A model for cyclic mechanical reinforcement

Mechanical force regulates a broad range of molecular interactions in biology. Three types of counterintuitive mechanical regulation of receptor–ligand dissociation have been described. Catch bonds are strengthened by constant forces, as opposed to slip bonds that are weakened by constant forces. The phenomenon that bonds become stronger with prior application of cyclic forces is termed cyclic mechanical reinforcement (CMR). Slip and catch bonds have respectively been explained by two-state models. However, they assume fast equilibration between internal states and hence are inadequate for CMR. Here we propose a three-state model for CMR where both loading and unloading regulate the transition of bonds among the short-lived, intermediate, and long-lived state. Cyclic forces favor bonds in the long-lived state, hence greatly prolonging their lifetimes. The three-state model explains the force history effect and agrees with the experimental CMR effect of integrin α5β1–fibronectin interaction. This model helps decipher the distinctive ways by which molecular bonds are mechanically strengthened: catch bonds by constant forces and CMR by cyclic forces. The different types of mechanical regulation may enable the cell to fine tune its mechanotransduction via membrane receptors.

on a bond prolongs its lifetime by orders of magnitude without changing the clamped force at which the bond lifetime is subsequently measured. Two different cyclic force waveforms were applied to the bond 26 : 1) a single loading-unloading cycle before clamping the force constant (single-cycled CMR, Fig. 1c), and 2) multiple loading-unloading cycles before clamping the force constant (multi-cycled CMR, Fig. 1e). The extent of bond lifetime prolongation increases with the amplitude of the cyclic force of single-cycled CMR (Fig. 1d) and the number of force cycles of multi-cycled CMR (Fig. 1f). This suggests that, in addition to the present level of force, the history of force application is memorized by the bond that later alters its lifetime. Other manifestations of force history effect have been observed and discussed in other systems 10,17,27,28 . However, most two-state catch bond models assume average lifetime as a single-valued function of force at the time when lifetime is measured without considering the prior history of force, which is inadequate for the CMR phenomenon. Recently, a two-state model was examined for the ability to account for force history effects 29 . This model was able to generate single-cycled CMR effect but the multi-cycled CMR was attributed to the pre-matured dissociation of short-lived bonds during loading-unloading cycles prior to lifetime measurement. However, this possibility was specifically ruled out experimentally in the original CMR study 26 . In this model, bond strengthening by CMR is not distinct from that by catch bonds, yet the biophysical mechanisms underlying these two phenomena may be conceptually different.
In this paper we describe a model for CMR that assumes bond dissociation from three states governed by the Bell model, with state occupancies regulated by force. The validity of the three-state model has been supported by its ability to fit several types of force regulation of molecular dissociation, including CMR, slip bond, and catch-slip bond. Importantly, in this model catch bond and CMR strengthen molecular interaction through different pathways, consistent with the proposal that catch bonds and CMR are based on distinct structural mechanisms 26 . This enables us to predict the circumstances under which a molecular interaction exhibits catch bond, slip bond, or CMR.

Results
Conceptualization and energy landscape. We constructed our model by conceptualizing an energy landscape. In previous two-state catch-slip bond models, molecular bonds may reside in one of two stable states and dissociate from either 16,30 . The two stable states are represented by two energy wells whose relative depths determine relative bond occupancies in the two states. Mechanical force tilts the energy landscape, alters the energy well depths, and shifts the bond occupancies from the short-lived to long-lived state to give rise to a catch bond. Our model adds an intermediate state that either connects or segregates the short-and long-lived states depending on force, generating two bidirectional transition pathways to link the three states. One pathway allows transition between the short-lived and intermediate states and the other pathway allows transition between the intermediate and long-lived states, without direct transition between the short-and long-lived states. Increasing force promotes transition from the short-lived state to the intermediate state but suppresses transitions in the other three directions. Decreasing force resumes all transitions and enables bond occupancies accumulated in the intermediate state during the loading phase to transition to the long-lived state. Therefore, the intermediate state stores bonds coming from the short-lived state during loading and dispatches them to the long-lived state during unloading.
