Clutch model for focal adhesions predicts reduced self-stabilization under oblique pulling

Cell-matrix adhesions connect the cytoskeleton to the extracellular environment and are essential for maintaining the integrity of tissue and whole organisms. Remarkably, cell adhesions can adapt their size and composition to an applied force such that their size and strength increases proportionally to the load. Mathematical models for the clutch-like force transmission at adhesions are frequently based on the assumption that mechanical load is applied tangentially to the adhesion plane. Recently, we suggested a molecular mechanism that can explain adhesion growth under load for planar cell adhesions. The mechanism is based on conformation changes of adhesion molecules that are dynamically exchanged with a reservoir. Tangential loading drives the occupation of some states out of equilibrium, which for thermodynamic reasons, leads to the association of further molecules with the cluster, which we refer to as self-stabilization. Here, we generalize this model to forces that pull at an oblique angle to the plane supporting the cell, and examine if this idealized model also predicts self-stabilization. We also allow for a variable distance between the parallel planes representing cytoskeletal F-actin and transmembrane integrins. Simulation results demonstrate that the binding mechanism and the geometry of the cluster have a strong influence on the response of adhesion clusters to force. For oblique angles smaller than about 40∘, we observe a growth of the adhesion site under force. However this self-stabilization is reduced as the angle between the force and substrate plane increases, with vanishing self-stabilization for normal pulling. Overall, these results highlight the fundamental difference between the assumption of pulling and shearing forces in commonly used models of cell adhesion.


