Home Search Collections Journals About Contact us My IOPscience Tension-induced binding of semiflexible biopolymers This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 New J. Phys. 16 113037 (http://iopscience.iop.org/1367-2630/16/11/113037) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 148.251.237.47 This content was downloaded on 06/02/2015 at 05:47 Please note that terms and conditions apply. Tension-induced binding of semiﬂexible biopolymers Panayotis Benetatos1, Alice von der Heydt2 and Annette Zippelius2,3 1 Department of Physics, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 702-701, Korea 2 Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 3 Max-Planck-Institut für Dynamik und Selbstorganisation, Am Faßberg 17, D-37077 Göttingen, Germany E-mail: pben@knu.ac.kr Received 21 May 2014, revised 29 September 2014 Accepted for publication 1 October 2014 Published 14 November 2014 New Journal of Physics 16 (2014) 113037 doi:10.1088/1367-2630/16/11/113037 Abstract We investigate theoretically the effect of polymer tension on the collective behavior of reversibly binding cross-links. For this purpose, we employ a model of two weakly bending wormlike chains aligned in parallel by a tensile force, with a sequence of inter-chain binding sites regularly spaced along the contours. Reversible cross-links attach and detach at the sites with an afﬁnity controlled by a chemical potential. In a mean-ﬁeld approach, we calculate the free energy of the system and ﬁnd the emergence of a free-energy barrier which controls the reversible (un)binding. The tension affects the conformational entropy of the chains which competes with the binding energy of the cross-links. This competition gives rise to a sudden increase in the fraction of bound sites as the tension increases. We show that this transition is related to the cross-over between weak and strong localization of a directed polymer in a pinning potential. The cross-over to the strongly bound state can be interpreted as a mechanism for force-stiffening which exceeds the capabilities of single-chain elasticity and thus available only to reversibly cross-linked polymers. Keywords: semiﬂexible polymers, phase transitions, cytoskeleton, theory Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 16 (2014) 113037 1367-2630/14/113037+12$33.00 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft New J. Phys. 16 (2014) 113037 P Benetatos et al 1. Introduction Cells and tissues are sensitive and responsive to mechanical forces. The constituent ﬁlament networks are able to adapt in a differentiated manner to a variety of strain or ﬂow conditions, substrates, and biological functions [1, 2]. Moreover, many cellular processes such as motion, adhesion, mitosis, and stress relaxation require a reorganization of the cytoskeleton. This remodeling can be achieved by transient or reversible cross-linkers that bind and unbind stochastically with characteristic on- and off-rates. Reversible unbinding provides biopolymer networks with subtle relaxation mechanisms and ultimately even allows them to ﬂow [3–5]. According to a model for the dynamics of the transient network [4], already a single time scale of cross-link unbinding gives rise to a broad spectrum of relaxation times. Experiments prove that thermal unbinding of cross-linkers affects substantially the viscoelastic properties of actin networks, and may result in cross-link-induced stiffening [6, 7]. Cytoskeletal ﬁlaments organize in branched or bundled structures with markedly different properties [8]. The stress ﬁbers are reversibly cross-linked actin bundles extending between focal adhesions which are the sites where cells adhere to the extracellular matrix. They have been the subject of signiﬁcant experimental and modeling activity in recent years [9–16]. Apart from thermal unbinding, which in vitro can be controlled by temperature, cross-link (un)binding prompted by mechanical forces is a potentially crucial mechanism to tune the ﬁber or network rigidity. Substantial effort is made to reproduce mechanical stimuli on cytoskeletal ﬁlaments in the laboratory [17], to unravel, e.g., the signaling route to stretch-induced reinforcement of actin stress ﬁbers [9, 11, 12, 18]. Some of the experiments that highlight the importance of biomechanical signaling in stress ﬁbers involve focused laser irradiation (laser nanoscissor surgery) [9, 15, 19], atomic force microscopy (AFM) probes [10, 16], or substrate stretching [11, 12]. Stretching forces can induce the adhesion of soft membranes (cells, vesicles) to a substrate [20–23], activate biochemical signals in the cytoskeleton [24], and even change gene expression [25]. For highly cross-linked microtubules, microrheological experiments have identiﬁed force-induced cross-link unbinding as the dominant contribution to viscoelasticity [26]. By studying a model of a reversibly bound actin ﬁlament bundle, Heussinger et al [27, 28] showed that unbinding is a cooperative effect, characterized by a free-energy barrier. This study aims to shed light on a collective path to stress-induced ﬁber (network) stiffening via reversible binding, by means of a minimal model. We analyze two reversibly cross-linked semiﬂexible polymers aligned in parallel by a longitudinal force. For small tensile forces, we expect that most cross-links are unbound, because constraining the transverse ﬂuctuations of the two ﬁlaments costs a substantial amount of entropy. Conversely, in a strongly stretched conﬁguration, many cross-links are expected to be in the bound state, because the cost of entropy is low. These expectations can be made quantitative within mean-ﬁeld theory: the system is found to undergo a discontinuous phase transition from a weakly bound state at low tension to a strongly bound state at high tension. The melting (base-pair opening) transition of double-stranded DNA is also known both to impact the bending ﬂexibility and to be affected by external forces [29]. Marenduzzo et al [30] consider DNA melting under stretching and determine the critical force as a function of temperature. In thermal equilibrium, so-called bubbles are present and a central issue of theoretical studies [31]. While our model bears some resemblance to and might be relevant for this transition, one should keep in mind that DNA is much more ﬂexible than the cytoskeletal 2 New J. Phys. 16 (2014) 113037 P Benetatos et al Figure 1. Model sketch for 11 reversible inter-chain cross-links. ﬁlaments, hence usually modeled as a self-avoiding random walk. Furthermore, torsion is believed to be essential for double-stranded DNA. This paper is organized as follows. In section 2, we introduce the model of two parallelstretched semiﬂexible chains with regularly spaced, reversible cross-links. In section 3, we introduce the average fraction of bound cross-links as an order parameter and calculate the free energy of our system in mean-ﬁeld theory. The analysis of the free energy yields a mean-ﬁeld ﬁrst-order transition from a weakly bound to a strongly bound state as the stretching force or the binding afﬁnity (controlled by the chemical potential) increases. In section 4, we discuss the cross-over between weak and strong localization of a directed polymer in a conﬁning transverse potential well, a system behaving similarly to that of the two cross-linked chains. Final remarks and conclusions are given in section 5. 2. Model The basis for our model is two identical semiﬂexible (locally inextensible) polymers which can reversibly bind to each other at equally spaced contour positions. Both polymers have contour length L and are aligned parallel along a given direction x by a tensile force f, see ﬁgure 1. For simplicity, we consider two spatial dimensions and use the weakly-bending approximation [32]. This model is a direct generalization of that with permanent cross-links employed in [33]. The effective Hamiltonian (elastic-energy functional) is given in terms of the transverse displacements {y j (s )}, s ∈ [0, L ], j = 1,2, by 2⎞ ⎛κ 2 2 f ∂s yj + ∂sy j ⎟ − 2fL ds ⎜ ⎝2 ⎠ 2 j=1 2 = ∑ ∫0 L ( ) ( ) 0 + g 2 N−1 ∑ n b ( y1 (bd ) − y2 (bd ) ) 2 , (1) b=1 where κ is the bending rigidity, 0 is the Hamiltonian of the two polymers without cross-links, and the last term accounts for the cross-links at contour sites bd of spacing d = L N . If cross-link b is bound, it acts as an harmonic spring of strength g and the binary variable nb assumes the value 1, otherwise nb = 0. We assume hinged-hinged boundary conditions, which suppress transverse displacements and curvatures at the end points, y j (0) = y j (L ) = 0 and ∂ 2s y j (0) = ∂ 2s y j (L ) = 0, for j = 1,2. In addition, we impose a no-slip condition, x1 (0) = x 2 (0). The total number of cross-links ﬂuctuates controlled by a chemical potential μ, so that the grand-canonical partition function reads 3 New J. Phys. 16 (2014) 113037 P Benetatos et al N−1 = ∫ [y1, y2 ] ∏ b=1 ⎛ 1 ⎞ ⎜ ∑ ⎟ e−β ⎜ ⎟ ⎝ nb =0 ⎠ ( − μ ∑ N−1 b=1 nb ), (2) where the functional integral ∫ [y1, y2 ] comprises all polymer conformations consistent with the boundary conditions, and β := 1 (k B T ). 