Order Statistics Theory of Unfolding of Multimeric Proteins

Dynamic force spectroscopy has become indispensable for the exploration of the mechanical properties of proteins. In force-ramp experiments, performed by utilizing a time-dependent pulling force, the peak forces for unfolding transitions in a multimeric protein (D)_N are used to map the free energy landscape for unfolding for a protein domain D. We show that theoretical modeling of unfolding transitions based on combining the observed first (f₁), second (f₂), …, N^th (f_N) unfolding forces for a protein tandem of fixed length N, and pooling the force data for tandems of different length, n₁ < n₂ < … < N, leads to an inaccurate estimation of the distribution of unfolding forces for the protein D, ψ_D(f). This problem can be overcome by using Order statistics theory, which, in conjunction with analytically tractable models, can be used to resolve the molecular characteristics that determine the unfolding micromechanics. We present a simple method of estimation of the parent distribution, ψ_D(f), based on analyzing the force data for a tandem (D)_n of arbitrary length n. Order statistics theory is exemplified through a detailed analysis and modeling of the unfolding forces obtained from pulling simulations of the monomer and oligomers of the all-β-sheet WW domain.     

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.     

Can non-mechanical proteins withstand force? Stretching barnase by atomic force microscopy and molecular dynamics simulation

Atomic force microscopy (AFM) experiments have provided intriguing insights into the mechanical unfolding of proteins such as titin I27 from muscle, but will the same be possible for proteins that are not physiologically required to resist force? We report the results of AFM experiments on the forced unfolding of barnase in a chimeric construct with I27. Both modules are independently folded and stable in this construct and have the same thermodynamic and kinetic properties as the isolated proteins. I27 can be identified in the AFM traces based on its previous characterization, and distinct, irregular low-force peaks are observed for barnase. Molecular dynamics simulations of barnase unfolding also show that it unfolds at lower forces than proteins with mechanical function. The unfolding pathway involves the unraveling of the protein from the termini, with much more native-like secondary and tertiary structure being retained in the transition state than is observed in simulations of thermal unfolding or experimentally, using chemical denaturant. Our results suggest that proteins that are not selected for tensile strength may not resist force in the same way as those that are, and that proteins with similar unfolding rates in solution need not have comparable unfolding properties under force.     

Complex Stability of Single Proteins Explored by Forced Unfolding Experiments

In the last decade atomic force microscopy has been used to measure the mechanical stability of single proteins. These force spectroscopy experiments have shown that many water-soluble and membrane proteins unfold via one or more intermediates. Recently, Li and co-workers found a linear correlation between the unfolding force of the native state and the intermediate in fibronectin, which they suggested indicated the presence of a molecular memory or multiple unfolding pathways (1). Here, we apply two independent methods in combination with Monte Carlo simulations to analyze the unfolding of α-helices E and D of bacteriorhodopsin (BR). We show that correlation analysis of unfolding forces is very sensitive to errors in force calibration of the instrument. In contrast, a comparison of relative forces provides a robust measure for the stability of unfolding intermediates. The proposed approach detects three energetically different states of α-helices E and D in trimeric BR. These states are not observed for monomeric BR and indicate that substantial information is hidden in forced unfolding experiments of single proteins.     

Analyzing forced unfolding of protein tandems by ordered variates, 2: dependent unfolding times

Statistical analyses of forced unfolding data for protein tandems, i.e., unfolding forces (force-ramp) and unfolding times (force-clamp), used in single-molecule dynamic force spectroscopy rely on the assumption that the unfolding transitions of individual protein domains are independent (uncorrelated) and characterized, respectively, by identically distributed unfolding forces and unfolding times. In our previous work, we showed that in the experimentally accessible piconewton force range, this assumption, which holds at a lower constant force, may break at an elevated force level, i.e., the unfolding transitions may become correlated when force is increased. In this work, we develop much needed statistical tests for assessing the independence of the unobserved forced unfolding times for individual protein domains in the tandem and equality of their parent distributions, which are based solely on the observed ordered unfolding times. The use and performance of these tests are illustrated through the analysis of unfolding times for computer models of protein tandems. The proposed tests can be used in force-clamp atomic force microscopy experiments to obtain accurate information on protein forced unfolding and to probe data on the presence of interdomain interactions. The order statistics-based formalism is extended to cover the analysis of correlated unfolding transitions. The use of order statistics leads naturally to the development of new kinetic models, which describe the probabilities of ordered unfolding transitions rather than the populations of chemical species.     

