# A generic self-stabilization mechanism for biomolecular adhesions under load

Andrea Braeutigam, Ahmet Nihat Simsek, Gerhard Gompper  Theoretical Physics of
Living Matter, Institute for Biological Information Processes,
Forschungszentrum Jülich, 52425 Jülich, Germany  Benedikt Sabass  Theoretical
Physics of Living Matter, Institute for Biological Information Processes,
Forschungszentrum Jülich, 52425 Jülich, Germany  Institute for Infectious
Diseases and Zoonoses, Department of Veterinary Sciences, Ludwig-Maximilians-
Universität München, 80752 Munich, Germany

###### Abstract

Mechanical loading generally weakens adhesive structures and eventually leads
to their rupture. However, biological systems can adapt to loads by
strengthening adhesions, which is essential for maintaining the integrity of
tissue and whole organisms. Inspired by cellular focal adhesions, we suggest
here a generic, molecular mechanism that allows adhesion systems to harness
applied loads for self-stabilization under non-equilibrium conditions –
without any active feedback involved. The mechanism is based on conformation
changes of adhesion molecules that are dynamically exchanged with a reservoir.
Tangential loading drives the occupation of some stretched conformation states
out of equilibrium, which, for thermodynamic reasons, leads to association of
further molecules with the adhesion cluster. Self-stabilization robustly
increases adhesion lifetimes in broad parameter ranges. Unlike for catch-
bonds, bond dissociation rates do not decrease with force. The self-
stabilization principle can be realized in many ways in complex adhesion-state
networks; we show how it naturally occurs in cellular adhesions involving the
adaptor proteins talin and vinculin.

##  I Introduction

Figure 1: Adhesion self-stabilization. a) Illustration of the basic adhesion
model which consists of unbound molecules $a$ and bonds $b$ that connect two
parallel rigid planes. A tangential load stretches all molecules in state $b$
as it shifts the upper boundary by $s$. Molecules can transition between
states $b$ and $a$ with extension-dependent rupture and binding rates given by
$\beta^{-}(h)$ and $\beta^{+}(h)$. Molecules can enter or leave the adhesion
cluster with rate constants given by $\gamma^{\pm}$. b) Mean number of bonds
in steady state as a function of loading force. Symbols show simulation
results and lines correspond to approximate analytical results. In the basic
model, an increased load on the adhesion reduces the number of bonds (red). In
contrast, an increased load produces a growth of the number of bonds in the
generalized model (blue). c) Illustration of the generalized adhesion model
incorporating molecule unfolding and refolding with rates
$\delta_{a,b}^{\pm}(h)$ as well as a molecule-exchange with the reservoir.
Mechanical load drives the system out of equilibrium, shifting the state
occupations. See also SI movies 1,2.

Multicellular organisms are held together by complex biomolecular adhesion
structures. For decades, cellular adhesions have been extensively studied,
both, as a paradigm for fundamental biophysical mechanisms, as well as to
understand their essential biological function. Yet, a very fundamental
property of cell-matrix adhesions remains mysterious – their ability to adapt
their size to the mechanical load. How is it possible – from a physics
perspective – that the strength of adhesions increases under load?

Cell-matrix adhesions, also called focal adhesions, are crucial for cell
physiology [1, 2], cell motility [3], cancer metastasis [4, 5], and
development [6, 7]. The structures consist of transmembrane integrins and
adaptor proteins that connect the force-generating actomyosin cytoskeleton
with the extracellular matrix. Accordingly, focal adhesions have been likened
to a “molecular clutch” [8, 9, 10, 11, 12, 13]. In a pioneering experiment by
Riveline et al. [14], it was shown that local application of centripetal
forces to adherent cells induces focal adhesion growth [14]. Moreover, the
size of focal adhesions is proportional to the load [15]. The regulatory
network associated with focal adhesions is complex and several biological
processes have been proposed to play a role for adhesion stability. These
include mechanosensitive activation of integrins [11], catch-bond behavior of
integrins and vinculin-actin binding [16], non-linear mechanical response of
unfolded proteins, and downstream signaling, e.g., mediated by the adaptor
protein p130Cas [17, 18]. During the past years, the pivotal role of the
adaptor protein talin for adhesion maturation has also been established [19,
20]. Talin directly transmits forces by binding with its globular head domain
to integrin, while its rod domain links to F-actin [21]. Under stretch,
conformation changes in talin occur, leading to an unfolding of protein
domains and to the exposure of cryptic binding sites for vinculin [22, 23,
24]. Vinculin, in turn, further recruits F-actin and thereby strengthens the
linkage [25, 26, 27]. Moreover, other adhesion types such as adherens
junctions are also capable of a load adaptation based on unfolding and
recruitment of further molecules [28]. In spite of the considerable amount of
theoretical approaches [29, 30, 31, 32, 33, 34, 35, 36, 37] and pioneering
work combining modeling and experiment [12, 38, 39], the physical principle
underlying load-adaptation of focal adhesions remains largely not understood.

We propose here a generic mechanism through which molecular adhesions can
harness mechanical load for adapting their size and stability without active
feedback. A minimal model is employed, which relies on a combination of
unfolding of adhesion molecules under force [39, 40], the dynamical exchange
of molecules with a reservoir [29, 30], and possibly the recruitment of
additional molecules that stabilize the unfolded conformations, all under the
constraint of thermodynamical consistency. The key property of our molecular
state-networks is that some states are separated from the reservoir and driven
out of equilibrium. Under tangential stress, the state occupations shift,
leading to a growth of the adhesion under increasing load, until the adhesion
ultimately fails at very high loads. This non-equilibrium mechanism of
adaptation is simple and robust. We show how the mechanism is naturally
realized by the talin-vinculin system at focal adhesions and perhaps in other
bioadhesions. Moreover, the proposed mechanism is general enough that it could
possibly be implemented in engineered biomimetic systems.

##  II Results

Molecular adhesion model. We consider a generic adhesion system consisting of
$N$ molecules that form harmonic bonds between two planar, parallel surfaces,
see Fig. 1. The molecule extension, i.e., the difference between actual length
and rest length, is denoted by $h$ and the spring constant by $\kappa$. The
bottom surface is fixed in space and a constant tangential loading force $F$
is exerted on the top, leading to a time-dependent tangential shift $s$. The
model is two-dimensional and forces normal to the surface are not considered.
The adhesion-molecule number $N$ can vary by exchange with a reservoir, as
determined by the rate constants $\gamma^{\pm}$. Individual molecules undergo
stochastic transitions between the adhesion cluster and the reservoir, and all
transition rates are chosen to fulfill detailed balance when $F=0$, i.e., in
thermal equilibrium, see Methods IV, which avoids unphysical energy injection
that could produce an apparent motor-like behavior.

In a first, basic model, see Fig. 1a, molecules from the reservoir associate
reversibly with the adhesion system via the state $a$, in which they have not
yet formed a bond between the upper and lower surfaces. The state where a bond
has formed is denoted by $b$. The bond formation rate is $\beta^{+}(h)$, which
is maximal when the extension of the adhesion molecule equals an optimal
binding distance $|h|=\ell_{\mathrm{b}}$. For bond formation, it is assumed
that the extension $h$ fluctuates thermally with magnitude
$\sigma=\sqrt{k_{\mathrm{B}}T/\kappa}$, where $k_{\mathrm{B}}T$ is the thermal
energy scale. For bond dissociation, we focus on slip-bond dynamics with
rupture rates $\beta^{-}(h)$ that increase exponentially with bond extension,
see Secs. IV A-B.

In a second, generalized model, see Fig. 1c, the molecules can undergo abrupt
conformational changes to unfolded states, denoted by a subscript $u$, such
that $a_{\mathrm{u}}$ are the unfolded, unbound states and $b_{\mathrm{u}}$
the unfolded, bound states. The overall number of bound molecules,
irrespective of their conformational state, is
$N_{B}=N_{b}+N_{b_{\mathrm{u}}}$, and the corresponding overall number of
unbound molecules is $N_{A}=N_{a}+N_{a_{\mathrm{u}}}$. Details of the reaction
rates are given in Methods IV A-C. We assume that mechanical relaxations occur
instantaneously and viscous damping is neglected, so that the sum of the
forces borne by the bonds equals the applied load $F$ at all times. Stochastic
bond dynamics is simulated with an exact algorithm, see Supplementary
Information II.

