Theory of Rapid Force Spectroscopy

ARTICLE

Received 22 Jan 2014 | Accepted 19 Jun 2014 | Published 31 Jul 2014 DOI: 10.1038/ncomms5463 OPEN Theory of rapid force spectroscopy

Jakob T. Bullerjahn1, Sebastian Sturm1 & Klaus Kroy1

In dynamic force spectroscopy, single (bio-)molecular bonds are actively broken to assess their range and strength. At low loading rates, the experimentally measured statistical distributions of rupture forces can be analysed using Kramers’ theory of spontaneous unbinding. The essentially deterministic unbinding events induced by the extreme forces employed to speed up full-scale molecular simulations have been interpreted in mechanical terms, instead. Here we start from a rigorous probabilistic model of bond dynamics to develop a uniﬁed systematic theory that provides exact closed-form expressions for the rupture force distributions and mean unbinding forces, for slow and fast loading protocols. Comparing them with Brownian dynamics simulations, we ﬁnd them to work well also at intermediate pulling forces. This renders them an ideal companion to Bayesian methods of data analysis, yielding an accurate tool for analysing and comparing force spectroscopy data from a wide range of experiments and simulations.

1 Universita¨t Leipzig, Institut fu¨r theoretische Physik, 04103 Leipzig, Germany. Correspondence and requests for materials should be addressed to K.K. (email: [email protected]).

NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463

n the realm of soft matter, structural stability often hinges on minimum and thus remains stable under weak perturbations. Its delicate intermolecular bonds that are easily broken apart thermal fluctuations can be represented, within an effective Ithrough external forces. The resulting malleability is char- picture, by those of a Brownian particle trapped in some one- acteristic of biological matter on practically all scales from single dimensional binding potential U(x), with the ‘reaction coordinate’ proteins1–3 over mesoscale cellular scaffold structures4 and x typically corresponding to the distance between two binding individual cells5 to whole tissues6. Thanks to the development partners. The function U(x) should have a stable minimum at of a variety of nanomanipulation methods, intermolecular x ¼ 0, surrounded by a region of attraction extending to some interactions can nowadays be investigated on a single-molecule finite coordinate xb40 beyond which U(x) either vanishes or level using a number of techniques commonly referred to as turns repulsive. As soon as the particle has left the basin of 7–12 ‘dynamic force spectroscopy’ , allowing the experimentalist to attraction, x(t)4xb, the bond can be considered broken. isolate single binding sites and probe their strength by quickly Excluding the possibility of quantum-mechanical tunnelling, and reliably inducing hundreds or thousands of unbinding events. which is negligible on a macromolecular scale, the dissociation The wealth of stochastic unbinding trajectories thus obtained is process is a tug of war between thermal and potential forces. The routinely analysed through a schematic but effective description thermal forces drive the bond coordinate x diffusively out of the 0 of molecular bonds in terms of the attraction range xb and the bound state and the deterministic force F(x) ¼ÀU (x) drags it activation energy E of the binding potential. To extract back to the origin. On a probabilistic level, the dissociation rate of quantitatively useful predictions from this simple model, any an ensemble of initially bound particles can then be deduced from theory of unbinding kinetics needs to provide a reasonable an associated Fokker–Planck equation27, a partial differential approximation to the underlying molecular dynamics. At the equation for the time-dependent distribution of particles that is relatively low loading rates that were conventionally realized in mathematically exact, but cannot be solved in closed form experiments, any transient effects arising from the finite without the help of further simplifying assumptions. In most relaxation time of the bond itself can be neglected. This has cases of practical interest, the activation energy E is large allowed for the development of a range of analytical theories of compared with the thermal energy scale, E44kBT. Dissociation forced bond breaking that greatly facilitate the analysis and then becomes a rare event, leaving sufficient time for any interpretation of dynamic force spectroscopy data13–18.In transients caused by the fast microscopic bond dynamics to die contrast, detailed molecular dynamics simulations2 usually out. In the steady-state situation that ensues, an accurate operate in the opposite limit of very high loading rates, where analytical solution of the Fokker–Planck equation is possible26, bond breaking becomes essentially deterministic19. Recently, also yielding a time-independent unbinding rate k that scales single-molecule assays are increasingly resolving the hitherto exponentially in the activation energy, kpexp( À E/kBT). elusive rapid dynamics of bond breaking and accompanying This provides a natural starting point for the analysis of force- macromolecular conformational changes, for example, in protein induced bond rupture. An external pulling force F acting along unfolding upon rapid loading20, or taut DNA recoil21 and the reaction coordinate x ‘tilts’ the internal binding potential U, supercoiling22. This provides a strong incentive to also push the which lowers the effective free energy barrier and increases the 24 existing theories to higher loading rates. Moreover, in spite of the unbinding rate. Within the Bell model , the reaction distance xb experimental progress and improved computational abilities23, is assumed to remain fixed under the external force, such that matching the loading rates used in experimental and simulation the force-dependent unbinding rate immediately follows as 20 studies remains challenging , making the development of a k(F) ¼ k(0)exp(Fxb/kBT). However, reaction rates are not unified analytical theory of dynamic force spectroscopy, covering measured directly in single-molecule experiments, where actual both fast and slow loading rates, all the more desirable. unbinding times may follow a broad statistical distribution. Also, To this end, we derive in the following a probabilistic theory of external forces are usually not constant in time. In a typical dynamic force spectroscopy in the spirit of Bell24, Evans & experiment, molecular bonds are pulled apart using retracting Ritchie13 and Dudko, Hummer & Szabo14 that becomes exact at actuators such as AFM cantilevers that exhibit Hookean high external forces (or, equivalently, high loading rates) and stretching behaviour and therefore exert forces growing linearly reduces to established results at low loading rates. We provide with their retraction distance. Assuming an external force F(t) explicit, closed-form analytical results for the most common that steadily increases in time, Evans & Ritchie13 derived from the experimental loading protocols. They agree with exact numerical force-dependent unbinding rate k(F) a general expression for the simulations of the microscopic bond model for all loading rates, experimentally measured distribution p(F) of rupture forces, save for a narrow region at the crossover from diffusion- Z dominated to deterministic dynamics. This makes them an ideal kðFÞ F kðf Þ : 25 pðFÞ¼ _ exp À _ df ð1Þ companion to Bayesian methods and a natural choice for the FðFÞ 0 Fðf Þ analysis of spectroscopy experiments and simulations alike. Moreover, we show that these results constitute the lowest- Here, the force dependence of the loading rate F_ is optional, order approximation to a rigorous mathematical formulation of but can be used to take into account nonlinear loading protocols escape kinetics, which opens the way of their systematic extension or nonlinearly elastic force transducers such as polymer to higher precision. linkers28,29. For a power-law binding potential and a cusp- shaped binding potential of finite range, Evans & Ritchie were furthermore able to improve upon the phenomenological Bell Results model by accounting for force-induced shifts in the reaction Theory. Molecular binding and unbinding transitions lie at the distance xb. Focusing on a linearly increasing external force, heart of every chemical reaction and have thus been thoroughly Dudko, Hummer & Szabo later derived analytic results for cusp- investigated long before the advent of single-molecule manip- shaped19 and linear-cubic30 binding potentials. These could ulation techniques. In 1940, Kramers laid down a comprehensive, finally be condensed into a uniform expression for the rupture analytically tractable theory of chemical reaction rates26 that has force distribution and its first two moments that contained a new since then become synonymous with reaction rate theory and can fit parameter to smoothly interpolate between different potential still be considered state of the art for most applications. The basic shapes14,15. A recent crop of theories further fleshed out the free idea is that a molecular bond corresponds to a local free energy energy landscape by taking into account the force fluctuations

