Probing static disorder in Arrhenius kinetics by single-molecule force spectroscopy

Identical sequences, different behaviors: Protein diversity captured at the single-molecule level

The classical "one sequence, one structure, one function" paradigm has shaped much of our intuition of how proteins work inside the cell. Partially due to the insight provided by bulk biochemical assays, individual biomolecules are often assumed to behave as identical entities, and their characterization relies on ensemble averages that flatten any conformational diversity into a unique phenotype. While the emergence of single-molecule techniques opened the gates to interrogating individual molecules, technical shortcomings typically limit the duration of these measurements, which precludes a complete characterization of an individual protein and, hence, capturing the heterogeneity among molecular populations. Here, we introduce an ultrastable magnetic tweezers design, which enables us to measure the folding dynamics of a single protein during several uninterrupted days with high temporal and spatial resolution. Thanks to this instrumental development, we fully characterize the nanomechanics of two proteins with a very distinct force response, the talin R3IVVI domain and protein L. Days-long recordings on the same protein individual accumulate thousands of folding transitions with submicrosecond resolution, allowing us to reconstruct their free energy landscapes and describe how they evolve with force. By mapping the nanomechanical identity of many different protein individuals, we directly capture their molecular diversity as a quantifiable dispersion on their force response and folding kinetics. By significantly expanding the measurable timescales, our instrumental development offers a tool for profiling individual molecules, opening the gates to directly characterizing biomolecular heterogeneity.

Use of DNA forceps to measure receptor-ligand dissociation equilibrium constants in a single-molecule competition assay

The ability of biophysicists to decipher the behavior of individual biomolecules has steadily improved over the past thirty years. However, it still remains unclear how an ensemble of data acquired at the single-molecule level compares with the data acquired on an ensemble of the same molecules. We here propose an assay to tackle this question in the context of dissociation equilibrium constant measurements. A sensor is built by engrafting a receptor and a ligand onto a flexible dsDNA scaffold and mounting this assembly on magnetic tweezers. This way, looking at the position of the magnetic bead enables one to determine in real-time if the two molecular partners are associated or not. Next, to quantify the affinity of the scrutinized single-receptor for a given competitor, various amounts of the latter molecule are introduced in solution and the equilibrium response of the sensor is monitored throughout the titration protocol. Proofs of concept are established for the binding of three rapamycin analogs to the FKBP12 cis-trans prolyl isomerase. For each of these drugs the mean affinity constant obtained on a ten of individual receptors agrees with the one previously determined in a bulk assay. Furthermore, experimental contingencies are sufficient to explain the dispersion observed over the single-molecule values.

A minimal kinetic model for the interpretation of complex catalysis in single enzyme molecules

Multi-exponential waiting-time distribution and randomness parameter greater than unity ascribe dynamic disorder in single-enzyme catalysis corroborated to the interplay of transforming conformers [English et al., Nat. Chem. Biol., 2006, 2, 87]. The associated multi-state model of enzymatic turnovers with statically heterogeneous catalytic rates misdescribes the non-linear uprising of the randomness parameter from unity in relation to the attributes of the fall-offs of the waiting-time distribution at different substrate concentrations. To resolve this crucial issue, we first employ a comprehensive stochastic reaction scenario and further rationalize and work out the minimal indispensable dynamic-disorder model that ensures the foregoing relationship upon comparison with the data. We elucidate that specific disregard for the transition rate coefficients in the multi-state model on account of the especially slow conformational transitions is the underlying reason for not achieving interrelation between the observables.

Electro-detachment of kinesin motor domain from microtubule in silico

Kinesin is a motor protein essential in cellular functions, such as intracellular transport and cell-division, as well as for enabling nanoscopic transport in bio-nanotechnology. Therefore, for effective control of function for nanotechnological applications, it is important to be able to modify the function of kinesin. To circumvent the limitations of chemical modifications, here we identify another potential approach for kinesin control: the use of electric forces. Using full-atom molecular dynamics simulations (247,358 atoms, total time ∼ 4.4 μs), we demonstrate, for the first time, that the kinesin-1 motor domain can be detached from a microtubule by an intense electric field within the nanosecond timescale. We show that this effect is field-direction dependent and field-strength dependent. A detailed analysis of the electric forces and the work carried out by electric field acting on the microtubule-kinesin system shows that it is the combined action of the electric field pulling on the β-tubulin C-terminus and the electric-field-induced torque on the kinesin dipole moment that causes kinesin detachment from the microtubule. It is shown, for the first time in a mechanistic manner, that an electric field can dramatically affect molecular interactions in a heterologous functional protein assembly. Our results contribute to understanding of electromagnetic field-biomatter interactions on a molecular level, with potential biomedical and bio-nanotechnological applications for harnessing control of protein nanomotors.