Figure 2:  Comparison of models I-IV for adhesions with and without molecule
unfolding and reservoir exchange. Steady-state quantities are plotted until
forces at which first complete adhesion ruptures occur. a) Averaged total
number of molecules $N$. b) Averaged number of bonds $N_{B}$ in steady state.
Only model IV shows self-stabilization where the mean number of bonds
increases initially with force. c) Continuous rupture-rebinding transitions
lead to a relative motion of the two planes bounding the adhesion. Self-
stabilization reduces the motion. d) Relative deviation of the average number
of unbound molecules $N_{A}$ from equilibrium. Note the increased molecule
accumulation in the $a$ states for self-stabilization. See Supplementary
Information I for parameters.

Self-stabilization of macromolecular adhesions. We first consider the basic
adhesion model, in which molecules do not change conformation, see Fig. 1a and
SI movie 1. Simulations reveal that a load $F$ can lead to a quasi-stationary
adhesive state, where perpetual rupture and binding events result in a
stationary state of persistent tangential sliding of the surfaces. For these
adhesions, the mean number $\langle N_{B}\rangle$ of bonds always decreases
monotonically with $F$, see Fig. 1b. Therefore, increasing load on adhesions
consisting of simple molecules promotes adhesion failure characterized by
rupture of all bonds. Next, we consider the generalized adhesion model
consisting of molecules that can undergo an unfolding transition under force,
see Fig. 1c and SI movie 2. For simplicity, we assume that unfolding only
entails an increase in the rest length, while the elastic properties remain
unchanged. Remarkably, the mean number $\langle N_{B}\rangle$ of bonds now
initially grows with increasing load $F$, see Fig. 1b, which prevents early
adhesion failure. This striking effect, which we call “self-stabilization”, is
the central finding of this work. We want to emphasize that this effect
crucially depends on the exchange of “folded” molecules (state $a$) with the
reservoir, so that the transitions $a\rightarrow b\rightarrow b_{\mathrm{u}}$
lead to a depletion of $a$ molecules, which can be replenished from the
reservoir, thereby leading to an increase of the adhesion size. This mechanism
is explained in detail below.

The simulation results can be corroborated with a mean-field approximation,
which can be solved analytically [33]. The stationary distributions of
molecules with extension $h$ in the bound and unbound states are denoted by
$n_{b}(h)$ and $n_{a}(h)$, respectively. For the basic adhesion model without
molecule unfolding, a drift-reaction equation is assumed where the average
sliding velocity of the adhesion $v=\langle\dot{s}\rangle$ stretches the
molecules, such that

$\partial_{t}n_{b}(h)+v\partial_{h}n_{b}(h)=\beta^{+}(h)n_{a}(h)-\beta^{-}(h)n_{b}(h)\,.$
(1)

Only stationary solutions with $\partial_{t}n_{b}(h)=0$ are considered. The
total number of molecules in the adhesion is obtained as
$N=N_{B}+N_{A}=\int_{-\infty}^{\infty}\\![n_{b}(h)+n_{a}(h)]\operatorname{d\\!}{h}$,
where the extension of the unbound molecules obeys a Gaussian distribution,
$n_{a}(h)\propto\mathcal{N}(0,\sigma_{a}^{2})$. The non-linear equations are
solved by expanding the distributions for small absolute values of
$\tilde{v}=v/(k_{\beta}\sigma_{b})$, where the binding constant $k_{\beta}$ is
employed as time unit and the extension variance of bound molecules
$\sigma_{b}^{2}$ as length unit, e.g.,
$n_{b}(h)=n_{b}^{*}(h)+\tilde{v}n_{b_{1}}(h)+\frac{1}{2}\tilde{v}^{2}n_{b_{2}}(h)+\mathcal{O}(\tilde{v}^{3})$.
The asterisk $(^{*})$ here and in the following denotes equilibrium quantities
calculated with $F=0$. Using the additional assumption that the optimal
molecule extension for binding, $\ell_{\mathrm{b}}$, is much smaller than the
typical length fluctuations,
$\tilde{\ell}_{\mathrm{b}}=\ell_{\mathrm{b}}/\sigma_{b}\ll 1$, we find

$N_{B}-N_{B}^{*}\approx-(2/\pi)^{1/2}\tilde{\ell}_{\mathrm{b}}N_{B}^{*}\,\tilde{v}^{2}\propto-F^{2}\,,$
(2)

and because, due to symmetry under reversal of the force direction, to leading
order $\tilde{v}\propto\\!F$. For the general case, where
$\ell_{\mathrm{b}}\ll\sigma_{b}$ does not hold, the first non-vanishing
correction to the equilibrium solution for the bonds $N_{B}$ can be shown to
still be of the order $\tilde{v}^{2}$ and strictly negative, see Supplementary
Information III. Thus, tangential force reduces the number of bonds and
thereby always destabilizes simple adhesions consisting of molecules that do
not undergo conformation changes, as expected intuitively.

To support the effect of self-stabilization in the generalized model shown in
Fig. 1c with analytical theory, we supplement Eq. (1) by two additional
equations for binding and unfolding transitions, see Supplementary Information
IV. For $|\tilde{v}|\ll 1$, the overall number of bound molecules,
$N_{B}=N_{b}+N_{b_{\mathrm{u}}}$, can be expanded as

$N_{B}-N_{B}^{*}\approx N_{B,1}\tilde{v}+N_{B,2}\,\tilde{v}^{2}/2\propto|F|.$
(3)

The leading contribution is linear in $\tilde{v}$ with a positive coefficient
$N_{B,1}$ (obtained from numerical analysis), as required for self-
stabilization! The second-order coefficient $N_{B,2}$ can be positive or
negative, see Supplementary Information Fig. S3. Hence, analytical models
confirm the existence of a self-stabilization regime where the number of
adhesion bonds initially increases with load.

Mechanism of self-stabilization. To obtain deeper insight into the necessary
ingredients for self-stabilization, and to demonstrate the essential
contribution of the reservoir, we compare several variants of the basic
models, see Fig. 2 — such as a simple adhesion model with fixed molecule
number (model I), a model comprising a molecule reservoir and therefore
adhesions of variable size (model II), a model with a fixed number of
molecules that can unfold under force (model III), and a model combining
unfolding molecules with a molecule reservoir (model IV). Some results from
models II and IV are also shown in Fig. 1. For models I-III, the mean number
of bonds $N_{B}$ decreases with force, see Fig. 2b. For model II with variable
system size, even the number of molecules in the adhesion decreases with
force, leading to an earlier adhesion failure on average. Thus, neither a
variable adhesion size nor molecule unfolding alone result in self-
stabilization. In model IV, which combines a variable adhesion size with
molecule unfolding, both the total number of molecules $N$ and the number of
bonds $N_{B}$ initially grow with increasing load on the adhesion, Figs. 2a,b.
The increased number of bonds improves load sharing among the molecules. One
consequence is a significant reduction of the sliding motion of the adhesion,
Fig. 2c. The inset in Fig. 2b shows that the bound fractions of molecules,
$N_{B}/N$, as a function of $F$ collapse onto a single master curve for all
models (I-IV). Hence, self-stabilization results from force-induced growth of
the adhesion and not from changes of the rupture properties of individual
molecules. This is the key difference to established catch-bond models, where
individual molecules exhibit an increase of bond lifetime within limited force
regimes.

Figure 2d illustrates the underlying mechanism of self-stabilization. The
tangential load $F$ causes a continuous molecular-state turnover with
recurring stretch, unfolding, and rupture of molecules along the transitions
$a\rightarrow b\rightarrow b_{\mathrm{u}}\rightarrow a_{\mathrm{u}}$. Overall,
the load increases the occupation of state $a_{\mathrm{u}}$. Meanwhile, the
state $a$, representing unbound, folded molecules, is in contact with the
reservoir and molecules are replenished here, which allows a concurrent
increase of the overall molecule number. The recruitment of molecules from the
reservoir crucially depends on the intermediate state $a_{\mathrm{u}}$ not to
be equilibrated, see Suppl. Figs. S6,S8, and to rebind and contribute to load
sharing again via $a_{\mathrm{u}}\rightarrow b_{\mathrm{u}}$. It is important
to note that without the reservoir, self-stabilization cannot occur because it
requires growth of the adhesion cluster. Generically, the principle behind
self-stabilization is that molecular-state occupation statistics are driven
out of equilibrium in a way that results in further influx of molecules from a
reservoir.