2 NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463 ARTICLE pertaining to stiff actuators16–18 (that is, actuators with a spring (potentially time-dependent) generalized unbinding rate k(t), constant k\E=x2). These theories nowadays provide the b kðtÞ¼ ÀS_ðtÞ=SðtÞ: ð5Þ standard tools used to convert rupture force histograms obtained from force spectroscopy experiments into estimates of In principle, we could now solve equation (3a) (or an the underlying binding potentials. equivalent integral formulation31,32) numerically33 to obtain the From a dynamical point of view, these approaches still rely on a time-dependent probability distribution W(x, t), calculate the strong separation between the short timescale of internal bond survival probability S(t) via equation (4) and feed the result into dynamics and the long timescale of bond dissociation. Although equation (5) to precisely compute the unbinding rate k(t) for a this approximation is well justified in the original Kramers known set of model parameters (E, xb, D). This procedure is, theory26 of a force-free bond, it is bound to break down as soon however, too computationally intensive for the inverse problem of as the external loading becomes rapid enough to effectively flatten obtaining E, xb and D from experimental data. To arrive at an out the energy barrier before rupture occurs. In that case, analytical approximation to the unbinding rate that is quickly unbinding becomes essentially deterministic as the bond is swept evaluated on a computer, we write down a first approximation to away ballistically by the external force. As shown by Hummer & W by ignoring the absorbing boundary, 19 ÂÃ Szabo for a harmonic cusp potential, one can accurately predict @ W ðx; tÞ¼ D@2 À @ Aðx; tÞ W ðx; tÞ; ð6aÞ the mean rupture force at high loading rates by neglecting t G x x G stochastic fluctuations altogether and afterwards obtain a global WGðx !Æ1; tÞ¼0: ð6bÞ approximation to the mean rupture force by manually interpolating between the high-force result and a conventional With the external driving potential V(x, t) in equation (3c), theory that accounts for the low-rate regime. which is at most parabolic, WG is strictly Gaussian for any In the following, we derive a probabilistic theory of forced localized initial distribution WG(x, t ¼ 0) ¼ d(x À x0). It is easily unbinding that includes fluctuations even at high pulling speeds evaluated analytically, 2 and becomes exact both in the limits of high and low loading wE½x À xðtÞÀx0CðtÞ expðÀ k Tx2½1 À C2ðtÞ Þ rates. Already the leading order results of this theory surpass W ðx; t jx Þ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB b ; ð7Þ G 0 2 2 previous work; moreover, it provides a systematic route for pkBTxb½1 À C ðtÞ=wE further improvement. We start with the same cusp-shaped where xtðÞ follows from a purely deterministic equation of binding potential as Hummer & Szabo, motion, 2 Eðx=xbÞ ; xoxb k T dxðtÞ 2E UðxÞ¼ : ð2Þ ; B @ ; ; À1; otherwise xð0Þ¼0 ¼À 2 x À xVðx tÞjx¼x ð8Þ D dt xb Formally, the potential becomes negatively infinite at x ¼ xb, C(t) denotes the positional autocorrelation function, corresponding to an absorbing boundary condition (thus À 2wDEt=k Tx2 ignoring rebinding events, which quickly become negligible CðtÞ¼e B b ; ð9Þ 1 14 under external load ). Although it can be argued that a and w the relative change in the system’s overall spring constant linear-cubic potential constitutes a more faithful approximation introduced by the externalÀÁ actuator; that is, w ¼ 1 for external to many real binding potentials, we will show in the following 2 2 fields or soft springs k E=xb , w ¼ 1 þ kxb=2E otherwise. that the practical difference between both models is often Going back to the time-dependent escape rate equation (5), we insignificant; what we lose in generality is more than made up note that the computation of S_(t) via equation (4) would be for by what we gain in terms of analytical tractability. needlessly complicated. Since escape events can only take place at To set the stage, we first summarize some basic definitions. The the absorbing boundary x , the outflow of bound particles into time-dependent probability W(x, t)dx to find any particle in the b the unbound state is equivalent to the probability flux j(xb, t) ¼ ensemble between x and x þ dx follows from the Fokker–Planck À S_(t) across x . It only depends on local properties of the equation27 b ÂÃ probability distribution W, via the continuity equation @ Wðx; tÞ¼ D@2 À @ Aðx; tÞ Wðx; tÞ; ð3aÞ t x x jðxb; tÞ¼Aðxb; tÞWðxb; tÞÀD@xWðx; tÞjx¼xb : ð10Þ