Introduction
Integrin-based focal adhesions mechanically connect cells to the surrounding extracelluar matrix.On the molecular level, focal adhesions form around clusters of heterodimeric transmembrane receptors called integrins that bind to extracellular proteins such as collagen, laminin, and fibronectin.Numerous intracellular adaptor proteins, including talin, vinculin, kindlin, and paxillin, co-localize with integrin clusters and together provide a link to the force-generating actomyosin cytoskeleton.Accordingly, focal adhesions have been likened to a 'molecular clutch' that regulates force transmission to the environment [1][2][3][4][5][6].Substantial research effort is being devoted to unraveling the molecular mechanisms that underlie the clutch function.
Since focal adhesions are at the interface between extracellular mechanical environments and the intracellular cytoskeleton, they act as signaling hubs translating mechanical stimuli into a biochemical response [7,8].In cells adhering to a planar substrate, focal adhesions grow under the application of an external tangential force and shrink when the force is decreased [9,10].Similarly, focal adhesions shrink under pharmacological inhibition of cytoskeletal contraction and the size of a focal adhesion is roughly proportional to the load that is applied to it [11].The interplay of mechanical forces and focal adhesion maturation plays a central role for the morphology of cells.For example, in migrating cells, growth of focal adhesions depends on their localization and large adhesions are formed in regions of high mechanical load [12].Moreover, the anisotropic intracellular F-actin structure, the localization of various adaptor proteins, and the shape of focal adhesions are connected to their sensitivity to directional mechanical cues [13,14].Although focal adhesions are most easily visualized in two-dimensional setups, adhesions in threedimensional matrices have been observed [15].In synthetic fibrous material, cells were also shown to recruit nearby fibers, dynamically increasing ligand density at the cell surface and promoting the formation of focal adhesions [16].
Numerous molecular processes play a role during the formation and maturation of focal adhesion under mechanical load.Transmembrane integrins can regulate their affinity for extracellular ligands by transitioning between inactive and active conformations in a mechanosensitive manner [4].On the intracellular side, integrin activation is promoted by the association of talin and kindlin.Talin directly transmits forces by binding with its N-terminal head domain to integrins, whereas its C-terminal rod domain links to F-actin [17,18].While talin and integrin alone form a weak bond with a short lifetime, the additional presence of kindlin-2 likely results in a talin-integrin bond with a longer lifetime, independent of an applied force [19].Actomyosin-generated forces that are subsequently transmitted via talin stabilize the link between integrins and extracelluar matrix ligands and result in an increase of bond lifetimes for both α V β 3 and α 5 β 1 integrins [20,21].This so-called catch-bond behavior, where load increases the bond lifetime, is also found, for instance, in the case of the link between the adaptor molecule vinculin and F-actin [22].
The molecule talin also acts by itself as a mechanosensor [23,24].Molecular dynamics simulations early on predicted that forces acting on talin inside a focal adhesion in vivo lead to a conformational change that unfolds the molecule and exposes cryptic vinculin-binding residues [25].This unfolded conformation of talin can facilitate the recruitment of vinculin, a critical early step in the process of focal adhesion stabilization under force.Experimentally, talin unfolding under stretch has been confirmed by single-molecule experiments [26,27], and vinculin binding to talin stabilizes the unfolded conformation of talin [28].The observed mechanosensitive localization of vinculin to binding sites on talin depends on an autoinhibition of vinculin in its native state [29,30].The occurrence of conformational changes in talin under mechanical load and subsequent vinculin recruitment has been confirmed in a number of in vivo experiments [17,[31][32][33].Recently, the talin-dependent mechanoresponse has also been suggested to play a role for loading-rate-dependent nuclear localization of yes-associated protein (YAP) [34].Other signaling pathways that transmit information about mechanical forces upstream include, e.g. a pathway mediated by the adaptor protein p130Cas [35,36] or a pathway involving force-dependent cytoskeletal association of the protein zyxin, which contains a Lin11, Isl-1, and Mec3 (LIM) domain and may participate in gene expression regulation [37,38].
The bulk of the work on focal adhesions has been carried out with cells on a planar substrate where adhesions are primarily exposed to forces that are tangential to the plane.However, some work has also considered the role of normal components of the force.Experimentally, it has been shown that α V β 3 integrins are coaligned in focal adhesions by the directionality of F-actin flow coupled to integrins via the adaptor protein talin, and data analysis suggests that integrins are markedly tilted relative to the planar substrate [39].A build-up of tangential forces in the adhesion plane can be prevented by anchoring extracellular ligands in supported lipid bilayers wherein they can freely move.Nano-fabricated physical barriers in the underlying substrate can then be used to tune the lateral resistive forces.With this setup, it has been observed that integrin clustering at ligands stimulates actin polymerization to form nascent adhesions which recruit talin, paxillin, and focal adhesion kinase irrespective of force generation.Vinculin recruitment, in contrast, requires tangential force build-up, which is consistent with the notion that this process is linked to talin unfolding [40].Normal forces at cellular scales can be measured by placing cells in between a pair of parallel microplates, where one of the plates can deform under the cell-generated forces [41].During the spreading of cells in this configuration, focal adhesions at the leading edge of the lamella appeared only when the cell body switched from convex to concave.This observation suggests that the transfer of tension in the apical cortex to the focal adhesion in the lamella depends on the configuration [41].The results, however, do not provide a quantitative relationship between the angle at which forces are transmitted to a focal adhesion and its growth under force.
Various models have been proposed over the past decades to explain the mesoscopic dynamics of individual adhesions.Originally, models of clusters of stochastic bonds were intended to represent membrane-to-surface adhesion where forces act in the normal direction to the substrate plane [42][43][44][45].Force-induced growth of adhesions under tangential load was first modeled using continuum models, where it was assumed that deformations locally trigger an unknown biochemical response inside focal adhesions, which drives molecule aggregation [46,47].A complementary thermodynamic reasoning predicts that elastic strain inside focal adhesions that are subjected to tangential forces decreases the chemical potential of the aggregated molecules, which could in turn drive growth of the structures [48].However, experimental data relating F-actin flow and rapidly exchanging adaptor proteins to traction force generation promote a dynamical view of the adhesion structure [2].Stochastic models relating bond dynamics to cytoskeletal motion under tangentially applied load have been employed to study the behavior of a molecular clutch that transmits traction forces [5,[49][50][51][52][53].Since cell adhesions are paramount for morphodynamics and for the dynamics of whole cells and tissue, clutch models can facilitate the elucidation of complex cellular processes [54,55].Stochastic clutch models have also been used to explain the behavior of mutant cells with impaired adhesion machinery [56], which further supported the central role of talin unfolding for adhesion growth.Building on these advances, we recently proposed a generic, thermodynamically consistent mechanism by which biomolecular adhesions can harness mechanical load for adapting their size and stability [57,58].The minimal model requires only a combination of unfolding of adhesion molecules under force, an exchange of molecules with a reservoir surrounding the focal adhesion, and possibly the recruitment of additional molecules, such as vinculin, which stabilize the unfolded conformation.Under tangential stress, the state occupations shift, leading to a growth of the adhesion under increasing load.At sufficiently large loads, this self-stabilization mechanism breaks down and the adhesion ultimately fails.The mechanism of adaptation is simple and robust when the typical clutch-model geometry consisting of a planar adhesion under tangential force is studied.
Here, we generalize the model proposed in [57] to the case of a force that is pulling at an oblique angle to the plane supporting the cell [58].The two planes that represent the interface between the cytoskeleton and the integrin transmembrane connection are assumed to remain parallel .Thereby, the probability of an adhesion molecule to form a bond remains independent of its location within the adhesion cluster.Simulation results reveal that the geometry of the adhesion cluster has a strong influence on its response to force.For traction forces exerted under oblique angles to the substrate plane roughly smaller than 40 • , we observe a growth of the adhesion site under force.However, this self-stabilization is reduced as the angle between the force and substrate plane increases.A reduction of self-stabilization at large angles is apparent in both the steady-state configuration and in the average time until complete cluster dissociation.

