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# Interpreting the widespread nonlinear force spectra of intermolecular bonds

Edited by Steven M. Block, Stanford University, Stanford, CA, and approved July 3, 2012 (received for review February 21, 2012)

### This article has a correction. Please see:

## Abstract

Single molecule force spectroscopy probes the strength, lifetime, and energetic details of intermolecular interactions in a simple experiment. A growing number of these studies have reported distinctly nonlinear trends in rupture force with loading rate that are typically explained in conventional models by invoking complex escape pathways. Recent analyses suggested that these trends should be expected even for simple barriers based on the basic assumptions of bond rupture dynamics and thus may represent the norm rather than the exception. Here we explore how these nonlinear trends reflect the two fundamental regimes of bond rupture: (*i*) a near-equilibrium regime, produced either by bond reforming in the case of a single bond or by asynchronized rupture of multiple individual bonds, and (*ii*) a kinetic regime produced by fast, non-equilibrium bond rupture. We analyze both single- and multi-bonded cases, describe the full evolution of the system as it transitions between near- and far-from-equilibrium loading regimes, and show that both interpretations produce essentially identical force spectra. Data from 10 different molecular systems show that this model provides a comprehensive description of force spectra for a diverse suite of bonds over experimentally relevant loading rates, removes the inconsistencies of previous interpretations of transition state distances, and gives ready access to both kinetic and thermodynamic information about the interaction. These results imply that single-molecule binding free energies for a vast number of bonds have already been measured.

Dynamic force spectroscopy (DFS) (1⇓–3) provides researchers the ability to characterize the microscopic basis of inter- and intra-molecular bonding. The technique is rooted in the early works of Eyring (4, 5) and Zhurkov (6) where the failure times of stressed materials were explained through Arrhenius-like models of thermal activation over energy barriers. This microscopic view was scaled down further by Bell (7), and later Evans (3), to analyze the failure of single-molecule bonds under stress. Evans and coworkers found (3, 8) that the typical rupture force of a single bond varies as the logarithm of the loading rate, *f* ∼ ln *r*, when it is driven far from equilibrium, where *r* = *df*/*dt* is the rate of increasing force with time. This model has served as a foundation for the modern analysis of force spectroscopy experiments, in which the data, arranged as a plot of the rupture force vs. log-loading rate (the force spectrum) are expected to form a straight line, at least for irreversible rupture of a single bond over a single energy barrier.

Although early works pointed out the consequences of reversibility and multiple bonds (8, 9), the irreversible single-bond model quickly became the standard for analyzing force spectra. Then, in a seminal experiment on the biotin-avidin system, more than one linear trend appeared in the force spectrum (2). Under the irreversible, linear model of single bond rupture, the logical conclusion was that these data were evidence of a succession of separate barriers along the rupture pathway. This interesting result seemed to support the remarkable strength of the biotin-avidin bond and appeared to reveal intermediates that were previously inaccessible experimentally. Surprisingly, as follow-up studies collected more data on a variety of different molecular systems, a peculiar commonality began to emerge from these experiments: Most showed nonlinear trends in the force spectra. Even a cursory search of the literature finds more than 55 publications that report nonlinear force spectra (see *SI Text* for details) and cover a large variety of intermolecular systems involving macromolecules (10), peptides (11), small molecules (12), inorganic probes and self-assembled monolayers (13), and binding in both aqueous and organic solvents (14).

A few exceptions notwithstanding, the vast majority of these studies interpreted the data as representing two barriers along the unbinding pathway. Thus the commonality of nonlinear force spectra suggests somewhat unnaturally that energy landscapes with well-resolved intermediate barriers, which might reasonably be expected in the case of complex macromolecules or high-affinity interactions, are instead widespread among all types of molecular species and experimental preparations. Moreover, a significant portion of those studies employing the two-barrier interpretation report a distance to the “inner” barrier that inexplicably—and nonphysically—fall orders of magnitude below 1 Å. These facts suggest that the common existence of two prominent barriers could be a misconception based on a false interpretation of these nonlinear trends, which instead capture some other aspects of the physics of bond rupture.

In this paper we present a self-consistent interpretation that accounts for the characteristic features of many—if not most—force spectroscopy experiments without invoking additional complexity to the escape pathway. We draw upon previous numerical and asymptotic analyses, which suggested that nonlinearity in force spectra arises either through the reforming of a single bond at slow loading rates (8, 9, 15) or as a consequence of asynchronous fluctuations of several independent interactions that comprise a multivalent or heterogeneous cluster (7, 8, 16⇓–18). In either case the force spectrum passes through two primary regimes—an equilibrium regime characterized by a finite force and a kinetic regime characterized by a weak dependence on loading rate, *f* ∼ ln *r*. While these primary loading regimes have each been analyzed in detail (15⇓–17), the lack of a tractable model spanning both regimes has excluded it as a competing theory to the multi-barrier model. Here we derive accurate interpolations between equilibrium and kinetic regimes in both the single- and multi-bond cases. Finally, we show that this approach accurately describes a large number of nonlinear force spectra collected from vastly dissimilar intermolecular systems.

