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Tension-induced binding of semiflexible biopolymers
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2014 New J. Phys. 16 113037
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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.
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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
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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.
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