# Motor-free contractility of active biopolymer networks

Sihan Chen1,2, Tomer Markovich2, Fred C. MacKintosh1,2,3,4 1Department of
Physics and Astronomy, Rice University, Houston, TX 77005
2Center for Theoretical Biological Physics, Rice University, Houston, TX 77005
3Department of Chemical and Biomolecular Engineering, Rice University,
Houston, TX 77005
4Department of Chemistry, Rice University, Houston, TX 77005

###### Abstract

Contractility in animal cells is often generated by molecular motors such as
myosin, which require polar substrates for their function. Motivated by recent
experimental evidence of motor-independent contractility, we propose a robust
motor-free mechanism that can generate contraction in biopolymer networks
without the need for substrate polarity. We show that contractility is a
natural consequence of active binding/unbinding of crosslinkers that breaks
the _Principle of Detailed Balance_ , together with the asymmetric force-
extension response of semiflexible biopolymers. We calculate the resulting
contractile velocity using both a simple coarse-grained model and a more
detailed microscopic model for a viscoelastic biopolymer network. Our model
may provide an explanation of recent reports of motor-independent
contractility in cells. Our results also suggest a mechanism for generating
contractile forces in synthetic active materials.

##  I Introduction

Living cells are known to be far from equilibrium. Powered by metabolic
components such as adenosine triphosphate (ATP), biophysical processes that
cannot happen in equilibrium systems take place in living cells, including
cell signaling Hernández-Lemus (2012), genetic transcription and replication
Murugan  _et al._ (2012) and active force generation Alberts  _et al._ (2017).
Most force generation in living cells is due to molecular motors, for example
myosin motors in animal cells Fletcher and Mullins (2010). These motors
perform directional motion on the substrate they bind to, thus generating
force in various forms, from muscle contraction Huxley (1969) to internal
stress of the cytoskeleton Ingber (2003); Marchetti  _et al._ (2013);
Markovich  _et al._ (2019). During cell division, motors are also believed to
be responsible for the contractile stress generated by the actomyosin ring
Mabuchi and Okuno (1977); Guo and Kemphues (1996); Straight  _et al._ (2003);
Glotzer (2005).

Recent experimental evidence, however, suggests that contractility may be
generated in the absence of motors Wloka  _et al._ (2013); Xue and Sokac
(2016). In Ref. Wloka  _et al._ (2013) it was found that myosin-II motors
become immobilized shortly before cytokinesis in budding yeast. It was also
shown that the contractility of the actomyosin ring in Drosophila embryo
remains unaffected even when motor activity is significantly inhibited Xue and
Sokac (2016). Both experiments imply the existence of an underlying
contractile mechanism other than molecular motors.

Figure 1: (a) Illustration of the coarse-grained model. One polymer segment
from a disordered network undergoes binding/unbinding process attaining a
steady-state length distribution, $P_{\rm on}$. The bound and unbound states
are denoted by dark and light blue, respectively. (b) Schematic plot (blue
line) of the equilibrium distribution of a semiflexible polymer end-to-end
length, $P_{\rm eq}$, and a sketch of a possible $P_{\rm on}$ (purple dashed
line). Activity usually broadens $P_{\rm on}$ as compared to $P_{\rm eq}$. (c)
Force-extension relation for inextensible semiflexible polymer
($\tau_{0}/\mu\to 0$ in Eq. (3)) and the two-spring PMF in Eq. (4) with
$K_{1}=2\tau_{0}/3\langle\Delta\ell\rangle$ and $K_{2}=10K_{1}$. Both PMFs
show strong asymmetry, being potential sources of broken spatial symmetry.

Microscopically, molecular motors work with a mechanism reminiscent of the
Smoluchowski-Feynman thermal ratchet Feynman (1963); Magnasco (1993); Prost
_et al._ (1994); Jülicher  _et al._ (1997). By consuming and dissipating
energy provided by metabolic components, motors undergo a directed cycle of
transitions among conformational states Spudich (2001), while actively
binding/unbinding in a way that violates the Principle of Detailed Balance
(DB) Boltzmann (1872); Jülicher  _et al._ (1997); Battle  _et al._ (2016);
Nardini  _et al._ (2017); Markovich  _et al._ (2021). However, to generate
directional motion or force, an energy consuming reaction itself is
insufficient. It also requires a broken spatial symmetry which gives rise to a
specific direction for the motion. This spatial asymmetry is usually due to
the polarity of the substrate such as actin or microtubules, with well-defined
plus and minus ends, to which a motor such as myosin or kinesin couples.
Theoretically, such asymmetry can be modeled as an asymmetric binding
potential Magnasco (1993); Prost  _et al._ (1994); Jülicher  _et al._ (1997).

Another spatial asymmetry present in living cells that has been studied
extensively, is the non-linear elasticity of the semiflexible biopolymers
Storm  _et al._ (2005); Gardel  _et al._ (2006); Mizuno  _et al._ (2007). The
biopolymers that form the cytoskeleton are usually much stiffer to bending
than most synthetic polymers, leading to a competition between the entropy and
the bending energy. As a result, the force-extension relation of the
biopolymers exhibits a strong asymmetry: when under stress, the polymer
stiffness nonlinearly increases up to $10^{3}$ times Bustamante  _et al._
(1994); Marko and Siggia (1994); Gardel  _et al._ (2004); Mizuno  _et al._
(2007); Gardel  _et al._ (2006); Koenderink  _et al._ (2009), a phenomenon
known as stress-stiffening. However, under compression a $\rm pN$ level force
is enough to buckle the polymer, and make its stiffness vanish Broedersz and
MacKintosh (2014). Such asymmetric force-extension relation, which originates
from the thermal fluctuations of bending deformations, exists in most
biopolymers including both polar filaments (e.g. actin and microtubule) and
apolar filaments (e.g. intermediate filaments (IF)). This asymmetry in
contraction vs expansion, is a potential source of broken spatial symmetry.

In Ref. Chen  _et al._ (2020), we have proposed a motor-independent
contractile mechanism based on two key elements: (i) active binding/unbinding
of non-motor crosslinkers which breaks DB and (ii) the asymmetric force-
extension relation of biopolymers that prefers contraction over expansion. We
have developed both a simple coarse-grained model and a detailed microscopic
model to demonstrate the mechanism for a viscous or elastic substrate. In this
paper, we extend our previous work Chen  _et al._ (2020) to include
viscoelastic substrates and discuss the interplay between the
binding/unbinding times and the substrate relaxation time. Our proposed
mechanism generates robust contractility for any viscoelastic substrates and
any asymmetric force-extension. This model may not only provide a basis for
understanding recent reports of myosin-independent contractility in cells, but
also suggests a mechanism that can be used for active force generation in
synthetic materials.

##  II Overview

In this work, we consider an unpolarized, disordered network formed by
crosslinked semiflexible polymers (Fig. 1(a)) in which molecular motors cannot
create contraction using their power stroke Jülicher  _et al._ (1997). For
simplicity, we focus on one particular polymer segment and treat the rest of
the network as a continuum viscoelastic substrate that the particular polymer
can bind (unbind) to (from). In the unbound state, there is no interaction
between the polymer and the substrate, while in the bound state, the polymer
exerts tension on the substrate, $\tau(\ell)=dU_{e}(\ell)/d\ell$, resulting in
an average contractile force

$\displaystyle\langle F_{s}\rangle_{\ell}=\int P_{\rm
on}(\ell)\tau(\ell)\mathop{}\\!\mathrm{d}\ell\,.$ (1)

Here $\ell$ is the polymer end-to-end length, $U_{e}$ is the potential of
mean-force (PMF) of the polymer, and $P_{\rm on}$ is the probability for the
polymer to have a given end-to-end distance $\ell$ in the bound state. If the
binding and unbinding processes are in equilibrium, DB is satisfied and
$P_{\rm on}(\ell)$ is governed by a Boltzmann distribution, leading to
vanishing contractile force. However, when the binding and/or unbinding
process are out of equilibrium, e.g., driven by consumption or catalysis of a
metabolic component such as ATP, $P_{\rm on}$ can be non-Boltzmann. For a
viscous substrate with drag coefficient $\gamma$, the contractile force
$\langle F_{s}\rangle_{\ell}$ creates an average contractile velocity,

$\displaystyle v=\langle F_{s}\rangle_{\ell}/\gamma\,.$ (2)

As discussed in the introduction, in order to have directed motion both time-
reversal symmetry and spatial symmetry have to be broken Feynman (1963);
Magnasco (1993, 1994). To see the manifestation of this principle within our
model it is instructive to view the active process as an additional non-
thermal noise on timescales longer than the binding/unbinding time MacKintosh
and Levine (2008); Gladrow  _et al._ (2016); Mura  _et al._ (2018), which
causes random expansion or contraction of the polymer segment and breaks time-
reversal symmetry. Such athermal noise usually tends to broaden the
distribution $P_{\rm on}$ 111we define the distribution width as the square
root of the variance, see e.g. Fig. 1(b). Notably, this active noise, which
breaks time-reversal symmetry, cannot induce contraction or any directed
motion without having some spatial symmetry broken Feynman (1963). This can be
seen from Eq. (1), where symmetric $\tau$ together with symmetric $P_{\rm on}$
(from symmetry arguments all directions are the same) must lead to vanishing
force. If $\tau$ (also the PMF, since $\tau$ is its derivative) is asymmetric,
and specifically, soft to compression and hard to extension, a symmetric or
nearly symmetric $P_{\rm on}$ naturally leads to a positive contractile force.
This can be understood intuitively: the polymer has equal chances to be
extended or compressed by the active process. It exerts a large contractile
force on the substrate when extended, while exerting a small expanding force
when compressed, which on average leads to contractility.

