Mechanical regulation of a molecular clutch defines force transmission and transduction in response to matrix rigidity

Cell function depends on tissue rigidity, which cells probe by applying and transmitting forces to their extracellular matrix, and then transducing them into biochemical signals. Here we show that in response to matrix rigidity and density, force transmission and transduction are explained by the mechanical properties of the actin–talin–integrin–fibronectin clutch. We demonstrate that force transmission is regulated by a dynamic clutch mechanism, which unveils its fundamental biphasic force/rigidity relationship on talin depletion. Force transduction is triggered by talin unfolding above a stiffness threshold. Below this threshold, integrins unbind and release force before talin can unfold. Above the threshold, talin unfolds and binds to vinculin, leading to adhesion growth and YAP nuclear translocation. Matrix density, myosin contractility, integrin ligation and talin mechanical stability differently and nonlinearly regulate both force transmission and the transduction threshold. In all cases, coupling of talin unfolding dynamics to a theoretical clutch model quantitatively predicts cell response.

Cell function 1 and major processes in cancer 2 and development 3 are driven by the mechanical rigidity of tissues, which cells probe through their contractile and adhesive molecular machinery. This machinery is composed of dynamic molecular bonds between the extracellular matrix (ECM), integrins, adaptor proteins and the force-generating actomyosin cytoskeleton, forming a mechanical link generally referred to as a 'molecular clutch' [4][5][6] . To sense and respond to rigidity, cells employ this molecular clutch first to transmit forces to their surrounding matrix 5,[7][8][9] , and then to transduce those forces into biochemical signals leading to transcriptional regulation in the nucleus 10 . The first step of force transmission has been modelled by introducing the dynamic properties of the clutch in computational simulations 4 , in a way that can predict the effects of adhesion mediated by different integrin types 5 . However, the fundamental prediction of such a clutch model is a biphasic force/rigidity relationship, which is in direct contradiction with the monotonically increasing curves observed in most systems [11][12][13][14][15] . Further, the key molecular clutch elements driving force transmission remain to be identified. The second step of force transduction is likely to be mediated by force-induced molecular conformational changes, which could occur at the level of ECM molecules, integrins, adaptor proteins and ion channels, among others 16,17 . Among all potential mechanosensing molecules, the adaptor protein talin is a particularly interesting candidate because it directly links integrins to actin, is stretched as cells transmit forces to the ECM 18,19 , and mediates cellular response to force 20,21 . Further, talin has been observed to unfold under force in vitro, exposing previously cryptic binding domains to vinculin 22 , which then binds and is likely to be activated 23 . However, if and how talin unfolding, or conformational changes in any other molecule, mediate rigidity sensing is unknown. Thus, how force transmission and transduction are coupled in response to rigidity remains unresolved. . Data show 1 out of 9, 9 and 3 independent experiments for control, talin 2 shRNA, and control + talin 1 head L325R, respectively. Lines are sigmoidal fits to experimental results. (f) Examples of YAP staining on cells plated on 5 and 29 kPa gels. In all quantifications (a,c,e), differences between control and both talin-depleted and control + talin 1 head L325R cells were significant only above 5 kPa (P = 0.006, P < 0.001, P < 0.001, two-way ANOVA). Scale bars, 20 µm. Grey dashed line in a,c,e marks the rigidity threshold. Error bars, s.e.m. a wild-type phenotype due to expression of talin 2 (refs 20,21)) and knocked down talin 2 levels using short hairpin RNA (shRNA). We first plated control and talin 2-depleted cells on polyacrylamide gels of different rigidities. Gels were coated with the ECM protein fibronectin, to which cells adhered specifically through α 5 β 1 and α v β 3 integrins ( Supplementary Fig. 1). Then, we measured cell-ECM force transmission using traction force microscopy. On the softer substrates, cellular forces increased with rigidity, and talin depletion had no effect (Fig. 1a,b). However, forces sharply diverged above a threshold rigidity of 5 kPa, increasing and decreasing for control cells and depleted cells, respectively (Fig. 1a,b). In control cells, this threshold coincided with the growth of focal adhesions rich in vinculin (Fig. 1c,d) and with the activation (nuclear translocation) of