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Department of Biological Sciences, Columbia University, New York, New York 10027
Correspondence: Address reprint requests to Jasna Bruji at her present address, Center for Soft Matter Research, Dept. of Physics, New York University, New York, NY 10003. E-mail: jb2929{at}nyu.edu; E-mail: jb2379{at}columbia.edu; or to J. M. Fernandez, E-mail: jfernandez{at}columbia.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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= 0.6 s–1 for all N. The remainder of the data that cannot be accounted for suggests alternative unfolding barriers in the energy landscape of the protein. This method investigates the statistical features of unfolding beyond the average measurement of a single rate constant, thus providing an attractive alternative for measuring kinetics by force-clamp spectroscopy. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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), including the function of ion channels in cell membranes (2,3) and the evoked synaptic transmission in neurons (4). With the advent of new single molecule techniques that can probe folding and unfolding of biomolecules over time (5–9), appropriate histogram fits and maximum-likelihood methods (10,11) to overcome problems of noise and artifacts are becoming increasingly important in the field. In these experiments, a single module protein fluctuates over time between different conformational states that can be observed and their lifetimes measured. If the reaction is modeled by the simplest two-state Markov process, as is often the case in protein (un)folding (12,13), the distribution of unfolding times can be fitted by a single exponential curve, yielding the rate constant of the reaction.
On the other hand, many modular proteins perform their function in tandem, such as the immunoglobulin modules in the muscle protein titin (14) or multiple ubiquitin modules in protein degradation (15). The influence of the proximity of multiple protein modules in a chain on the unfolding process is a biologically relevant question, since it may introduce correlations via domain swapping (16), module-module interactions (17), aggregation (18) or energy storage through the molecular spring (19). A wealth of information about the underlying kinetics of the process can be inferred from the distribution of unfolding times of the individual protein modules. However, the number of modules in the chain, N, and the order number of each unfolding event in the chain, k, must first be taken into account in the statistical analysis. Such an analysis can then test the system's dynamics, as purely stochastic (random) or including memory (correlations), as well as the presence of alternative unfolding energy barriers.
Force-clamp spectroscopy using atomic force microscopy (AFM) unambiguously follows single polyprotein unfolding and refolding trajectories under a stretching force as a function of protein end-to-end length (13,20). Identical protein modules in the chain serve as a firm signature of the single molecule in the experiment. These measurements then yield the precise dwell time, t, defined as the time it takes for each module in the chain to unfold from the moment the force is applied, which is marked by a step increase in length in the force-clamp trajectories. Measuring the dwell times to a large number of individual unfolding events allows for a statistical analysis of the system's kinetics. Previously, we have shown that the average of just a few such unfolding trajectories, analogous to bulk measurements (12), can be fitted with a single exponential curve, suggesting that the process at a given force appears stochastic (Markovian) (13). Moreover, the average rate of unfolding, , was shown to be exponentially dependent on the stretching force (21).
Here, with a large statistical pool of 1647 unfolding events of ubiquitin, we are able to perform thorough tests of Markovian behavior, the effects of the chain length N on the kinetics of unfolding, as well as the validity of the two-state model for unfolding. In the AFM, the cantilever picks up single molecule chains at random points on the surface, implying that any number of protein modules within a chain, N 
12, may be picked up in a pulling experiment. In single polyprotein chains, if the unfolding events are truly independent of one another, the number of modules in the chain N should not influence the time course of the average unfolding trajectory. On the other hand, histograms of dwell times can be fitted with a probability density function that is dependent on N. In this study we demonstrate that ubiquitin modules unfold independent of one another in each protein chain, refuting arguments of module-module correlations. Finally, the assumption of two-state unfolding is explored, by testing how successful a unique unfolding rate constant is in fitting all the dwell times using the derived stochastic model. We conclude that a single rate constant is unable fit to all the obtained dwell-time distributions, which is indicative of other pathways in the free energy landscape. We thus provide a statistical analysis for data obtained by the force-clamp technique, while probing important questions as to the microscopic mechanism of unfolding of ubiquitin.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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) to form 12 identical domains in the polyprotein chain. The 12-mer ubiquitin is cloned into the pQE30 (Qiagen, Valencia, CA) expression vector and transformed into the BLR(DE3) Escherichia coli expression strain. This construct has a C-terminus His-tag and has two additional residues (arginine and serine) between each module in the chain. The proteins were purified by histidine metal-affinity chromatography with Talon resin (Clontech, Palo Alto, CA) and by gel-filtration using a Superdex 200 HR column (GE Biosciences). The length of the individual folded protein is 3.8 nm from the PDB and the step length at 110 pN is measured to be 20.4 ± 0.7 nm, totaling 24.2 ± 0.7 nm for the fully extended length of the unfolded protein. This corresponds to the unraveling of all the 78 amino acid residues in the protein structure.
