Focal adhesion kinase - the reversible molecular mechanosensor
BBiophysical Journal Volume: 00 Month Year 1–13 1
Focal adhesion kinase – the reversible molecular mechanosensor
S. Bell and E. M. Terentjev
Abstract
Sensors are the first element of the pathways that control the response of cells to their environment. After chemical, the nextmost important cue is mechanical, and protein complexes that produce or enable a chemical signal in response to a mechanicalstimulus are called mechanosensors. There is a sharp distinction between sensing an external force or pressure/tension appliedto the cell, and sensing the mechanical stiffness of the environment. We call the first mechanosensitivity of the 1st kind, andthe latter mechanosensitivity of the 2nd kind. There are two variants of protein complexes that act as mechanosensors of the2nd kind: producing either a one-off or a reversible action. The latent complex of TGF- β is an example of the one-off action:on the release of active TGF- β signal, the complex is discarded and needs to be replaced. In contrast, focal adhesion kinase(FAK) in a complex with integrin is a reversible mechanosensor, which initiates the chemical signal in its active phosphory-lated conformation, but can spontaneously return to its closed folded conformation. Here we study the physical mechanismof the reversible mechanosensor of the 2nd kind, using FAK as a practical example. We find how the rates of conformationchanges depend on the substrate stiffness and the pulling force applied from the cell cytoskeleton. The results compare wellwith the phenotype observations of cells on different substrates.Insert Received for publication Date and in final form [email protected] Cells exist within a complex and varying environment. Tofunction effectively, cells must collect information abouttheir external environment, and then respond appropriately.Cell environment has a profound effect on cell migration andcell fate. It is also a major factor in metastasis of certaincancers (1, 2).Sensing is the first part of the chain of events that consti-tute the cell response to external stimuli. Cells respond to avariety of cues; both chemical and mechanical stimuli mustbe transduced inside the cell. Mechanosensors are proteincomplexes that produce responses to mechanical inputs (3,4). There are two distinct types of mechanosensing: reactingto an external force, or sensing the viscoelastic propertiesof the cell environment. We call the first mechanosensitivityof the 1st kind, and the latter mechanosensitivity of the 2ndkind.Mechanosensitive ion channels (MSC), such as alame-thicin (5), are an example of mechanosensors of the firstkind. MSCs exist in all cells and provide a non-specificresponse to stress in a bilayer membrane (6, 7). Localmechanical forces could be produced by many external fac-tors, but MSC operation appears to be universal and quitesimple. The ion channel is closed at low tension, openingas the tension exceeds a certain threshold, allowing ions to cross the membrane. Traditionally, MSC operation is under-stood as a two-state model. There is a balance of energygain on expanding the ‘hole’ under tension, and the energypenalty on increasing the hydrophobic region on the innerrim of the channel exposed to water on opening. Thesetwo-state systems (open/closed, or bonded/released) withthe energy barrier between the states depending on appliedforce, are common in biophysics (8, 9). Rates of transitionin these systems are often calculated using the ‘Bell for-mula’ (10), which has them increasing exponentially withthe force. This is just the classical result of Kramers andSmoluchowski (11, 12), but it is invalid in the limit of highforces or weak barriers.Mechanosensitivity of the 2nd kind is different in nature.The sensor has to actively measure the response coefficient(stiffness in this case, or matrix viscosity in the case of bacte-rial flagellar motion). On macroscopic scales (in engineeringor rheometry) we can do this with two separate measure-ments: of force (stress) and of position (strain), or we couldcontrast two separate points of force application. One couldalso use inertial effects, such as impact or oscillation, to mea-sure the stiffness or elastic constant of the element. None ofthese options are available on a molecular scale. The singlesensor complex cannot measure relative displacements in thesubstrate, and the overdamped dynamics prevents any roleof inertia. As a result, important biophysical work on focal © 2015 The Authors a r X i v : . [ q - b i o . S C ] F e b adhesion complexes (13) had to resort to an idea of dynam-ically growing force (or force-dependent velocity) appliedto the proximal side of the two-spring sensor. Other impor-tant work (14) also relies on the dynamics of applied forcewith an elaborate construct of ‘catch-bonds’ whose stabilityincreases with pulling force. In reality, the cell cytoskeletalfilaments exert a pulling force that is constant on the time-scales involved. Further, the internal observable needed tosense stiffness in the catch-bond model (namely, fraction ofunbound integrin-ECM bonds at a focal adhesion) does nothave a clear downstream measurement process associatedwith it.In an earlier study (15), we addressed the problem of howa mechanosensor of the 2nd kind should work, by developinga physical mechanism with a similar action to the two-springmodel of Schwarz et al. (13, 16). That work focussed on thelatent complex of TGF- β (17–19), which is an irreversibleone-off sensor: after the latent complex is ‘broken’ and activeTGF- β released, the whole construct has to be replaced.Here we apply these ideas to a reversible mechanosensor:protein tyrosine-kinase, now called focal adhesion kinase(FAK) (20–23). As the name suggests, FAK is abundant inthe regions of focal adhesions (21), which are developed inthe cells on more stiff substrates, often also associated withfibrosis: the development of stress fibers of bundled actinfilaments connecting to these focal adhesions and deliver-ing a substantially higher pulling force. FAK is also presentin cells of soft substrates in spite of the lack of any focaladhesions, and also in the lamellipodia during cell motility(22, 24, 25). Phosphorylation of tyrosine residues of FAK iswell known as the initial step of at least two signalling path-ways of mechanosensing (26), leading to the cell increasingproduction of smooth muscle actin, and eventually fibrosis. To achieve our aim of developing a self-consistent physicalmodel of a molecular mechanosensor, we first need to havean extensive overview of the biological system, or the seriesof elements transmitting force between the cytoskeleton andthe ECM. The system starts with the activated integrins bind-ing to ECM, and follows to a group of cytoplasmic proteinsthat bridge between integrin and the cytoskeletal actin fila-ments. These in turn are assumed to provide the pulling forceby the action of myosin. Among these proteins is FAK (alongwith talin, paxillin and vinculin), which we discuss in greaterdetail – identifying the conformational transition associatedwith its activation, and how the effective free energy of sucha protein must evolve on conformational change.We then proceed to the main focus of this work – toconstruct the physical model that includes the viscoelasticresponse of the ECM and the thermally activated responseof the protein mechanosensor. It turns out that both thermalactivation (thermal noise) and viscous damping are essential in both elements of the mechanical chain. We outline how to‘solve’ this physical problem, that is, derive the effective rateof FAK opening and activation (as well as the reverse rate ofits auto-inhibition) using the methods of stochastic Kramerstheory. We present and test these rates in the Results section.