Model
We consider a cluster of molecules between two parallel planes that extend along the abscissa of a Cartesian coordinate system, see sketch in figure 1(a).While the lower plane is fixed at y = 0, the upper plane can undergo translational motion along the x− and y−axis and the displacement relative to its original position is denoted by s = (s x , s y ) ⊤ .Mechanically, the molecules are modeled as linear, elastic bonds with spring coefficient κ.The stretch of a molecule with index i is denoted by h = (h x,i , h y,i ) ⊤ .
The molecules inside the cluster can be in four distinct states, see the state diagram in figure 1(b).We model the formation of bonds between the two planes as well as a conformational change of the molecules, which is motivated by the unfolding transition of talin.State a represents folded molecules that are associated with the cluster, but are not bonded to the upper plane.A molecule reservoir is connected to state a, representing an exchange of adhesion molecules with the environment.State b represents molecules that have formed a bond between the two planes, but are still in the folded conformation.State b u represents bonds between the two planes that have unfolded.Finally, state a u represents unfolded molecules that are not bonded to the upper plane.We denote the time-dependent number of molecules in state q ∈ {a, b, a u , b u } by N q .The overall number of bonds formed by the molecules in an adhesion cluster is denoted by N B = N b + N bu and the overall number of unbound molecules is denoted by N A = N a + N au .Due to an exchange of unbound, folded molecules with a reservoir, the overall number of molecules in the adhesion cluster, N = N B + N A , is variable.The molecules undergo stochastic, reversible transitions between the states, compare illustrations in figure 1(c).The whole system is only driven out of equilibrium by the exerted force F that changes the transition rates.Below, we will denote equilibrium quantities with an asterisk (*) superscript, e.g.N * B is the overall number of bound molecules in equilibrium when F = 0.

Transition rates
An overview of the expressions used for the transition rates is provided in table 1.The molecule reservoir is modeled as a constant chemical potential.Folded molecules are exchanged  Transition-rate expressions used in this work.The parameter ℓ b is not the rest length of the molecules but determines an optimal stretch at which bond formation is maximal.Bonds in the unfolded state experience a reduced stretch hu = h − ∆ compared to folded bonds with extension h.The parameters ∆ 1 and ∆ 2 can be thought of as the distances between the energetic minima representing folded or unfolded state and the maximum of the energetic barrier that separates the states.Accordingly, ∆ 1 + ∆ 2 = ∆.For simplicity, we also assume

Transition
Rate coefficient between state a and the reservoir at rates determined by the constants γ + and γ − .Note that these constants alone fix the steady-state occupation number of a since the state is directly connected to the reservoir.
Bonds that connect both planes, denoted by state b, are formed by unbound molecules, denoted by state a. Formation of bonds is governed by the rate coefficient β + (h).Binding rates depend on an effective binding affinity ϵ b and on the , where h ⩾ 0 if h x , h y ⩾ 0 and h ⩽ 0 if h x , h y ⩽ 0. The maximum value of the probability for binding is given at |h| = ℓ b , where ℓ b is the optimal binding length.This formulation can represent, e.g. a binding pocket that favors binding at a finite, but small distance ℓ b .The distance is assumed to be smaller than typical thermal length fluctuations σ = k B T/κ > ℓ b , where k B denotes Boltzmann's constant and T the system temperature, taken to be at 298 K.If the distance between the planes is larger than the molecule rest length ℓ 0 , we assume that molecules in the unbound state a align orthogonally to the surfaces and then form a bond with an initial stretch h = s y − ℓ 0 ⩾ 0. For s y − ℓ 0 < 0, we fix the range of binding stretches h ∈ [s y − ℓ 0 , |s y − ℓ 0 |] so that no energy is artificially injected into the system.The maximum compression that a bond can experience is s y − ℓ 0 and we cap the maximum extension at |s y − ℓ 0 | to maintain symmetry.Given a fixed distance between the planes, the binding stretch determines the binding angle α via sin(α) = s y /(ℓ 0 + h), which allows in general two solutions, 0 ⩽ α 1 ⩽ 90 • and α 2 = 180 • − α 1 .In biological adhesions, clusters elongate in the direction of the applied force [9,11].To introduce this bias in the model, the second solution is only considered if the newly formed bond subtends less than 90 • with the force vector, i.e. if α 2 − θ < 90 • .The total binding propensity is obtained by integration of the binding probability over the whole binding range and is denoted by β + .For already unfolded molecules, the binding rate is defined analogously, with the molecule extension given by h u = h − ∆.More details regarding the binding rates are provided in appendix B.
The rupture of bonds in the states b and b u are determined by the rate coefficients β − (h) and β − u (h), which are closely related to the Dembo-Bell rates [42].Both, compression and stretch increase the rupture probability per unit time.A prefactor k β exp(−ℓ 2 b /2σ 2 ) appearing in β − (h) is chosen such that the ratio of the binding to unbinding rates obeys a Boltzmann distribution as with energy difference E b = κh 2 /2 − ϵ b .Thus, basic thermalequilibrium conditions are satisfied.
Molecule unfolding occurs with a rate δ + b (h) or δ + a for bound or unbound molecules, respectively, and increases the rest length of the molecule by ∆.The transitions between native and unfolded states are modeled as thermally assisted jumps over a single energy barrier.We assume that the two distances between the minima and the barrier maximum are given by ∆ 1 = ∆/2 and ∆ 2 = ∆/2.A parameter ϵ f > 0 accounts for the stretch-independent amount of energy that is required for unfolding.The ratio of the unfolding rate to the folding rate for bound molecules is given by where the energy difference is given by Unfolding transitions also occur between the states a and a u .For these unbound molecules, the molecule length fluctuates around its rest length.The distribution of the resulting molecule stretch is assumed to be a zero-centered Gaussian with variance σ 2 as p a (h) ∼ N (0, σ 2 ).The average unfolding rate for every unbound molecule in state a is given by The average refolding rate per unbound, unfolded molecule is obtained similarly as δ − a = k δ .Hence, the ratio of the averaged unfolding to folding rate for unbound molecules is given by δ a = exp(−ϵ f /k B T).The extension-dependent rates are constructed such that Kolmogorov's cycle condition is respected.This condition means here that when a single molecule goes through a cycle of states in absence of driving forces, e.g. a → b → b u → a u → a, the product of transition rates in one direction equals the product of reverse transition rates as This thermodynamic constraint follows from the principle of microscopic reversibility and ensures that our molecules do not act as molecular motors, but rather as passive adhesions.
Once an external force is applied to the adhesion, the stretch of any particular molecule may change during a cycle and the condition can break down.

