Full distance-resolved folding energy landscape of one single protein molecule

Hidden dynamics in the unfolding of individual bacteriorhodopsin proteins

Protein folding occurs as a set of transitions between structural states within an energy landscape. An oversimplified view of the folding process emerges when transiently populated states are undetected because of limited instrumental resolution. Using force spectroscopy optimized for 1-microsecond resolution, we reexamined the unfolding of individual bacteriorhodopsin molecules in native lipid bilayers. The experimental data reveal the unfolding pathway in unprecedented detail. Numerous newly detected intermediates-many separated by as few as two or three amino acids-exhibited complex dynamics, including frequent refolding and state occupancies of <10 μs. Equilibrium measurements between such states enabled the folding free-energy landscape to be deduced. These results sharpen the picture of the mechanical unfolding of membrane proteins and, more broadly, enable experimental access to previously obscured protein dynamics.

Protein unfolding mechanisms and their effects on folding experiments

In this review, I discuss the various methods researchers use to unfold proteins in the lab in order to understand protein folding both in vitro and in vivo. The four main techniques, chemical-, heat-, pressure- and force-denaturation, produce distinctly different unfolded conformational ensembles. Recent measurements have revealed different folding kinetics from different unfolding mechanisms. Thus, comparing these distinct unfolded ensembles sheds light on the underlying free energy landscape of folding.

When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches

Protein folding research stalled for decades because conventional experiments indicated that proteins fold slowly and in single strokes, whereas theory predicted a complex interplay between dynamics and energetics resulting in myriad microscopic pathways. Ultrafast kinetic methods turned the field upside down by providing the means to probe fundamental aspects of folding, test theoretical predictions and benchmark simulations. Accordingly, experimentalists could measure the timescales for all relevant folding motions, determine the folding speed limit and confirm that folding barriers are entropic bottlenecks. Moreover, a catalogue of proteins that fold extremely fast (microseconds) could be identified. Such fast-folding proteins cross shallow free energy barriers or fold downhill, and thus unfold with minimal co-operativity (gradually). A new generation of thermodynamic methods has exploited this property to map folding landscapes, interaction networks and mechanisms at nearly atomic resolution. In parallel, modern molecular dynamics simulations have finally reached the timescales required to watch fast-folding proteins fold and unfold in silico. All of these findings have buttressed the fundamentals of protein folding predicted by theory, and are now offering the first glimpses at the underlying mechanisms. Fast folding appears to also have functional implications as recent results connect downhill folding with intrinsically disordered proteins, their complex binding modes and ability to moonlight. These connections suggest that the coupling between downhill (un)folding and binding enables such protein domains to operate analogically as conformational rheostats.

Hidden Markov Modeling with Detailed Balance and Its Application to Single Protein Folding

Hidden Markov modeling (HMM) has revolutionized kinetic studies of macromolecules. However, results from HMM often violate detailed balance when applied to the transitions under thermodynamic equilibrium, and the consequence of such violation has not been well understood. Here, to our knowledge, we developed a new HMM method that satisfies detailed balance (HMM-DB) and optimizes model parameters by gradient search. We used free energy of stable and transition states as independent fitting parameters and considered both normal and skew normal distributions of the measurement noise. We validated our method by analyzing simulated extension trajectories that mimicked experimental data of single protein folding from optical tweezers. We then applied HMM-DB to elucidate kinetics of regulated SNARE zippering containing degenerate states. For both simulated and measured trajectories, we found that HMM-DB significantly reduced overfitting of short trajectories compared to the standard HMM based on an expectation-maximization algorithm, leading to more accurate and reliable model fitting by HMM-DB. We revealed how HMM-DB could be conveniently used to derive a simplified energy landscape of protein folding. Finally, we extended HMM-DB to correct the baseline drift in single-molecule trajectories. Together, we demonstrated an efficient, versatile, and reliable method of HMM for kinetics studies of macromolecules under thermodynamic equilibrium.