Biological system
A sensor is a device that detects or measures a physicalproperty and records, indicates, or otherwise responds to it.An important characteristic of any sensor is its proportionalresponse to the input signal; in this aspect, a sensor is nota relay, which is a device that switches on/off response onreceiving a sufficient level of input signal. In the case ofmechanosensors of the 2nd kind, the property that we needto measure is the stiffness (elastic modulus) of the extra-cellular matrix (or other environment the cell is immersedin). To probe the modulus of a medium, a force has to beapplied to it, either as a local point source, or as distributedstress. In the cell the source of this force is the actin-myosinactivity of cytoskeleton. Therefore, we need to trace theseries of connected devices, from the point of force origin(F-actin) to the point of its application at ECM.Figure 1 illustrates this force chain, which has beenreproduced in a large number of important publications inthis field (27–31). As well as FAK, there are several impor-tant players that we should also consider: integrins, talin,paxillin, and the cytoskeleton. How do these componentseach contribute to the function of the complex?The integrin family of transmembrane proteins linkthe extracellular matrix (ECM) to the intracellular actincytoskeleton via a variety of protein-tyrosine kinases, one ofwhich is FAK (32). Integrins are aggregated in focal adhe-sions, and they mediate the cell interaction with ECM (3).Activation of integrins is required for adhesion to the sub-strate; active integrins acquire ligand affinity and bind tothe proteins of the ECM. It is well established that integrinactivation and clustering leads to FAK activation and the sub-sequent signalling chain of mechanosensing and cytoskele-tal remodelling, e.g. see the review by Parsons (33). Thereis a large body of literature on integrins, with definitivereviews by Hynes (34, 35) explicitly stating that integrins
ECM F - a c t i n cy t op l a s m t a l i n paxillin F AK f Figure 1: The chain of force transduction from the F-actinterminators of the cytoskeleton, through several associatedproteins, passed on to the activated β integrin binding toligands of the deformable ECM. Biophysical Journal 00(00) 1–13 are the mechanosensors. However, activated integrins pos-sess no further catalytic activity of their own, and so cancan only act in isolation as a switch, which is not the pro-portional response required for sensor design. A good sum-mary by Giancotti (36), while talking about ‘integrin sig-nalling’, in fact, shows schemes where FAK is the nearestto cytoskeletal actin filaments. The important work by Guanet al. (37, 38) establishes a clear correlation chain of extra-cellular fibronectin – transmembrane integrins – intracellularFAK, but offers no reason to assume that integrin is thesensing device on this chain.This lack of clarity regarding the specifics of integrinengagement and FAK activation arises from the lack of adetailed physical model: we simply do not understand indetail at a molecular/physical level how FAK is activated.One possibility, explored by U. Schwartz (13, 16), is thatclusters of activated integrins always activate FAK and gen-erate the mechanosensing signal that leads to the increasingF-actin pulling force. As some of the integrins are broken offtheir ECM attachment, the associated FAK signal reduces,regulating the further force increase – and that is the actionof the focal adhesion mechanosensor complex.In this paper we propose a different mechanism, wherethe activation of FAK is dependent on cytoskeletal tensionand ECM stiffness, and the integrin (along with other mem-bers of the force chain in Fig. 1) is merely playing a roleof force transducer. Of course, without the activated inte-grin there would be no force transduction to ECM, and nomechanosensing. Here, we look at each individual integrin-FAK sensor, as opposed to exploring the role of clustering.This is clearly a shortcoming, as clustering is definitely animportant aspect of the process: allostery of integrins (andassociated FAK) must have a role in the signalling process,as in chemotaxis (39, 40). This will have to be a topic of fur-ther study, while the present paper focuses on the physicalmodel of individual FAK sensor operation.There is a clear indication that phosphorylation of FAKis a key step in the mechanosensing process (4). Indeed,Schaller et al. (21) state that FAK phosphorylation is the ini-tial step of signalling, and show evidence that crosslinkingintegrins and ECM (i.e. making the ‘substrate’ stiffer) leadsto an enhanced FAK phosphorylation, while conversely, adamage to integrin is connected with a reduced activation ofFAK.In the native folded state of FAK, the FERM domain (theN-terminal of the protein) is physically bonded to the cat-alytic domain (kinase) (23, 41); we call this closed state, [c].A conformational change occurs, which we shall call a tran-sition to an open state, [o], when this physical bond is dis-rupted and the kinase separates from the FERM domain, seeFig. 2. Note that because there is a peptide chain link ([362-411] segment) between the FERM and kinase domains,they remain closely associated even after the conformationalchange – this is what makes FAK a reversible mechanosen-sor. The activation of the catalytic domain occurs in two paxillin P f [c] u integrin ECM [o] u max [a] x cellmembrane F E R M F A T F - a c t i n k i na s e f F E R M F A T F - a c t i n k i na s e s r c P f F E R M F A T F - a c t i n k i na s e PP Tyr397
Figure 2: Schematic representation of FAK conformations.The FERM domain of FAK is associated with the integrin-talin assembly, near the cell membrane. while the FATdomain is associated the actin binding site (31). The pullingforce is transmitted through this chain to the FERM-kinasephysical bond. In the closed state [c] the kinase domain isinactive and the whole FAK protein is in its native low-energy state. Once the physical bond holding the FERMdomain and the kinase together is broken, the protein adoptsthe open conformation [o]. In the open state, first the Tyr397site spontaneously phosphorylates, which in turn allowsbinding of Src and further phosphorylation of the kinase -turning it into the active state [a], see (41, 45, 46).steps: first the Tyr397 residue phosphorylates, which thenallows binding of the Src kinase (42), which in turn pro-motes phosphorylation of several other sites of the catalyticdomain (Tyr407, 576, 577, 861 and 925), making FAK fullyactivated. There is also a process involving p130cas, actingas a kinase substrate, involved in generating the response ofactivated FAK (43, 44). We shall call this state [a] in thesubsequent discussion.Recent work (31) explicitly confirms the critical roleof tension, delivered from the actin cytoskeleton toFAK/integrin and involved in mechanosensing. A key rolein this system is played by talin. There are many papersinvestigating the correlation of talin (as well as paxillin) with β -integrin and FAK, but recent advances clearly show thattalin is capable of high stretching by a tensile force (47, 48),implying a function similar to that of titin in muscle cells(acting as an extension-limiter). It is also now clear that theimmobile domain at the N-terminal of talin is associated withintegrin, as well as the FERM domain of FAK (47, 48), whilethe C-terminal of talin is associated with paxillin and the Biophysical Journal 00(00) 1–13 2 METHODS opened u u U(u) G o closed max G o f= [c] active G a f f K ( f , )+K - (a)(b) u [o] U(u)
