Structural Basis and Kinetics of Force-Induced Conformational Changes of an αA Domain-Containing Integrin

Background Integrin αLβ2 (lymphocyte function-associated antigen, LFA-1) bears force upon binding to its ligand intercellular adhesion molecule 1 (ICAM-1) when a leukocyte adheres to vascular endothelium or an antigen presenting cell (APC) during immune responses. The ligand binding propensity of LFA-1 is related to its conformations, which can be regulated by force. Three conformations of the LFA-1 αA domain, determined by the position of its α7-helix, have been suggested to correspond to three different affinity states for ligand binding. Methodology/Principal Findings The kinetics of the force-driven transitions between these conformations has not been defined and dynamically coupled to the force-dependent dissociation from ligand. Here we show, by steered molecular dynamics (SMD) simulations, that the αA domain was successively transitioned through three distinct conformations upon pulling the C-terminus of its α7-helix. Based on these sequential transitions, we have constructed a mathematical model to describe the coupling between the αA domain conformational changes of LFA-1 and its dissociation from ICAM-1 under force. Using this model to analyze the published data on the force-induced dissociation of single LFA-1/ICAM-1 bonds, we estimated the force-dependent kinetic rates of interstate transition from the short-lived to intermediate-lived and from intermediate-lived to long-lived states. Interestingly, force increased these transition rates; hence activation of LFA-1 was accelerated by pulling it via an engaged ICAM-1. Conclusions/Significance Our study defines the structural basis for mechanical regulation of the kinetics of LFA-1 αA domain conformational changes and relates these simulation results to experimental data of force-induced dissociation of single LFA-1/ICAM-1 bonds by a new mathematical model, thus provided detailed structural and kinetic characterizations for force-stabilization of LFA-1/ICAM-1 interaction.


Introduction
Integrins are a family of heterodimeric transmembrane receptors composed of an a and a b subunit that involve in a wide variety of physiological processes such as cell adhesion, cell migration and immunoresponse [1]. They usually bear forces upon binding to ligands in cell-cell and cell-extracellular matrix adhesions, which are crucial to mechanosensing and mechnotransduction of cells [2,3]. Of the 24 known human integrins, 10 of them, including the integrin a L b 2 or lymphocyte functionassociated antigen 1 (LFA-1) studied here, have an additional aA (or aI) domain inserted in the b-propeller domain of the a subunit, where the ligand binding site resides [4]. By binding intercellular adhesion molecule 1 (ICAM-1), LFA-1 mediates adhesion of leukocytes to the blood vessel wall or antigen presenting cells (APC), and sustains forces generated by the blood flow or the cell's motile machinery [1,5].
In response to various biochemical [3,4,6] and mechanical signals [7,8], integrins change conformations and ligand binding affinities. In physiological condition, they may assume a bent conformation and have a low ligand binding affinity. Inside-out signaling or changes in the metal ion conditions from Ca 2+ /Mg 2+ to Mn 2+ result in integrin conformational change to an extended form, with a closed or swung-out hybrid domain, accompanied by a higher ligand binding affinity (Fig. 1A, 1B) [3,4,9].
In addition to global conformational changes in the whole ectodomain and in the hybrid domain, the aA domain conformation also controls the affinity of aA-containing integins such as LFA-1 [5,9]. Several aA domains, including that of LFA-1, have been crystallized [10][11][12][13][14][15], revealing as many as three conformations termed closed, intermediate and open, depending on the position of the C-terminal a 7 -helix [5] (Fig. 1C). As measured by surface plasmon resonance [5] and micropipette adhesion frequency assay [6], LFA-1 with the aA domain locked in the intermediate and open conformations have hundreds and thousands folds higher affinities for ICAM-1, respectively, than that locked in the closed conformation. A study of molecular dynamics (MD) simulations of aA domains with implicit water suggested that the fractions of these three conformation states are sensitive to the force applied to the C-terminus of their a 7 -helix [16]. Using a biomembrane force probe (BFP), single LFA-1/ICAM-1 bonds are found to dissociate from three states with distinct apparent off-rates and associated fractions [9]. The short-lived fraction (with the greatest apparent off-rate) is dominant at zero force and the fractions of intermediate-and long-lived states increase with the tensile force applied to the bond. The force-dependent transitions among these three fractions of bond states give rise to the LFA-1/ ICAM-1 catch bond behavior in which the bond lifetimes are prolonged by tensile force in a certain regime [9].
