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Department of Physics and Astronomy, University of California at Los Angeles
Correspondence: Address reprint requests to Robijn Bruinsma, Tel.: 310-825-8539; E-mail: bruinsma{at}physics.ucla.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONA TWO-STATE MODEL FOR...RESULTS AND DISCUSSIONAPPENDIX A: ELASTIC STRESS...APPENDIX B: NUMERICAL VALUESAPPENDIX C: LIST OF...ACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONA TWO-STATE MODEL FOR...RESULTS AND DISCUSSIONAPPENDIX A: ELASTIC STRESS...APPENDIX B: NUMERICAL VALUESAPPENDIX C: LIST OF...ACKNOWLEDGEMENTSREFERENCES |
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). Adhesion sites are micron-size protein clusters linking the cytoskeleton to the extracellular matrix (ECM) that fills the space between cells. Forces generated inside cells along F-Actin filaments by Myosin II motor proteins are transmitted to the ECM via these adhesion sites (2). Adhesion sites pass through different stages of development. Initial adhesions or IAs (3) form at the outer edge of the lamellipodium of a cell. They are relatively simple structures consisting of integrin transmembrane proteins linked outside the cell to specific ECM ligands (such as fibronectin) and inside the cell to the adaptor protein Talin I that also binds actin filaments (see Fig. 1 and Refs. 4,5). The link between an IA and the actin filaments is weak and can slip under an applied force in the picoNewton (pN) range (6), an effect sometimes referred to as "a slipping clutch". IAs can mature to focal complexes (FCs), located somewhat further from the edge (we will refer to IAs and FCs together as "nascent" adhesion sites). FCs are bound to the cytoskeleton by a reinforced link—the clutch has "engaged"—and contain a larger number of constituent components, such as Vinculin (see Fig. 1), and apply an appreciable force, in the nanoNewton (nN) range (7), to the substrate. FCs provide purchasing points for traction forces exerted by migrating cells on the substrate (8). Finally, fully mature adhesion sites (focal adhesions or FAs) are large, complex structures that are important centers of cell signaling.
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). It is well known that the size of mature FAs reversibly increases and decreases as a function of applied force (1). The force regulation of nascent adhesion sites (IAs and FCs) takes a different form: nascent adhesion sites match the force they exert to the stiffness of the substrate (10,11). Increased substrate stiffness stimulates development of IAs to FCs and causes a large increase in the force level applied by the cell to the substrate, from picoNewtons to nanoNewtons. This dependence of the force level on substrate rigidity is, for instance, responsible for durotaxis (12): a cell navigating over a heterogeneous substrate is guided by substrate elastic rigidity. It also provides a mechanism for the guidance of cell development by mechanical interactions with the surrounding tissue. Unlike the force regulation mechanism of mature FAs, the reinforcement of nascent adhesion sites does not depend on changes in size of the adhesion site. Optical trap assays using micron-sized, ligand-covered beads of fixed surface area exhibit the same reinforcement phenomena as actual adhesion sites (10).
Adhesion site development and elastic stress
The activation and reinforcement of IAs has been found to depend on chemomechanical activity localized to the adhesion site (10). The specific pathway is not fully known, but it is believed to involve phosphatase activity triggered by a force-induced conformational change of the integrin proteins (13,14). From a physical viewpoint it is quite surprising that substrate rigidity can regulate the reinforcement of an adhesion site. Adhesion sites are either stationary with respect to the substrate or motile (15). Newton's action-reaction principle seems to demand that the reaction force exerted by the ECM on a stationary adhesion site has to be equal and opposite to the traction force, the total external force applied by actin/Myosin contractile activity on the adhesion site. This latter force is generated inside the cell, far from the adhesion site, and a priori, should not depend on the elastic rigidity of the ECM. Substrate deformations generated by stationary adhesion sites have been measured using patterned, deformable gels. The results are consistent with the predictions of continuum elasticity theory for the strain of a point source subject to an applied force (16). Now, as discussed in Appendix A, it follows from continuum elasticity theory that the stress distribution of an elastic medium subject to a localized external force exerted on the surface does not depend on the rigidity of the medium. This means that the reaction forces exerted by the substrate on a stationary adhesion site (and its proteins) do not depend on the substrate rigidity, so the activation of a stationary adhesion site containing force-sensitive proteins cannot be regulated by substrate rigidity.