the mechanosensitive transcriptional regulator YAP 10 ( Fig. 1e,f). In contrast, talin-depleted cells spread on gels, but did not develop focal adhesions or localize YAP to the nucleus at any rigidity ( Fig. 1c-f). To confirm those results, we blocked talin function in control cells by a mechanism alternative to shRNA. We transfected cells with a dominant negative talin head mutant (L325R) that displaces endogenous talin for integrin binding, but does not activate integrins or link them to the cytoskeleton 24 . On stiff substrates, increasing levels of talin head L325R expression progressively reduced force transmission to the levels of talin-depleted cells (Fig. 1a,b and Supplementary Fig. 2), and abrogated adhesion growth and YAP nuclear translocation as expected ( Fig. 1c-f). In contrast and consistently with shRNA results, talin head L325R expression had no effect on soft substrates (Fig. 1a-f and Supplementary Fig. 2). Further reinforcing the importance of the talin-dependent threshold, it also marked the recruitment of integrins and phosphorylated FAK to adhesions (Fig. 2a-d and Supplementary Fig. 3), and the association of adhesions to stress fibres (Fig. 2c,d). Those results provide important insights into the two steps required for rigidity sensing, force transmission and force transduction. Regarding force transmission, we show that the core long-standing prediction of the clutch adhesion model (a biphasic force/rigidity curve, so far only partially observed in neuronal filopodia 4 ) is correct, but fully unveiled only in the absence of talinmediated reinforcement. Regarding force transduction, we show that it is triggered above a rigidity threshold in a talin-dependent manner, leading to adhesion growth, downstream biochemical signalling and YAP activation.

Regulation of talin unfolding by a molecular clutch can explain force transmission and transduction in response to rigidity
We then assessed whether the effects of rigidity and talin could be mediated by known force regulators such as myosin phosphorylation and cell spreading. Whereas myosin phosphorylation levels slightly fluctuated as a function of substrate stiffness and talin depletion ( Supplementary Fig. 4), those fluctuations did not correlate with transmitted forces. As reported previously 5 , cell areas increased with rigidity, reaching a plateau at about 5 kPa ( Supplementary Fig. 4). However, no significant differences were measured between control and talin-depleted cells. Further, the threshold rigidity leading to adhesion growth, YAP activation and force decrease in talin-depleted cells was not associated with cell spreading changes. Thus, neither myosin phosphorylation nor cell spreading could account for the effects of talin, or the rigidity threshold. Alternatively, the rigidity threshold may result from the regulation of talin unfolding by the ECM-integrin-talin-actin clutch. To   investigate this, we compared how force affects the unfolding time of single talin molecules (previously measured 25 ) versus the unbinding time of single fibronectin-integrin bonds (both for α 5 β 1 integrins, previously measured 26 , and α v β 3 integrins, measured here in Fig. 3a). For clutches mediated by either integrin type, at low forces integrins unbind faster, releasing force transmission and preventing talin unfolding. However, unfolding becomes faster than unbinding above a threshold force (Fig. 3b). This threshold force for unfolding could be further modulated by unbinding events in integrin-talin-actin bonds, which would decrease overall clutch unbinding times, or by load sharing between talin and other adaptor molecules, which would decrease the load on talin and slow unfolding. Independently of its specific value, a force threshold for talin unfolding thus emerges, which may mediate the rigidity threshold observed in Fig. 1.
To evaluate this possibility, we developed a computational approach to couple talin unfolding to a clutch model 5 (see Supplementary Note for details). This model considers a given number of myosin motors progressively pulling on an actin fibre, which is bound to a deformable substrate through molecular clutches formed by adaptor proteins (such as talin), integrins and fibronectin (Fig. 3c). Once the clutches bind according to a given binding rate, fibre contraction deforms the substrate and results in progressive force loading, which is slow or fast on soft or stiff substrates, respectively. This force loading leads to either clutch unbinding or talin unfolding. If talin unfolds, vinculin binds, leading to adhesion reinforcement and growth 20 and integrin recruitment. In the absence of talin unfolding, integrins are not recruited, but force is still assumed to be transmitted between integrins and actin through other adaptor proteins 27 .