Entropic forces drive self-organization and membrane fusion by SNARE proteins

SNARE proteins are the core of the cell's fusion machinery and mediate virtually all known intracellular membrane fusion reactions on which exocytosis and trafficking depend. Fusion is catalyzed when vesicle-associated v-SNAREs form trans-SNARE complexes ("SNAREpins") with target membrane-associated t-SNAREs, a zippering-like process releasing ∼65 kT per SNAREpin. Fusion requires several SNAREpins, but how they cooperate is unknown and reports of the number required vary widely. To capture the collective behavior on the long timescales of fusion, we developed a highly coarse-grained model that retains key biophysical SNARE properties such as the zippering energy landscape and the surface charge distribution. In simulations the ∼65-kT zippering energy was almost entirely dissipated, with fully assembled SNARE motifs but uncomplexed linker domains. The SNAREpins self-organized into a circular cluster at the fusion site, driven by entropic forces that originate in steric-electrostatic interactions among SNAREpins and membranes. Cooperative entropic forces expanded the cluster and pulled the membranes together at the center point with high force. We find that there is no critical number of SNAREs required for fusion, but instead the fusion rate increases rapidly with the number of SNAREpins due to increasing entropic forces. We hypothesize that this principle finds physiological use to boost fusion rates to meet the demanding timescales of neurotransmission, exploiting the large number of v-SNAREs available in synaptic vesicles. Once in an unfettered cluster, we estimate ≥15 SNAREpins are required for fusion within the ∼1-ms timescale of neurotransmitter release.

Direct Observation of Folding Energy Landscape of RNA Hairpin at Mechanical Loading Rates

By applying a controlled mechanical load using optical tweezers, we measured the diffusive barrier crossing in a 49 nt long P5ab RNA hairpin. We find that in the free-energy landscape the barrier height (G^‡) and transition distance (x^‡) are dependent on the loading rate (r) along the pulling direction, x, as predicted by Bell. The barrier shifted toward the initial state, whereas ΔG^‡ reduced significantly from 50 to 5 kT, as r increased from 0 to 32 pN/s. However, the equilibrium work (ΔG) during strand separation, as estimated by Crook's fluctuation theorem, remained unchanged at different rates. Previously, helix formation and denaturation have been described as two-state (F ↔ U) transitions for P5ab. Herein, we report three intermediate states I₁, I, and I₂ located at 4, 11, and 16 nm respectively, from the folded conformation. The intermediates were observed only when the hairpin was subjected to an optimal r, 7.6 pN/s. The results indicate that the complementary strands in P5ab can zip and unzip through complex routes, whereby mismatches act as checkpoints and often impose barriers. The study highlights the significance of loading rates in force-spectroscopy experiments that are increasingly being used to measure the folding properties of biomolecules.

Implementation and Tuning of an Optical Tweezers Force-Clamp Feedback System

Feedback systems can be used to control the value of a system variable. In optical tweezers, active feedback is often implemented to either keep the position or tension applied to a single biomolecule constant. Here, we describe the implementation of the latter: an optical force-clamp setup that can be used to study the motion of processive molecular motors under a constant load. We describe the basics of a software-implemented proportional-integral-derivative (PID) controller, how to tune it, and how to determine its optimal feedback rate. Limitations, possible feed-forward applications, and extensions into two- and three-dimensional optical force clamps are discussed. The feedback is ultimately limited by thermal fluctuations and the compliance of the involved molecules. To investigate a particular mechanical process, understanding the basics and limitations of the feedback system will be helpful for choosing the proper feedback hardware, for optimizing the system parameters, and for the design of the experiment.

Single-Molecule Protein Folding Experiments Using High-Precision Optical Tweezers

How proteins fold from linear chains of amino acids to delicate three-dimensional structures remains a fundamental biological problem. Single-molecule manipulation based on high-resolution optical tweezers (OT) provides a powerful approach to study protein folding with unprecedented spatiotemporal resolution. In this method, a single protein or protein complex is tethered between two beads confined in optical traps and pulled. Protein unfolding induced by the mechanical force is counteracted by the spontaneous folding of the protein, reaching a dynamic equilibrium at a characteristic force and rate. The transition is monitored by the accompanying extension change of the protein and used to derive conformations and energies of folding intermediates and their associated transition kinetics. Here, we provide general strategies and detailed protocols to study folding of proteins and protein complexes using optical tweezers, including sample preparation, DNA-protein conjugation and methods of data analysis to extract folding energies and rates from the single-molecule measurements.

Single-Molecule Chemo-Mechanical Spectroscopy Provides Structural Identity of Folding Intermediates

Single-molecule force spectroscopy has emerged as a powerful tool for studying the folding of biological macromolecules. Mechanical manipulation has revealed a wealth of mechanistic information on transient and intermediate states. To date, the majority of state assignment of intermediates has relied on empirical demarcation. However, performing such experiments in the presence of different osmolytes provides an alternative approach that reports on the structural properties of intermediates. Here, we analyze the folding and unfolding of T4 lysozyme with optical tweezers under a chemo-mechanical perturbation by adding osmolytes. We find that two unrelated protective osmolytes, sorbitol and trimethylamine-n-oxide, function by marginally decelerating unfolding rates and specifically modulating early events in the folding process, stabilizing formation of an on-pathway intermediate. The chemo-mechanical perturbation provides access to two independent metrics of the relevant states during folding trajectories, the contour length, and the solvent-accessible surface area. We demonstrate that the dependence of the population of the intermediate in different osmolytes, in conjunction with its measured contour length, provides the ability to discriminate between potential structural models of intermediate states. Our study represents a general strategy that may be employed in the structural modeling of equilibrium intermediate states observed in single-molecule experiments.