Replacing W by WG in equation (10) defines the corresponding Wðxb; tÞ0; ð3bÞ R t ; 0 0 approximate flux jG, survival function SG ¼ 1 À 0 jGðÞxb t dt , D E and escape rate kG(t) ¼ jG(xb, t)/SG(t). Aðx; tÞ À @xVðx; tÞþ2 x : ð3cÞ Note that the Gaussian expressions become asymptotically k T x2 B b exact for pulling forces greater than the critical force needed to Here, D denotes the diffusion coefficient and V(x, t) an external flatten out the energy barrier, F44Fc ¼ 2E/xb (corresponding to _ 2 pulling force potential that may either represent some prescribed high loading rates F 44 DFc=xb in a constant-rate experiment), force protocol F(t)ÀqxV(x, t) or a moving external spring, if due to an essentially ballistic decay of the bound state. Intuitively, V(x, t)k[x À y(t)]2/2. The spatially integrated distribution the probability density W does not have time to ‘sense’ (and Z G xb react to) the presence of the absorbing boundary before it is SðtÞ¼ Wðx; tÞdx ð4Þ pulled across xb by the deterministic drift term A(x, t)WG(x, t). À1 For the escape rate, we thus find the exact asymptotic limit is the proportion of bound particles or ‘survival function’ at any F_ !1 given time t. A constant survival function S(t) ¼ const corre- kðtÞkGðtÞ¼jGðxb; tÞ=SGðtÞ: ð11Þ sponds to zero escape events and thus a vanishing unbinding rate The situation is slightly more complicated for slow loading, k ¼ 0. Likewise, a positive escape rate k40 amounts to an where deterministic driving and stochastic dynamics interfere. exponentially decaying survival function S(t) ¼ S(0)exp( À kt). In Consider first the limiting case of a freely diffusing particle, that the general case, S(t) may be neither constant nor exponential, is, if the binding potential, the external forcing and the absorbing but an arbitrarily complex function of time. However, it is always boundary condition are neglected. There is a purely diffusive net 0 monotonously decaying and nonnegative and thus defines a flux jdiff: ¼ÀDWGðÞxb; t at the position xb, half of which is

NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463 randomly backscattered (see Fig. 1). It is suggestive that inserting obtain the closed analytical result a perfect absorber at x ¼ xb would suppress the backscattering ð Þ 2 k0 F 2x F Eð1 À½1 À xðFÞ=xb Þ=kBT and therefore double the diffusive ﬂux across xb. Adding the pðFÞ 1 þ À e F_ Fc xb deterministic drift back in, we get hið17Þ 2 kBTk0 Eð1 À½1 À F=Fc Þ=kBT F_ !0 Ã Âexp À e À 1 ; jðxb; tÞj ðxb; tÞð12aÞ Fx_ b

where k0 is the associated force-free Kramers rate for the cusp Ã j ðxb; tÞAðxb; tÞWGðxb; tÞÀ2D@xWGðx; tÞjx¼xb : ð12bÞ potential, rffiffiffiffiffiffiffiffiffiffiffi This expression for the flux can indeed be understood as the 2DE E k ¼ e À E=kBT ; ð18Þ first-order approximation to an exact theory, as discussed in the 0 k Tx2 pk T Ã B b B Methods section. There we also show that j (xb, t) provides an asymptotically exact expression for the escape rate at low loading and xFðÞis the force-dependent coordinate of the maximum of W (x,F), rates, G _ hi _ F kBTFxb À 2DEF=k T Fx_ 2 F!0 Ã B b : kðtÞj ðxb; tÞ=1: ð13Þ xðFÞ¼xb À 2 1 À e ð19Þ Fc DFc

We now observe that the additional factor of two in front of In the limit of small loading rates we obtain xðFÞFxb=Fc Ã the diffusive component in j compared with jG is irrelevant and equation (17) reduces to the expression derived by Dudko, for the limit of large pulling rates, which is dominated by Hummer & Szabo (DHS) in ref. 14 for n ¼ 1/2. In the Methods the deterministic drift term A(xb, t)WG(xb, t), suggesting that section, we provide a compilation of analogous results for stiff Ã equation (11) still holds with jG replaced by j . Moreover, note force transducers, arbitrary driving protocols or arbitrary initial that for high energy barriers E44kBT and low pulling rates conditions W(x, t ¼ 0) (Table 1), as well as a closed expression for _ 2 F oo DFc=xb, the Gaussian survival probability SG(t) remains the mean rupture force /FS. close to 1 since WG extends only negligibly beyond xb for all Some remarks about the comparison of data and theory may be relevant times t, that is, useful, here. First, note that, in any case, a comparison of data and Z 1 theory for the distribution p(F) is substantially more informative _ ÀðE À FtxbÞ=ðkBTÞ / S 1 À SGðtÞ¼ WGðx; tÞdx ¼ O e 1: than merely fitting F . Yet, every analytical approximation to xb the true rupture force distribution depends on at least three ð14Þ different parameters (binding energy E, attraction range xb and diffusivity D). Direct fits of experimentally obtained rupture force Our two asymptotic results can therefore be combined into the histograms are therefore prone to get trapped in some local unified approximation optimum in parameter space, thus often missing the best possible jÃðx ; tÞ solution. One way to obtain more reliable results is to first kðtÞ b ; ð15Þ aggregate several data sets obtained under different loading rates SGðtÞ and subsequently perform a ‘global’ fit using a single set of which is exact both in the limits of low and high loading rates. parameters only. Conventionally, this is done by either perform- ing a least-squares fit of several histograms at once or by fitting Together with equation (1), this yields the rupture force 13–15,19 distribution the mean rupture force as a function of loading rate . Both Z approaches can be improved upon systematically using a 1 d t jÃðx ; t0Þ maximum-likelihood approach25 that, instead of optimizing for b 0 : pðFÞ exp À 0 dt ð16Þ some arbitrary measure of fit quality (such as the squared F_ ðFÞ dt SGðt Þ 0 residual), employs Bayesian analysis to select those model Evaluating p(F) in its general form equation (16) still requires parameters most likely to underlie the given rupture force some computational effort, since it depends in a potentially histograms. Compared with conventional fitting methods, the complicated way on experimental details such as the driving maximum-likelihood approach straightforwardly extends to protocol, the initial distribution of particles within the bound heterogeneous data sets encompassing different linker state and the stiffness of the force actuator. However, for the stiffnesses or loading protocols. Moreover, it has proven common case of a linearly increasing pulling force F ¼ Ft_ (exerted significantly more robust with respect to small ensemble sizes25, 2 by an external field or a soft spring with stiffness k E=xb)we a potentially crucial advantage in the analysis of real-world

a b

W (x, t) W (x, t)

jdiff. 2jdiff. j j * jdrift jdrift

x b x b

Figure 1 | Flux into an absorber at xb. (a), Given some statistical distribution W(x, t) of particles, these particles can be driven across the position xb, either ballistically by external forces (jdrift) or diffusively through random thermal noise (jdiff.). (b) Inserting an absorber at xb effectively eliminates diffusive Ã backscattering, thus doubling the diffusive contribution to the resultant probability ﬂux j into xb.