If talin unfolding is not considered, we recapitulate the fundamental prediction of the clutch model 4 , that is, a biphasic force/rigidity curve with an optimal rigidity of maximum force transmission (Fig. 3d, blue line). Below the optimal rigidity, force loads so slowly that clutches unbind from the ECM before exerting significant forces. Above the optimal rigidity, force loading is so fast that clutches unbind from the ECM before other clutches have time to bind, reducing cooperativity and decreasing total force transmission. The presence of talin does not affect the curve at low rigidities, where forces are too low to allow unfolding. However, above a rigidity threshold, force loading becomes fast enough to allow talin unfolding before integrin unbinding. This leads to vinculin binding and integrin recruitment, increasing integrin binding and force transmission, and eliminating the biphasic relationship ( Fig. 3d, red line). In all cases, force transmission resists actin contraction, leading to a negative correlation between force and actin flow (Fig. 3e). The model closely reproduced our force measurements (see Supplementary Table 1 and Supplementary Note for model parameters and assumptions). Confirming the validity of the clutch hypothesis, actin flows correlated negatively with forces, and could also be reproduced by the model using the same parameters ( Fig. 3e,f, and Supplementary Videos 1 and 2). Thus, control of talin unfolding by force transmitted through the ECM-actin clutch can explain how rigidity regulates both force transmission, and the threshold for mechanotransduction.

The rigidity threshold is mediated by talin unfolding under force and subsequent vinculin binding
We then carried out several experiments to validate this mechanism molecularly. First, we rescued talin 2-depleted cells with either fulllength (FL) talin 1 or two separate fragments, the talin 1 rod and the talin 1 head. FL talin 1 rescued rescue force generation, adhesion growth and YAP localization to control levels, confirming that talin 1 and 2 had the same effect 20,21 . However, neither the talin rod nor the head rescued the phenotype of depleted cells ( Fig. 4a-d).
As the talin head is sufficient to activate integrins 21,28 , this shows that integrin activation without force transmission through talin was not sufficient to trigger a rigidity response. Confirming this, FL talin mutants that do not bind integrins (W359A) or bind but do not activate integrins (L325R) 24 did not rescue force generation or YAP localization above the rigidity threshold ( Fig. 4e,   We then evaluated whether the effect of talin unfolding was mediated by vinculin binding. We transfected control cells with VD1, a vinculin fragment that is dominant over endogenous vinculin for talin binding 29 but prevents normal vinculin function 30,31 owing to the lack of remaining functional domains. Blocking vinculin function through VD1 transfection had the same effect as talin depletion, that is, forces decreased at high rigidities, and YAP remained cytosolic (Fig. 5e,f and Supplementary Fig. 5). VD1 formed large focal adhesions above but not below the rigidity threshold (Fig. 5g), confirming that vinculin binding was specifically triggered above the threshold. As a negative control, transfection of a VD1 mutant (A50I) with reduced affinity for talin had no effect ( Fig. 4e-g). Neither VD1 nor VD1 A50I had any effect on talin-depleted cells ( Supplementary Fig. 5). Collectively, those data show that force unfolds talin above a rigidity threshold, leading to vinculin binding, adhesion growth and YAP translocation to the nucleus.
The molecular determinants of the clutch regulate force transmission and transduction Finally, we analysed the role of different clutch molecular determinants predicted to regulate force transmission 32,33 . First, if available clutches are decreased (by reducing substrate fibronectin coating density, see Supplementary Fig. 6), overall force transmission should be reduced. However, force loading per clutch increases because actomyosin contractility is distributed among fewer clutches, triggering talin unfolding, reinforcement and YAP translocation at a lower rigidity threshold. As the rigidity corresponding to peak force transmission in depleted cells is also determined by force loading 32,33 , it also shifts to a lower value. If fibronectin coating is increased, the inverse effect is expected. Second, reducing the binding rate of integrins (by partially blocking integrins with the GPen peptide) should reduce the number of bound clutches, leading to effects similar to those of reducing fibronectin coating. However, in this case  integrins are partially blocked, impairing reinforcement and adhesion growth on talin unfolding. This counters the effect of the increased loading per integrin, leading to reduced overall force transmission and a shift to lower rigidities of the peak force in depleted cells, but no major change in the threshold rigidity for reinforcement and YAP translocation. Third, decreasing myosin contractility (by using different concentrations of blebbistatin) should reduce force loading, increasing the rigidity threshold for talin unfolding and YAP activation. Similarly, the force peak in depleted cells should shift to higher rigidities, and reduce its height. All of those predictions were verified experimentally (Fig. 6). Although model predictions did not always provide an exact quantitative match, they consistently predicted the shifts in overall forces, the rigidity threshold at which forces diverge between control and depleted cells, and the position of the force peak in depleted cells. Those predictions were obtained by adjusting only the relevant parameters in each case: number of fibronectin molecules (n f ) for fibronectin coating (Fig. 6a-c), integrin binding and recruitment rates (k ont and d add ) for GPen (Fig. 6d-f), and number of myosin motors (n m ) for blebbistatin (Fig. 6g-i). In all cases, YAP localized to the nucleus at the same rigidity threshold where measured forces diverged between control and depleted cells (Fig. 6j,k,l). Thus, force transmission was systematically modulated by the molecular determinants of the clutch, leading to mirror shifts in the thresholds for talin unfolding and YAP activation.