The homemade AFM is constructed as described elsewhere (20). Using analog electronics based on a proportional-integral-derivative amplifier whose output is fed to the piezoelectric positioner, the AFM can be operated under force-clamp conditions, in which the cantilever is kept at a constant deflection (force) for a few seconds. The drift of the cantilever introduces an error in the force calibration and is important to address. There are mainly two sources of drift: mechanical relaxation of the whole experimental setup, and the bending of the cantilever because of temperature changes in the room, changes in adsorption on the surface, and the presence of air bubbles. The drift manifests itself in the length trajectories as a nonstationary value of the length over time at constant force, and can therefore be quantified. We have a procedure that rejects any trajectory with an error in the cantilever position of ±5 pN over the course of the experiment (several seconds), thus avoiding any influence of drift on the theoretical model and its results. This corresponds to a change in cantilever position of 
0.33 nm. Also, to avoid drift, we only start taking data when the experiment is equilibrated at room temperature to ensure constant force conditions.
The typical feedback response is set to 4–6 ms. The used cantilevers (Veeco, MLCT-AUHW, Woodbury, NY) are individually calibrated using the equipartition theorem (23) yielding spring constant values in the range of 12–18 pN/nm. As a result, the force-clamp technique maintains the calibrated cantilever in a constant position, such that the molecule is under a fixed stretching force for the time of the experiment. Typical measurements of protein length yield peak-to-peak noise of ±0.5 nm. The polyprotein is suspended in phosphate buffer solution on a gold cover slide.
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RESULTS |
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= 0.90–1.57 s–1). This narrow range and the absence of a trend as a function of N demonstrates that the average unfolding rate is independent of N. Because the length of the chain does not influence the average rate of unfolding, it would seem that the events are independent of one another. The average over all the data (i.e., chain lengths of N = 3–12, black curve) naturally passes in between the rest of the curves and is fit with a single exponential (dashed gray curve) with = 1.03 s–1. This result is in remarkable agreement with the previous estimation of the unfolding rate at 110 pN, = 1.0 ± 0.1 s–1, obtained from a fit of a much reduced unfolding data set as a function of force (13). Nevertheless, this large statistical pool of data reveals significant deviations from the single exponential fit, indicating the presence of other unfolding pathways, which are discussed in detail elsewhere (22).
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, the average time course of the normalized protein length, 
, as in Fig. 1, would then predict
 | (1) |
therefore agrees with the assumption of probabilistic Markovian unfolding, yet the deviations from the single exponential fits are indicative of violations of the two-state model for unfolding. Next, we probe the stochastic dynamics of the forced unfolding trajectories. We directly measure the dwell times to the unfolding events of individual protein modules in each polyubiquitin chain under a stretching force. The time it takes to observe multiple unfolding events as a function of chain length is then statistically investigated. If each module unfolds probabilistically with the same rate constant, it should take longer (on average) to observe three events out of three than out of a pool of 12 modules, given that there are more choices. Several such trends become apparent from the length versus time trajectories shown in Fig. 1, b and c. In agreement with the previous remark, the longer the chain length N, the shorter the dwell times to the same number of consecutive events, k, as observed in Fig. 1 b. Second, multiple events k in chains of the same length, e.g., N = 5, are shown to unfold at progressively longer times in Fig. 1 c. Because the unfolding process is stochastic and not deterministic, these observations are only true on average, and can be quantitatively shown in the dwell-time histograms of all the unfolding data. The dwell times of all the events that occur first in the sequence, k = 1, for different chain length ranges N in Fig. 3 a confirm faster unfolding times for longer chains. Also, dwell times to multiple events (k = 1, 4, 7) occurring in sequences of the same N range in Fig. 3 b show not only longer dwell times for more events, but also a change in the shape of the distributions from exponential for a single event to broad distributions for large k. These distributions indicate that the unfolding is probabilistic and that the dwell-time distributions depend on both N and k, which we next investigate theoretically.
