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Dynamics of unbinding of cell adhesion molecules: Transition from catch to slip bonds

Edited by Bruce J. Berne, Columbia University, New York, NY, and approved December 21, 2004 (received for review September 17, 2004)
Abstract
The unbinding dynamics of complexes involving celladhesion molecules depends on the specific ligands. Atomic force microscopy measurements have shown that for the specific Pselectin–Pselectin glycoprotein ligand (sPSGL1) the average bond lifetime 〈t〉 initially increases (catch bonds) at low (≤10 pN) constant force, f, and decreases when f > 10 pN (slip bonds). In contrast, for the complex with G1 antiPselectin monoclonal antibody 〈t〉 monotonically decreases with f. To quantitatively map the energy landscape of such complexes we use a model that considers the possibility of redistribution of population from one forcefree state to another forcestabilized bound state. The excellent agreement between theory and experiments allows us to extract energy landscape parameters by fitting the calculated curves to the lifetime measurements for both sPSGL1 and G1. Surprisingly, the unbinding transition state for Pselectin–G1 complex is close (0.32 nm) to the bound state, implying that the interaction is brittle, i.e., once deformed, the complex fractures. In contrast, the unbinding transition state of the Pselectin–sPSGL1 complex is far (≈ 1.5 nm) from the bound state, indicative of a compliant structure. Constant f energy landscape parameters are used to compute the distributions of unbinding times and unbinding forces as a function of the loading rate, r_{f} . For a given r_{f} , unbinding of sPSGL1 occurs over a broader range of f with the most probable f being an order of magnitude less than for G1. The theory for cell adhesion complexes can be used to predict the outcomes of unbinding of other protein–protein complexes.
Formation and breakage of noncovalent protein–protein interactions are crucial in the functions of celladhesion complexes. Adhesive interactions between leukocytes and blood vessel walls involve a dynamic competition between bond formation and breakage (1). Under physiological conditions of blood circulation, the hydrodynamic force of the flow is applied to the linkage between leukocytes and endothelium. Rolling of cells requires transient tethering of the cell to the substrate and subsequent dissociation at high shear rates that are generated by the hydrodynamic flow field. Because of the requirement of adhesive interaction and the breakage of such bonds to facilitate rolling, only a certain class of molecules is involved in the recognition process. The remarkable rolling function is mediated by Ca^{2+}dependent specific bonds between the family of L, E, and Pselectin receptors and their specific ligands such as ESL1, podocalyxin, and PSGL1 (2–6). Specific interactions of Pselectins, expressed in endothelial cells or platelets, with PSGL1 (Pselectin glycoprotein ligand 1) enable leukocytes to roll on vascular surfaces during the inflammatory response by transient interruption of cell transport (tethering) in blood flow under constant wall shear stress. These interactions have been used extensively to probe tethering and rolling of leukocytes on vascular surfaces in flow channel experiments (2–15). Experiments show that the dissociation rates (also referred to as offrates), which govern cell unbinding kinetics, increase with increasing shear stress or equivalently the applied force.
It is generally believed that the applied force lowers the freeenergy barrier to bond rupture and, thus, shortens bond lifetimes (16). In contrast, Dembo et al. (17, 18) hypothesized that force could also prolong bond lifetimes by deforming the adhesion complexes into an alternative locked or bound state. These two distinct dynamic responses to external force are referred to as slip and catch bonds (17, 18). Whereas the dynamics of slip bonds has been extensively studied (5, 6, 13, 19–22), up until recently, evidence for catch bonds has been lacking. Using atomic force microscopy (AFM), Marshall et al. (1) measured the force dependence of lifetimes of Pselectin with two forms of PSGL1, namely, the monomeric and dimeric ligands sPSGL1 and PSGL1, which form, respectively, a single and double bond with Pselectin, and with G1, a blocking antiPselectin monoclonal antibody. The bond lifetimes were measured at values of forces that are lower than the level of their fluctuations by averaging over a large number of single lifetimeforce trajectories (1). The average bond lifetime of the highly specific Pselectin interaction with PSGL1 initially increased with force, indicating catch bonds (1). Beyond a critical force, the average lifetime decreased with force, as expected for slip bonds (1). In contrast to the behavior for specific Pselectin–PSGL1 complexes, Pselectin–G1 bond lifetimes decreased exponentially with force in accordance with the predictions of the Bell model (16). Marshal et al. (1) also found that both Pselectin–PSGL1 and Pselectin–G1 bond lifetimes measured at a fixed force appeared to follow a Poissonian distribution.