The PMF of semiflexible polymers has exactly these required properties: it
stiffens under extension and softens under compression, see Fig.1(c). The
force-extension relation for a semi-flexible polymer is [Fig. 1(c)] MacKintosh
_et al._ (1995); Odijk (1995); Wilhelm and Frey (1996); Storm  _et al._
(2005); Broedersz and MacKintosh (2014):

$\displaystyle\frac{\ell(\tau)-\ell_{0}}{\langle\Delta\ell\rangle}=\frac{\ell_{0}}{\langle\Delta\ell\rangle}\frac{\tau}{\mu}+\varepsilon\left(\frac{\tau}{\tau_{0}}\left(1+\frac{\tau}{\mu}\right)\right)\,,$
(3)

where $\tau_{0}\equiv\pi^{2}k_{B}T/(6\langle\Delta\ell\rangle)$ is a
characteristic tension, and
$\langle\Delta\ell\rangle\simeq\ell_{0}^{2}/(6\ell_{p})$ is the mean end-to-
end thermal contraction of a stiff polymer of rest length $\ell_{0}$ and
persistence length $\ell_{p}$. Here, $\mu$ is the enthalpic stretch modulus,
$k_{B}$ is the Boltzmann constant, $T$ is the temperature and
$\varepsilon(\phi)=1-3\frac{\pi\sqrt{\phi}\coth(\pi\sqrt{\phi})-1}{\pi^{2}\phi}$
Broedersz and MacKintosh (2014). As discussed above, the robustness of the
contractility does not depend on the specific form of the asymmetric PMF. To
illustrate this we will also use a simple and analytically tractable two-
spring PMF

$\begin{aligned} \tau(\ell)=\left\\{\begin{aligned}
K_{1}(\ell-\ell_{0})\qquad(\ell<\ell_{0})\\\
K_{2}(\ell-\ell_{0})\qquad(\ell\geq\ell_{0})\end{aligned}\right.\end{aligned}\,,$
(4)

where $K_{1}<K_{2}$ are two different spring constants under compression and
extension.

The paper is organized as follows. We first detail our calculation of the
contractile velocity for the coarse-grained model with a viscous substrate
(Sec. III), which has been discussed in our previous work Chen  _et al._
(2020). We then introduce a microscopic model which accounts for the details
of the binding/unbinding process, and discuss the contractile velocity for a
substrate characterized by the Maxwell model (Sec. IV). We show that, a
viscoelastic substrate is equivalent to a viscous substrate with a modified
binding potential (Sec. IV). Next, we discuss the conditions in which our
microscopic model reduces to the coarse-grained one (see Sec. IV). In Sec. V
we draw our conclusions.

##  III A Simple Coarse-grained Model

We begin with a simple coarse-grained model to demonstrate how contractility
can originate in an apolar substrate. As explained above, we consider a single
polymer that binds to/unbinds from the substrate with rates $\omega_{\rm
on}$/$2\omega_{\rm off}$ (the factor of $2$ accounts for the fact that both of
the two polymer ends can unbind). When DB is obeyed the binding and unbinding
rates are related by $\omega_{\rm on}=2\omega_{\rm off}\exp(-U_{b})$, where
$U_{b}$ is the binding potential which is usually rugged Nishizaka  _et al._
(1995); Jülicher  _et al._ (1997). The rugged binding potential suggests that
in DB $\omega_{\rm on}$ and/or $\omega_{\rm off}$ should vary according to the
binding and unbinding position. Here we break DB in the simplest way by
assuming that both $\omega_{\rm on}$ and $\omega_{\rm off}$ are constants
Jülicher  _et al._ (1997). To start, consider a single binding/unbinding event
that is divided into the following steps: (i) in the unbound state the polymer
length $\ell_{u}$ is assumed to relax fast to an equilibrium distribution with
rest length $\ell_{0}$, $P_{\rm
eq}(\ell_{u})=\exp[-U_{e}(\ell_{u})/k_{B}T]/Z$, where
$Z=\int\mathop{}\\!\mathrm{d}\ell_{u}\exp[-U_{e}(\ell_{u})/k_{B}T]$ is the
partition function (Fig. 1(b)), (ii) the polymer binds to the substrate at
rate $\omega_{\rm on}$; at the same time, its initial end-to-end length
changes from $\ell_{u}$ to $\ell_{b}$ due to the binding, with probability
$P_{b}(\ell_{b})$, (iii) once the polymer binds, it contracts the viscous
substrate due to its PMF, $U_{e}$, and (iv) the polymer actively unbinds at
constant rate $2\omega_{\rm off}$. Here, we neglect thermal aspects of
unbinding that we assume to be dominated by active processes.

In this coarse-grained model, the details of the substrate binding potential
are neglected. The effect of the binding process is modeled as an immediate
change of the polymer length from $\ell_{u}$ (before binding) to $\ell_{b}$
(after binding). Such a change in length is a result of the rugged binding
potential of biopolymers Nishizaka  _et al._ (1995); Jülicher  _et al._
(1997). The probability that the polymer length has changed from $\ell_{u}$ to
$\ell_{b}$ due to binding is denoted by $P_{c}(\ell_{b},\ell_{u})$, such that
the binding probability (the polymer length distribution just after binding to
the substrate) is:

$\displaystyle P_{b}(\ell_{b})=\int\mathop{}\\!\mathrm{d}\ell_{u}P_{\rm
eq}(\ell_{u})P_{c}(\ell_{b},\ell_{u})\,.$ (5)

In general the distribution $P_{c}$ depends on the form of both the binding
potential and the elastic PMF. In the limit where the binding potential is
much larger than PMF, $P_{c}$ is dominated by the substrate binding potential.
Then, for a homogeneous substrate $P_{c}$ is only a function of the length
change (translational symmetry) Chen  _et al._ (2020),
$P_{c}(\ell_{u},\ell_{b})=P_{c}(\ell_{u}-\ell_{b})$. Further assuming a
symmetric binding potential, which is the case for e.g. isotropic substrate or
a substrate consisting of apolar filaments, we find $P_{c}$ is symmetric
around $\ell_{u}=\ell_{b}$. Note that we choose a symmetric binding potential
as a ‘worst-case scenario’ in which common motor activity is inhibited,
although our mechanism does not rely on this symmetry. The simplest form of
the binding probability is characterized by a single length scale $d$ that can
be associated with the typical spacing between binding sites:

$\displaystyle P_{c}(|\ell_{b}-\ell_{u}|)=\frac{1}{d}\qquad{\rm
for}\qquad|\ell_{b}-\ell_{u}|<\frac{d}{2}\,,$ (6)

and $P_{c}=0$ otherwise. In our model $P_{c}$ is a uniform distribution on
$(-d/2,d/2)$, but any symmetric (or even slightly asymmetric) $P_{c}$ leads to
contractility.

Figure 2:  Numerical results for the contractile velocity. (a) Contractile
velocity as function of the typical binding-site spacing for
$\tilde{\gamma}\to\infty$. For the semiflexible PMF, we use $\mu=4.37\times
10^{-8}\rm N$, $\tau_{0}=0.68{\rm pN}$, $\ell_{0}=1\rm\mu m$,
$\delta\ell=6.2{\rm nm}$, while for the _two-spring_ PMF we choose
$K_{2}=\mu/\ell_{0}$ and $K_{1}=0.35\tau_{0}/\delta\ell$ (both PMFs are chosen
to have the same $\delta\ell$). The solid purple line is the large-$d$
analytical solution of Eq. (12) and the solid red line is the small-$d$
analytical solution of Eq. (16). In (b) and (c) we plot the force-velocity
relation for the two-spring and the semiflexible PMFs, respectively. The
dimensionless quantities used are: $\tilde{d}=d/\delta\ell$,
$\tilde{\gamma}=2\gamma\omega_{\rm off}\delta\ell/\tau_{0}$,
$\tilde{v}=v/(2\omega_{\rm off}\delta\ell)$ and $\tilde{F}=F/\tau_{0}$.
Parameters used: (b) $\tilde{d}=0.1$ , and (c) $\tilde{d}=10$. Both
$\tilde{F}$ and $\tilde{\gamma}\tilde{v}$ are further rescaled according to
their $d$ dependence (quadratic for large $\tilde{d}$ and linear for small
$\tilde{d}$). In (a), (b) and (c) $\omega_{\rm on}\gg\omega_{\rm off}$.