Figure 3:  Rupture behavior and lifetimes of adhesion clusters. a) The
reservoir-exchange rates $\gamma^{\pm}$ control the association and
dissociation of molecules with the adhesion. b) Exemplary force-response of
adhesion clusters without reservoir connection and with strong reservoir
connection. A force jump amplifies molecule unfolding. Without self-
stabilization, the cluster does not reach a non-equilibrium steady state but
dissociates shortly after force application. c) Average lifetimes of the
adhesion clusters with $N^{*}\approx 10$ for different values of
$\gamma^{\pm}$ with $\gamma=\gamma^{+}/\gamma^{-}$ held constant. A strong
reservoir connection results in an adhesion lifetime maximum at finite, non-
vanishing external forces. d) Average cluster lifetimes as a function of force
and the unfolding length $\Delta$ for $k_{\delta}=0.1,\,k_{\beta}=1/t_{0}$ and
$\gamma^{\pm}=0$ (i) and $\gamma^{\pm}=100/t_{0}$ (ii). See Supplementary I
for other parameter values.

Self-stabilization is a robust mechanism. To be an efficient and universally
applicable mechanism, self-stabilization must be able to compensate load
changes in non-stationary conditions and should not depend on a fine-tuning of
parameters. To investigate these aspects, we consider step-like load changes
for different model parameters. After a load jump from $F_{1}=0$ to $F_{2}=F$,
adhesion clusters either dissociate quickly or reach a new non-equilibrium
steady state. Self-stabilization after a load jump depends on the strength of
the molecule exchange with the reservoir, which is controlled by the values of
$\gamma^{+}$ and $\gamma^{-}$, Fig. 3a. Exemplary trajectories for the number
of bonds in the folded and unfolded states, $b$ and $b_{\mathrm{u}}$, are
shown in Fig. 3b. The most-likely rupture forces are higher in self-
stabilizing than in non-self-stabilizing adhesions, and grow with increasing
reservoir-exchange rates ($\gamma^{\pm}>k_{\beta}$), see Supplementary Fig.
S4, because adhesion molecules can quickly be recruited from or released into
the reservoir. The self-stabilization mechanism can thus also work under
dynamic load conditions.

To measure adhesion lifetimes, we simulate systems consisting of few
molecules, because lifetimes increase rapidly with the number of molecules.
Load-jump simulations are carried out for a reservoir-exchange rate ratio
$\gamma=1$, which leads to adhesions with $N^{*}\approx 10$ molecules in
equilibrium. Lifetime is measured as the time from the force jump,
$F_{1}\rightarrow F_{2}$, to the rupture of the last adhesion bond. While
lifetimes of adhesions with no reservoir coupling decrease monotonically with
force, i.e., show a pure slip-bond behavior, lifetimes of self-stabilizing
adhesions exhibit a maximum at non-vanishing forces, see Fig. 3c. This
lifetime maximum becomes more pronounced for increasing $\gamma^{+}$ and also
depends on the rate-constant ratio $k_{\delta}/k_{\beta}$, see Supplementary
Fig. S5. To further assess the robustness of the adhesion-lifetime increase to
parameter choices, we vary the unfolding length $\Delta$, which determines the
width of the energy barrier between the native and the unfolded molecule
state. For adhesions without reservoir coupling, $\gamma^{\pm}=0$, the
unfolding length $\Delta$ does not have a large impact on the adhesion
lifetime, see Fig. 3d(i). However, a significant increase in adhesion
lifetimes is observed for a broad range of unfolding lengths $\Delta>0$ if the
reservoir coupling is strong, see Fig. 3d(ii). Self-stabilization is less
effective for very small or very large values of $\Delta$, where native and
unfolded state become indistinguishable or the unfolded state becomes
inaccessible, respectively.

Figure 4:  Exemplary realization of self-stabilization in focal adhesions. a)
Talin binds with its globular head to integrin and with its rod to actin
filaments. The first domain to unfold under force is the R3 domain with two
vinculin-binding sites. b) State diagram for talin molecules. c-d) Averaged
steady-state simulation results for different linking ratios
$\lambda=\lambda^{+}/\lambda^{-}$. See Supplementary I for parametrization. c)
Vinculin recruitment produces self-stabilization. d) The relative motion of
the top plane modeling actin fibers is reduced through vinculin-based self-
stabilization.

Cell-matrix adhesions. The principle of self-stabilization can be realized in
a large variety of molecular-state networks — as long as they allow for non-
equilibrated states that are increasingly populated under mechanical stress.
As stated in the introduction, talin unfolding and its interaction with other
focal-adhesion proteins have been identified as major contributors to adhesion
formation. We study here a model of self-stabilization that incorporates the
binding of an adaptor protein such as vinculin to talin, see Fig. 4a. The
talin rod domain contains 11 cryptic vinculin binding sites. Under load,
subdomains of the rod successively unfold, thereby enabling vinculin
recruitment which blocks talin refolding and promotes focal-adhesion growth
[41, 42]. Talin unfolding typically starts at forces around
$5\text{\,}\mathrm{pN}$ [22, 41, 25] with the R3 domain, by which two vinculin
binding sites are exposed. In our model, we focus on this first unfolding
transition. Six additional vinculin-bound states $b_{\mathrm{u},}$ and
$a_{\mathrm{u},}$ are introduced as both sites in the R3 domain can be
occupied independently, see Fig. 4b. Rate constants for binding and unbinding
of vinculin to unfolded talin are denoted by $\lambda^{\pm}$ and are assumed
to be the same for all transitions. Other model parameter values are estimated
according to experimental results [21, 23, 40], see Supplementary Information
I.

In the vinculin-bound states, talin refolding is blocked. Thus, vinculin
binding generates talin states that are not in direct contact with the
reservoir and can therefore be driven out of equilibrium. Simulation results
for different values of the parameter $\lambda=\lambda^{+}/\lambda^{-}$ are
shown in Fig. 4c. For $\lambda=0$, all unbound states are equilibrated with
the reservoir and no self-stabilization occurs. For $\lambda>1$, the number of
bonds increases with force, which not only stabilizes the adhesion but also
translates into a reduction of adhesion sliding, see Fig. 4d. Given the high
affinity with a dissociation constant in the range
$${10}^{-7}$-${10}^{-8}\text{\,}\mathrm{M}$$ of unfolded talin for vinculin,
we expect the vinculin binding constant to be larger than unity [43, 44, 45,
40]. Moreover, we conjecture that the remaining cryptic binding sites in talin
that open at higher forces extend the demonstrated self-stabilization effect
to larger loads. In summary, the minimal adhesion model can be applied to
specific biological systems like the integrin adhesome, where vinculin binding
to unfolded talin domains results in a reaction network containing molecular
states that are populated by force application — and thus leads to self-
stabilization. Analytical results of a simplified, corresponding mean-field
model is given in Supplementary Information VI.

##  III Discussion

Our theoretical model reveals a strikingly simple mechanism for a counter-
intuitive load-response of adhesions, in which tangential mechanical load can
result in adhesion enhancement — instead of the ubiquitous adhesion weakening
and rupture. This self-stabilization relies on molecular-conformation state-
networks that are driven out of equilibrium by a mechanical load. By shifting
the state occupations, the load causes a net influx of adhesion molecules from
a surrounding reservoir. Notably, this self-stabilization does not require
extra chemical energy, but the non-equilibrium conditions produced by the load
suffice.

The primary motivation for our theoretical work are experimental observations
on integrin-based focal adhesions, which display size adaptation to the
applied load in planar cell cultures [14, 15]. Different focal adhesion
stabilization mechanisms presumably act in parallel, including actin
polymerization, transcription regulation, integrin activation, and
conformation changes of the adaptor protein talin. Contrasting to this
complexity, we predict that adhesion self-stabilization emerges naturally in
systems that merely incorporate the unfolding transition of the adhesion
molecules, like talin, and a mechanism which prevents rapid bulk-exchange of
unfolded states, e.g., due to vinculin binding. Since the proposed models
respect basic physical constraints, such as the detailed balance in thermal
equilibrium, no energy is artificially injected into the system. Additional
chemical driving, e.g., through the Rap1-GTP–interacting adaptor molecule [46]
or phosphorylation of vinculin or paxillin [47, 48], can provide an additional
layer of biological control over the suggested adhesion-stabilization
mechanism.