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 | (2) |
Because the probability of a module to unfold in the time range [t, t + dt] is dt, it follows that p(t) = 1 – e–t. Thus the probability of the kth module to unfold in the time range [t, t + dt], is then just (N – (k – 1)) dt, since there are N – (k – 1) modules that remain folded. Summing over all the unfolding events, the probability P(t, N, k), that exactly k – 1 modules out of N unfold until time t and then the kth module unfolds in the time range [t, t + dt], is then
 | (3) |
This equation is normalized from zero to infinity over t.
We then used Monte Carlo simulations to test the probability theory. The simulations were based on a Markovian (stochastic) process, with a single unfolding rate constant of = 1.0 s–1, obtained for the average trajectory over all the data shown in Fig. 2. Each simulation generated 1000 length trajectories over time for each chain length N, while marking the unfolding events by an increase in length of 20 nm, mimicking the experiments. The resulting distribution of dwell times for multiple unfolding events k = 1, 3, 5, 7 for N = 7 is shown in Fig. 4. The success of the fits of Eq. 3 with = 1.0 s–1 confirms that the probability theory we derived captures the stochastic two-state dynamics.
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t 
for each event k and chain length N that are obtained from Eq. 3, according to
 | (4) |
For each N, the average dwell times are successfully fitted with Eq. 4, as shown in Fig. 5, with a relatively narrow distribution of rate constants, = 0.87 ± 0.13 s–1, shown in the inset. These variations in the average rates indicate discrepancies from the two-state model, but are also affected by the following experimental artifacts.
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) fits the majority of the data for each chain length N.
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-value and we solve Eq. 3 to obtain N as the maximum in the probability P. We therefore find the most likely N for each single molecule trajectory. We then determine the average rate constant that best fits all the dwell-time data, segregated by both k and the allocated N, as in the inset of Fig. 5. This average rate output from the data is then input into the binomial distribution in Eq. 3 in the next step of the iteration procedure. When the input and the output rates converge, we consider this to be the best possible rate constant that fits the data. Using this iteration procedure, we find the data to converge to = 0.6 s–1 from values far above and below it, but only 81% of the data is allocated an N that is within the possible experimental range. These data are shown in Fig. 6 a, beautifully fitting the binomial distribution across the available range of N and k with the predetermined single rate constant. The remainder of the data is allocated an unrealistic N value in this procedure. Comparing the normalized average trajectories of the data selected by the iteration procedure and the outliers, shown in Fig. 6 b, we observe a fourfold faster average rate constant for the outliers. Previous analysis presented in Fig. 5 included all the data, as it did not assume a single best-fitting rate constant for all N, such that the 19% representing the outliers are accounted for in the variations of the fitting rate constants.
These results suggest that there is a subpopulation of modules unfolding much faster than the most probable pathway (the most likely rate constant). Our statistical analysis reveals this heterogeneity at the chain level when the faster outlier modules are a majority. It is a coarse grained approach, which implies an even larger population of outliers at the single module level. One explanation for the outlier population is that there is at least one alternative pathway in the protein free energy landscape. However, even the populations separated by our iteration method significantly deviate from the single exponential fits in Fig. 6 b, which is suggestive of an even more complex unfolding landscape. Although we cannot rule out a small contribution from misfolded proteins on the surface with our technique, we have previously found that they occur in <2% of the cases and are accompanied by extensions that do not correspond to the signature in length of the protein (24). The outliers are therefore highly unlikely to arise from misfolding. We next investigate whether the noise in the stretching force arising from the thermal noise of the cantilever could be the source of the outlier population.