4 NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463 ARTICLE experiments. In general, the larger the range of applicability of a formalism, we do not actually absorb particles as they cross the given theory of force spectroscopy, the better the results obtained absorbing boundary at xb but merely count them. After leaving using the maximum-likelihood approach. This fact should work the potential well, they build up a ‘phantom population’ that may to our advantage, as our model covers a much wider range of later produce an unphysical backflow into the bound state. In loading rates than the predominant Bell–Evans ‘standard principle, this deficiency could be rectified, at least for the cusp model’13 and its recent extensions by DHS14 and others16–18. potential, by extending the systematic integral equation approach In the remainder and in the Supplementary Note 1, we illustrate outlined in the Methods to higher orders. Yet, the range of the data fitting procedure with some examples and provide some applicability of equation (16) already covers the vast majority of practical tools and protocols for optimized data analysis. unbinding events. By simply truncating p(F) after its first zero crossing, we obtain an analytical expression for the rupture force distribution that works well at all loading rates, except for a Application to simulation data. Although our theory becomes _ 2 narrow range close to a critical value Fc ¼ DFc=xb. exact in the limits of high and low external loading rates, the Although we find the derivation of an analytical, global quality of our approximation at finite loading rates is a priori approximation to the rupture force distribution p(F) satisfying in unclear. To assess the practical applicability of our method, we itself, its practical utility lies first and foremost in the accurate and have thus generated synthetic rupture force distributions using efficient analysis of experimental unbinding data. To estimate Brownian Dynamics simulations of the underlying microscopic how well our model stacks up against the prevailing methods of bond model (see Methods). As Fig. 2 exemplifies, our theory is as data analysis, we have thus analysed numerically obtained rupture good as the approach by Hummer & Szabo19 in capturing the / S force histograms, see Supplementary Fig. 1. We considered both mean rupture force F as a function of loading rate. Moreover, the cusp-shaped potential and a linear-cubic binding potential whereas the Hummer–Szabo model builds on an athermal (Supplementary Figs 2 and 3), using local and global (maximum- treatment of the underlying microscopic equation of motion, likelihood) fits both of equation (16) and of the rupture force we fully account for thermal fluctuations. These can be significant 14 17 _ 2 distributions derived by DHS and Maitra & Arya (as well as a even within the ‘ballistic’ regime F4DFc=xb, as Fig. 3 and the 20 ‘cusp-optimized’ counterpart to their results, detailed in experimental data by Rico et al. show. Hence, our approach Supplementary Note 2) for soft and stiff linkers, respectively. allows us to perform a complete and systematic Bayesian analysis Furthermore, we emulated conventional experimental procedures (see Methods and ref. 25) of rupture force histograms (as opposed by fitting the mean rupture force F, using both the analytical to mean rupture forces), thus making full use of the available expressions derived by Friddle15, associated to the DHS-model14, experimental data. As apparent from Fig. 3, our approximation and Maitra & Arya17, as well as the numerical interpolation still breaks down at intermediate loading rates, because, in our formula derived by Hummer & Szabo19. In the Supplementary Note 1, the interested reader will find a comprehensive discussion 500 of all these approaches, the gist of which is as follows. At high loading rates, unsurprisingly, our theory proves superior to existing probabilistic theories13,14,16,17. More importantly, 400 〉 however, the combination of the maximum-likelihood analysis F

〈 and our globally valid approximation is powerful enough to

log compete with the best conventional models even for a linear- 300 cubic binding potential while requiring neither prior knowledge

(pN) of the potential shape nor additional fit parameters. The only 〉 F 〈 200 other global model to date, the numerical interpolation formula obtained by Hummer & Szabo, yields similarly good results as our own approach if supplied with sufficient data, although the 100 Current experimental limit increased computational cost of numerical integration leads to longer fitting times. Using a smaller range of loading rates (that is, 3–6 decades in F_ instead of 12), the higher information content of 100 102 104 106 108 our full-histogram description can produce significantly better results. Moreover, as our approach inherits the greater generality F˙ (pN s–1) and robustness of the Bayesian maximum-likelihood analysis, this Figure 2 | Mean rupture force /FS as a function of loading rate. Circles: gap is bound to widen with small ensemble sizes or heterogeneous /FS as determined from our numerical simulations (see Fig. 3). Crosses: data sets. Since the maximum-likelihood method is somewhat 15 2 À 1 best DHS fit (E ¼ 9.49 Â kBT, xb ¼ 1.01 nm, D ¼ 574 nm s ). Triangles: more difficult to implement in practice than traditional methods, 19 2 À 1 best Hummer–Szabo fit (E ¼ 10.2 Â kBT, xb ¼ 1.0 nm, D ¼ 1015 nm s ). we wrote (and provide for free use) a ready-made Mathematica Squares: /FS determined from p(F) (see Methods section), using the fit notebook that covers some of the most common force parameters obtained in Fig. 3. Pentagons: asymptotic analytical spectroscopy setups, see Supplementary Data 1. approximation equation (40) to /FS, using the fit parameters obtained in Fig. 3. Inset shows the same data in double-logarithmic coordinates. The mean rupture force converges onto the Hummer–Szabo19 asymptote F_ 1/2 at Discussion 2 Our theory provides an analytically tractable generic model of large loading rates, F_ DFc=x , and onto the logarithmic DHS asymptote b dynamic force spectroscopy that reproduces previously known, /FSB[ln F_]1/2 (ref. 14) at intermediate loading rates, k Tk =x F_ B 0 b accurate results14,17 at low loading rates and improves upon the DF =x2 (dashed red lines, shifted upwards for better visibility). In the limit c b results derived by Hummer & Szabo for high loading rates19 by _ F-0, external forces are too small to induce rupture, yielding as the including stochastic fluctuations. It provides excellent results measured rupture force the force at the time of spontaneous unbinding, also at intermediate loading rates, thus constituting an ideal / SB_ F F/k0. For our choice of system parameters, the best currently companion to Bayesian methods of data analysis. Apart from available experimental setup20 would already be able to access the ballistic serving as a systematic and convenient replacement for current regime where /FS increases with F_ 1/2. theories of force spectroscopy, our approach may prove crucial to

NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463

a b c 100 104 107 101 105 108 102 106 109 103 107 1010 4 11 ) 10 10 F ( p

0 10 20 30 0 100 200 300 102 103 104 F [pN] F [pN] F [pN]