An energy landscape that displays the aforementioned properties can be constructed (Fig. 2a). The x-y plane maps to the space spanned by the reaction coordinates. The x-axis is along the direction of force. The y-axis is perpendicular to force. Three energy wells depicted by solid contours represent the three stable bound states, and one energy well depicted by dashed contours represents the dissociated state. The two transition pathways connecting the short-lived state to the intermediate state and then to the long-lived state are illustrated by the solid curves. The three dissociation pathways connecting the three bound states to the dissociated state are illustrated by the dashed curves. For clarity, we hid the dissociated state and dissociation pathways in Fig. 2a, and plotted the energy landscape for state transition only in a 3-D view with the vertical axis representing free energy E (Fig. 2b). The depths of energy wells are indicated by color coding. The heights of the energy barriers determine the force-free transition rates between the neighboring states. The two transition pathways connecting the three states are highlighted by the blue and red curves in the top view.
To illustrate how force tilts the energy landscape, we split the energy landscape based on the two pathways colored in blue and red, and project them onto the x-E plane (Fig. 2c) or look into the origin along the diagonal line x = y = E (Fig. 2d). The dashed curves in Fig. 2c   The first term on the right-hand side of each equation accounts for dissociation at rates of k off i (i, j = 1-3), whereas the other terms account for the inner-exchange of bond occupancies at rates of k ij (i, j = 1-3). The transition rates are assumed to follow the Bell equation 4 ij B , where k ij 0 is the force-free transition rate, f is force, Δ x ij represents the distance from the bottom of the ith energy well to the point of transition to the jth energy well, k B is Boltzmann constant, and T is absolute temperature. Dissociation rates are also assumed to follow the Bell equation: i B , where the parameters are of similar definitions as the transition rates. The average lifetime can be calculated by: represents the fraction of bonds dissociated from the ith state and satisfies 31 :

Comparison to integrin CMR experiment.
Previous experiments demonstrating the CMR phenomenon were performed with three different loading waveforms 26 : 1) force-clamp, 2) single-cycled CMR, and 3) multi-cycled CMR ( Fig. 3a,b insets). While force-clamp elicits catch bonds 11 , both single-cycled and multi-cycled CMRs further prolong bond lifetime by orders of magnitude 26 . Bond lifetimes increase with increasing peak force in single-cycled CMR (symbols in Fig. 3a) and increasing cycle number in multi-cycled CMR (symbols in Fig. 3b), before reaching a plateau. The lifetimes distribute multi-exponentially (Fig. 3c,d), suggesting that bonds dissociate from multiple states along multiple pathways. Lifetime prolongation is due to bond occupancies shifting from the short-lived state to the long-lived state (symbols in Fig. 3c,d and Supplementary Fig. S1) 26 .
To compare with data, we assumed that all bonds occupy the short-lived state initially and are subjected to specific force cycles (see Methods). By integrating the master equations over time along the given force history, we obtained the fraction of bonds dissociated from each state (ω i ), and average time (< t> ) of total bond survival, i.e., bond lifetime as measured from experiment ( Fig. 3, Supplementary Fig. S1, and Supplementary Information). Using Monte Carlo Least-Squares fitting 32 , we obtained the best-fit parameters by comparing the model solutions with two sets of data, one from single-cycled (Fig. 3a,c) and the other from multi-cycled (Fig. 3b,d) CMR experiments. Each set contains multiple experiments with a range of peak forces or cycle numbers. Data from each experiment are plotted as a distribution of bond lifetimes and their average (Fig. 3).