Force balance.
An external force F = (F x , F y ) ⊤ is exerted on to the upper plane and the force is transmitted to the lower plane via the molecules that form a bond.The angle between force vector and the adhesion plane is denoted by θ ∈ [0, 90 • ].Each bound molecule has its own angle α i relative to the substrate plane, where different molecules are denoted by the index i.For parallel bond alignment we assume α i = θ but in general α i ̸ = θ.The force balance conditions are written as Simulations of stochastic state transitions are performed with a Gillespie algorithm.We assume that mechanical equilibration occurs instantaneously after every state transition.Therefore, the upper plane is shifted in the x− and y−directions after a transition to ensure a force balance according to equations ( 5) and (6).Details of the implementation of the force-balance condition are found in appendix C.

Non-dimensionalization.
We employ the thermal energy scale k B T as our unit of energy.The thermal fluctuation length of the molecules σ = k B T/κ is employed as the unit length.The value of σ dictates the range of distances that a molecule can bridge during bond formation.For a molecule with a stiffness of κ = 1 pN nm −1 , the thermal length fluctuations are in the range of ∼2 nm.Together, k B T and σ set a force scale as F 0 = k B T/σ.With the previously estimated values, we have that F 0 ≈ 2 pN.This force scale is consistent with experimentally observed forces at adhesion proteins [31,[59][60][61].If not stated otherwise, the unit time is given by the inverse of the rate constant for bond formation t 0 = k −1 β .In the following, dimensionless quantities will be denoted with a tilde, e.g.F = F/F 0 .

Parallel bond alignment
In a first model, we assume that all bonds are aligned in parallel with the applied force, i.e. α i = θ for all bond indices i.In this case, only the absolute value of the displacement s = s(cos θ, sin θ) ⊤ , where s = ±(s 2 x + s 2 y ) 1 2 , can vary and the model is effectively one-dimensional.In the following, only the limiting cases of θ = 90 • and θ = 0 • are presented to demonstrate the different effects of normal and tangential forces.

Normal force only.
For θ = 90 • , all bonds align orthogonally to the adhesion planes, see figure 2(a)(i).Models with this bond configuration and geometry, but without the unfolding transition, have been studied before, compare [43,[62][63][64].When molecules cannot unfold, all bonds are stretched by an equal amount h and the force balance reads in non-dimensional form F = F/(κσ) = N B h.The displacement of the upper adhesion surface is given by s y = ℓ 0 + h.In the present model, the different lengths of the folded and unfolded conformations result in a different force-balance equation F = N B h − N bu ∆, because unfolded bonds experience a reduced stretch h u = h − ∆.Consequently, the elastic energy of the bonds is always non-zero if both conformations are present simultaneously, i.e. if N b > 0 and N bu > 0. In simulations, we observe that the cluster switches between two energetically more favorable configurations with N b ≈ 0 or N bu ≈ 0, so that the distance between the adhesion surfaces s y is either close to ℓ 0 or ℓ 0 + ∆, see figure 2

(a)(i)-(iii). At vanishing force, the average number of bonds is either given by N
With increasing force, the distance between the surfaces increases.As a result, unbound molecules need to stretch more in order to form a bond, leading to an increase in the rupture probability.The average time until the last bond ruptures decreases monotonically with force, see figure 2(a)(iv).Thus, a force acting vertically on the adhesion upper plane leads to a destabilization of the adhesion, regardless of the presence of unfolding transitions.