Mechanical Characterization of Protein L in the Low-Force Regime by Electromagnetic Tweezers/Evanescent Nanometry

Mechanical manipulation at the single molecule level of proteins exhibiting mechanical stability poses a technical challenge that has been almost exclusively approached by atomic force microscopy (AFM) techniques. However, due to mechanical drift limitations, AFM techniques are restricted to experimental recordings that last less than a minute in the high-force regime. Here we demonstrate a novel combination of electromagnetic tweezers and evanescent nanometry that readily captures the forced unfolding trajectories of protein L at pulling forces as low as 10 ∼ 15 pN. Using this approach, we monitor unfolding and refolding cycles of the same polyprotein for a period of time longer than 30 min. From such long-lasting recordings, we obtain ensemble averages of unfolding step sizes and rates that are consistent with single-molecule AFM data obtained at higher stretching forces. The unfolding kinetics of protein L at low stretching forces confirms and extends the observations that the mechanical unfolding rate is exponentially dependent on the pulling force within a wide range of stretching forces spanning from 13 pN up to 120 pN. Our experiments demonstrate a novel approach for the mechanical manipulation of single proteins for extended periods of time in the low-force regime.     

Unfolding the A2 Domain of Von Willebrand Factor with the Optical Trap

Von Willebrand factor (VWF) is a multimeric plasma glycoprotein involved in both hemostasis and thrombosis. VWF conformational changes, especially unfolding of the A2 domain, may be required for efficient enzymatic cleavage in vivo. It has been shown that a single A2 domain unfolds at most probable unfolding forces of 7-14 pN at force loading rates of 0.35-350 pN/s and A2 unfolding facilitates A2 cleavage in vitro. However, it remains unknown how much force is required to unfold the A2 domain in the context of a VWF multimer where A2 may be stabilized by other domains like A1 and A3. With the optical trap, we stretched VWF multimers and a poly-protein (A1A2A3)₃ that contains three repeats of the triplet A1A2A3 domains at constant speeds of 2000 nm/s and 400 nm/s, respectively, which yielded corresponding average force loading rates of 90 and 22 pN/s. We found that VWF multimers became stiffer when they were stretched and extended by force. After force increased to a certain level, sudden extensional jumps that signify domain unfolding were often observed. Histograms of the unfolding force and the unfolded contour length showed two or three peaks that were integral multiples of ∼21 pN and ∼63 nm, respectively. Stretching of (A1A2A3)₃ yielded comparable distributions of unfolding force and unfolded contour length, showing that unfolding of the A2 domain accounts for the behavior of VWF multimers under tension. These results show that the A2 domain can be indeed unfolded in the presence of A1, A3, and other domains. Compared with the value in the literature, the larger most probable unfolding force measured in this study suggests that the A2 domain is mechanically stabilized by A1 or A3 although variations in experimental setups and conditions may complicate this interpretation.     

Dephosphorylation of r Factor by Protein Phosphatase 2A in Synaptosomal Cytosol Fractions, and Inhibition by Aluminum

When the synaptosomal cytosol fraction from rat brain was chromatographed on a DEAE-cellulose column and assayed for protein phosphatases for τ factor and histone H1, two peaks of activities, termed peak 1 (major) and peak 2 (minor), were separated. Each peak was in a single form on Sephacryl S-300 column chromatography. Both peaks 1 and 2 dephosphorylated τ factor phosphorylated by Ca²⁺/calmodulin-dependent protein kinase II and the catalytic subunit of cyclic AMP-dependent protein kinase. The K_m values were in the range of 0.42–0.84 μM for τ factor. There were no differences in kinetic properties of dephosphorylation between the substrates phosphorylated by the two kinases. The phosphatase activities did not depend on Ca²⁺, Mn²⁺, and Mg²⁺. Immunoprecipitation and immunoblotting analysis using polyclonal antibodies to the catalytic subunit of brain protein phosphatase 2A revealed that both protein phosphatases are the holoenzymic forms of protein phosphatase 2A. Aluminum chloride inhibited the activities of both peaks 1 and 2 with IC₅₀ values of 40–60 μM. These results suggest that dephosphorylation of r factor in presynaptic nerve terminals is controlled mainly by protein phosphatase 2A and that the neurotoxic effect of aluminum seems to be related mostly to inhibition of dephosphorylation of τ factor     

