Theory of rapid force spectroscopy

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.


〈F 〉HS
c linear-cubic potential 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 To evaluate how well our theory fares in practice, we have performed Brownian Dynamics simulations of particles bound either via a cusp-shaped (a, b) or linear-cubic potential (c, d) and acted upon by an external field/soft spring (a, c) or by a stiff spring (b, d). The rupture force histograms thus obtained were analyzed "locally" (performing a different fit for each loading rate) and "globally" (performing a single fit for several decades inḞ) using both our own theory and the best available steady-state ("slow") theories (specifically, "slow cusp/cubic" denotes the DHS [14] (a, c)/Maitra-Arya [17] model (b, d), evaluated either for a cusp-shaped or for a linear-cubic binding potential, see Supplementary Note 1 for further details). We have furthermore followed standard operating procedures by alternatively fitting the mean rupture force F as a function of loading rate, using the analytical results of the DHS [15] (a, c)/Maitra-Arya model (b, d), again specialized either to a cusp-shaped or linear-cubic binding potential, and using the numerical interpolation derived by Hummer & Szabo [19] ("HS") that extends to arbitrarily high loading rates. Each single fit is represented by a box split in half along its vertical axis, with the box width indicating the range of external loading rates taken into account and the two halves containing the best-fit value for the attraction range x fit b , measured in multiples of its true value x b = 1 nm and the best-fit value E fit for the energy barrier, also measured in multiples of its true value E = 10 k B T . As an example, we have highlighted in green in a the results of a 6-decade fit to simulations of a cusp-shaped binding potential under a linearly increasing external force field. The entry spans 6 decades, from 1 to 10 5 pN s −1 and belongs to the category "slow/cusp", indicating that a Maximum-Likelihood analysis of rupture force data obtained under 6 different loading rates has been performed, using the "cusp" (ν = 1/2) DHS [14] model and yielding the following results: we have generated rupture force histograms analogous to those shown in fig. 3 in the main text. Solid lines show a global fit of our theory with µ as a free parameter (see Table 1). The best-fit parameters thus obtained are (E = 11.2 k B T, x b = 1.13 nm, D = 774 nm 2 s −1 , µ = 5.75) current experimental limit  At low pulling speeds (red curve), the bound state remains stable up to the time of bond rupture; on an experimental timescale, the subsequent decrease in pulling force appears instantaneous. Within the ballistic regime, the bound state is no longer stabilized by a free energy barrier; still, a pronounced change in intermolecular binding forces −U (x ≈ x b ) manifests itself in a force maximum at x b for intermediate pulling speeds (γv − κ[y(t rupture ) − x b ] − F R < 0, green curve), whereas at even larger pulling speeds the force continues to rise towards its ultimate limit F(t → ∞) = vγ − F R , but exhibits a kink at x = x b (blue curve). Hence, the characteristic signature of bond rupture is found to vanish gradually beyond the critical pulling force, disappearing completely only in the ultimate limit v = ∞.
Supplementary Note 1: Benchmarking. To assess the practical performance of our theory in comparison to established models of dynamic force spectroscopy, we generate synthetic rupture force histograms over a wide range of pulling speeds ranging from 1 to 10 11 pN s −1 (using a direct integration of the underlying Langevin equation, as explained in the Methods section of the main text). In addition to the cusp-shaped binding potential described in the main text, we also consider a linear-cubic binding potential In contrast to the cusp potential, the linear-cubic potential lacks an absorbing boundary, such that rebinding must be prevented by strong repulsive forces within the unbound state. We thus only consider the bond broken once x has passed 1.5 x b . For each of the four different experimental setups (cusp/field, linear-cubic/field, cusp/spring, linear-cubic/spring), we obtain a set of 12 different rupture force histograms, each of which we first analyze separately. Apart from our own theory, we also apply the corresponding results by DHS [14] and Maitra & Arya [17] (and eq. (4-S)), depending on which best applies to the experimental setup at hand. Furthermore, we generate "global" fits by subsuming histograms obtained under a range of different pulling speeds and analyzing them all at once using the maximum-likelihood method proposed in [25]. Since we do not want to bias the results by our choice of starting value, we use a global optimization method (the NMinimize optimizer integrated into Mathematica), only restricting the range of allowed parameters to a rather large region (3 k B T < E < 30 k B T , 0.3 nm < x b < 3 nm, 300 nm 2 s −1 < D < 3000 nm 2 s −1 , 0.3 < µ < 9). For each fit, "global" and "local", 1600 measured rupture forces per loading rate were used for the analysis.