Figure 3 | Analytical theory compared with simulations. Using a single set of model parameters, our theory (equation (17), solid lines) provides an 2 accurate global approximation to both slow (a) and fast (c) external force protocols (as compared with the intramolecular relaxation timescale, xb=D ¼ _ 5 À 1 1 ms in our case), apart from a narrow range (b) close to a critical loading rate (FcE10 pN s for our choice of parameters). The ‘experimental’ rupture 2 À 1 force histograms have been generated by direct stochastic integration (see Methods section), using E ¼ 10 Â kBT, T ¼ 300 K, xb ¼ 1 nm, D ¼ 1,000 nm s _ 11 À 1 2 À 1 and F ¼ 1y10 pN s . Our best-fit parameters obtained with equation (17) are E ¼ 10.15 Â kBT, xb ¼ 0.98 nm, D ¼ 976 nm s . Since the mean rupture force varies by orders of magnitude at large loading rates, we use double-logarithmic scaling for F_4107 pN s À 1 (c). the analysis of future high-speed force spectroscopy setups and Inserting this expression into equation (21b) and then integrating over the interval ( À N, x ] results in full-scale MD simulations. Finally, we note that at low enough b Z x ð22Þ @ b loading rates, the effect of a polymeric force transducer onto the jðx ; tÞ¼À Wðy; tÞdy b @ resultant rupture force distributions can be approximated by t À1Z t _ 28,29 ð21bÞ choosing F(t) and F(t) accordingly . At higher loading rates, ¼ jGðxb; tÞÀ jGðxb; t jxb; tÞjðxb; tÞdt ð25Þ the dynamics of force propagation within the polymer21,34,35 may 0 Z 1 xb become relevant, which could be an interesting subject for future þ jðxb; tÞÀ sðy; tÞdy: theoretical developments based on our theory. 2 À1 We now want to determine the sink term, s(x,t). We shall therefore make the Methods ansatz Alternative derivation of equation (12b). The (non-Gaussian) probability dis- sðx; tÞ¼ À2gðtÞWðx; tÞdðx À xbÞ; ð26Þ tribution function W(x, t) is, in the presence of an absorbing boundary at x ¼ xb,in general not analytically tractable. However, it is related to its Gaussian counterpart which, by introducing the effective flux c(xb, t) ¼ jG(xb, t) þ g(t)WG(x, t), reduces 32,33 WG(x, t) in a simple manner , equation (25) to Z Z t t jðx ; tÞ¼2cðx ; tÞÀ2 cðx ; t jx ; tÞjðx ; tÞdt ð27Þ Wðx; tÞ¼WGðx; tÞÀ WGðx; t jxb; tÞjðxb; tÞdt; ð20aÞ b b b b b 0 0 Equation (27) was originally derived by Buonocore et al. in ref. 37, where it was Wðx xb; tÞ¼0; ð20bÞ noted that the function g(t) can be uniquely determined using 0 where j(x, t) is the associated probability flux, satisfying the continuity equation lim cðxb; t jxb; t Þ¼0; ð28Þ t0!t @Wðx; tÞ @jðx; tÞ ¼À ; ð21aÞ @t @x a criterion that follows from the fact that for all t, with the boundary conditions W(x, t)|x- À N ¼ W(xb, t) ¼ 0. The Gaussian WGðx; t j xb; tÞ¼dðx À xbÞ: ð29Þ distribution WG(x, t) and flux jG(x, t) also satisfy the continuity equation (21a) for 36 W (x, t)| - N ¼ 0. Alternatively , the absorbing boundary condition in Solving equation (28), we obtain g(t) ¼ÀA(xb, t)/2 (ref. 38) and thus G x ± equation (20b) can be interpreted as a sink term s(x, t)o0 8t, resulting in the @ ; 1 WGðx tÞ modified equation cðxb; tÞ¼ Aðxb; tÞWGðxb; tÞÀD x¼xb : ð30Þ 2 @x @Wðx; tÞ @jðx; tÞ ¼À þ sðx; tÞ; ð21bÞ Formally, equation (27) can be solved iteratively, @t @x Z t where W(x, t)|x-±N ¼ 0. Using the fact that the flux vanishes at infinity, jðx ; tÞ¼2cðx ; tÞÀ4 cðx ; t jx ; t Þcðx ; t Þdt Æ ... - b b b b 1 b 1 1 j(x À N, t) ¼ 0, we integrate equation (21a) to obtain Z 0 Z Z t t1 @ xb ÀðÀ2Þn þ 1 cðx ; t jx ; t Þ cðx ; t jx ; t Þ ð31Þ Wðy; tÞdy ¼Àjðx ; tÞ; ð22Þ b b 1 b 1 b 2 @t b Z 0 0 À1 tn À 1 _ Â cðx ; t jx ; t Þjðx ; t Þdt ... dt dt ; or equivalently, S(t) ¼Àj(xb, t). b n À 1 b n b n n 2 1 Taking the time derivative of equation (20a) gives 0 Z t t t ? t t -N @ ; ðÞ@ ; where 0o no n À 1o o 1ot and j(xb, n) vanishes as n . Truncating the Wðx tÞ 20a WGðx tÞ d 0 ¼ À WGðx; t jxb; tÞjðxb; tÞdt above result after the first term yields a first-order approximation j (x , t)tothe @t @t dt b Z 0 flux that coincides with equation (12b), @ ; t @ ; t jGðx tÞ jGðx t jxb Þ 0 À þ jðxb; tÞdt ð23Þ jðxb; tÞj ðxb; tÞ2cðxb; tÞ @x 0 @x ¼ Aðxb; tÞWGðxb; tÞÀ2D@xWGðx; tÞjx¼xb ð32Þ À WGðx; t jxb; tÞ jðxb; tÞ; |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} Ã ¼ j ðxb; tÞ: ¼dðx À xb Þ which can be compared with equation (21a), leading to Z t Asymptotic exactness at low loading rates. Our escape rate k(F) and rupture @jðx; tÞ @jGðx; tÞ @jGðx; t jxb; tÞ ¼ À jðxb; tÞdt þ dðx À xbÞjðxb; tÞ: ð24Þ force distribution p(F) both reduce to the results of DHS (equations 3 and 4 in @ @ @ _ - Ã x x 0 x ref. 14 for n ¼ 1/2), in the limit F 0. The flux j (xb, F) in equation (12b) is a sum