Building from the above studies, we used steered molecular dynamics (SMD) simulations with explicit water to study the forceinduced transitions of conformations of the LFA-1 aA domain. We also constructed a mathematical model to describe the interstate transitions integrin and their coupling with ligand dissociation. Using this model, we re-analyzed our previous data on single LFA-1/ICAM-1 bonds lifetimes measured from biomembrane force probe (BFP) force-clamp experiments [9], and estimated interstate transition rates that govern the time courses for activation of the liganded LFA-1 under force [9].

SMD-simulated force-induced conformation transitions of LFA-1 aA domain
To study the force-induced conformational transitions of the LFA-1 aA domain, we used constant-force SMD simulations to pull the C-terminus of its a 7 -helix, as the position of the tensionbearing a 7 -helix determines the aA domain conformation [5,16].
Unlike the previous implicit water simulations [16], our simulations included physiologically relevant water molecules. To observe the sequential transitions of the a 7 -helix position, we quantified the root mean square distance (RMSD) between the simulated structure and its initial ''up'' position, which corresponds to the ''closed'' conformation of the aA domain [16]. Pulling the a 7 -helix C-terminus in the first 3.6 ns only increased the RMSD slightly, indicating the stability of the ''up'' position ( Fig. 2A, 2B). A sudden increase of the RMSD from 3 to 6 Å was then observed during 3.4-4 ns simulations, suggesting state transitions. Zooming in this transition phase with a magnified time scale, a stable ''intermediate'' a 7 -helix position with a 4.5-Å RMSD was observed ( Fig. 2A inset, 2C). This ''intermediate'' a 7helix position is linked to the ''intermediate'' conformation of the aA domain. After two abrupt increments, the RMSD was stabilized at around 8 Å for the next 10 ns, corresponding to a ''down'' position of the a 7 -helix and the ''open'' conformation of the aA domain ( Fig. 2A, 2D). After the pulling force was removed at the 15 ns time point, the a 7 -helix returned back from the ''down'' position to the ''up'' position in a few nanoseconds and remained up within the next 20-ns simulations (Video S1).
Besides the a 7 -helix position, another remarkable difference between the open and closed conformation of LFA-1 aA domain revealed by structural studies is the metal ion position at the metal ion dependent adhesion site (MIDAS). It was observed that in the open conformation, the MIDAS metal ion underwent inward movement for about 2 Å . Previous implicit water molecular dynamics simulations suggested that the movement of a 7 -helix and that of the MIDAS metal ion were coupled. Hence, we measured the RMSD of the MIDAS metal ion and other important residues between the simulated structures and the open or closed conformations ( Figure S1). These included residues S139, S141, T206, and D239 that coordinated the MIDAS metal ion and residues L289, F292, and L295 that formed a ''ratchet''-like structure to define the position of the a 7 -helix. In the simulations, although the pulling force induced movements of the a 7 -helix, no movements of the MIDAS metal ion were observed ( Figure S1B), nor were their coordinating residues (Figure S1C-F). Nevertheless, we did observe the relevant ''ratchet''-like movements on residues L289, F292, and L295 (Figure S1G-I), which followed the movements of the a 7 -helix.
