Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference

Accurate Memory Kernel Extraction from Discretized Time-Series Data

Memory effects emerge as a fundamental consequence of dimensionality reduction when low-dimensional observables are used to describe the dynamics of complex many-body systems. In the context of molecular dynamics (MD) data analysis, accounting for memory effects using the framework of the generalized Langevin equation (GLE) has proven efficient, accurate, and insightful, particularly when working with high-resolution time series data. However, in experimental systems, high-resolution data are often unavailable, raising questions about the impact of the data resolution on the estimated GLE parameters. This study demonstrates that direct memory extraction from time series data remains accurate when the discretization time is below the memory time. To obtain memory functions reliably, even when the discretization time exceeds the memory time, we introduce a Gaussian Process Optimization (GPO) scheme. This scheme minimizes the deviation of discretized two-point correlation functions between time series data and GLE simulations and is able to estimate accurate memory kernels as long as the discretization time stays below the longest time scale in the data, typically the barrier crossing time.

Direct observation of chemo-mechanical coupling in DnaK by single-molecule force experiments

Protein allostery requires a communication channel for functional regulation between distal sites within a protein. In the molecular chaperone Hsp70, a two-domain enzyme, the ATP/ADP status of an N-terminal nucleotide-binding domain regulates the substrate affinity of a C-terminal substrate-binding domain. Recently available three-dimensional structures of Hsp70 in ATP/ADP states have provided deep insights into molecular pathways of allosteric signals. However, direct mechanical probing of long-range allosteric coupling between the ATP hydrolysis step and domain states is missing. Using laser optical tweezers, we examined the mechanical properties of a truncated two-domain DnaK(1-552ye) in apo/ADP/ATP- and peptide-bound states. We find that in the apo and ADP states, DnaK domains are mechanically stable and rigid. However, in the ATP state, substrate-binding domain (SBD)∗ye is mechanically destabilized as the result of interdomain docking followed by the unfolding of the α-helical lid. By observing the folding state of the SBD, we could observe the continuous ATP/ADP cycling of the enzyme in real time with a single molecule. The SBD lid closure is strictly coupled to the chemical steps of the ATP hydrolysis cycle even in the presence of peptide substrate.

Unraveling the Mechanics of a Repeat-Protein Nanospring: From Folding of Individual Repeats to Fluctuations of the Superhelix

Tandem-repeat proteins comprise small secondary structure motifs that stack to form one-dimensional arrays with distinctive mechanical properties that are proposed to direct their cellular functions. Here, we use single-molecule optical tweezers to study the folding of consensus-designed tetratricopeptide repeats (CTPRs), superhelical arrays of short helix-turn-helix motifs. We find that CTPRs display a spring-like mechanical response in which individual repeats undergo rapid equilibrium fluctuations between partially folded and unfolded conformations. We rationalize the force response using Ising models and dissect the folding pathway of CTPRs under mechanical load, revealing how the repeat arrays form from the center toward both termini simultaneously. Most strikingly, we also directly observe the protein's superhelical tertiary structure in the force signal. Using protein engineering, crystallography, and single-molecule experiments, we show that the superhelical geometry can be altered by carefully placed amino acid substitutions, and we examine how these sequence changes affect intrinsic repeat stability and inter-repeat coupling. Our findings provide the means to dissect and modulate repeat-protein stability and dynamics, which will be essential for researchers to understand the function of natural repeat proteins and to exploit artificial repeats proteins in nanotechnology and biomedical applications.

Unraveling the Mechanics of a Repeat-Protein Nanospring: From Folding of Individual Repeats to Fluctuations of the Superhelix

Tandem-repeat proteins comprise small secondary structure motifs that stack to form one-dimensional arrays with distinctive mechanical properties that are proposed to direct their cellular functions. Here, we use single-molecule optical tweezers to study the folding of consensus-designed tetratricopeptide repeats (CTPRs), superhelical arrays of short helix-turn-helix motifs. We find that CTPRs display a spring-like mechanical response in which individual repeats undergo rapid equilibrium fluctuations between partially folded and unfolded conformations. We rationalize the force response using Ising models and dissect the folding pathway of CTPRs under mechanical load, revealing how the repeat arrays form from the center toward both termini simultaneously. Most strikingly, we also directly observe the protein's superhelical tertiary structure in the force signal. Using protein engineering, crystallography, and single-molecule experiments, we show that the superhelical geometry can be altered by carefully placed amino acid substitutions, and we examine how these sequence changes affect intrinsic repeat stability and inter-repeat coupling. Our findings provide the means to dissect and modulate repeat-protein stability and dynamics, which will be essential for researchers to understand the function of natural repeat proteins and to exploit artificial repeats proteins in nanotechnology and biomedical applications.