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and computed from pÃ(F) the mean rupture force via a V (x, t ) R F (t ) 1 FpÃðFÞdF hiF ¼ R0 : ð36Þ 1 pÃðFÞdF (x/x )2 0 b Alternatively, we can (for constant F_ and m ¼ 1) provide a rough analytical 〈 〉 x slow estimate that is exact in the limits of small and large loading rates, respectively, Z t 1 hiF ¼ fpðf Þdf hFi 2 þhFi 2 : ð37Þ F_ DFc =x F_ DFc =x mol. 0 b b (y (t ) − x )2 For large loading rates the mean rupture force evaluates to 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x y (t ) b _ 1 pwEFxb À wbE=2 1 wbE wbE hiF F_ DF =x2 e Â 1 þ I1=4 þ I À 1=4 c b 2 2D wbE 2 2 b 〈 〉 sffiffiffiffiffiffiffiffiffiffi x G wbE wbE 2Fx_ ÀÁ þ I þ I wbE 1 b þ O ðÞwbE À 1 ; 3=4 2 5=4 2 Db ð38aÞ

where In(z) is the modified Bessel function of the first kind. This result coincides with the deterministic mean rupture force originally derived in ref. 19. For small loading rates FooFc, we arrive at a result previously obtained from 0 〈 〉 xb y (t ) 39 x slow the Bell–Evans model by Gergely et al. , qX e E1ðqXÞ hFi 2 : ð38bÞ F_ DFc =x c b wbxb 〈x 〉 G 1/2 _ Here, q ¼ exp(bE[1 À w]), X ¼ w k0/bFxb and E1(z) denotes the exponential integral, Z 1 e À s E1ðzÞ¼ ds ; ð39Þ z s Inserting equation (38a) and (38b) into equation (37), we thus end up with sffiffiffiffiffiffiffiffiffiffi qX _ 0 xb 〈 〉 y (t ) e E1ðqXÞ 2Fxb x slow hFi þ : ð40Þ wbxb Db Figure 4 | Measuring rupture forces with a stiff transducer. (a) As long It should be noted that we provide /FS only for the sake of completeness. In as the bond remains intact, the combined bond-transducer system can be fact, it is wasteful to discard experimentally measured force fluctuations instead of seen as two harmonic springs connected in series. At low pulling speeds, directly fitting the full histograms p(F) and we thus advise against the use of equation (40) (as much as any other available predictions for /FS) for analysing positional fluctuations within the bound state translate into force the experimental data. fluctuations that can be smoothed out via a time-moving average (see inset). (b) As long as the effective free energy barrier is still large compared Simulations. To generate rupture force histograms across a large range of external with kBT, the probability distribution W(x, t) closely approximates a loading rates, we have used a stochastic Euler scheme to directly integrate the Gaussian centred within the bound state and /xSslow(t) virtually coincides equivalent Langevin equation for the Fokker–Planck equation (3a), with /xS (t). ( ) At high pulling forces, the static force-balance argument 0 G c gx_ðtÞ¼ ÀU ðxÞÀ@xVðx; tÞþxðtÞ; ð41Þ a fails, as it yields an equilibrium position /xS (t) beyond x . The slow b where x(t) denotes Gaussian, white noise, improved approximation /xS (t) instead is always bounded above by x . G b 0 0 hixðtÞ ¼ 0; xðtÞxðt Þ¼2kBTgdðt À t Þð42Þ and g the bond friction coefficient, g ¼ k T/D. External force is applied either of the Gaussian flux jGðÞ¼xb; F ½AxðÞÀb; F D@x WGðÞjx; F and the diffusive B x¼xb through an external field V(x,t) ¼ÀxFt_ (corresponding to a deterministic force) flux jdiff:ðÞ¼Àxb; F D@xWGðÞjx; F x¼x . Due to their asymptotic behaviour, b or through a moving external spring centred at y(t) ¼ yt_ . Initial particle positions 2 2 jGðxb; FÞ OðFx_ =DF Þ; F_ DF =x are drawn from a Boltzmann distribution; once the particle position x(t) crosses xb, b c c b ; ð33aÞ WGðxb; FÞ DbF; otherwise we consider the bond broken and record the corresponding rupture force. Although one might argue that prior knowledge of the barrier position x allowed b 2 us an unrealistic advantage over actual experiments, we note that even beyond the jdiff:ðxb; FÞ wDbF ð1 À F=F Þ; F_ DF =x c c c b ; ð33bÞ critical pulling force, strong variations in the intramolecular binding potential W ðx ; FÞ wDbFc; otherwise G b U around xb manifest themselves in experimentally detectable force signatures, see Supplementary Note 3 and the accompanying Supplementary Fig. 4. with Fc ¼ 2E/xb and F(t) defined according to equation (53) for each limit, respectively, the diffusive flux dominates over the Gaussian flux at low loading rates. Using the expressions found in Table 1, we can determine the escape rate Maximum-likelihood method. Following ref. 25, the likelihood P of a given set of E _ - under the assumption that S(F) 1 for F 0, parameters E, xb, D is computed as follows, YM YN kðFÞj_ = 2 jdiff ðxb; FÞj_ = 2 ÀÁÈÉi F DFc xb F DFc xb PFfgjðÞE; xb; D ; F_ 1; ... ; F_ M ¼ pðFj jðE; xb; DÞ; F_ iÞ; ð43Þ F 2 ð34Þ k w3=2 1 À ebE½1 À wð1 À F=Fc Þ ; i¼1 j¼1 0 F c where {F} denotes the totality of rupture forces taken into consideration and ÀÁ where k0 denotes the Kramers rate (equation 18). For w ¼ 1 (that is, an external 2 pFj jðÞE; xb; D ; F_ i ð44Þ field or a soft spring, k E=xb) the rate above coincides with equation 3 in ref. 14 2 for n ¼ 1/2. For an external spring we have w ¼ 1 þ kxb=2E and obtain equation the rupture force distribution p given by equation (16), evaluated at the measured _ 2 _ _ (3-S) of the Supplementary Note 2. Similarly, for F DFc=xb it can be shown that rupture forces F ¼ Fj, with model parameters (E, xb, D) and F ¼ Fi. The logarithm p(F) reduces to equation 4 in ref. 14 for n ¼ 1/2 in the case of an external field of the right-hand side is then maximized using the ‘NMaximize’ method provided (F(t) ¼ Ft_ ) and to equation (4-S) for springs (y(t) ¼ yt_ ), respectively. by Wolfram Mathematica. Since the maximum-likelihood method is somewhat more involved than conventional fitting procedures, we provide a ready-made Mathematica notebook that allows the user to analyse her data through a simple Mean rupture force. To compute the mean rupture force /FS shown in Fig. 2, graphical interface (Supplementary Data 1). we have eliminated the spurious zero crossings of p(F) at intermediate loading rates _ 2 F DFc=xb by setting Nonconstant loading rates and arbitrary initial conditions. In deriving pðFÞ; pðFÞ40 pÃðFÞ¼ ð35Þ equation (17), we have started from a localized initial condition W(x,t ¼ 0) ¼ 0; otherwise d(x À x0), calculated the corresponding distribution of first passage times and in