## Results and Discussion

We begin with the all-important single-bond system that illustrates the main physical assumptions of our model. Evans and Williams discussed the difficulties in measuring a true single-bond when blindly allowing molecules to connect between the probe and substrate (8). Their main conclusion was that, unless additional knowledge is used to determine bond valency (19), the actual experiment is more likely to measure multiple bonds than single bonds. Thus, we then follow with a description of the multiple bond case, because this is the most probable configuration measured in the laboratory. Finally, we compare the models with numerical simulations, our own experimental data, and 10 other datasets taken from the literature.

### Single Bond.

In a DFS experiment the force transducer [atomic force microscope (AFM) cantilever, biomembrane force probe, or an optical trap] typically exhibits a Hookean response to displacement, and thus if a single bond is characterized by a single barrier, the combined bond/transducer system comprises two states: the bound state, *b*, when the bond is formed, and the unbound state, *u*, when the bond is broken and the system fluctuates in the parabolic well of the transducer. If we assume that the time scale of intra-well relaxation is faster than any other time scales of the system, then the probability of finding the system in the bound state, *p*_{b}(*t*), is described by the master equation of a two-state Markov process (8), [1]where the instantaneous unbinding, *k*_{u}(*t*), and binding, *k*_{b}(*t*), rates are time-dependent to reflect their dependence on force.

The transition rates used in Eq. **1** must adequately describe the dynamics, yet they do not need to be complicated for modeling typical laboratory experiments. Several recent publications (20⇓⇓–23) showed that it is possible to produce nonlinearities in the force spectrum at very fast loading rates under an assumption of a force-dependent transition state (which can result from certain model-based potentials). However, the loading rate required to explore this regime of nonlinearity is 4 to 7 orders of magnitude greater than the upper end of loading rates reached in the laboratory (see *SI Text* for details). Therefore, for this work we chose to use the simpler unbinding rate given by the Bell model (7), which estimates the effect of force on the barrier to be linear and depends on a minimal number of parameters [Eq. **2**]. In this model, the rebinding rate is approximately the natural frequency of the transducer, , scaled by the Boltzmann-weighted energy of a spring extended between the barrier location *x*_{t} and the relative displacement of the spring minimum, *f*/*k*_{c} (see Fig. 1*A*). Thus the unbinding and binding transition rates are given by, [2][3]where is the intrinsic unbinding rate of the molecular complex (where the probe-molecule connection may contribute additional fluctuations and a modified damping), β^{-1} = *k*_{B}*T* is Boltzmann’s constant times the temperature, and defines the equilibrium free energy Δ*G*_{bu} = *G*_{u} - *G*_{b} of the bond *G*_{b}, relative to the free cantilever *G*_{u}.

The two rates in Eqs. **2** and **3** cross at a unique force, *f*_{eq}, given by (1, 24) [4]which defines the equilibrium force for the bond/transducer system. It can also be shown that for βΔ*G*_{bu}≫1 the mean force in the limit of vanishing loading rate is also equal to *f*_{eq}. The existence of this limiting force—the lowest force required to break a bond for a given force transducer stiffness—has been predicted by a number of theoretical studies (8, 9, 15), and we have recently presented an experimental demonstration of this phenomenon (24). On a more fundamental level, the existence of this force reflects major differences between the dynamics of free bonds and bonds attached to a force probe that adds a confining potential to the system and creates a defined unbound state. Note that even polymer linkers between the molecule and the probe behave like an entropic spring over large fractional extensions when the contour length is significantly larger than the persistence length (25). This effect results in an effectively spring-like potential for small loads where near-equilibrium effects are important. The major underlying assumption of the linear first-passage model is that a free noncovalent bond will break on its own in absence of external force if left alone for a sufficiently long time. However, when the same bond is attached to a force transducer, the presence of the transducer potential precludes the bond from breaking until a minimal force is applied, at which point a second state is created. Given the existence of the two states, another profound consequence is that the overall dynamics of the system should now be dependent on the shape of the second well and thus on the transducer stiffness *k*_{c} in all regimes of bond rupture (26⇓–28).

