A Theoretical Model for the Mechanical Unfolding of Repeat Proteins

We consider the mechanical stretching of a polypeptide chain formed by multiple interacting repeats. The folding thermodynamics and the interactions among the repeats are described by the Ising model. Unfolded repeats act as soft entropic springs, whereas folded repeats respond to a force as stiffer springs. We show that the resulting force-extension curve may exhibit a pronounced force maximum corresponding to the unfolding of the first repeat. This event is followed by the unfolding of the remaining repeats, which takes place at a lower force. As the protein extension is increased, the force-extension curve of a sufficiently long repeat protein displays a plateau, where the force remains nearly constant and the protein unfolds sequentially so that the number of unfolded repeats is proportional to the extension. Such a sequential mechanical unfolding mechanism is displayed even by the repeat proteins whose thermal denaturation is highly cooperative, provided that they are long enough. By contrast, the unfolding of short repeat progressions can be cooperative.     

Dwell-time distribution analysis of polyprotein unfolding using force-clamp spectroscopy

Using the recently developed single molecule force-clamp technique we quantitatively measure the kinetics of conformational changes of polyprotein molecules at a constant force. In response to an applied force of 110 pN, we measure the dwell times of 1647 unfolding events of individual ubiquitin modules within each protein chain. We then establish a rigorous method for analyzing force-clamp data using order statistics. This allows us to test the success of a history-independent, two-state model in describing the kinetics of the unfolding process. We find that the average unfolding trajectory is independent of the number of protein modules N in each trajectory, which varies between 3 and 12 (the engineered protein length), suggesting that the unfolding events in each chain are uncorrelated. We then derive a binomial distribution of dwell times to describe the stochastic dynamics of protein unfolding. This distribution successfully describes 81% of the data with a single rate constant of alpha = 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.     

Spectrin domains lose cooperativity in forced unfolding

Spectrin is a multidomain cytoskeletal protein, the component three-helix bundle domains are expected to experience mechanical force in vivo. In thermodynamic and kinetic studies, neighboring domains of chicken brain alpha-spectrin R16 and R17 have been shown to behave cooperatively. Is this cooperativity maintained under force? The effect of force on these spectrin domains was investigated using atomic force microscopy. The response of the individual domains to force was compared to that of the tandem repeat R1617. Importantly, nonhelical linkers (all-beta immunoglobulin domains) were used to avoid formation of nonnative helical linkers. We show that, in contrast to previous studies on spectrin repeats, only 3% of R1617 unfolding events gave an increase in contour length consistent with cooperative two-domain unfolding events. Furthermore, the unfolding forces for R1617 were the same as those for the unfolding of R16 or R17 alone. This is a strong indication that the cooperative unfolding behavior observed in the stopped-flow studies is absent between these spectrin domains when force is acting as a denaturant. Our evidence suggests that the rare double unfolding events result from misfolding between adjacent repeats. We suggest that this switch from cooperative to independent behavior allows multidomain proteins to maintain integrity under applied force.     

Coarse-Grained Simulations of Topology-Dependent Mechanisms of Protein Unfolding and Translocation Mediated by ClpY ATPase Nanomachines

Clp ATPases are powerful ring shaped nanomachines which participate in the degradation pathway of the protein quality control system, coupling the energy from ATP hydrolysis to threading substrate proteins (SP) through their narrow central pore. Repetitive cycles of sequential intra-ring ATP hydrolysis events induce axial excursions of diaphragm-forming central pore loops that effect the application of mechanical forces onto SPs to promote unfolding and translocation. We perform Langevin dynamics simulations of a coarse-grained model of the ClpY ATPase-SP system to elucidate the molecular details of unfolding and translocation of an α/β model protein. We contrast this mechanism with our previous studies which used an all-α SP. We find conserved aspects of unfolding and translocation mechanisms by allosteric ClpY, including unfolding initiated at the tagged C-terminus and translocation via a power stroke mechanism. Topology-specific aspects include the time scales, the rate limiting steps in the degradation pathway, the effect of force directionality, and the translocase efficacy. Mechanisms of ClpY-assisted unfolding and translocation are distinct from those resulting from non-allosteric mechanical pulling. Bulk unfolding simulations, which mimic Atomic Force Microscopy-type pulling, reveal multiple unfolding pathways initiated at the C-terminus, N-terminus, or simultaneously from both termini. In a non-allosteric ClpY ATPase pore, mechanical pulling with constant velocity yields larger effective forces for SP unfolding, while pulling with constant force results in simultaneous unfolding and translocation.     

