Force-clamp analysis techniques give highest rank to stretched exponential unfolding kinetics in ubiquitin

Abstract

Force-clamp spectroscopy reveals the unfolding and disulfide bond rupture times of single protein molecules as a function of the stretching force, point mutations, and solvent conditions. The statistics of these times reveal whether the protein domains are independent of one another, the mechanical hierarchy in the polyprotein chain, and the functional form of the probability distribution from which they originate. It is therefore important to use robust statistical tests to decipher the correct theoretical model underlying the process. Here, we develop multiple techniques to compare the well-established experimental data set on ubiquitin with existing theoretical models as a case study. We show that robustness against filtering, agreement with a maximum likelihood function that takes into account experimental artifacts, the Kuiper statistic test, and alignment with synthetic data all identify the Weibull or stretched exponential distribution as the best fitting model. Our results are inconsistent with recently proposed models of Gaussian disorder in the energy landscape or noise in the applied force as explanations for the observed nonexponential kinetics. Because the physical model in the fit affects the characteristic unfolding time, these results have important implications on our understanding of the biological function of proteins.

Figure 1

(A) A typical force-clamp unfolding trajectory of a single ubiquitin polyprotein pulled with a constant stretching force of 110 pN. The beginning of the plateau that precedes the staircase of unfolding events marks time zero t₀ as the moment when the molecule is held taught under the applied force. The dwell times are then measured as the time interval between t₀ and each of the unfolding steps. Finally, the molecule detaches at t_d. The stepwise unfolding is illustrated in the schematic diagram. (B) Unfolding dwell times from the staircases are plotted on a semilog scale in the order that they are collected and show a broad and homogeneous distribution of times.

Figure 2

(A) The unfolding probability $F \left(t\right)$ for four models proposed in the literature is used to fit the same empirical CDF of dwell times. The normalization of each $F \left(t\right)$ leads to different timescales on which the data unfold. The inset shows the corresponding conditional $P \left(t\right)$ on a log-log plot to emphasize the goodness of fit at short times. (B) Changing the time window from 5 s in (A) to t_c shows the variability in the characteristic unfolding time between the different models. They span more than two orders of magnitude and only the Weibull and the exponential distribution settle to a given value. The inset shows how the number of data points changes as the time window is expanded.

Figure 3

Estimate of the fitting parameters in the Weibull in (A) and the Gaussian disorder distribution in (B) as a function of the experimental time window. Bayesian sampling shows that the fluctuations around the mean of the parameters diminish as the time window increases. The constant solid lines are the parameter values obtained from the maximum likelihood function in Eq. 3 and the dashed lines are their standard deviation.

Figure 4

A Kuiper statistic of 1 signifies a perfect match between the experimental data and the proposed distribution. Deviations from the line at 1 quantify the disagreement between the maximum likelihood function estimate for the four models and the experimental data set as a function of the experimental time window t_c.

Figure 5

Comparison between synthetic data sets generated using the parameters in the maximum likelihood function for the Weibull and Gaussian disorder distribution and the experimental data set of ubiquitin. The constant solid lines are the parameter values obtained from the maximum likelihood function in Eq. 3 and the dashed lines are their standard deviation. Fitting the three data sets using the Weibull distribution gives the fluctuations in parameter a in (A) and b in (B) and using the Gaussian disorder distribution gives k_F in (C) and σ in (D). The ubiquitin data and the synthetic Weibull distribution behave similarly above t_c = 3 s in all cases, but the synthetic Gaussian distribution is significantly different.