Role of ergodicity, aging, and Gaussianity in resolving the origins of biomolecule subdiffusion

The internal motions of biomolecules are essential to their function. Although biological macromolecules conventionally show subdiffusive dynamics, only recently has subdiffusion been associated with non-ergodicity. These findings have stimulated new questions in biophysics and statistical mechanics. Is non-ergodic subdiffusion a general strategy shared by biomolecules? What underlying mechanisms are responsible for it? Here, we performed extensive molecular dynamics (MD) simulations to characterize the internal dynamics of six different biomolecules, ranging from single or double-stranded DNA, a single domain protein (KRAS), two globular proteins (PGK and SHP2), to an intrinsically disordered protein (SNAP-25). We found that the subdiffusive behavior of these biomolecules falls into two classes. The internal motion of the first three cases is ergodic subdiffusion and can be interpreted by fractional Brownian motion (FBM), while the latter three cases involve non-ergodic subdiffusion and can be modeled by mixed origins of continuous-time random walk (CTRW) and FBM.

Connecting conformational stiffness of the protein with energy landscape by a single experiment

The structure-function dynamics of a protein as a flexible polymer is essential to describe its biological functions. Here, using single-molecule magnetic tweezers, we have studied the effect of ionic strength on the folding mechanics of protein L, and probed its folding-associated physical properties by re-measuring the same protein in a range of ammonium sulfate concentrations from 150 mM to 650 mM. We observed an electrolyte-dependent conformational dynamics and folding landscape of the protein in a single experiment. Salt increases the refolding kinetics, while decreasing the unfolding kinetics under force, which in turn modifies the barrier heights towards the folded state. Additionally, salt enhances the molecular compaction by decreasing the Kuhn length of the protein polymer from 1.18 nm to 0.58 nm, which we have explained by modifying the freely jointed chain model. Finally, we correlated polymer chain physics to the folding dynamics, and thus provided an analytical framework for understanding compaction-induced folding mechanics across a range of ionic strengths from a single experiment.

Direct observation of chaperone-modulated talin mechanics with single-molecule resolution

Talin as a critical focal adhesion mechanosensor exhibits force-dependent folding dynamics and concurrent interactions. Being a cytoplasmic protein, talin also might interact with several cytosolic chaperones; however, the roles of chaperones in talin mechanics remain elusive. To address this question, we investigated the force response of a mechanically stable talin domain with a set of well-known unfoldase (DnaJ, DnaK) and foldase (DnaKJE, DsbA) chaperones, using single-molecule magnetic tweezers. Our findings demonstrate that chaperones could affect adhesion proteins’ stability by changing their folding mechanics; while unfoldases reduce their unfolding force from ~11 pN to ~6 pN, foldase shifts it upto ~15 pN. Since talin is mechanically synced within 2 pN force ranges, these changes are significant in cellular conditions. Furthermore, we determined that chaperones directly reshape the energy landscape of talin: unfoldases decrease the unfolding barrier height from 26.8 to 21.7 k_BT, while foldases increase it to 33.5 k_BT. We reconciled our observations with eukaryotic Hsp70 and Hsp40 and observed their similar function of decreasing the talin unfolding barrier. Quantitative mapping of this chaperone-induced talin folding landscape directly illustrates that chaperones perturb the adhesion protein stability under physiological force, thereby, influencing their force-dependent interactions and adhesion dynamics.

Chakraborty et al. uses single-molecule magnetic tweezers to investigate the chaperone-modulated talin protein mechanics. The results showed that chaperones are involved in the regulation of talin folding/unfolding under mechanical force with some chaperones stabilizing talin and increasing the force, whereas others destabilize it and reduce the force.

Non-ergodicity of a globular protein extending beyond its functional timescale

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble. Here, by performing single-molecule fluorescence resonance energy transfer (smFRET) experiments and molecular dynamics (MD) simulations of a multi-domain globular protein, cytoplasmic protein-tyrosine phosphatase (SHP2), we demonstrate that the functional inter-domain motion is observationally non-ergodic over the time spans 10⁻¹² to 10⁻⁷ s and 10⁻¹ to 10² s. The difference between observational non-ergodicity and simple non-convergence is discussed. In comparison, a single-strand DNA of similar size behaves ergodically with an energy landscape resembling a one-dimensional linear chain. The observed non-ergodicity results from the hierarchical connectivity of the high-dimensional energy landscape of the protein molecule. As the characteristic time for the protein to conduct its dephosphorylation function is ∼10 s, our findings suggest that, due to the non-ergodicity, individual, seemingly identical protein molecules can be dynamically and functionally different.

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble.

Protein nanomechanics in biological context

How proteins respond to pulling forces, or protein nanomechanics, is a key contributor to the form and function of biological systems. Indeed, the conventional view that proteins are able to diffuse in solution does not apply to the many polypeptides that are anchored to rigid supramolecular structures. These tethered proteins typically have important mechanical roles that enable cells to generate, sense, and transduce mechanical forces. To fully comprehend the interplay between mechanical forces and biology, we must understand how protein nanomechanics emerge in living matter. This endeavor is definitely challenging and only recently has it started to appear tractable. Here, I introduce the main in vitro single-molecule biophysics methods that have been instrumental to investigate protein nanomechanics over the last 2 decades. Then, I present the contemporary view on how mechanical force shapes the free energy of tethered proteins, as well as the effect of biological factors such as post-translational modifications and mutations. To illustrate the contribution of protein nanomechanics to biological function, I review current knowledge on the mechanobiology of selected muscle and cell adhesion proteins including titin, talin, and bacterial pilins. Finally, I discuss emerging methods to modulate protein nanomechanics in living matter, for instance by inducing specific mechanical loss-of-function (mLOF). By interrogating biological systems in a causative manner, these new tools can contribute to further place protein nanomechanics in a biological context.