The complex dynamical response of the Pselectin–PSGL1 complex to force can be used to map the energy landscape of interaction between the macromolecules (23). For complexes, whose forcedependent behavior can be described by the Bell model, the unbinding involves escape from a single bound state. The observed behavior in Pselectin–PSGL1 complex requires an energy landscape model with at least two bound states, one of which is preferentially stabilized by force. Such a model has already been proposed for a complex involving GTPase Ran, a small protein that regulates transport of macromolecules between the cell nucleus and cytoplasm, and the nuclear import receptor importin β1 (24). Unbinding studies by AFM reveals that this complex fluctuates between two conformational states at different values of the force. The purpose of the present work is to show that the observed catch–slip behavior in specific protein–protein complexes in general and Pselectin–PSGL1 in particular can be captured by using an energy landscape that allows for just two bound states. The lifetime associated with bound states of the complex are assumed to be given by the Bell model (16). Although the Bell model is only approximate (25), it describes well the dissociation of single Lselectin bonds over a broad range of loading rates (26). Using the twostate model, we show that the experimental results for Pselectin–PSGL1 complex can be quantitatively explained by using parameters that characterize the energy landscape. In accord with experiments, we also find that the application of the same model to the unbinding of the ligand from Pselectin–G1 complex shows the absence of the second bound state. Thus, a unified description of specific and nonspecific protein–protein interaction emerges by comparing theory with experiments.
Theory and Methods
The Model. We use a twostate model (Fig. 1) for the energy landscape governing Pselectin–ligand interaction, in which a single Pselectin receptor (R) forms an adhesion complex (LR) with a ligand (L). The complex LR undergoes conformational fluctuations between states LR _{1} and LR _{2} with rates r _{12} = r _{10}exp[–F _{12}/k _{B} T] and r _{21} = r _{20}exp[–F _{21}/k _{B} T] for transitions LR _{1} → LR _{2} and LR _{2} → LR _{1} with barrier height F _{12} and F _{21}, respectively. The attempt frequencies r _{10} and r _{20} depend on the shape of the freeenergy landscape characterizing LR _{1} ⇄ LR _{2} transitions. In the absence of force, f, the equilibrium constant, K _{eq}, between LR _{1} and LR _{2} is given by K _{eq} ≡ r _{12}/r _{21} = (r _{10}/r _{20}) e^{–F/kBT}, where F is the free energy of stability of LR _{1} with respect to LR _{2} (Fig. 1). In the presence of f, K _{eq} becomes , where σ = x _{2} – x _{1}, the conformational compliance, is the distance between the minima. Force alters the freeenergy landscape of Pselectin–ligand unbinding (Fig. 1) and, thus, alters the bond breakage rates k _{1}(f) and k _{2}(f), which, according to the Bell model, are given by k _{1} = k _{10} e ^{y1f/kBT} and k _{2} = k _{20} e ^{y2f/kBT} (16). The prefactors k _{10} and k _{20} are the forcefree bondbreakage rates, and y _{1}, y _{2} are the minimal adhesion bond lengths at which the complex becomes unstable [distances between energy minima of states LR _{1} and LR _{2} and their respective transition states (Fig. 1)]. We assume that in the presence of f, the probability of rebinding is small. The dynamics of the adhesion complex in freeenergy landscape, which is set by the parameters σ, y _{1}, and y _{2}, can be inferred by using lifetime measurements of Pselectin–ligand bonds subject to a pulling force. We consider an experimental setup in which the applied force is either constant or ramped up with a constant loading rate r_{f} = κv _{0}, where κ is a cantilever spring constant and v _{0} is the pulling speed.