The aim of the article is to propose a tractable physical model that describes the transition from IAs to FCs for dynamic adhesion sites (of fixed area). The model and its assumptions are described in A Two-State Model for Nascent Adhesion Sites. The analytical treatment is given in Methods, below, for readers with a background in statistical mechanics. Results and Discussion describes the results of the Methods section and addresses the question of how dynamical adhesion sites can be activated by substrate rigidity, followed by predictions that can be tested by micromechanical methods.
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A TWO-STATE MODEL FOR NASCENT ADHESION SITES |
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TOPABSTRACTINTRODUCTIONA TWO-STATE MODEL FOR...RESULTS AND DISCUSSIONAPPENDIX A: ELASTIC STRESS...APPENDIX B: NUMERICAL VALUESAPPENDIX C: LIST OF...ACKNOWLEDGEMENTSREFERENCES |
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), such that the reaction free energy includes the mechanical work performed by the applied force during the conformational change.
The adhesion site is assumed to be subject to three forces: 1), a force by the link potential that connects the adhesion site to the filament bundle; 2), a viscoelastic force exerted by the ECM on the site; and 3), a random thermal force. ECMs have complex rheological properties, as do actin networks. We will assume a very simple case, namely a substrate that can be treated as a Voigt-Kelvin body. For the present case, this means that the substrate exerts an elastic restoring force on the adhesion site as well as a Newtonian viscous drag. The mechanical properties of the filament bundle also are described in the simplest possible terms. The bundle is assumed to be subject to four forces: 1), the constant traction force; 2), a Newtonian viscous drag by the cytoplasm; 3), the link force of the adhesion site; and 4), a random thermal force. Finally, an important assumption of the model is that the equilibration time for activation, which has been measured to be of the order of 1–10 s (10), is due to mechanical relaxation and not chemical equilibration.
Under these assumptions, it is possible to obtain an expression for the typical force level exerted on the substrate by the adhesion site using the methods of statistical mechanics as discussed in Methods, below. In Appendix B we discuss the validity of the assumptions and typical parameter values of the model.
Methods
Denote the two adhesion site states as S = –1 (passive) and S = 1 (active). Let 
G be the Gibbs free energy difference between the two states in the absence of applied force. G is proportional to the number of integrins of the site and, in general, can depend on the concentration of reactants involved in the activation reaction. Let F be the force applied to the adhesion site by the link between the adhesion site and the cytoskeleton. We shall see that, in general, F need not equal the traction force T for a dynamic adhesion site. The thermodynamic work by the external force during the conformational change of the integrins is then F d*, which is where d* is a length scale characterizing the molecular displacement of the integrins during the conformational change. Under conditions of chemical equilibrium, the expectation value of the site variable is given by
 | (1) |
). In the opposite case, e.g., if the activation reactions of the different integrin/phosphatase complexes of an adhesion site are not correlated, then the argument of Eq. 1 should be divided by the number of integrins of the site.
The adhesion site can move along the filament bundle direction (x direction), and the position of the site is denoted by X(t). The filament position along the x direction is denoted by Z(t), with X(t = 0) = Z(t = 0) = 0. The filament bundle is also exposed to a traction force T along the x direction. The link between bundle and site is described by a potential of mean force U(
,S) with (t) the relative displacement Z(t)–X(t) between bundle and site. The potential energy also depends on the site variable S and has the following general properties: 1), the absolute minimum of U(,S) with respect to is located at = 0, and 2), U(,S)
0 for large ||. The range of the potential is denoted by f. The activation energy for escape out of the potential well is denoted by U for S = –1 and by U for S = 1. We will assume that U/f << T, which means that in the passive S = –1 state, the force of the link potential is weak compared to the traction force T. The activation barrier U of the active S = 1 is assumed large compared to U.
Turning to the equation of motion of the adhesion site, the substrate—a Voigt-Kelvin body by assumption—exerts a harmonic restoring force –kX and a viscous drag 
on the adhesion site. The spring constant equals k 
Ya, with Y the Young's Modulus of the substrate—the rigidity that is supposed to be measured by the nascent adhesion site—and with a as the adhesion site diameter. The friction coefficient equals 
B 
a, with the substrate viscosity. The equation of motion of the site is then
 | (2) |
 | (3) |
The equation of motion of the bundle is obtained in a similar way by setting the sum of the forces on the bundle equal to the viscous drag on the bundle,
 | (4) |
,S) = 0 (i.e., no link) then the filaments are dragged along with a velocity VR. We identify VR with the (measured) retrograde flow velocity of actin filaments inside the lamellipodium, so R = T/VR.
