Theory of rapid force spectroscopy

Abstract

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.

Figure 1. Flux into an absorber at x_b.

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

Figure 2. Mean rupture force ‹F› as a function of loading rate.

Circles: ‹F› as determined from our numerical simulations (see Fig. 3). Crosses: best DHS fit (=9.49 × k_BT, x_b=1.01 nm, D=574 nm² s⁻¹). Triangles: best Hummer–Szabo fit (=10.2 × k_BT, x_b=1.0 nm, D=1015, nm² s⁻¹). Squares: ‹F› determined from p(F) (see Methods section), using the fit parameters obtained in Fig. 3. Pentagons: asymptotic analytical approximation equation (40) to ‹F›, 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–Szabo asymptote ^1/2 at large loading rates, , and onto the logarithmic DHS asymptote ‹F›~[ln ]^1/2 (ref. 14) at intermediate loading rates, (dashed red lines, shifted upwards for better visibility). In the limit →0, external forces are too small to induce rupture, yielding as the measured rupture force the force at the time of spontaneous unbinding, ‹F›~/k₀. For our choice of system parameters, the best currently available experimental setup would already be able to access the ballistic regime where ‹F› increases with ^1/2.

Figure 3. Analytical theory compared with simulations.

Using a single set of model parameters, our theory (equation (17), solid lines) provides an accurate global approximation to both slow (a) and fast (c) external force protocols (as compared with the intramolecular relaxation timescale, in our case), apart from a narrow range (b) close to a critical loading rate (_c≈10⁵ pN s⁻¹ for our choice of parameters). The ‘experimental’ rupture force histograms have been generated by direct stochastic integration (see Methods section), using =10 × k_BT, T=300 K, x_b=1 nm, D=1,000 nm² s⁻¹ and =1…10¹¹ pN s⁻¹. Our best-fit parameters obtained with equation (17) are =10.15 × k_BT, x_b=0.98 nm, D=976 nm² s⁻¹. Since the mean rupture force varies by orders of magnitude at large loading rates, we use double-logarithmic scaling for >10⁷ pN s⁻¹ (c).

Figure 4. Measuring rupture forces with a stiff transducer.

(a) As long as the bond remains intact, the combined bond-transducer system can be seen as two harmonic springs connected in series. At low pulling speeds, positional fluctuations within the bound state translate into force 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 with k_BT, the probability distribution W(x, t) closely approximates a Gaussian centred within the bound state and ‹x›_slow(t) virtually coincides with ‹x›_G(t). (c) At high pulling forces, the static force-balance argument a fails, as it yields an equilibrium position ‹x›_slow(t) beyond x_b. The improved approximation ‹x›_G(t) instead is always bounded above by x_b.