Residue D239 coordinated directly with the MIDAS metal ion in the closed conformation as observed in the crystal structures [11,12]. On the other hand, in the open conformation, D239 might not coordinate with metal ion directly but through a water molecule. In our pulling simulation, it seemed that the strong ionic interaction between D239 and the metal ion constrained the metal ion at its closed (outward) position, thus preventing the inward movement from being observed within the short timescale of the simulation. To test this hypothesis, we performed a set of three simulations. These simulations started from the structures generated from the above pulling simulations. The snapshots at 0, 3.7 and 16 ns were taken as the respective new starting points. Among them, the 0 ns configuration represented the ''up'' position of the a 7 -helix, the 3.7 ns configuration represented the ''middle'' position and the 16 ns one represented the ''down'' position. In these free dynamics simulations, the applied force was released. To prevent the a 7 -helix from returning back to the ''up'' position in the simulations starting from 3.7 and 16 ns snapshots, we constrained the Ca atoms of the a 7 -helix in addition to the original constraint residues. Firstly, 30 ns free dynamics simulations were performed followed by 20 ns free dynamics simulations with the point charges of the two oxygen atoms of D239 carboxyl group reduced by 0.5e each. As shown in Fig. 3 Figure S1 and Video S1. doi:10.1371/journal.pone.0027946.g002 tions successively by applied force (Fig. 2). To incorporate such conformational change kinetics into the kinetics of force-induced ligand dissociation, we constructed a mathematical model for the BFP force-clamp experiment in which single LFA-1/ICAM-1 bonds were pulled with a constant force until rupture [9]. This simple model considers two interstate transition steps: from state C 1 to state C 2 and from state C 2 to state C 3 (Fig. 4) as well as three ligand dissociation steps from each of the three states. Each of these steps is assumed irreversible, which seems reasonable under force, as force drives unidirectionally both the interstate transition and ligand dissociation.
The model results in a set of coupled, linear, first-order, ordinary differential equations (Equations 1-3, Materials and Methods) governing the changes of the probabilities of the LFA-1/ ICAM-1 bond in the three states in time, with constant coefficients (functions of force but not time): two interstate transition rates, k 12 and k 23 , as well as three reverse-rates k r1 , k r2 and k r3 . The equations were solved analytically (Equations 7-9, Materials and Methods). The solution was fit to the data of the BFP force-clamped experiments [9] to obtain three apparent dissociation rate constants k 1 , k 2 , k 3 and their associated apparent fractions v 1 , v 2 , v 3 (summarized in Tables S1,S2,S3 Analysis of force-dependent ICAM-1 dissociation reveals characteristics of three LFA-1 states The intrinsic reverse-rates, k r1 -k r3 , of ICAM-1 dissociation from the three LFA-1 states were plotted versus force in Fig. 5 in the range analyzed. They follow trends similar to the apparent offrates determined previously [9], but are quantitatively different. Interestingly, ICAM-1 dissociated from state C 1 with the highest but least force-sensitive reverse-rate k r1 (Fig. 5A), from state C 2 with an intermediate reverse-rate k r2 that has an intermediate force sensitivity (Fig. 5B), and from state C 3 with the lowest but most force-sensitive reverse-rate k r3 (Fig. 5C). Although the model assumes that all bonds start from state C 1 and then proceed successively to states C 2 and C 3 , the k r1 -k r3 values were evaluated from data without assuming a priori their relative values and relative sensitivities to force. It is therefore gratifying that our analysis of the previous BFP experimental data [9] with the present model returns the results that state C 1 is short-lived, state  (Fig. 2).
The force dependencies of all these intrinsic reverse-rates follow the Bell model [17], as indicated by the linear reverse-rates vs. force semi-log plots. They were indifferent to cation conditions Ca 2+ /Mg 2+ Mg 2+ /EGTA or Mn 2+ , suggesting that the initial  A LFA-1/ICAM-1 bond is assumed to dissociate irreversibly from one of three states -C 1 , C 2 and C 3 -with respective reverse-rates k r1 , k r2 and k r3 . This is coupled with one-way sequential transitions from C 1 to C 2 and from C 2 to C 3 with respective interstate transition rates k 12 and k 23 . The corresponding mathematical model is Equations 1-3 in Materials and Methods. doi:10.1371/journal.pone.0027946.g004 global conformation of the LFA-1 before it was liganded and stressed did not affect the intrinsic dissociation rates.