After the polymer binds to the substrate with initial length $\ell_{b}$ at
$t=0$, it contracts the substrate due to its elastic energy. The polymer
length in the bound state, $\ell_{f}$, thus changes over time. For a viscous
substrate with drag coefficient $\gamma$, the survival probability for
$\ell_{f}$, $P(\ell_{f};t)$ (probability that at time $t$ the polymer is still
bound to the substrate and has length $\ell_{f}$), follows a 1-variable
Fokker-Planck equation (FPE)

$\displaystyle\partial_{t}{P}(\ell_{f};t)+\partial_{\ell_{f}}{\bm{J}(\ell_{f};t)}=-2\omega_{\rm
off}{P}$ $\displaystyle{P}(\ell_{f};t=0)=P_{b}(\ell_{b}=\ell_{f})\,,$ (7)

where $J=-(k_{B}T/\gamma)\partial P/\partial\ell_{f}-{P}\partial
U_{e}/\partial\ell_{f}$. The constant unbinding rate $2\omega_{\rm off}$
allows us to write $P$ as $P=P_{s}\exp(-2\omega_{\rm off})$, where the term
$\exp(-2\omega_{\rm off})$ is the probability that the polymer is still bound
at time $t$. The distribution $P_{s}$ then satisfies a standard FPE:

$\displaystyle\partial_{t}{P}_{s}(\ell_{f};t)+\partial_{\ell_{f}}{\bm{J_{s}}(\ell_{f};t)}=0$
$\displaystyle{P_{s}}(\ell_{f};t=0)=P_{b}(\ell_{b}=\ell_{f})\,,$ (8)

where $J_{s}=-(k_{B}T/\gamma)\partial{P_{s}}/\partial\ell_{f}-{P_{s}}\partial
U_{e}/\partial\ell_{f}$. Solving Eq. (8) gives the distribution $P_{s}$, from
which the steady state distribution $P_{\rm on}$ is calculated:

$\displaystyle{P_{\rm on}}(\ell)=C_{\rm on}\int\mathop{}\\!\mathrm{d}t\cdot
2\omega_{\rm off}P_{s}(\ell_{f}=\ell;t)e^{-2\omega_{\rm off}t},$ (9)

with $C_{\rm on}=\omega_{\rm on}/(\omega_{\rm on}+2\omega_{\rm off})$ being
the probability to be in the bound state. We then find the contractile force,
$\langle F_{s}\rangle_{\ell}$ through Eq. (1), and the average contractile
velocity through Eq. (2).

In Fig. 2(a) we numerically calculate the contractile velocity for both the
semiflexible and the two-spring PMFs. We find positive contractile velocity
for both PMFs, suggesting that our contractile mechanism is robust for any
asymmetric elastic PMF that is hard to stretch and soft to compress. For both
PMFs, the $d$-dependences of the velocity are similar. The velocity increases
monotonically with $d$, with different scaling dependences in two regimes
separated by a characteristic lengthscale $\delta\ell$. Here, $\delta\ell$ is
defined to be the root-mean-square fluactuations of the polymer end-to-end
length under its equilibrium distribution. Its value is
$\delta\ell=\sqrt{(\pi-2)k_{B}T/(\pi K_{1})}$ for two-spring PMF (assuming
$K_{1}\ll K_{2}$) and $\delta\ell=\ell_{0}^{2}/(\sqrt{90}\ell_{p})$ for
semiflexible PMF (see supplementary material of Ref. Chen  _et al._ (2020)).
For $d\ll\delta\ell$, a quadratic $d$-dependence is observed, while for
$d\gg\delta\ell$ we find that $v\sim d$. The different scaling exponents in
these two limits originate in different profiles of $P_{\rm on}$: In the
large-fluctuation limit ($\delta\ell\gg d$), $P_{\rm on}$ is only slightly
perturbed from the equilibrium distribution $P_{\rm eq}$, while in the small-
fluctuation limit ($\delta\ell\ll d$), $P_{\rm on}$ is almost independent of
$P_{\rm eq}$. Below we discuss in detail these two limits.

###  III.1 Small-fluctuation limit

One can always write the corresponding Langevin equation of the FPE Eq. (8)
using standard methods Gardiner (2009). For $\delta\ell\ll d$, thermal
fluctuations are small enough that we can neglect the thermal term in the
Langevin equation. In this case, for a given binding length $\ell_{b}$, the
polymer length in the bound state is uniquely determined by a trajectory
$\ell_{f}^{*}(\ell_{b};t)$ which follows a Langevin equation that corresponds
to Eq. (8):

$\displaystyle\gamma\frac{\mathop{}\\!\mathrm{d}{\ell}_{f}^{*}(\ell_{b};t)}{\mathop{}\\!\mathrm{d}t}$
$\displaystyle=-\tau\left(\ell_{f}^{*}\right)$
$\displaystyle{\ell}_{f}^{*}(\ell_{b};t=0)$ $\displaystyle=\ell_{b}\,,$ (10)

where $\tau(\ell_{f})=U_{e}^{\prime}(\ell_{f})$ is the polymer tension. In Eq.
(8), for a given binding length $\ell_{b}$, i.e., an initial condition
$P_{s}(\ell_{f};t=0)=\delta(\ell_{f}-\ell_{b})$, the trajectory in Eq. (10)
gives a solution of Eq. (8), $P_{s}=\delta(\ell_{f}-\ell_{f}^{*})$, where
$\delta(x)$ is the Dirac $\delta$-function. Therefore, the resulting $P_{s}$
from the binding probability $P_{b}(\ell_{b})$, is the average probability
distribution of trajectories starting with all possible $\ell_{b}$

$\displaystyle
P_{s}(\ell_{f};t)=\int\mathop{}\\!\mathrm{d}\ell_{b}P_{b}(\ell_{b})\delta\left[\ell_{f}-\ell_{f}^{*}(\ell_{b};t)\right]\,.$
(11)

The steady-state distribution in the bound state is then calculated by solving
Eq. (11) and substituting $P_{s}$ into Eq. (9). Then, the contractile velocity
can be found with the help of Eq. (1) and Eq. (2). For the two-spring PMF,
this velocity assumes a simple form:

$\displaystyle v=$ $\displaystyle\frac{dC_{\rm on}\gamma\omega_{\rm
off}^{2}(K_{2}-K_{1})}{8(K_{1}+\gamma\omega_{\rm off})(K_{2}+\gamma\omega_{\rm
off})}\,,$ (12)

which depends linearly on $d$. This expression is in perfect agreement with
the numerical solution of the FPE, Eq. (8), see Fig. 2(a).

The contractile velocity for the semiflexible potential cannot be found
analytically, because the potential does not have an explicit analytical
expression. However, in the $\delta\ell\ll d$ limit, the force-extension
relation of the semiflexible PMF has two asymptotic limits, for large stretch
or compression Broedersz and MacKintosh (2014):

$\displaystyle\tau(\ell)\approx\left\\{\begin{aligned}
&\frac{\mu}{\ell_{0}}(\ell-\ell_{0})\qquad(\ell-\ell_{0}\gg\delta\ell)\\\
&-\tau_{0}\qquad\qquad\,\,(\ell_{0}-\ell\gg\delta\ell)\,.\end{aligned}\right.$
(13)

Therefore, if we approximate the semiflexible PMF by a two-spring PMF with
$K_{2}=\mu/\ell_{0}$ and $K_{1}\ll K_{2}$, the two PMFs generate the same
average force just after binding ($K_{1}$ does not affect the average force as
long as $K_{1}\ll K_{2}$). In Fig. 2(a) we show that the contractile velocity
of the semiflexible PMF coincides with that of the two-spring PMF in the
small-fluctuation (large-$d$) limit, given the proper choice of the two-spring
parameters. To be specific, we let the two PMFs have the same $\delta\ell$.
For this aim we set
$K_{1}=3.32\tau_{0}\ell_{p}/\ell_{0}^{2}=0.35\tau_{0}/\delta\ell$ such that
the two spring PMF gives the same $\delta\ell$ with the semiflexible PMF.

###  III.2 Large-fluctuation limit

For $\delta\ell\gg d$, thermal noise cannot be neglected, but the change in
length due to the binding is relatively small. Then, $P_{b}$ is only slightly
perturbed from the equilibrium distribution in the unbound state. Expanding
$P_{b}$ of Eq. (5) to second order in $d$ gives

$\displaystyle P_{b}(\ell_{b})\simeq P_{\rm
eq}(\ell_{b})\left[1+\frac{d^{2}}{24}\left(\frac{U_{e}^{\prime
2}(\ell_{b})}{(k_{B}T)^{2}}-\frac{U_{e}^{\prime\prime}(\ell_{b})}{k_{B}T}\right)\right]\,,$
(14)

where
$f^{\prime}(\ell_{b})=[\mathop{}\\!\mathrm{d}f(\ell)/\mathop{}\\!\mathrm{d}\ell]_{\ell=\ell_{b}}$
and
$f^{\prime\prime}(\ell_{b})=[\mathop{}\\!\mathrm{d}^{2}f(\ell)/\mathop{}\\!\mathrm{d}\ell^{2}]_{\ell=\ell_{b}}$.
Since both the FPEs of Eq. (8) and Eq. (9) are linear, the resulting $P_{\rm
on}$ is similarly perturbed around the equilibrium distribution, where the
deviation scales as $d^{2}$. Thus, from Eq. (1) we have $\langle
F_{s}\rangle_{\ell}\sim d^{2}$, and the contractile velocity shows quadratic
dependences on $d$, as is seen in Fig. 2(a). To show this explicitly, let us
consider a nearly rigid substrate, i.e., $\tilde{\gamma}\rightarrow\infty$. In
this case the polymer is not relaxing in the bound state (i.e., $P_{\rm
on}(\ell)=C_{\rm on}P_{b}(\ell_{b}=\ell)$), and produces an average
contractile force (on the substrate) of

$\displaystyle\langle F_{s}\rangle_{\ell}=\frac{C_{\rm
on}d^{2}}{24}\int\mathop{}\\!\mathrm{d}\ell_{b}P_{\rm
eq}(\ell_{b})U_{e}^{\prime\prime\prime}(\ell_{b})\,,$ (15)

where we have used Eqs. (1) and (14). This force is positive for any potential
with positive $U_{e}^{\prime\prime\prime}(\ell)$. Note that the above result
is obtained for constant on/off rates. In case that the on/off rates obey
detailed balance, $P_{\rm on}$ would remain the equilibrium distribution even
for finite $d$ and $\langle F_{s}\rangle_{\ell}$ would vanish. For the two-
spring PMF, the contractile force calculated from Eq. (15) leads to a
contractile velocity:

$\displaystyle v=\frac{C_{\rm on}(K_{2}-K_{1})\sqrt{K_{1}}d^{2}}{12\sqrt{2\pi
k_{B}T}\gamma}\,,$ (16)

which agrees perfectly to the numerical results in the small-$d$ limit, see
Fig. 2(a). For the semiflexible PMF, numerical result gives $v\approx
0.24d^{2}\tau_{0}/(\delta\ell^{2}\gamma)$. Comparing this result with Eq.
(16), we find that by setting $K_{1}=0.35\tau_{0}/\delta\ell$ and
$K_{2}=13\tau_{0}/\delta\ell$, the two-spring PMF gives the same contractile
velocity with the semiflexible PMF in the small $d$ limit, which is confirmed
by our numerical results, see Fig. 2(a). Here, the value of $K_{1}$ is chosen
to be the same as that in Sec. III.1, such that the two-spring PMF has the
same $\delta\ell$ with the semiflexible PMF. We then set the value of $K_{2}$
such that Eq. (16) gives the same velocity as the semiflexible PMF. Together
with previous discussion on the small-fluctuation limit, this suggests that
given the proper parameter choice, the two-spring PMF can mimic the
semiflexible PMF both in the small and large-fluctuation limits.