Inspection of Eqs. **3** and **4** reveals that the rebinding rate can be expressed as a function of the unbinding rate and the equilibrium force *f*_{eq}. Surprisingly, in this case the complete two-state process described by Eq. **1** includes only one additional free parameter, *f*_{eq}, to the usual unknowns , and *x*_{t}. To understand the significance of *f*_{eq} consider the evolution of the transition rates with force (Fig. 1*B*). For forces less than *f*_{eq} the rebinding rate dominates the dynamics, acting to keep the system in the bound state. At forces beyond *f*_{eq} the rebinding rate falls off rapidly due to the quadratic term -*f*^{2}/2*k*_{c} in the exponent, leaving the unbinding rate as the only significant contributor to the dynamics. Note that, although the distribution of rupture forces extends below *f*_{eq}, it is impossible to observe a mean force below *f*_{eq}. This important feature of the bond rupture process allows us to simplify the two-state master equation (Eq. **1**) by starting the bond loading process at *f* = *f*_{eq}, instead of starting at *f* = 0 [Seifert (15) used the same approach for analyzing the two-state problem through a mean-field approach]. With this simplification we truncate Eq. **1** to, [5]and inserting Eq. **2** in Eq. **5** we find the mean rupture force , [6]where *f*_{β} = *k*_{B}*T*/*x*_{t} is the thermal force scale, and is the exponential integral. We can also produce an analytical approximation to Eq. **6** by using the relationship *e*^{z}*E*_{1}(*z*) ≅ ln(1 + *e*^{-γ}/*z*), where γ = 0.577... is Euler’s constant. The most significant feature of Eq. **6** is that it extends over the two primary regimes of the single-bond dynamic force spectrum: a linear regime in the slow loading limit, , and a nonlinear regime in the fast loading limit, , leading to the characteristic ln *r* dependence expected for the irreversible rupture of single bonds. Indeed Eq. **6** is remarkably accurate: If we compare its predictions with the numerical solution of Eq. **1** we see negligible differences over a variety of parameters and loading rates (Fig. 1*C*), with the largest error of roughly 5% occurring at the “elbow” of the spectrum where *R*(*f*_{eq}) ∼ 1.

### Multiple Bonds.

We now turn to the system that comprises *N* bonds arranged in parallel between the probe and substrate. As we discussed in the introduction, this is arguably the common state of affairs in the laboratory (8). In a continuum approach, the number of formed parallel bonds *N*_{b}, of which *N*_{t} total bonds are allowed to independently unbind and rebind, is represented by the differential equation (7, 16⇓–18) [7]where *k*_{u}(*f*) is given by Eq. **2** and *k*_{a} is the constant rebinding rate of an individual molecule acting over a short distance from its retracted position while the overall cluster remains bound. This multiple bond model has been extensively examined numerically and by asymptotic analysis (7, 16⇓–18, 29⇓–31). Again, just as with the single bond case, two primary regimes emerge. At slow loading rates, fluctuations in the number of closed bonds is relatively fast (Fig. 2), and the system establishes a meta-stable number of bonds, *N*(*f*) = *N*_{t}/(1 + *k*_{u}(*f*/*N*)/*k*_{a}). From here we define *N* as the equilibrated number of formed bonds at zero force, . Stability of the cluster is completely lost when the force exceeds the equilibrium force (7, 17, 30) [8]where *W*(*x*) is the Lambert *W*-function, which is defined as the solution for *W* to the equation *We*^{W} = *x*, and defines the equilibrium free energy Δ*G*_{ba} = *G*_{a} - *G*_{b} of the closed bond *G*_{b} relative to an open bond *G*_{a} within the bound cluster. In systems where fluctuations in *N*_{b} are large, rupture can occur long before stability in *N* is lost, leading to a more gradual decrease in rupture force with decreasing loading rate than predicted here. The second primary regime appears at fast loading rates where the rebinding rate *k*_{a}, and hence fluctuations in *N*_{b}, become less important. Here the force reaches higher loads and the bonds rupture rapidly after the first bond fails, leading to the irreversible rupture force, (16, 17). Note that the force spectrum in this regime follows the same classic *f* ∼ ln *r* trend, therefore the resulting force spectrum will look qualitatively very similar to the single bond case. We obtain an interpolation between these two regimes (see *SI Text* for details): [9]Again, this equation is remarkably accurate in describing the force spectrum over the complete range of loading rates. When we compared the spectrum predicted by Eq. **9** to Monte Carlo simulation of the complete model in Eq. **7** over seven decades in loading rate (Fig. 2) using the Gillespie algorithm (32), we found that this model accurately captured the two primary trends, as well as the crossover region between the two loading rate regimes.