Supplementary Table S1 lists the best-fit parameters. k 21 0 and k 23 0 are significantly faster than k 12 0 and k 32 0 , suggesting that the intermediate state is the least stable in the absence of force. k 12 0 is the slowest, indicating short-lived state is the most stable state without force. Fitting returned one positive (1 → 2) and three negative (2 → 1, 2 → 3, and 3 → 2) Δ x ij values for inner-exchanges of states (Supplementary Table S1), indicating that one of these transition directions is along and the others are opposite to the force direction, consistent with the conceptual energy landscape. The absolute values of Δ x ij vary from 1.35-6.80 nm, indicating that the inner-exchanges of states are highly sensitive to force. The best-fit Δ x i values are all positive. The Δ x i for dissociation from the short-lived and intermediate states are 1.08 and 1.31 nm, respectively, again indicating the high force sensitivity of the dissociation rates. The high force sensitivity of these rates may be understood in terms of force-induced integrin conformational changes, which may propagate to the ligand binding site to regulate dissociation kinetics. Integrin is a large protein, with a ~10 and ~20 nm linear dimension in the bent and extended conformations 33 that can change back and forth reversibly under force 34,35 . Such conformational changes could have a leverage effect, resulting in higher force sensitivity of rate coefficients, manifested as nanometer range transition and dissociation distances. It is worth noting that the Δ x of dissociation from the long-lived state is 0.03 nm, indicating that the dissociation pathway is nearly perpendicular to the force direction. Further analysis of transition directions confirmed that these directions are the most appropriate combination for CMR ( Supplementary Fig. S2). The solutions compare well with bond lifetime distributions from both single-cycled (Fig. 3c) and multi-cycled (Fig. 3d) CMR experiments. The predicted average bond lifetimes also follow the changing trends of the data with the increasing amplitude (Fig. 3a) and number (Fig. 3b) of force cycles. In addition, the simulated fraction of bonds dissociated from each state follows similar trend of that estimated from experimental data ( Supplementary  Information and Fig. S1).
Parametric analysis. Parametric analysis reveals that the multi-cycled CMR solution is quite robust, as ± 0.5-pN change around the 10-pN clamped force generates nearly indistinguishable average post-cycle bond lifetime curves (Fig. 3b). By comparison, the single-cycled CMR solutions are very sensitive to the clamped force, as a ± 0.5-pN change around the 5 pN value results in a ~10-s variation in the average post-cycle bond lifetime curve (Fig. 3a). Considering the experimental difficulties to precisely control the clamped force and loading/ unloading patterns at low forces and the fact no freely adjustable parameter is used in this comparison, our model does a good job to predict the data trends for both single-cycled and multi-cycled CMR experiments. The modest deviation between model solution and the low peak force data from the single-cycled CMR experiments is considered acceptable. Thorough parametric analysis was carried out to show how k ij 0 , Δ x ij , and loading/unloading rates influence the model predictions ( Supplementary Information and Figs S3 and S4). Interestingly, the slower the loading or unloading, the longer the bond lasts. The frequency of the cyclic force is determined by both loading and unloading rates. Slower loading/unloading rates lead to lower cyclic frequency. It has been reported that cells in focal adhesions generate ~0.1 Hz dynamic forces 36 , much lower than the ~0.5 Hz cyclic force in our previous multi-cycled CMR experiment 26 , suggesting that cells may use cyclic forces to strengthen their focal adhesions to the ECM (see Supplementary Information).
We assumed that initially the integrin resides in the short-lived state because crystallographic and EM studies suggest that the bent conformation is more favorable for integrins in the absence of ligand. To evaluate the effects of this assumption, we examined the dependence of the average bond lifetime on the initial condition. As the initial bonds shifted from the short-lived state to the long-lived state, the calculated average lifetimes increase in both single-cycled and multi-cycled loading cases, but the CMR effects become less and less pronounced ( Supplementary Fig. S5). The reduced CMR effect is due to the fact that bonds initially in the long-lived state have already assumed the longest lifetimes even without cyclic force exerting on them.