Having illustrated the coarse-grained model in two extreme limits, let us
consider a property that is commonly measured for molecular motors, the force-
velocity relation. This relation describes a motor’s ability to do work under
an external load. Since our non-motor mechanism creates motor-like
contraction, it is also useful to compute the force-velocity relation in our
model. To calculate this relation, we exert a constant tension $F$ on the two
ends of the polymer. The constant force modifies the elastic potential of Eqs.
(7) and (8) from $U_{e}(\ell_{f})$ to $U_{e}(\ell_{f})-F\ell_{f}$. In Fig.
2(b)-(c) we plot force-velocity curves for the small and large $d$ limits,
respectively. In Fig. 2(b) we use the two-spring PMF, while in Fig. 2(c) we
use the semiflexible polymer PMF. In both cases, the velocity is reduced by an
applied load, in a way similar to molecular motors Derényi and Vicsek (1996).
We find that decreasing the dimensionless viscosity $\tilde{\gamma}$ results
in lower velocities and correspondingly lower stall forces, due to the
increased compliance and stress relaxation of the substrate.

##  IV Microscopic Model

Figure 3: (a) Illustration for the microscopic model. Two ends of a polymer
actively bind to and unbind from two regions of a Maxwell substrate. Each of
the substrate regions is connected to the rest of the network via a viscous
damper and an elastic spring. Each of the polymer ends binds to the substrate
region with a triangular binding potential (upper right). (b) Profile of the
modified binding potential $U_{b}^{v}$ for various rescaled substrate spring
constants, $\tilde{K}=Kd^{2}/\Delta E$. The height of $U_{b}^{v}$ increases
with $\tilde{K}$ and reaches $U_{b}$ when $\tilde{K}\to\infty$. (c) Rescaled
contractile velocity, $\tilde{v}=v/(\omega_{\rm off}\delta\ell)$, as function
of $\tilde{K}$, for various ratios between the substrate relaxation time
$\tau_{s}=1/(Km_{y})$ and the unbinding time $\tau_{\rm off}$. The velocity
increases monotonically with $\tilde{K}$ and approaches its maximum as
$\tilde{K}\gg 1$. The PMF is the same as the semiflexible PMF used in Fig. 2.
In (c) $d=10\rm nm$.

Having demonstrated our mechanism using a simple coarse-grained model, let us
switch to a more complete microscopic model. We will later show how the
coarse-grained model can be derived as a limit of this microscopic model. In
this model, we still study the active binding/unbinding of a semiflexible
segment on a viscoelastic substrate where the details of the binding potential
will be considered.

As sketched in Fig. 3(a), we consider a semiflexible segment whose two ends, A
and B, bind and unbind to two regions on a viscoelastic substrate denoted as
$\rm S_{A}$ and $\rm S_{B}$. Each of the substrate regions represents a small
part of the entire substrate, and they can move independently to each other or
the rest of the substrate. The substrate regions are assumed to be rigid and
their positions are denoted by $y_{A}$ and $y_{B}$, respectively. We consider
a Maxwell substrate by connecting each substrate region with the rest of the
substrate with a viscous damper of mobility $m_{y}$ and a spring of spring
constant $K$ in series. Let $y_{A}^{e}$ and $y_{A}^{v}$ be the lengths of the
elastic and viscous parts of the substrate region $\rm S_{A}$, and similarly
define $y_{B}^{e}$ and $y_{B}^{v}$ (note that these lengths can be negative as
they are measured relative to the initial positions). These lengths satisfy
$y_{A,B}=y_{A,B}^{e}+y_{A,B}^{v}$.

The two polymer ends (crosslinkers) can be thought as particles moving within
the substrate with effective mobility $m_{x}$, whose positions are denoted by
$x_{A}$ and $x_{B}$, respectively. The polymer end A (B), successively binds
to and unbinds from the substrate region $\rm S_{A}$ ($\rm S_{B}$). When A (B)
is unbound, there is no interaction between the polymer end A (B) and the
substrate region $\rm S_{A}$ ($\rm S_{B}$). When A (B) is bound to $\rm S_{A}$
($\rm S_{B}$), the polymer end interacts with the substrate region via an
effective binding potential, $U_{b}(x_{A}-y_{A})$ ($U_{b}(x_{B}-y_{B})$). The
binding potentials of biopolymers are usually rugged due to the microscopic
structure of the monomers. Since we assume the substrate is apolar, these
potentials should be symmetric. For simplicity, we assume $U_{b}$ to be a
periodic triangular potential of depth $\Delta E$ and period $d$, although
periodicity is not essential. Lastly, the PMF of the polymer is
$U_{e}(x_{B}-x_{A})$.

The binding and unbinding rates of the two polymer ends are denoted by
$\omega_{\rm on}$ and $\omega_{\rm off}$, respectively. As discussed in the
beginning of Sec. III, constant binding/unbinding rates break DB, which
together with a rugged binding potential and asymmetric elasticity can lead to
a steady state substrate contraction.

The polymer can only exert tension on the substrate when its two ends are
bound to the substrate, and we define this state as the bound state. The
unbound state is thus the state in which one or both polymer ends are unbound.
We are interested in how the substrate length, defined as $y_{B}-y_{A}$,
changes during the bound state. For that aim we consider a bound state which
starts at $t=0$. Just before $t=0$, one of the two ends must be bound to the
substrate region while the other being unbound, so the system could enter the
bound state. We assume the polymer end A is bound while B is unbound (this is
equivalent to the case in which B is bound while A is unbound). At $t=0$, B
binds to $S_{B}$, and the system enters the bound state. While in the bound
state, the polymer interacts with the substrate and deforms it. The evolution
of the system is described by a survival probability ${\cal P}^{\rm
ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v};t\right)$ (each variable with
subscript $A,B$ stands for two variables with substripts $A$ and $B$,
respectively). The bound state ends at $t=t_{e}$, when one of the polymer ends
unbinds from the substrate. The distribution of $t_{e}$ is exponential with
unbinding rate $2\omega_{\rm off}$ (derivation is as in the paragraph before
Eq. (8)), where the factor of two is due to the fact that the unbinding of
either polymer ends terminates the bound state.

During a single binding/unbinding event, the average contraction of the
substrate is defined as

$\displaystyle\Delta y=\langle y_{B}^{v}-y_{A}^{v}\rangle_{t=0}-\langle
y_{B}^{v}-y_{A}^{v}\rangle_{t=t_{e}}\,,$ (17)

where

$\displaystyle\langle y_{B}^{v}-y_{A}^{v}\rangle_{t=0}$ $\displaystyle=\int
DxDy\,\,(y_{B}^{v}-y_{A}^{v})$ $\displaystyle\times{\cal P}^{\rm
ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v};t=0\right)\,$ $\displaystyle\langle
y_{B}^{v}-y_{A}^{v}\rangle_{t=t_{e}}$ $\displaystyle=2\omega_{\rm
off}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}t_{e}\int
DxDy\,(y_{B}^{v}-y_{A}^{v})\,\,$ $\displaystyle\times{\cal P}^{\rm
ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v};t=t_{e}\right)\,\,$ (18)

are the average values of $y_{B}^{v}-y_{A}^{v}$ at the start and at the end of
the bound state. Here
$Dx=\mathop{}\\!\mathrm{d}x_{A}\mathop{}\\!\mathrm{d}x_{B}$ and
$D_{y}=\mathop{}\\!\mathrm{d}y_{A}^{e}\mathop{}\\!\mathrm{d}y_{B}^{e}\mathop{}\\!\mathrm{d}y_{A}^{v}\mathop{}\\!\mathrm{d}y_{B}^{v}$.
In Eq. (18) we only consider the viscous parts of the substrate, because the
elastic parts relax immediately to their rest lengths in the unbound state,
and the lengths of the viscous parts are the only lengths that show changes
due to the relaxation in the bound state.