Distributions of Bond Lifetime at Constant Force. When f is constant, the populations P _{1}(t) and P _{2}(t) of states LR _{1} and LR _{2} can be calculated by solving the system of equations subject to initial conditions P _{1}(0) = 1/(K _{eq} + 1) and P _{2}(0) = K _{eq}/(K _{eq} + 1). In the AFM experiments, f fluctuates slightly around a constant value. The smoothness of the dependence of the lifetimes on f suggests that these fluctuations are not significant. The solution to Eq. 1 is where and D = (k _{1} + k _{2} + r _{12} + r _{21})^{2} – 4(k _{1} k _{2} + k _{1} r _{21} + k _{2} r _{12}). The ensemble average nth moment of the bond lifetime is where the distribution of lifetimes, P(t) = P _{1}(t) + P _{2}(t), is given by the sum of contribution from states LR _{1} and LR _{2}. In the limit of slow conformational fluctuations (i.e., when r _{12}, r _{21} ≪ k _{1}, k _{2}), P(t) = P _{1}(0)exp[–k _{1} t] + P _{2}(0)exp[–k _{2} t], whereas P(t) = exp[–(k _{1} + k _{2})t] in the opposite case.
Distributions of Unbinding Times and Forces for TimeDependent Force. When the pulling force is ramped up with the loading rate, i.e., f(t) = r_{f}t, the rate constants k _{1} k _{2}, r _{12} and r _{21} become timedependent and P _{1}(t) and P _{2}(t) are computed by numerically solving Eq. 1. The distribution of unbinding times, p_{t} (t), is p_{t} (t) ≡ k _{1}(t)P _{1}(t) + k _{2}(t)P _{2}(t) and the distribution of unbinding forces, p_{f} (f), can be computed by rescaling (t, p_{t} (t)) → (r_{f}t, p_{f} (f)), where p_{f} = (1/r_{f} )[k _{1}(f)P _{1}(f/r_{f} ) + k _{2}(f)P _{2}(f/r_{f} )]. The typical rupture force vs. loading rate, f*(r_{f} ), is obtained from p_{f} (f) by finding extremum, (d/dt)p_{f} _{} _{f} _{=} _{f} _{*} = 0 (5, 6).
Results
Unbinding Under Constant Force. We calculated the distribution of bond lifetimes, P(t), average lifetimeforce characteristics, 〈t(f)〉, and lifetime fluctuations, 〈t ^{2}〉 – 〈t〉^{2}. The model parameters of the energy landscape were obtained by fitting the theoretical curves of 〈t〉 vs. f to the experimental data (1) for Pselectin adhesion complexes with monomeric form sPSGL1 and antibody G1 (see Fig. 3 in ref. 1). The lifetimeforce data were adjusted to exclude experimental noise. The results displayed in Fig. 2 were obtained by using the model parameters given in Table 1 (all calculations were performed at room temperature). Since K _{eq} ≪ 1 for sPSGL1, in the absence of force, binding of Pselectins with sPSGL1 stabilizes LR _{1} of the Pselectin. For the antibody G1, K _{eq} = 1(k _{10} = k _{20} and y _{1} = y _{2}), indicating that both states are equally stable, leading to a landscape with one minimum. Pselectins form a stronger adhesion complex with G1 compared to sPSGL1: k _{10} for G1 is smaller than k _{10} for sPSGL1, and y _{1} is smaller than y _{1} or y _{2} for sPSGL1. This finding implies that adhesion complexes with G1 are less sensitive to the applied force.
