Theory, analysis, and interpretation of single-molecule force spectroscopy experiments

Abstract

Dynamic force spectroscopy probes the kinetic and thermodynamic properties of single molecules and molecular assemblies. Here, we propose a simple procedure to extract kinetic information from such experiments. The cornerstone of our method is a transformation of the rupture-force histograms obtained at different force-loading rates into the force-dependent lifetimes measurable in constant-force experiments. To interpret the force-dependent lifetimes, we derive a generalization of Bell's formula that is formally exact within the framework of Kramers theory. This result complements the analytical expression for the lifetime that we derived previously for a class of model potentials. We illustrate our procedure by analyzing the nanopore unzipping of DNA hairpins and the unfolding of a protein attached by flexible linkers to an atomic force microscope. Our procedure to transform rupture-force histograms into the force-dependent lifetimes remains valid even when the molecular extension is a poor reaction coordinate and higher-dimensional free-energy surfaces must be considered. In this case the microscopic interpretation of the lifetimes becomes more challenging because the lifetimes can reveal richer, and even nonmonotonic, dependence on the force.

Fig. 1.

Unzipping of DNA hairpins in a nanopore. Experimental data from ref. 28. (A) Unzipping voltage histograms at various voltage-ramp speeds. Black lines are the predicted voltage histograms with parameters extracted from the least-squares fit of Eq. 5 to the collapsed histograms, as shown in B. (B) Lifetime τ(V) of the DNA hairpin as a function of the applied voltage V. The lifetime is obtained by transforming the rupture-voltage histograms (filled symbols, colors as in A) according to Eq. 2. Note the excellent agreement with the lifetimes measured directly at constant voltage (open circles)

Fig. 2.

Unfolding of a protein with AFM. Experimental data from ref. . (A) Rupture-force histograms at various pulling speeds. Black lines are the predicted rupture-force distributions, Eq. 1, with the parameters of the least-squares fit of Eq. 5 to the collapsed histograms in B. (B) Lifetime τ(F) as a function of the applied force F, obtained by transforming the force histograms in A according to Eq. 2 (filled symbols; colors as in A) with the loading rate Ḟ(F) given by Eq. 4. Open symbols show the transformed histograms without linker correction.