DISCUSSION
Our results unveil the mechanisms by which microenvironment rigidity regulates both force transmission and transduction, and reconcile previous findings. Indeed, even though cell-ECM adhesion is widely accepted to be mediated by a molecular clutch mechanism, its predicted biphasic force/rigidity relationship is inconsistent with monotonically increasing trends observed in most systems [11][12][13][14] . Here we show that the biphasic force/rigidity relationship is normally masked by talin-mediated reinforcement and adhesion growth, and is fully unveiled only on talin depletion. This depletion allowed us to test the molecular determinants of force transmission, revealing that its regulation by rigidity, ECM coating density, cell contractility or integrin activity fully abides by the clutch model first proposed eight years ago 4,32,33 . This leads to interesting and counter-intuitive results, such as that decreasing ECM coating enhances the mechanical response (Fig. 6j), or that mild myosin inhibition (5-15 µM) increases cell-ECM force transmission in talin-depleted cells for a specific rigidity range (10-15 kPa, Fig. 6g,h). This is because although overall contractility is reduced, the force peak is shifted to higher rigidities. Whereas in this work we focused on fibronectin substrates bound to cells through α 5 β 1 -and α v β 3 -mediated catch bonds, we note that the emergence of a threshold for talin unfolding does not require catch bonds. Even if integrins behaved as slip bonds, a rigidity threshold would occur if the force/unbinding and force/unfolding curves crossed at a given force. Given the extremely steep decay in talin unfolding times as a function of force (Fig. 3a), this is likely to happen in most scenarios. Further, talin unfolding at low forces would also be prevented by very fast refolding rates, a factor that was also included in our modelling. Thus, the force and rigidity threshold for talin unfolding is likely to apply in many physiological scenarios, and could be regulated by several factors. First, clutch unbinding events at the level of integrin-talin-actin bonds could increase overall clutch unbinding rates, displacing the threshold to higher forces/rigidities. This effect would alter the specific values of model output, but would not modify the overall trends of the force/rigidity curves with and without talin, the presence of a rigidity threshold for unfolding, or the regulation of this threshold by the different factors. Second, load sharing between talin and other adaptor proteins could reduce the force experienced by individual talin molecules, also increasing the threshold. Indeed, the best fit of our model was obtained by setting the fraction of force on talin to 7.3% (see Supplementary Table 1), suggesting that talin experiences only a small fraction of the load transmitted by integrins (of the order of a few piconewtons). This is consistent with recently measured tension levels across single talin molecules within cells 18 , and with the observation that talin depletion did not affect force transmission at low rigidities (Fig. 1). Collectively, those data support the notion that the soft properties of talin 16 are optimized to allow unfolding at low forces, thereby detecting force levels without impairing force transmission (which may be mediated by other molecules such as α-actinin 27 ).
An open question arising from our work is how vinculin binding to talin leads to adhesion growth and YAP translocation. The mechanisms involved are likely to include talin-induced integrin clustering [34][35][36] , signalling triggered by vinculin activation on talin binding 37 , vinculin-actin binding to reinforce the mechanical clutch 29,38 , and the relay of mechanical forces to the nucleus through stress fibres 39,40 . Nevertheless and independently of downstream events, our study clarifies how rigidity regulates force transmission, and how force transmission is in turn converted into a biochemical signal. Given the myriad physiological and pathological processes associated with tissue stiffening 41 and YAP signalling 42 , this understanding may also open the door to further fundamental discoveries in biology, and new therapeutic strategies.

Methods and any associated references are available in the online version of the paper.