Mechanosensitive conformation changes of adhesion-linked proteins and
subsequent recruitment of additional molecules are recurring motifs in many
fundamental adhesion structures besides focal adhesions, for instance adherens
junctions [49, 50, 41] and hemidesmosomes [51]. Motor proteins also undergo
mechanosensitive conformation changes and can form dynamical ensembles.
Therefore, we expect that the suggested non-equilibrium mechanism for self-
stabilization can help to decipher many physiological and pathophysiological
processes controlled by mechano-chemical factors, and may even allow novel
designs of bio-inspired, artificial adhesion systems.

##  IV Model and Methods

###  IV.1 Binding

Different states are allowed to have their own spring constants and
corresponding quantities are denoted by subscripts, e.g., $\kappa_{a}$ and
$\sigma_{a}=\sqrt{k_{\mathrm{B}}T/\kappa_{a}}$. The probability per unit time
for an unbound molecule to bind at an extension $h$ is given by the function

$\beta^{+}(h)\frac{n_{a}(h)}{N_{a}}=\frac{k_{\beta}}{\sqrt{2\pi}\sigma_{b}}\mathrm{e}^{\textstyle-\frac{(|h|-\ell_{\mathrm{b}})^{2}}{2\sigma_{b}^{2}}+\frac{\epsilon_{\mathrm{b}}}{k_{\mathrm{B}}T}}\,,$
(4)

where $k_{\beta}$ is the intrinsic binding rate constant and
$\epsilon_{\mathrm{b}}$ is a constant. Similar binding rate expressions have
been used previously [52]. The total binding rate is obtained by integration
over $h$ so that

$\beta^{+}=k_{\beta}\left(1+\mathrm{Erf}\left(\tilde{\ell}_{\mathrm{b}}/\sqrt{2}\right)\right)\mathrm{e}^{\tilde{\epsilon}_{\mathrm{b}}}\,,$
(5)

where $\mathrm{Erf}(x)$ denotes the error function and the dimensionless
quantities $\tilde{\ell}_{\mathrm{b}}=\ell_{\mathrm{b}}/\sigma_{b}$ and
$\tilde{\epsilon}_{\mathrm{b}}=\epsilon_{\mathrm{b}}/(k_{\mathrm{B}}T)$ are
used. Binding of unfolded molecules via $\beta_{\mathrm{u}}^{+}(h)$ is defined
analogously with the extension $h_{\mathrm{u}}$ of unfolded molecules.

###  IV.2 Unbinding

The rupture rate $\beta^{-}(h)$ is defined by the detailed-balance condition
in thermal equilibrium [53],

$\beta^{+}(h)/\beta^{-}(h)=\mathrm{e}^{\textstyle-h^{2}/(2\sigma_{b}^{2})+h^{2}/(2\sigma_{a}^{2})+\overline{\epsilon}_{\mathrm{b}}}\,,$
(6)

where
$\overline{\epsilon}_{\mathrm{b}}=\tilde{\epsilon}_{\mathrm{b}}+\ln(\sigma_{a}/\sigma_{b})$
is the effective binding affinity. The rupture rate results as

$\beta^{-}(h)=k_{\beta}\mathrm{e}^{\textstyle(2|h|\ell_{\mathrm{b}}-\ell_{\mathrm{b}}^{2})/(2\sigma_{b}^{2})}\,.$
(7)

The rupture rate of unfolded bonds, $\beta_{\mathrm{u}}^{-}(h)$, is defined
analogously with the extension $h_{\mathrm{u}}$ of unfolded molecules.

###  IV.3 Unfolding and refolding

The unfolding and refolding reaction is modeled as the transition between two
local energy minima separated by a single barrier. The distance to the barrier
is denoted by $\Delta_{1}$ for unfolding and by $\Delta_{2}$ for refolding.
Their sum is equal to the total unfolding length $\Delta$. The unfolding rates
are thus defined as

$\delta_{a,b}^{+}(h)=k_{\delta}\mathrm{e}^{\textstyle(2\Delta_{1}h-\Delta_{1}^{2})/(2\sigma_{a,b}^{2})-\tilde{\epsilon}_{\mathrm{f}}}\,,$
(8)

where $\tilde{\epsilon}_{\mathrm{f}}=\epsilon_{\mathrm{f}}/(k_{\mathrm{B}}T)$
is a constant energy contribution for the conformation change. The reverse
rates are given by

$\delta_{a,b}^{-}(h)=k_{\delta}\mathrm{e}^{\textstyle(-2\Delta_{2}h-\Delta_{2}^{2})/(2\sigma_{a,b}^{2})}\,.$
(9)

The ratio of unfolding to refolding rate of bonds is given by the energy
change of a bond going from extension $h$ to $h-\Delta$ as

$\delta_{b}^{+}(h)/\delta_{b}^{-}(h-\Delta)=\mathrm{e}^{\textstyle(2h\Delta-\Delta^{2})/(2\sigma_{b}^{2})-\tilde{\epsilon}_{\mathrm{f}}}\,.$
(10)

For unbound molecules, the total unfolding probability per time and bond is
given by
$\delta_{a}^{+}=k_{\delta}\exp\left(-\tilde{\epsilon}_{\mathrm{f}}\right)\,$.
The total refolding rate is $\delta_{a}^{-}=k_{\delta}\,$.

###### Acknowledgements.

BS acknowledges funding by the European Research Council (g.a.No. 852585)

## Author Contributions

A.B. and B.S. designed and performed the research; A.B., A.S., G.G. and B.S
discussed results and wrote the manuscript.

## Competing Interests statement

The authors declare no competing interests.

##  Supplementary Discussion I Parameters

The minimal model for self-stabilization presented in the main text is
motivated by focal cell-matrix adhesions. Model parameters were chosen
accordingly. Table S1 contains all model parameters and their numerical values
employed for the simulations if not stated otherwise in the text.

Variable |  Description |  sim. model |  sim. talin |  unit |
---|---|---|---|---|---
$k_{\mathrm{B}}T$ |  thermal energy |  $4.114$ |  $4.114$ |  $\mathrm{pN}\text{\,}\mathrm{nm}$ |
$\kappa_{a}$ |  spring constant |  $0.25$ |  $0.5$ |  $\mathrm{pN}\text{\,}{\mathrm{nm}}^{-1}$ |
$\kappa_{b}$ |  spring constant |  $0.25$ |  $0.5$ |  $\mathrm{pN}\text{\,}{\mathrm{nm}}^{-1}$ |
$\sigma$ |  thermal fluctuation length |  $4.057$ |  $2.868$ |  $\mathrm{nm}$ |
$k_{\beta}$ |  rate, binding |  $1$ |  $1$ |  $1/t_{0}$ |
$\ell_{\mathrm{b}}$ |  binding distance |  $1$ |  $1$ |  $\mathrm{nm}$ |
$\epsilon_{\mathrm{b}}$ |  energy, binding |  $1.50$ |  $3.00$ |  $k_{\mathrm{B}}T$ |
$k_{\delta}$ |  rate, folding |  $1$ |  $100$ |  $1/t_{0}$ |
$\epsilon_{\mathrm{f}}$ |  energy, folding |  $0.50$ |  $5.83$ |  $k_{\mathrm{B}}T$ |
$\Delta$ |  unfolding length |  $10$ |  $12$ |  $\mathrm{nm}$ |
$\Delta_{1}$ |  transition state distance, unfolding |  $5$ |  $7$ |  $\mathrm{nm}$ |
$\Delta_{2}$ |  transition state distance, refolding |  $5$ |  $5$ |  $\mathrm{nm}$ |
$\lambda^{+}$ |  rate, linking |  $\\{0,1,2\\}$ |  $\\{0.0,0.5,1.0,2.0\\}$ |  $1/t_{0}$ |
$\lambda^{-}$ |  rate, unlinking |  $1$ |  $1$ |  $1/t_{0}$ |
$\gamma^{+}$ |  rate, addition from reservoir |  $\\{20.0,14.5,11.4\\}$ |  $\\{1.,0.996,0.991,0.977\\}$ |  $1/t_{0}$ |
$\gamma^{-}$ |  rate, removal to reservoir |  $1$ |  $0.1$ |  $1/t_{0}$ |
Table S1: Model parameters and values employed for simulations.