Force-dependent kinetics of LFA-1 transitions from shortto intermediate-to long-lived states and inhibition by XVA143
Interestingly, our kinetic analysis found that the transition rate k 12 from the short-to intermediate-lived states of LFA-1/ICAM-1 bonds (Fig. 6) was zero at zero force but increased with force in the range studied (Fig. 6A). Force also enhanced the transition rate k 23 from the intermediate-to long-lived states of LFA-1/ICAM-1 bonds from its zero value at zero force (Fig. 6B), but to a lesser extent (compare the two Fig. 6 panels). In the force regime studied (,20 pN), the force-dependent interstate transition rates were indifferent to the cation conditions Ca 2+ /Mg 2+ , Mg 2+ /EGTA or Mn 2+ , thus were not affected by the initial global conformation of the LFA-1 molecule before it was liganded and stressed.
With XVA143, a small molecule antagonist that blocks the interaction between the aA and bA domains [18,19], the forcedependent k 12 was suppressed (Fig. 6A, red). The transition from the intermediate-to long-lived states of LFA-1 was nearly completely blocked by XVA143, as shown by the zero k 23 in the force range studied (Fig.6B, red). A possible explanation for this result may be that the force applied on the a 7 -helix to induce the conformational changes has to be transmitted through the Figure 5. Force-dependent reverse-rates of three states under different cation conditions. Intrinsic reverse-rates k r1 (A), k r2 (B) and k r3 (C) of ICAM-1 dissociating from respective LFA-1 states C 1 , C 2 , and C 3 (see Fig. 3) were estimated by fitting the experimental data from Ref. [9] with our kinetic model (equations [22][23][24] and plotted versus force at indicated cation conditions. doi:10.1371/journal.pone.0027946.g005 Force decelerates LFA-1 dissociation from ICAM-1 by accelerating LFA-1 activation With the intrinsic parameters k 12 , k 23 , and k r1 -k r3 estimated, we used equations 1-3 to study the dynamic evolution of LFA-1/ ICAM-1 bonds and of individual conformation states and their overall behavior. As shown with representative model predictions for the Mg 2+ /EGTA condition, ligand dissociation manifests as decrease in time of the total survival probability of an LFA-1/ ICAM-1 bond in all three states (Fig. 7 A and B). The decay of the curve is decelerated by force from 0 to 5.9 pN (Fig. 7A). This is not surprising since this force range corresponds to the experimentally observed catch-bond regime where the bond lifetimes are prolonged by force [9]. As force further increases, the decay of the bond survival probability is accelerated by force (Fig. 7B), corresponding to the slip-bond regime where the bond lifetimes are shortened by force, also observed experimentally [9]. Similar trends are predicted for other cation conditions (data not shown).
Remarkably, our analysis predicts that as force increases, the probability vs. time curves of the long-lived LFA-1/ICAM-1 bonds (C 3 state) are left-shifted, as the slope of the initial phase is increased and the time needed to reach the maximal probability is shortened by 10-folds, from .3 s to ,0.3 s (Fig. 7C). Since LFA-1 with an open aA domain binds ligand with the highest affinity [5] and the lowest reverse-rate (Fig. 5C), this result indicates that force accelerates the activation of LFA-1/ICAM-1 bond by increasing the interstate transition rates.

Discussion
As primary force-bearing molecules governing cell-cell and cellmatrix adhesions [2,3], integrins are tightly regulated biochemically [3,4,6] and mechanically [7,8] [5]. The distribution among these conformations has been observed by MD simulations to depend on force [16]. The present work has added to this body of literature by defining the sequential process of the force-induced conformational changes of the LFA-1 aA domain and modeling the coupled kinetics of interstate transition between, and ligand dissociation from, different LFA-1 states.
Unlike the previous implicit water SMD study that analyzed the cluster distribution of aA domain conformations at the end of force application [16], our explicit water SMD simulations have observed the sequential transitions of the aA domain under force: Upon pulling the LFA-1 aA domain C-terminus, the a 7 -helix successively moved from the up to middle and down positions (Fig. 2). Our reduced charge simulations suggest that when a 7helix stays in middle or down position, the MIDAS ion has a strong tendency to move inward to its open position, which binds ligand with high affinity. These simulations indicate that applied force results in successive changes from the closed to intermediate and open conformations.