Note: Supplementary Information is available in the online version of the paper METHODS Cell culture constructs, and transfection. Talin 1 −/− mouse embryonic fibroblasts were described previously 20,21 , and cultured in DMEM 1× (Life Technologies, 41965), supplemented with 15% FBS. Wild-type mouse embryonic fibroblasts were also described previously 20 , and cultured in DMEM supplemented with 10% FBS. All cells tested negative for mycoplasma contamination. All transfections were carried out using the Neon transfection device according to the manufacturer's instructions. To deplete talin levels, cells were transfected with talin 2 shRNA, which contained puromycin resistance (previously described 21 ). One day after transfection, cells were incubated with 2 µg ml −1 puromycin for four days to select for transfected cells. Resulting transfection efficiency was of 52%, with a standard deviation of 11% (see Supplementary Fig. 3). EGFP-talin 1 was a gift from D. Critchley (University of Leicester, UK) and described previously 43 , EGFP-talin 1 IVVI was prepared in-house from EGFP-talin 1 by introducing four point mutations (T809I/T833V/T867V/T901I). EGFP-talin 1 head (Addgene plasmid no. 32856) and EGFP-talin 1 rod (Addgene plasmid no. 32855) were obtained from A. Huttenlocher (University of Wisconsin-Madison, USA) 44 . EGFP-VD1 (Addgene plasmid no. 46270, described as pEGFPC1/GgVcl 1-258) and EGFP-VD1 A50I (Addgene plasmid no. 46271, described as pEGFPC1/GgVcl 1-258 A50I) were obtained from S. Craig (Johns Hopkins School of Medicine, USA) 30 . EGFP-FL talin 1 W359A and EGFP-talin 1 L325R (both FL and head fragment) were gifts from M. Ginsberg's laboratory (UC San Diego, USA) and described previously 24 . Lifeact-GFP was described previously 5 . For talin 2 shRNA experiments, cells were transfected with talin 2 shRNA+corresponding plasmid five days before experiments. For control cells, transfections were made the day before experiments.
Preparation of polyacrylamide gels. Polyacrylamide gels were prepared as previously described 5 . Briefly, glass-bottom dishes (Mattek) were activated with a solution of 3-(trimethoxysilyl)propyl methacrylate (Sigma), acetic acid and ethanol (1:1:14), washed three times with ethanol and air-dried for 10 min. To generate gels of different stiffness, different concentrations of acrylamide and bis-acrylamide were mixed (see Supplementary Table 2) in a solution containing 0.5% ammonium persulfate, 0.05% tetramethylethylenediamine (Sigma), 0.4% fluorescent red carboxylated nanobeads (Invitrogen), and 4.8 mg ml −1 NH-acrylate. Ten microlitres of this solution was then placed on the centre of glass-bottom dishes and covered with 12-mmdiameter glass coverslips. After gel polymerization, top coverslips were removed and gels were incubated with fibronectin (Sigma) overnight at 4 • C. After washing gels with PBS, cells were then trypsinized and plated on gels. Experiments were carried out 4-8 h after cell seeding. To compare fibronectin coating densities on the gels, fibronectin used for coating was previously labelled with an Alexa Fluor 488 protein labelling kit according to the manufacturer's instructions (A-10235, Thermo Fisher Scientific). Then, fibronectin coating densities at the gel surface were measured by acquiring epifluorescence images with a 20× objective (NA 0.45), and quantifying resulting fluorescence intensity levels.
Polyacrylamide gel stiffness measurements. The stiffness (Young's modulus) of polyacrylamide gels was measured by atomic force microscopy as previously described 47 . Briefly, measurements were made with a custom-built atomic force microscope attached to an inverted optical microscope (Nikon TE200). Silicon nitride pyramidal tips with an effective half-angle θ of 20 • and a nominal spring constant of k = 0.01-0.03 N m −1 were used (MLCT, Bruker). The actual spring constant was calibrated by thermal tuning using the simple harmonic oscillator model 48 . The Young's modulus was measured by recording 10 force-displacement curves with a peak-to-peak amplitude of 6 µm and a frequency of 1 Hz. Three points near the gel centre were selected in each gel, separated 5 µm from each other. For each stiffness, ≥6 gels produced in two batches were measured. To compute the Young's modulus (E), the Hertz model equation for pyramidal tips was fitted to the force-displacement curves. The equation was fitted for an effective indentation of 1,000 nm.