After addition of Eqs. 2 and 4 and integration, one can express the site displacement X(t) in terms of the relative displacement (t),
 | (5) |
, with = (R + B)/k the mechanical relaxation time, then Eq. 5 reduces to
 | (6) |
RVR = T. The next steps simplify for B << R (the opposite case does not present new features). Inserting Eq. 6 in Eq. 5, and using B << R, we obtain the following equation of motion for the relative displacement (t) if t << ,
 | (7) |
(t) moving in a potential well U(,S) subject to a constant force BVR, a harmonic restoring force and a time-dependent force Fex(t) = kVRt. The effective potential energy 
has the form of a double-well potential with one minimum (1) near = 0 and a second minimum (2) near = VRt. The first minimum corresponds to an adhesion site bound to the filament and the second minimum to a dissociated site. As Fex(t) grows with time, the second minimum becomes more dominant and, eventually, a transition from the first to the second minimum must take place. For the S = –1 passive case, the characteristic timescale for the transition between these two states must be less than . The reason is that the applied force Fex(t = R/k) = VRR at time t = for a site in the first minimum = 1 equals the traction force T. By assumption, the traction force T significantly exceeds the maximum restoring force, of order U/f, that the potential well can exert in the passive state. This means that the adhesion site must have dissociated before t = . If, on the other hand, the maximum restoring force U/f of the active case, with S = 1, significantly exceeds T, then an S = 1 site is not expected to dissociate before t = . The filament bundle is arrested in this case and the full traction force T is exerted on the adhesion site.
We are interested in the most likely force 
F(k)
exerted on the adhesion site at the moment of dissociation. If we assume that, following dissociation, the adhesion site will rebind to actin filaments, then this F(k) can be considered as the typical maximum force that is exerted on the adhesion site during a dissociation/rebinding cycle. Applying Kramer's method for computing escape probabilities out of a potential well (19) to Eq. 7, we find for F(k),
 | (8) |
is the thermal force for bond dissociation, whereas J0 is the attempt rate for escape out of the well. This attempt rate depends on the second derivatives of U() at = 0 and at = f, on the friction coefficient B, and on temperature, but we will not require an explicit expression. Finally, U is the expectation value of the activation energy of the potential during dissociation. If the relaxation time of the coordinate (t) is sufficiently long compared to the chemical equilibration time of the site variable S, then U is the thermal average of the passive and active activation energies for S = –1, respectively, S = 1,
 | (9) |
F(k) of Eq. 8, then Eqs. 1, 8, and 9 constitute a set of coupled equations. They can be combined into a single self-consistency condition for F(k), which is simplified by introducing dimensionless quantities. Using a force scale fß = kBT/f, a stiffness scale 
and an energy scale kBT, the self-consistency condition reads
 | (10) |
G/d* is the force level such that the free energies of the active and passive states are degenerate. Dimensionless quantities are marked with an accent. |
RESULTS AND DISCUSSION |
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These two scenarios are illustrated graphically in Fig. 2, which shows the solution of the self-consistency condition, Eq. 10. The stiffness parameter k of the horizontal axis is proportional to the rigidity of the substrate (i.e., the Young's Modulus). The vertical axis is the most likely force at the moment of dissociation of the adhesion site from the filament bundle. The force level of the lower branch of Fig. 2, which corresponds to an IA, exhibits a logarithmic dependence on the Young's Modulus for highly deformable substrates (small k-values). This logarithmic dependence has the same physical origin as the logarithmic dependence on force loading-rate of the well-known Bell-Evans expression for molecular dissociation forces (20). As the substrate rigidity increases, the dissociation force increases and for increasing k-value a threshold kc is reached where the dissociation force level diverges. The threshold rigidity is 
Note that the threshold rigidity depends exponentially on the force level at which the energies of the active and passive states are degenerate. The upper branch of Fig. 2 corresponds to an activated adhesion site with a dissociation force that is not dependent on substrate rigidity. If the traction force T is less than the maximum restoring force of the reinforced potential, then this branch corresponds to a stationary site linked to an arrested filament bundle.