Let us discuss the kinetic mechanism of transition from catch to slip bonds for unbinding of sPSGL1. At forces below ≈3pN, r _{12} ≈ r _{10}, r _{21} ≈ r _{20}, and k _{1} ≈ k _{10}, k _{2} ≈ k _{20}. In this regime, unbinding occurs from state LR _{1} (P*_{1}(0) ≫ P*_{2}(0)). In the intermediate force regime, 3 < f ≤ 10 – 12pN, k _{1} ≫ k _{2}, r _{12} ≫ r _{21}, and, hence, P*_{1}(0) ≪ P*_{2}(0) (k _{1} ≪ r _{12} due to y _{1} ≪ σ, see Table 1). In this limit, the unbinding dynamics is dominated by decay from state LR _{2} with the smallest eigenvalue z _{1} (corresponding to the longest time scale 1/z _{1}), which is , where . Expanding in power of and retaining only the first order term, we see that the distribution of bond lifetimes is determined by the unbinding rate At low forces, k _{eff} is dominated by the first term in Eq. 4 so that k _{eff} is given by the catch rate constant, k _{eff} = k _{catch} ≈ k _{1}/K*_{eq}, decreasing with f due to the increase in K*_{eq}. For f greater than a critical force f _{c} ≈ 10 pN, unbinding occurs from state LR _{2} with rate k _{eff} = k _{slip} ≈ k _{2}, which increases with f. As a result, the dual behavior is observed in the average lifetime, 〈t〉, which grows sharply at low f reaching a maximum at (f _{c}, 〈t*〉) ≈ (10 pN, 0.7s). For f > f _{c}, 〈t〉 decays to zero, indicating the transition from catch to slip bonds (Fig. 2a ). In contrast, 〈t〉 for a complex with G1 starts off at ≈5 s for f ≈ 5 pN (data not shown) and decays to zero at higher values of f (Fig. 2b ). There is also qualitative difference in the lifetime fluctuations for sPSGL1 and G1. For sPSGL1, has a peak at (f _{c}, 〈t*〉). However, for G1, δ(f) is peaked at lower f and undergoes a slower decay at large f compared with sPSGL1 (Fig. 2).
For binding to sPSGL1, increase of f to ≈10 pN, results in the redistribution of P(t) around longer lifetimes (compare curves for f = 2, 5, and 10 pN in Fig. 2a ). When f exceeds 10 pN, P(t) shifts back toward shorter lifetimes. In contrast, P(t) for complexes with G1 is Poissonian, ≈e ^{–tk1(f)} and the growth of k _{1} with f favors shorter bond lifetimes as f is increased. Stretching of complexes with sPSGL1 couples conformational relaxation and unbinding in the range 0–10 pN and leads to unbinding only when f > 10 pN. Thus, force plays two competing roles: It facilitates unbinding and funnels the Pselectin population into a forcestabilized bound state, LR _{2}. At low forces redistribution of initial (forcefree) population of bound states [P _{1} = 1/(K _{eq} + 1) > P _{2} = K _{eq}/(K _{eq} + 1)] into forcedependent population competes with unbinding. When f exceeds a critical force ≈10 pN, the dynamics of unbinding is determined by the bond breakage from maximally populated state LR _{2}. In this force regime, the distribution of lifetimes becomes again Poissonian, , and narrows at shorter lifetimes for large f (Fig. 2).