Electric-field-stimulated protein mechanics

The internal mechanics of proteins-the coordinated motions of amino acids and the pattern of forces constraining these motions-connects protein structure to function. Here we describe a new method combining the application of strong electric field pulses to protein crystals with time-resolved X-ray crystallography to observe conformational changes in spatial and temporal detail. Using a human PDZ domain (LNX2^PDZ2) as a model system, we show that protein crystals tolerate electric field pulses strong enough to drive concerted motions on the sub-microsecond timescale. The induced motions are subtle, involve diverse physical mechanisms, and occur throughout the protein structure. The global pattern of electric-field-induced motions is consistent with both local and allosteric conformational changes naturally induced by ligand binding, including at conserved functional sites in the PDZ domain family. This work lays the foundation for comprehensive experimental study of the mechanical basis of protein function.

Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension

Some colloidal suspensions contain two types of particles-small and large particles-to improve the lubricating ability, light absorptivity, and so forth. Structural and chemical analyses of such colloidal suspensions are often performed to understand their properties. In a structural analysis study, the observation of the number density distribution of small particles around a large particle (g_LS) is difficult because these particles are randomly moving within the colloidal suspension by Brownian motion. We obtain g_LS using the data from a line optical tweezer (LOT) that can measure the potential of mean force between two large colloidal particles (Φ_LL). We propose a theory that transforms Φ_LL into g_LS. The transform theory is explained in detail and tested. We demonstrate for the first time that LOT can be used for the structural analysis of a colloidal suspension. LOT combined with the transform theory will facilitate structural analyses of the colloidal suspensions, which is important for both understanding colloidal properties and developing colloidal products.

Structure-Based Derivation of Protein Folding Intermediates and Energies from Optical Tweezers

Optical tweezers (OTs) measure the force-dependent time-resolved extension of a single macromolecule tethered between two trapped beads. From this measurement, it is possible to determine the folding intermediates, energies, and kinetics of the macromolecule. Previous data analysis generally has used the extension as a reaction coordinate to characterize the observed folding transitions. Despite its convenience, the extension poorly describes folding in the absence of force. Here, we chose the contour length of the unfolded polypeptide as a reaction coordinate and modeled the extensions of protein structures along their predicted folding pathways based on high-resolution structures of the proteins in their native states. We included the extension in our model to calculate the total extensions, energies, and transition rates of the proteins as a function of force. We fit these calculations to the corresponding experimental measurements and obtained the best-fit conformations and energies of proteins in different folding states. We applied our method to analyze single-molecule trajectories of two representative protein complexes responsible for membrane fusion, the HIV-1 glycoprotein 41 and the synaptic SNARE proteins, which involved transitions between two and five states, respectively. Nonlinear fitting of the model to the experimental data revealed the structures of folding intermediates and transition states and their associated energies. Our results demonstrate that the contour length is a useful reaction coordinate to characterize protein folding and that intrinsic extensions of protein structures should be taken into account to properly derive the conformations and energies of protein folding intermediates from single-molecule manipulation experiments.

Mechanical Folding and Unfolding of Protein Barnase at the Single-Molecule Level

The unfolding and folding of protein barnase has been extensively investigated in bulk conditions under the effect of denaturant and temperature. These experiments provided information about structural and kinetic features of both the native and the unfolded states of the protein, and debates about the possible existence of an intermediate state in the folding pathway have arisen. Here, we investigate the folding/unfolding reaction of protein barnase under the action of mechanical force at the single-molecule level using optical tweezers. We measure unfolding and folding force-dependent kinetic rates from pulling and passive experiments, respectively, and using Kramers-based theories (e.g., Bell-Evans and Dudko-Hummer-Szabo models), we extract the position of the transition state and the height of the kinetic barrier mediating unfolding and folding transitions, finding good agreement with previous bulk measurements. Measurements of the force-dependent kinetic barrier using the continuous effective barrier analysis show that protein barnase verifies the Leffler-Hammond postulate under applied force and allow us to extract its free energy of folding, ΔG0. The estimated value of ΔG0 is in agreement with our predictions obtained using fluctuation relations and previous bulk studies. To address the possible existence of an intermediate state on the folding pathway, we measure the power spectrum of force fluctuations at high temporal resolution (50 kHz) when the protein is either folded or unfolded and, additionally, we study the folding transition-path time at different forces. The finite bandwidth of our experimental setup sets the lifetime of potential intermediate states upon barnase folding/unfolding in the submillisecond timescale.