In focal adhesions, mechanical force is transmitted along single or multiple
proteins linked in series. In the latter case, the overall spring constant
$\kappa_{b}$ will be dominated by the element with the smallest spring
constant. The stiffness of cellular adhesion proteins lies in the order of
$\mathrm{pN}\text{\,}{\mathrm{nm}}^{-1}$ [54, 55]. The employed value of
$0.25\text{\,}\mathrm{pN}\text{\,}{\mathrm{nm}}^{-1}$ is similar to values
used for previous adhesion models [56, 57, 58, 59]. The values of
$\kappa_{a,b}$ used for the talin simulations were chosen to match the
experimentally measured unfolding and refolding behavior, as explained below.
The thermal energy scale $k_{\mathrm{B}}T$ and the spring constant $\kappa$
determine the mean thermal fluctuation length $\sigma$, which is used as a
length unit.
Experimentally, turnover of talin is significantly faster than turnover of
integrin [60]. The binding and unbinding rates of talin are determined by the
intrinsic rate $k_{\beta}$, the optimal binding distance $\ell_{\mathrm{b}}$,
and the constant energy contribution $\epsilon_{\mathrm{b}}$. The intrinsic
binding rate constant $k_{\beta}$ sets the time unit $t_{0}$. The value of the
binding distance $\ell_{\mathrm{b}}$ agrees with the value employed in Ref.
[58] and is smaller than the fluctuation length. The strength of individual
bonds is determined by
$F_{0}=k_{\mathrm{B}}T/\ell_{\mathrm{b}}\approx$4\text{\,}\mathrm{pN}$$, as
the rupture rate depends on force $F$ as $\propto\exp(F/F_{0})$. The constant
$\epsilon_{\mathrm{b}}$ can be understood as an effective affinity parameter.
Its value is chosen to be rather low to fix the adhesion cluster size at
equilibrium, which is proportional to
$\exp(\epsilon_{\mathrm{b}}/k_{\mathrm{B}}T)$, compare Refs. [52, 61].
The unfolding and refolding rates depend on $k_{\delta}$,
$\epsilon_{\mathrm{f}}$, $\Delta$ and $\Delta_{1,2}$ with
$\Delta=\Delta_{1}+\Delta_{2}$. The energy contribution
$\epsilon_{\mathrm{f}}$ determines the ratio between folded and unfolded
molecules in equilibrium. It shifts the energy barrier between both states
that are separated by the distances $\Delta_{1}$ and $\Delta_{2}$. For the
simulations of talin molecules, the unfolding and refolding rate constants are
chosen in accordance with the experimental results in Ref. [23]. For a time
unit $t_{0}=$1\text{\,}\mathrm{s}$$, unfolding occurs at a rate of
$0.015\text{\,}{\mathrm{s}}^{-1}$ at zero force. The folding and unfolding
rates intersect at a force of $5\text{\,}\mathrm{pN}$, the value at which
talin R3 domain unfolding and refolding is observed [41, 23, 40].
The cross-linking by additional adaptor proteins is described by making use of
two rate constants $\lambda^{\pm}$. In the general simulation model, the value
of these constants lies within the same order of magnitude as the intrinsic
rate constants $k_{\delta}$ and $k_{\beta}$. For integrin-based adhesions,
this reaction is realized by the association of vinculin to unfolded talin
domains. The molecular interactions of talin and vinculin are highly complex.
At low forces, vinculin binding at unfolded talin domains strengthens the
adhesion [41, 62, 63]. The recruitment of vinculin is proposed to act as a
negative feedback loop that stabilizes the force acting on the complex [40].
Additionally, the vinculin-talin interaction also depends on the direction of
forces [64]. In our model, we assume linking-rate constants with a value
comparable to the intrinsic binding rate $k_{\beta}$. In recent experiments
addressing the association of vinculin with the talin R3 domain, vinculin
binding has been observed with a rate on the order of
${10}^{-1}\text{\,}{\mathrm{s}}^{-1}$ when sufficient tension was applied to
talin to induce unfolding [40].
The exchange of molecules with a reservoir is governed by the two rate
constants $\gamma^{+}$ and $\gamma^{-}$. Their ratio $\gamma$ is proportional
to the number of molecules in the adhesion-cluster at equilibrium, i.e., when
$F=0$. For comparability of the results of different models, the rate for
adding molecules was increased from $\gamma^{+}$ to
$\tilde{\gamma}^{+}=\gamma^{+}(1+\exp(-\epsilon_{\mathrm{f}}/k_{\mathrm{B}}T))$
for the model without unfolding, and similarly when linking is included. This
choice ensures the same number of molecules and bonds at $F=0$. As described
in the main text, the molecule-exchange with the reservoir plays a fundamental
role for the self-stabilization mechanism. A strong connection to the
reservoir allows the adhesion to grow with increasing force.

The construction of molecular state models for talin in integrin-based
adhesions poses a challenge due to the large number of states and due to the
fact that addition and removal of an adhesion molecule requires several steps
in vivo [65, 66]. Furthermore, several pathways have been suggested for the
recruitment of talin to integrin-based adhesions [67]. Values for the halftime
of fluorescence recovery after photobleaching (FRAP) experiments for talin in
focal adhesions are on the order of seconds or tens of seconds, depending on
the cell type and substrate stiffness [68, 48]. The exchange of single
molecules can therefore be expected to occur on timescales of seconds or even
faster. Hence, the values for $\gamma^{\pm}$ are chosen such, that they ensure
a frequent exchange of molecules with the reservoir, but still allow to
observe the processes within the cluster.

##  Supplementary Discussion II Stochastic simulations

Molecule-state trajectories are simulated with the Gillespie algorithm [69,
70]. Its basis is provided by the stochastic formulation of chemical kinetics
where the probability that a reaction $i$ will happen in the infinitesimal
time interval $\delta\tau$ is determined by the product of the available
reactants or reactant pairs $N_{i}$ and a parameter $\mu_{i}$. Assuming
$j\in[1,\dots,m]$ available reactions, the function $P(i,\tau)$ describes the
probability that the reaction $i$ is the first one to occur after a waiting
time $\tau$ in the next infinitesimal time interval. For $0\leq\tau<\infty$ we
have

$\displaystyle
P(i,\tau)=N_{i}\mu_{i}\exp\left(-\sum\limits_{j=1}^{m}N_{j}\mu_{j}\tau\right)\,.$
(11)

Starting from an initial configuration, the time for the next reaction and the
type of reaction are drawn according to $P(i,\tau)$ repeatedly. Force balance
is restored instantaneously after each event.
In response to the tangential external force, adhesion clusters either reach a
quasi-steady-state or dissociate. The time- and ensemble averages are
calculated for trajectories that do not lead to complete dissociation during
the simulation time. If not stated otherwise, $50$ trajectories are tracked
for $>${10}^{6}$$ single-bond transitions for the measurements of steady-state
quantities, during which the tangential force is held constant. To measure
adhesion lifetime, more than 200 cluster trajectories are simulated until
their dissociation.

##  Supplementary Discussion III Adhesion model without unfolding

The following mean-field equations are used to approximate the dynamics of the
stretch-dependent state-occupation numbers in the basic adhesion model without
molecule unfolding

$\displaystyle\diffp{}{t}N_{a}$
$\displaystyle=-\beta^{+}N_{a}+\int\limits_{-\infty}^{\infty}\\!\beta^{-}(h)n_{b}(h)\operatorname{d\\!}{}h-\gamma^{-}N_{a}+\gamma^{+}\,,$
(12) $\displaystyle\diffp{}{t}n_{b}(h)$
$\displaystyle=-v\diffp{}{h}n_{b}(h)-\beta^{-}(h)n_{b}(h)+\beta^{+}(h)n_{a}(h)\,,$
(13)

where we omitted explicit $t$-dependence in our notation. Since the unbound
molecules in state $a$ are assumed to relax quickly mechanically, their
extensions obey a Gaussian distribution. Equation (12) describes the mean
total number of unbound molecules. The mean number of bonds $n_{b}(h)$ evolves
according to Eq. (13). The drift term in (13) accounts for the average
relative velocity $v=\langle\dot{s}\rangle$ between the adhesion planes that
is due to the tangential force $F$. An expansion of $n_{b}(h)$ as