Identification and correction of miscalibration artifacts based on force noise for optical tweezers experiments

Single-molecule force spectroscopy using optical tweezers continues to provide detailed insights into the behavior of nanoscale systems. Obtaining precise measurements of their mechanical properties is highly dependent on accurate instrument calibration. Therefore, instrumental drift or inaccurate calibration may prevent reaching an accuracy at the theoretical limit and may lead to incorrect conclusions. Commonly encountered sources of error include inaccuracies in the detector sensitivity and trap stiffness and neglecting the non-harmonicity of an optical trap at higher forces. Here, we first quantify the impact of these artifacts on force-extension data and find that a small deviation of the calibration parameters can already have a significant downstream effect. We then develop a method to identify and remove said artifacts based on differences in the theoretical and measured noise of bead fluctuations. By applying our procedure to both simulated and experimental data, we can show how effects due to miscalibration and trap non-linearities can be successfully removed. Most importantly, this correction can be performed post-measurement and could be adapted for data acquired using any force spectroscopy technique.

Slow Transition Path Times Reveal a Complex Folding Barrier in a Designed Protein

De-novo designed proteins have received wide interest as potential platforms for nano-engineering and biomedicine. While much work is being done in the design of thermodynamically stable proteins, the folding process of artificially designed proteins is not well-studied. Here we used single-molecule force spectroscopy by optical tweezers to study the folding of ROSS, a de-novo designed 2x2 Rossmann fold. We measured a barrier crossing time in the millisecond range, much slower than what has been reported for other systems. While long transition times can be explained by barrier roughness or slow diffusion, we show that isotropic roughness cannot explain the measured transition path time distribution. Instead, this study shows that the slow barrier crossing of ROSS is caused by the population of three short-lived high-energy intermediates. In addition, we identify incomplete and off-pathway folding events with different barrier crossing dynamics. Our results hint at the presence of a complex transition barrier that may be a common feature of many artificially designed proteins.

Propagation and attenuation of mechanical signals in ultrasoft 2D solids

The propagation of elastic waves in soft materials plays a crucial role in the spatiotemporal transmission of mechanical signals, e.g., in biological mechanotransduction or in the failure of marginal solids. At high Reynolds numbers Re ≫ 1, inertia dominates and wave propagation is readily observed. However, mechanical cues in soft and biological materials often occur at low Re, where waves are overdamped. Overdamped waves are not only difficult to observe experimentally, also theoretically their description remains incomplete. Here, we present direct measurements of the propagation and attenuation of mechanical signals in colloidal soft solids, induced by an optical trap. We derive an analytical theory for low Re wave propagation and damping, which is in excellent agreement with the experiments. Our results present both a previously unexplored method to characterize damped waves in soft solids and a theoretical framework showing how localized mechanical signals can provoke a remote and delayed response.

Bioorthogonal protein-DNA conjugation methods for force spectroscopy

Accurate and stable site-specific attachment of DNA molecules to proteins is a requirement for many single-molecule force spectroscopy techniques. The most commonly used method still relies on maleimide chemistry involving cysteine residues in the protein of interest. Studies have consequently often focused on model proteins that either have no cysteines or with a small number of cysteines that can be deleted so that cysteines can then be introduced at specific sites. However, many proteins, especially in eukaryotes, contain too many cysteine residues to be amenable to this strategy, and therefore there is tremendous need for new and broadly applicable approaches to site-specific conjugation. Here we present bioorthogonal approaches for making DNA-protein conjugates required in force spectroscopy experiments. Unnatural amino acids are introduced site-specifically and conjugated to DNA oligos bearing the respective modifications to undergo either strain-promoted azidealkyne cycloaddition (SPAAC) or inverse-electron-demand Diels-Alder (IE-DA) reactions. We furthermore show that SPAAC is compatible with a previously published peptide-based attachment approach. By expanding the available toolkit to tag-free methods based on bioorthogonal reactions, we hope to enable researchers to interrogate the mechanics of a much broader range of proteins than is currently possible.