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Table 1 | Analytical approximations to the rupture force distribution p(F). Ã 0 Ã 2 wE½x À xFðÞÀx CFðÞ w x xFÃ 2 exp À exp À E½À ðÞ k Tx2 1 À C2ðÞFÃ k Tx2MFðÞÃ B b ½ wE kBT B b General definitions W x FÃ x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , W x FÃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , GðÞ¼; j 2 2 Ã GðÞ; 2 Ã pkBTxb ½1 À C ðÞF =wE pkBTxb MFðÞ=wE "# FRÃ Ã Ã Ã 2wDE Ã Ã xb F F ðÞ0 Ã 1 Ã Ã CFðÞ¼ exp À 2 tFðÞ, xFðÞ¼ À CFðÞÀ CfðÞdf , kBTx w Fc Fc Fc b FÃðÞ0 hi hi _ 2 2E Ã 1 2 Ã Ã wD FÃ Ã wFtðÞ; F DFc=xb Fc ¼ , MFðÞ¼1 þ À 1 C ðÞF , AxðÞ¼b; F À Fc , F ¼ xb m kBT w FtðÞ; otherwise

External ﬁeld setup External spring setup 2 2 V(x, t) ¼ÀF(t)x, w ¼ 1 V(x, t) ¼ k[x À y(t)] /2, w ¼ 1 þ kxb=2E; F(t) deﬁned in equation (53) R x R b Ã Ã F_ Ã FÃ FÃ W0ðyÞ½Aðxb;f ÞÀ2D@x WGðx;f j yÞdy pðFÞÀð Þ @ exp À À1 F_ðFÞ @FÃ FÃð0Þ F_ ÃðfÃÞ R F(t) x 1 F(t) R b Ã Ã À fÃ W0ðzÞ½Aðxb;j ÞÀD@x WGðx;j j zÞdz Â 1 À À1 djÃ dfÃ E FÃð0Þ F_ ÃðjÃÞ x¼xb E

arbitrary initial conditions W0ðxÞ xb xb Arbitrary force F(t) External moving spring, centred at y(t)

R hi _ Ã Ã Ã Ã Ã F ðF Þ @ F Aðxb;f Þ 2wDbFc xðf Þ pðFÞÀ Ã exp À Ã þ 1 À F_ðFÞ @F F ð0Þ F_ ÃðfÃÞ F_ ÃðfÃÞMðfÃÞ xb h hi y(t) R Ã Ã Ã Ã f Aðxb ;j Þ wDbFc xðj Þ ˙ ÂWGðxb; f Þ 1 À Ã þ 1 À F(t) F ð0Þ F_ ÃðjÃÞ F_ ÃðjÃÞMðjÃÞ xb i E À 1 E Ã Ã Ã ÂWGðxb; j Þdj df x x b equilibrium initial conditions W0ðxÞ/expðÀmwUðxÞ=kBTÞ b External moving spring, centred at y(t) Arbitrary force F(t)

ÀÁÂÃ F_Ãx 1 C FÃ 2 b½ À ð Þ À 2 À 1 Aðx ;FÃÞ W ðx ;FÃÞ y(t) p F 1 wFcMðFÃÞ MðFÃÞ b G b yt˙ ð ÞF_ pffiffiffiffiffiffiÂÃ Ã 2 exp wDFc wE e À wEð1 À F =wFcÞ =kBT À e À wE=kBT F_Ãx2 pkBT b hi Ã _ Ã E Ã 2wDE Ã Ã F kBTF xb Ã E CðF Þ¼exp À 2 F ; xðF Þ¼xb À 2 2 ½1 À CðF Þ _ Ã wFc w DF kBTF xb c equilibrium initial conditions W x exp mwU x =k T xb 0ð Þ/ ðÀ ð Þ B Þ xb m¼1 Linear force ramp FðtÞ¼Ft_ ; ! Linearly moving spring, centred at y(t) ¼ y_(t) equation (17)

We provide here closed formulas for various common experimental situations, as well as generic expressions for nonlinear loading protocols and arbitrary initial conditions. All scenarios assume a monotonously increasing (or decreasing) external load, which allows us to unambiguously convert rupture forces into the time domain via the functional inverse t(F) of the time-dependent rupture force F(t) (that is, for a linearly increasing force ramp F(t) ¼ Ft_ , t follows as t(F) ¼ F/F_). Notice that equation (17) is a special case of the bottom entry of this table for w ¼ m ¼ 1.

the end averaged over all possible starting points x0oxb using an equilibrium Stiff force transducers. For stiff transducers, that is, external spring constants \ 2 Boltzmann distribution. This choice of initial condition should be appropriate to k E=xb, the relation between rupture time and rupture force depends to a certain most real-world scenarios, but since our theory readily extends to nonequilibrium degree on the experimental time resolution. Instead of a deterministic time- initial conditions (corresponding, for example, to a particle held fixed by an dependent pulling force F(t), one measures then a force within the transducer that auxiliary trapping potential that is turned off at t ¼ 0), Table 1 provides a generic fluctuates with the bond coordinate x(t). These fast fluctuations are typically rupture force distribution for arbitrary choices of W(x, t ¼ 0)W0(x). Importantly, smoothed out using a time-moving average, which in a conventional low-speed for a thermally stable, smoothly varying binding potential U the initial distribution experiment corresponds to an equilibrium average, with respect to the time- is generally Gaussian around the energy minimum at x ¼ 0, dependent combined potential U(x) þ V(x, t) (molecular bond þ transducer), yielding thus as the measured rupture force ½U00ðÞ0 þ V00ðÞ0; t ¼ 0 x2 W0ðÞx; 0 exp À ; ð45Þ 2kBT F ¼ k½yðtÞÀxðtÞ: ð47Þ but its width may deviate from the value obtained from our cusp-shaped model As long as the effective free energy barrier is still large compared with kBT _ = 2 potential. To better account for other types of binding potentials (such as the often (which can safely be assumed in the limit of low loading rates F DFc xb), the used linear-cubic potential), we can thus treat actual distribution of bound particles is essentially a Gaussian centred within the bound state. Hence, the anharmonicity of the molecular binding potential beyond 00 00 00 00 / S U ðÞþ0 V ðÞ0; t ¼ 0 U ðÞþ0 V ðÞ0; t ¼ 0 xb can be neglected and the equilibrium position x of bound particles follows m 17 00 00 ¼ 2 00 ð46Þ from a simple force-balance argument (see Fig. 4a), UcuspðÞþ0 V ðÞ0; t ¼ 0 2E=xb þ V ðÞ0; t ¼ 0 k k ; as an additional fit parameter, in analogy to the parameter n used by Dudko, mol:hxislowðtÞ¼ ½yðtÞÀhix slowðtÞ ð48aÞ Hummer & Szabo14 (although in contrast to n, our fit parameter m is relevant only _ 2 at high loading rates F DFc=xb since at low loading rates the memory of the kyðtÞ initial condition fades away long before rupture occurs). hix slowðtÞ¼ ; ð48bÞ kmol: þ k In Table 1, we also provide a generic expression for arbitrary force protocols 00 2 2 F(t) that do not necessarily scale linearly in time. where kmol: ¼ Ucuspð0Þ¼2E=xb for the cusp potential. Setting w ¼ 1 þ kxb=2E,