Energy Landscape of Ubiquitin Is Weakly Multidimensional

Single molecule pulling experiments report time-dependent changes in the extension (X) of a biomolecule as a function of the applied force (f). By fitting the data to one-dimensional analytical models of the energy landscape, we can extract the hopping rates between the folded and unfolded states in two-state folders as well as the height and the location of the transition state (TS). Although this approach is remarkably insightful, there are cases for which the energy landscape is multidimensional (catch bonds being the most prominent). To assess if the unfolding energy landscape in small single domain proteins could be one-dimensional, we simulated force-induced unfolding of ubiquitin (Ub) using the coarse-grained self-organized polymer-side chain (SOP-SC) model. Brownian dynamics simulations using the SOP-SC model reveal that the Ub energy landscape is weakly multidimensional (WMD), governed predominantly by a single barrier. The unfolding pathway is confined to a narrow reaction pathway that could be described as diffusion in a quasi-1D X-dependent free energy profile. However, a granular analysis using the P_fold analysis, which does not assume any form for the reaction coordinate, shows that X alone does not account for the height and, more importantly, the location of the TS. The f-dependent TS location moves toward the folded state as f increases, in accord with the Hammond postulate. Our study shows that, in addition to analyzing the f-dependent hopping rates, the transition state ensemble must also be determined without resorting to X as a reaction coordinate to describe the unfolding energy landscapes of single domain proteins, especially if they are only WMD.

Correlations within polyprotein forced unfolding dwell-times introduce sequential dependency

Polyproteins, comprised from proteins arrayed in tandem, respond to mechanical loads through partial unfolding and extension. This response to tension that enables their physiological function is related to the ability to dynamically regulate their elasticity. The unique arrangement of their individual mechanical components (proteins and polymeric linkers), and the interactions between them eventually determines their performance. The sequential unfolding-times within a polyprotein are inherently assumed to be independent and identically distributed (iid), thus expected to follow an exponential distribution. Nevertheless, a large body of literature using single molecule force spectroscopy (SMFS) provides evidence that forced unfolding-times of N proteins within a polyprotein do not follow the exponential distribution. Here we use SMFS with Atomic Force Microscopy to measure the unfolding kinetics of Poly-(I91)₈ at 180 pN. The unfolding time-intervals were statistically analysed using three common approaches, all exhibiting an N-effect: hierarchical behavior with non-identical unfolding time distributions. Using continuous time random walk approach indicates that the unfolding times display subdiffusive features. Put together with free-energy reconstruction of the whole unfolding polyprotein, we provide physical explanation for this nontrivial behavior, according to which the elongating polypeptide chain with each unfolding event intervenes with the sequential unfolding probabilities and correlates them.

Mechanical Unfolding of Single Polyubiquitin Molecules Reveals Evidence of Dynamic Disorder

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.

Non-Markov bond model for dynamic force spectroscopy

Single-molecule force spectroscopy data are conventionally analyzed using a schematic model, wherein a molecular bond is represented as a virtual particle diffusing in a one-dimensional free-energy landscape. However, this simple and efficient approach is unable to account for the "anomalous" bond-breaking kinetics increasingly observed in force spectroscopy experiments and simulations, e.g., in the form of non-exponential distributions of bond lifetimes under constant load. Here, we show that such characteristic traits arise naturally in a rigorous extension of the one-dimensional theory that accounts for the transient dynamics of a generic set of coupled degrees of freedom. These "hidden modes" affect the reaction dynamics in various ways, depending on their relaxation spectrum and the loading protocol, giving rise, in particular, to apparent static and dynamic disorder. In two complementary asymptotic limits, we are able to find exact analytical expressions for pertinent experimental observables, such as the mean rupture force and the rupture-force distribution. Intriguingly, our asymptotic results become unconditionally exact at high loading rates, thus providing us with a microscopically consistent theory of rapid force spectroscopy that avoids the usual Markov assumption.

History, rare, and multiple events of mechanical unfolding of repeat proteins

Mechanical unfolding of proteins consisting of repeat domains is an excellent tool to obtain large statistics. Force spectroscopy experiments using atomic force microscopy on proteins presenting multiple domains have revealed that unfolding forces depend on the number of folded domains (history) and have reported intermediate states and rare events. However, the common use of unspecific attachment approaches to pull the protein of interest holds important limitations to study unfolding history and may lead to discarding rare and multiple probing events due to the presence of unspecific adhesion and uncertainty on the pulling site. Site-specific methods that have recently emerged minimize this uncertainty and would be excellent tools to probe unfolding history and rare events. However, detailed characterization of these approaches is required to identify their advantages and limitations. Here, we characterize a site-specific binding approach based on the ultrastable complex dockerin/cohesin III revealing its advantages and limitations to assess the unfolding history and to investigate rare and multiple events during the unfolding of repeated domains. We show that this approach is more robust, reproducible, and provides larger statistics than conventional unspecific methods. We show that the method is optimal to reveal the history of unfolding from the very first domain and to detect rare events, while being more limited to assess intermediate states. Finally, we quantify the forces required to unfold two molecules pulled in parallel, difficult when using unspecific approaches. The proposed method represents a step forward toward more reproducible measurements to probe protein unfolding history and opens the door to systematic probing of rare and multiple molecule unfolding mechanisms.