Figure 3: Schematic potential energy of different FAK con-formations. (a) The force-free molecule has its native foldedstate [c], compare with Fig. 2. The binding free energy ∆ G o has to be overcome to separate the kinase from the FERMdomain, after which there is a range of conformations ofroughly the same energy is achieved by further separatingthese two domains in the open state [o]. At full separation(distance u max ) the Src binding and kinase phosphorylationlead to the active state [a] of the protein, with the free energygain ∆ G a . (b) When a pulling force is applied to this sys-tem ( f > f > ) the potential energy profile distorts, sothat both [o] and [a] states shift down in energy by the sameamount of − f · u max .focal adhesion targeting (FAT) domain (C-terminal) of FAK.The actin filaments of the cytoskeleton exert a pulling forceon this zone. Talin, therefore, acts as a scaffold for other pro-teins to arrange around, but more importantly, allows forceto be transmitted from the cytoskeleton to the ECM, via inte-grins. All of these established facts are consistent with themodel of conformational change in FAK sketched in Fig. 2,where the integrin is the bridging element to the ECM, withthe FERM domain localised near the cell membrane and N-terminal of talin. At the opposite end, the FAT domain canbe pulled away by an applied force. This model is supportedby the recent computational analysis (49) showing that theclosed and the open states of FAK are reversibly reached byincreasing and decreasing of pulling force.Since we shall not consider the cell motility, one hasto assume that FERM domain remains fixed with respectto the ECM/integrin reference frame. That is, if there is adeformation in (soft) ECM, then this point will move accord-ingly, with the integrin and the local cell membrane alljoined together. In our model, to achieve the large displace-ment associated with the [c] → [o] conformational transition of FAK, in the crowded intracellular environment, a mechan-ical work is expended. This mechanical energy can onlycome from the active cytoskeletal forces, delivered via actinfilaments.We can now record these conformational changes in theFAK structure in the schematic plot of the ‘unfolding freeenergy’, which will play the role of potential energy U ( u ) for the subsequent stochastic analysis of the sensor action,illustrated in Fig. 3. The concept of such unfolding freeenergy is becoming quite common (50), when one identifiesan appropriate reaction coordinate and discovers that a deepfree energy minimum exists in the native folded state, with abroad range of intermediate conformations having a ragged,but essentially flat free energy profile – before the final fullunfolding rises the energy rapidly. Figure 3(a) needs to belooked at together with the conformation sketches in Fig. 2:the native state [c] needs a substantial free energy ( ∆ G o )to disrupt the physical bonds holding the kinase and FERMdomains together. However, once this is achieved, there areonly very minor free energy changes due to the small bend-ing of the [362-411] segment (23), when the kinase andFERM domains are gradually pulled apart. This change ismeasured by the relative distance, which we label u in thesketch and the plot. If one insists on further separation ofthe protein ends, past the fully open conformation [o] at u = u max , the protein will have to unfold at a great cost tothe free energy. Binding of Src and phosphorylation (i.e. con-verting the [o] state into the [a] state) lowers the free energyof the fully open conformation by an amount ∆ G a . Note thatthere is no path back to the closed state, once the kinase isactivated: one can only achieve ‘autoinhibition’ (23) via the[a] → [o] → [c] sequence.If we accept the basic form of the protein potential pro-file, as shown in Fig. 3(a), the effect of the pulling force f applied to FAK from the actin cytoskeletal filaments isreflected by the mechanical work: U ( u ) − f · u . If we takethe reference point u = 0 as the closed native conformation,then the opening barrier reduces by: ∆ G o − f · u . Similarly,the free energy of the fully open state [o] becomes lower by: ∆ G o − f · u max , see Fig. 3(b). Since the binding free energyof Src and phosphorylation does not depend on the appliedforce, the energy level of the active state [a] lowers by thesame amount of ∆ G a relative to the current [o] state. Stochastic two-spring model
The two-spring model discussed in detail by Schwarz et al.(13, 16) and often reproduced afterwards (26) is a correctconcept, except that it needs to take into account that boththe viscoelastic substrate and the sensor, described by thepotential energy U ( u ) , experience independent thermal exci-tations. This is inevitable at the molecular level, since we areconsidering the mechanical damping in the substrate (as wemust) and in the sensor (as we will). In the overdamped limit Biophysical Journal 00(00) 1–13 sub c f U ( x -x ) Substrate FAK x x F A T k i n ase F E R M Figure 4: A scheme of the 2-spring model used to produceequations (1). The viscoelastic substrate is characterisedby its elastic stiffness and stress-relaxation time given by γ sub /κ . The conformational change of FAK is described bya potential U ( u ) , see Fig. 3, and the associated relaxationtime determined by the damping constant γ c .all forces must balance along the series of connected ele-ments, and only thermal fluctuations – independent in thetwo elements – can create a relative displacement in themiddle of this series (i.e. on the sensor). It is this relativedisplacement that one needs to ‘measure’ the stiffness.Following the logic outlined in greater detail in earlierwork (15), we introduce two independent stochastic vari-ables. The first is x = x , which measures the displacementof the substrate with respect to its undeformed referencestate, and therefore also marks the position of the FERMdomain (or the origin of the length u ). The second is x that measures the displacement of the far end of the kinasedomain: the point of application of the pulling force f , seeFig. 