# On artifacts in single-molecule force spectroscopy

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Contributed by Attila Szabo, October 5, 2015 (sent for review September 15, 2015; reviewed by Michael Hinczewski and Emanuele Paci)

## Significance

The response of single molecules to applied forces can be probed using atomic force microscopes and laser tweezers. In these experiments, a biomolecule is attached, for example, to a mesoscopic (approximately micron size) bead that is trapped in the focus of a laser, via a long intervening polymer. To interpret such experiments, one must know the extent that the measured data reflect the behavior of the molecule of interest. Here, we develop an analytically tractable theory that can be used to determine the influence of the pulling device (apparatus) on the measured quantities, such as the rates of conformational changes. Our theory allows one to make the appropriate corrections when necessary.

## Abstract

In typical force spectroscopy experiments, a small biomolecule is attached to a soft polymer linker that is pulled with a relatively large bead or cantilever. At constant force, the total extension stochastically changes between two (or more) values, indicating that the biomolecule undergoes transitions between two (or several) conformational states. In this paper, we consider the influence of the dynamics of the linker and mesoscopic pulling device on the force-dependent rate of the conformational transition extracted from the time dependence of the total extension, and the distribution of rupture forces in force-clamp and force-ramp experiments, respectively. For these different experiments, we derive analytic expressions for the observables that account for the mechanical response and dynamics of the pulling device and linker. Possible artifacts arise when the characteristic times of the pulling device and linker become comparable to, or slower than, the lifetimes of the metastable conformational states, and when the highly anharmonic regime of stretched linkers is probed at high forces. We also revisit the problem of relating force-clamp and force-ramp experiments, and identify a linker and loading rate-dependent correction to the rates extracted from the latter. The theory provides a framework for both the design and the quantitative analysis of force spectroscopy experiments by highlighting, and correcting for, factors that complicate their interpretation.

The mechanical manipulation of single molecules by laser tweezers or atomic force microscopes has led to remarkable insights into how biomolecules respond to external forces. In conceptually the simplest experiment, a single molecule is connected by a flexible polymer linker to a bead, say a micrometer in diameter, that is trapped in the focus of a laser beam. The position of this beam is adjusted so that the force on the construct is kept constant (see Fig. 1*A*). Suppose the molecule of interest can exist in folded and unfolded states, and that the applied force is chosen so that the populations of these states are about equal. Then the total extension hops between short (i.e., folded) and long (i.e., unfolded) values (see Fig. 1*B*). From such trajectories, one can extract the folding and unfolding rates. These observables are commonly analyzed using the phenomenological Bell–Evans model (1, 2), which ignores the possible influence of the “apparatus” that we define here as both the linker and the bead. The same is true of the simplest microscopic descriptions of force-induced transitions, where the dynamics is described as diffusion along the molecular (not total) extension, in the presence of a force-dependent free energy profile (3⇓⇓⇓–7). However, such a description would be rigorously valid only when the response of the apparatus is much faster than the fluctuations of the molecular extension. In reality, this is far from being true, and, in principle, the observed transition rates could be artifacts due to the slow response of the pulling device.

The influence of the apparatus has been considered by a number of groups, using a variety of approaches (8⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–28). The conclusions vary from one extreme, where the effect is negligible when the linkers are sufficiently “soft,” to the other, where the observed rates have little to do with the dynamics of the molecule of interest. In this paper, we will provide a quantitative answer to the question of how much the measured rates differ from what would be observed if force could be applied directly to the biomolecule (i.e., in the absence of linkers and beads/tips). We denote this rate by

The key feature of this model is that the diffusion coefficient along the molecular extension can be much larger than that along the measured extension. As a result, there are now three timescales in play (see Fig. 1*A*). The first describes the fluctuations of the molecular extension denoted here by

## Results and Discussion

### Constant Force Experiments.

The simplest free energy surface that describes the force-clamp experiment is*q* and *x* are the measured and molecular extensions, respectively, *l*, and *F* is the applied force. Such a free energy surface is shown in Fig. 2 for a force chosen so that the population of the folded and unfolded states are equal (