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The qualitative features of the simple two-state model appear to be surprisingly similar to the experimental observations as summarized in the Introduction: rigidity-dependent force regulation by adhesion sites is not in violation of fundamental physical principles. It should be clear, however, that the assumptions underlying the two-state model, as described in A Two-State Model for Nascent Adhesion Sites, do not do justice to the rich phenomenology of actual adhesion sites. For instance, the assumption of there being just two adhesion-site states is too simplistic to describe the complexity of the Src kinase/phosphatase and Rho-GTPase signaling pathways that are known to be involved in actual adhesion site activation (1), nor does the model account for the complex rheological properties of the ECM or the cytoplasm.
It is proposed, however, that certain general features are not expected to be sensitively dependent on these assumptions, and it is these features that should be verified experimentally. The claim that stationary adhesion sites of fixed area should not be involved in cellular force-regulation determined by substrate rigidity is such a general result—relying essentially only on the action-reaction principle—and it could be directly verified. The forces exerted by stationary adhesion sites have been measured by the bed-of-nails method (21) and—for larger adhesion sites—were found to be proportional to the adhesion site area. The measured slope is proportional to the force per integrin (see Appendix B). The prediction is that this slope should not be sensitive to the bending stiffness of the vertical columns, provided the sites are stationary. Note, though, that stationary adhesion sites very well could respond to changes in externally applied forces since that would increase the forces on the adhesion site proteins.
Next, the predicted logarithmic dependence of the slip force on substrate rigidity of slipping IAs is expected to be a robust feature that could be verified by optical trap assays of ligand-covered beads (see below). The most interesting general prediction of the model concerns, however, the reversibility of IA activation. Even though the chemical reaction describing transitions between active and passive states was assumed to be reversible, and in fact to be close to chemical equilibrium, the model predicts that the mechanical system as a whole is not reversible. Once an IA has been promoted to an FA under an increase in substrate rigidity, the adhesion site should, according to the model, remain reinforced after the rigidity is reduced. This is in contrast to the size regulation of mature FAs under external forces, which is reversible.
Physically, this irreversibility is a consequence of the fact that once the clutch has engaged, i.e., with the adhesion site locked to the filament bundle, the stationary site is now fully subject to the external traction T. For a stationary site, the reaction force must remain equal to T, even during a quasistatic reduction of the rigidity. This prediction could be directly tested in an optical bead assay. The stiffness constant of an optical trap—controlled by laser intensity—corresponds to the stiffness constant of the harmonic force and thus to the horizontal axis of Fig. 2. Assume that a ligand-covered optical bead is placed on the lamellipodium of a cell and captured by a stiff optical trap. After a waiting period of the order of the mechanical relaxation time, the force on the bead should start to rise as the adhesion molecules are getting activated and a link is established with the cytoskeleton. At this point the bead is released from the trap and carried along by the retrograde motion of the actin filaments. The two-state model now predicts that if this bead is captured once again by the trap, then the adhesion site still should be in the activated state and able to immediately apply the same external traction force to the bead. This prediction is at least qualitatively consistent with the observations reported in Ref. 10.
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APPENDIX A: ELASTIC STRESS DISTRIBUTION OF STATIONARY ADHESION SITES |
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p. The substrate rigidity is determined by the Young's Modulus. The boundary conditions of the surface—the x,y plane—are that no external force is applied to the surface apart from a single site located at the origin. Away from the origin, the components zi = iz of the stress tensor ij are thus 0 at the surface (njij is the force per unit area, along the i direction, applied to an area element with normal nj). The traction force on the site is assumed to be directed along the x axis and is characterized by a local stress distribution zx = 0 (x,y) whose area integral equals the traction force T.
