize, directly, the all-atom AMBER [36] Coulomb and van der Waals potentials used to mimic hydrogen bonds and stacking [27] . Model details Assuming no sequence variability along the strand, e.g. poly(A)-poly(T), such double helix can be considered as a quasi-one-dimensional crystal with the elementary cell being one nucleotide pair of the double helix. In the ground (minimum energy) state each successive nucleotide pair is obtained from its predecessor by translation along the z-axis by step l followed by a rotation around the same axis through helical step ∆φ. (Here we use l instead of ∆l = l − l 0 , because, unlike in the case of a "simple" structure such as the 2D zigzag, the equilibrium value of the DNA longitudinal step l 0 in our model is not known a priori, and is obtained by solving the corresponding minimization problem.) Thus, the energy of the ground state is a function of 38 variables: 36 Cartesian coordinates of 12 grains in the first nucleotide pair, and ∆φ, l. Model details To find the minimum energy (ground) state of the homogeneous (that is no sequence variability along the strand) double helix under tension, we solve the following minimization problem over 37 variables Model details under the fixed value of the longitudinal step l. The summation is taken over only one base pair and neighboring base pairs are obtained from it by rotation and translation. Conjugate gradient is used to find the minimum; the initial condition corresponds to all-atom B-form DNA. The minimization yields the effective site energy (per one base pair) E(l) as a function of the longitudinal step l -see Fig. 19 . The energy minimum is reached for the longitudinal step l 0 = 3.352Å, which corresponds to the B-form of dsDNA. Model details Since the effective site energy is non-convex, we predict two-phase stretching for dsDNA (see Figs. 17, 18 ) based on the general mechanism described in the main part. The DNA structure and its model potentials are most complex among the three polymer models analyzed in this work. We have therefore chosen the DNA to further test our general predictions through molecular dynamics simulations at 300K (for the 2D zig-zag and the alpha-helix only purely mechanical stretching without thermal fluctuations was considered). Origins of the non-convex shape of the effective site deformation for double-stranded DNA Variation of the base stacking and hydrogen bond components of the effective site deformation energy as a function of the chain extension for the double-helix DNA is shown in Fig. 19 (a) . In this computation, thermal fluctuations are not considered. As the tension grows, the base stacking weakens, the corresponding energy
{'paper_id': '10231306', 'title': 'Two-phase stretching of molecular chains', 's2_url': 'https://api.semanticscholar.org/CorpusID:10231306', 'source': 'S2ORC'}