Pulling Speed Dependence of Unbinding Times and Forces. The excellent agreement between theory and experiment, which allows us to extract the parameters that characterize the energy landscape (Fig. 1) of the adhesion complexes, validates the model. By fixing these parameters, we have obtained predictions for p_{t} (t), p_{f} (f), and f* as a function of r_{f} for sPSGL1 and G1 unbinding from Pselectins. Because G1 possesses a higher affinity to Pselectins (compare k _{10}, k _{20}, and y _{1}, y _{2} in Table 1), p_{t} (t) computed for G1 exhibits an order of magnitude slower decay compared with p_{t} (t) for sPSGL1. For a given r_{f}, p_{t} for G1 has a peak that is smeared somewhat out at smaller r_{f} , whereas p_{t} for sPSGL1 starts to develop a peak only at r_{f} > 0.3 nN/s (Fig. 3). The peak position of p_{t} approaches zero and the width decreases as r_{f} is increased implying faster unbinding for both ligands. In contrast to p_{t} (0), p_{f} (0) decreases and f* increases as r_{f} is increased for both G1 and sPSGL1 (see Fig. 4). This finding implies that in contrast to unbinding times, increasing r_{f} favors unbinding events occurring at larger forces (5, 6). Comparison of p_{t} (t) and p_{f} (f) for G1 and sPSGL1 at a given r_{f} shows that, although Pselectin forms a tighter adhesion complex with G1, a linear increase of the applied force affects the stability of the complex with G1 more profoundly compared with sPSGL1. The presence of forcestabilized bound state LR _{2} for sPSGL1 facilitates a dynamical mechanism for alleviating the applied mechanical stress with higher efficiency, compared with singlestate Michaelis–Menten kinetics, L + R ⇋ LR for G1. This is illustrated in the Insets of Fig. 4, where we compared f* as a function of log(r_{f} ) for sPSGL1 and G1. f* is a straight line for G1. Due to dynamic disorder (27, 28), f*(r_{f} ) for sPSGL1 is convex up with initial and final slopes signifying two distinct mechanisms of Pselectin–sPSGL1 bond rupture.
Discussion and Conclusions
To account for the transition between catch and slip bonds of Pselectin–PSGL1 complex in the forced unbinding dynamics, we have considered a minimal kinetic model that assumes that Pselectins may undergo conformational fluctuations between the two states. Both fluctuations and Pselectin–ligand bond breaking are modulated by the applied force that not only enhances the unbinding rates but also alters the thermodynamic stability of the two states (Fig. 1). Using four parameters, namely, the rates r _{12}, r _{21} of conformational fluctuations and k _{1}, k _{2} of unbinding and the Bell model, we computed the distribution of bond lifetimes, the ensemble average bond lifetime, and lifetime fluctuations. The calculations are in excellent agreement with the experimental data on the unbinding of celladhesion complexes at constant force (1). The parameters, extracted by fitting the theoretical curves to experiment, allow us to obtain quantitatively the energy landscape characteristics. The fitted parameters show that the dual catch–slip character of the Pselectin–sPSGL1 complex can only be explained in terms of two bound states. In the forcefree regime, Pselectin–sPSGL1 exists predominantly in one conformational state with higher thermodynamic stability (K _{eq} ≪ 1). The release of sPSGL1 is much faster compared with the unstable state (k _{10} ≫ k _{20}). In contrast, using the same model, we found that G1 forms a tighter adhesion complex with Pselectin compared with sPSGL1. The two states are equally stable when Pselectins bind to form a tighter complex (compared with binding with sPSGL1) with antibody G1 (K _{eq} = 1). For G1, these states are kinetically indistinguishable both in the forcefree regime (k _{10} = k _{20}) and when the force is applied (y _{1} ≈ y _{2}), implying a single bound state.