The contractile velocity is calculated using $v=\Delta y/{\cal T}$, where
${\cal T}$ is the average time between two binding/unbinding events. By
definition, ${\cal T}$ is the sum of the average lifetimes in the bound and
unbound states. The average lifetime of the polymer in the bound state (both
ends are bound) is $\tau_{\rm off}=1/2\omega_{\rm off}$. The average lifetime
in the unbound state is more complicated as it is composed of two different
states: (a) both ends are unbound, and (b) only one end is unbound. The
probabilities to be in these two states can be written as, $C_{a}=(1-C_{\rm
on})^{2}$, $C_{b}=2C_{\rm on}(1-C_{\rm on})$, respectively. Here $C_{\rm
on}=\omega_{\rm on}/(\omega_{\rm on}+\omega_{\rm off})$ is the fraction of
time in which one of the polymer ends is bound to the substrate (regardless of
the other polymer end), see also Eq. (9). Therefore, when the polymer is in
the bound state, the probabilities to be in states (1) and (2) are,
$P_{a}=C_{a}/(C_{a}+C_{b})$ and $P_{b}=C_{b}/(C_{a}+C_{b})$, respectively.
Since the unbound state can only be ended when the system is in state (2), the
net binding rate, given that the polymer is in the unbound state, is
$\omega^{*}_{\rm on}=P_{b}\omega_{\rm on}$, and the average lifetime of the
unbound state is $\tau_{\rm on}=1/\omega^{*}_{\rm off}$. Taken together, we
have ${\cal T}=\tau_{\rm off}+\tau_{\rm on}=1/(2C_{\rm on}^{2}\omega_{\rm
off})$.

In the bound state, the evolution of the system can be described by six
variables: $x_{A,B}$, $y_{A,B}^{e}$ and $y_{A,B}^{v}$, with the total
potential energy $W^{\rm ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v}\right)$:

$\displaystyle W^{\rm ve}$
$\displaystyle=U_{e}(x_{B}-x_{A})+U_{b}(x_{B}-y_{B}^{e}-y_{B}^{v})$
$\displaystyle+U_{b}(x_{A}-y_{A}^{e}-y_{A}^{v})+\frac{K}{2}(y_{A}^{e})^{2}+\frac{K}{2}(y_{B}^{e})^{2}\,.$
(19)

Note that we set the rest lengths of both $y_{A}^{e}$ and $y_{B}^{e}$ to zero
for simplicity.

In general, the survival probability of the system should also be described by
these 6 variables, ${\cal P}^{\rm
ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v};t\right)$. However, we can eliminate
$y_{A}^{e}$ and $y_{B}^{e}$ since they obtain their deformed values
instantaneously, such that they can always be considered as fast variables.
Therefore, we can describe the system dynamics using a 4-variable survival
probability ${\cal P}^{v}$ Magnasco (1994):

$\displaystyle{\cal P}^{\rm ve}$ $\displaystyle=\frac{\exp[-W^{\rm
ve}/k_{B}T]}{Z^{v}\left(x_{A,B},y_{A,B}^{v}\right)}{\cal
P}^{v}\left(x_{A,B},y_{A,B}^{v};t\right)\,.$ (20)

with
$Z^{v}=\int\mathop{}\\!\mathrm{d}y_{A}^{e}\mathop{}\\!\mathrm{d}y_{B}^{e}\,{\exp({-W^{\rm
ve}/k_{B}T})}$.

The evolution of ${\cal P}^{v}$ satisfies a FPE,

$\displaystyle\partial_{t}{\cal P}^{v}+\nabla\cdot{{\bm{J}^{v}}}=-2\omega_{\rm
off}{\cal P}^{v}\,,$ (21)

with $J_{\alpha}=-m_{\alpha}(k_{B}T\partial_{\alpha}{\cal P}^{v}+{\cal
P}^{v}\partial_{\alpha}W^{v})$ without the summation convention. Here
$m_{\alpha}=m_{x}$ for $\alpha=x_{A,B}$ and $m_{\alpha}=m_{y}$ for
$\alpha=y_{A,B}^{v}$. Notably, this FPE is governed by an effective potential
$W^{v}\left(x_{A,B},y_{A,B}^{v}\right)$:

$\displaystyle W^{v}$
$\displaystyle=\int\mathop{}\\!\mathrm{d}y_{A}^{e}\mathop{}\\!\mathrm{d}y_{B}^{e}\frac{\exp({-W^{\rm
ve}/k_{B}T})}{Z^{v}\left(x_{A,B},y_{A,B}^{v}\right)}$ $\displaystyle\times
W^{\rm ve}\left(x_{A,B},y_{A,B}^{e},y_{A,B}^{v}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!=U_{e}(x_{B}-x_{A})+U_{b}^{v}(x_{B}-y_{B}^{v})+U_{b}^{v}(x_{A}-y_{A}^{v})\,.$
(22)

Surprisingly, we find this effective potential to have the same form as the
original total potential energy of a pure viscous substrate (see discussion
later on the viscous substrate), but with a modified binding potential

$\displaystyle U_{b}^{v}(x)$
$\displaystyle=\int\mathop{}\\!\mathrm{d}y\frac{\exp\left[-[U_{b}(x-y)+\frac{K}{2}y^{2}]/k_{B}T\right]}{Z_{b}(x)}$
$\displaystyle\times\left[U_{b}(x-y)+\frac{K}{2}y^{2}\right]\,,$ (23)

where
$Z_{b}(x)=\int\mathop{}\\!\mathrm{d}y\,{\exp\left[-[U_{b}(x-y)+Ky^{2}/2]/k_{B}T\right]}$.
This suggests that a viscoelastic substrate is equivalent to a viscous
substrate with a modified binding potential $U_{b}^{v}$.

The effective potential $U_{b}^{v}$ couples the original binding potential
$U_{b}$ and the elastic energy of the substrate. Same as the bare binding
potential, it is periodic with period $d$, but the profile of each potential
well is flattened by the elastic energy (see Fig. 3(b)). For $K\gg\Delta
E/d^{2}$, the substrate shows almost a pure viscous response, where
$U_{b}^{v}\approx U_{b}$. For smaller $K$, $U_{b}^{v}$ deviates from $U_{b}$,
and for $K\ll\Delta E/d^{2}$, $U_{b}^{v}$ becomes entirely flat. This is
because when $K\to\infty$ the Maxwell substrate reduces to a pure viscous
substrate, and when $K$ is small the elastic compliance of the substrate is
weak and almost all the stress can be relaxed by the deformation of the
elastic part, while the deformation of the viscous part is negligible.
Therefore, in order to generate non-trivial contractility during the
binding/unbinding events, the substrate rigidity $K$ should be large enough
($K\geq\Delta E/d^{2}$).

To determine the initial condition of Eq. (21), we need to specify the meaning
of $y_{A}^{v}$ and $y_{B}^{v}$. Since the substrate regions are defined to be
rigid, their positions can be represented by any point within them. Here, for
simplicity we choose $y_{A}^{v}$ and $y_{B}^{v}$ to be the positions of the
nearest binding sites (bottoms of the potential wells in $U_{b}^{v}$) of A and
B at $t=0$, respectively. Hence, at $t=0$ we have $|x_{A,B}-y_{A,B}^{v}|\leq
d/2$. Assuming the system relaxes fast in the unbound state, the initial
condition is given by

$\displaystyle{\cal
P}^{v}\left(x_{A,B},y_{A,B}^{v};t=0\right)=\frac{\chi\left(x_{A,B},y_{A,B}^{v}\right)}{Z}$
$\displaystyle\times\exp{\Big{(}-[U_{e}(x_{B}-x_{A})+U_{b}^{v}(x_{A}-y_{A})]/k_{B}T\Big{)}}\,,$
(24)

where $Z=\int DxDy^{v}{\cal P}^{v}$ is the partition function
($Dx=\mathop{}\\!\mathrm{d}x_{A}\mathop{}\\!\mathrm{d}x_{B}$ and
$Dy^{v}=\mathop{}\\!\mathrm{d}y_{A}^{v}\mathop{}\\!\mathrm{d}y_{B}^{v}$) and

$\displaystyle\chi\left(x_{A,B},y_{A,B}^{v}\right)$
$\displaystyle=\Theta\left(d/2-\left|x_{A}-y_{A}^{v}\right|\right)$
$\displaystyle\times\Theta\left(d/2-\left|x_{B}-y_{B}^{v}\right|\right)\,,$
(25)

gives the boundaries of the initial condition ($\Theta(x)$ is the Heaviside
function). $\chi$ appears because we define the values of $y_{A}^{v}$ and
$y_{B}^{v}$ at $t=0$ to be the positions of the nearest binding sites of $\rm
A$ and $\rm B$. Note that $U_{b}^{v}$ for the polymer end B is taken into
account in the initial condition, because it is assumed to be bound and
immobile during the binding of A.

The survival probability at time $t$ is then derived from Eq. (21) and (24),
which can be used to calculate the contractile velocity. The large number of
variables in the FPE prevents intuitive understanding of the dynamics, and
further introduces difficulties in the numerical solution. Below we will apply
two physical conditions that reduces the 4-variable FPE to a 1-variable
equation.

First, we assume $m_{x}\gg m_{y}$ because $m_{x}$ is the mobility of one end
of a single polymer, while $m_{y}$ stands for the mobility of a substrate
region which is a collection of multiple polymers that feels larger friction
than a single polymer end.

