A catch bond mechanism with looped adhesive tethers for self-strengthening materials

The lifetime of chemical bonds shortens exponentially with force. Oddly, some protein-ligand complexes called catch bonds exhibit a sharp increase in lifetime when pulled with greater force1. Inventing catch bond interfaces in synthetic materials would enable force-enhanced kinetics or self-strengthening under mechanical stress. We present a molecular design that recapitulates catch bond behavior between nanoparticles tethered with macromolecules, consisting of one looped and one straight tether linking particles with weak adhesion. We calibrate the loop stiffness such that it opens around a target force to enable load-sharing among tethers, which facilitates a sequential to coordinated failure transition that reproduces experimental catch bond force-lifetime curve characteristics. We derive an analytical relation validated by molecular simulations to prove that loop and adhesion interactions can be tailored to achieve an unprecedented spectrum of catch bond lifetime curves with this simple design. Our predictions break new ground towards designing tunable, catch-bond inspired self-strengthening materials


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The lifetime of chemical bonds shortens exponentially with force. Oddly, some protein-ligand complexes called catch bonds exhibit a sharp increase in lifetime when pulled with greater force 1 . Inventing catch bond interfaces in synthetic materials would enable force-enhanced kinetics or self-strengthening under mechanical stress. We present a molecular design that recapitulates catch bond behavior between nanoparticles tethered with macromolecules, consisting of one looped and one straight tether linking particles with weak adhesion. We calibrate the loop stiffness such that it opens around a target force to enable load-sharing among tethers, which facilitates a sequential to coordinated failure transition that reproduces experimental catch bond force-lifetime curve characteristics. We derive an analytical relation validated by molecular simulations to prove that loop and adhesion interactions can be tailored to achieve an unprecedented spectrum of catch bond lifetime curves with this simple design. Our predictions break new ground towards designing tunable, catch-bond inspired self-strengthening materials.
The force-enhanced lifetime behavior of catch bonds, which occurs only at small scales where thermal forces are relevant, starkly contrasts with ordinary chemical interactions called 'slip bonds', which exhibit an exponential lifetime decay with force, as predicted by Kramers 2 and Bell 3 , and Zhurkov's earlier work on fracture 4 . Experiments and simulations on various catch bond proteins (P-selectin/L-selectin 5 , pili adhesin FimH 6 , integrin 7 , actin 8 , von Willebrand factor 9 and kinetochore 10 ) have revealed that catchbonds are common and play critical roles in many physiological processes. Different phenomenological models 11 , such as the two-pathway model 12 and structural models, i.e. allosteric 13 and the sliding rebinding models 14 , have been developed to t the nonmonotonic lifetime curve of catch bonds with empirical parameters obtained from experiments. Phenomenological and structural models on proteins excel at tting biophysical lifetimes curves, but do not illustrate the dynamic molecular mechanism of a catch-bond in action 11 . The complex 3D structure and dynamics of proteins makes it di cult to understand which features are necessary for catch bond formation, and existing molecular simulation approaches offer limited resolution into probing conformational dynamics of a catch bond protein in action.
The self-strengthening and the reversible nature of the catch bonds makes them uniquely important for materials science because a discontinuous transition to high resistance to failure is often desired in extreme loading scenarios that arise in applications spanning sports equipment, garments, adhesives, protective gear and structural components. Recent modeling efforts by the authors have established that it is possible to design shape-changing nanoparticles that form catch bonds with each other, speci cally by creating soft switches that facilitate force-induced a nity changes around a target force 15,16 . Unlike shear jamming seen in colloids and granular matter, these systems don't require volume preserving deformations to exhibit strengthening. Instead, they utilize instabilities to trigger an a nity switch that works even in pure tension. However, designing such instabilities with nanoparticles is challenging synthetically. Alternatively, it has been hypothesized that nanoparticle networks held together by polymer tethers 17 that exhibit catch bonds 18-20 could improve mechanical properties such as toughness beyond a threshold strain-rate or force level. Yet, macromolecular designs that reproduce a catch bond mechanism in such polymer-grafted nanoparticle networks are yet to be theoretically proposed and synthesized.