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this simplifies to 12. Neuman, K. C. & Nagy, A. Single-molecule force spectroscopy: optical ÂÃkytðÞ kytðÞ tweezers, magnetic tweezers and atomic force microscopy. Nat. Meth. 5, FslowðÞ¼t k ytðÞÀhix slowðÞt ¼ : ð49Þ 491–505 (2008). 1 þ k=kmol: w / S 13. Evans, E. & Ritchie, K. Dynamic strength of molecular adhesion bonds. Large external forces, on the other hand, may shift x slow(t) beyond xb, Biophys. J. 72, 1541–1555 (1997). rendering the static force-balance argument obviously inadequate. We obtain a 14. Dudko, O. K., Hummer, G. & Szabo, A. Intrinsic rates and activation free reasonable generalization of equation (48b) by instead truncating the time- energies from single-molecule pulling experiments. Phys. Rev. Lett. 96, 108101 dependent approximate probability distribution W (x, t|x )atx ¼ x and using G 0 b (2006). that to compute /xS(t), 15. Friddle, R. W. Unified model of dynamic forced barrier crossing in single hiðx tÞhix ðtÞ; molecules. Phys. Rev. Lett. 100, 138302 (2008). R G x 16. Freund, L. B. Characterizing the resistance generated by a molecular b xW ðx; t jx Þdx hix ðtÞ¼ RÀ1 G 0 bond as it is forcibly separated. Proc. Natl Acad. Sci. USA 106, 8818–8823 G xb W ðx; t jx Þdx À1 G pffiffiffiffiffiffiffiffi0 ð50Þ (2009). 2 ðÞx À xtðÞÀx CtðÞ 2=ps ðtÞ À b 0 17. Maitra, A. & Arya, G. Model accounting for the effects of pulling-device hix 2s2 ðÞt ¼ xðtÞÀ e x ; stiffness in the analyses of single-molecule force measurements. Phys. Rev. Lett. erf xb À pxðffiffitÞÀx0CðtÞ þ 1 2sx ðtÞ 104, 108301 (2010). 18. Friddle, R. W., Noy, A. & De Yoreo, J. J. Interpreting the widespread non-linear with mean xðtÞ, variance s2ðtÞ and the autocorrelation function C(t) defined as x force spectra of intermolecular bonds. Proc. Natl Acad. Sci. USA 109, usual (note that the particle position x0 at t ¼ 0 is irrelevant in the low-force limit and is eliminated in the high-force limit by averaging over the initial distribution of 13573–13578 (2012). 19. Hummer, G. & Szabo, A. Kinetics from nonequilibrium single-molecule pulling particle positions W(x0, t ¼ 0)), Z experiments. Biophys. J. 85, 5–15 (2003). kx t 20. Rico, F., Gonzalez, L., Casuso, I., Puig-Vidal, M. & Scheuring, S. High-speed b _ 0 0 0; xðtÞ¼ À Cðt À t Þyðt Þdt ð51aÞ force spectroscopy unfolds titin at the velocity of molecular dynamics wFc 0 simulations. Science 342, 741–743 (2013). 2 2 21. Otto, O., Sturm, S., Laohakunakorn, N., Keyser, U. F. & Kroy, K. Rapid internal kBTx ½1 À C ðtÞ s2ðtÞ¼ b ; ð51bÞ contraction boosts DNA friction. Nat. Commun. 4, 1780 (2013). x 2wE 22. van Loenhout, M. T. J., de Grunt, M. V. & Dekker, C. Dynamics of DNA supercoils. Science 338, 94–97 (2012). 2wDEt 23. Lee, E. H., Hsin, J., Sotomayor, M., Comellas, G. & Schulten, K. CðtÞ¼exp À 2 : ð51cÞ kBTxb Discovery through the computational microscope. Structure 17, 1295–1306 In the limit of low pulling forces, equation (50) reduces to the established result (2009). 24. Bell, G. I. Models for the specific adhesion of cells to cells. Science 200, 618–627 equation (48b), hix GðtÞxðtÞhix slowðtÞ. In the high-force limit, however, an equilibrium average is no longer warranted as a force spectroscopy setup with (1978). sufficient time resolution should instead measure the transducer force at the very 25. Getfert, S. & Reimann, P. Optimal evaluation of single-molecule force moment of rupture, spectroscopy experiments. Phys. Rev. E 76, 052901 (2007). 26. Kramers, H. A. Brownian motion in a field of force and the diffusion model of FfastðtÞ¼k½yðtÞÀxb: ð52Þ chemical reactions. Physica 7, 284–304 (1940). This equation coincides with the operational definition of ‘rupture force’ used 27. Risken, H. The Fokker-Planck Equation (Springer-Verlag, 1984). 19 by Hummer & Szabo . Since in the limit of high loading rates, xb is generally 28. Friedsam, C., Wehle, A. K., Ku¨hner, F. & Gaub, H. E. 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NATURE COMMUNICATIONS | 5:4463 | DOI: 10.1038/ncomms5463 | www.nature.com/naturecommunications 9 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5463

Author contributions Reprints and permission information is available online at http://npg.nature.com/ J.T.B. and S.S. developed the theory and analysed the simulation data. S.S. and K.K. wrote reprintsandpermissions/ the paper. J.T.B. and S.S. contributed equally to the study. All authors discussed the How to cite this article: Bullerjahn, J. T. et al. Theory of rapid force spectroscopy. results and commented on the manuscript. Nat. Commun. 5:4463 doi: 10.1038/ncomms5463 (2014).

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