The Power of Force: Insights into the Protein Folding Process Using Single-Molecule Force Spectroscopy

One of the major challenges in modern biophysics is observing and understanding conformational changes during complex molecular processes, from the fundamental protein folding to the function of molecular machines. Single-molecule techniques have been one of the major driving forces of the huge progress attained in the last few years. Recent advances in resolution of the experimental setups, aided by theoretical developments and molecular dynamics simulations, have revealed a much higher degree of complexity inside these molecular processes than previously reported using traditional ensemble measurements. This review sums up the evolution of these developments and gives an outlook on prospective discoveries.

The physics of pulling polyproteins: a review of single molecule force spectroscopy using the AFM to study protein unfolding

One of the most exciting developments in the field of biological physics in recent years is the ability to manipulate single molecules and probe their properties and function. Since its emergence over two decades ago, single molecule force spectroscopy has become a powerful tool to explore the response of biological molecules, including proteins, DNA, RNA and their complexes, to the application of an applied force. The force versus extension response of molecules can provide valuable insight into its mechanical stability, as well as details of the underlying energy landscape. In this review we will introduce the technique of single molecule force spectroscopy using the atomic force microscope (AFM), with particular focus on its application to study proteins. We will review the models which have been developed and employed to extract information from single molecule force spectroscopy experiments. Finally, we will end with a discussion of future directions in this field.

Connecting thermal and mechanical protein (un)folding landscapes

Molecular dynamics simulations supplement single-molecule pulling experiments by providing the possibility of examining the full free energy landscape using many coordinates. Here, we use an all-atom structure-based model to study the force and temperature dependence of the unfolding of the protein filamin by applying force at both termini. The unfolding time-force relation τ(F) indicates that the force-induced unfolding behavior of filamin can be characterized into three regimes: barrier-limited low- and intermediate-force regimes, and a barrierless high-force regime. Slope changes of τ(F) separate the three regimes. We show that the behavior of τ(F) can be understood from a two-dimensional free energy landscape projected onto the extension X and the fraction of native contacts Q. In the low-force regime, the unfolding rate is roughly force-independent due to the small (even negative) separation in X between the native ensemble and transition state ensemble (TSE). In the intermediate-force regime, force sufficiently separates the TSE from the native ensemble such that τ(F) roughly follows an exponential relation. This regime is typically explored by pulling experiments. While X may fail to resolve the TSE due to overlap with the unfolded ensemble just below the folding temperature, the overlap is minimal at lower temperatures where experiments are likely to be conducted. The TSE becomes increasingly structured with force, whereas the average order of structural events during unfolding remains roughly unchanged. The high-force regime is characterized by barrierless unfolding, and the unfolding time approaches a limit of ∼10 μs for the highest forces we studied. Finally, a combination of X and Q is shown to be a good reaction coordinate for almost the entire force range.

Protein folding and unfolding under force

The recent revolution in optics and instrumentation has enabled the study of protein folding using extremely low mechanical forces as the denaturant. This exciting development has led to the observation of the protein folding process at single molecule resolution and its response to mechanical force. Here, we describe the principles and experimental details of force spectroscopy on proteins, with a focus on the optical tweezers instrument. Several recent results will be discussed to highlight the importance of this technique in addressing a variety of questions in the protein folding field.

Ligand-induced changes of the apparent transition-state position in mechanical protein unfolding

Force-spectroscopic measurements of ligand-receptor systems and the unfolding/folding of nucleic acids or proteins reveal information on the underlying energy landscape along the pulling coordinate. The slope Δx(‡) of the force-dependent unfolding/unbinding rates is interpreted as the distance from the folded/bound state to the transition state for unfolding/unbinding and, hence, often related to the mechanical compliance of the sample molecule. Here we show that in ligand-binding proteins, the experimentally inferred Δx(‡) can depend on the ligand concentration, unrelated to changes in mechanical compliance. We describe the effect in single-molecule, force-spectroscopy experiments of the calcium-binding protein calmodulin and explain it in a simple model where mechanical unfolding and ligand binding occur on orthogonal reaction coordinates. This model predicts changes in the experimentally inferred Δx(‡), depending on ligand concentration and the associated shift of the dominant barrier between the two reaction coordinates. We demonstrate quantitative agreement between experiments and simulations using a realistic six-state kinetic scheme using literature values for calcium-binding kinetics and affinities. Our results have important consequences for the interpretation of force-spectroscopic data of ligand-binding proteins.