Nucleotide-Dependent Dimer Association and Dissociation of the Chaperone Hsp90

Hsp90 is an essential molecular chaperone, which has to be in a dimeric form for its correct function. While the affinity of the dimer has previously been measured, little is known about how it associates and dissociates and the factors that influence this. We perform an in-depth single molecule characterization of the C-terminal association and dissociation of Hsp90. We find more than one dissociation rate, indicating that the dimer has a stable and an unstable state. Furthermore, we find that the stability of the C-terminal association is dependent on the presence of ATP, despite the C-terminal dimerization interface being distal to the catalytic site.

Nanomechanics of the substrate binding domain of Hsp70 determine its allosteric ATP-induced conformational change

Owing to the cooperativity of protein structures, it is often almost impossible to identify independent subunits, flexible regions, or hinges simply by visual inspection of static snapshots. Here, we use single-molecule force experiments and simulations to apply tension across the substrate binding domain (SBD) of heat shock protein 70 (Hsp70) to pinpoint mechanical units and flexible hinges. The SBD consists of two nanomechanical units matching 3D structural parts, called the α- and β-subdomain. We identified a flexible region within the rigid β-subdomain that gives way under load, thus opening up the α/β interface. In exactly this region, structural changes occur in the ATP-induced opening of Hsp70 to allow substrate exchange. Our results show that the SBD's ability to undergo large conformational changes is already encoded by passive mechanics of the individual elements.

Single-molecule dissection of stacking forces in DNA

We directly measured at the single-molecule level the forces and lifetimes of DNA base-pair stacking interactions for all stack sequence combinations. Our experimental approach combined dual-beam optical tweezers with DNA origami components to allow positioning of blunt-end DNA helices so that the weak stacking force could be isolated. Base-pair stack arrays that lacked a covalent backbone connection spontaneously dissociated at average rates ranging from 0.02 to 500 per second, depending on the sequence combination and stack array size. Forces in the range from 2 to 8 piconewtons that act along the helical direction only mildly accelerated the stochastic unstacking process. The free-energy increments per stack that we estimate from the measured forward and backward kinetic rates ranged from -0.8 to -3.4 kilocalories per mole, depending on the sequence combination. Our data contributes to understanding the mechanics of DNA processing in biology, and it is helpful for designing the kinetics of DNA-based nanoscale devices according to user specifications.

α-Actinin/titin interaction: A dynamic and mechanically stable cluster of bonds in the muscle Z-disk

Stable anchoring of titin within the muscle Z-disk is essential for preserving muscle integrity during passive stretching. One of the main candidates for anchoring titin in the Z-disk is the actin cross-linker α-actinin. The calmodulin-like domain of α-actinin binds to the Z-repeats of titin. However, the mechanical and kinetic properties of this important interaction are still unknown. Here, we use a dual-beam optical tweezers assay to study the mechanics of this interaction at the single-molecule level. A single interaction of α-actinin and titin turns out to be surprisingly weak if force is applied. Depending on the direction of force application, the unbinding forces can more than triple. Our results suggest a model where multiple α-actinin/Z-repeat interactions cooperate to ensure long-term stable titin anchoring while allowing the individual components to exchange dynamically.

Advances in the microrheology of complex fluids

New developments in the microrheology of complex fluids are considered. Firstly the requirements for a simple modern particle tracking microrheology experiment are introduced, the error analysis methods associated with it and the mathematical techniques required to calculate the linear viscoelasticity. Progress in microrheology instrumentation is then described with respect to detectors, light sources, colloidal probes, magnetic tweezers, optical tweezers, diffusing wave spectroscopy, optical coherence tomography, fluorescence correlation spectroscopy, elastic- and quasi-elastic scattering techniques, 3D tracking, single molecule methods, modern microscopy methods and microfluidics. New theoretical techniques are also reviewed such as Bayesian analysis, oversampling, inversion techniques, alternative statistical tools for tracks (angular correlations, first passage probabilities, the kurtosis, motor protein step segmentation etc), issues in micro/macro rheological agreement and two particle methodologies. Applications where microrheology has begun to make some impact are also considered including semi-flexible polymers, gels, microorganism biofilms, intracellular methods, high frequency viscoelasticity, comb polymers, active motile fluids, blood clots, colloids, granular materials, polymers, liquid crystals and foods. Two large emergent areas of microrheology, non-linear microrheology and surface microrheology are also discussed.