To introduce the second condition we consider three key timescales in the
system. The first one is the time for a single polymer end to relax within one
potential well, $\tau_{r}=\tau_{d}\left({k_{B}T}/{\Delta E^{v}}\right)$, where
$\tau_{d}={d^{2}}/{2k_{B}Tm_{x}}$ and $\Delta E^{v}$ is the difference between
the maximum and minimum of $U_{b}^{v}$. The complete derivation of $\tau_{r}$,
which is the so-called intrawell relaxation time Derényi and Astumian (1999)
is provided in the Appendix. The second timescale is the time for the polymer
end to hop to another potential well, $\tau_{\rm
hop}=\tau_{d}\exp\left({\Delta E^{v}}/{k_{B}T}\right)$ (see Appendix for
derivation). Lastly, the third timescale is the average lifetime in the bound
state, $\tau_{\rm off}=1/(2\omega_{\rm off})$. In biopolymers $\Delta E$ is
usually much larger than $k_{B}T$, and we assume $\Delta E^{v}$ is also large
enough such that $\tau_{r}\ll\tau_{\rm off}\ll\tau_{\rm hop}$. This requires
the network rigidity $K$ to be comparable or larger than $\Delta E/d^{2}$.
Because the value of $K$ is in principle proportional to the network density,
the assumption is relevant for densely-crosslinked network. The relation
between the three timescales implies that during the bound state the polymer
end relaxes quickly within the initial potential well and unbinds before it
can hop to another potential well.

Figure 4: Numerical results for the microscopic model. (a) Profile of the
modified effective interaction $W^{v*}$, for various rescaled substrate spring
constants, $\tilde{K}=Kd^{2}/\Delta E$. $W^{v*}$ becomes more asymmetric for
increasing $\tilde{K}$ and approaches $W^{*}$ of the viscous substrate limit
(see (c)) when $\tilde{K}\to\infty$. (b) Rescaled contractile velocity,
$\tilde{v}=v/(2\omega_{\rm off}\delta\ell)$, as function of rescaled binding
site spacing $\tilde{d}$, for various $\tilde{K}$ values. The dependence on
$\tilde{d}$ is non-monotonic. In the small $\tilde{d}$ limit the microscopic
model reduces to the coarse-grained model. The maximum velocity is reached at
$\tilde{d}\sim 1$. Dashed yellow line shows a realistic value $d=10\rm nm$
with $\delta\ell=6\rm nm$, which is close to the maximum-velocity location.
(c) Profile of the effective interaction $W^{*}$ in the viscous limit, for
various $\tilde{d}$ values. $W^{*}$ is more asymmetric for small $d$ and
approaches $U_{b}$ when $\tilde{d}\to\infty$. (d) Rescaled contractile
velocity as function of rescaled binding site spacing for various rescaled
substrate mobility, $\tilde{m}_{y}=m_{y}\tau_{0}/(2\omega_{\rm
off}\delta\ell)$ values in the viscous substrate limit. Similarly to the
viscoelastic case (see (b)), the dependence on $d$ is non-monotonic. PMF
parameters are the same as that in Fig. 2. The binding potential is the same
as that in Fig. 3.

With these two physical conditions we simplify the FPE as follows. Since
$\tau_{\rm off}\ll\tau_{\rm hop}$, the probability that either of the polymer
ends hops before unbinding is small, and both polymer ends are trapped in the
potential wells to which they bind, indicating that ${\cal
P}^{v}\sim\chi\left(x_{A,B},y_{A,B}^{v}\right)$. Furthermore, because
$\tau_{r}\ll\tau_{\rm off}$ and $m_{x}\ll m_{y}$, we can treat $x_{A}$ and
$x_{B}$ as fast variables, such that

$\displaystyle\\!{\cal P}^{v}\left(x_{A,B},y_{A,B}^{v};t\right)$
$\displaystyle\\!\\!=\\!\frac{\exp{\left[-W^{v}\left(x_{A,B},y_{A,B}^{v}\right)\\!\\!/k_{B}T\right]}}{Z_{Y}\left(y_{A,B}^{v}\right)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\chi\left(x_{A,B},y_{A,B}^{v}\right){\cal
P}^{v}_{Y}\left(y_{A,B}^{v};t\right)\,,$ (26)

with
$Z_{Y}=\int\mathop{}\\!\mathrm{d}x_{A}\mathop{}\\!\mathrm{d}x_{B}\chi\,{e^{-W^{v}/k_{B}T}}$.
The distribution ${\cal P}^{v}\left(x_{A,B},y_{A,B}^{v};t\right)$ can thus be
determined by its marginal distribution ${\cal
P}^{v}_{Y}\left(y_{A,B}^{v};t\right)$. Substituting Eq. (26) into Eq. (21), we
have the reduced FPE for ${\cal P}^{v}_{Y}\left(y_{A,B}^{v};t\right)$

$\displaystyle\partial_{t}{\cal
P}^{v}_{Y}\left(y_{A,B}^{v};t\right)+\nabla\cdot{\bm{J^{*}}\left(y_{A,B}^{v};t\right)}=-2\omega_{\rm
off}{\cal P}^{v}_{Y}\,,$ (27)

with $J^{*}_{\alpha}=-m_{y}(k_{B}T\partial_{\alpha}{\cal P}^{v}_{Y}+{\cal
P}^{v}_{Y}\partial_{\alpha}W^{v*})$ without the summation convention, and
$\alpha=y^{v}_{A,B}$. Here

$\displaystyle W^{v*}(y_{B}^{v}-y_{A}^{v})$
$\displaystyle\\!=\\!\\!\\!\int\\!\\!\mathop{}\\!\mathrm{d}x_{A}\mathop{}\\!\mathrm{d}x_{B}\frac{\chi\left(x_{A,B},y_{A,B}^{v}\right)\,}{Z_{Y}\left(y_{A,B}^{v}\right)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times
W^{v}\left(x_{A,B},y_{A,B}^{v}\right)\exp({-W^{v}/k_{B}T})\,$ (28)

is the effective interaction between the two substrate regions $S_{A}$ and
$S_{B}$. Note that $W^{v*}$ is a function of $(y_{B}^{v}-y_{A}^{v})$ because
the original interaction potential $W$ is a function of relative positions,
such that the substrate is translational invariant. Due to this symmetry, it
is instructive to perform a variable substitution:
$(y_{A}^{v},y_{B}^{v})\to(y_{+},y_{-})$, where $y_{+}=y_{A}^{v}+y_{B}^{v}$ and
$y_{-}=y_{B}^{v}-y_{A}^{v}$. It is straightforward to verify that $y^{+}$
follows diffusional dynamics with mobility $m_{y}$, while the survival
probability of $y_{-}$ satisfies the following 1-variable FPE,

$\displaystyle\partial_{t}{\cal
P}_{-}(y_{-};t)+\partial_{y_{-}}{\bm{J_{-}}(y_{-};t)}=-2\omega_{\rm off}{\cal
P}_{-}\,,$ (29)

where $J_{-}=-2m_{y}k_{B}T\partial_{y_{-}}{\cal P}_{-}+{\cal
P}_{-}\partial_{y_{-}}W^{v*}$. Equation (29) suggests that the distance
between the two substrate regions follows the same equation of motion as a
particle diffusing in 1D under the influence of the potential $W^{v*}$. The
effective interaction $W^{v*}$ is essentially the averaged $W$ in the
equilibrium distribution of $x_{A}$ and $x_{B}$. As shown in Fig. 4(a),
$W^{v*}$ becomes asymmetric for any rescaled substrate spring constant,
$\tilde{K}=Kd^{2}/\Delta E$, and its shape becomes more asymmetric for
increasing $\tilde{K}$. When $\tilde{K}\to\infty$, the substrate reduces to a
pure viscous substrate and the profile of $W^{v*}$ approaches that of $W^{*}$,
the effective interaction for the viscous substrate (see discussion later in
this section). In the $\tilde{K}\to 0$ limit in which the substrate is
extremely soft, both $U_{b}^{v}$ and $W^{v*}$ become flat and there is no
interaction between the two substrate regions, resulting in vanishing
contractility. However, for any finite value of $\tilde{K}$, the profile of
$W{{}^{*}}$ will always be asymmetric for contraction and expansion, such that
a positive contractile velocity is expected. Interestingly, a particle moving
in a periodic $W^{v*}$ is mathematically equivalent to a motor binding on a
polar filament Jülicher  _et al._ (1997), with the crucial distinction that
the motor directional motion is dictated by the filament polarity, while the
motion here is always contractile in character.

After obtaining the effective potential, we use Eq. (29) to calculate the
survival probability for $y_{-}$ and $\Delta y$ of Eq. (17) is

$\displaystyle\Delta y=\langle y_{-}\rangle_{t=0}-\langle
y_{-}\rangle_{t=t_{e}}\,,$ (30)

where the average at $t=0$ and $t=t_{e}$ follows the same definition as in Eq.
(18), with the average being respect to ${\cal P}_{-}$ instead of ${\cal
P}^{\rm ve}$.

In Fig. 3(c) we numerically calculate the contractile velocity as function of
the rescaled substrate spring constant $\tilde{K}$. The velocity increases
monotonically with $\tilde{K}$, as a result of the more asymmetric $W^{v*}$
for increasing $\tilde{K}$. For $K\gg\Delta E/d^{2}$, the velocity
asymptotically converges to the velocity profile for a viscous substrate.
Therefore, to reach a maximum contractile velocity, the substrate rigidity
should be much larger than $\Delta E/d^{2}$.

The Maxwell substrate also introduces an intrinsic substrate relaxation time,
$\tau_{s}=1/(Km_{y})$. If $\tau_{s}\ll\tau_{\rm off}$, where $\tau_{\rm
off}=1/(2\omega_{\rm off})$ is the average lifetime of the bound state, the
substrate relaxes completely before the polymer unbinds, leading to a small
contractile velocity. A monotonic dependence between the contractile velocity
and the ratio $\tau_{s}/\tau_{\rm off}$ is observed in Fig. 3(c). Clearly,
$\tau_{s}/\tau_{\rm off}$ must be large enough to reach a considerable
contraction. If $\tau_{s}/\tau_{\rm off}\gg 1$, the substrate behaves almost
like an elastic substrate with spring constant $K$, with $v/m_{y}$ being the
contractile force exerted on the substrate. In this case the substrate
relaxation can be neglected during a single binding/unbinding event, so the
contractile force is not reduced by the substrate relaxation, leading to
maximized contractile force.

