Theoretical analysis of single-molecule force spectroscopy experiments: heterogeneity of chemical bonds

Abstract

We show that the standard theoretical framework in single-molecule force spectroscopy has to be extended to consistently describe the experimental findings. The basic amendment is to take into account heterogeneity of the chemical bonds via random variations of the force-dependent dissociation rates. This results in a very good agreement between theory and rupture data from several different experiments.

FIGURE 1

Schematic illustration of dynamic AFM force spectroscopy: a single chemical bond, e.g., in a ligand-receptor complex, is connected via two flexible linker molecules with the tip of an AFM cantilever and a piezoelectric element. The latter pulls down the attached linker molecule at some constant velocity υ. The resulting elastic reaction force of the cantilever can be determined from the deflection of a laser beam. The main quantity of interest is the force value at the moment when the bond dissociates.

FIGURE 2

Force-extension curves of four representative single-molecule pulling experiments, two with pulling velocity υ = 100 nm/s (solid) and two with υ = 5000 nm/s (dashed), obtained by dynamic AFM force spectroscopy for the DNA fragment expE1/E5 and the regulatory protein ExpG (23). The abrupt drop of f(t) indicates dissociation of the chemical bond between expE1/E5 and ExpG. Apart from noise effects, the forces f(t) before dissociation (relevant in Eq. 1) collapse quite well to a single force-extension master-curve F(s); see Eq. 2. It can be noted that for a given pulling velocity, the rupture forces are distributed over a considerable range and that larger pulling velocities result in larger average dissociation forces.

FIGURE 3

(Symbols) The functions for different pulling velocities υ, obtained according to Eq. 4 from the same experiment (23) as in Fig. 2. Each depicted point corresponds to one rupture event at f = f_n and hence a step of the piecewise constant function Eq. 4. Only f_n above f_min = 20 pN have been taken into account; see discussion below Eq. 3. The number N_υ of experimental data points for the six different velocities υ in Eq. 4 are N_50nm/s = 20; N_100nm/s = 44; N_500nm/s = 179; N_1000nm/s = 208; N_2000nm/s = 108; and N_5000nm/s = 253. A few very small or large f_n are omitted in this plot for the sake of better visibility of the remaining symbols. (Solid lines) Theoretical functions for the same pulling velocities υ as the symbols, using Eqs. 9, 11–16, and 21. For more details, see section Heterogeneity of Chemical Bonds.

FIGURE 4

Same symbols as in Fig. 3 but for dynamic AFM force spectroscopy data by Bartels et al. (23) for the dissociation of the DNA fragment expG1/G4 from the regulatory protein ExpG. (Symbols) The functions for different pulling velocities υ, obtained according to Eq. 4 and taking into account only f_n above f_min = 10 pN. (Solid lines) Theoretical functions for the same pulling velocities υ as the symbols, using Eqs. 9, 11–16, and 20.

FIGURE 5

Same as in Fig. 3 but for dynamic AFM force spectroscopy data by Eckel et al. (24) for the dissociation of the PhoB peptide (wild-type) of E. coli from the DNA target sequence. (Symbols) The functions for different pulling velocities υ, obtained according to Eq. 4 and taking into account only f_n above f_min = 20 pN. (Solid lines) Theoretical functions for the same pulling velocities υ as the symbols, using Eqs. 9, 11–16, and 21.

FIGURE 6

Same as in Fig. 3 but for dynamic AFM force spectroscopy data by Eckel et al. (25) for the dissociation of a calixaren host molecule (resorc[4]arene) from a cationic guest (ammonium). (Symbols) The functions for different pulling velocities υ, obtained according to Eq. 4 and taking into account only f_n above f_min = 25 pN. (Solid lines) Theoretical functions for the same pulling velocities υ as the symbols, using Eqs. 9, 11–16, and 22.

FIGURE 7

Same as in Fig. 3 but for micropipette-based force probe data by Nguyen-Duong et al. (19) for the dissociation of immunoglobulin of type G from protein A. (Symbols) The functions for different loading rates r; see below Eq. 6, taking into account only f_n above f_min = 15 pN. (Solid lines) Theoretical functions for the same loading rates r as the symbols, using Eqs. 7, 9, 11–16, and 23.

FIGURE 8

Same as in Fig. 3 except that in panel a, only f_n above f_min = 50 pN and in panel b, only f_n above f_min = 100 pN have been taken into account. The solid lines are the corresponding theoretical functions using Eqs. 9, and 11–19. (Dashed lines) Same as solid lines but after refitting the parameters k₀, α_m, and σ to the given data subset, resulting in k₀ = 0.000020 s⁻¹, α_m = 0.19 pN⁻¹, σ = 0.095 pN⁻¹ for panel a, and in k₀ = 0.017 s⁻¹, α_m = 0.091 pN⁻¹, σ = 0.040 pN⁻¹ for panel b.

FIGURE 9

Same symbols as in Fig. 3 but for synthetic rupture data, sampled numerically according to Eqs. 9, and 11–19. The velocities υ and the number of rupture events N_υ for each υ are identical to those in Fig. 3.

FIGURE 10

(Symbols) Same experimental data as in Fig. 3. (Solid lines) Theoretical functions for the same pulling velocities υ as the symbols, using Eqs. 8, 9, 11, 16, 34, and 36.

FIGURE 11

Sketch of the relevant dissociation rates of a chemical bond whose reaction coordinate x experiences a reaction potential U(x) with an intermediate energy barrier.

FIGURE 12

(Histograms) Same experimental rupture data as in Fig. 3 but represented as rupture force distributions. (Solid lines) Theoretical curves according to Eqs. 9 and 11–19. (Dotted lines) Same but for k₀ = 0.011 s⁻¹, α_m = 0.14 pN⁻¹, σ = 0 pN⁻¹, f_min = 0 (see main text), and with divided by a factor 3 for better visibility of the other curves.

FIGURE 13

Most probable rupture force f* versus pulling velocity υ (logarithmic scale) by solving Eq. 37 numerically with Eq. 9, f_min = 0, Eqs. 11–16, k₀ and α_m from Eq. 19, and five different values of σ. For f_min > 0, the maximum of the plotted curves and f_min yield f*.

FIGURE 14

Same as Fig. 13, except that in Eq. 17 rather than α_m in Eq. 16 has been kept fixed to the value 0.135 pN⁻¹.

FIGURE 15

Same as Fig. 14, except that f* was not determined according to Eq. 37 but rather by fitting the rupture force distributions by Gaussian Eq. 39. (The fit was performed on the interval 0 ≤ f ≤ 250 pN.)