The resulting elastic displacement of the substrate in the z > 0 half-space can be obtained by solving the classical equations of continuum elasticity theory. Specifically, the displacement ux of the surface of the substrate along the x direction (i.e., the traction direction) at a distance 
from the adhesion site can be shown to be (22)
 | (A1) |
The surface integral extends here over the area surrounding the origin where the force is applied. The key point is that ux is inversely proportional to Y, which is true also for the y and z components of the displacement profile. According to Eq. A1, the displacement u0 of the center of the adhesion site along the traction direction is of the order of
 | (A2) |
0 the spatial average of the externally applied stress over the area of the adhesion site, i.e., 0 = T/A. The substrate elastic reaction force on the site acts as a harmonic spring with a spring constant k Ya proportional to the Young's Modulus. Note that the adhesion-site displacement measured in units of a is of the order of the dimensionless ratio 0/Y of the applied stress and the Young's Modulus. One can obtain the value of the traction force T from a measurement of the elastic displacement field of the substrate, which is in fact a feasible procedure (16).
Geometrical deformations of the elastic medium are characterized by the strain tensor 
which can be obtained from the displacement field. The components of the strain tensor clearly are again inversely proportional to Y. The forces exerted by the substrate on the adhesion site are, however, determined by the stress tensor ij, which is related to the strain tensor by
 | (A3) |
Because the components of the strain tensor are inversely proportional to Y, it follows that the substrate stress tensor is independent of the Young's Modulus of the substrate. For example, far from the adhesion site, the xx component of the stress tensor drops with distance as
 | (A4) |
Equation A4 can be understood in terms of dimensional analysis. The substrate stress at a given location could depend on the externally applied stress 0, the elastic modulus Y of the substrate, the spatial distance r from the adhesion site, and the size a of the site. In view of the linearity of the equations of elasticity, the substrate stress must be proportional to the external stress 0 multiplied by some dimensionless function of the ratio (a/r). Far from the adhesion site, the stress should be proportional to the adhesion site area, again in view of the linearity, so this should be a quadratic function. It is not possible to construct a separate dimensionless factor that depends on the Young's Modulus Y and not on 0, so the stress must be independent of Y. In a more complex model, one might describe an adhesion site as a thin elastic plaque with a separate elastic modulus Y*. In that case, the substrate stress, in principle, could depend on a separate dimensionless ratio (Y/Y*). However, by applying the condition of stress continuity to the plaque, it follows that this dependence is absent.
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APPENDIX B: NUMERICAL VALUES |
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The traction force T transmitted by adhesion sites has been measured for a variety of deformable substrates. The traction force of larger adhesion sites scales with the area A of the site (21) as T = 0 A. The mean stress level 0 is of order 1–5 nN/µ2, 
1% of the stress of the actin/Myosin bundles of animal muscles. Assuming that the traction force is proportional to the number N of actin filaments linked to the site, then T = N fM where fM is the traction force per filament. This unit traction must be of the order of the tension per unit area 0 times the area per filament. The spacing a0 between the filaments of an F-Actin bundle cross-linked by 
-actinin—an important component of FAs—is 30 nm, whereas 30 nm also happens to be the spacing for both integrin and actin binding sites along the elongated Talin I adapter protein (see Fig. 1). For a0 equal to 30 nm and an area per filament of order 
the unit force fM is of the order of a few picoNewtons. The total number N of filaments per site, and presumably also the number of integrins, is then of the order of 103–104 for an adhesion site area of the order of 10 µm2.
Although the rheological properties of model gels vary considerably, certain collagen gels indeed can be described as a Voigt-Kelvin body (23) with an elastic modulus Y in the range of kPa and an effective viscosity in the range of 106 Pa s. This is a fairly typical Y value for the model gels used in the adhesion assays, although the rheological properties of actual ECMs vary greatly and normally are non-Newtonian. The dimensionless site displacement u/a = 0/Y of Appendix B, the typical substrate strain level near an adhesion site, would then be of the order of 1 and the spring constant k would be in the range of pN/nm for a 1-µm site. This is also the order of magnitude of the stiffness of optical traps. The friction coefficient B a for a 1-µm adhesion site would be of order 104 pN s/nm. For an optical trap in aqueous medium, the friction coefficient of the bead would be negligible compared to R, which is estimated as follows. The actin retrograde flow velocity VR is in the range of µm/min, so the force loading rate kVR is in the range of 15 pN/s, and therefore the effective friction coefficient per filament R = fM/VR should be of the order of 0.1 pN s/nm. The mechanical relaxation time = R/k would be of the order of 0.1-s times the number of filaments. The assumption R >> B we made for convenience holds for optical beads but may fail for gels.