Single-molecule unfolding force distributions reveal a funnel-shaped energy landscape

The protein folding process is described as diffusion on a high-dimensional energy landscape. Experimental data showing details of the underlying energy surface are essential to understanding folding. So far in single-molecule mechanical unfolding experiments a simplified model assuming a force-independent transition state has been used to extract such information. Here we show that this so-called Bell model, although fitting well to force velocity data, fails to reproduce full unfolding force distributions. We show that by applying Kramers' diffusion model, we were able to reconstruct a detailed funnel-like curvature of the underlying energy landscape and establish full agreement with the data. We demonstrate that obtaining spatially resolved details of the unfolding energy landscape from mechanical single-molecule protein unfolding experiments requires models that go beyond the Bell model.     

Predicting the order in which contacts are broken during single molecule protein stretching experiments

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force-displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments.     

A novel pattern recognition algorithm to classify membrane protein unfolding pathways with high-throughput single-molecule force spectroscopy

MOTIVATION: Misfolding of membrane proteins plays an important role in many human diseases such as retinitis pigmentosa, hereditary deafness and diabetes insipidus. Little is known about membrane proteins as there are only very few high-resolution structures. Single-molecule force spectroscopy is a novel technique, which measures the force necessary to pull a protein out of a membrane. Such force curves contain valuable information on the protein structure, conformation, and inter- and intra-molecular forces. High-throughput force spectroscopy experiments generate hundreds of force curves including spurious ones and good curves, which correspond to different unfolding pathways. Manual analysis of these data is a bottleneck and source of inconsistent and subjective annotation. RESULTS: We propose a novel algorithm for the identification of spurious curves and curves representing different unfolding pathways. Our algorithm proceeds in three stages: first, we reduce noise in the curves by applying dimension reduction; second, we align the curves with dynamic programming and compute pairwise distances and third, we cluster the curves based on these distances. We apply our method to a hand-curated dataset of 135 force curves of bacteriorhodopsin mutant P50A. Our algorithm achieves a success rate of 81% distinguishing spurious from good curves and a success rate of 76% classifying unfolding pathways. As a result, we discuss five different unfolding pathways of bacteriorhodopsin including three main unfolding events and several minor ones. Finally, we link folding barriers to the degree of conservation of residues. Overall, the algorithm tackles the force spectroscopy bottleneck and leads to more consistent and reproducible results paving the way for high-throughput analysis of structural features of membrane proteins.     

An analysis of the influence of protein intrinsic dynamical properties on its thermal unfolding behavior

The influence of the protein topology-encoded dynamical properties on its thermal unfolding motions was studied in the present work. The intrinsic dynamics of protein topology was obtained by the anisotropic network model (ANM). The ANM has been largely used to investigate protein collective functional motions, but it is not well elucidated if this model can also reveal the preferred large-scale motions during protein unfolding. A small protein barnase is used as a typical case study to explore the relationship between protein topology-encoded dynamics and its unfolding motions. Three thermal unfolding simulations at 500 K were performed for barnase and the entire unfolding trajectories were sampled and partitioned into several windows. For each window, the preferred unfolding motions were investigated by essential dynamics analysis, and then associated with the intrinsic dynamical properties of the starting conformation in this window, which is detected by ANM. The results show that only a few slow normal modes imposed by protein structure are sufficient to give a significant overlap with the preferred unfolding motions. Especially, the large amplitude unfolding movements, which imply that the protein jumps out of a local energy basin, can be well described by a single or several ANM slow modes. Besides the global motions, it is also found that the local residual fluctuations encoded in protein structure are highly correlated with those in the protein unfolding process. Furthermore, we also investigated the relationship between protein intrinsic flexibility and its unfolding events. The results show that the intrinsic flexible regions tend to unfold early. Several early unfolding events can be predicted by analysis of protein structural flexibility. These results imply that protein structure-encoded dynamical properties have significant influences on protein unfolding motions.     

Comparing proteins by their unfolding pattern

Single molecule force spectroscopy has evolved into an important and extremely powerful technique for investigating the folding potentials of biomolecules. Mechanical tension is applied to individual molecules, and the subsequent, often stepwise unfolding is recorded in force extension traces. However, because the energy barriers of the folding potentials are often close to the thermal energy, both the extensions and the forces at which these barriers are overcome are subject to marked fluctuations. Therefore, force extension traces are an inadequate representation despite widespread use particularly when large populations of proteins need to be compared and analyzed. We show in this article that contour length, which is independent of fluctuations and alterable experimental parameters, is a more appropriate variable than extension. By transforming force extension traces into contour length space, histograms are obtained that directly represent the energy barriers. In contrast to force extension traces, such barrier position histograms can be averaged to investigate details of the unfolding potential. The cross-superposition of barrier position histograms allows us to detect and visualize the order of unfolding events. We show with this approach that in contrast to the sequential unfolding of bacteriorhodopsin, two main steps in the unfolding of the enzyme titin kinase are independent of each other. The potential of this new method for accurate and automated analysis of force spectroscopy data and for novel automated screening techniques is shown with bacteriorhodopsin and with protein constructs containing GFP and titin kinase.     

