Direct quantification of the attempt frequency determining the mechanical unfolding of ubiquitin protein

Abstract

Understanding protein dynamics requires a comprehensive knowledge of the underlying potential energy surface that governs the motion of each individual protein molecule. Single molecule mechanical studies have provided the unprecedented opportunity to study the individual unfolding pathways along a well defined coordinate, the end-to-end length of the protein. In these experiments, unfolding requires surmounting an energy barrier that separates the native from the extended state. The calculation of the absolute value of the barrier height has traditionally relied on the assumption of an attempt frequency, υ(‡). Here we used single molecule force-clamp spectroscopy to directly determine the value of υ(‡) for mechanical unfolding by measuring the unfolding rate of the small protein ubiquitin at varying temperatures. Our experiments demonstrate a significant effect of the temperature on the mechanical rate of unfolding. By extrapolating the unfolding rate in the absence of force for different temperatures, varying within the range spanning from 5 to 45 °C, we measured a value for the activation barrier of ΔG(‡) = 71 ± 5 kJ/mol and an exponential prefactor υ(‡) ∼4 × 10(9) s(-1). Although the measured prefactor value is 3 orders of magnitude smaller than the value predicted by the transition state theory (∼6 × 10(12) s(-1)), it is 400-fold higher than that encountered in analogous experiments studying the effect of temperature on the reactivity of a protein-embedded disulfide bond (∼10(7) M(-1) s(-1)). This approach will allow quantitative characterization of the complete energy landscape of a folding polypeptide from highly extended states, of capital importance for proteins with elastic function.

FIGURE 1.

Unfolding the ubiquitin polyprotein in a force-clamp at varying temperatures. A, schematics of the experimental force-clamp setup, provided with a Peltier element that lies just below the gold cover slide onto which single proteins are deposited. const., constant; Temp. control, temperature control. B, unfolding the ubiquitin polyprotein under a constant force of 130 pN results in a staircase-like elongation, where the unfolding of each monomer in the chain occurs stochastically at a time Δt after the application of force, eliciting steps of ∼20 nm in length. The time course of unfolding is accelerated with the temperature at which the experiments are conducted. PID, (proportional integral differential) force feedback.

FIGURE 2.

The combined effect of force and temperature on the kinetics of the mechanical unfolding of ubiquitin. A, four averaged and normalized polyubiquitin unfolding time courses obtained at a constant temperature of 25 °C at different constant stretching forces: 110, 130, 150, and 170 pN. Discontinuous black lines correspond to single exponential fits with associated rate constants presented as circles in Fig. 3. B, average time course of unfolding at a constant force of 130 pN at varying temperatures: 15, 25, and 35 °C. Discontinuous black lines correspond to single exponential fits, yielding the values for the associated unfolding rate constants presented in Fig. 3.

FIGURE 3.

The effect of temperature on the force dependence of polyubiquitin unfolding. Semilogarithmic plot of the rate of unfolding of ubiquitin, k(F, T), as a function of the pulling force at: 5 °C (triangles), 15 °C (inverted triangles), 25 °C (circles), 35 °C (diamonds), and 45 °C (squares). The solid colored lines in each case represent fits to the Arrhenius/Bell term (Equation 3 under “Results”) to the experimental data. From these fits, we obtained the associated Δx and k₀ value for each probed temperature. The range of forces probed for each particular temperature is chosen such that they do not compromise the measurement due to limited feedback response at fast unfolding rates and cantilever drift for slow unfolding kinetics.

FIGURE 4.

The value of the Δx is slightly increased upon raising the temperature in the range of 5–45 °C. The increase in Δx, although small, is significant (Student's t test for a 95% confidence). From the Arrhenius/Bell fit to the data (Fig. 3 and Equation 3 under “Results”), we obtained the Δx value for each given temperature. The obtained value ranges from 2.0 ± 0.3 Å at 5 °C to 2.7 ± 0.4 Å at 45 °C. Linear fit to the data (dashed line) yields a slope of (1.6 ± 0.5) × 10⁻³ nm/°C.

FIGURE 5.

The dependence of k₀ with 1/T allows calculation of the unfolding energy barrier and the collision frequency factor. From the Arrhenius/Bell fit shown in Fig. 3, we obtained the value of the unfolding rate in the absence of force k₀ for each temperature probed: 5 °C (triangle), 15 °C (inverted triangle), 25 °C (circle), 35 °C (diamond), and 45 °C (square). The solid black line represents the fit of Equation 1 to the data, giving rise to values for the activation barrier of ΔG^‡ = 71 ± 5 kJ/mol and for exponential prefactor, υ^‡ = ∼10^{9.6 ± 0.99} (±S.D.), corresponding to a mean value of υ^‡ = 4 × 10⁹ s⁻¹.