Hidden multiple bond effects in dynamic force spectroscopy

Abstract

In dynamic force spectroscopy, a (bio-)molecular complex is subjected to a steadily increasing force until the chemical bond breaks. Repeating the same experiment many times results in a broad distribution of rupture forces, whose quantitative interpretation represents a formidable theoretical challenge. In this study we address the situation that more than a single molecular bond is involved in one experimental run, giving rise to multiple rupture events that are even more difficult to analyze and thus are usually eliminated as far as possible from the further evaluation of the experimental data. We develop and numerically solve a detailed model of a complete dynamic force spectroscopy experiment including a possible clustering of molecules on the substrate surface, the formation of bonds, their dissociation under load, and the postprocessing of the force extension curves. We show that the data, remaining after elimination of obvious multiple rupture events, may still contain a considerable number of hidden multiple bonds, which are experimentally indistinguishable from true single bonds, but which have considerable effects on the resulting rupture force statistics and its consistent theoretical interpretation.

Figure 1

(A–D) Typical force extension curves from four single runs of a dynamic force spectroscopy experiment by AFM. Ideally, the force F(s) steadily grows with increasing distance s until the chemical bond ruptures (A). In panels B and C, more than one force dip (i.e., downward jump of F(s)) is clearly visible, indicating that more than a single bond was involved. Curves like in panel D would commonly still be attributed to a single bond rupture within the given noise level and resolution limit. The depicted experimental data in panels A–D for a protein-DNA complex (PhoB mutant and target sequence, pulling velocity of 2000 nm/s, cantilever stiffness 13 pN/nm, linker length 30 nm) have been kindly provided by A. Bieker and D. Anselmetti (Bielefeld University). (E and F) Typical histograms of experimental rupture force distributions for two different pulling velocities v (7). After eliminating all the experimental force extension curves with clearly visible multiple bond signatures as those in panels B and C, the number of remaining apparent single-bond rupture events contributing to panel E was 202, and 151 for panel F. The main features are a pronounced first peak (most probable rupture force), vague indications of possible secondary peaks, and a long tail.

Figure 2

(A) Schematic sketch of dynamic force spectroscopy by AFM: a chemical bond of interest, e.g., in a ligand-receptor complex, is connected via two linker molecules with the tip of an AFM cantilever and a piezoelectric element at distance s. The latter is employed for pulling down the attached linker molecule at some constant velocity v that in turn leads to an elastic reaction force F(s) of the cantilever, determined from the deflection of a laser beam. (B) Illustration of the model for multiple parallel bonds. The AFM tip is modeled as a half-sphere and forces f_i act on the ligand-receptor bonds.

Figure 3

(Dashed lines) Six realizations of force extension curves for double bonds (N = 2), numerically simulated as described in the Model section (see main text). Receptors were uniformly distributed on the sample surface with density ρ_rec = 10⁻³ nm⁻². For a better comparability with the experimental curves from Fig. 1, A–D, we sampled the force extension curves in regular time steps of Δt = 0.1 ms and added a Gaussian (thermal) noise with standard deviation ${\sigma }_f=\sqrt{\kappa k_{\text{B}}T}=6.4$ pN (2). After that, a running average over 0.5 ms was calculated, imitating the effect of an experimental low-pass filter, and resulting in the solid lines. The distance Δs between the rupture of the two bonds is indicated in each figure. (Dotted line, bottom-right panel) One realization of a force extension curve for a single bond (N = 1; compare to Fig. 1A). The somewhat larger fluctuations observed in Fig. 1, A–D, can be attributed to instrumental noise on top of the thermal noise.

Figure 4

(A) (Red) Probability of formation (and rupture) of at least one bond (i.e., N ≥ 1) within one force distance cycle. (Black) Probability of observing a false single bond (i.e., an apparent single bond is, de facto, a hidden multiple bond). (Blue) Fraction of false single bonds among all multiple bonds. All three probabilities are presented for various values of the density ρ_rec of uniformly distributed receptors and have been obtained as detailed in the Model section, and Uniformly Distributed Receptors. (Error bars) Statistical spread (standard deviation) due to our sampling of 100 different tips (see main text). (B) The corresponding mean rupture forces 〈f〉. (C) Representative rupture force distribution for one AFM tip (see main text) and ρ_rec = 2 × 10⁻⁴ nm⁻². (D) Same for ρ_rec = 10⁻³ nm⁻².

Figure 5

Same as Fig. 4, A and B, but for N_lin = 5 linkers in panels A and B, and N_lin = 15 linkers in panels C and D.

Figure 6

(A) Fraction of false single bonds among all multiple bonds versus resolution limit Δs_max for maximum binding lengths d_max = 8 nm (solid), d_max = 12 nm (dotted), and d_max = 16 nm (dashed). For further details regarding the employed receptor clustering model, see main text. (B) Probability of observing a false single bond (i.e., an apparent single bond is, de facto, a hidden multiple bond). (C) Rupture force distributions for Δs_max = 1 nm. For reasons of better visibility, the distribution for d_max = 12 nm is not shown.

Figure 7

(A–D) Rupture force distributions for different pulling velocities v, assuming clustering of receptors. For further simulational details, see main text. (E) The corresponding survival probabilities according to Eq. 11. Velocities increase in the direction indicated (arrow). (F) The most probable rupture force f^∗ from panels A–D versus logarithm of the loading rate λ. (Solid line) Best linear fit. For more details, see main text.