Mechanical Unfolding of Single Polyubiquitin Molecules Reveals Evidence of Dynamic Disorder

Abstract

Mechanical unfolding of single polyubiquitin molecules subjected to a constant stretching force showed nonexponentiality in the measured probability density of unfolding (waiting time distribution) and the survival probability of the folded state during the course of the measurements. These observations explored the relevance of disorder present in the system under study with implications for a static disorder approach to rationalize the experimental results. Here, an approach for dynamic disorder is presented based on Zwanzig’s fluctuating bottleneck (FB) model, in which the rate of the reaction is controlled by the passage through the cross-sectional area of the bottleneck. The radius of the latter undergoes stochastic fluctuations that in turn is described in terms of the end-to-end distance fluctuations of the Rouse-like dynamics using a non-Markovian generalized Langevin equation with a memory kernel and Gaussian colored noise. Our results are comprised of analytical expressions for the survival probability and waiting time distribution, which show excellent agreement with the experimental data throughout the range of the applied forces. In addition, by fitting the survival probabilities at different stretching forces, we quantify two system parameters, namely, the average free energy ΔG av and the average distance to the transition state Δx av, both perfectly recovered the experimental estimates. These agreements validate the present model of polymer dynamics, which captures the very essence of dynamic disorder in single-molecule pulling experiments.

An Exactly Solvable Stochastic Kinetic Theory of Single-Molecule Force Experiments

“…A similar trend was observed previously. 18,19 A few more points need to be addressed to wind up the above discussion. Our work provides a better analysis of the experiment when compared with the Bell model; the latter predicts exponential f(t), f(t) = k u exp[−k u t], and S(t), S(t) = exp[−k u t], and therefore not suitable to satisfy the measured data.…”

A Revisit to Turnover Kinetics of IndividualEscherichia coliβ-Galactosidase Molecules

“…We encountered similar equations in the recent studies of force unfolding of a protein molecule 22 and the escape of ssDNA molecules from an α-hemolysin channel 21 under the force clamp and voltage clamp conditions, respectively. Starting from eq 14, details of the steps to get an expression for the survival probability in the Laplace domain are outlined in these references.…”

“…Utilizing Zwanzig’s indirect approach, a detailed calculation then yields the Smoluchowski equation for the noise averaged survival probability with a quadratic sink, stated as follows where the time-dependent diffusion coefficient . We encountered similar equations in the recent studies of force unfolding of a protein molecule and the escape of ssDNA molecules from an α -hemolysin channel under the force clamp and voltage clamp conditions, respectively. Starting from eq , details of the steps to get an expression for the survival probability in the Laplace domain are outlined in these references.…”

“…In the second case, we avoid the mechanistic details of product formation and delineate the structural transitions of the biocatalyst with the temporal fluctuations of the spatial distance between the different locations along a coarse-grained polymer chain . Such an idea appeared suitable in a number of recent and earlier studies, where excellent recovery of the observed results produced by the seemingly different experiments was established. − In accordance with our present view of the reaction, it is sufficient only to regard for substrate to product conversion, which in turn allows us to determine the probability of survival of the substrate molecule, S ( t ), that the latter survives up to time t . The first derivative of S ( t ) yields the waiting time distribution of the enzymatic turnovers.…”

Nonexponential kinetics captured in sequential unfolding of polyproteins over a range of loads

“…However, several studies have shown that the resulting empirical distributions did not follow the Poisson distributions ( Brujic et al., 2006 , 2007 ; Garcia-Manyes et al., 2007 ). This observation led to an effort to explain the measured deviation from exponentiality by means of static and dynamic disorders ( Kuo et al., 2010 ; Chatterjee and Cherayil, 2011 ; Zheng et al., 2014 ; Costescu et al., 2017 ; Kundu et al., 2020 ), corrugated energy landscapes ( Brujic et al., 2006 ; Lannon et al., 2012 ), and as consequence of the polymeric nature of the proteins ( Bell and Terentjev, 2016 ). In their pioneering computational work, Bura et al., showed how unfolding in polyproteins become less correlated when reducing the applied force from 88 to 66 ​pN ( Bura et al., 2007 , 2008 ).…”