4 for an illustration. These two variables satisfy a pairof coupled overdamped Langevin equations: γ sub ˙ x = − κx + d U d( x − x ) + (cid:112) k B T γ sub · ζ ( t ) ,γ c ˙ x = − d U d( x − x ) + f + (cid:112) k B T γ c · ζ ( t ) , (1)where κ is the elastic stiffness and γ sub the damping con-stant of viscoelastic substrate (ECM), while γ c is the (com-pletely independent) damping constant for the conforma-tional changes in FAK structure; the base stochastic process ζ ( t ) is assumed to be Gaussian and normalised to unity. Notethat it is the difference in independent position coordinates u = x − x , that affects the sensor potential U ( u ) . Theproblem naturally reduces to a 2-dimensional Smoluchowskiequation for the variables x = x ( t ) for the substrate,and u ( t ) for the FAK conformations, with the correspond-ing diffusion constants D i = k B T /γ i , and the Cartesiancomponents of diffusion current: J i = − k B Tγ i e − V eff /k B T ∇ i (cid:16) e V eff /k B T P (cid:17) , (2) u x max u sensor: s ub s t r a t e : Figure 5: The 2D contour plot of the effective potential V eff ( x, u ) at a certain value of pulling force applied. Theposition of substrate anchoring has moved from x = 0 to ¯ x = f /κ , and the depth of the energy well of the [o] statehas lowered to ∆ G o − f u max . The dashed line shows thetrajectory of the system evolution that leads to the openingof the [c] state.where P ( x, u ; t ) is the probability distribution of the pro-cess, and V eff ( x , x ) = 12 κx − f x + U ( x − x )= 12 κx − f x + U ( u ) − f u (3)represents the effective potential landscape over which thesubstrate and the mechanosensor complex move, subject tothermal excitation and the external constant force f .The effective Kramers problem of escape over the bar-rier has been solved many times over the years (8, 11, 12,51, 52). The multi-dimensional Kramers escape problem,with the potential profile not dissimilar to that in Fig. 5was also solved many times (53, 54). Unlike many previousapproaches, we will not allow unphysical solutions by mis-treating the case of very low/vanishing barrier. In the casewhen the effective potential barrier is not high enough topermit the classical Kramers approach of steepest descentintegrals, one of several good general methods is via Laplacetransformation of the Smoluchowski equation (12, 55). Thecompact answer for the mean time of first passage from theclosed state [c] to the top of the barrier of height Q a distance ∆ u away is: τ + = ∆ u D (cid:34)(cid:18) k B TQ (cid:19) (cid:16) e Q/k B T − (cid:17) − k B TQ (cid:35) . (4)This is a key expression, which gives the standard Kramersthermal-activation law when the barrier is high (which isalso the regime when the ‘Bell formula’ (10) is valid), butin the limit of low barrier it correctly reduces to the simplediffusion time across the distance ∆ u . Biophysical Journal 00(00) 1–13 3 RESULTS AND DISCUSSION
Estimates of material parameters
In order to make plots with parameter values correspondingto a real cell, let us start with the strength of the bond holdingthe FERM and kinase domain in the closed (autoinhibited)state. The MD simulation study (49) estimated the energybarrier for FAK opening as ∆ G o /k B T ≈ , which is ca. kcal/mol at room temperature. This value appears too high,since it is known that interdomain hydrophobic interaction insuch proteins is usually low-affinity (56). A reasonable valuefor this interdomain bonding is ca. kcal/mol, or ∼ k B T .We shall present the results and quantitative model predic-tions using this assumed magnitude of the energy barrier ∆ G o .We can also take the position of the barrier from thesame study: u = 0 . nm, again, a reasonable value forthe protein domain structure. This makes the critical force F c = 3∆ G o / u ≈ pN. This is a high force that is likelyto unfold most proteins, and is also unlikely to be generatedby a single actin filament of a cell cytoskeleton. For com-parison, the force to fully unfold integrin is quoted as 165pN (57). Buscemi et al. (57) also quoted 40 pN as the forcerequired to unlock the physical bond of the latent complexof TGF- β
1. Other reports investigate the force required todisrupt the fibronectin-integrin-cytoskeleton linkage, findingthe value of only 1-2 pN (58, 59). For a force f = 5 pN, thescaled non-dimensional value ¯ f ≈ . .We also need to estimate values of substrate stiffness.For reference, the elastic modulus of a typical collagen-richmammalian tendon is . GPa (60), of a collagen/elastic lig-ament: . Mpa (61), and of an aorta wall: . MPa (62).Synthetic rubber has a modulus around kPa (63). If ahalf-space occupied by an elastic medium (e.g. gel sub-strate or glass plate) with the Young modulus Y , and a pointforce F is applied along the surface (modelling the pullingof the integrin-ECM junction, Fig. 2), the response coeffi-cient (spring constant) that we have called the stiffness isgiven by κ = (4 / πY ξ , where ξ is a short-distance cutoff:essentially the mesh size of the substrate. This is a classi-cal relation going as far back as Lord Kelvin (64). For aweak gel with Y = 10 kPa, and a characteristic networkmesh size ξ = 10 nm, we obtain κ = 4 . · − N/m, andthe scaled non-dimensional parameter ¯ κ ≈ . . On a stiffmineral glass with Y = 10 GPa, we must take the charac-teristic size to be a ‘cage’ size (slightly above the size of amonomer), ξ = 1 nm, which gives κ = 42 N/m, and thenon-dimensional parameter ¯ κ ≈ . A typical stiff plastichas a value about 10 times smaller. So a large spectrum ofvalues ¯ κ could be explored by living cells.Finally, we need estimates of the damping constants. Thesimulation study (49) determined a reasonable value for thediffusion constant of the FAK complex: D = k B T /γ c ≈ · − m s − . At room temperature, this gives the dampingconstant: γ c = 7 · − kg s − . Then, the overall scale (‘baremagnitude’) of the rate K + is (∆ G o /u γ c ) ≈ · s − , which means a time scale of ca. 12 ns, cf. equation (7). This‘bare’ time scale is compatible with available data and simu-lations on full and partial protein unfolding (65); naturally, atgiven bonding energy and low pulliung force the actual rateof FAK opening/activation would be much lower: the plotsindicate tens of microseconds to milliseconds range.To estimate the damping constant of the viscoelastic sub-strate, we assessed the characteristic time of its stress relax-ation, which is the ratio γ sub /κ in our parameter notation.The order of magnitude of stress relaxation time in gelsis quite long, up to s. Using the values of κ for gelsgiven above, the typical damping constant is calculated as: γ sub ≈ .