In Fig. 4(b) we plot the relation between the contractile velocity and the
binding site spacing $d$. Interestingly, we find that in the small-$d$ limit
the velocities calculated from various $\tilde{K}$ values collapse on a single
curve, which shows a quadratic dependence on $d$. This is not surprising
because the microscopic model reduces to the coarse-grained model in this
limit. In fact, when we analyze Eq. (28) we find that the total energy $W^{v}$
is composed of two interactions, $U_{e}$ and $U_{b}^{v}$. The fluctuation
lengthscale of $U_{e}$ is $\delta\ell$, while the distance between $x_{A}$
($x_{B}$) and $y_{A}^{v}$ ($y_{B}^{v}$) is smaller than $d/2$, implying that
in the $d\ll\delta\ell$ limit we can approximate $U_{e}(x_{B}-x_{A})$ with
$U_{e}(y_{B}^{v}-y_{A}^{v})$. Furthermore, when $\Delta E^{v}\gg k_{B}T$ we
have

$\begin{aligned} \exp(-U_{b}^{v}(x)/k_{B}T)\approx\left\\{\begin{aligned}
1\qquad(x=nd)\\\ 0\qquad(x\neq nd)\end{aligned}\right.\end{aligned}\,,$ (31)

where $n$ is any integer. Substituting Eq. (31) and
$U_{e}(x_{B}-x_{A})=U_{e}(y_{B}^{v}-y_{A}^{v})$ into Eq. (28) gives
$W^{v*}=U_{e}$. This suggests that the 1-variable FPE in Eq. (29) is identical
to Eq. (7) in the coarse-grained model, with $y^{-}$ and $2m_{y}$ in Eq. (29)
replaced by $\ell_{f}$, $1/\gamma$ in Eq. (7). Therefore, the microscopic
model can be reduced to the coarse-grained model in the small-$d$ limit (when
$\Delta E^{v}\gg k_{B}T$). In this case, the details of the effective binding
potential $U_{b}^{v}$ are not important, and the contractile velocity of the
microscopic model for any $K$ value is the same as that of the coarse-grained
model.

We also find that the contractile velocity shows a non-monotonic dependence on
$d$ in Fig. 4(b). The velocity increases with $d$ for $d\ll\delta\ell$,
decreases with $d$ for $d\gg\delta\ell$, and reaches its maximum at
$d\sim\delta\ell$. When $d$ is large, the effective binding potential
$U_{b}^{v}$ essentially becomes flat, weakening its ability to stretch or
compress the polymer when it binds. This also explains the different
predictions of the microscopic model and the coarse-grained model in the
large-$d$ limit. The coarse-grained model is valid only when the change in
$U_{e}$ is smaller than the binding potential, see discussion after Eq. (5).
Since the change in the polymer length is of the order of $d$, the change in
$U_{e}$ can be approximated by $U_{e}(\ell_{0}+d)$, which increases
dramatically with $d$ when $d>\delta\ell$. Therefore, for $d\gg\delta\ell$ we
have $U_{e}\geq\Delta E^{v}$, which suggests that the effective binding
potential does not have sufficient energy to deform the polymer, and the
maximum velocity is always reached at $d\sim\delta\ell$. We also find that the
value of $d$ at the maximum velocity is larger for increasing $\tilde{K}$
(Fig. 4(b)), due to the fact that $\Delta E^{v}$ increases with $\tilde{K}$
(Fig. 3(b)).

Fig. 4(b) implies that the values of $d$ and $\delta\ell$ needs to be close
enough for considerable contraction. Biologically, the value of $d$ is fixed
for a given substrate filament, which is about the same order of magnitude as
monomer size, e.g., $d\approx 10\rm nm$ for actin filaments. $\delta\ell$, on
the other hand, has a strong dependence on the segment rest length $\ell_{0}$
(crosslinking distance), $\delta\ell\sim\ell_{0}^{2}/\ell_{p}$. Therefore, we
expect the contractile velocity of a crosslinked network to be maximized under
an appropriate crosslinking distance. For actin cytoskeleton, for instance, a
typical value of the crosslinking distance is $\ell_{0}\approx 1\rm\mu m$,
corresponding to $\delta\ell=6.2\rm nm$ and $\tilde{d}=1.6$. Surprisingly, we
find this $\tilde{d}$ value is very close to the maximum-velocity position,
see dashed yellow line in Fig. 4(b). Such coincidence implies that the
crosslinking distance in cytoskeleton may be the result of evolution that
optimizes contractility.

###  IV.1 The limit of a viscous substrate

Having discussed the microscopic model with a Maxwell substrate, let us switch
to a simpler case in which the substrate is viscous, i.e., the $K\gg\Delta
E/d^{2}$ limit.

Visually, a viscous substrate is obtained by removing the two springs in Fig.
3(a). In this case, the lengths of the springs $y_{A}^{e}$ and $y_{B}^{e}$ do
not appear as variables of the system state, and the total potential energy is
$W^{v}$ defined in Eq. (22), with $U_{b}^{v}$ replaced by $U_{b}$. The
survival probability, ${\cal P}^{v}\left(x_{A,B},y_{A,B}^{v}\right)$, is
governed by the same FPE of Eq. (21), with the initial condition Eq. (24).

Following the same steps of variable elimination in Eqs. (20-22), the
4-variable FPE is reduced to a 1-variable equation governed by the potential
energy $W^{*}(y_{B}-y_{A})$, which has the same form as
$W^{v*}(y_{B}^{v}-y_{A}^{v})$ in Eq. (28), with $U_{b}^{v}$ replaced by
$U_{b}$. As shown in Fig. 4(c), $W^{*}$ becomes asymmetric for finite $d$, and
its shape becomes more asymmetric for small $d$. In fact, in the small $d$
limit ($d\ll\delta\ell$), $W^{*}$ has the same form as the polymer PMF
$U_{e}$, because in this limit the microscopic model reduces to the coarse-
grained model (see discussion below Eq. (31)).

In Fig. 4(d) we numerically calculate the contractile velocity and show its
dependence on $d$. It shows similar non-monotonically $d$-dependence as the
viscoelastic substrate. This is because the viscoelastic substrate is
equivalent to the viscous substrate with a modified binding potential, as we
point out in Eq. (22). We also find that the contractile velocity decreases
for increasing $\tilde{m}_{y}$ (or decreasing $\omega_{\rm off}$), similar to
that in the coarse-grained model (see Fig. 2 (c)).

##  V Discussion

We have presented a detailed (microscopic) model that demonstrates how
contractility can be produced in the absence of molecular motors. Our model
considers the details of the binding/unbinding process of one polymer on a
viscoelastic network, which is assumed to be a Maxwell material. We find that
contractility naturally results from the active (un)binding that violates DB,
together with the asymmetric force-extension relation that breaks spatial
symmetry. Notably, both of these features are generic for biopolymer networks
such as the cytoskeleton. In our model, the key lengthscale determining the
contractile force, is the binding site spacing $d$. Taking $d\simeq 10$nm, of
order the size of a globular protein or the spacing of binding sites on a
cytoskeletal filament, and $\delta\ell\simeq 6$nm, corresponding to an actin
filament of length $1\mu$m, our model predicts a maximum contractile force
$\sim 0.5\rm pN$. As a comparison, single myosin molecule produces a force
$\sim 3\rm pN$ Finer  _et al._ (1994), suggesting that our mechanism generates
a weaker but comparable force relative to motors. It is also worth emphasizing
that our mechanism is additive in that it can generate such contractility
among multiple filaments even in disordered networks.

We have explored various limits of our detailed model. Of special importance
are the limits in which the details of the binding process are neglected (the
so-called coarse-grained model) which greatly simplifies the calculations and
allow for intuitive understanding of the contractile mechanism. Such
simplification is valid at $d\leq\delta\ell$, when our coarse-grained model
and microscopic model predict similar contractile velocities. However, their
predictions differ when $d\gg\delta\ell$: For the microscopic model there
exists an optimal $d$ value and the velocity decreases for $d$ exceeding that
optimal value, while for the coarse-grained model the velocity always
increases with $d$. This difference is due to the large elastic energy when
stretching the polymer in the large-$d$-limit. In the coarse-grained model,
the binding process deforms the polymer length with a binding probability,
regardless of the energy required for the deformation. In the microscopic
model, on the other hand, the deformation of the polymer length is limited by
the binding potential, and the change in the elastic energy cannot exceed
$\Delta E$. Therefore, in the large-$d$-limit the coarse-grained model fails.
Similar to molecular motors, a large enough $\Delta E$ is required to generate
considerable directional motion Jülicher  _et al._ (1997).

In our microscopic model, we show that the binding/unbinding on a Maxwell
substrate, is equivalent to the binding/unbinding on a viscous substrate with
a modified binding potential. The reduction naturally originates from the
definition of the Maxwell material, which is composed of a pure elastic part
and a pure viscous part. Since the elastic part responds to external stress
immediately, it can always be regarded as a fast variable, allowing us to only
consider the relaxation of the viscous part. Therefore, this result does not
rely on any specific assumption, nor is it limited to the particular model
considered in this paper. Rather, it can be applied to any binding/unbinding
events on Maxwell substrates, whether the binding/unbinding is in or out of
equilibrium. Moreover, the binding site spacing of the modified binding
potential remains unchanged for any substrate elastic rigidity. Because the
binding site spacing is the only parameter associated with the substrate
binding potential in the coarse-grained limit, when $d\leq\delta\ell$ the
microscopic model should be reduced to the coarse-grained model with a viscous
substrate, regardless of the original substrate elastic rigidity.