3. Mean-ﬁeld treatment of many reversible cross-links Following a mean-ﬁeld approach analogous to [27], we replace the individual cross-link degrees of freedom, nb, with their average value n= N−1 N˜ 1 = ∑ nb, N−1 N − 1 b=1 (3) N˜ = n (N − 1) denoting the total number of bound cross-links. This way, we ﬁnd for the grandcanonical partition function in mean-ﬁeld approximation, relative to that of the polymers without binding sites, N−1 mf = ⎛ ⎞ ∑ ⎜⎝ N N−˜ 1⎟⎠ exp (βμN˜ ) N − 1 (ng). (4) N˜ = 0 Here, N − 1 (ng) is the relative canonical partition function of two weakly bending chains with exactly N − 1 irreversible crosslinks of effective strength ng (substitute n b → n in equation (1)). This partition function has been computed exactly in [33] and is given by N−1 N − 1 (ng ) = ∏ Z l (n), l=1 ⎧ ngd ψl (0) − ψl δ f Z l (n) = ⎨1 + f ⎩ sinh δ f ψl δ f = , δ f cosh δ f − cos φl ( ( ) ( )) ( ⎫−1 2 ⎬ , ⎭ ) (5) with the cross-link spacing d = L/N, the force parameter δf = d f κ, (6) and φl = πl N . Using the Stirling approximation for N ≫ N˜ ≫ 1, we obtain for the mean-ﬁeld free energy per cross-link site, (N − 1) βGmf := −ln mf βGmf = n ln n + (1 − n) ln (1 − n) − βμn + 1 2(N − 1) N−1 ∑ ln Z l (n). (7) l=1 The equilibrium value of n, which acts as the order parameter for the binding-unbinding transition, is the one that minimizes the free energy Gmf at ﬁxed values of the other parameters of our model. These are: the effective force strength δ f2 = d 2f κ , the effective cross-link 4 New J. Phys. 16 (2014) 113037 P Benetatos et al Figure 2. Mean-ﬁeld free energy Gmf (n ) for very small n; parameters are chosen as N = 20 (19 possible binding sites), βμ = −0.1, cross-link strength w = 10 4 and three different force parameters δf ; a minimum for very small n, corresponding to a weakly bound state, exists for all δf. Figure 3. Free energy Gmf (n ); the second minimum becomes globally stable at δ f , c = 76.0; parameters as in ﬁgure 2. strength 2w = gd 3 κ (in equation (5), gd f = 2w δ f2), the chemical potential βμ; the systems under consideration are additionally parameterized by the number of cross-link sites, N − 1. At least for ﬁnite N, the behavior of Gmf (n ) for very small n is dominated by the entropic contribution n ln (n ), implying a negative slope of Gmf (n ) for very small n independently of the choice of the other parameters, and hence a weakly bound state with a very small fraction of bound cross-links. A blow-up of this region is shown in ﬁgure 2, where n m* , c denotes the bound cross-link fraction at this free-energy minimum, whose value itself is extremely close to (below) zero. Whether or not a markedly bound state with a substantial fraction of bound cross-links exists in mean-ﬁeld theory, depends on the parameters of our model. In ﬁgure 3, we show βGmf (n ) over a larger range of n-values, revealing the formation of a second minimum of Gmf at a value of n considerably larger than zero. We can locate the transition to the markedly or strongly bound state by computing for each given set of parameters the fraction n* > n m* , c which minimizes Gmf (with Gmf (n*) ⩾ 0 ), and varying the parameters until n* = nc* 5 New J. Phys. 16 (2014) 113037 P Benetatos et al Figure 4. Phase diagram in the plane of effective force and cross-link strength; symbols: ﬁrst-order transition data obtained from mean-ﬁeld theory; dashed line: ﬁt with scaling relation derived from model of directed polymer in a pinning potential (section 4); the chosen parameters are N = 300 and βμ = −0.1. Figure 5. Phase diagram in the plane of effective force and chemical potential; symbols: ﬁrst-order transition obtained from mean-ﬁeld theory; dashed line: ﬁt with scaling relation from model of directed polymer in a pinning potential (section 4); parameters are N = 300 and w = 1000. additionally fulﬁls Gmf (nc* ) = 0 . Thus, the transition is ﬁrst-order in mean-ﬁeld theory with a discontinuous jump in the order parameter from a very small to a sensibly large value between zero and one. Locating the binding-unbinding transition allows us to map out the phase diagram in several parameter planes, e.g., in the plane spanned by force and cross-link strength, or in the plane of force and chemical potential. The cross-link strength against force parameter at the transition is shown in ﬁgure 4 (symbols indicate data obtained by minimization of Gmf ). For all parameter values, we clearly observe a sharp transition between a weakly bound and a strongly bound state. As the force (or δf) is increased, the bound state