$n_{b}(h)=\sum\limits_{j=0}^{\infty}\frac{1}{j!}\,n_{b_{j}}(h)\,\tilde{v}^{j},$
(14)

with $\tilde{v}=v/(k_{\beta}\sigma_{b})$ solves the steady state conditions
$\diffp{}{t}N_{a}=0$ and $\diffp{}{t}n_{b}(h)=0$ if

$\displaystyle
n_{b_{0}}(h)=n_{b}^{*}(h)=\frac{\gamma^{+}}{\gamma^{-}}\frac{\beta^{+}(h)}{\beta^{-}(h)}\,p_{a}(h)\,,\quad
n_{b_{j}}(h)=-j\frac{n_{b_{j-1}}^{\prime}(h)}{\beta^{-}(h)}\quad\text{for}\
j>0\,,$ (15)

where $p_{a}(h)$ is a Gaussian function with zero mean and variance
$\sigma_{a}^{2}$ and the prime denotes the derivative with respect to the
extension $h$. The first correction $n_{b_{1}}(h)$ is an odd function, so that
its integral vanishes. The first non-vanishing correction is given by the term
for $j=2$. Up to second order, the integrated steady state solution for the
mean number of bonds $N_{B}=N_{b}$ is given by

$\displaystyle
N_{B}=\int\limits_{-\infty}^{\infty}\\!n_{b}(h)\mathrm{d}h=N_{B}^{*}\left(1-\sqrt{\frac{2}{\pi}}\tilde{\ell}_{\mathrm{b}}\mathrm{e}^{\tilde{\ell}_{\mathrm{b}}^{2}}\left(1-\sqrt{2\pi}\tilde{\ell}_{\mathrm{b}}\mathrm{e}^{2\tilde{\ell}_{\mathrm{b}}^{2}}\mathrm{Erfc}\left(\sqrt{2}\tilde{\ell}_{\mathrm{b}}\right)\right)\tilde{v}^{2}\right)\quad\mathrm{with}\
N_{B}^{*}=\frac{\gamma^{+}}{\gamma^{-}}\mathrm{e}^{\overline{\epsilon}_{\mathrm{b}}}\,,$
(16)

where $\tilde{v}=v/(\sigma_{b}k_{\beta})$ and
$\tilde{\ell}_{\mathrm{b}}=\ell_{\mathrm{b}}/\sigma_{b}$ have been used and
$\mathrm{Erfc}(x)$ denotes the complementary error function. This first
correction $\propto v^{2}$ is negative since the complementary error function
for an argument $x>0$ is bounded by

$\mathrm{Erfc}(\sqrt{2}\ell_{\mathrm{b}})\leq\frac{2}{\sqrt{\pi}}\frac{\mathrm{e}^{-2\tilde{\ell}_{\mathrm{b}}^{2}}}{\sqrt{2}\tilde{\ell}_{\mathrm{b}}+\sqrt{2\tilde{\ell}_{\mathrm{b}}^{2}+4/\pi}}<\frac{\mathrm{e}^{-2\tilde{\ell}_{\mathrm{b}}^{2}}}{\sqrt{2\pi}\tilde{\ell}_{\mathrm{b}}}\,.$
(17)

The force-balance equation
$F=\int_{-\infty}^{\infty}n_{b}(h)\kappa_{b}h\,\mathrm{d}h$ connects the
external force with the resulting mean velocity $v$. The lowest-order
contribution ($j=0$) to the force balance vanishes because $n_{b_{0}}(h)$ is
symmetric. For the first non-vanishing correction, we find

$\tilde{F}=\frac{F\sigma_{b}}{k_{\mathrm{B}}T}=N_{B}^{*}\mathrm{e}^{\tilde{\ell}_{\mathrm{b}}^{2}/2}\left(-\frac{2\tilde{\ell}_{\mathrm{b}}}{\sqrt{2\pi}}+\left(1+\tilde{\ell}_{\mathrm{b}}^{2}\right)\mathrm{e}^{\tilde{\ell}_{\mathrm{b}}^{2}/2}\mathrm{Erfc}\left(\frac{\tilde{\ell}_{\mathrm{b}}}{\sqrt{2}}\right)\right)\tilde{v}\,.$
(18)

The bracketed term on the right hand side of Eq. (18) is strictly positive.
The approximations for $\tilde{\ell}_{\mathrm{b}}\ll 1$ are given in the main
text.

Figure S2:  The average steady-state behavior of basic adhesions without
unfolding. Symbols: Simulation results. Solid lines: analytical approximations
using an expansion in $\tilde{v}$ up to second order. Dash-dotted lines
without symbols: Approximation of the expansion for
$\tilde{\ell}_{\mathrm{b}}\ll 1$. a) The analytical approximation slightly
underestimates the mean number of bonds in steady state. b) The force-balance
condition connects the external force $F$ with the drift $v$.

##  Supplementary Discussion IV Adhesion model with unfolding

The numbers of unbound molecules obey the following approximate rate equations

$\displaystyle\diffp{}{t}N_{a}$
$\displaystyle=-\beta^{+}N_{a}+\int\limits_{-\infty}^{\infty}\\!\beta^{-}(h)n_{b}(h)\operatorname{d\\!}{}h-\delta_{a}^{+}N_{a}+\delta_{a}^{-}N_{a_{\mathrm{u}}}-\gamma^{-}N_{a}+\gamma^{+}\,,$
(19) $\displaystyle\diffp{}{t}N_{a_{\mathrm{u}}}$
$\displaystyle=-\beta_{\mathrm{u}}^{+}N_{a_{\mathrm{u}}}+\int\limits_{-\infty}^{\infty}\\!\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u}}}(h)\operatorname{d\\!}{}h-\delta_{a}^{-}N_{a_{\mathrm{u}}}+\delta_{a}^{+}N_{a}\,.$
(20)

The distributions of molecule numbers with extension $h$ experience a drift
due to the velocity $v$ of the upper boundary and obey

$\displaystyle\diffp{}{t}n_{b}(h)$
$\displaystyle=-v\diffp{}{h}n_{b}(h)-\beta^{-}(h)n_{b}(h)+\beta^{+}(h)n_{a}(h)-\delta_{b}^{+}(h)n_{b}(h)+\delta_{b}^{-}(h)n_{b_{\mathrm{u}}}(h)\,,$
(21) $\displaystyle\diffp{}{t}n_{b_{\mathrm{u}}}(h)$
$\displaystyle=-v\diffp{}{h}n_{b_{\mathrm{u}}}(h)-\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u}}}(h)+\beta_{\mathrm{u}}^{+}(h)n_{a_{\mathrm{u}}}(h)-\delta_{b}^{-}(h)n_{b_{\mathrm{u}}}(h)+\delta_{b}^{+}(h)n_{b}(h)\,.$
(22)

The steady-state solution for $v=0$, i.e., the equilibrium solution, is given
by

$N_{a}^{*}=\frac{\gamma^{+}}{\gamma^{-}}\,,\qquad
N_{a_{\mathrm{u}}}^{*}=\frac{\gamma^{+}}{\gamma^{-}}\frac{\delta_{a}^{+}}{\delta_{a}^{-}}\,,\qquad
n_{b}^{*}(h)=\frac{\gamma^{+}}{\gamma^{-}}\frac{\beta^{+}(h)}{\beta^{-}(h)}p_{a}(h)\,,\qquad
n_{b_{\mathrm{u}}}^{*}(h)=\frac{\gamma^{+}}{\gamma^{-}}\frac{\beta_{\mathrm{u}}^{+}(h)}{\beta_{\mathrm{u}}^{-}(h)}p_{a_{\mathrm{u}}}(h)\,.$
(23)

Integration yields the total number of bonds in equilibrium as

$N_{B}^{*}=\int\limits_{-\infty}^{\infty}\\!\Big{[}n_{b}^{*}(h)+n_{b_{\mathrm{u}}}^{*}(h)\Big{]}\operatorname{d\\!}{}h=\frac{\gamma^{+}}{\gamma^{-}}\mathrm{e}^{\overline{\epsilon}_{\mathrm{b}}}\left(1+\mathrm{e}^{-\tilde{\epsilon}_{\mathrm{f}}}\right)\,.$
(24)

To investigate the case $F>0$, the state distributions are expanded in powers
of the average velocity $\tilde{v}$ as above, see Supplementary Discussion
III. The connection of the state $a$ with the reservoir enforces $N_{a_{j}}=0$
for $j>0$. The solutions of the above equations cannot be found with an exact
iterative formula. Instead, a numerical approach is used to find the
corrections $N_{a_{\mathrm{u},j}}$, which can then be inserted back into the
rate equations to solve for $n_{b_{j}}(h)$ and $n_{b_{\mathrm{u},j}}(h)$. Up
to first order in $\tilde{v}$, the mean number of bonds remains unaffected or
increases with the velocity, as demonstrated for different unfolding steps
$\Delta$ and optimal binding lengths $\ell_{\mathrm{b}}$ in Fig. S3a. The
first-order correction vanishes for $\Delta\to 0$ and $\Delta\to\infty$. The
second-order correction can be both positive or negative, Fig. S3b. Strong
self-stabilization is found for $\tilde{\ell}_{\mathrm{b}}\ll 1$.