With this premise in mind, we choose to take a minimalistic approach to designing interfaces that replicate catch bond features. Instead of complex topologies, we utilize linear polymer tethers covalently grafted onto a nanoparticle and interacting with another particle with weak adhesive interactions (slip bonds) that we call an adhesin interaction (Fig. 1A). While both tethers have the same contour length (L I = L II ), one of the tethers can form a loop stabilized by an ordinary slip bond that will 'store' some of its length, whereas the other lacks the interaction that facilitates a loop. These design elements, consisting of weak adhesin and loop interactions, are quite universal. An adhesin interaction can be formed simply from any non-covalent secondary interaction, such as hydrogen bonds or electrostatics 21  Looped or hairpin conformations are also attainable by DNA 23 and RNA molecules 24 . In foldamers 25 and DNA containing gels 26 , it has been possible to make synthetic looped polymers as well 27 . To our knowledge, the use of loops to create interfaces with catch bond kinetics have not been previously reported.
In our model, we consider two adhesin interactions to be identical and have different bond characteristics from the loop interaction. As shown in Fig. 1A, in equilibrium, the minimum energy con guration of this system is such that the folded tether keeps the two surfaces together under thermal undulations, while the tether without the loop is slack. When a constant tensile force is applied to pull the two surfaces apart, the tether with the loop (tether II) becomes taut and carries most of the load. Thus, its adhesin will dissociate before the adhesin of the tether without the loop (tether I). Conceptually, there are two primary kinetic pathways through which the interface can fail. If the tether II adhesin dissociates before its loop opens, the entire load will be transferred to tether I. Since adhesins are sequentially exposed to the entire load and fail one after the other, this will result in sequential failure. Alternatively, if the loop opens before tether II adhesin dissociates, the load is shared between the adhesins as both tethers become stretched. The overall lifetime will be greatly extended as each adhesin is exposed to half of the load, and the system exhibits coordinated failure. These failure mechanisms are schematically illustrated in Fig. 1C, also showing the load distribution between the tethers. It should be noted that the existence of these two failure mechanisms is analogous to low and high a nity states of proteins, or failure pathways of different resistance often described in two-state or two-pathway phenomenological models of catch bonds 28 .
The central question that must be addressed is how the energy landscapes of loop and tether interactions should be designed to facilitate a transition from one failure mode to the other as the magnitude of the force increases. To answer this question, we derived an analytical expression for lifetime of the interface, which denotes the time required for the dissociation of both adhesin interactions. To describe the kinetics of the loop and adhesin interactions in a simple way, we use Bell's theorem 3 , which can describe rate of bond dissociation under external force f as 1 where is the natural vibration frequency, of the bond and is the energy barrier, Δx is the transition state distance, and is the thermal energy respectively. It is convenient to combine the natural frequency and exponential of the energy barrier by de ning a thermal off rate , which is the rate of bond dissociation in the absence of force. Hence, adhesin dissociation rate and loop opening rate have the following forms 2 3 Note that controls the force dependence of , and with the larger will be more force-sensitive and result in a sharper decrease in bond lifetime with force.
The timeline of the adhesin dissociation versus loop opening events on the looped tether governs which failure pathway a single pulling trial will follow. Thus, the probability of adhesin breaking before loop opening, or vice versa, dictates the system lifetime. If we call average sequential trial lifetime as and average coordinated lifetime as the average lifetime of the system, will have the form of The essential design principle to ensure catch bond behavior in our system can be stated as follows. At lower forces, the tethers should predominantly fail sequentially. As the force increases, the system should become more likely to exhibit coordinated failure, which will prolong the lifetime. Analytically, this translates into transitioning sigmoidally from 1 ( ) to 0 ( ) with increasing ( Fig. 2A). Hence, should be chosen greater than and should be chosen as less than .