is increasingly preferred, because the entropy loss due to cross-linking is reduced. Increasing g (or w) implies a decrease in the range of the cross-link potential, hence a stronger localization and a more pronounced reduction of entropy for each bound cross-link, giving rise to a larger region of weakly bound states. Instead of increasing the force f, we can alternatively increase the chemical potential βμ (the afﬁnity for binding) to control the transition from the weakly bound to the strongly bound state. The phase diagram in the plane of f and βμ is shown in ﬁgure 5. As expected, the transition 6 New J. Phys. 16 (2014) 113037 P Benetatos et al Figure 6. Fraction of bound cross-links as a function of force for parameters N = 500, βμ = −0.1 and w = 10 4 . occurs at smaller f, if the afﬁnity for cross-link binding is large. Furthermore, the fraction nc* of bound cross-links increases monotonically with βμ as indicated by two values in ﬁgure 5. Beyond these qualitative considerations, it is actually possible to derive scaling relations for the transition parameters, the ﬁts to which are represented as dashed lines in ﬁgures 4 and 5. The scaling relations will be derived in section 4. Eventually, ﬁgure 6 visualizes the jump of the order parameter, the mean-ﬁeld fraction of bound cross-links, as a function of stretching force (the weakly bound state cannot be resolved on this scale). 4. Directed polymer in a transverse potential well In this section, we shall discuss a related model of a single directed polymer in a continuous well potential. First, we give a plausibility argument, why this model should be comparable to our system of two reversibly cross-linked polymers. Then, we brieﬂy review the main features of the model, which has been analyzed [34] in the context of magnetic ﬂux lines in a high-Tc superconductor being pinned by columnar defects. Transferring these results to our model allows us to derive and test scaling predictions for the transition lines in the phase diagram (section 4.1). First, we observe that reversible cross-links provide an effective, short-range attractive potential for the two connected polymers. This is most easily seen by considering a single crosslink site. Integrating out the cross-link degree of freedom, we obtain an effective potential [35] as a function of the inter-polymer distance Δy = y1 (L 2) − y2 (L 2) at the cross-link site, ⎡ ⎧ ⎛ g ⎞⎫⎤ βVeff (Δy) = −ln ⎢ 1 + exp ⎨ β ⎜ μ − Δy 2⎟ ⎬ ⎥ , ⎠⎭⎦ ⎩ ⎝ ⎣ 2 (8) shown in ﬁgure 7 for two representative values of βμ. Assuming μ < 0 , the effective potential is approximated by an inverted Gaussian potential ⎛ Δy 2 ⎞ βVeff (Δy) ≈ −e βμ exp ⎜ − 2 ⎟ , ⎝ ac ⎠ (9) 7 New J. Phys. 16 (2014) 113037 P Benetatos et al Figure 7. Effective cross-link potential as a function of the inter-polymer distance Δy. whose range is ac = 2k B T g and whose amplitude or depth is U0 = k B T exp(βμ) (in general, the depth is k B T ln (1 + e βμ)). To further ease the setup of the analogy with a ﬂexible directed polymer in a square well, one may approximate the effective potential by Veff (Δy ) = −U0 for −ac < Δy < ac and Veff (Δy ) = 0 otherwise. Furthermore, for many equally spaced cross-link sites, the effective potential acts approximately continuously along the polymer length—like the potential well extending continuously in the direction of the tensile force. Finally, we assume that the long wavelength transverse excitations are dominated by the second term of 0 in equation (1), so that we can treat one of the polymers as a ﬂexible, directed polymer in a short-range potential well that constrains the transverse displacements. This assumption holds in the strong-stretching limit, for which the ‘memory’ length l m := κ f is much smaller than the site spacing d, l m ≪ d . In this limit, the off-boundary, transverse mean-square ﬂuctuations of a stretched semiﬂexible polymer are known to scale as ∼k B TL f , i.e., to become independent of the bending rigidity κ [36]. The partition function of a ﬂexible directed chain, subject to a tensile force with boundary condition y (0) = y (L ) = 0 and to a well potential V(y) continuous in tension direction, relative to the free (noninteracting) chain, is given by the path integral ⎡ ⎣ y (L ) = 0 rel = y (L ) = 0 ∫y (0) = 0 L dy 2 ds L ⎤ ( ) ds − β ∫ V [y (s) ] ds⎥⎦ . ⎡ ⎤ y (s ) exp ⎢ − ∫ ( ) ds ⎥ ⎣ ⎦ f ∫y (0) = 0 y (s) exp ⎢− β2 ∫0 βf 2 0 L 0 dy 2 ds (10) When the polymer is conﬁned within the well, there is a competition between energy gain and entropy loss, the latter depending on the tensile