04 kg s − , and the ratio ζ = γ c /γ sub ∼ · − .For stiff substrates, we need the vibration damping timein a solid glass (one must not confuse this with the creepstress relaxation, extensively studied in glasses (66) but notrelated to our viscoelastic response). The characteristic timewe are looking for is closer to the β -relaxation time of the‘cage’ motion (67), and the literature gives values in therange of . s (68). Combining the corresponding valueof stiffness κ discussed above gives the damping constant γ sub ≈ . − , and the ratio ζ = γ c /γ sub ∼ − orless. We have established a physical model for the opening ofFAK under tension. Let us now apply the generic expressionmean first passage time of equation (4) to V eff ( x , x ) to findthe rates of this conformational change. Having establishedrealistic parameter values, we can plot the behaviour of ourmodel, and test its predictions against what is observed inthis biological system. Rate of [c]-[o] transition: K + There are many complexities regarding choosing an optimalpath across the potential landscape V eff ( x, u ) , some of whichare discussed in (53, 54), but we are aiming for the quickestway to a qualitatively meaningful answer. As such, we shallassume that the reaction path consists of two ‘legs’: fromthe origin down to the minimum of the potential, which isshifted to ¯ x = f /κ due to the substrate deformation, andfrom this minimum over the saddle (barrier) into the openstate of FAK conformation. The average time along the firstleg is given by the equation (4) with the distance ∆ u = ¯ x and the negative energy level E = − f / κ , with the dif-fusion constant determined by the damping constant of thesubstrate: τ sub = 2 γ sub κ + 4 γ sub k B Tf (cid:16) e − f / κk B T − (cid:17) . (5) Biophysical Journal 00(00) 1–13 open i ng r a t e K [ / s ] + (a) (b)
20 6040 pulling force f [pN] pulling force f [pN] . . . . . . . Figure 6: The rate constant of FAK opening K + ( f, κ ) is plotted as a function of the pulling force f , for several values of givensubstrate stiffness labelled on the plot. Here we take the bond strength of the FERM-kinase link ∆ G o = 11 k B T , u = 0 . nm, and the ratio of damping constants γ c /γ sub = 10 − (see the discussion in text about the representative values of param-eters). The plot (a) illustrates the overall nature of this response, while the plot (b) zooms in the region of small forces whichare biologically relevant.Here the ratio γ sub /κ is the characteristic stress-relaxationtime of the viscoelastic substrate (69), which will play a sig-nificant role in our results. Naturally, τ sub = 0 when there isno pulling force and the minimum is at (0 , .In the region between the minimum of V eff and the poten-tial barrier, a number of earlier papers (15, 52, 54) haveused the effective cubic potential to model this portion of U ( u ) . In this case, when the pulling force is applied, the bar-rier height is reducing as: E = ∆ G o (1 − f u / G o ) / ,while the distance between the minimum [c] and the max-imum at the top of the barrier is reducing as: ∆ u = u (1 − f u / G o ) / . Substituting these values intoequation (4), we find the mean passage time over the barrier: τ esc = − γ c u ∆ G o (cid:16) − fu G o (cid:17) / (6) + γ c k B T u ∆ G (cid:16) − fu G o (cid:17) (cid:18) e ∆ G o ( − fu G o ) / /k B T − (cid:19) . In the limit of high barrier ∆ G o (cid:29) k B T and small force thisexpression becomes proportional to e − (∆ G o − F u ) /k B T , i.e.recovers the ‘Bell formula’ that people use widely. When theforce increases towards the limit F c = 3∆ G o / u , this time τ esc reduces to zero: there is no barrier left to overcome, andthe minimum of V eff shifts to coincide with the entrance tothe [o] state.The overall rate constant of ‘escape’ K + (the transition[c] → [o]) is then determined as the inverse of the total time: K + = ( τ sub + τ esc ) − . From examining equations (5) and (6)it is evident that the rate of FAK opening is a strong functionof the pulling force f , but more importantly: it changes dra-matically with the substrate stiffness κ . The important expo-nential factor e f / κk B T appears in τ sub ; it was discussedat length in (15) where it has emerged in a very differentapproach to solving a similar problem, and interpreted as an effective ‘enzyme effect’ of the system being confined at thebottom of the potential well before jumping over the barrier.In order to analyse and plot it, we need to scale therate constant K + to convert it into non-dimensional val-ues. First, we can identify a characteristic time scale of theFAK conformational change: u γ c / ∆ G o . The two controlparameters defining the opening rate K + are also made non-dimensional: scaling the force by the natural value of theFERM-kinase holding potential, ∆ G o /u , and scaling thesubstrate stiffness by ∆ G o /u . After these transformations,and some algebra, we obtain: K + = (cid:18) ∆ G o u γ c (cid:19) g ¯ f (cid:0) − f / (cid:1) ζ (cid:0) − f / (cid:1) Ψ [ f ] + ¯ f ζ Ψ [ f ] , (7)with shorthand notations Ψ [ f ] = exp[ − g ¯ f / κ ] + g ¯ f / κ − , Ψ [ f ] = exp[ g (cid:0) − f / (cid:1) / ] − g (cid:0) − f / (cid:1) / − , where the non-dimensional abbreviations stand for: theenergy barrier g = ∆ G o /k B T , the force ¯ f = f · u / ∆ G o ,the substrate stiffness ¯ κ = κ · u / ∆ G o , and the ratio ofdamping constants ζ = γ c /γ sub . There are several keyeffects predicted by this expression for the rate of FAK open-ing under force, while attached to a viscoelastic base, whichwe can examine by plotting it.The effectiveness of FAK as a mechanosensor of the 1stkind, i.e. responding to an increase of applied force witha conformation change, is illustrated in Fig. 6. Figure 6(a)highlights the rapid increase in the rate that FAK opens(and its subsequent phosphorylation) on stiffer substrates.For the complex to actively probe the substrate stiffness(mechanosensitivity of the second kind), we posit that thecell remodels itself in response to FAK activation, increasingthe pulling force. This increases the level of FAK activa-tion until a maximum rate is reached. Any increase in force Biophysical Journal 00(00) 1–13 3 RESULTS AND DISCUSSION open i ng r a t e K [ / s ] + . . . . Young modulus Y [Pa] pN 9 pN1.4 pN0.23 pN 13 pN9 pN1.4 pN0.23 pN 20 pN s en s i t i v i t y [ a . u . ] Young modulus Y [Pa] (a) (b)