Based on this design principle, we heuristically propose an original set of parameters for and , which are listed in Table 1. The units are chosen to match the typical force and lifetime ranges of existing catch bond curves. For these parameters, Fig. 2B shows that the lifetime of the interface is nonmonotonic with force, showing a positive slope and peak at an intermediate range of forces. This proves our original conjecture that the dual tether interfaces with loops can exhibit catch bond behavior for a calibrated set of parameters. For interpreting and generalizing the lifetime curves, we specify four quantities of interest (Fig. 2C), which are the maximum value of the lifetime curve after the initial slip regime, , the critical force at which it occurs, and gain in lifetime, , which is the ratio of and and normalized force range of the catch bond peak, ,which is de ned as the difference between the force values of at and divided by the critical force and are determined by numerically computing the zeros of the derivative of Eq. 6. These two points mark the slip-to-catch and catch-to-slip transitions, respectively. Systems that have > 1 show catch bond behavior. Figure 2D shows how varying while keeping Δx values constant governs catch bond performance (black line). As grows, increase in reaches a plateau. The same observation is true for , i.e. the width of the lifetime behavior approaches to limit 0.255 with increasing (red line). On the other hand, according to Fig. 2E, is maximum when the is 1.93, going above or below this value decreases the gain till catch bond behavior is no longer possible, i.e. there is a narrow range of ratios of parameters that catch bond behavior is possible. Similarly has a maximum, it reaches 0.255 when is 1.71. The relations depicted in Fig. 2D-E, are true for all values of and Δx that gives the same and ; thus, we conclude that the relative values of the parameters chie y govern the existence of catch bond behavior, and their magnitudes are less important.
In addition, it should be noted that our lifetime curve is triphasic (slip-catch-slip behavior 29 ), which is also seen in various biological catch bonds 7,30 . and values we nd for near-optimal systems we report here are comparable to what's seen in some of these biological systems.
To validate these analytical predictions, we carry out Markov chain Monte Carlo (MCMC) simulations to calculate lifetime from rate parameters. This method is commonly used to describe bond rupture and formation events 31,32 , where rate constants are used to simulate probabilistic events with acceptance checks that account for both thermal effects and force contributions. As illustrated in Fig. 2B, the simulation results (black diamonds) are in excellent agreement with the analytical expression.
After establishing the catch bond behavior in our two-tether system, we systematically vary kinetic parameters to examine the factors that affect and Fig. 3 shows that can be decreased by increasing and and increased by increasing .
shows a more complex relationship, where value of has a peak. On the other hand, can be increased by increasing and and decreased by increasing and . The important takeaway is that we can target certain and values by varying and . However, there is a tradeoff between maximizing and tuning when varying kinetic parameters. We emphasize that the plots we obtain is speci c to the system at hand, but nevertheless the conceptual design framework we have established here can be applied to diverse systems with similar soft switch mechanisms that regulate low to high a nity transitions.
Next, we test our ndings in a computational experiment using molecular dynamics (MD) simulations, to illustrate that the theoretical concepts do hold for realistic molecular systems. A coarse-grained representation of the system is shown in Fig. 4A. To generate landscapes that qualitatively have similar kinetics as the analytical study, we represent and with Morse potentials, which have the following general form: Here, is the depth of the energy well, is the equilibrium bond distance and is the parameter that controls the width of the well (the smaller is, the broader the well), and x is the bond length.
The base interaction parameters that we used in the MD simulations are listed in Table 2. In methods section, we discuss our justi cation of and parameters.
As in the case of the theoretical model, we explore how the lifetime curve characteristics depend on the energy landscape parameters. Figure 4C demonstrates the effect of , where increasing its value lowers and increases . This means that overly stable and stiff loops will require a larger force to be opened, and that naturally causes a reduction in the lifetime at both low and high force levels.
Conversely, Fig. 4D shows that increasing brings the whole lifetime curve up considerably, given the exponential dependence of the lifetimes on the adhesin strength. As shown in Fig. 4E-F, increasing lowers and increases . Increasing has an opposite effect, since stiffer adhesin interactions concentrate the work more greatly on the loops, triggering a transition to the coordinated failure around a lower force threshold.
To compare the analytical and MD results, we need to establish relations between the kinetic and interaction parameters. According to Bell's theorem, is proportional to . In the Morse potential, is equivalent to the energy barrier , thus we plot Fig. 5A-B with respect to . Since the Morse potential does not have a transition state, we do not have a de nite to de ne the width of the energy landscape. However, we can use the in ection point of the Morse potential and infer that the width is inversely proportional to . Thus, we plot Fig. 5C-D with respect to Overall, Fig. 5 follows the same trends as Fig. 3, i.e. shift in and value of depend similarly in both parameter space. Although there are differences in Fig. 3B and 5B, where the right shift in is sharper in MD simulations and Fig. 3C and 5C, where the increase in is concave in analytical equations and convex in MD simulations. These minor differences, which don't change any of our main conclusions, can be attributed to the nonlinearity of the energy landscape or additional rebinding events that we have not considered in this iteration of the analytical model. Bell's theorem is the simplest description of forcedependence in dissociation kinetics, and may not accurately describe trends in nonlinear systems.