The simplest model of the dynamics is 2D diffusion on *q*, is governed by the diffusion of the mesoscopic bead that is typically in the range of ^{2}/s. For an unfolded protein, ^{2}/s (29, 30). When the response of the apparatus is so slow that many conformational transitions occur before the measured extension *q* changes (technically in the *q*, *q*, and then taking the logarithm of this distribution. Applying this procedure to the trajectory in Fig. 1*B*, one obtains the blue curve in Fig. 2. The corresponding rate, denoted by

In the opposite limit, when *x*, with diffusion constant **2** (26, 33). Interestingly, for a harmonic linker, this problem is the same as finding

We are now faced with the dilemma that the measured rate, denoted here by *q* is much faster than that along the “hidden” molecular extension *x*, i.e., *λ* is the positive root of **1**) is given in Eq. **16**. For our purposes, it is sufficient to know that when *molecule is stiffer than the linker*, then

This analysis seems to suggest that the measured rate **4**, which depends on both *q* (blue curve in Fig. 2), and is proportional to

These results provide a practical way of determining whether the rate extracted from the measured trajectory along the total extension, *q*, is close to the molecular rate and, hence, not an artifact. To do this, one must estimate *q* in a single conformational state (either in the folded or unfolded well). This can be found by first calculating the autocorrelation function

In this regime, where *q*. Thus, one cannot estimate a molecular diffusion constant by equating the Kramers rate corresponding to *x* at fixed *q*) between two linker free energy profiles corresponding to the folded (*f*) and unfolded (*u*) states of the molecule. These profiles as a function of *q* are approximately given by *q* are determined by the relaxation of the system after a hop occurs, and their duration is governed by

If **7** to estimate *q* in the state from which the transition occurs [i.e., *x*. However, when *x*. This has been previously discussed (26) in the context of deconvoluting the results of molecular simulations of a leucine zipper.

We now illustrate and validate the above theoretical considerations by using results from Brownian dynamics simulations of anisotropic diffusion on a 2D surface. The details are given in *Materials and Methods*. The 2D free energy surface is shown in Fig. 2 for the force *q* is shown in Fig. 1*B*. The barrier height of the 2D surface is *x*, *q*, **16**) and the analytical expression from Eq. **6**, calculating **5** is valid for a wide range of

Thus, when the ratio of the linker and molecular force constants is small, the observed rate is essentially the same as the molecular rate, as long as *q*,

An alternative way of estimating the molecular diffusion coefficient, *Inset*, typical transition paths along *q* are compared with a true, but experimentally inaccessible, transition path along *x*. When

### Constant Velocity/Trap Separation Experiments.

We now consider two kinds of experiments on constructs containing a single molecule where the measured force fluctuates. The first kind generates a “hopping” trajectory qualitatively similar to Fig. 1*B*, but now it is the position of the laser trap that is held constant (39). The corresponding rate, which we will denote by

The simplest free energy surface appropriate for these two experiments is*z* is held fixed in the trap experiment and moved at constant velocity *v* in the ramp experiment with *q*, unless

In these experiments, the instantaneous force *q* fluctuates. Thus, care must be taken in defining the rupture force in a force-ramp experiment where *z* moves with a constant velocity. The rupture force is commonly found from the extrapolated average force at rupture. To closely reflect the way the force is obtained experimentally, we define the rupture force as**9** and **10** with respect to time, it can be shown that the force loading rate is

To find *Materials and Methods* that**11** with

If the molecular curvatures are large, then **12** can be rewritten in terms of the loading rate as**12** and **13** are key results of this paper that quantify the difference in the apparent force-dependent rates in constant force (**12** and **13** is on the scale of nanometers, one expects this activation free energy to be less than