Traction force measurements. Traction force measurements were performed as described previously 5 . Briefly, cells seeded on gels were placed on an inverted microscope (Nikon Eclipse Ti). Phase contrast images of single cells and fluorescence images of the embedded nanobeads were obtained with a 40× objective (NA 0.6). At the end of the measurements, cells were trypsinized and an image of bead position in the relaxed state of the gel was acquired. By comparing bead positions with and without cells, a map of gel deformations caused by cells was first obtained using custom particle imaging velocimetry software 49 . Then, after assuming that gel displacements were caused by forces exerted by cells in the cell-gel contact area, the corresponding map of cell forces was calculated using a previously described Fourier transform algorithm 39,50 . The average forces per unit area exerted by each cell were then calculated. To calculate the minimum detectable force levels for each rigidity, we followed the same procedure in cell-free gel areas, and calculated the resulting forces. Phase contrast images were also used to calculate average cell spreading areas as a function of substrate stiffness.
Immunostaining. For fluorescence staining, cells were fixed with 4% paraformaldehyde, permeabilized with 0.1% Triton X-100, and labelled first with primary antibodies (1 h, room temperature), and then with Alexa-conjugated secondary antibodies (Invitrogen; 1 h, room temperature). Phalloidin was added with the secondary antibody. Fluorescence images were then acquired with a 60× oil-immersion objective (NA 1.40) using a spinning-disc confocal microscope (Andor). The length of adhesions was assessed by measuring the length of bright vinculin, β 3 integrin or pFAK stainings at the cell edge. Integrin density was assessed as described previously 5 . The area containing pFAK-positive adhesions was calculated after segmenting adhesions as previously described 51 . The degree of YAP nuclear localization was assessed by calculating the ratio between YAP fluorescence in the nuclear region and the cytoplasmic region immediately adjacent. Nuclear and cytoplasmic regions were previously determined by co-staining the nucleus with Hoechst 33342.

Rearward flow measurements.
To measure actin rearward flow, cells were transfected with Lifeact-GFP. Cells were then plated on gels of varying rigidity, and imaged every second for 2 min with a 60× oil-immersion objective (NA 1.40) with a spinning-disc confocal microscope (Andor). For each cell, kymographs were obtained at the cell periphery, and actin speed was measured from the slope of actin features observed in the kymographs. In cells plated on 0.6 kPa gels, actin features were so diffuse that no reliable slopes could be measured in kymographs.

Western blots.
For western blotting of talin, myosin light chain, and phosphorylated myosin light chain, cells were directly incubated with 1× Laemli and boiled at 95 • C for 5 min. Cell lysates were loaded on 4-20% polyacrylamide gels (Bio-Rad), and electrophoresis proteins were then transferred to a nitrocellulose membrane (Whatman, GE Healthcare Life Sciences), which was blocked with 5% drymilk-Tris buffer saline-0.2% Tween. The membrane was incubated first with primary antibodies (overnight, 4 • C), and then with horseradish-peroxidase-coupled secondary antibodies (1 h, room temperature). Bands were revealed using the LumiLight kit (Roche) and quantified using ImageJ software.