Additionally, our analytical theory does not account for adhesin rebinding to occur, and in the MD simulations, some of the cases do exhibit rebinding. MD analysis of rebinding effect on lifetime curves is provided in the Supplementary Information. We note that there are models to study the force-dependent kinetics of parallel bonds 33 including rebinding effects, however, in our case utilizing parallel tethers alone doesn't result in catch bond behavior.
Lastly, we investigate the effects of the tether length on the catch bond behavior using MD simulations. For these simulations, we vary the number of segments in our two-tether system. As an example, the tethers in Fig. 4A have 5 segments. Figure 6 shows that as the number of segments increases, the peak lifetime of the system starts to decrease, ultimately transitioning to slip behavior in 15-unit tether. This behavior is expected because longer tethers exhibit lower stiffness, and the stiffness of a polymer can play an important role in transmitting the force to the bond. It has been shown in the literature 34 that for polymer tethers with adhesive tips, softer tethers will result in a higher off rate, i.e. shorter lifetime.
However, we rely on our established insights to re-calibrate the adhesin and loop kinetics. According to Fig. 5, we modi ed the MD interaction parameter to see if a catch bond behavior can be obtained with long tether systems. This data is shown as 15* in Fig. 6, showing re-emergence of the catch bond behavior. The is increased to 12 from 10 and is reduced to 1.54 from 2. This improves the gain and thus facilitate a slip-catch transition, without considering tether elasticity. This demonstrates that our theoretical and computational simulations still provide valuable insight into systems with longer chains, but also that the loop and adhesin kinetics would need to be tailored for speci c tether characteristics.
In conclusion, this work demonstrates a novel catch bond mechanism that could be employed in polymer or macromolecule grafted nanoparticles. The essential feature of the tethers is to control the lifetime of the interface by tuning the opening kinetics of a loop that facilitates load-sharing by revealing hidden length in one of two tethers interconnecting the particles. This work establishes an analytical theory validated by both Monte Carlo and MD simulations, which provides insights into how in such a model system one could precisely dictate the shape of the force vs. lifetime curve by tuning loop and adhesin kinetics. By tailoring of the stiffness and strength of a loop in one of the tethers, it is possible to control sequential vs. coordinated failure of the adhesive tethers, which can be used to engineer forcedependence of the kinetic pathways of interfacial dissociation. Tuning the energy landscapes or rate parameters clearly revealed that a target peak lifetime, gain, and the normalized force range over which catch bond behavior occurs can be programmed. While we have not exhaustively explored all possible designs, this versatile strategy opens the possibility of creating catch bonds that exhibit lifetime characteristics reaching beyond the feasible range seen in biological proteins (typically pN forces and seconds in peak lifetimes). This has major implications for mechanosensitive materials, nanocomposites, and drug delivery systems that could utilize mechanical cues for tailoring interfacial strength. Moreover, the fact that polymer-like tethers, loops, and weak bonds can be generated even in macroscopic thermalized magneto-mechanical systems 35 , we envision that it is possible to create catch bonds in a scale-invariant fashion. Lastly, using molecular dynamics simulations, we generated reproduceable lifetime curves reminiscent of highly complex catch bond proteins. While our design is unprecedented and distinct from biological catch bonds, our study also raises the questions as to whether catch bonds can occur through load-sharing and programmed unfolding of multiple adhesion proteins, where soft fold and stiff ligand interactions might hold the key to achieving force enhanced lifetimes in diverse biomolecular systems. Future studies may build on these ndings to study mechanical properties of catch-bond capable networks of nanoparticles or investigate modi cations to the design to further optimize performance metrics such as the lifetime gain and normalized force range of synthetic catch bonds. In summary, this work presents a versatile strategy for making man-made catch bond mechanisms, suggests that catch bond behavior can occur readily in interfaces consisting of which event happens rst, the simulation branches into two paths. If adhesin dissociates rst, the simulation will calculate the lifetime for sequential failure, where event times S1 and S2 are recorded (Fig. 1C). If loop opens rst, the simulation will calculate lifetime for coordinated failure, where event times C1, C2 and C3 are recorded (Fig. 1C). The accumulated time steps are the lifetime of the of the twotether system in that simulation run, which is repeated 100,000 times for a given force to obtain an ensemble of exponentially distributed lifetimes and their average at that force. The four model parameters ( , , and ) are chosen to be the same as the analytical theory for comparison.