Figure S3:  Numerical results for the coefficients of the first- and second-
order corrections to the equilibrium solution for the number of bonds. The
adhesion cluster connected to a molecule reservoir and results for different
values for the molecule unfolding length $\Delta$ and the binding parameter
$\ell_{\mathrm{b}}$ are shown. The first-order correction $\propto v$ is
positive for $\ell_{\mathrm{b}}<\sigma_{b}$ and vanishes in the limits
$\Delta\rightarrow 0$ and $\Delta\gg\sigma_{b}$. The second-order correction
$\propto v^{2}$ can be both positive and negative.

###  IV.i Rupture behavior and adhesion cluster lifetimes

Application of a sudden force jump from $F_{1}=0$ to $F_{2}>0$ either leads to
a quick dissociation of the adhesion or the adhesion system relaxes to a non-
equilibrium steady state. For large adhesion clusters ($N>20$), the lifetimes
of those adhesions that reach the non-equilibrium steady-state usually exceeds
the finite simulation time. The fraction of those adhesion clusters that
rupture almost immediately, $\Phi_{\mathrm{rupt}}$, is shown in Fig. S4a as a
function of $F_{2}$ for different reservoir rates $\gamma^{+}$. Without
reservoir connection, when no self-stabilization occurs, the fraction of
ruptured adhesion clusters sharply increases at a threshold force. Self-
stabilization broadens the curves and increases the typical forces at which
rupture occurs. The numerical results are well-fitted by a shifted and scaled
error function.

The fraction of ruptured adhesion clusters after sufficiently long simulation
times represents the cumulative probability distribution for cluster
dissociation after a sudden force application. Therefore, the force value
$F_{2}$ at which $\Phi_{\mathrm{rupt}}(F_{2})=0.5$ corresponds to the highest
rupture probability, see Fig. S4b. The most probable rupture force increases
with increasing reservoir rate $\gamma^{+}$.

Figure S4:  Fraction of ruptured adhesions after a sudden load-jump from
$F_{1}=0$ to $F_{2}$ for different reservoir-exchange rates $\gamma^{\pm}$.
The ratio $\gamma=\gamma^{+}/\gamma^{-}=20$ is fixed. Systems were simulated
for $5\text{\cdot}{10}^{5}$ reaction steps after the force jump. a) The
fraction of ruptured adhesion clusters as a function of the magnitude of the
force step. The strength of the reservoir connection, determined by
$\gamma^{\pm}$, is varied while the ratio $\gamma=\gamma^{+}/\gamma^{-}$ is
held constant. For comparison, the fraction of ruptured adhesion clusters in
simulations without reservoir connection is shown with black bullets. A
reservoir connection implies here self-stabilization. Due to the large system
size ($N^{*}\approx 176$), most systems that did not rupture initially after
the load jump remained stable for the rest of the simulation time. Lines show
fits to error functions. b) Force $F_{2}$ at which rupture is most likely for
different reservoir rate values $\gamma^{+}$. Values are extracted from the
fit curves in a). The black horizontal line shows the most probable rupture
force for $\gamma^{+}=0$.  Figure S5:  Lifetimes of small adhesions
($N^{*}\approx 10$) for different values of the rate prefactors $k_{\beta}$
and $k_{\delta}$ and the reservoir-exchange rates $\gamma^{\pm}$. The ratio
$\gamma=\gamma^{+}/\gamma^{-}=1$ is fixed. a) $k_{\delta}$ = 0, no unfolding
b) $k_{\delta}=k_{\beta}$ c) $k_{\delta}=0.01k_{\beta}$. Only the lifetime
results for the adhesion model IV, which exhibits self-stabilization, display
a maximum at finite, non-vanishing forces. The lifetime of the adhesions is
substantially increased for intermediate forces if $\gamma^{+}$ and
$k_{\beta}$ are larger than $k_{\delta}$, such that molecules coming from the
reservoir establish new bonds frequently.

Fig. S5 shows adhesion lifetimes, defined as the time until first complete
dissociation of all bonds in an adhesion after application of a force jump.
Small adhesion clusters ($N^{*}\approx 10$) with different values for the
reservoir rates $\gamma^{\pm}$ are studied. The small molecule-numbers allow a
direct measurement of the lifetimes by simulating the clusters until the last
bond dissociates. For an adhesion system without molecule unfolding, realized
by setting $k_{\delta}=0$, the lifetime decreases monotonically with
increasing force for any value of $\gamma^{\pm}$, see Fig. S5a. The
equilibrium lifetime is largest for adhesion clusters without reservoir
connection, realized by $\gamma^{\pm}=0$. Figs S5b,c display lifetimes of
adhesion models with unfolding molecules. The lifetime of adhesions without
molecule exchange with a reservoir decrease monotonically with increasing
force (black bullets). However, when the system is coupled to a molecule
reservoir, so that the self-stabilization mechanism takes effect, the adhesion
lifetime curves have a maximum at non-zero, finite forces. A substantial
lifetime increase through self-stabilization is realized when both the
reservoir rates and the intrinsic binding rate $k_{\beta}$ are large compared
to the unfolding rate $k_{\delta}$.

###  IV.ii Special case: no unfolding in the unbound state $a$

To investigate how cyclic fluxes along the state network $a-b-
b_{\mathrm{u}}-a_{\mathrm{u}}-a$ affect self-stabilization, we set
$\delta_{a}^{\pm}=0$. Thereby, the cycle in the single-molecule transition-
diagram is broken. Physically, this modification means that unfolding is only
allowed when the molecule is bound between both planes. It should be
emphasized that this model variant still allows the emergence of cyclic fluxes
in the high-dimensional continuous state space spanned by the extensions of
the molecules.
In Fig. S6, results from the new model variant are compared with results from
model IV defined in the main text. For $\delta_{a}^{\pm}=0$, the increase of
the adhesion molecule number with force is significantly stronger than for the
model IV with cyclic flux. Thus, self-stabilization is enhanced in the absence
of cyclic flux, see Fig. S6a,b. The relative velocity of the two planes
bounding the adhesion is reduced accordingly, see Fig. S6c. The enhanced self-
stabilization can be attributed to a stronger accumulation of molecules in the
state $a_{\mathrm{u}}$, from which molecules can only escape via state
$b_{\mathrm{u}}$ in this model variant. Note that the force value at which
first rupture events are observed does not increase greatly, see black,
vertical lines in Fig. S6a. A comparison with the inset in Fig. S6b shows,
that the force per bond, $F/N_{B}$, is strongly reduced because of the
increased number of bonds $N_{B}$.

Figure S6:  Steady state results of adhesion model IV (compare Fig. 2) and
adhesion clusters with $\delta_{a}^{\pm}=0$. When unbound unfolding and
refolding is inhibited, the self-stabilization effect is more pronounced.