Frequency modulation atomic force microscopy reveals individual intermediates associated with each unfolded I27 titin domain

In this study, we apply a dynamic atomic force microscopy (AFM) technique, frequency modulation (FM) detection, to the mechanical unfolding of single titin I27 domains and make comparisons with measurements made using the AFM contact or static mode method. Static mode measurements revealed the well-known force transition occurring at 100-120 pN in the first unfolding peak, which was less clear, or more often absent, in the subsequent unfolding peaks. In contrast, some FM-AFM curves clearly resolved a force transition associated with each of the unfolding peaks irrespective of the number of observed unfolded domains. As expected for FM-AFM, the frequency shift response of the main unfolding peaks and their intermediates could only be detected when the oscillation amplitudes used were smaller than the interaction lengths being measured. It was also shown that the forces measured for the dynamical interaction of the FM-AFM technique were significantly lower than those measured using the static mode. This study highlights the potential for using dynamic AFM for investigating biological interactions, including protein unfolding and the detection of novel unfolding intermediates.     

Mechanical Unfolding of Acylphosphatase Studied by Single-Molecule Force Spectroscopy and MD Simulations

Single-molecule manipulation methods provide a powerful means to study protein transitions. Here we combined single-molecule force spectroscopy and steered molecular-dynamics simulations to study the mechanical properties and unfolding behavior of the small enzyme acylphosphatase (AcP). We find that mechanical unfolding of AcP occurs at relatively low forces in an all-or-none fashion and is decelerated in the presence of a ligand, as observed in solution measurements. The prominent energy barrier for the transition is separated from the native state by a distance that is unusually long for α/β proteins. Unfolding is initiated at the C-terminal strand (β_T) that lies at one edge of the β-sheet of AcP, followed by unraveling of the strand located at the other. The central strand of the sheet and the two helices in the protein unfold last. Ligand binding counteracts unfolding by stabilizing contacts between an arginine residue (Arg-23) and the catalytic loop, as well as with β_T of AcP, which renders the force-bearing units of the protein resistant to force. This stabilizing effect may also account for the decelerated unfolding of ligand-bound AcP in the absence of force.     

The nanomechanics of polycystin-1 extracellular region

Recent evidence suggests that polycystin-1 (PC1) acts as a mechanosensor, receiving signals from the primary cilia, neighboring cells, and extracellular matrix and transduces them into cellular responses that regulate proliferation, adhesion, and differentiation that are essential for the control of renal tubules and kidney morphogenesis. PC1 has an unusually long extracellular region ( approximately 3000 amino acids) with a multimodular structure. Proteins with a similar architecture have structural and mechanical roles. Based on the structural similarities between PC1 and other modular proteins that have elastic properties we hypothesized that PC1 functions mechanically by providing a flexible and elastic linkage between cells. Here we directly tested this hypothesis by analyzing the mechanical properties of the entire PC1 extracellular region by using single molecule force spectroscopy. We show that the PC1 extracellular region is highly extensible and that this extensibility is mainly caused by the unfolding of its Ig-like domains. Stretching the native PC1 extracellular region results in a sawtooth pattern with equally spaced force peaks that have a wide range of unfolding forces (50-200 pN). By combining single-molecule force spectroscopy and protein engineering techniques, we demonstrate that the sawtooth pattern in native PC1 extracellular region corresponds to the sequential unfolding of individual Ig-like domains. We found that Ig-like domains refold after mechanical unfolding. Hence, the PC1 extracellular region displays a dynamic extensibility whereby the resting length might be regulated through unfolding/refolding of its Ig-like domains. These force-driven reactions may be important for cell elasticity and the regulation of cell signaling events mediated by PC1.     