Single fibronectin-α v β 3 bond lifetime measurements. The lifetime of single fibronectin-α v β 3 bonds was measured using a previously described biomembrane force probe (BFP) technique 52 . Biotinylated red blood cells (RBCs) for BFP experiments were collected abiding a Georgia Institute of Technology IRB-approved protocol, and prepared as previously described 52 . Target beads were first covalently linked with anti-Penta His (histidine) antibody (catalogue no. 34660, Qiagen), and then further covered with Hexa-His tagged recombinant α v β 3 ectodomain, which was a gift from J. Takagi, Osaka University, Japan 53 . Probe beads were first functionalized with streptavidin through covalent linkage and then partially covered with biotinylated fibronectin module III, domain 7-10 (FN III7-10 , a generous gift from A. Garcia, Georgia Tech, USA). To provide maximum integrin activation, experiments were carried out in the presence of 2 mM Mn 2+ . In a BFP experiment, the probe bead was glued through biotin-streptavidin interaction onto the apex of the RBC, which was aspirated by a micropipette and acted as a force transducer. A second opposing micropipette grabbed the target bead and drove it to repeatedly impinge the probe bead, contact for 2 s and then retract (ramping). Displacement of the probe bead was tracked in real time, which reflected the force exerted on it. If an adhesion event occurred, meaning that one bond or more was formed between the two bead surfaces, ramping resulted in a tensile force signal of the probe bead that pulled on and elongated the RBC. The ramping was then paused at a preset force level (clamping) to wait for bond dissociation, manifested by a backward deformation of the RBC and a sudden force drop to 0 pN. To ensure that most adhesion events (>90%) were single molecular interactions, the frequency of adhesion occurrence was adjusted to be low (<20%) by titrating the coating densities on both beads 54 . The time that each adhesion survived during clamping is termed the lifetime, which was collected under a range of positive forces. To derive lifetimes under zero force, the ramping was paused at 0 pN and held for 20 s. Sudden drops/increases in the thermal fluctuation signal of the probe bead were used to judge the bond association/dissociation, given that bonding suppresses thermal fluctuation 55 . The average lifetimes were then plotted against the corresponding forces to form a 'lifetime versus force' curve 56 . To confirm binding specificity, control experiments were performed by either adding a α V β 3 blocking antibody (clone LM609, EMD Millipore) or coating beads only with streptavidin instead of FN III7-10 . Both controls yielded rare binding (∼3%).      For 5 kPa, p=0.033 between 1µg/ml and 10 µg/ml and p=0.033 between 1µg/ml and 100 µg/ml. For 11 kPa, p<0.001 between 100µg/ml and, both, 1 µg/ml and 10 µg/ml. exponent and the other with negative exponent. The parameters of the two exponentials were obtained after fitting the curve to experimental data from either the fibronectin--α5β1 bond 3 or the fibronectin--αvβ3 bond (Fig. 2a, lifetime data in the figure correspond to the inverse of k off ).
In both cases, we took experimental data corresponding to the case of maximum activation (with Mn 2+ ions). Fits shown in figures were obtained by using fibronectin--α5β1 data, but equivalently good fits could be obtained with fibronectin--αvβ3 data after minor modifications of the parameters. To best fit experimental results, the resulting k off (f) curve was multiplied by a scaling factor of 0.9, possibly reflecting minor differences in experimental conditions between single molecule measurements and integrins within adhered live cells. Additionally, we further increased the unbinding rates of integrin--Fn bonds at very low forces (< 1 pN) with respect to experimental values. Introducing this change only increased the slope at low rigidities, and did not alter any of the other model predictions. The presence of a very high unbinding rate (low affinity) at low forces may reflect a low activation state in integrins not submitted to cyclic mechanical stretch 4 . k unf (f) was modeled as a simple slip bond (one exponential with a positive exponent) by fitting the curve to previously measured data 5 .
Because the load on each integrin may be shared between talin and other adaptor molecules, the force used to calculate unfolding was corrected by a factor FR, corresponding to the fraction of integrin--transmitted force experienced by talin. Then, unbinding and unfolding times were determined stochastically according to k off and k unf . If unfolding time was shorter than unbinding time, and fell within the time step window, then talin was allowed to either bind to vinculin (according to a force--independent binding rate k vin ) or refold (according to a refolding rate k fold ). k fold was also modelled as a simple exponential (with negative exponent) fitted to measured data 5 . If vinculin binding occurred before refolding, adhesion reinforcement was assumed to occur, and integrin density increased by d add integrins/μm 2 . If integrins unbound before talin unfolding or vinculin binding, integrin density decreased by d add , reflecting the fact that adhesions shrink if force application is decreased 6,7 . However, integrin density was not allowed to decrease below the basal d int level.
Parameters n f , k ont , n m , d add , and FR were adjusted during simulations to fit the different experimental conditions tested. For those parameters, both 95% confidence intervals and sensitivities S were calculated. Confidence intervals were calculated by using the nlparci Matlab function taking as an input the experimental data and the Jacobian matrix of the fitted model function. The sensitivity S was calculated following a previously described approach 8 according to the following expression: Where p are the different parameters, and k threshold is the threshold substrate spring constant (rigidity) that leads to talin unfolding, defined as the point in which integrin density increased by 10% with respect to baseline values. S values can be interpreted as the fold change in k threshold induced by a fold change in parameter value. To calculate S values, k threshold was calculated for p values around the optimal fitted values. Then, k threshold was plotted against p in a log--log scale, and S was taken as the slope of a linear fit to the plot.