Conventionally, global fits are often obtained not by fitting the rupture force distributions themselves, but by analyzing the mean rupture force F [39] (or the most probable rupture force [13]) as a function of the external loading rate. We have thus included this method as well, but only using the conventional (quasistatic) expressions for F [15,17] and the extrapolation by Hummer & Szabo [19], though one may derive analogous results from our theory, if so desired (see the Methods section of the article).
Supplementary Fig. 1 provides an overview of the fit parameters thus obtained and their relative deviations from the true model parameters.
Supplementary Note 2: Microscopic Kramers rate for a cusp potential in stiff spring limit. In [17], the work of DHS [14] and Friddle [15] is extended to explicitly account for a harmonic force transducer, thus relaxing the commonly made assumption that the pulling device is much softer than the intramolecular bond. Only a linear-cubic binding potential (i.e. the ν = 2/3 case in [14]) is considered in [17], but not the ν = 1/2 "cusp" scenario. To compare our results to the optimized version of the conventional steady-state approximation, we provide here a short derivation of a "cusp-optimized" counterpart to the results given in [17]. The thermal escape of a particle, moving in an energy landscape over an effective energy barrier of height χβ E 1, χ = 1 + κx 2 b /2E , can be described using Kramers theory [14,26], where the rate of escape is given by F(t) = κy(t)/χ, F c = 2E /x b and k 0 is the associated Kramers rate in absence of the force transducer. The rate expression above can be used to compute the accompanying RFD via eq. (1). For a force ramp, i.e. y(t) =ẏt and thus F(t) =Ḟt withḞ = const, the resulting RFD reads Finally, the mean and the variance of the distribution can be deduced along the lines of [15], where q, X and E 1 (z) are defined as in the Methods section of the main text.
Supplementary Note 3: Experimental detection of "ballistic rupture events". During the peer-review process the question was raised as to how one might measure, or even define, a "rupture" event when there is no longer an effective free energy barrier to stabilize the bound state. A step-like transition between two discrete, stable equilibrium positions (i.e., the time-dependent energy minimum x within the bound state and the location y(t) of the external force actuator) can arise only from the metastable free energy landscape pertaining to subcritical pulling forces F < F c . Yet, even as the effective free energy barrier vanishes, the intramolecular binding potential U still underlies the combined effective potential U +V , and with it the strong variation in intramolecular forces F(x) = −U (x) characteristic of a well-defined transition region around x b . The question of how to best relate experimentally measured force traces to the first-passage time distribution at x b is an (albeit important) implementation detail to be decided by the practitioner. Still, to illustrate the matter and to provide at least one possible working definition of "ballistic bond rupture", we consider a non-singular, but sharp transition region characterized by finite slopes −U (x ↑ x b ) = F L , −U (x ↓ x b ) = F R . Evaluating the athermal dynamics of x(t), ≡ −γv(t) + F R + κδ x(t) , (8b-S) we find that, as we enter the ballistic regime (i.e., the pulling force at the time of rupture exceeds F c = F L ), a friction-limited dynamic equilibrium ensues, with the steady-state extension of the external force actuator saturating at F = γv − F R . Depending on F R , this implies either a force maximum at x = x b or an ongoing increase in the pulling force that is, however, preceded by a detectable kink as x crosses the barrier position. Hence, the characteristic "rupture signature" seen in the time-resolved pulling force F(t) does not disappear immediately as loading rates increase beyond the critical loading rate, but instead vanishes gradually (as sketched in Supplementary Fig. 4), thus in principle allowing for experimental detection for all but infinite loading rates v = ∞.