Molecular Dynamics. The MD simulations were performed using the LAMMPS software package 36 . The simulations were run in the NVT ensemble at 50 K using a Langevin thermostat with a damping factor of 100 time steps. A lower temperature is used to reduce noise while ensuring thermal uctuations give rise to stochastic bond breaking events. The adopted time step of 1 fs was found to be su ciently small to ensure accuracy of the lifetime measurements. Initial con guration of the system is shown in Fig. 4. Each simulation starts with 50,000 time step equilibration process, which is adequate given the simple nature and small size of our system. After equilibration, an instantaneous constant force is applied perpendicular to the top surface, while the bottom surface is held stationary. We measure lifetimes under forces ranging from 160 to 380 pN. Both surfaces are treated as rigid bodies that are constrained to move only in the direction normal to their planes. The exible polymer tethers consist of harmonic bond interactions with no angle terms. The mass of the tether beads and the surface beads are set as 1000 g/mol. Each tether has 6 beads that are connected with 2Å long harmonic bonds with stiffness 1000 kcal/molÅ 2 stiffness.
In line with the two-state model representation of catch bonds, the mean lifetime of the system must increase when a tensile force is applied, which occurs if the force facilitates a propensity for the system to transition into the high-a nity state. At small forces, most trials should result in failure in the lowa nity state, which in this study corresponds to the sequential failure path. At large forces, most trials should result in transition to a high-a nity state, which in this study corresponds to the coordinated failure path. Interaction energies between the adhesins and the bottom plate are measured. To obtain meaningful statistics for each force value, 10,000 trials were performed. The interface lifetime was recorded when the interactions energies for both adhesins drop below where we considered adhesins to be dissociated. Animations of sample simulation runs that end with sequential and coordinated failure are provided in the Supplementary Information as Movie S1 and Movie S2 respectively. It should be noted that the temperature and the force ranges used in the simulations are arbitrary and do not aim to reproduce any particular experimental setup. We note that goal here is to prove the existence of catch bond behavior in these systems in a generic way and quantify its dependence on parameters in a theory-driven fashion, rather than matching a particular experimental observation.
Bootstrapping. The events we simulated all followed exponential distributions, where the mean and the standard deviation of the lifetime data were equal. Since our data were far from the normal distribution, we decided to use the bootstrapping method instead of using standard deviations, to calculate the variability of of the data. For each force value, a histogram of means was created from 10000 bootstrap samples with the size of the original data set both for MCMC and MD simulations. A con dence interval of at a 95% con dence level is calculated and plotted as error bars.
Interaction energy landscapes. Based on our previous works on catch bond systems 15,16 , we hypothesized that the loop interaction, which serves as a force-sensitive switch, should have a deeper and a broader energy well compared to adhesin energy landscape to enable catch bond formation, as shown in Fig. 4B. The work done on the bonds by force tilts loop and adhesin energy landscapes , and the resulting reduction in the energy barrier ( is sensitive to the distance to the transition state . Therefore, in Eq. 7, ensuring that has a lower (broader landscape) compared to results in the of the folded interaction to be reduced greatly when the applied force is adequately large. According to Kramer's theory 2 , is directly proportional to , thus the loop lifetime declines more sharply when subjected to force compared to the adhesin lifetime. In addition, since has a deeper barrier compared to , loop lifetime is greater than the adhesin in the absence of force. However, since has larger reduction of under force, crosses over and becomes shorter beyond a threshold force value. This facilitates the transition to coordinated failure through loop opening at large forces.

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