Effects of changes in state occupancies. In experiment and model, bond lifetime is defined from the moment when the force reaches the clamped level to the moment of bond dissociation. A prior force history exerts its effect by changing the state occupancies of bonds at the "initial" time from which their lifetimes are measured. To investigate how a prior force history affects the initial occupancies, we tracked real time transition rates and real time state occupancies during loading-unloading along three specific force histories: a single ramp to 10-pN clamped force, 2.5-cycles of loading-unloading between 0 and 10 pN followed by clamping at 10 pN, and a single-cycle of loading-unloading with 22-pN peak force and 5-pN clamped force. By integrating the master equations, the transition rates and state occupancies were obtained in these prior force histories (Fig. 4).
As shown in Fig. 4a-c, for all three prior force histories, loading and unloading cycle(s) promote bond redistribution among the three states, using the intermediate state as a relay station. In the force clamp phase, the inner exchange between short-lived and intermediate state reaches equilibrium and that between intermediate and long-lived state is suppressed. Therefore, bonds dissociate from the states that they occupied at the time without transitioning to other states. The fraction of bonds dissociated from the intermediate state is low because of its low stability (cf. its shallow energy well in Fig. 2). Because of its two orders of magnitude faster transition rate to the long-lived than to the short-lived state at low or zero forces (Fig. 4e,f) Similarities and differences among slip bond, catch-slip bond, and CMR. To further investigate how prior force histories affect the state occupancies, we generated a phase diagram by simulating multi-cycled CMR over a force vs. cycle number space ranging from 1-40 pN and 0.5-3.5 cycles using the parameter set that best-fits the integrin α 5 β 1 -FN CMR data. Each point in this force-cycle number plane is color-coded with RGB values according to the fractions of bonds dissociated from the three states. This phase space was then divided into three regions according to the dominant state from which most bonds dissociate (Fig. 5a). At low forces and low cycle numbers, most bonds dissociate from the short-lived state. At high forces with a single force ramp, most bonds dissociate from the intermediate state. In the remaining region, most bonds dissociate from the long-lived state. This phase diagram reveals differential effects of amplitude and cycle number of dynamic forces on the state occupancy of bonds, which is a key mechanism for CMR. It also suggests different biophysical mechanisms for catch bonds and CMR: A constant force with a single ramp promotes bonds transition to the intermediate state to give rise to catch bonds, whereas cyclic forces promote bond transition to the long-lived state to generate CMR effects.
The distinctive biophysical mechanisms for catch bonds and CMR afford the possibility of different types of mechanical regulation of molecular bonds by constant vs. cyclic forces. For instance, if < < k k k off 3 off 1 off 2 , increasing levels of constant force would drive bond occupancy to the intermediate state with accelerated dissociation to produce a slip bond. In comparison, increasing the number of loading-unloading cycles would drive bond occupancy to the long-lived state with decelerated dissociation to exhibit CMR. To illustrate this, we ran four sets of single ramp force (Fig. 5b) or multi-cycled cyclic force (Fig. 5c) simulations with different off-rates (Supplementary Table S2). As predicted,   Our model fits the bond lifetime vs. clamped force data well for both peptides of high biological activities, which behave as catch-slip bonds, and peptides of low biological activities, which behave as slip-only bonds 13 (Fig. 6

Discussion
Possible source of fitting errors. In some cases our model fits deviate from the experimental lifetime distributions, especially for the short lifetimes in single-cycled CMR with larger peak forces and in multi-cycled CMR with higher cycle numbers (Fig. 3c,d). Possible causes for such discrepancies may include: 1) Short lifetime distributions are determined by fast-dissociating bonds from the short-lived and intermediate states. The fractions of bonds dissociated from the intermediate state are low in all cases ( Supplementary Fig. S1), preventing us from obtaining a reliable fitting value of the off-rate for the intermediate state. 2) All initial bonds were assumed to reside in the short-lived state, which may result in more short-lived bonds dissociated from that state. 3) Bonds dissociation during loading-unloading was constrained to avoid pre-matured bond dissociation. Since the shortlived bonds dissociated the fastest, this constraint may result in higher fraction of bond dissociated from the short-lived state after force arrived at the clamped level for bond lifetime measurement. 4) Since the unloading is a non-linear process, finding the precise endpoint of the unloading in experiment is difficult, making the starting point of the lifetime inaccurate, which preferentially impacts the accuracy of the short lifetime measurements.
