Probing static disorder in Arrhenius kinetics by single-molecule force spectroscopy

Abstract

The widely used Arrhenius equation describes the kinetics of simple two-state reactions, with the implicit assumption of a single transition state with a well-defined activation energy barrier DeltaE, as the rate-limiting step. However, it has become increasingly clear that the saddle point of the free-energy surface in most reactions is populated by ensembles of conformations, leading to nonexponential kinetics. Here we present a theory that generalizes the Arrhenius equation to include static disorder of conformational degrees of freedom as a function of an external perturbation to fully account for a diverse set of transition states. The effect of a perturbation on static disorder is best examined at the single-molecule level. Here we use force-clamp spectroscopy to study the nonexponential kinetics of single ubiquitin proteins unfolding under force. We find that the measured variance in DeltaE shows both force-dependent and independent components, where the force-dependent component scales with F(2), in excellent agreement with our theory. Our study illustrates a novel adaptation of the classical Arrhenius equation that accounts for the microscopic origins of nonexponential kinetics, which are essential in understanding the rapidly growing body of single-molecule data.

Fig. 1.

Measuring the survival probability of ubiquitin proteins unfolding under a stretching force. (A) A single polyubiquitin molecule is picked up from the surface by the cantilever tip and stretched under a constant force. (B) Stretching a ubiquitin polyprotein at a constant force of 110 pN results in a series of 20 nm stepwise increments in the polyprotein length, marking the unfolding of individual ubiquitins in the chain. We measure the dwell time to unfolding, t_i, for each unfolding event. (C) A histogram of 2799 unfolding events measures the probability density of unfolding p(t) at 110 pN. At short dwell times the distribution deviates significantly from a single exponential (black trace).

Fig. 2.

Survival probability for ubiquitin unfolding under force is well described by static disorder theory. Plot of ln[- ln 〈S(t)〉] versus ln t at 90 pN (filled circles), 110 pN (open circles), 130 pN (filled squares), 150 pN (open squares), 170 pN (filled triangles), and 190 pN (open triangles), respectively. The slopes of all the curves are less than 1, indicating the nonexponential survival probability measured at all forces. The solid lines represent the fits of the static disorder survival probability (Eq. 7) to the data at each force, with the unfolding rate of crossing the average barrier height k_F and the variance of the barrier heights σ² as fit parameters. The values of these parameters are compiled in Table 1. The errors in the fit parameters were estimated using the bootstrap method.

Fig. 3.

The unfolding rate of crossing the average barrier height, k_F, depends exponentially on the pulling force. A linear dependence between ln k_F and the applied force reveals the remarkable result that the most probable unfolding rate, k_F, follows the simple Arrhenius law. Fitting ln k_F with the Arrhenius equation (solid line) yields the average barrier height in the absence of force ΔG_avg = 85.1 pN nm and the average distance to the transition state Δx_avg = 0.23 nm.

Fig. 4.

Force dependency of the variance in the barrier heights to unfolding. Plot of the measured variance of the barrier heights, σ², as a function of the square of the pulling force. The solid line corresponds to a fit of the data from 90 pN to 170 pN with (Eq. 4). The fit gives and . The measured values of σ² increase linearly with F² in this range of forces, in agreement with the prediction of our model.