Tangential force only.
For θ = 0 • , all bonds align parallel to the adhesion plane, see figure 2(b)(i).The vertical distance s y between the planes vanishes.Cell adhesion models with such bond configurations are typically referred to as clutch models, see [57] and [5,49,56,65,66].Since there are no preferred binding sites along the planes, molecules can form a bond at any extension.The distance between the planes does not affect the competition between folded and unfolded conformations b and b u .At vanishing force, the average number of bonds is given by N * B = γβ(0)(1 + δ a ).If the molecules can unfold, i.e. k δ > 0, and if the unbound state a is connected to a molecule reservoir, then molecules accumulate in the unfolded, unbound state a u , so that more molecules are available for rebinding.As a result, the total number of bound molecules N B increases with force, which we refer to as self-stabilization, see figure 2(b)(ii).We note that self-stabilization is still observed even when the unfolding of bonded molecules does not result in a relaxed stretch h − ∆, i.e. self-stabilization does not result from a change of the mechanical properties of the bonds upon unfolding.Under constant load, cycles of molecule binding at small stretches and rupture at large stretches lead to a non-zero average sliding velocity v x between the two planes of the adhesion in a non-equilibrium steady-state, see figure 2(b)(iii).The average time until complete cluster dissociation exhibits a peak at non-vanishing forces, see figure 2(b)(iv).Details regarding this self-stabilization mechanism are provided in [57].

Arbitrary bond alignment
In the preceding section, clusters of parallel bonds were studied either under a pulling or a tangential force.Selfstabilization, i.e. growth of adhesions under force, is only possible for tangential force.To explore if this result persists when bonds experience a mixture of pulling and shearing forces in a two-dimensional geometry, the restriction of parallel bond alignment is lifted and clusters under arbitrary force directions are considered.

Steady-state configurations and self-stabilization.
Simulation results reveal a strong dependence of the steadystate configuration on the force angle θ with a threshold between θ = 30 • and θ = 40 • , see figure 3.As illustrated in figure 3(a), the low surface separation under predominantly shearing forces leads to a cluster with bonds that are largely aligned in the direction of the force.For predominantly pulling forces, the bond alignment is strongly reduced.This qualitative change can be explained by referring to the steady-state separation between the boundaries s y , as seen in figure 3(d).For θ ⩾ 40 • under small forces, the steady-state separation s y is close to, but smaller than, the molecule rest length l 0 .This means that the stretches h ∈ [s y − l 0 , |s y − l 0 |] have a relatively small range, and so the angles α 1 and α 2 deviate slightly around 90 • .Resultantly, an acute angle α 1 and an obtuse angle α 2 are typically both viable for selection.At smaller angles θ ⩽ 30 • , the vertical separation of the planes decreases, which results in a stronger bias for the selection of acute angles α 1 .Consequently, the steady-state configurations for clusters under predominantly tangential forces differ strongly to those under predominantly normal forces.
Adhesion self-stabilization, which implies an increase in the average number of bonds N B with force, is only observed for angles θ ⩽ 30 • , see figure 3(b).Thus, self-stabilization does not occur for predominantly pulling forces in this model.Since the bond rupture rate increases with the load, the average number of unbound molecules N A increases for all bond angles under load, see figure 3(c).If the force has a non-vanishing component in the x−direction, the two planes slide against each other with an average velocity v x , see figure 3(e).This sliding velocity is low at small angles when self-stabilization results in large adhesion sites.For 40 • ⩽ θ ⩽ 80 • , the sliding velocity increases strongly with force.As the angle is increased even more, the tangential component of the force decreases so that the sliding motion is reduced.At θ = 90 • , the average velocity v x vanishes.
Figure 3(f) shows the average elastic energy per bond E/N B .For all force angles, the cluster finds an energetically favorable configuration with E/N B ≈ 0 at low forces for F → 0. The increase of the elastic energy with force is lowest for θ = 30 • and strongest for θ = 40 • , which is again explained by a rather sudden transition from shearing to pulling.In the pulling regime, for θ ⩾ 40 • , the relative velocity in x direction is largest for θ = 40 • and decreases with larger angles, see figure 3(e), leading to a decrease of transiently stored elastic energy.For shearing, when θ ⩽ 30 • , self-stabilization significantly reduces the force per bond.Here, the elastic energy per bond is lowest for θ = 30 • and increases as the angle is reduced.This effect can possibly be explained by bond alignment because, around θ = 30 • , some oppositely orientated molecules are able to relax during relative motion of the planes while, as θ → 0, all bonds are continuously stretched until rupture.