Revealing Abrupt and Spontaneous Ruptures of Protein Native Structure under picoNewton Compressive Force Manipulation

Manipulating protein conformations for exploring protein structure-function relationship has shown great promise. Although protein conformational changes under pulling force manipulation have been extensively studied, protein conformation changes under a compressive force have not been explored quantitatively. The latter is even more biologically significant and relevant in revealing protein functions in living cells associated with protein crowdedness, distribution fluctuations, and cell osmotic stress. Here we report our experimental observations on abrupt ruptures of protein native structures under compressive force, demonstrated and studied by single-molecule AFM-FRET spectroscopic nanoscopy. Our results show that the protein ruptures are abrupt and spontaneous events occurred when the compressive force reaches a threshold of 12-75 pN, a force amplitude accessible from thermal fluctuations in a living cell. The abrupt ruptures are sensitive to local environment, likely a general and important pathway of protein unfolding in living cells.

Effects of non-equilibrium angle fluctuation on F1-ATPase kinetics induced by temperature increase

F₁-ATPase (F₁) is an efficient rotary protein motor, whose reactivity is modulated by the rotary angle to utilize thermal fluctuation. In order to elucidate how its kinetics are affected by the change in the fluctuation, we have extended the reaction-diffusion formalism [R. Watanabe et al., Biophys. J., 2013, 105, 2385] applicable to a wider range of temperatures based on experimental data analysis of F₁ derived from thermophilic Bacillus under high ATP concentration conditions. Our simulation shows that the rotary angle distribution manifests a stronger non-equilibrium feature as the temperature increases, because ATP hydrolysis and P_i release are more accelerated compared with the timescale of rotary angle relaxation. This effect causes the rate coefficient obtained from dwell time fitting to deviate from the Arrhenius relation in P_i release, which has been assumed in the previous activation thermodynamic quantities estimation using linear Arrhenius fitting. Larger negative correlation is also found between hydrolysis and P_i release waiting time in a catalytic dwell with the increase in temperature. This loss of independence between the two successive reactions at the catalytic dwell sheds doubt on the conventional dwell time fitting to obtain rate coefficients with a double exponential function at temperatures higher than 65 °C, which is close to the physiological temperature of the thermophilic Bacillus.

Nonexponential kinetics of ion pair dissociation in electrofreezing water

Temporally- or spatially-heterogeneous environments can participate in many kinetic processes, from chemical reactions and self-assembly to the forced dissociation of biomolecules. Here, we simulate the molecular dynamics of a model ion pair forced to dissociate in an explicit, aqueous solution. Triggering dissociation with an external electric field causes the surrounding water to electrofreeze and the ion pair population to decay nonexponentially. To further probe the role of the aqueous environment in the kinetics, we also simulate dissociation events under a purely mechanical force on the ion pair. In this case, regardless of whether the surrounding water is a liquid or already electrofrozen, the ion pair population decays exponentially with a well-defined rate constant that is specific to the medium and applied force. These simulation data, and the rate parameters we extract, suggest the disordered kinetics in an electrofreezing medium are a result of the comparable time scales of two concurrent processes, electrofreezing and dissociation.

Force-Dependent Folding and Unfolding Kinetics in DNA Hairpins Reveals Transition-State Displacements along a Single Pathway

Biomolecules diffusively explore their energy landscape overcoming energy barriers via thermally activated processes to reach the biologically relevant conformation. Mechanically induced unfolding and folding reactions offer an excellent playground to feature these processes at the single-molecule level by monitoring changes in the molecular extension. Here we investigate two-state DNA hairpins designed to have the transition states at different locations. We use optical tweezers to characterize the force-dependent behavior of the kinetic barrier from nonequilibrium pulling experiments by using the continuous effective barrier approach (CEBA). We introduce the mechanical fragility and the molecular transition-state susceptibility, both useful quantities to characterize the response of the transition state to an applied force. Our results demonstrate the validity of the Leffler-Hammond postulate where the transition state approaches the folded state as force increases, implying monotonically decreasing fragility with force and a non-negative transition state susceptibility at all forces.