Since decreasing the initial occupancy of short-lived bond would reduce the CMR effect ( Supplementary Fig. S5), we only tested if removing the constraint of dissociation during loading-unloading could improve the fitting. Reduced chi squares indicate that a better fitted lifetime distribution could indeed be obtained, returning a new best-fit Δ x 12 value of 0.76 nm (Fig. 3 and Supplementary Fig. S6). However, to avoid the influence of pre-matured bond dissociation, we chose to constrain the dissociation during loading-unloading.

Physical analogies.
To better understand the three-state model, we used the laser-pump (Fig. 7a) and air-pump (Fig. 7b) systems as metaphors to illustrate how it works. Atoms in a laser-pump and air in an air-pump are analogous to receptor-ligand bonds, all of which have three states. The operations of both pumps are governed by similar master equations to those of our three-state model. The state transitions behave similarly under force or electric current. For instance, atoms in the laser-pump reside in the ground, excited, and metastable states. Electric current, analogous to mechanical force, inputs an energy ∆ E, which lowers the free energy barrier from the ground state to the excited states. Therefore, electric current increases atom excitation rate k 12 from the ground state to the excited state by a factor of ∆ e E k T / B , just like mechanical force increases bond transition rate from the short-lived state to the intermediate state by a factor of ∆ e f x k T / B . Stopping the electric current resumes the transition rate, and allows the excited atoms to rapidly decay to the metastable state, just like the unloading force allows bonds to rapidly transition from the intermediate state to the long-lived state. Therefore, a stronger electric current or multiple electric current cycles can boost more atoms to the excited state and eventually more atoms in the metastable state. Similarly, force inputs mechanical energy to the air-pump system and the one-way valves guide the air flow from the upper chamber to the lower chamber and to the air balloon. Pulling harder or repeated pulling-pushing cycles input more energy, thus pumping more air to the lower chamber, and eventually to the air balloon.

Potential relation between integrin states and conformations.
Electron microscopic studies reveal the existence of multiple stable conformations for several integrin ectodomains, e.g., bent ectodomain with a closed headpiece and connected legs, extended ectodomain with a closed headpiece and joined legs, and extended ectodomain with an open headpiece and separate legs (Fig. 7c) [40][41][42][43] . Integrins in the bent conformation assume a resting state with a low affinity for ligands 40 . Extended integrins with a closed headpiece and joined legs assume an intermediate state, which is predicted to have an intermediate affinity for ligands 41,42 . Extended integrins with an open headpiece and separate legs assume an activated state with a high affinity for ligands 40,44,45 . It is therefore of interest to relate the force-regulated transitions among the three states in our model to the force-regulated changes in integrin conformations. Three major domain movements are involved in the conformational change from a bent conformation with a closed headpiece and connected legs to an extended conformation with an open headpiece and separate legs: integrin extension, headpiece opening and hybrid domain swing-out, and separation of the legs. The precise order of these movements in integrin conformational changes is still speculative. Xiao et al.