## Concluding Remarks

In this paper, we have quantitatively shown how the measured rates are influenced by the ever-present linkers and the relatively slow dynamics of a bead or a cantilever in a single-molecule pulling device. For an experiment at constant force, the molecular (**7**. For soft linkers and slow conformational changes, the difference in the two rates is negligible. However, for “fast-folding” proteins or when an anharmonic (e.g., worm-like chain) linker is fully extended, substantial deviations may arise. In constant force experiments on polyprotein constructs, in addition to the familiar statistical factors, the rates also increase with the number of unfolded units, because the linkers become softer. This effect would cause nonexponentiality in the aggregated lifetime distribution obtained by lumping together times from different modules.

In Eq. **12**, we have also established the relationship between the force-dependent rate

In this paper, we have made the simplifying assumption that the molecular free energy, along the extension, has a barrier. When this is not the case, the molecular free energy surface for a system with metastable states must itself be multidimensional (11, 12, 19, 40). As long as this surface has two basins, our analysis can be readily extended, leading to essentially the same conclusions. Finally, it should be emphasized that the molecular rates considered here are those that would be observed in the hypothetical case where forces could be directly applied to the molecule (i.e., without linker and pulling device). The problem of whether such rates, when extrapolated to zero force, agree with the rates obtained, say, in bulk denaturation experiments, is beyond the scope of this paper.

## Materials and Methods

### Kramers and Langer Theory.

For a 1D metastable free energy profile

where **4**. In one dimension, **4** reduces to Eq. **14**.

### Constant Force Experiments.

Let us now use Langer theory to calculate the rate corresponding to the 2D free energy surface given in Eq. **1**. The minima (

Combining these equations, one sees that **14**. If we assume that *l*, as expected for a polymer linker, it follows from Eq. **15b** that **14**. Calculating the Hessians and then solving for *λ*, one can relate

with **5**, which breaks down if the diffusion is sufficiently anisotropic (35). For stiff linkers, **16** is accurate for all values of the diffusion anisotropy, and *k*_{L}→ σ *k*_{M}/(γ − 1) as σ → 0. In the limits of small *D*_{q} or large *D** _{x}*,

*k*

_{L}hence becomes proportional to

*D*

_{q}instead of the molecular diffusion coefficient

*D*

_{x}.

### Constant Velocity/Trap Separation Experiments.

Let us now calculate the rate for the 2D surface given in Eq. **8**. using Langer’s formula from Eq. **4**. The extrema are solutions of Eq. **10**. Now, because we defined the rupture force *F* in terms of **9**), it follows from Eqs. **10** and **15** that the minima for the force-ramp free energy surface are identical to those in the constant force case (**9** can be rewritten as **10** at the maxima can be written as

Let us assume that the maxima at constant velocity are close to those at constant force, *δ* to first order. Then, using Eq. **15** at

These linear equations can be solved for ε and *δ*, which will then be used to relate the activation barrier to the molecular one. Here, the free energy of activation is *δ* and

which turns out to be remarkably simple. Here, **11**, with *δ*. Finally, we evaluate the determinants, and solve for *λ* when **4**, we recover Eq. **12**.

### Free Energy Surface.

Eq. **1** describes the 2D free energy surface for force-clamp experiments. Fig. 2 shows an example of this surface,

### Brownian Dynamics Simulations.

We performed anisotropic Brownian dynamics simulations, using displacements

of molecular and total extensions *x* and *q*, respectively. *B* shows an example of the measured extension as a function of time when

## Acknowledgments

P.C. and G.H. were supported by the Max Planck Society. A.S. was supported by the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases of the National Institutes of Health.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: pilar.cossio{at}biophys.mpg.de, gerhard.hummer{at}biophys.mpg.de, or attilas{at}nih.gov.

Author contributions: P.C., G.H., and A.S. designed research, performed research, analyzed data, and wrote the paper.

Reviewers: M.H., Case Western Reserve University; and E.P., University of Leeds.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1519633112/-/DCSupplemental.

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