In this work, we have focused on the Maxwell substrate, which behaves as solid
at short times and fluid at long times. Another well-known viscoelastic model
is the Kelvin-vorgit substrate, which can be applied to our model as well. The
Kelvin-vorgit substrate behaves as fluid in short time and solid in long time.
Therefore, at short times we expect the Kelvin-Vorgit substrate to deform in
the same way as the viscous substrate that has been discussed in Sec. IV,
while at long times the substrate stops deformation due to its own elasticity.
We choose not to study in detail the Kelvin-Vorgit substrate because most
biomaterials, including the cytoskeleton and the extracellular matrix, are
known to fludize at long times Broedersz  _et al._ (2010); Ijima  _et al._
(2018). Therefore, the Maxwell model is more appropriate for the substrate we
have in mind that are formed by biomaterials. However, the Maxwell model does
impose an instataneous elastic response of the substrate, while in general a
viscoelastic material, such as the Burgers model, needs a finite time to build
up its elastic stress. In our model we assume the elastic relaxation time of
the substrate to be much smaller than the unbinding time such that we can
consider it to be instataneous.

We have only discussed the Maxwell substrate with a single relaxation time for
simplicity, however, our mechanism should also generate contraction in Maxwell
substrate with multiple relaxation times Wiechert (1889); Broedersz  _et al._
(2010); Chen  _et al._ (2021). Our discussion about the relaxation time (see
Sec. IV) suggests that contractility will be observed as long as there exists
one relaxation time that is larger than or comparable to the unbinding time.

The contractile mechanism proposed in our model requires activity that breaks
time-reversal symmetry together with a spatial asymmetry that directs the
active motion. Such Brownian-ratchet-like mechanism is similar to the
enzymatic cycle of molecular motors, which is fueled by the hydrolysis of ATP
Jülicher  _et al._ (1997). However, there exists a distinct difference between
the motor and non-motor mechanisms, which is the origin of the broken spatial
symmetry. The asymmetry for motors comes from the geometrical polarity of the
substrate filaments they bind to, which directs the motor motion from well
defined plus (minus) end to the other end Magnasco (1993). Because the motor
motion is not directly related to contractility, it may require a specific
network architecture to generate contraction Ennomani  _et al._ (2016). On the
contrary, the non-motor mechanism in this paper does not rely on substrate
polarity. Instead, it relies on the mechanical asymmetry that is generic for
any semiflexible biopolymer. In addition to semiflexible polymers, polymers
with small persistence length may also possess similar mechanical asymmetry in
the opposite limit of semiflexibility, e.g., DNA Marko and Siggia (1994). Such
asymmetry leads to a directed motion that is always towards the _contractile
direction_ , resulting in contractility regardless of network architecture.
Moreover, the non-motor mechanism may even generate contractility on apolar
filaments, such as intermediate filaments, which has not been thought
possible.

Interestingly, although in this paper we have only considered the mechanical
asymmetry on substrate filaments, the crosslinking proteins may also be
responsible for such asymmetry if they are soft enough, e.g., filamin.
Contractility has been observed in disordered/apolar actin bundles when the
network is crosslinked with filamin Weirich  _et al._ (2017), although the
active, non-equilibrium aspect of this system is unclear. It is even possible
that other apolar filaments like septin may be responsible for force
generation, since they play a role in the contractile ring Mavrakis  _et al._
(2014); Valadares  _et al._ (2017).

Despite the different origins of spatial asymmetry, the motor and non-motor
mechanisms are not mutually exclusive. Rather, the non-motor mechanism should
be considered as an additional mechanism to the motor mechanism. The non-motor
mechanism is able to generate contractility on both polar and apolar
substrates. In our model, we have assumed apolar substrates as a worst-case
scenario in which motors do not work. However, considering a polar substrate
within our model is straightforward, where one should replace the symmetric
binding potential in Sec. IV with an asymmetric one, in which case both the
motor and non-motor mechanisms can take place simultaneously. In such a case,
the non-motor mechanism enhances the contractility of the motor mechanism. The
effects of the two mechanisms may be distinguished from their different
$\tilde{d}$ dependence. The motor-driven contractility usually requires the
polymer to buckle e Silva  _et al._ (2011); Lenz  _et al._ (2012), which only
happens when the tension exceeds $\tau_{0}$. Because
$\tau_{0}\sim\delta\ell^{-1}$, the motor-driven contractility strengthens with
decreasing $\tilde{d}=d/\delta\ell$. On the other hand, the contraction of
non-motor mechanism, as we have shown for both coarse-grained and microscopic
models, increases with increasing $\tilde{d}$ when $d\leq\delta\ell$, which
shows opposite $\tilde{d}$-dependence to motor mechanism.

To conclude, our model may provide an explanation for recent observations of
non-myosin-dependent dynamics of the contractile ring during cell division or
cellularization Wloka  _et al._ (2013); Xue and Sokac (2016). It is possible
that ATP-dependent crosslinking by myosin in disordered actin networks or
other structures may generate contractility Hatwalne  _et al._ (2004); Shen
and Wolynes (2006); Wang  _et al._ (2011); e Silva  _et al._ (2011); Alvarado
_et al._ (2013); Tjhung  _et al._ (2013); Markovich  _et al._ (2019), without
the need for a motor _power stroke_. Our model suggests a generic, steady-
state mechanism for contraction even in apolar or fully disordered structures.

###### Acknowledgements.

_Acknowledgments_ : This work was supported in part by the National Science
Foundation Division of Materials Research (Grant No. DMR-1826623) and the
National Science Foundation Center for Theoretical Biological Physics (Grant
No. PHY-2019745). The authors acknowledge helpful discussions with A. Abdalla,
M. Gardel, C. Schmidt and J. Theriot.

##  Appendix A Estimation of $\tau_{r}$ and $\tau_{\rm hop}$

In this appendix we derive the expressions of $\tau_{r}$ and $\tau_{\rm hop}$
that are used in Sec. IV. $\tau_{r}$ and $\tau_{\rm hop}$ are two
characteristic timescales of the modified binding potential $U_{b}^{v}$, whose
height is $\Delta E^{v}$. Since $U_{b}^{v}$ does not have an analytical
expression, to obtain an estimation of the two timescales we approximate
$U_{b}^{v}$ by a triangular potential with height $\Delta E^{v}$ and
periodicity $d$. $\tau_{r}$ is the time required for the polymer end to relax
within one binding site. Let the binding site be within $(-d/2,d/2)$ and
consider the diffusion of a particle with mobility $m_{x}$ within a triangular
binding potential. We use the “intrawell relaxation time” introduced in Ref.
Derényi and Astumian (1999) to estimate $\tau_{r}$. It is defined as the
average mean-first-passage time of the particle from any fixed initial
position $x_{0}$, to a final position $x$ that is sampled from a Boltzmann
distribution governed by $U_{b}^{v}$:

$\displaystyle\tau_{r}$
$\displaystyle=\frac{1}{k_{B}TZm_{x}}\int_{-d/2}^{d/2}\mathop{}\\!\mathrm{d}x\int_{x}^{d/2}\mathop{}\\!\mathrm{d}y\int_{y}^{d/2}\mathop{}\\!\mathrm{d}z$
$\displaystyle\times\exp\Big{[}\left[-U_{b}^{v}(x)+U_{b}^{v}(y)-U_{b}^{v}(z)\right]/k_{B}T\Big{]}$
$\displaystyle\approx\frac{d^{2}}{2k_{B}Tm_{x}}\left(\frac{k_{B}T}{\Delta
E^{v}}\right)\,,$ (A1)

where $Z=\int_{-d/2}^{d/2}\mathop{}\\!\mathrm{d}x\exp[-U_{b}^{v}(x)]$ is the
partition function. In Eq. (A1), the integrals over $\mathop{}\\!\mathrm{d}y$
and $\mathop{}\\!\mathrm{d}z$ calculate the mean-first-passage time from
$x_{0}$ to $x$, and the integral over $\mathop{}\\!\mathrm{d}x$ calculates the
average mean-first-passage time with the Boltzmann weight of $U_{b}^{v}(x)$.
Note that $\tau_{r}$ is independent of the initial position $x_{0}$.

$\tau_{\rm hop}$ is the average time for the polymer end to hop to another
binding site. The time required to escape from a potential well can be
estimated by the mean-first passage time from the bottom of the well to the
top of the well, which is Gardiner (2009):

$\displaystyle\tau_{\rm hop}$
$\displaystyle=\frac{1}{k_{B}Tm_{x}}\int_{0}^{d/2}\\!\\!\\!\mathop{}\\!\mathrm{d}y\int_{-d/2}^{y}\\!\\!\\!\mathop{}\\!\mathrm{d}z\exp\Big{[}[U_{b}^{v}(y)-U_{b}^{v}(z)]/k_{B}T\Big{]}$
$\displaystyle\approx\frac{d^{2}}{2k_{B}Tm_{x}}\exp\left(\frac{\Delta
E^{v}}{k_{B}T}\right)\,.$ (A2)

We then conclude that for $\Delta E^{v}\gg k_{B}T$, we have
$\tau_{r}\ll\tau_{\rm off}\ll\tau_{\rm hop}$. It can be understood
intuitively: in the large $\Delta E^{v}$ limit, the potential well is steep
enough thus driving a fast relaxation within the potential well, while the
high potential barrier prevents hopping towards another binding site.

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