A key assumption of the theory is that the chemical equilibration time should be less than the mechanical relaxation time, i.e., the rate-limiting step for activation must be mechanical relaxation and not chemical equilibration. From optical bead model studies (10), it is known that the rate-limiting step for IA to FC switching is of the order of 1–10 s. The predicted mechanical relaxation time = R/k is of order 0.1-s times the number of filaments or complexes, using the earlier estimates. For 102–103 filaments per site, this mechanical relaxation time is of the order of 10–100 s, which is in reasonable agreement with the measured relaxation time. It is thus at least consistent to assume that the mechanical degrees of freedom indeed are slow compared to the chemical equilibration rates.
Values for the most likely dissociation force F of single molecular links can be obtained from single-molecule micromechanics. A recent single-molecule study of the interaction between individual 4ß1 integrins and VCAM ligands (E. Evans, unpublished) reports that the Bell-Evans relation holds for that case with a thermal force scale fß = kBT/
f of 13 pN and a zero-force dissociation rate J (i.e., off-rate) in the range of 0.1–1/s depending on condition. For a loading rate of 10 pN/s, the dissociation force is then in the 1–10 pN range. Assuming an attempt rate Jo for dissociating weak protein/protein links to be in the (usual) range of 109 Hz would mean that the single-link activation energy U of an integrin/VCAM pair is in the range of 20 kBT. For a reinforced integrin dimer, the dissociation force was indeed found to be much larger, in the range of 50–100 pN, and relatively independent of the force loading-rate (E. Evans, unpublished), consistent with the two-state model. This would correspond to a reinforced binding energy of the order of 100 kBT. Single-molecule studies of the strength of the integrin/Talin link are currently not available. The generalization from single-molecule studies to clusters of links is complicated by the fact that this depends sensitively on whether the links act mechanically in parallel or in series (24). For N cooperative links acting in parallel, the thermal force level should be that of a single link, whereas the off-rate should be 
in terms of the single-molecule quantities. Note that for large N this would greatly increase the dissociation force. For a noncooperative, zipper-type bond dissociation, the thermal force in Eq. 11 should be N times the single-molecule force level fß, whereas 
In view of these, and other uncertainties, we refrain from making quantitative estimates of the critical force for unbinding and the critical Young's Modulus. It clearly would be extremely useful if the model studies of the slip state could be repeated for different stiffness constants of the trap to check whether the Bell-Evans relation holds for slipping IAs.
The zero-force free energy difference G of an adhesion site between the active and passive states can be estimated by recalling that the characteristic force level fp exerted on a reinforced complex is of the order of pN per filament. To estimate the mechanical work during activation, we need to know the conformational changes of actin-bound adhesion site integrins during activation, which currently are not known. Conformational changes of integrins upon ligand binding were examined in solution studies (17), who found that the ß-integrin tail, which is linked to the Talin adapter in adhesion sites, can undergo a large rotation between competing "open" and "closed" conformers. The associated displacement of the tail end is unusually large, 10 nm. If rotation of the ß-integrin tail is indeed the conformational change that is linked to integrin activation, then the mechanical work fp d* per site available for activation would be of the order of 10 kBT. This must exceed the zero-force reaction free energy G per integrin. It follows that the integrin/phosphatase complex must be a very sensitive mechanosensor as it apparently operates barely above the thermal noise level.
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APPENDIX C: LIST OF SYMBOLS |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONA TWO-STATE MODEL FOR...RESULTS AND DISCUSSIONAPPENDIX A: ELASTIC STRESS...APPENDIX B: NUMERICAL VALUESAPPENDIX C: LIST OF...ACKNOWLEDGEMENTSREFERENCES |
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Submitted on June 24, 2004; accepted for publication April 5, 2005.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONA TWO-STATE MODEL FOR...RESULTS AND DISCUSSIONAPPENDIX A: ELASTIC STRESS...APPENDIX B: NUMERICAL VALUESAPPENDIX C: LIST OF...ACKNOWLEDGEMENTSREFERENCES |
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with J the zero-force escape rate of the adhesion site in the slip state and with VR the retrograde velocity. The lower branch corresponds to a slipping adhesion site, terminating at a critical stiffness. The upper branch corresponds to a reinforced state that fully transmits traction to the substrate. Once the system is reinforced, it will not return to the slip state.