Capturing transition paths and transition states for conformational rearrangements in the ribosome

To reveal the molecular determinants of biological function, one seeks to characterize the interactions that are formed in conformational and chemical transition states. In other words, what interactions govern the molecule's energy landscape? To accomplish this, it is necessary to determine which degrees of freedom can unambiguously identify each transition state. Here, we perform simulations of large-scale aminoacyl-transfer RNA (aa-tRNA) rearrangements during accommodation on the ribosome and project the dynamics along experimentally accessible atomic distances. From this analysis, we obtain evidence for which coordinates capture the correct number of barrier-crossing events and accurately indicate when the aa-tRNA is on a transition path. Although a commonly used coordinate in single-molecule experiments performs poorly, this study implicates alternative coordinates along which rearrangements are accurately described as diffusive movements across a one-dimensional free-energy profile. From this, we provide the theoretical foundation required for single-molecule techniques to uncover the energy landscape governing aa-tRNA selection by the ribosome.

Subnanometre enzyme mechanics probed by single-molecule force spectroscopy

Enzymes are molecular machines that bind substrates specifically, provide an adequate chemical environment for catalysis and exchange products rapidly, to ensure fast turnover rates. Direct information about the energetics that drive conformational changes is difficult to obtain. We used subnanometre single-molecule force spectroscopy to study the energetic drive of substrate-dependent lid closing in the enzyme adenylate kinase. Here we show that in the presence of the bisubstrate inhibitor diadenosine pentaphosphate (AP5A), closing and opening of both lids is cooperative and tightly coupled to inhibitor binding. Surprisingly, binding of the substrates ADP and ATP exhibits a much smaller energetic drive towards the fully closed state. Instead, we observe a new dominant energetic minimum with both lids half closed. Our results, combining experiment and molecular dynamics simulations, give detailed mechanical insights into how an enzyme can cope with the seemingly contradictory requirements of rapid substrate exchange and tight closing, to ensure efficient catalysis.

Adenylate kinase catalyses the interconversion of adenosine phosphates, and plays a crucial role in maintaining cellular energy homeostasis. Here, the authors use single molecule optical tweezers to understand how the enzyme's conformation dynamics modulates catalysis.

Folding and assembly of the large molecular machine Hsp90 studied in single-molecule experiments

Folding of small proteins often occurs in a two-state manner and is well understood both experimentally and theoretically. However, many proteins are much larger and often populate misfolded states, complicating their folding process significantly. Here we study the complete folding and assembly process of the 1,418 amino acid, dimeric chaperone Hsp90 using single-molecule optical tweezers. Although the isolated C-terminal domain shows two-state folding, we find that the isolated N-terminal as well as the middle domain populate ensembles of fast-forming, misfolded states. These intradomain misfolds slow down folding by an order of magnitude. Modeling folding as a competition between productive and misfolding pathways allows us to fully describe the folding kinetics. Beyond intradomain misfolding, folding of the full-length protein is further slowed by the formation of interdomain misfolds, suggesting that with growing chain lengths, such misfolds will dominate folding kinetics. Interestingly, we find that small stretching forces applied to the chain can accelerate folding by preventing the formation of cross-domain misfolding intermediates by leading the protein along productive pathways to the native state. The same effect is achieved by cotranslational folding at the ribosome in vivo.

Hydrodynamic fluctuation-induced forces in confined fluids

We study thermal, fluctuation-induced hydrodynamic interaction forces in a classical, compressible, viscous fluid confined between two rigid, planar walls with no-slip boundary conditions. We calculate hydrodynamic fluctuations using the linearized, stochastic Navier-Stokes formalism of Landau and Lifshitz. The mean fluctuation-induced force acting on the fluid boundaries vanishes in this system, so we evaluate the two-point, time-dependent force correlations. The equal-time correlation function of the forces acting on a single wall gives the force variance, which we show to be finite and independent of the plate separation at large inter-plate distances. The equal-time, cross-plate force correlation, on the other hand, decays with the inverse inter-plate distance and is independent of the fluid viscosity at large distances; it turns out to be negative over the whole range of plate separations, indicating that the two bounding plates are subjected to counter-phase correlations. We show that the time-dependent force correlations exhibit damped temporal oscillations for small plate separations and a more irregular oscillatory behavior at large separations. The long-range hydrodynamic correlations reported here represent a "secondary Casimir effect", because the mean fluctuation-induced force, which represents the primary Casimir effect, is absent.