Investigating protein unfolding kinetics by pulse proteolysis

Investigation of protein unfolding kinetics of proteins in crude samples may provide many exciting opportunities to study protein energetics under unconventional conditions. As an effort to develop a method with this capability, we employed “pulse proteolysis” to investigate protein unfolding kinetics. Pulse proteolysis has been shown to be an effective and facile method to determine global stability of proteins by exploiting the difference in proteolytic susceptibilities between folded and unfolded proteins. Electrophoretic separation after proteolysis allows monitoring protein unfolding without protein purification. We employed pulse proteolysis to determine unfolding kinetics of E. coli maltose binding protein (MBP) and E. coli ribonuclease H (RNase H). The unfolding kinetic constants determined by pulse proteolysis are in good agreement with those determined by circular dichroism. We then determined an unfolding kinetic constant of overexpressed MBP in a cell lysate. An accurate unfolding kinetic constant was successfully determined with the unpurified MBP. Also, we investigated the effect of ligand binding on unfolding kinetics of MBP using pulse proteolysis. On the basis of a kinetic model for unfolding of MBP•maltose complex, we have determined the dissociation equilibrium constant (K_d) of the complex from unfolding kinetic constants, which is also in good agreement with known K_d values of the complex. These results clearly demonstrate the feasibility and the accuracy of pulse proteolysis as a quantitative probe to investigate protein unfolding kinetics.     

Studies of the unfolding of an unstable mutant of staphylococcal nuclease: Evidence for low temperature unfolding and compactness of the high temperature unfolded state

Fluorescence and circular dichroism data as a function of temperature were obtained to characterize the unfolding of nuclease A and two of its less stable mutants. These spectroscopic data were obtained with a modified instrument that enables the nearly simultaneous detection of both fluorescence and CD data on the same sample. A global analysis of these multiple datasets yielded an excellent fit of a model that includes a change in the heat capacity change, ΔC_p, between the unfolded and native states. This analysis gives a ΔC_p of 2.2 kcal/mol/·K for thermal unfolding of the WT protein and 1.3 and 1.8 kcal/mol/K for the two mutants. These ΔC_p values are consistent with significant population of the cold unfolded state at ∼0°C. Independent evidence for the existence of a cold unfolded state is the observation of a separately migrating peak in size exclusion chromatography. The new chromatographic peak is seen near 0°C, has a partition coefficient corresponding to a larger hydrodynamic radius, and shows a red-shifted fluorescence spectrum, as compared to the native protein. Data also indicate that the high-temperature unfolded form of mutant nuclease is relatively compact. Size exclusion chromatography shows the high temperature unfolded form to have a hydrodynamic radius that is larger than that for the native form, but smaller than that for the urea or pH-induced unfolded forms. Addition of chemical denaturants to the high-temperature unfolded form causes a further unfolding of the protein, as indicated by an increase in the apparent hydrodynamic radius and a decrease in the rotational correlation time for Trp140 (as determined by fluorescence anisotropy decay measurements). Proteins 28:227–240, 1997 © 1997 Wiley-Liss Inc.     

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.     

A novel peak detection approach with chemical noise removal using short‐time FFT for prOTOF MS data

Peak detection is a pivotal first step in biomarker discovery from MS data and can significantly influence the results of downstream data analysis steps. We developed a novel automatic peak detection method for prOTOF MS data, which does not require a priori knowledge of protein masses. Random noise is removed by an undecimated wavelet transform and chemical noise is attenuated by an adaptive short‐time discrete Fourier transform. Isotopic peaks corresponding to a single protein are combined by extracting an envelope over them. Depending on the S/N, the desired peaks in each individual spectrum are detected and those with the highest intensity among their peak clusters are recorded. The common peaks among all the spectra are identified by choosing an appropriate cut‐off threshold in the complete linkage hierarchical clustering. To remove the 1 Da shifting of the peaks, the peak corresponding to the same protein is determined as the detected peak with the largest number among its neighborhood. We validated this method using a data set of serial peptide and protein calibration standards. Compared with MoverZ program, our new method detects more peaks and significantly enhances S/N of the peak after the chemical noise removal. We then successfully applied this method to a data set from prOTOF MS spectra of albumin and albumin‐bound proteins from serum samples of 59 patients with carotid artery disease compared to vascular disease‐free patients to detect peaks with S/N≥2. Our method is easily implemented and is highly effective to define peaks that will be used for disease classification or to highlight potential biomarkers.     