proposed several conformational transition pathways 41 induced by different stimulations. Inside-out signaling is predicted to lead to leg separation first. Ligand binding is predicted to stimulate the head piece opening first (outside-in signaling). Additional evidence has been found supporting the inside-out and outside-in activation mechanism [46][47][48][49][50] . However, it less clear what biochemical signal would lead to integrin extension. Steered molecular dynamics (SMD) simulations suggest that mechanical force may induce hybrid domain swing-out and integrin extension 51,52 . This force-induced extension has been observed in single-molecule experiments 34,35 . Crystallographic 40,41 and SMD studies have suggested a downward movement of α 7 helices in the α A and/or βA domains. This pistol-like movement has been proposed to relay hybrid domain swing-out to conformational changes at the ligand binding site to result in the high affinity state 41,53 . To this end, we propose a conceptual model of integrin conformational changes induced by dynamic forces (Fig. 7c). In this mechanical model, force exerted via an engaged ligand tends to extend the integrin from the bend to extended conformation. Loading promotes the integrin to transition from the short-lived state to the intermediate state, which assumes an extended conformation. Unloading tends to relax the integrin headpiece and allow the downward pistol-like movement of the α 7 helices in the α A and/or β A domains to induce the swing-out of the hybrid domain and the separation of the legs. In other words, unloading promotes integrin to transition from the intermediate state to the long-lived state. Increasing the amplitude or number of loading-unloading cycles increases the total flux through the intermediate state, relaying more bond occupancies from the short-lived state to the long-lived state cumulatively, thereby prolonging bond lifetime (Fig. 7c).
Comparison to two-state model. Chen 29 . In their model, both time elapses of T 0 , starting from the initial time of force application, and T 1 , starting from the first instant of force clamping, were simulated. Thus, T 0 includes, but T 1 excludes, the short-lived bonds dissociated during loading-unloading cycles prior to when the experimental bond lifetime measurement begins. The authors were able to generate single-cycled CMR using the same physical mechanism as catch bonds. However, they found that only T 1 , but not T 0 , exhibited both single-and multi-cycled CMR effects. The apparent prolongation of T 1 in multi-cycled CMR was due to pre-selection of long-lived bonds by cyclic forces, an experimental artifact rather than a real biophysical phenomenon. We note that the possible effect of cyclic forces biasing measurement was ruled out by specifically designed control experiment ( Supplementary Fig. S3 in ref. 26). Theoretically two-state models lack the ability to produce multi-cycled CMR effect in T 0 . Nor is it possible to generate multi-cycled CMR effect in T 1 without pre-selection of long-lived bonds. Therefore, the need for a three state model stems from the difficulty for any two-state model to account for the multi-cycled CMR. This theoretical assertion can be derived from the use of the Bell equation 4 to model bond transition between states, which was assumed by Chen et al. 29 and also in the present work (see justification in Supplemental Information). In our model, the intermediate state cyclically relays the transition of short-lived bonds to the long-lived state, which accumulate over repetitive force cycles (Fig. 4), thus prolonging the bond lifetime. Therefore, our three-state model provides a new biophysical mechanism for strengthening of molecular bonds that is distinct from that for catch bonds, which can account for the more complex CMR data. Future studies will examine whether and how our three-state model may describe other forms of force history-dependent mechanical regulation of molecular bond dissociation.

Methods
We assume that all bonds initially reside in the short-lived state. The master equation was numerically integrated along different prior force histories to calculate bond lifetimes resulted in the force-clamp phase. Three prior force histories were considered, corresponding three experimental assays: force-clamp, single-cycled CMR, and multi-cycled CMR. In the loading phase, force increased linearly with time, with a loading rate of 1000 pN/s. To mimic the experiments, unloading of single-cycled and multi-cycled CMRs was controlled by a proportionalintegral-derivative algorithm: Here u is the increment for the current time step, ε is the difference between the aim value and the current value.
The parameters K C , T i , and T d are set to be 0.0001, 6 s, and 0 s, respectively. Since high forces lead to fast transition of bond occupancies between states, the time step during loading-unloading was set to be 0.01 μ s. To save computational time, the time step in the low force clamping phase was set to be 1 μ s. To avoid pre-matured bond dissociation, we constrained the dissociation during loading and unloading. Computed bond survival distributions and force-lifetime relationships were compared to the experimental data (Fig. 3) by Monte Carlo Least-Squares fitting 32 to obtain the best-fit model parameters. With best-fit parameters, we further calculated the fraction of bonds dissociated from each state,