Extracellular rigidity sensing by talin isoform-specific mechanical linkages

The ability of cells to adhere and sense differences in tissue stiffness is crucial for organ development and function. The central mechanisms by which adherent cells detect extracellular matrix compliance, however, are still unknown. Using two single-molecule-calibrated biosensors that allow the analysis of a previously inaccessible but physiologically highly relevant force regime in cells, we demonstrate that the integrin activator talin establishes mechanical linkages following cell adhesion, which are indispensable for cells to probe tissue stiffness. Talin linkages are exposed to a range of piconewton forces and bear, on average, 7-10 pN during cell adhesion depending on their association with F-actin and vinculin. Disruption of talin's mechanical engagement does not impair integrin activation and initial cell adhesion but prevents focal adhesion reinforcement and thus extracellular rigidity sensing. Intriguingly, talin mechanics are isoform specific so that expression of either talin-1 or talin-2 modulates extracellular rigidity sensing.

Nucleotides regulate the mechanical hierarchy between subdomains of the nucleotide binding domain of the Hsp70 chaperone DnaK

The regulation of protein function through ligand-induced conformational changes is crucial for many signal transduction processes. The binding of a ligand alters the delicate energy balance within the protein structure, eventually leading to such conformational changes. In this study, we elucidate the energetic and mechanical changes within the subdomains of the nucleotide binding domain (NBD) of the heat shock protein of 70 kDa (Hsp70) chaperone DnaK upon nucleotide binding. In an integrated approach using single molecule optical tweezer experiments, loop insertions, and steered coarse-grained molecular simulations, we find that the C-terminal helix of the NBD is the major determinant of mechanical stability, acting as a glue between the two lobes. After helix unraveling, the relative stability of the two separated lobes is regulated by ATP/ADP binding. We find that the nucleotide stays strongly bound to lobe II, thus reversing the mechanical hierarchy between the two lobes. Our results offer general insights into the nucleotide-induced signal transduction within members of the actin/sugar kinase superfamily.

Forced protein unfolding leads to highly elastic and tough protein hydrogels

Protein-based hydrogels usually do not exhibit high stretchability or toughness, significantly limiting the scope of their potential biomedical applications. Here we report the engineering of a chemically cross-linked, highly elastic and tough protein hydrogel using a mechanically extremely labile, de novo-designed protein that assumes the classical ferredoxin-like fold structure. Due to the low mechanical stability of the ferredoxin-like fold structure, swelling of hydrogels causes a significant fraction of the folded domains to unfold. Subsequent collapse and aggregation of unfolded ferredoxin-like domains leads to intertwining of physically and chemically cross-linked networks, entailing hydrogels with unusual physical and mechanical properties: a negative swelling ratio, high stretchability and toughness. These hydrogels can withstand an average strain of 450% before breaking and show massive energy dissipation. Upon relaxation, refolding of the ferredoxin-like domains enables the hydrogel to recover its massive hysteresis. This novel biomaterial may expand the scope of hydrogel applications in tissue engineering.

Microrheology of colloidal systems

Microrheology was proposed almost twenty years ago as a technique to obtain rheological properties in soft matter from the microscopic motion of colloidal tracers used as probes, either freely diffusing in the host medium, or subjected to external forces. The former case is known as passive microrheology, and is based on generalizations of the Stokes-Einstein relation between the friction experienced by the probe and the host-fluid viscosity. The latter is termed active microrheology, and extends the measurement of the friction coefficient to the nonlinear-response regime of strongly driven probes. In this review article, we discuss theoretical models available in the literature for both passive and active microrheology, focusing on the case of single-probe motion in model colloidal host media. A brief overview of the theory of passive microrheology is given, starting from the work of Mason and Weitz. Further developments include refined models of the host suspension beyond that of a Newtonian-fluid continuum, and the investigation of probe-size effects. Active microrheology is described starting from microscopic equations of motion for the whole system including both the host-fluid particles and the tracer; the many-body Smoluchowski equation for the case of colloidal suspensions. At low fluid densities, this can be simplified to a two-particle equation that allows the calculation of the friction coefficient with the input of the density distribution around the tracer, as shown by Brady and coworkers. The results need to be upscaled to agree with simulations at moderate density, in both the case of pulling the tracer with a constant force or dragging it at a constant velocity. The full many-particle equation has been tackled by Fuchs and coworkers, using a mode-coupling approximation and the scheme of integration through transients, valid at high densities. A localization transition is predicted for a probe embedded in a glass-forming host suspension. The nonlinear probe-friction coefficient is calculated from the tracer's position correlation function. Computer simulations show qualitative agreement with the theory, but also some unexpected features, such as superdiffusive motion of the probe related to the breaking of nearest-neighbor cages. We conclude with some perspectives and future directions of theoretical models of microrheology.