The force-induced transition of the three aA domain conformations observed in our simulations correlates with the forcedependent three-state dissociation observed in our previous BFP experiment [9]. Another interesting simulation result is that the a 7helix relaxed back to the up position after force removal ( Fig. 2A), suggesting that force is required to maintain its intermediate and down conformations under the simulation conditions. This also correlates with the experimental observation that the LFA-1/ ICAM-1 reverse-rate at zero-force was indifferent to changes in cation conditions and XVA143 treatment [9]. These correlations support our hypothesis that while the ICAM-1 association on-rate depends on the global conformations of LFA-1, the ligand dissociation off-rate is primarily determined by the aA domain conformation, which has been supported by experiment [9].
We constructed a mathematical model to further test this hypothesis, by examining how the three aA domain conformational transition may be related to the three-state dissociation kinetics. The model assumes force-induced successive transitions from C 1 to C 2 and C 3 states (Fig. 4), in accordance with the SMD results. Comparing to the previous phenomenological treatment, which fitted the force-dependent lifetime distributions by three apparent off-rates and their associated static fractions [9], the present mechanistic model treats the coupled kinetics of both interstate transition and ligand dissociation. This new model advances our knowledge in several aspects.
First, analyzing the previous BFP experiments [9] with this model has shown that the stability of LFA-1/ICAM-1 bonds are lowest at C 1 , intermediate at C 2 , and highest at C 3 states, suggesting a correspondence of the short-, intermediate-and longlived states with the closed, intermediate, and open conformations, respectively. Incorporating other forms of integrin conformational changes and relating them to functionality will be an important subject of future studies.
Second, the previously proposed allosteric mechanism for the LFA-1/ICAM-1 catch-slip bond [9] can be fully accounted for using the newly evaluated intrinsic parameters. Indeed, although the force-dependent dissociation of ICAM-1 from each of the three states behaves as slip bonds (Fig. 5), force accelerates transition from C 1 to C 2 more than it does dissociation from C 1 to R+L (compare Figs. 5A and 6A). Force also increases transition rate k 23 from C 2 to C 3 comparably to it does dissociation rate k r2 from C 2 to R+L (compare Figs. 5B and 6B). This interplay between force-accelerated interstate transition and dissociation gives rise to the LFA-1/ICAM-1 catch bond at low forces (Fig. 7A) and slip bond at higher forces (Fig. 7B), as observed experimentally [9]. Third, our model reveals that XVA143 suppresses the transition from C 1 to C 2 and inhibits the transition from C 2 to C 3 without altering the intrinsic reverse-rates k r1 -k r2 for dissociation from the three LFA-1/ICAM-1 bond states. This result has elucidated the mechanism for XVA143 to covert the LFA-1/ICAM-1 catch-slip bond to slip-only bond. Because both interstate transitions are induced by force (Fig. 6), our data indicate that XVA143 significantly weakens the force transmission from the aA to bA domains by blocking the binding of the intrinsic ligand of the aA domain a 7 -helix to the bA domain MIDAS [18,19]. This finding supports the hypothesis that the three-state dissociations of LFA-1/ ICAM-1 bonds are tightly regulated by the three-conformation transition of the LFA-1 aA domain.
Fourth, the new model has allowed us to estimate the time scale for integrin activation by force. Integrin activation has been suggested to be almost instantaneous [3], but data from different experiments are variable. Binding of fluorochrome-labled ligands to integrin a IIb b 3 reveals fast reversible formation of an integrin/ ligand precomplex followed by a stable irreversible complex, during which the affinity upregulation occurs in a time scale of 10 seconds [20,21]. Conversion from selectin-mediated rolling to integrin-mediated firm adhesion of leukocytes on endothelium and the detachment followed thereafter are used as criteria for integrin activation and deactivation [3,22,23]. Chemokine-triggered full activation of LFA-1 mediates arrest of rolling lymphocytes on high endothelial venules within 1 second under flow conditions similar to those in the circulation [3,24]. The conversion of rolling to stationary adhesion after the initial attachment of a neutrophil is induced by IL-1 in as little as 0.24 s in the presence of 1 dyn/cm 2 shear stress [22]. Force has been shown to facilitate the affinity upregulation at the cellular level. Our work provided the first estimates at the single-molecule level for the time scales of forceinduced integrin activation from the reciprocal interstate transition rates, 1/k 12 and 1/k 23 , which range from tens of milliseconds to several seconds (Fig. 6). Thus, the activation times estimated herein are in accordance with the previous reports. In addition, the interstate transition rates increase with increasing force (Fig. 6), indicating that force accelerates LFA-1 activation (Fig. 7C) These results further extend the model for activation of aA domain-containing integrins that we proposed previously [9]. Our molecular dynamics simulations show that applying forces shifted the equilibrium of different conformations of integrin aA domain, which is also supported by the agreement between our mathematical model fits and the experimental data, which indicates that force enhances the transition rates. Without force, the up position of the a 7 -helix in the aA domain is the favored conformation, where the MIDAS ion tends to stay at the outward position, and the ligand binding affinity is low. When force is applied, the equilibrium of the a 7 -helix position is shifted to middle and down; as a result, the MIDAS metal ion tends to stay at the inward position, and the ligand binding affinity is high.