Fig. 7 a shows a typical length trajectory of a polyprotein unfolding and the corresponding constant force trace in Fig. 7 b. A histogram of forces experienced by the protein due to thermal noise of the cantilever is shown in Fig. 7 c, with an average force 
pN and a standard deviation of 
f = 6.82 pN. We investigate the influence of this force noise on the rate of unfolding and show that it is negligible compared to the deviations we measure with the outliers.
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 | (5) |
is the rate of unfolding, 0 is the rate of unfolding in the absence of force, 
x is the distance to the transition state, and kBT is the thermal energy. Because of Brownian motion the applied force is not constant, and has a probability distribution given by:
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,
 | (7) |
. This probability distribution has an expected value of
 | (8) |
Using the experimental values of f and 
in Fig. 7 we obtain an expected value of the narrow distribution of rates to be within 10% of the rate measured without taking the force noise into account. On the other hand, we find a four times higher average value of the rate of unfolding of the "outlier" population, which shows that this effect is negligible compared to the experimentally observed discrepancy.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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,17). Another signature of Markovian behavior was shown to be the binomial distribution of unfolding dwell times, in which the average fitting rate constants in Fig. 5 were also independent of N. The experimental data also proved that the dwell times to the unfolding events are dependent on N since N is small. For example, the longer the chain, the shorter one has to wait for the same number of events to occur. On the other hand, the last event in each chain takes place at later times for longer chains.
While the Markovian assumption was shown to hold well, we also discovered significant deviations from the two-state model, which implies a single rate constant for unfolding. First, the single exponential fits to the average trajectories in Figs. 2 and 6 b clearly strayed from the experimental data. Also, there were significant variations in the rate constant fits of the average dwell times as a function of N in the inset of Fig. 5. Furthermore, the single best fitting rate constant for the data according to the binomial model did not account for all the dwell times in Fig. 6 a. Instead, the two separated populations in Fig. 6 b had vastly different average rate constants and still could not be fitted with single exponentials. All these discrepancies suggest that multiple unfolding barriers do exist in the energy landscape of ubiquitin. Indeed, a theoretical analysis at the single molecule level facilitated the discovery of a distribution of unfolding pathways in the glassy landscape of ubiquitin (22). Our results are consistent with that finding, since the "selected" data in Fig. 6 b corresponds to the most probable pathway at the peak of the distribution in Fig. 4 of Oberhauser et al. (24), while the "outliers" account for the tail of the observed broad distribution.
In our experiments we are probing the lifetimes of different conformational states of the single protein molecules explored under a constant stretching force. Arguably, this is a much reduced phase space as compared to the molecules that are free in solution, which is determined by the pulling direction and the magnitude of the pulling force. We are perturbing the system away from its native state in solution to deduce the mechanical stability of the molecules. These states are relevant in nature as the cell is an environment that is often under stress (25). A protein's mechanical stability and conformational diversity under force are therefore relevant parameters in nature.
Bulk kinetic measurements of ubiquitin show both single pathway mechanisms and intermediates that appear as a function of unfolding conditions (26–32). However, they are unable to distinguish between the pathways of individual molecules that are blurred in the ensemble. Single molecule data has the important advantage of distinguishing between the individual molecules, which reveals a wealth of new information. We have demonstrated how measuring dwell-time distributions can produce kinetic signatures of the conformations of ubiquitin at any given pulling force. Such a capability is useful, particularly for future studies of how the unfolding path depends on thermodynamic variables, such as the temperature. With the advent of new statistical energy landscape theories of protein folding and insight into the complexity of the dynamics of folded protein structures, there is an increasing need for more sophisticated statistical analysis of protein unfolding/folding measurements.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Work on this project was supported by National Institutes of Health grants R01 HL66030 and R01 HL61228 to J.M.F. J.B., PhD., holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund.
Submitted on October 13, 2006; accepted for publication December 27, 2006.
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REFERENCES |
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2 = 40. The average trace over all the data is also shown with 
and a mean ± SD of