Protein-fluctuation-induced water-pore formation in ion channel voltage-sensor translocation across a lipid bilayer membrane

We have applied a combined fluorescence microscopy and single-ion-channel electric current recording approach, correlating with molecular dynamics (MD) simulations, to study the mechanism of voltage-sensor domain translocation across a lipid bilayer. We use the colicin Ia ion channel as a model system, and our experimental and simulation results show the following: (1) The open-close activity of an activated colicin Ia is not necessarily sensitive to the amplitude of the applied cross-membrane voltage when the cross-membrane voltage is around the resting potential of excitable membranes; and (2) there is a significant probability that the activation of colicin Ia occurs by forming a transient and fluctuating water pore of ∼15 Å diameter in the lipid bilayer membrane. The location of the water-pore formation is nonrandom and highly specific, right at the insertion site of colicin Ia charged residues in the lipid bilayer membrane, and the formation is intrinsically associated with the polypeptide conformational fluctuations and solvation dynamics. Our results suggest an interesting mechanistic pathway for voltage-sensitive ion channel activation, and specifically for translocation of charged polypeptide chains across the lipid membrane under a transmembrane electric field: the charged polypeptide domain facilitates the formation of hydrophilic water pore in the membrane and diffuses through the hydrophilic pathway across the membrane; i.e., the charged polypeptide chain can cross a lipid membrane without entering into the hydrophobic core of the lipid membrane but entirely through the aqueous and hydrophilic environment to achieve a cross-membrane translocation. This mechanism sheds light on the intensive and fundamental debate on how a hydrophilic and charged peptide domain diffuses across the biologically inaccessible high-energy barrier of the hydrophobic core of a lipid bilayer: The peptide domain does not need to cross the hydrophobic core to move across a lipid bilayer.

On the improvement of free-energy calculation from steered molecular dynamics simulations using adaptive stochastic perturbation protocols

The potential of mean force (PMF) calculation in single molecule manipulation experiments performed via the steered molecular dynamics (SMD) technique is a computationally very demanding task because the analyzed system has to be perturbed very slowly to be kept close to equilibrium. Faster perturbations, far from equilibrium, increase dissipation and move the average work away from the underlying free energy profile, and thus introduce a bias into the PMF estimate. The Jarzynski equality offers a way to overcome the bias problem by being able to produce an exact estimate of the free energy difference, regardless of the perturbation regime. However, with a limited number of samples and high dissipation the Jarzynski equality also introduces a bias. In our previous work, based on the Brownian motion formalism, we introduced three stochastic perturbation protocols aimed at improving the PMF calculation with the Jarzynski equality in single molecule manipulation experiments and analogous computer simulations. This paper describes the PMF reconstruction results based on full-atom molecular dynamics simulations, obtained with those three protocols. We also want to show that the protocols are applicable with the second-order cumulant expansion formula. Our protocols offer a very noticeable improvement over the simple constant velocity pulling. They are able to produce an acceptable estimate of PMF with a significantly reduced bias, even with very fast perturbation regimes. Therefore, the protocols can be adopted as practical and efficient tools for the analysis of mechanical properties of biological molecules.

Evidence of disorder in biological molecules from single molecule pulling experiments

Heterogeneity in biological molecules, resulting in molecule-to-molecule variations in their dynamics and function, is an emerging theme. To elucidate the consequences of heterogeneous behavior at the single molecule level, we propose an exactly solvable model in which the unfolding rate due to mechanical force depends parametrically on an auxiliary variable representing an entropy barrier arising from fluctuations in internal dynamics. When the rate of fluctuations--a measure of dynamical disorder--is comparable to or smaller than the rate of force-induced unbinding, we show that there are two experimentally observable consequences: nonexponential survival probability at constant force, and a heavy-tailed rupture force distribution at constant loading rate. By fitting our analytical expressions to data from single molecule pulling experiments on proteins and DNA, we quantify the extent of disorder. We show that only by analyzing data over a wide range of forces and loading rates can the role of disorder due to internal dynamics be quantitatively assessed.

Stretching of single poly-ubiquitin molecules revisited: dynamic disorder in the non-exponential unfolding kinetics

A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.

Manipulating and probing enzymatic conformational fluctuations and enzyme–substrate interactions by single-molecule FRET-magnetic tweezers microscopy

To investigate the critical role of the enzyme–substrate interactions in enzymatic reactions, the enzymatic conformation and enzyme–substrate interaction at a single-molecule level are manipulated by magnetic tweezers, and the impact of the manipulation on enzyme–substrate interactions are simultaneously probed by single-molecule FRET spectroscopy.

On the mechanism of recombination hotspot scanning during double-stranded DNA break resection

Double-stranded DNA break repair by homologous recombination is initiated by resection of free DNA ends to produce a 3'-ssDNA overhang. In bacteria, this reaction is catalyzed by helicase-nuclease complexes such as AddAB in a manner regulated by specific recombination hotspot sequences called Crossover hotspot instigator (Chi). We have used magnetic tweezers to investigate the dynamics of AddAB translocation and hotspot scanning during double-stranded DNA break resection. AddAB was prone to stochastic pausing due to transient recognition of Chi-like sequences, unveiling an antagonistic relationship between DNA translocation and sequence-specific DNA recognition. Pauses at bona fide Chi sequences were longer, were nonexponentially distributed, and resulted in an altered velocity upon restart of translocation downstream of Chi. We propose a model for the recognition of Chi sequences to explain the origin of pausing during failed and successful hotspot recognition.