Figure 7: (a) The rate constant of the [c] → [o] transition K + ( f, κ ) plotted against the substrate stiffness (on logarithmic scale),for values of the pulling force f corresponding to the position of the peaks in Fig. 6(a). As in Fig. 6, we take γ c /γ sub = 10 − for illustration. The arrows point at the inflection point on each curve, i.e. the region of maximum sensitivity. (b) The plot of‘sensitivity’ dK + /dκ for the same parameters, illustrating the maximum sensitivity range at each level of pulling force. Notethat the peak in sensitivity corresponds roughly with the corresponding stiffnesses used to generate Fig. 6, indicating that thesensor is adaptable.beyond this point decreases the rate of FAK opening. Thiswould act as a mechanism for negative feedback, which set-tles the cell tension in homeostasis. The stiffer the substrate,the higher the rate of FAK activation and, accordingly, themore α -SMA stress fibers one would find in this adjustedcell (leading to morphological changes such as fibroblast-myofibroblast transition, or the fibrosis of smooth musclecells). The plot 6(b) zooms in to the region of small forcesand highlights the effect of soft substrates. On substrateswith sufficiently small κ there is no positive force that givesa maximum in the opening rate. Thus, any pulling force onthe FAK-integrin-ECM chain has the effect of lowering theactivation of FAK relative to the untensioned state, and sothe cell does not develop any great tension in the cytoskele-ton. This is consistent with the observation that cells do notdevelop focal adhesions on soft gels.Figure 7 presents the same rate of FAK opening, butfocuses on the effect of substrate stiffness. As we haveshown, the possible range of parameter κ is large, and so weplot the axis of stiffness in logarithmic units. The rate of FAKactivation has a characteristic (generic) form of any sensor inthat it undergoes a continuous change between the ‘off’ and‘on’ states. The latter is a state of high rate of FAK openingand the subsequent phosphorylation, initiating the signal-ing chains leading to more actin production and increase ofstress fibers. For each cell, characterised by a specific level ofpulling force, the substrate could be ‘too soft’, meaning thatFAK does not activate at all – and also ‘too stiff’, where therate of activation reaches a plateau and no longer respondsto further stiffening. Between these two limits, there is arange of maximum sensitivity where the rate of activationdirectly reflects the change of substrate stiffness. Figure 7(b)highlights this by presenting the ‘sensitivity’ directly as thevalue of the derivative dK + /dκ . We see that cells with ahigher pulling force (i.e. with high actin-myosin activity anddeveloped stress fibers) are sensitive to the substrates in the stiff range. In contrast, cells that exert a low pulling force(i.e. no stress fibers, low actin-myosin activity) are mostlysensitive to soft substrates. This is in good agreement withbroad observations about the cell mechanosensitivity of the2nd kind, and their response to substrate stiffness. Stress relaxation in substrate regulates K + There are many indications in the literature that not onlythe substrate stiffness, but also the degree of viscoelasticity(often measured by the characteristic time of stress relax-ation) have an effect on cell mechanosensitivity (69). It isactually irrelevant what particular viscoelastic model oneshould use for the substrate, and certainly impossible to havea universal model covering the highly diverse viscoelasticityof gels, filament networks, and disordered solids like plasticand glass. In the spirit of our ultimately simplified viscoelas-tic model expressed in equations (1), the single parametercharacterising viscoelasticity could be the characteristic timescale γ sub /κ : this could be a measure of the actual stressrelaxation time of different substrates. This would be a veryshort timescale in stiff solids, while complex disordered fila-ment networks like a typical ECM would have this time mea-sured in minutes or hours. We now find that the rate of FAKopening is strongly affected by the viscoelastic relaxationproperties of the substrate.We assume that the damping constant γ c of the FAKcomplex remains the same. In that case, Fig. 8 shows howchanging the damping constant of the substrate γ sub (or theassociated loss modulus of the viscoelastic material) can reg-ulate the FAK mechanosensor. All curves retain exactly thesame topology and amplitude, but the range of sensitivityshifts in either direction. The red curve for ζ = γ c /γ sub =10 − is the same as the red curve for f = 4 . pN in bothplots in Fig. 7. We find that for substrates with greater stressrelaxation (i.e. greater loss modulus, or γ sub , leading to the Biophysical Journal 00(00) 1–13