Crosstalk elimination in the detection of dual-beam optical tweezers by spatial filtering

In dual-beam optical tweezers, the accuracy of position and force measurements is often compromised by crosstalk between the two detected signals, this crosstalk leading to systematic and significant errors on the measured forces and distances. This is true both for dual-beam optical traps where the splitting of the two traps is done by polarization optics and for dual optical traps constructed by other methods, e.g., holographic tweezers. If the two traps are orthogonally polarized, most often crosstalk is minimized by inserting polarization optics in front of the detector; however, this method is not perfect because of the de-polarization of the trapping beam introduced by the required high numerical aperture optics. Here we present a simple and easy-to-implement method to efficiently eliminate crosstalk. The method is based on spatial filtering by simply inserting a pinhole at the correct position and is highly compatible with standard back focal plane photodiode based detection of position and force. Our spatial filtering method reduces crosstalk up to five times better than polarization filtering alone. The effectiveness is dependent on pinhole size and distance between the traps and is here quantified experimentally and reproduced by theoretical modeling. The method here proposed will improve the accuracy of force-distance measurements, e.g., of single molecules, performed by dual-beam optical traps and hence give much more scientific value for the experimental efforts.

Spectral Rate Theory for Two-State Kinetics

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

Calibrating optical tweezers with Bayesian inference

We present a new method for calibrating an optical-tweezer setup that does not depend on input parameters and is less affected by systematic errors like drift of the setup. It is based on an inference approach that uses Bayesian probability to infer the diffusion coefficient and the potential felt by a bead trapped in an optical or magnetic trap. It exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy.

Ultrafast folding kinetics and cooperativity of villin headpiece in single-molecule force spectroscopy

In this study we expand the accessible dynamic range of single-molecule force spectroscopy by optical tweezers to the microsecond range by fast sampling. We are able to investigate a single molecule for up to 15 min and with 300-kHz bandwidth as the protein undergoes tens of millions of folding/unfolding transitions. Using equilibrium analysis and autocorrelation analysis of the time traces, the full energetics as well as real-time kinetics of the ultrafast folding of villin headpiece 35 and a stable asparagine 68 alanine/lysine 70 methionine variant can be measured directly. We also performed Brownian dynamics simulations of the response of the bead-DNA system to protein-folding fluctuations. All key features of the force-dependent deflection fluctuations could be reproduced: SD, skewness, and autocorrelation function. Our measurements reveal a difference in folding pathway and cooperativity between wild-type and stable variant of headpiece 35. Autocorrelation force spectroscopy pushes the time resolution of single-molecule force spectroscopy to ∼10 µs thus approaching the timescales accessible for all atom molecular dynamics simulations.

Rigid DNA beams for high-resolution single-molecule mechanics

Bridging the gap: Rigid DNA linkers (blue, see picture) between microspheres (green) for high-resolution single-molecule mechanical experiments were constructed using DNA origami. The resulting DNA helical bundles greatly reduce the noise generated in studies of conformation changes using optical tweezers and were applied to study small DNA secondary structures.

From mechanical folding trajectories to intrinsic energy landscapes of biopolymers

In single-molecule laser optical tweezer (LOT) pulling experiments, a protein or RNA is juxtaposed between DNA handles that are attached to beads in optical traps. The LOT generates folding trajectories under force in terms of time-dependent changes in the distance between the beads. How to construct the full intrinsic folding landscape (without the handles and beads) from the measured time series is a major unsolved problem. By using rigorous theoretical methods--which account for fluctuations of the DNA handles, rotation of the optical beads, variations in applied tension due to finite trap stiffness, as well as environmental noise and limited bandwidth of the apparatus--we provide a tractable method to derive intrinsic free-energy profiles. We validate the method by showing that the exactly calculable intrinsic free-energy profile for a generalized Rouse model, which mimics the two-state behavior in nucleic acid hairpins, can be accurately extracted from simulated time series in a LOT setup regardless of the stiffness of the handles. We next apply the approach to trajectories from coarse-grained LOT molecular simulations of a coiled-coil protein based on the GCN4 leucine zipper and obtain a free-energy landscape that is in quantitative agreement with simulations performed without the beads and handles. Finally, we extract the intrinsic free-energy landscape from experimental LOT measurements for the leucine zipper.