The Janus-faced role of external forces in mechanochemical disulfide bond cleavage

Recent force microscopy measurements on the mechanically activated cleavage of a protein disulfide bond through reaction with hydroxide ions revealed that for forces greater than 0.5 nN, the acceleration of the reaction rate is substantially reduced. Here, using ab initio simulations, we trace this 'reactivity switch' back to a dual role played by the mechanical force, which leads to antagonistic effects. On the one hand, the force performs work on the system, and thereby accelerates the reaction. On the other hand, the force also induces a conformational distortion that involves the S-S-C-C dihedral angle, which drives the disulfide into a conformation that is shielded against nucleophilic attack because of steric hindrance. The discovery of force-induced conformational changes that steer chemical reactivity provides a new key concept that is expected to be relevant beyond this specific case, for example in understanding how 'disulfide switches' regulate protein function and for the rational design of mechanoresponsive materials.

Force dependency of biochemical reactions measured by single-molecule force-clamp spectroscopy

Here we describe a protocol for using force-clamp spectroscopy to precisely quantify the effect of force on biochemical reactions. A calibrated force is used to control the exposure of reactive sites in a single polyprotein substrate composed of repeated domains. The use of polyproteins allows the identification of successful single-molecule recordings from unambiguous mechanical unfolding fingerprints. Biochemical reactions are then measured directly by detecting the length changes of the substrate held at a constant force. We present the layout of a force-clamp spectrometer along with protocols to design and conduct experiments. These experiments measure reaction kinetics as a function of applied force. We show sample data of the force dependency of two different reactions, protein unfolding and disulfide reduction. These data, which can be acquired in just a few days, reveal mechanistic details of the reactions that currently cannot be resolved by any other technique.

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

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.

Dynamic prestress in a globular protein

A protein at equilibrium is commonly thought of as a fully relaxed structure, with the intra-molecular interactions showing fluctuations around their energy minimum. In contrast, here we find direct evidence for a protein as a molecular tensegrity structure, comprising a balance of tensed and compressed interactions, a concept that has been put forward for macroscopic structures. We quantified the distribution of inter-residue prestress in ubiquitin and immunoglobulin from all-atom molecular dynamics simulations. The network of highly fluctuating yet significant inter-residue forces in proteins is a consequence of the intrinsic frustration of a protein when sampling its rugged energy landscape. In beta sheets, this balance of forces is found to compress the intra-strand hydrogen bonds. We estimate that the observed magnitude of this pre-compression is enough to induce significant changes in the hydrogen bond lifetimes; thus, prestress, which can be as high as a few 100 pN, can be considered a key factor in determining the unfolding kinetics and pathway of proteins under force. Strong pre-tension in certain salt bridges on the other hand is connected to the thermodynamic stability of ubiquitin. Effective force profiles between some side-chains reveal the signature of multiple, distinct conformational states, and such static disorder could be one factor explaining the growing body of experiments revealing non-exponential unfolding kinetics of proteins. The design of prestress distributions in engineering proteins promises to be a new tool for tailoring the mechanical properties of made-to-order nanomaterials.

A tensegrity structure is one composed of members that are permanently under either tension or compression, and the balance of these tensile and compressive forces provides the structure with its mechanical stability. Macroscale tensegrity structures, which include Buckminster Fuller's geodesic domes, achieve exceptional structural integrity with a minimal use of resources. The question we address in this work is whether nature makes use of molecular-scale tensegrity in the design of proteins. Using Molecular Dynamics simulations of the protein ubiquitin, we measure the network of pairwise forces connecting the amino acid residues and show that this network does indeed have the character of a tensegrity structure. Furthermore, we find that the arrangement of tensile and compressive forces is such that hydrogen bonds in the protein's beta sheet, which are crucial for bearing mechanical loads, are compressed. This pre-compression is enough to significantly lengthen the lifetime of a bond under a given force, and thus should be an important factor in determining the protein's mechanical strength. The rational design of molecular prestress networks promises to be a new avenue for the engineering of proteins with made-to-order mechanical properties, for applications in medicine, materials and nanotechnology.

Interpreting single turnover catalysis measurements with constrained mean dwell times

Observation of a chemical transformation at the single-molecule level yields a detailed view of kinetic pathways contributing to the averaged results obtained in a bulk measurement. Studies of a fluorogenic reaction catalyzed by gold nanoparticles have revealed heterogeneous reaction dynamics for these catalysts. Measurements on single nanoparticles yield binary trajectories with stochastic transitions between a dark state in which no product molecules are adsorbed and a fluorescent state in which one product molecule is present. The mean dwell time in either state gives information corresponding to a bulk measurement. Quantifying fluctuations from mean kinetics requires identifying properties of the fluorescence trajectory that are selective in emphasizing certain dynamic processes according to their time scales. We propose the use of constrained mean dwell times, defined as the mean dwell time in a state with the constraint that the immediately preceding dwell time in the other state is, for example, less than a variable time. Calculations of constrained mean dwell times for a kinetic model with dynamic disorder demonstrate that these quantities reveal correlations among dynamic fluctuations at different active sites on a multisite catalyst. Constrained mean dwell times are determined from measurements of single nanoparticle catalysis. The results indicate that dynamical fluctuations at different active sites are correlated, and that especially rapid reaction events produce particularly slowly desorbing product molecules.