103 10 . . . . open i ng r a t e K [ / s ] + Young modulus Y [Pa] = c sub -6 -8 -9 -10 -7 f= Figure 8: The rate of the [c] → [o] transition K + ( f, κ ) plot-ted against the substrate stiffness for a fixed (low) value ofpulling force and a set of changing stress-relaxation prop-erties of the substrate measured by the ratio γ c /γ sub , cf.equations (1) and (7). The range of maximum sensitivityshifts to the effectively stiffer substrate range for materialswith higher damping constant γ sub .ratio ζ becoming smaller), the FAK sensor will activate athigher stiffnesses. In other words, in stiffer substrates, stressrelaxation suppresses the response of a sensor with respectto a strictly elastic substrate.We also see the strong effect of substrate viscoelasticityon the absolute value of rate of FAK activation K + . Figure9 shows a tough rubber with the Young modulus of ∼ MPa (not a completely rigid glass). A range of γ sub is tested,and here we see how the material with a higher loss factor(i.e. lower ratio ζ ) has a reduced response at a lower fullingforce. This is essentially analogous to the substrate appear-ing ‘softer’. This might appear counter to the conclusion onedraws from Fig. 8 (where the range of sensitivity shifted tostiffer substrate), but one must remember that we are explor-ing different aspects of the same expression K + ( f, κ ) : theinformation conveyed by Fig. 6 is exactly the same as that inFig. 7(a). Rate of [o]-[c] transition: K − The free energy profile of the conformation change lead-ing to the [o] → [c] transition (i.e. the spontaneous return ofFAK to its native folded conformation: the autoinhibition)is essentially described by the linear potential, see Fig. 3.From the reference point of [o] state, the energy barrier is E = f ( u max − u ) , and we should assume that the physicaldistance the FERM domain needs to travel remains constant:it is determined by the extent of the protein structure (23, 45).This process also does not depend on the substrate stiffness.As a result, the rate of the folding transition is the inverse of
60 MPa . . . . open i ng r a t e K [ / s ] + . . pulling force f [pN] = c sub -6-8 -7 -9 Figure 9: The rate of the [c] → [o] transition K + ( f, κ ) plottedagainst the pulling force f (in the biologically relevant rangeof small forces) for a set of values γ c /γ sub labelled on theplot representing the change in stress-relaxation characteris-tics of the substrate. The Young modulus of the substrate is Y ≈ MPa.the mean first-passage time (4) with these parameters: K − ( f ) = fγ c ∆ u (cid:18) k B Tf ∆ u (cid:104) e f ∆ u/k B T − (cid:105) − (cid:19) − , (8)with the shorthand notation ∆ u = ( u max − u ) . Whenthe force is high, and the [o] state has a deep free-energyminimum generated by this external mechanical work, seeFig. 3(b), this rate reduces exponentially with pulling force: K − ≈ ( f /γ c k B T ) e − f ∆ u/k B T . This reflects the increas-ing stability of the [o] state when FAK is pulled with a highforce, even before it phosphorylates and further stabilises inthe active state [a]. On the other hand, at vanishing force: f → , this rate becomes K − ≈ k B T /γ c ∆ u , which isthe free-diffusion time over the distance ( u max − u ) , or thenatural time of re-folding of the force-free open state.We must mention several factors that would make theprocess of auto-inhibition more complicated, and its rate K − deviate from the simple expression (8). First of all, the [o]state will in most cases be quickly phosphorylated, whichmeans there will be an additional binding energy ∆ G a stabil-ising this conformation – making the effective rate of autoin-hibition much lower. On the other hand, there is an effectof extension-elasticity of talin (47, 48) that would providean additional returning force acting on the FAK complex:this would make the low/zero force case fold back faster,at a higher rate K − . While these are interesting and impor-tant questions that need to be investigated, at the moment wewill focus on the simplest approximation to understand theuniversal qualitative features of FAK sensor dynamics.In order to be able to compare different expressions, andplot different versions of transition rates, we must identifythe non-dimensional scaling of K − . Factoring the same nat-ural time scale as we used for K + , the expression takes the Biophysical Journal 00(00) 1–130 4 CONCLUSIONS K r a t e s K and K [ / s ] + - pulling force f [pN] . . . . . . Figure 10: Comparison of the opening and closing rates, K + and K − , for several different substrate stiffnesses. As in sev-eral previous plots, ∆ G o = 11 k B T , u = 0 . nm, and thedamping constant ratio ζ = 10 − . When the cytoskeletonpulling force is too low, the rate of autoinhibition rapidlyincreases and one does not expect strong phosphorylationand positive feedback of mechanosensor.form: K − = (cid:18) ∆ G o u γ c (cid:19) g ¯ f (cid:0) e g ¯ fλ − (cid:1) − g ¯ f λ , (9)where, as before: the force is scaled as f = ¯ f ∆ G o /u ,the opening energy barrier g = ∆ G o /k B T , and the ratioof two length scales (in [c] and [o] states) is labelled bythe parameter λ = ( u max − u ) /u (see Fig. 2). We don’thave direct structural information about the physical extentof FAK opening. However, taking the structural data on theseparate FAK domains from the work of Eck, Schaller andGuan (23, 45, 46), we make an estimate that u max ≈ . nm,essentially determined by the double of the size of foldedkinase domain, cf. Fig. 2. This gives λ ≈ and lets us plotthe comparison of the two transition rates, ¯ K + and ¯ K − .Figure 10 gives the transition rates, K + ( f, κ ) and ¯ K − ( f ) , plotted as a function of increasing pulling force.The rate of closing, ¯ K − , does not depend on the substrateparameters and is rapidly increasing when the [c]-[o] rangeof protein potential energy is flat, cf. Fig. 3. In this rangeof parameters, the product g ¯ f λ in the equation (9) is large,and the expression decays exponentially: exp[ − g ¯ f λ ] . Thisimplies that the transition from the strongly autoinhibitedpopulation of FAK sensors to the largely activated sensorsis rather sharp. We find that the crossover force at which K + ≈ K − is a relatively universal prediction, giving anestimate for the order of magnitude force required to keepthe FAK conformation open as f ∗ ≈ pN.One might be tempted, in the traditional way, to interpretthe ratio of the ‘on’ and ‘off’ rates K + /K − as an equilibriumconcentration of closed and open/activated states. However,we must remember that this process of mechanosensing isinherently non-equilibrium, even though it might be steady-state on the time scale of sensor response. Even in the regime of very low pulling forces, when K − (cid:29) K + , the few FAKmolecules that are spontaneously open would provide therequired (low) level of signal to the cell pathways. It is sim-ply an