In summary, this study defines the structural basis for mechanical regulation of the kinetics of LFA-1 aA domain conformational changes and relates these simulation results to experimental data of force-induced dissociation of single LFA-1/ ICAM-1 bonds by a new mathematical model. Future studies may include simulations to compare aA domains of other integrins and model refinements to add reverse transitions among the three conformational states.

Molecular dynamics simulations
The LFA-1 aA domain was modeled from the crystal structure 1LFA (residues 128-292) [12] except for the distorted a 7 -helix (residues 293-308), which was from another crystal structure 1ZON [11]. The MIDAS Mg 2+ and all crystallized waters in 1LFA were retained. The modeled structure was soaked in an 80680680 Å 3 water box with periodic boundary conditions, which included 3 Na + and 2 Cl 2 to neutralize the system. The NAMD package [25] and CHARMM22 all-atom force field [26] were used for energy minimization and molecular dynamics simulations. A 12-Å cutoff was used for van der Waals interactions and Particle Mesh Ewald summation was used to calculate the electrostatic interactions. Energy was minimized in multi-steps with careful treatments of the interactions to avoid any clashes between the a 7 -helix and other portion of the aA domain. The energy-minimized system was then equilibrated for 6 ns with temperature controlled at 310 K by Langevin dynamics with damping coefficient ,1 ps 21 and pressure controlled at 1 atm by Lagevin piston method. At the end of equilibration, the RMSD of the system converged and the a 7 -helix reached a position that aligned well with that observed in the up position of the Mac-1 aA domain structure 1JLM [10]. A 15-ns free dynamics simulation was performed with the equilibrated structure to generate initial conformations for SMD simulations. Two constant-force SMD simulations were performed, starting respectively from 10 and 15 ns of the free dynamics simulations, with the Ca atoms of residues 131-135, 167-172, 177-181 and 232-234 of the b 1 -b 4 strands harmonically constrained by springs with a spring constant ,140 pN/Å . A 250-pN force was applied at the C-terminal residue Val308 to pull the a 7 -helix along its axis to the down position suggested by the Mac-1 aA domain structure 1IDO [10]. The backbone hydrogen-bonding atoms in the a 7 -helix were constrained to prevent it from unfolding such that the constraint forces would be added if the distance between the hydrogen-bond pair exceeded 3.5 Å through a spring with a spring constant of ,700 pN/Å .
With the snapshots obtained from the SMD simulations at 0, 3.7, and 16 ns as respective starting points, we performed additional 50-ns free dynamics simulations for each case, with the Ca atoms of the a 7 -helix residues constrained. At 30 ns, the PSF input file for NAMD was modified such that the point charges of the two carboxyl oxygen atoms of residue D239 were changed from 20.76e to 20.26e, and the point charge of one Na + atom far away from the protein was changed to 0 to maintain charge neutral of the system.

Mathematical modeling
We constructed a mathematical model to describe the coupled kinetics of force-induced successive interstate transitions from the three states of LFA-1/ICAM-1 bonds and dissociation from these states (Fig. 4). The three states are denoted as C 1 , C 2 and C 3 , with interstate transition rates k 12 and k 23 (Fig. 4). Under tensile force, each transition step is assumed to be unidirectional and irreversible, for there was no observable reverse transition of the a 7 -helix position when pulling force was applied (Fig. 2).