Mechanical unfolding studies of protein molecules

Atomic force microscopy (AFM) enables the pick up of a single protein molecule to apply a mechanical force. This technique, called "force spectroscopy," provides unique information about the intermediates and free energy landscape of the mechanical unfolding of proteins. In this review, we introduce the AFM-based single molecule force spectroscopy of proteins and describe recent studies that answer some fundamental questions such as "is the mechanical resistance of proteins isotropic?", "what is the structure of the transition state in mechanical unfolding?", and "is mechanical unfolding related to biological functions?"

A conditional gating mechanism assures the integrity of the molecular force-sensor titin kinase

As more and more recent investigations point out, force plays an important role in cellular regulation mechanisms. Biological responses to mechanical stress are often based on force-induced conformational changes of single molecules. The force sensor, titin kinase, is involved in a signaling complex that regulates protein turnover and transcriptional adaptation in striated muscle. The structural architecture of such a force sensor determines its response to force and must assure both activity and mechanical integrity, which are prerequisites for its function. Here, we use single-molecule force-clamp spectroscopy to show that titin kinase is organized in such a way that the regulatory domains have to unfold before secondary structure elements that determine the overall fold and catalytic function. The stepwise unfolding over many barriers with a topologically determined sequence assures that the protein can react to force by conformational changes while maintaining its structural integrity.

Efficient free-energy-profile reconstruction using adaptive stochastic perturbation protocols

The Jarzynski-relation-based free-energy calculation is limited by the very slow convergence of the estimate when dissipation is high. We present two novel perturbation protocols able to significantly improve the quality of the potential of mean force (PMF) calculation by reducing the estimate's bias without increasing the number of samples. The protocols are directly applicable with both numerical simulations and real-life experiments. The improvement is achieved through the intentional but controlled widening of the distribution of the external work used to perturb a given system. Our protocols can achieve the same accuracy in PMF estimation as the normal constant velocity pulling with less than 10% of the required samples.

Unfolding proteins with an atomic force microscope: force-fluctuation-induced nonexponential kinetics

We show that in experimental atomic force microscopy studies of the lifetime distribution of mechanically stressed folded proteins the effects of externally applied fluctuations cannot be distinguished from those of internally present fluctuations. In certain circumstances this leads to artificially nonexponential lifetime distributions, which can be misinterpreted as a signature of protein complexity. This work highlights the importance of fully characterizing and controlling external sources of fluctuation in mechanical studies of proteins before drawing conclusions on the physics at play on the molecular level.

Direct quantification of the attempt frequency determining the mechanical unfolding of ubiquitin protein

Understanding protein dynamics requires a comprehensive knowledge of the underlying potential energy surface that governs the motion of each individual protein molecule. Single molecule mechanical studies have provided the unprecedented opportunity to study the individual unfolding pathways along a well defined coordinate, the end-to-end length of the protein. In these experiments, unfolding requires surmounting an energy barrier that separates the native from the extended state. The calculation of the absolute value of the barrier height has traditionally relied on the assumption of an attempt frequency, υ(‡). Here we used single molecule force-clamp spectroscopy to directly determine the value of υ(‡) for mechanical unfolding by measuring the unfolding rate of the small protein ubiquitin at varying temperatures. Our experiments demonstrate a significant effect of the temperature on the mechanical rate of unfolding. By extrapolating the unfolding rate in the absence of force for different temperatures, varying within the range spanning from 5 to 45 °C, we measured a value for the activation barrier of ΔG(‡) = 71 ± 5 kJ/mol and an exponential prefactor υ(‡) ∼4 × 10(9) s(-1). Although the measured prefactor value is 3 orders of magnitude smaller than the value predicted by the transition state theory (∼6 × 10(12) s(-1)), it is 400-fold higher than that encountered in analogous experiments studying the effect of temperature on the reactivity of a protein-embedded disulfide bond (∼10(7) M(-1) s(-1)). This approach will allow quantitative characterization of the complete energy landscape of a folding polypeptide from highly extended states, of capital importance for proteins with elastic function.

Single-molecule force-clamp spectroscopy: dwell time analysis and practical considerations

Single-molecule force-clamp spectroscopy has become a powerful tool for studying protein folding/unfolding, bond rupture, and enzymatic reactions. Different methods have been developed to analyze force-clamp spectroscopy data on polyproteins to obtain kinetic parameters characterizing the mechanical unfolding of proteins, which are often modeled as a two-state process (a Poisson process). However, because of the finite number of domains in polyproteins, the statistical analysis of the force-clamp spectroscopy data is different from that of a classical Poisson process, and the equivalency of different analysis methods remains to be proven. In this article, we show that these methods are equivalent and lead to accurate measurements of the unfolding rate constant. We also demonstrate that distinct from the constant-pulling-velocity experiments, in which the unfolding rate extracted from the data is dependent on the number of protein domains in the polyproteins (the N effect), force-clamp experiments do not show any N effect. Using a simulated data set, we also highlighted important practical considerations that one needs to take into account when using the single-molecule force-clamp spectroscopy technique to characterize the unfolding energy landscape of proteins.