NMR‐monitored titration of acid‐stress bacterial chaperone HdeA reveals that Asp and Glu charge neutralization produces a loosened dimer structure in preparation for protein unfolding and chaperone activation

HdeA is a periplasmic chaperone found in several gram‐negative pathogenic bacteria that are linked to millions of cases of dysentery per year worldwide. After the protein becomes activated at low pH, it can bind to other periplasmic proteins, protecting them from aggregation when the bacteria travel through the stomach on their way to colonize the intestines. It has been argued that one of the major driving forces for HdeA activation is the protonation of aspartate and glutamate side chains. The goal for this study, therefore, was to investigate, at the atomic level, the structural impact of this charge neutralization on HdeA during the transition from near‐neutral conditions to pH 3.0, in preparation for unfolding and activation of its chaperone capabilities. NMR spectroscopy was used to measure pK_a values of Asp and Glu residues and monitor chemical shift changes. Measurements of R₂/R₁ ratios from relaxation experiments confirm that the protein maintains its dimer structure between pH 6.0 and 3.0. However, calculated correlation times and changes in amide protection from hydrogen/deuterium exchange experiments provide evidence for a loosening of the tertiary and quaternary structures of HdeA; in particular, the data indicate that the dimer structure becomes progressively weakened as the pH decreases. Taken together, these results provide insight into the process by which HdeA is primed to unfold and carry out its chaperone duties below pH 3.0, and it also demonstrates that neutralization of aspartate and glutamate residues is not likely to be the sole trigger for HdeA dissociation and unfolding.     

On the mechanism of protein fold‐switching by a molecular sensor

Alternate frame folding (AFF) is a mechanism by which conformational change can be engineered into a protein. The protein structure switches from the wild‐type fold (N) to a circularly‐permuted fold (N′), or vice versa, in response to a signaling event such as ligand binding. Despite the fact that the two native states have similar structures, their interconversion involves folding and unfolding of large parts of the molecule. This rearrangement is reported by fluorescent groups whose relative proximities change as a result of the order–disorder transition. The nature of the conformational change is expected to be similar from protein to protein; thus, it may be possible to employ AFF as a general method to create optical biosensors. Toward that goal, we test basic aspects of the AFF mechanism using the AFF variant of calbindin D_9k. A simple three‐state model for fold switching holds that N and N′ interconvert through the unfolded state. This model predicts that the fundamental properties of the switch—calcium binding affinity, signal response (i.e., fluorescence change upon binding), and switching rate—can be controlled by altering the relative stabilities of N and N′. We find that selectively destabilizing N or N′ changes the equilibrium properties of the switch (binding affinity and signal response) in accordance with the model. However, kinetic data indicate that the switching pathway does not require whole‐molecule unfolding. The rate is instead limited by unfolding of a portion of the protein, possibly in concert with folding of a corresponding region. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

The effect of specific proline residues on the kinetic stability of the triosephosphate isomerases of two trypanosomes

The effect of specific residues on the kinetic stability of two closely related triosephosphate isomerases (from Trypanosoma cruzi, TcTIM and Trypanosoma brucei, TbTIM) has been studied. Based on a comparison of their β‐turn occurrence, we engineered two chimerical enzymes where their super secondary β‐loop‐α motifs 2 ((βα)₂) were swapped. Differential scanning calorimetry (DSC) experiments showed that the (βα)₂ motif of TcTIM inserted into TbTIM (2Tc) increases the kinetic stability. On the other hand, the presence of the (βα)₂ motif of TbTIM inserted into TcTIM (2Tb) gave a chimerical protein difficult to purify in soluble form and with a significantly reduced kinetic stability. The comparison of the contact maps of the (βα)₂ of TbTIM and TcTIM showed differences in the contact pattern of residues 43 and 49. In TcTIM these residues are prolines, located at the N‐terminal of loop‐2 and the C‐terminal of α‐helix‐2. Twelve mutants were engineered involving residues 43 and 49 to study the effect over the unfolding activation energy barrier (E_A). A systematic analysis of DSC data showed a large decrease on the E_A of TcTIM (ΔE_A ranging from 468 to 678 kJ/mol) when the single and double proline mutations are present. The relevance of Pro43 to the kinetic stability is also revealed by mutation S43P, which increased the free energy of the transition state of TbTIM by 17.7 kJ/mol. Overall, the results indicate that protein kinetic stability can be severely affected by punctual mutations, disturbing the complex network of interactions that, in concerted action, determine protein stability. Proteins 2017; 85:571–579. © 2016 Wiley Periodicals, Inc.