Correcting molecular transition rates measured by single-molecule force spectroscopy for limited temporal resolution

Equilibrium free-energy-landscape parameters governing biomolecular folding can be determined from nonequilibrium force-induced unfolding by measuring the rates $k$ for transitioning back and forth between states as a function of force F. However, bias in the observed forward and reverse rates is introduced by limited effective temporal resolution, which includes the mechanical response time of the force probe and any smoothing used to improve the signal-to-noise ratio. Here we use simulations to characterize this bias, which is most prevalent when the ratio of forward and reverse rates is far from unity. We find deviations in $k$(F) at high rates, due to unobserved transitions from short- to long-lived states, and at low rates, due to the corresponding unobserved transitions from long- to short-lived states. These missing events introduce erroneous curvature in log($k$) vs F that leads to incorrect landscape parameter determination. To correct the measured $k$(F), we derive a pair of model-independent analytical formulas. The first correction accounts for unobserved transitions from short- to long-lived states, but does surprisingly little to correct the erroneous energy-landscape parameters. Only by subsequently applying the second formula, which corrects the corresponding reverse process, do we recover the expected $k$(F) and energy-landscape quantities. Going forward, these corrections should be applied to transition-rate data whenever the highest measured rate is not at least an order of magnitude slower than the effective temporal resolution.

Figure 1

Effective time resolution $\tau$ biases observed transition rates $k_{\mathrm{obs}}$. (a) Diagram showing unfolding transitions modeled as thermal activation along a one-dimensional free-energy landscape characterized by the height of the transition barrier $\mathrm{\Delta }{G}^{‡}$ and the distance to the barrier from stable states $\mathrm{\Delta }{x}^{‡}$. The subscript U refers to unfolding and F to folding. (b) A simulated rate map shows $\log \left(k_{\mathrm{obs}}\right)$ vs F at different $\tau$ for unfolding (pink) and folding (blue). Curvature introduced by limited $\tau$ can be misinterpreted as experimentally constraining $\mathrm{\Delta }{G}^{‡}$. (c) Illustration of how changes in the molecular coordinate (protein extension) are imperfectly reflected in the observed coordinate (AFM cantilever deflection). The three sketches sequentially show a folded protein experiencing high force from a bent cantilever, an unfolded protein prior to motion of the cantilever, and finally an unfolded protein rendered detectable by the relaxation of a cantilever having a response time $\tau$.

Figure 2

Simulations explore the effect of limited time resolution on observed transition kinetics. (a) Force-extension curve simulated with a rapid 0.1-μs time resolution (thin red line). The inset force-vs-time plot shows the same simulated trajectory at full time resolution and at a $\tau$ representative of an advanced assay (thick black line, 50 µs). Finite time resolution was imposed by eliminating state occupancies shorter than $\tau$. (b) Transition rates (× or ○) calculated from all $\tau =0.1\mu \mathrm{s}$ data agree with the expected force dependence based on the parameters used in the simulation (lines). Rates were calculated using either the method of Zhang and Dudko [33] [Eq. (3)), ×] or the present method [Eq. (4)), ○]. (c) Observed rate maps as a function of $\tau$. The force at which folding and unfolding rates intersect, $F_{1/2}$, is the force at which the folded and unfolded states are equally populated. Error bars are standard errors of the mean derived from a bootstrap analysis. (d) Plot of $F_{1/2}$ as a function of effective time resolution.

Figure 3

Energy-landscape parameter values before and after time-response corrections were applied. Values of (a) $k_0$, (b) $\mathrm{\Delta }{x}_{\mathrm{U}}^{‡}$, and (c) $\mathrm{\Delta }{G}_{\mathrm{U}}^{‡}$ as a function of $\tau$ determined by fitting the Dudko model [Eq. (2)) with $\nu =2/3$] to unfolding transition rates before correction (black circles), after applying the correction of Eq. (7)) (blue squares), and after applying both corrections (red diamonds). The dashed green line denotes the input parameter used in the simulation. For some $\tau$, the full Dudko model could not be reliably fit to the corrected data because the uncertainty in $\mathrm{\Delta }{G}_{\mathrm{U}}^{‡}$ was larger than its value (gray region). In (c), note that the vertical scale is different above the axis break. Values of (d) $k_0$ and (e) $\mathrm{\Delta }{x}_{\mathrm{U}}^{‡}$ as a function of $\tau$ determined by fitting the Bell model [Eq. (2)) with $\nu =1$] to the same data. Note that while this fit extended to high $\tau$ without breaking down, the parameters systematically disagreed with the underlying quantities. Error bars are standard deviations from the fitting function; some error bars are smaller than the plotted symbols. (f) Plot of the expected rate maps generated from the Dudko and Bell models for the simulation parameters. The disagreement between these maps in the vicinity of $F_{1/2}$ explains the systematic disagreement of the Bell-model fits.

Figure 4

Plot giving the product of $k_{\mathrm{mol}}$ and $\tau$ for which a certain fraction of the molecular transitions are experimentally observable. The dashed green line corresponds to 90% of transitions detected. Transition rates for systems lying above this line must be corrected.

Figure 5

Accurate rate map recovered after applying a pair of analytic corrections. (a) Uncorrected rates as a function of F calculated from $\tau =50\mu \mathrm{s}$ simulated data (circles) compared with the expected rate map based on the simulation parameters (lines). (b) Rate map after application of the correction for unobserved occupancy of short-lived states (squares). (c) The rate map after the correction for unobserved transitions from long- to short-lived states was also applied (diamonds). Open symbols denote rates for which the number of missing transitions is greater than tenfold the number of observed transitions, causing the second correction to break down (see Fig. 6).

Figure 6

Correction factor for unobserved transitions into short-lived states plotted as a function of force. The correction factor Γ is the ratio of the number of transitions added by the second analytic correction to the number present in the original data. Points associated with $\mathrm{\Gamma }>10$ (dashed green line) failed to yield correct rate values after applying Eqs. (7)) and (8) and are therefore recommended to be excluded from further analysis.