Adhesion lifetime.
Self-stabilization under tangential loading not only implies growth of the adhesion, but also a longer average time until the adhesion dissociates under force, which we call lifetime [57].To investigate adhesion stability under oblique loading, we measure the time until complete cluster dissociation for multiple simulation runs and calculate the lifetime.To accumulate sufficient full-dissociation events in simulations, we focus on small adhesion sites with N * ≈ 10 molecules in steady state.In figure 4(a), the lifetime is shown as a function of force for different values of the unfolding length ∆.A non-monotonous dependence of lifetime on force is observed for small angles, θ = 0 • and θ = 30 • , and finite unfolding lengths ∆ ∈ {2, 4, 6}.Thus, an increase of lifetime with force is possible if the unfolding length is larger, but comparable to the thermal fluctuation length.Selfstabilization does not occur for ∆ = 0 because the unfolding and refolding rates become stretch-independent.For large unfolding lengths ∆, the high energy barrier between the folded and unfolded state suppresses the conformational change, so that transitions between folded, bound and unfolded, bound states hardly occur.At large forces, lifetime decreases exponentially with force.This lifetime decrease is almost independent of the parameters ∆ and θ since at large forces, all bonds rupture in rapid succession and rebinding hardly occurs.The angle-dependence of adhesion lifetime is illustrated more explicitly in figure 4(b).Long lifetimes under finite force only occur for θ ≲ 40 • .For larger angles, the force results in a separation of the planes that suppresses bond formation and selfstabilization.

Concluding discussion
A minimal model for adhesion self-stabilization, proposed in [57], is generalized here to the case of a force that is pulling at an oblique angle to the plane supporting the cell.An adhesion is composed of elastic spring-like molecules that connect two planes loaded by an external force.Molecules undergo reversible stochastic transitions between states that represent the formation or breakage of bonds and the unfolding or folding of the molecules.All transition rates fulfill thermodynamic constraints guaranteeing microscopic reversibility in equilibrium, i.e. the system is only driven by the applied mechanical force.The results demonstrate that an unfolding of adhesion molecules results in qualitatively different system behaviors under predominantly pulling and predominantly tangential forces.In the case of a normal force pulling at the adhesion plane, bonds are aligned orthogonally to the surface so that the unfolding transition increases the distance that can be bridged by the molecular bonds.With every rupture or unfolding transition, the load on the remaining bonds increases and the distance between the planes grows.The larger distance reduces the likelihood of a subsequent formation of new bonds and prevents self-stabilization of the adhesion.In the alternative case of pure tangential shearing, a broad distribution of bond stretches emerges.The force drives continuous cycles of bond binding, unfolding, and rupture, which lead to a relative sliding between the planes.As the force is increased, the unfolded states become accessible and more molecules are drawn from the reservoir into the adhesion.The number of bonds increases despite increasing forces.An intuitive, although hardly rigorous explanation of self-stabilization is that the unfolded states that become populated on force application represent an 'additional phase space volume' to be filled by additional molecules from the reservoir.Self-stabilization does not result from a change of the mechanical properties of molecules upon unfolding.Folded and unfolded states have the same spring constant and self-stabilization still occurs in simulations if the length of the reaction coordinate for unfolding, ∆, is not added to the bond rest length after unfolding.
For oblique pulling with angles between the force and the plane up to θ ≃ 30 • , the average occupation numbers of molecule states stay largely unaltered compared to purely tangential pulling.The separation between the two bounding surfaces stabilizes at a value that lies slightly above F sin θ/κ for low forces.All bonds are aligned in the direction of the force with a narrow distribution of bond angles, which reduces the average stretch per bond and thus the elastic energy per molecule.As a result, the relative motion of the two surfaces is low.For force angles θ ⩾ 40 • , a very different picture emerges.The distance that is bridged by the cluster stabilizes at a value close to the folded bonds' rest length ℓ 0 , so that the distribution of folded bonds is centered around the vertical binding position at 90 • .Due to their longer rest length, the average distribution of unfolded bonds' angles is shifted towards lower values with an average median of arcsin(ℓ 0 /(ℓ 0 + ∆)) ≈ 65 • .Although unbound molecules accumulate in the cluster with increasing force, the average number of bonds stays below its equilibrium value.The relative velocity of the two planes increases stronger with force at high angles than at small angles.In the limiting case θ = 90 • , there is no net motion; the number of bonds decreases monotonously with force, and no self-stabilization occurs.
Overall, the presented generalization of the adhesion model for self-stabilization [57] to two dimensions reveals that the binding mechanism and the geometry of the molecule cluster have a strong influence on the response of adhesions to force.For biological focal adhesions, the geometric constraints assumed for vertical pulling may, however, not apply as strictly as modeled here.For example, it is likely that the very long extension of talin anchored within the three-dimensional structure of the focal adhesion acts as a mechanical buffer to allow polymerization and turnover to counter rupture events.Further modeling efforts are required to describe such effects.
indices i, the stretch that is required for binding is determined by the vertical distance between the planes s y .For folded molecules, the only possible binding stretch is then given by For unfolded bonds, the vertical stretch component is given by h u,y = h y − ∆ sin θ.Once a molecule binds, its binding point stays fixed.Therefore all shifts of the upper boundary due to bond transitions like unbinding, unfolding or folding, will take place in the direction of the force vector.The system is effectively one-dimensional.