indication of sensor reversibility: Fig. 10 predicts thatas soon as the force reduces below f ∗ , most of the FAKmolecules would fold back and autoinhibit their action. Figures 6 and 7 each contain lots of information, but whenwe link the two we uncover the true nature of this reversiblemechanosensor. If we place a cell on a substrate of givenstiffness κ f (or Young modulus Y f ), then according to ourmodel, the mechanosensors will generate a positive feed-back loop: increasing the rate of FAK activation, whichleads (via the Erk or Rho GTPase pathways (26)) to theincreased production of actin. Assembling more F-actin, thecell will increase tension in the cytoskeleton, which willfurther increase the rate of FAK activation on stiffer sub-strates. This positive feedback goes until approximately thepeak position of the curves in Fig. 6a, after which the furtherincrease of tension shuts down the FAK activation response.The resulting negative feedback loop returns the cell to itshomeostatic level of the cytoskeleton tension f ( κ f ) , cor-responding to the given substrate stiffness. Importantly, onvery soft substrates (gels or soft tissues), the FAK signallingfeedback is always negative and FAK autoinhibition on itsown would lead to a very low cytoskeletal tension – no focaladhesions or stress fibers are formed on such substrates. Itis likely that other mechanosensors become more relevanton very soft substrates (and in planktonic suspension), suchas the TGF β latent complex (15): after all, the very nameof FAK suggests its relation to focal adhesions, which onlyoccur on stiff substrates. This idea corresponds very wellwith experimental work showing that cells on sufficientlysoft substrates do not form stable focal adhesions (70).Returning back to the homeostatic tension, if we look atthe sensitivity of the FAK complex at this fixed level of force, f ( κ f ) , we find the maximum sensitivity (identified with thepeaks in Fig. 7b) is very close to the actual substrate stiffness, κ f . So, this physical model describes a naturally adaptivesensor: not only does cytoskeletal tension adjust according tothe substrate stiffness, but this remodelling adapts the sensorresponse so that it remains most sensitive to its immedi-ate surroundings – small changes in the substrate stiffnesswill give large changes in the activation rate of FAK acti-vation once the positive and negative feedback rebalance inhomeostasis.This is desirable behaviour in a biological sensor, and itis remarkable that it is produced in our model with no priorstipulation. We initially only required that the cell be respon-sive to changes in the stiffness – how big these changes were,or if they were optimal, was not close to the front of our Biophysical Journal 00(00) 1–131 minds. For such a simple model to predict useful adaptivesensing behaviour is exciting to us.Our model of a single focal adhesion kinase is obviouslynot the whole story. There have been several experimen-tal works showing that FAK dimerisation is an importantinitiator of FAK autophosphorylation. We did not attemptto capture any collective effects in the present model, andacknowledge that there is significant ground to be gained inexpanding our model to a one describing the allosteric cou-pling. Nevertheless, one can easily see how such collectiveeffects might be generated within our model.Phosphorylated FAK acts on several important signallingmolecules, such as Rho and Rac. If these molecules act toincrease the tension in actin filaments in the broad vicin-ity, rather than strictly for filaments attached to active FAKmolecules, then it is obvious that there will be a cooperativeeffect – once a single focal adhesion kinase autophosphory-lates, the tension in surrounding filaments will increase, andthis increases the probability of a second opening event, andso on.The dependence of the FAK opening rate on stress relax-ation partly explains results obtained in experimental workon cell spreading with different viscoelastic substrates (69).Chaudhuri et al. saw suppression of cell spreading (associ-ated with lower FAK activation) on substrates with signif-icant stress relaxation, compared with purely elastic sub-strates of nominally the same storage modulus. We shouldnote that we fail to capture the behaviour Chaudhuri et al.observed at very low stiffness (1.4kPa). On such a soft sub-strate, they saw that the number of cells with stress fiberswas actually enhanced on substrates with stress relaxation –the opposite trend to stiffer substrates. This isn’t surprising;our model deals with mature focal adhesion complexes thatwould not develop on substrates of ∼ kPa stiffness.In summary, this work develops a theoretical model ofthe physical mechanism that a reversible mechanosensor ofthe 2nd kind should use. We focus all our discussion on thefocal adhesion kinase, in association with integrin and talin,connecting the force-providing cytoskeletal F-actin and thevarying-stiffness ECM. However, the fundamental principlesof the model apply to all reversible molecular complexes thatmay be represented by the two-spring model of Fig. 4. In thiscase, instead of FAK, it could instead (or simultaneously)be any of the other big molecules with autocatalytic activityinvolved in the force chain in Fig. 1. The obvious alternativewould be talin, which has the confirmed connection betweenintegrin and actin (31) and the large conformational changeunder applied force (48). The next steps are to link the mainresult of this work (the rate of opening K + ) with the non-linear dynamics of one or several signalling pathways thatproduce the morphological response of the cell to the signalthe mechanosensor generates. Author contribution
All authors contributed to carrying out research and writingthe paper.
Acknowledgments
We have benefited from many useful discussions and supportof G. Fraser, K. Chalut, T. Alliston, X. Hu, and D. C. W. Foo.This work has been funded by EPSRC EP/M508007/1.
References
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