Molecular dynamics simulations of a protein model in uniform and elongational flows

We present a molecular dynamics study of the conformational deformation of a minimalist beta-barrel protein model in two different types of hydrodynamic flows: uniform and elongational. We investigate the characteristics of protein stretching, paying special attention to the unfolding intermediate states and their relationship to the protein folding/unfolding problem. In the uniform flow simulations, one end of the modeled protein was tethered to a fixed point in space and the forced unfolding process was observed. The unfolding takes place via a few stages involving one or two intermediate states, depending on which end is tethered. The calculated force-extension curves show plateau regimes and hysteresis as the protein is stretched and refolded, in qualitative agreement with the experimental measurements. The physical behavior observed in our numerical simulations of the forced unfolding in an elongational flow is very different from that in uniform flow. The protein unfolds abruptly from the globular state to a fully stretched state without going through any observable intermediate states. From these observation, we stress that protein unfolding pathways under the influence of an external force are highly dependent on the mechanism of the exerted force.     

Repetitive pulling catalyzes co-translocational unfolding of barnase during import through a mitochondrial pore

We present a computational study of barnase unfolding during import into mitochondria through a model translocon. In contrast to thermal (or chemical) unfolding, the major intermediates of co-translocational unfolding are mainly mediated by non-native interactions accompanying the protein configurations induced by pulling forces. These energy contributions, combined with backbone topological constraints imposed by the model pore, result in milestones along the unfolding pathways which are significantly different not only from those experienced during thermal (or chemical) denaturation, but also from those observed in single-molecule pulling by both ends without pore constraints. Two on-pathway major translocation intermediates trapped in long-lived states by significantly high unfolding barriers are identified. A fraction of these pathways can, however, skip such local kinetic traps and result in extremely fast translocations, leading to a dramatic kinetic partitioning spanning approximately four orders of magnitude. The fraction of fast translocation events is shown to increase upon switching the pull off and on, when compared to pulling at constant force. This suggests a "catalytic" mechanism by which the mitochondrial import machinery regulates this partitioning by repetitively pulling in cycles. A number of mutation sites that alter the kinetic "flow" of the unfolding trajectories are suggested and tested.     

Molecular force modulation spectroscopy revealing the dynamic response of single bacteriorhodopsins

Recent advances in atomic force microscopy allowed globular and membrane proteins to be mechanically unfolded on a single-molecule level. Presented is an extension to the existing force spectroscopy experiments. While unfolding single bacteriorhodopsins from native purple membranes, small oscillation amplitudes (6-9 nm) were supplied to the vertical displacement of the cantilever at a frequency of 3 kHz. The phase and amplitude response of the cantilever-protein system was converted to reveal the elastic (conservative) and viscous (dissipative) contributions to the unfolding process. The elastic response (stiffness) of the extended parts of the protein were in the range of a few tens pN/nm and could be well described by the derivative of the wormlike chain model. Discrete events in the viscous response coincided with the unfolding of single secondary structure elements and were in the range of 1 microNs/m. In addition, these force modulation spectroscopy experiments revealed novel mechanical unfolding intermediates of bacteriorhodopsin. We found that kinks result in a loss of unfolding cooperativity in transmembrane helices. Reconstructing force-distance spectra by the integration of amplitude-distance spectra verified their position, offering a novel approach to detect intermediates during the forced unfolding of single proteins.     

Efficient unfolding pattern recognition in single molecule force spectroscopy data

Background

Single-molecule force spectroscopy (SMFS) is a technique that measures the force necessary to unfold a protein. SMFS experiments generate Force-Distance (F-D) curves. A statistical analysis of a set of F-D curves reveals different unfolding pathways. Information on protein structure, conformation, functional states, and inter- and intra-molecular interactions can be derived.

Results

In the present work, we propose a pattern recognition algorithm and apply our algorithm to datasets from SMFS experiments on the membrane protein bacterioRhodopsin (bR). We discuss the unfolding pathways found in bR, which are characterised by main peaks and side peaks. A main peak is the result of the pairwise unfolding of the transmembrane helices. In contrast, a side peak is an unfolding event in the alpha-helix or other secondary structural element. The algorithm is capable of detecting side peaks along with main peaks. Therefore, we can detect the individual unfolding pathway as the sequence of events labeled with their occurrences and co-occurrences special to bR's unfolding pathway. We find that side peaks do not co-occur with one another in curves as frequently as main peaks do, which may imply a synergistic effect occurring between helices. While main peaks co-occur as pairs in at least 50% of curves, the side peaks co-occur with one another in less than 10% of curves. Moreover, the algorithm runtime scales well as the dataset size increases.