Contrasting the individual reactive pathways in protein unfolding and disulfide bond reduction observed within a single protein

Identifying the dynamics of individual molecules along their reactive pathways remains a major goal of modern chemistry. For simple chemical reactions, the transition state position is thought to be highly localized. Conversely, in the case of more complex reactions involving proteins, the potential energy surfaces become rougher, resulting in heterogeneous reaction pathways with multiple transition state structures. Force-clamp spectroscopy experimentally probes the individual reaction pathways sampled by a single protein under the effect of a constant stretching force. Herein, we examine the distribution of conformations that populate the transition state of two different reactions; the unfolding of a single protein and the reduction of a single disulfide bond, both occurring within the same single protein. By applying the recently developed static disorder theory, we quantify the variance of the barrier heights, σ(2), governing each distinct reaction. We demonstrate that the unfolding of the I27 protein follows a nonexponential kinetics, consistent with a high value of σ(2) ∼ 18 (pN nm)(2). Interestingly, shortening of the protein upon introduction of a rigid disulfide bond significantly modulates the disorder degree, spanning from σ(2) ∼ 8 to ∼21 (pN nm)(2). These results are in sharp contrast with the exponential distribution of times measured for an S(N)2 chemical reaction, implying the absence of static disorder σ(2) ∼ 0 (pN nm)(2). Our results demonstrate the high sensitivity of the force-clamp technique to capture the signatures of disorder in the individual pathways that define two distinct force-induced reactions, occurring within the core of a single protein.

First passage time distribution in stochastic processes with moving and static absorbing boundaries with application to biological rupture experiments

We develop and investigate an integral equation connecting the first passage time distribution of a stochastic process in the presence of an absorbing boundary condition and the corresponding Green's function in the absence of the absorbing boundary. Analytical solutions to the integral equations are obtained for three diffusion processes in time-independent potentials which have been previously investigated by other methods. The integral equation provides an alternative way to analytically solve the three diffusion-controlled reactive processes. In order to help analyze biological rupture experiments, we further investigate the numerical solutions of the integral equation for a diffusion process in a time-dependent potential. Our numerical procedure, based on the exact integral equation, avoids the adiabatic approximation used in previous analytical theories and is useful for fitting the rupture force distribution data from single-molecule pulling experiments or molecular dynamics simulation data, especially at larger pulling speeds, larger cantilever spring constants, and smaller reaction rates. Stochastic simulation results confirm the validity of our numerical procedure. We suggest combining a previous analytical theory with our integral equation approach to analyze the kinetics of force induced rupture of biomacromolecules.

Atomistic evidence of how force dynamically regulates thiol/disulfide exchange

The intricate coupling of mechanical force and chemical reactivity has been increasingly revealed in recent years by force spectroscopy experiments on the thiol/disulfide exchange reaction. We here aimed at elucidating the underlying dynamic effects of force on the reaction center for the case of disulfide bond reduction by dithiothreitol at forces of 200-2000 pN, by combining transition path sampling and quantum/classical mechanical simulations. Reaction rates and their dependence on force as quantified by Δx(r), the distance between reactant and transition state, are in good agreement with experiments but indicate a shift of the transition state structure at high forces. Indeed, while an associate S(N)2 mechanism prevails, force causes a move of the transition state to a longer length of the cleaving bond and a shorter length of the forming disulfide bond. Our results highlight the distribution of force into various degrees of freedom, which implies that care must be taken when correlating Δx(r) with a single order parameter of the reaction.

Water's role in the force-induced unfolding of ubiquitin

In atomic force spectroscopic studies of the elastomeric protein ubiquitin, the β-strands 1-5 serve as the force clamp. Simulations show how the rupture force in the force-induced unfolding depends on the kinetics of water molecule insertion into positions where they can eventually form hydrogen bonding bridges with the backbone hydrogen bonds in the force-clamp region. The intrusion of water into this region is slowed down by the hydrophobic shielding effect of carbonaceous groups on the surface residues of β-strands 1-5, which thereby regulates water insertion prior to hydrogen bond breakage. The experiments show that the unfolding of the mechanically stressed protein is nonexponential due to static disorder. Our simulations show that different numbers and/or locations of bridging water molecules give rise to a long-lived distribution of transition states and static disorder. We find that slowing down the translational (not rotational) motions of the water molecules by increasing the mass of their oxygen atoms, which leaves the force field and thereby the equilibrium structure of the solvent unchanged, increases the average rupture force; however, the early stages of the force versus time behavior are very similar for our "normal" and fictitious "heavy" water models. Finally, we construct six mutant systems to regulate the hydrophobic shielding effect of the surface residues in the force-clamp region. The mutations in the two termini of β-sheets 1-5 are found to determine a preference for different unfolding pathways and change mutant's average rupture force.