force on the chain. We notice that the path integrals in the numerator and the denominator correspond to density matrix elements of a ﬁctitious quantum particle in a potential V(y) and a free one, respectively [37]. The mapping is as follows: f ↔ m , β ↔ −1, s ↔ t , L ↔ β˜ , where m, t, β˜ are mass, imaginary time, and inverse temperature parameter (1 k B T˜ ) for the ﬁctitious quantum particle. The thermodynamic limit (L → ∞) for the polymer corresponds to the zero temperature limit ( β˜ → ∞) for the ﬁctitious quantum particle. In this limit, the density matrix of a quantum particle is dominated by the ground state: 8 New J. Phys. 16 (2014) 113037 ( P Benetatos et al ) ( ) ρ y, y′; β˜ ≈ ψ0 (y) ψ0* (y′) exp −β˜ E0 , (11) where ψ0 (y ) is the ground-state eigenfunction and E0 the ground-state energy. Using the aforementioned correspondence, we obtain: rel ⎛ 2πLk B T ⎞1 2 =⎜ ⎟ ψ0 (0) ψ0 (0) exp (−βLE0 ), f ⎝ ⎠ (12) where the prefactor arises from the density matrix of the free particle. In the limit L → ∞, the binding free energy per unit length can be extracted from equation (12) using rel = exp(−βLG ), and we obtain G = E0. We point out that in (1 + 1) dimensions, as well as in (1 + 2) dimensions, the ﬁctitious quantum particle always has a bound state, implying that the corresponding directed polymer is always bound, in agreement with our data set presented in ﬁgure 2. However, there is a clear cross-over between a strongly bound and a weakly bound state, and it is this cross-over which is captured by the mean-ﬁeld analysis. For a strong stretching force, the entropic contribution to the free energy is small compared to the energy of the potential well which can be treated as inﬁnitely deep. In this case, the corresponding quantum problem is that of a particle in an inﬁnitely deep well. From elementary quantum mechanics [38], we obtain the strong-conﬁnement free energy per unit length 2 π2 ( kBT) Gs = −U0 + . 4 2fa c2 (13) The second term in the rhs of the above equation is the entropy loss due to the conﬁnement in the effective potential, which in the strong-stretching case is much smaller than the ﬁrst term. In effect, the particle is localized within the width of the effective potential well, ac. In the opposite limit of weak localization, the stretching force is smaller and allows the polymer to perform ﬂuctuations far beyond the range of the potential well, which can be treated as a delta-function: V1 (y ) = −U0 2ac δ (y ). Using the ground state of the corresponding quantum problem, the weak-conﬁnement free energy per unit length becomes Gw = − f 2( kb T ) 2 (U02a c)2 . (14) In this case, it is known that the wave function of the ﬁctitious quantum particle is ∝ exp(−| y | l⊥ ), with l⊥ = (k BT )2 (fU0 2ac ). Hence, the crossover between strong and weak binding occurs at a force fc such that ( k B T )2 2fc a c2 = U0. (15) 4.1. Scaling predictions for our system Exploiting the analogy of our system with a directed polymer in a square well, we can extract from equation (15) scaling relations between the parameters at the binding-unbinding transition (for transparency, we omit the label c here). The force f (∝ δ f2 at constant site spacing d, 9 New J. Phys. 16 (2014) 113037 P Benetatos et al bending rigidity κ, and β) is expected to increase linearly with the inverse squared potential width 1 ac2 (∝ w at constant d, κ, and β), i.e. δ f2 ∝ w. (16) Inspection of ﬁgure 5 shows that the computed transition obeys this scaling relation with a high precision. In fact, this scaling should be inherent to our model, since the cross-link strength enters the free energy, equation (7), only via the ratio w δ f2 . On the basis of ﬁgure 5, we test the expected linear scaling of the force with the inverse potential depth 1 U0 at the transition, see the parameter mapping following equation (8), viz −1 δ f2 ∝ (βU0 ) −1 = ⎡⎣ln 1 + e βμ ⎤⎦ . ( ) (17) We notice that the scaling of force with potential depth is reasonably good, but not as convincing as that with potential width. One reason is that the points which are markedly off the scaling curve correspond to small stretching forces, for which the analogy with the (ﬂexible) directed polymer breaks down. Another reason appears to be that particularly for positive βμ, identifying the square-well depth with the minimum of the parabola-shaped, effective potential, see ﬁgure 7, is a less accurate approximation. 