Interestingly, the multi-bond model of Eq. **9** shows that, if we combine the number of bonds *N* into an apparent force scale, [10]along with a generic definition of an equilibrium force *f*_{eq}, then the multi-bond force spectrum takes the same mathematical form as the single-bond model in Eq. **6**. Eq. **10** shows that the multiple-bond model modifies the single-bond model simply through the parameter *N*, which factors inversely to the transition state *x*_{t} everywhere in the function. Significantly, this observation explains the unreasonably small transition state distances reported in many force spectroscopy experiments. When an *N*-bonded system is analyzed in the kinetic regime using the common-practice single-bond model, the fitted apparent transition state, , will be *N* times smaller than the true distance *x*_{t}.

### Comparison to Experimental Data.

These models provide a theoretical basis for extracting a consistent set of interaction parameters of a bond over the broader context of loading conditions, not just those that satisfy the far-from-equilibrium condition. To test the validity of our model experimentally, we measured rupture forces for the nickel-mediated bond between nitrilotriacetic acid (NTA) and poly-Histidine (His_{6}) over four decades of loading rates (see *SI Text*). As Fig. 3*A* demonstrates, the mean force with log-loading-rate displays the expected two-regime behavior. The fit of Eq. **6** to the data is excellent, and we obtain reasonable values for all parameters: *x*_{t} = 0.92 ± 0.15 *Å*, and . The kinetic unbinding rate of 4.3 s^{-1} is of the same order as 1.8 ± 0.40 s^{-1} found for a single Ni-NTA unbinding from His_{6} by time-resolved fluorescence (33). The same system was previously investigated by another group (34), with the only difference that the probe-surface linkages of His_{6} and NTA functionalities were inverted. Those measurements (Fig. 3*A*) appear to diverge from ours. Attempts to corroborate the two data sets using the multi-bond model, through varying *N*, were unsuccessful. However, when we replace the value for the spring constant in our experiments (38 pN/nm) with that used in the previous study (10 pN/nm) and use exactly the same parameters obtained from our data, the force spectrum predicted by the single-bond model of Eq. **6** provides an excellent fit to these data points. This result highlights the significance of the transducer stiffness on the resulting force spectrum (9, 24, 26, 27) and demonstrates that proper analysis can unify the force spectra collected under different experimental conditions.

The deeper implications of the model in Eq. **9** become clearer when we apply it to DFS data for a wide range of systems. In addition to our data, we have compiled force spectra from 10 different studies, each probing a different intermolecular bond and extending over a sufficient span of loading rates to exhibit the two dynamic regimes. Eq. **9** fits the full range of all data sets very well, even for those datasets in which the lowest value of the pulling rate is still too large to fully reach the plateau of the near-equilibrium regime (Fig. 3; see also Table 1 for the fitted parameter values). Moreover, all data collapse onto a single universal line when plotted using the natural coordinates of that equation (Fig. 3*C*; and see Eq. **S3**). We also compare our model to the outstanding analysis of DFS measurements for multiples of biotin-avidin bonds by Teulon et al. (35). To be valid, the parameters predicted by Eq. **9** should remain constant regardless of the number of bonds, *N*. In Fig. 3*D* we present global fits to the three curves of Teulon et al. (assuming *N* = 1, 2, and 3, and keeping *x*_{t} and unchanged for all datasets) and find good agreement with the data for all three curves.

As predicted by the multiple bond analysis, a number of spectra we examined produce apparent transition state distances that are unrealistically small, for example 6.3 × 10^{-4} *Å*, 1.3 × 10^{-3} *Å*, 1.5 × 10^{-2} *Å*, and 1.4 × 10^{-2} *Å* from refs. 13 and 36⇓–38, respectively, with several on the order of 10^{-2} *Å* (see Table S1 for complete list of parameters). Note that these values are very similar to those given in the original publications using the linear *f* ∼ ln *r* model on the high-force regime. Potential of mean force calculations generally find the distance from the minimum to the primary barrier to be on the order of 1 Å (39). With this rough estimate the apparent distances found on the order of reflect bond numbers around *N* = 100, which is the same number of bonds predicted when assuming thiol monolayers on Au-coated probes with 100 nm tip radii using JKR contact mechanics (40). We note that if our model is to be used to extract true transition state distances from experimental data from a multi-bond system, the number of bonds will need to be determined independently. Alternatively, independent methods of ensuring that measurements probe only single bond interactions (8) can be used prior to applying the single-bond model to the data. However, the intrinsic unbinding rate of a single bond , and the single-bond free energy between the bound and closely unbound states Δ*G*_{ba}, can both be determined without knowledge of *N*. Further discussion of the correlation between equilibrium force and transition state as it applies to bond valency is provided in *SI Text* (see Fig. S1).