Appendix B. Bond formation at an arbitrary angle
In a two-dimensional system, the bond angle has to be variable.In general, possible bond angles lie in the range α ∈ The angle under which the molecule binds depends on its current stretch h and the current vertical distance between the planes s y .To demonstrate how the binding position is obtained from h and s y , we assume that a folded adhesion molecule is connected to the lower plane at the origin.For a displacement of the upper plane (s x , s y > 0), the binding point r = (r x , r y ) along the upper plane can be obtained from the overall length of the molecule ℓ 0 + h as For unfolded molecules, ℓ 0 has to be replaced with ℓ 0 + ∆.If s y < |ℓ 0 + h| in equation (B1), two values are possible for the x−coordinate of r, which results in two possible bond angles α 1 ⩽ 90 • and α 2 = 180 • − α 1 .If the distance between the plane is smaller than the rest length, s y < ℓ 0 , the range of possible binding stretches is limited by the value h ⩾ h min = s y − ℓ 0 < 0. The minimal value h min is the point where the bond is compressed most, which is the case for α = 90 • .We assume that a molecule with stretch h can only form a bond if h ∈ [h min , |h min |].This choice ensures that molecule binding occurs symmetrically around h = 0 and no energy is artificially injected.The total binding rate in this case is obtained as the integral over the stretch-dependent binding rate β + (h) as The normalization factor A(h min ) accounts for a limited stretch range for the equilibrium distribution p * b (h).It is given by For a vanishing vertical distance between the two planes, s y = 0, the threshold for the binding stretches can be extended to the limit |h min | → ∞, such that lim |h min |→∞ A(h min ) = 1.Thus, the chosen normalization correctly reproduces the onedimensional clutch model with only tangential forces [57].
If s y ⩾ ℓ 0 in equation (B1), the vertical distance between the planes is bigger than the rest length.In this case, it is assumed that folded molecules can only bind vertically at (0, s y ) with α = 90 • and a fixed stretch h = s y − ℓ 0 .Thus, the range of possible stretch values is reduced to a single value and h min approaches zero.For a small but finite stretch range ϵ, the normalization factor is approximated as (B5) In the last step, the error function is expanded until first order in ϵ.Using this result, the total binding rate for the case s y ⩾ ℓ 0 is given by (B6) which recovers, in the limit ϵ → 0, the classical model for parallel bonds when the force is oriented in the normal direction to the planes.If only the geometry determined the binding probabilities, the system would be invariant under parity along the x−axis.Such binding rates could result in a situation where the overall load gets balanced by the interplay of compression of some molecules and stretch of others.In biological adhesions, however, forces are likely transmitted by stretched molecules and parity is also broken by the molecular architecture, e.g. by the intrinsic orientation of actin filaments.Therefore, a final modification is introduced to break the symmetry between opposite binding angles.In the case that s y < ℓ 0 , i.e. if two binding angles are possible, the second angle is only allowed if it encloses not more than 90 • with the force vector.Hence, at θ = 0, binding is restricted to the angles [0 • , 90 • ], while at θ = 90 • , the full range [0 • , 180 • ] is allowed.

