Reversible bond kinetics from single-molecule force spectroscopy experiments close to equilibrium

Analysis of bond rupture data from single-molecule force spectroscopy experiments commonly relies on the strong assumption that the bond dissociation process is irreversible. However, with increased spatiotemporal resolution of instruments it is now possible to observe multiple unbinding-rebinding (or unfolding-refolding) events in a single pulling experiment. Here we augment the theory of force-induced unbinding by explicitly taking into account rebinding kinetics and provide approximate analytic solutions of the resulting rate equations. Furthermore, we use a short-time expansion of the exact kinetics to construct numerically efficient maximum likelihood estimators for the parameters of the force-dependent unbinding and rebinding rates, which pair well with and complement established methods, such as the analysis of rate maps. The estimators are independent of the applied force protocol and can therefore be used to globally analyze multiple force-extension traces for both force-ramp and force-clamp experiments. We provide an open-source implementation of the theory, evaluated for Bell-like rates, which we apply to synthetic data generated by a Gillespie stochastic simulation algorithm for time-dependent rates.

Figure 1

Schematic of SMFS experiments using dynamic force protocols. (a) Idealized force-extension curve for a linear force protocol, defined by a constant loading rate $\dot{F}=\kappa v$ and a state-dependent force offset $F^{*}=\kappa y\left(s\right)$. Here $\kappa$ and $v$ denote the spring constant and speed of the force transducer, respectively. (b) More realistic force-extension curve, where the bound ($s=b$) and unbound ($s=u$) force branches are modeled by wormlike chains with different contour lengths. In this case, the loading rate $\dot{F}\left[F\right]$ is a nonlinear function of force. (c) Comparison of RFDs for reversible (solid blue line) and irreversible unbinding (dashed red line). The unbinding forces in (a) are distributed according to $p\left[F\right]$, which is much broader than the distribution $p_{\text{irr}}\left[F\right]$ for strictly irreversible unbinding.

Figure 2

Validity range of approximations. (a) Relative population in the bound state as a function of the applied force $F=\dot{F}t$, plotted for different loading rates $\dot{F}$. The quasiequilibrium approximation [Eq. (14), red dashed line] is independent of $\dot{F}$ and captures well the low-loading-rate behavior of the exact solution [Eq. (9), blue solid lines], whereas the instantaneous rebinding approximation [Eq. (11), green dashed-dotted lines] only becomes applicable at high loading rates. (b) Average number of unbinding events as a function of the loading rate for $t^{\prime }=0$. The two approximations [Eq. (15), red dotted line, and Eq. (6) evaluated for $B_{\text{inst}}\left(t\right)$, green dashed-dotted line] correctly reproduce the asymptotic limits of $\langle N_{\mathrm{off}}\rangle$ (blue solid line), respectively. Their point of intersection (open black circle) lies just below $\langle N_{\mathrm{off}}\rangle =3$. The hybrid expression in Eq. (18) (orange dashed-double-dotted line) interpolates between the two limiting regimes and holds for all loading rates. All equations were evaluated using the Bell-like rates of Eq. (17) with ${\beta }^{-1}=4\mathrm{pN}\mathrm{nm}$, $\mathrm{\Delta }{x}_{\mathrm{off}}=0.5\mathrm{nm}$, $k_{\mathrm{off}}^0\approx 0.18{\mathrm{s}}^{-1}$, $\mathrm{\Delta }{x}_{\mathrm{on}}=1.0\mathrm{nm}$, and $k_{\mathrm{on}}^0\approx 14.9{\mathrm{s}}^{-1}$.

Figure 3

Single replica MLEs of linear force-extension curves and their sample statistics. Each replica simulation ($\dot{F}=100\mathrm{p}\mathrm{N}{\mathrm{s}}^{-1}$, other simulation parameters as specified in Appendix pp4) generated a data set of $M=10$ time series, which was analyzed using the estimators in Eqs. (25, 26, 27, 28) at a fixed time step $\mathrm{\Delta }t=0.1\mu \mathrm{s}$. The individual parameter estimates (gray open circles) scatter around their respective sample average (solid lines), resulting in a sample standard deviation (shaded regions) of comparable size as the corresponding standard error estimate [Eq. (29) with $\delta F^2\equiv 0$, error bars]. Slight deviations between sample averages and ground-truth values of the parameters (dashed lines) are due to the use of a finite time step, and therefore vanish as $\mathrm{\Delta }t\to 0$ (see Fig. 4).

Figure 4

Estimating relative bias in the rate parameter MLEs with respect to the loading rate $\dot{F}$ and time step $\mathrm{\Delta }t$. To imitate the experimental situation of limited instrumental time resolution, we lengthen the time step between subsequent observations in our data sets and analyze the resulting time series using Eqs. (25, 26, 27, 28). Generally, the accuracy of the parameter estimates decreases with increasing $\mathrm{\Delta }t$. We observe this trend at three different loading rates, (a) $\dot{F}=10$, (b) $\dot{F}=100$, and (c) $\dot{F}=1000\mathrm{p}\mathrm{N}{\mathrm{s}}^{-1}$, in exactly the same fashion, thus indicating that the effect is $\dot{F}$-independent. Note that we did not exclude any outliers, so the whiskers of the box plots mark the maximum and minimum values of each sample.

Figure 5

Comparing rate map predictions to MLEs at limited response times. A single set of force-extension curves ($\dot{F}=100\mathrm{p}\mathrm{N}{\mathrm{s}}^{-1}$, $M=10$) was analyzed, on the one hand, using the rate-map method of Ref. [28] (gray circles and squares, where error bars represent one standard deviation) and, on the other hand, via the estimators in Eqs. (25, 26, 27, 28) (red dashed lines). The MLEs are as follows: (a) $\mathrm{\Delta }{x}_{\text{off}}=(0.411\pm 0.007)\mathrm{nm}$, $k_{\text{off}}^0=(2.5\pm 0.3){\mathrm{s}}^{-1}$, $\mathrm{\Delta }{x}_{\text{on}}=(0.227\pm 0.006)\mathrm{nm}$, $k_{\mathrm{on}}^0=(5200\pm 500){\mathrm{s}}^{-1}$ for $\mathrm{\Delta }t=10\mu \mathrm{s}$, and (b) $\mathrm{\Delta }{x}_{\text{off}}=(0.441\pm 0.009)\mathrm{nm}$, $k_{\text{off}}^0=(1.8\pm 0.2){\mathrm{s}}^{-1}$, $\mathrm{\Delta }{x}_{\text{on}}=(0.265\pm 0.007)\mathrm{nm}$, and $k_{\mathrm{on}}^0=(7700\pm 800){\mathrm{s}}^{-1}$ for $\mathrm{\Delta }t=100\mu \mathrm{s}$. Deviations from the exact rates (blue solid lines) increase as the time resolution gets poorer.