The dissociation of the LFA-1/ICAM-1 bond can occur at any of the C 1 , C 2 and C 3 states, with intrinsic reverse-rates k r1 , k r2 and k r3 , respectively. Dissociation from each state is also assumed unidirectional and irreversible. This is reasonable because in the BFP force-clamped experiments [9], once a bond was rupture by tensile force, its component receptor and ligand were pulled apart and no longer able to rebind under the applied force.
Let p 1 , p 2 and p 3 denote the respective probabilities of ICAM-1 bound with LFA-1 at C 1 , C 2 and C 3 states, respectively. The kinetic equations governing the time evolution of the system can be formulated as: Equations 1-3 can be expressed in a matrix form: Let k 1 , k 2 , and k 3 be the eigen-values of A with corresponding eigen-vectors v 1 , v 2 and v 3 , respectively. It can be found that: k 2~kr2 zk 23 ð5Þ and v 21~v31~v32~0 , where v ij is the jth component of the vector v i . Therefore, the general solution of equations 1-3 can be expressed as: p 2~a2 exp({k 1 t)zb 2 exp({k 2 t) ð8Þ where a 1 , a 2 , a 3 , b 2 , b 3 , c 3 are nonzero constants. By substituting equations 7-9 into equations 1-3 and compare the corresponding coefficients, we have: In addition, because both the experimental data [9] and our SMD simulations (Fig. 2) showed that the transition from C 1 to C 2 and C 3 did not happen without force applied, the initial condition can be set as: Applying this initial condition to equations 7-9, we have: Taking equations 10-15 together, each of a 1 , a 2 , a 3 , b 2 , b 3 , c 3 can be solved as a function of k 12 , k 23 , k 1 , k 2 and k 3 . From this, by letting v 1~a1 za 2 za 3 , v 2~b2 zb 3 , v 3~c3 , and taking equations 4-6 into account, we got: v 3~k 12 k 23 (k 3 {k 1 )(k 3 Summing equations 7-9 yields: The left hand side of the equation 19 is the total survival probability of the LFA-1/ICAM-1 bond in all states, which corresponds to the measurements from the BFP force-clamped experiments [9]. The format at the right-hand side indicated that k 1 , k 2 , k 3 should be the apparent off-rates and v 1 , v 2 , v 3 should be the associated apparent fractions of the three bond states analyzed from the experimental data [9]. With the apparent off-rates k 1 , k 2 , k 3 and the apparent associated fractions v 1 , v 2 , v 3 obtained from fitting the experimental data (summarized in Table S1,S2,S3,S4,S5,S6) [9], the intrinsic kinetic parameters k 12 , k 23 , k r1 , k r2 and k r3 can be obtained by solving equations 4-6 and 16-18 and expressed as functions of the known apparent kinetic parameters: . The RMSD between the simulated a 7 -helix structure and the equilibrated structure shown in Fig. 2A is redrawn in (A). S139, S141, T206 and D239 are key residues that coordinate the metal ion. L289, F292 and L295 are ''ratchet'' residues that locate on b6-a7 loop or on a 7 -helix. They have been proposed to be important to the a 7 -helix position.
(TIF)    Video S1 SMD simulation of pulling the a 7 -helix of the LFA-1 aA domain. The simulated structures were shown in cyan with the a 7 -helix shown in green. The equilibrated a 7 -helix at the up (blue) and down (red) positions are superimposed for comparison. The Mg 2+ ion is shown as golden spheres. A 250-pN force was applied to the Ca atom of the residue 308 at the Cterminal of the a 7 -helix. At 15 ns, the force was released to allow the system to relax. The Ca atoms of residues 131-135, 167-172, 177-181 and 232-234 of the b 1 -b 4 strands were constrained to prevent the rigid body motion of the aA domain. The backbone hydrogen-bonding atoms in the a 7 -helix were constrained to prevent it from unfolding. (MPG)