5. Conclusions We have studied two reversibly cross-linked, semiﬂexible ﬁlaments under tension and shown that the two ﬁlaments are always in a (weakly) bound state. Within mean-ﬁeld theory, there is a discontinuous phase transition as a function of applied stretching force from a weakly bound state at small force to a strongly bound state at large force. The critical force as a function of either cross-link strength or chemical potential displays a scaling, which can be derived from a model of a single polymer in a conﬁning potential. Our analysis has been restricted to two dimensions, but neglecting entanglement effects and twist, our results can be generalized to (1 + 2) dimensions, since in the weakly-bending approximation, the two transverse directions decouple. The sharp transition obtained in mean-ﬁeld theory is expected to be replaced by a crossover, if ﬂuctuations are taken into account—similar to the cross-over observed in the model of a single stretched polymer in a conﬁning potential. Nonetheless, the tension-induced binding cross-over from a weakly to a strongly bound state predicted by our model can be interpreted as a versatile microscopic mechanism of force-stiffening. It provides an extra contribution to supramolecular nonlinear elasticity insofar as reversibly binding polymers are enabled to tune their mechanical properties beyond those of a single semiﬂexible polymer: a larger tensile force causes more cross-links to bind, and a larger number of bound cross-links results in a larger tensile stiffness of the system [33]. Thereby, the polymer pair responds to an increasing external force with a larger modulus to diminish the effect of this force. The main experimental signature that arises from our theoretical results would be a cross-link regulated tension stiffening of semiﬂexible ﬁlament bundles accompanied by an increase in the number of bound cross-links. In the model of [3], softening of the cytoskeletal network due to cross-link unbinding (due to bending) competes with inherent stiffening due to a single ﬁlamentʼs or cross-linkʼs nonlinear elasticity. It is an interesting open question to be clariﬁed further by experiments whether cells 10 New J. Phys. 16 (2014) 113037 P Benetatos et al make use of the stiffening predicted by our model to adapt the mechanical properties of the cytoskeleton to the temporary task or substrate. The described mechanism might also give clues to the formation and stress-induced strengthening of stress ﬁbers. It is known that stretching induces mobilization of the protein zyxin from focal adhesions to actin ﬁlaments and the stress ﬁbers become thicker in a zyxindependent manner [12]. It is worth investigating whether this stress-ﬁber strengthening is actually a collective binding of reversible cross-links similar to that described in our model. Single stress-ﬁber experiments similar to those described in [14] might be useful in this direction. This study could be extended in several directions, some of which are work in progress. First, it would be desirable to ﬁnd a tractable description to compute not only the mean-ﬁeld fraction of bound cross-links, but also the spatial correlation of bound cross-link sites. Longitudinal tension and boundary conditions imply that transient cross-links are more likely to bind close to the polymer ends, an effect which might weaken the binding transition we spotted here. Second, the thermodynamic limit of continuous binding sites and its impact on the crossover remains to be analyzed further. An obvious step following the observation of the binding cross-over would be to compute the force-extension relation for transient binding and compare it to the irreversibly cross-linked case [33]. The stretching elasticity of bundles of many parallelaligned semiﬂexible polymers [39, 40] with reversible cross-links is another challenging problem. Finally, semiﬂexible polymers with more complicated binding modes, such as crosslinks aligning (welding) the corresponding ﬁlament segments in the stretching direction [41, 42] or nonlocal binding [43], should be investigated in the case of reversible cross-links. Acknowledgments We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) via grant SFB-937/A1. Also, we thank Claus Heussinger for useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] Pritchard R H, Huang Y Y S and Terentjev E M 2014 Soft Matter 10 1864 Picu R C 2011 Soft Matter 7 6768 Heussinger C 2012 New J. Phys. 14 095029 Broedersz C P, Depken M, Yao N Y, Pollak M R, Weitz D A and MacKintosh F C 2010 Phys. Rev. Lett. 105 238101 Åström J A, Kumar P B S, Vattulainen I and Karttunen M 2008 Phys. Rev. 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