Conclusions

Our algorithm satisfies the requirements of an automated methodology that combines high accuracy with efficiency in analyzing SMFS datasets. The algorithm tackles the force spectroscopy analysis bottleneck leading to more consistent and reproducible results.     

Mechanically unfolding protein L using a laser-feedback-controlled cantilever

Force spectroscopy using the atomic force microscope (AFM) can yield important information on the strength and lifetimes of the folded states of single proteins and their complexes when they are loaded with force. For example, by mechanically unfolding concatenated proteins at different velocities, a dynamic force spectrum can be built up that allows reconstruction of the energy landscape that the protein traverses during unfolding. To characterize fully the unfolding landscape, however, it is necessary both to explore the entire force spectrum and to characterize each species populated during unfolding. In the conventional AFM apparatus, force is applied to the protein construct through a compliant cantilever. This limits the dynamic range of the force spectrum that can be probed, and the cantilever recoil after unfolding may mask the presence of metastable intermediates. Here, we describe to our knowledge a new technique—constant-deflection AFM—in which the compliance of the AFM cantilever is removed. Using this technique, we show that protein L exhibits a more complex unfolding energy landscape than previously detected using the conventional technique. This technique is also able to detect the presence of a refolding intermediate whose formation is otherwise prevented by cantilever recoil.     

Analyzing forced unfolding of protein tandems by ordered variates, 1: Independent unfolding times

Most of the mechanically active proteins are organized into tandems of identical repeats, (D)N, or heterogeneous tandems, D1-D2-...-DN. In current atomic force microscopy experiments, conformational transitions of protein tandems can be accessed by employing constant stretching force f (force-clamp) and by analyzing the recorded unfolding times of individual domains. Analysis of unfolding data for homogeneous tandems relies on the assumption that unfolding times are independent and identically distributed, and involves inference of the (parent) probability density of unfolding times from the histogram of the combined unfolding times. This procedure cannot be used to describe tandems characterized by interdomain interactions, or heteregoneous tandems. In this article, we introduce an alternative approach that is based on recognizing that the observed data are ordered, i.e., first, second, third, etc., unfolding times. The approach is exemplified through the analysis of unfolding times for a computer model of the homogeneous and heterogeneous tandems, subjected to constant force. We show that, in the experimentally accessible range of stretching forces, the independent and identically distributed assumption may not hold. Specifically, the uncorrelated unfolding transitions of individual domains at lower force may become correlated (dependent) at elevated force levels. The proposed formalism can be used in atomic force microscopy experiments to infer the unfolding time distributions of individual domains from experimental histograms of ordered unfolding times, and it can be extended to analyzing protein tandems that exhibit interdomain interactions.     

Protein Unfolding by Biological Unfoldases: Insights from Modeling

The molecular determinants of the high efficiency of biological machines like unfoldases (e.g., the proteasome) are not well understood. We propose a model to study protein translocation into the chamber of biological unfoldases represented as a funnel. It is argued that translocation is a much faster way of unfolding a protein than end-to-end stretching, especially in a low-force regime, because it allows for a conformational freedom while concentrating local tension on consecutive regions of a protein chain and preventing refolding. This results in a serial unfolding of the protein structures dominated by unzipping. Thus, pulling against the unfoldase pore is an efficient catalyst of the unfolding reaction. We also show that the presence of the funnel makes the tension along the backbone of the substrate protein nonuniform even when the protein gets unfolded. Hence, the stalling force measured by single-molecule force spectroscopy techniques may be smaller than the traction force of the unfoldase motor.     

Protein Unfolding by Biological Unfoldases: Insights from Modeling

The molecular determinants of the high efficiency of biological machines like unfoldases (e.g., the proteasome) are not well understood. We propose a model to study protein translocation into the chamber of biological unfoldases represented as a funnel. It is argued that translocation is a much faster way of unfolding a protein than end-to-end stretching, especially in a low-force regime, because it allows for a conformational freedom while concentrating local tension on consecutive regions of a protein chain and preventing refolding. This results in a serial unfolding of the protein structures dominated by unzipping. Thus, pulling against the unfoldase pore is an efficient catalyst of the unfolding reaction. We also show that the presence of the funnel makes the tension along the backbone of the substrate protein nonuniform even when the protein gets unfolded. Hence, the stalling force measured by single-molecule force spectroscopy techniques may be smaller than the traction force of the unfoldase motor.