Appendix C. Implementation of the force-balance condition
It is assumed that the system relaxes instantaneously to mechanical equilibrium after every stochastic state transition.During the simulations, the force-balance condition is used to determine the new position of the upper plane s ′ after every state transition from the position s before the transition.For two dimensions, s ′ = s + x with the displacement x cannot be determined analytically.Therefore, a Newton-Raphson method is implemented.The algorithm starts from an initial guess for x 0 and approaches the correct solution x by means of the iteration x j+1 = x j − J −1 (x j ) g (x j ) , (C1) where J denotes the Jacobi-matrix of the equation system The iteration stops when the root of g, that is the vector x R such that g(x R ) = 0, is found within the precision goal, i.e. we find a vector x R ′ such that g 2 (x R ′ ) < ϵ 1 for some precision ϵ.As a precautionary measure, so that the iteration indeed follows the direction of the gradient towards the root, the condition g(x j+1 ) • g(x j+1 ) < g(x j ) • g(x j ) has to be met.If the proposed next value, which is obtained with equation (C1) , leads to an increase of the square value of g, only a fraction 0 < µ < 1 of the step is proposed, x j+1 = x j − µ J −1 (x j ) g (x j ) . (C3) Close to the root of g, the method converges quadratically.Rarely, but especially for clusters with only few bonds, the Newton-Raphson method does not converge to the correct solution that restores force balance.In such a case, the trajectory is rejected and the simulation is restarted.

Figure 1 .
Figure 1.(a) Sketch of the two-dimensional adhesion model.A force F is exerted on the upper plane at an oblique angle θ and the bonded molecules subtend at an angle α i with the two planes.(b) State diagram of the system.The arrows denote possible transitions between states.(c) Illustrations of the possible transitions that occur within the model.

Figure 2 .
Figure 2. Clusters with parallel receptor-ligand bonds for normal loading θ = 90 • (a), and tangential loading θ = 0 • (b).(a)(i) Illustration of parallel molecules for θ = 90 • .(a)(ii) In steady state, the adhesion molecule cluster can switch between an unfolded configuration with N b ≈ 0 and N bu > 0 and an folded configuration with N bu ≈ 0 and N b > 0. Dots show simulation results where systems are either in a folded or unfolded configuration.Lines show the corresponding, different solutions of the macroscopic master equations, compare appendix E. The number of bonds in each state is plotted relative to N * B = N B (F = 0), which is the expected overall number of bound molecules when no force is applied.(a)(iii) The two steady-state configurations correspond to bond stretches h ≈ 0 and hu = h − ∆ ≈ 0. (a)(iv) Average time until complete cluster dissociation for clusters with γ chosen as {2.678, 1.786, 0.893, 0.447}, which results in small clusters with N * B = N B (F = 0) ∈ {12, 8, 4, 2} bonds in equilibrium.(b)(i) Illustration of parallel molecules for θ = 0 • .(b)(ii) In steady-state, both bond conformations are present simultaneously and the number of unfolded bonds initially increases with load F. (b)(iii) Cycles of binding and rupture lead to an average constant sliding velocity vx between the adhesion surfaces.(b)(iv) The average cluster lifetime exhibits a peak at non-vanishing forces.Results are shown for γ ∈ {1.667, 1.112, 0.556, 0.278}, which leads to N * B ∈ {12, 8, 4, 2}.Other parameter values are given in appendix D.

Figure 3 .
Figure 3. Simulation results for adhesion clusters in steady state where force is exerted at different angles θ with arbitrary bond alignment.Data points are shown until complete cluster dissociation is observed within the simulation time.(a) Sketches of a cluster under predominantly shearing forces (θ ⩽ 30 • ) and a cluster under low shear (θ ⩾ 40 • ).(b) The total number of bonds N B grows strongly with force for θ ⩽ 30 • in a qualitatively similar manner.At larger force angles, N B decreases or increases only marginally and cluster dissociation is observed at lower forces.(c) The number of unbound molecules increases with force for all force angles θ with a qualitatively similar dependence on force for θ ⩽ 30 • and θ ⩾ 40 • , respectively.(d) The distance sy between the adhesion reaches a plateau sy < ℓ 0 for θ ⩽ 30 • .For θ ⩾ 40 • , the distance is close to the native molecule rest length ℓ 0 .(e) The relative sliding velocity between the adhesion surfaces vx vanishes for θ = 90 • .For force angles 40 • ⩽ θ ⩽ 80 • , vx increases strongly with the external force F. Low velocities are found for θ ⩽ 30 • .(f) The average elastic energy per bond increases strongest with F for θ = 40 • , while the lowest values at a constant force are found for θ = 30 • .Parameter values are given in appendix D.

Figure 4 .
Figure 4. Average time until complete dissociation for the two-dimensional cluster model with unfolding and a reservoir rate ratio γ = γ + /γ − = 1.14.Clusters are initialized with N = 10 ≈ N * folded bonds.(a) Lifetimes for a range of unfolding lengths ∆ and force angles.(b) Heatmap chart of lifetimes for ∆ = 2.5.The dashed line indicates the critical angle θ = 40 • , below which a significant increase of lifetime with load is observed.Parameter values are given in table 2.

Table 2 .
Parameter values for simulations of the two-dimensional adhesion cluster model with binding, rupture and conformational changes.