What other arguments support or refute our interpretation? It is logical to assume that if rebinding plays a significant role in the bond rupture at slow loading rates then we should be able to observe these events in the raw data. However, this is not a trivial phenomenon to observe, because the rare sojourns in the unbound state may not be long enough to be observed with typical sampling rates on the order of 1 kHz. Hence fast sampling rates should be used at slow loading rates to confirm or discard near-equilibrium effects. In the multi-bonded scenario the question regarding rebinding events becomes irrelevant, because macroscopic rebinding from the completely unbound state is not required to establish the quasi-steady-state number of bonds and therefore the equilibrium force.

Finally, we consider the conditions required to observe nonlinearity within this model—that is, the loading rate that marks the crossover between near-equilibrium to kinetic regimes. Inspection of Figs. 1 and 2 shows—as a consequence of the exponential integral appearing in Eq. **9**—a useful relationship for benchmarking where this crossover occurs is when *R*(*f*_{eq}/*N*)/*N* = 1, or, [11]Using the extremes of experimentally observed values for *f*_{eq} ∼ 10–200 pN, , with nominal *x*_{t} ∼ 1 *Å* and *N* ∼ 1–50, we find crossover loading rates encompass a range *r*_{x} ∼ 1–10^{5} pN/s. Remarkably, this range brackets the typical loading rates explored in a laboratory, as shown by the shaded area in the inset to Fig. 3*B*. Likewise, we show in *SI Text* that the theoretical equilibrium unbinding force for a single-bond falls in the range of *f*_{eq} ∼ 40 to 108 pN. Not only is this range above the noise limits of widely used AFM systems, but these forces can be quite high, and are remarkably consistent with the experimentally measured low-force regime data (see inset to Fig. 3*B*).

## Conclusions

The findings presented in this work raise the general question of whether the multi-barrier model is an appropriate interpretation of force spectroscopy experiments. Unfortunately, the multi-barrier model imposes any arbitrary number of barriers along the bond potential to match the number of linear trends observed in the experimental force spectrum. If fitting is the only criterion, then the multi-barrier hypothesis can never be rejected, because this segmented approach can be tailored to fit any force spectrum. Further, it introduces at least twice as many free parameters as the number of perceived linear trends. In contrast, the single-barrier model presented here is based on a simpler potential energy surface, is governed by only three free parameters, and provides tangible, testable requirements, such as the cross-over loading rate and equilibrium force, that can be calculated based on physical parameters and compared against experiment. Thus, while the validity of a multi-barrier model is difficult to prove or disprove, researchers can easily test whether the data behave according to the predictions of the complete single-barrier model. Nonetheless, while the model presented here provides a comprehensive description of forced bonds comprising single-barrier interactions, we emphasize that experimental data not accounted for by such a description can arise for any number of reasons. It is up to the researcher to determine whether such deviations from the single-barrier description are an experimental artifact or a signature of a more complex energy landscape.

Our findings imply that both near- and far-from-equilibrium regimes are readily accessible in a DFS measurement, and these regimes are both inherent and ubiquitous. The results also show that, in most force spectroscopy experiments, multiple connections between the probe and surface cannot be treated simply as an effective correlated single bond, because conformational freedom in these systems permits microscopic unbinding between closely situated intermolecular pairs, establishing the equilibrium threshold for rupture. More importantly, if we abandon the assumption of strictly irreversible bond rupture inherent in the standard analysis, we find that both kinetic and equilibrium properties can be determined from a single dataset and the bond can be described by just three parameters: the transition state distance, the unbinding rate, and the value of *f*_{eq}. The surprising consequence of this conclusion is that the data required for determining equilibrium free energies corresponding to single- and multiple-bond scenarios are already available in the literature for a large number of intermolecular systems.

## Acknowledgments

This work was supported by the Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract DE-AC52-07NA27344. Measurements were performed at the Molecular Foundry, Lawrence Berkeley National Laboratory, with support from the Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract DE-AC02-05CH1123. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. Work at Lawrence Livermore National Laboratory was performed under the auspices of the U.S. Department of Energy by under Contract DE-AC52-07NA27344.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. E-mail: jjdeyoreo{at}lbl.gov or rwfridd{at}sandia.gov.

Author contributions: R.W.F., A.N., and J.J.D.Y. designed research; R.W.F. performed research; R.W.F. analyzed data; and R.W.F., A.N., and J.J.D.Y. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1202946109/-/DCSupplemental.

Freely available online through the PNAS open access option.

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