##  Supplementary Discussion V Adhesion model with unfolding and catch bonds

In the models considered so far, a slip-bond behavior is assumed, i.e., the
increase of the single-bond rupture rate with the applied extension $h$ is
monotonic, see sec. IV B. Thus, so far, single-bond lifetimes decreased when
tension increased. Bonds that become longer-lived when tension increases are
called catch bonds [45]. A number of molecular bonds in cellular adhesions
have been described as catch-slip bonds [71, 72, 73, 74]. These interactions
behave like a catch bond up to some force threshold while the slip-bond
behavior takes effect at higher forces. To further test the generality of the
self-stabilization mechanism, simulations of different adhesion models are
performed where the bonds behave like pure catch bonds. The binding and
unbinding rate of folded bonds are changed to

$\beta^{+}(h)\frac{n_{a}(h)}{N_{a}}=\frac{k_{\beta}}{\sqrt{2\pi}\sigma_{b}}\exp\left(-\frac{\left(|h|+\ell_{\mathrm{b}}\right)^{2}}{2\sigma_{b}^{2}}+\frac{\epsilon_{\mathrm{b}}}{k_{\mathrm{B}}T}\right)\,,\qquad\beta^{-}(h)=k_{\beta}\exp\left(\frac{-2|h|\ell_{\mathrm{b}}-\ell_{\mathrm{b}}^{2}}{2\sigma_{b}^{2}}\right)\,.$
(25)

The rates for unfolded bonds are defined analogously with the extension
$h_{\mathrm{u}}$. Note that the difference to the slip bond rates given in
sec. IV A-B lies only in one sign in each exponential function. Thus, detailed
balance still holds in equilibrium at $F=0$. The steady-state simulation
results are shown in Fig. S7. Again, only for model IV with a combination of
unfolding and association of new molecules from the reservoir, a pronounced
increase in the number of bonds is observed for small forces, see Fig. S7b.
Figure S7d shows that the self-stabilization is caused by an accumulation of
unbound molecules. Beyond this regime, at high forces, the pure catch-bond
dynamics leads to a separation of the bond distribution into two
subpopulations. Firstly, one has few, rather static molecules that carry most
of the tension. Therefore, these molecules have a large extension and long
lifetime. Secondly, a large number molecules form transient bonds with low,
symmetrically distributed extensions. As a result, the cluster stops moving,
as can be seen in the velocity-force curve in Fig. S7c for high forces.

Figure S7:  Simulation results for adhesion models in which single molecules
behave like catch bonds. a) Averaged total number of molecules $N$ in steady
state. b) Averaged number of bonds $N_{B}$ in steady state. Adhesion clusters
with a reservoir connection (dark red and dark blue) show a regime, where the
mean number of bonds increases with force. The self-stabilizing mechanism is
particularly pronounced for the model IV (dark blue) in which molecules can
unfold and are in exchange with a reservoir. c) The continuous binding and
rupture leads to an on average constant relative shift of the boundaries per
time. Adhesion models with bond unfolding show an increased velocity. d) The
relative deviation of the average number of unbound molecules $N_{A}$ from
equilibrium shows that the self-stabilization mechanism is connected to an
increased accumulation of adhesion molecules in unbound states (dark blue).
The parameter values are given in Supplementary Discussion I.

##  Supplementary Discussion VI Adhesion model with unfolding and cross-
linking

We consider an extended adhesion model that also contains the effect of
adaptor protein binding. Figure S8a shows the corresponding single-molecule
state-transition diagram with linking rates $\lambda^{\pm}$. Linking the load-
bearing molecules with adaptor proteins to the adhesion prevents a
dissociation of the molecules from the adhesion. Thereby, the linking
generates states that are not directly connected to the molecule reservoir.
For an approximate analytical description of the system, the set of equations
from Supplementary Discussion IV is extended to account for the linked states.
The unbound states obey

$\displaystyle\diffp{}{t}N_{a}$
$\displaystyle=-\beta^{+}N_{a}+\int\limits_{-\infty}^{\infty}\\!\beta^{-}(h)n_{b}(h)\operatorname{d\\!}{}h-\delta_{a}^{+}N_{a}+\delta_{a}^{-}N_{a_{\mathrm{u}}}-\gamma^{-}N_{a}+\gamma^{+}\,,$
(26) $\displaystyle\diffp{}{t}N_{a_{\mathrm{u}}}$
$\displaystyle=-\beta_{\mathrm{u}}^{+}N_{a_{\mathrm{u}}}+\int\limits_{-\infty}^{\infty}\\!\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u}}}(h)\operatorname{d\\!}{}h-\delta_{a}^{-}N_{a_{\mathrm{u}}}+\delta_{a}^{+}N_{a}-\gamma^{-}N_{a_{\mathrm{u}}}+\gamma^{+}\frac{\delta_{a}^{+}}{\delta_{a}^{-}}-\lambda^{+}N_{a_{\mathrm{u}}}+\lambda^{-}N_{a_{\mathrm{u},1}}\,,$
(27) $\displaystyle\diffp{}{t}N_{a_{\mathrm{u},1}}$
$\displaystyle=-\beta_{\mathrm{u}}^{+}N_{a_{\mathrm{u},1}}+\int\limits_{-\infty}^{\infty}\\!\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u},1}}(h)\operatorname{d\\!}{}h-\lambda^{-}N_{a_{\mathrm{u},1}}+\lambda^{+}N_{a_{u}}\,.$
(28)

The equations determining the evolution of the extension-dependent state
distributions read

$\displaystyle\diffp{}{t}n_{b}(h)$
$\displaystyle=-v\diffp{}{h}n_{b}(h)-\beta^{-}(h)n_{b}(h)+\beta^{+}(h)n_{a}(h)-\delta_{b}^{+}(h)n_{b}(h)+\delta_{b}^{-}(h)n_{b_{\mathrm{u}}}(h)\,,$
(29) $\displaystyle\diffp{}{t}n_{b_{\mathrm{u}}}(h)$
$\displaystyle=-v\diffp{}{h}n_{b_{\mathrm{u}}}(h)-\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u}}}(h)+\beta_{\mathrm{u}}^{+}(h)n_{a_{\mathrm{u}}}(h)-\delta_{b}^{-}(h)n_{b_{\mathrm{u}}}(h)+\delta_{b}^{+}(h)n_{b}(h)$
$\displaystyle\quad-\lambda^{+}n_{b_{\mathrm{u}}}(h)+\lambda^{-}n_{b_{\mathrm{u},1}}(h)\,,$
(30) $\displaystyle\diffp{}{t}n_{b_{\mathrm{u},1}}(h)$
$\displaystyle=-v\diffp{}{h}n_{b_{\mathrm{u},1}}(h)-\beta_{\mathrm{u}}^{-}(h)n_{b_{\mathrm{u},1}}(h)+\beta_{u}^{+}(h)n_{a_{\mathrm{u},1}}(h)-\lambda^{-}n_{b_{\mathrm{u},1}}(h)+\lambda^{+}n_{b_{u}}(h)\,.$
(31)

Figure S8:  Model for adhesion clusters with self-stabilization realized by
linking of auxiliary molecules. a) Single-molecule state transition diagram.
Unfolded molecules can establish an additional link to the adhesion that
prevents one-step dissociation of the molecule from the adhesion. b)
Comparison between simulation results (symbols connected by dashed lines) and
the expansion of bond states up to second order in the speed $|\tilde{v}|$
(solid lines) for different linking ratios $\lambda$. Results for an adhesion
model without the unfolding transition are shown for comparison. Parameter
values are given in Supplementary Discussion I.  Figure S9:  Averaged steady-
state simulation results for adhesion models with the unfolding-linking
mechanism and two reservoir connections. Black and blue symbols show results
for different linking ratios $\lambda=\lambda^{+}/\lambda^{-}$. Red: Adhesions
without unfolding for comparison. a) Only adhesions with a positive linking
ratio show a self-stabilization regime where the number of bonds increases
with force. b) Relative motion of the upper plane and the adhesion. Insets:
The fraction of bonds and the velocity as a function of force per bond show
the same behavior for all systems. The parameter values are given in
Supplementary Discussion I.

The steady-state results for the number of molecules in state $a$, $b$,
$b_{\mathrm{u}}$, $a_{\mathrm{u}}$ are given by Eq. 23. The additional linked
states fulfill

$n_{b_{\mathrm{u},1}}^{*}(h)=\frac{\lambda+}{\lambda^{-}}n_{b_{\mathrm{u}}}^{*}(h)\,,\quad
N_{a_{\mathrm{u},1}}^{*}=\frac{\lambda+}{\lambda^{-}}N_{a_{\mathrm{u}}}^{*}\,.$
(32)

As before, an expansion of the distributions for small speeds $|\tilde{v}|$
yields the first corrections to the equilibrium distribution. Figure S8 shows
a comparison between steady-state results obtained in simulations and in the
analytical approximation. Figure S9 shows simulation results for a range of
parameters.

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