The conformational compliance, σ, which leads to a decrease, σ f, of the freeenergy barrier separating the two freeenergy minima, are similar for sPSGL1 and G1. Bound and unbound Pselectin states are more separated in the freeenergy landscape when bound to sPSGL1. For sPSGL1, the LR _{1} and LR _{2} bond starts to break when the bond length exceeds 1.5 nm (y _{1}) and 1.1 nm (y _{2}), respectively. For G1 the distance from the only bound state to the transition state is only 0.32 nm, implying that the transition state is close to the bound state. The freeenergy difference F between states LR _{1} and LR _{2} of Pselectin–sPSGL1, which is obtained by equating r _{12} and r _{21} for sPSGL1, is of the order of 2k _{B} T. From the assumption that when P _{1} ≪ P _{2} the freeenergy barrier for transition LR _{1} → LR _{2} disappears, we found that the barrier height is F _{21} ≈ 5 – 6k _{B} T (see Fig. 1). Because of the presence of a more thermodynamically stable conformational state at higher values of f (for sPSGL1), the average Pselectin–sPSGL1 complex lifetime exhibits an initial increase at 0 ≤ f ≤ 10 pN (catch bond). After the force exceeds a critical force f _{c} ∼ 10 pN, the bond breakage rate of the forcestabilized state becomes nonnegligible and the bond lifetime decreases (slip bond). In both catch and slip regimes, the dynamics of unbinding can be characterized by the catch and slip bond rates k _{catch} and k _{slip}, respectively. The transition from catch to slip regime allows Pselectins to dynamically regulate their activity toward specific ligands such as sPSGL1 by means of extending the bond lifetime within a physiologically relevant range of mechanical stress and differentiate them from other biological molecules such as antibody G1 with k _{1} ≈ k _{2}. Because of this, force profiles of bond lifetime for unbinding of G1 and sPSGL1 are both qualitatively and quantitatively different. The microscopic mechanisms for dissipating external perturbation induced by mechanical stress or hydrodynamic flow are distinctly different for sPSGL1 and G1. In the case of G1, a mechanical stress breaks the Pselectin–G1 bond. However, in the case of sPSGL1, at low values of force the mechanical stress is dissipated by Pselectin conformational relaxation rapidly attaining a new equilibrium (P _{1} > P _{2}) → (P*_{1} < P*_{2}) as force is increased. When f = f _{c} ≈ 10 pN, the population of the locked state reaches a maximum (P*_{2} ≈ 1), and only at higher forces, f > f _{c}, does unbinding occur.
We have used our model to obtain testable experimental predictions for the distributions of unbinding times, p_{t} (t), unbinding forces, p_{f} (f), and typical rupture force, f*, at finite pulling speeds. These quantities can be directly accessed through experiment in which a pulling force is ramped up following a linear dependence on time, i.e., f = r_{f}t. These calculations further confirm that Pselectin forms a tighter adhesion complex with antibody G1 that lives (on average) 10–20 times longer compared with a complex with sSPGL1. In contrast to p_{t} (t) for which the average lifetime is inversely proportional to r_{f} for both ligands, the peak position of p_{f} (f) increases with pulling speed. This tendency is slower for a complex with sPSGL1; a 10fold increase of r_{f} from 0.1 nN/s to 1.0 nN/s shifts p_{f} by 30 pN for G1 and only by 3 pN for sPSGL1. We directly compared the most probable rupture force f* vs. r_{f} for G1 and sPSGL1 in the range 0 < r_{f} < 1.2 nN/s and observed an increase of f* from 0 to 80 pN in the case of G1 and only a marginal change from 0 to 3.5 pN in the case of sPSGL1 (Fig. 4). Our findings demonstrate that a twostate Pselectin system with an increasingly more stable (at large forces) slow ligand releasing locked state may serve as an effective molecular device that can relieve mechanical stress with a surprisingly high efficiency. The resulting dual response to stretching provides a simple mechanokinetic mechanism for regulating cell adhesion under physiological conditions of varying shear force. The theory described here can also be used to analyze forceinduced unfolding of protein–protein complexes. More generally, the model in conjunction with mechanical unfolding experiments can be used to map the characteristics of the energy landscape of complexes involving biological macromolecules.
Acknowledgments
This work was supported by the National Science Foundation.
Footnotes

↵ § To whom correspondence should be addressed. Email: thirum{at}glue.umd.edu.

Author contributions: V.B. and D.T. designed research; V.B. performed research; V.B. analyzed data; and V.B. and D.T. wrote the paper

This paper was submitted directly (Track II) to the PNAS office.

Note Added in Proof. After this article was accepted, we became aware of a related article (29).
 Copyright © 2005, The National Academy of Sciences
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