Single-molecule folding mechanisms of the apo- and Mg(2+)-bound states of human neuronal calcium sensor-1

Neuronal calcium sensor-1 (NCS-1) is the primordial member of a family of proteins responsible primarily for sensing changes in neuronal Ca(2+) concentration. NCS-1 is a multispecific protein interacting with a number of binding partners in both calcium-dependent and independent manners, and acting in a variety of cellular processes in which it has been linked to a number of disorders such as schizophrenia and autism. Despite extensive studies on the Ca(2+)-activated state of NCS proteins, little is known about the conformational dynamics of the Mg(2+)-bound and apo states, both of which are populated, at least transiently, at resting Ca(2+) conditions. Here, we used optical tweezers to study the folding behavior of individual NCS-1 molecules in the presence of Mg(2+) and in the absence of divalent ions. Under tension, the Mg(2+)-bound state of NCS-1 unfolds and refolds in a three-state process by populating one intermediate state consisting of a folded C-domain and an unfolded N-domain. The interconversion at equilibrium between the different molecular states populated by NCS-1 was monitored in real time through constant-force measurements and the energy landscapes underlying the observed transitions were reconstructed through hidden Markov model analysis. Unlike what has been observed with the Ca(2+)-bound state, the presence of Mg(2+) allows both the N- and C-domain to fold through all-or-none transitions with similar refolding rates. In the absence of divalent ions, NCS-1 unfolds and refolds reversibly in a two-state reaction involving only the C-domain, whereas the N-domain has no detectable transitions. Overall, the results allowed us to trace the progression of NCS-1 folding along its energy landscapes and provided a solid platform for understanding the conformational dynamics of similar EF-hand proteins.

Diffusion Monte Carlo study on temporal evolution of entropy and free energy in nonequilibrium processes

A computational scheme to describe the temporal evolution of thermodynamic functions in stochastic nonequilibrium processes of isothermal classical systems is proposed on the basis of overdamped Langevin equation under given potential and temperature. In this scheme the associated Fokker-Planck-Smoluchowski equation for the probability density function is transformed into the imaginary-time Schrödinger equation with an effective Hamiltonian. The propagator for the time-dependent wave function is expressed in the framework of the path integral formalism, which can thus represent the dynamical behaviors of nonequilibrium molecular systems such as those conformational changes observed in protein folding and ligand docking. The present study then employs the diffusion Monte Carlo method to efficiently simulate the relaxation dynamics of wave function in terms of random walker distribution, which in the long-time limit reduces to the ground-state eigenfunction corresponding to the equilibrium Boltzmann distribution. Utilizing this classical-quantum correspondence, we can describe the relaxation processes of thermodynamic functions as an approach to the equilibrium state with the lowest free energy. Performing illustrative calculations for some prototypical model potentials, the temporal evolutions of enthalpy, entropy, and free energy of the classical systems are explicitly demonstrated. When the walkers initially start from a localized configuration in one- or two-dimensional harmonic or double well potential, the increase of entropy usually dominates the relaxation dynamics toward the equilibrium state. However, when they start from a broadened initial distribution or go into a steep valley of potential, the dynamics are driven by the decrease of enthalpy, thus causing the decrease of entropy associated with the spatial localization. In the cases of one- and two-dimensional asymmetric double well potentials with two minimal points and an energy barrier between them, we observe a nonequilibrium behavior that the system entropy first increases with the broadening of the initially localized walker distribution and then it begins to decrease along with the trapping at the global minimum of the potential, thus leading to the minimization of the free energy.

Characterization of the Stiffness of Multiple Particles Trapped by Dielectrophoretic Tweezers in a Microfluidic Device

Characterization of the stiffness of multiple particles trapped by tweezers-based force spectroscopy is a key step in building simple, high-throughput, and robust systems that can investigate the molecular interactions in a biological process, but the technology to characterize it in a given environment simultaneously is still lacking. We first characterized the stiffness of multiple particles trapped by dielectrophoretic (DEP) tweezers inside a microfluidic device. In this characterization, we developed a method to measure the thermal fluctuations of the trapped multiple particles with DEP tweezers by varying the heights of the particles in the given environment at the same time. Using the data measured in this controlled environment, we extracted the stiffness of the trapped particles and calculated their force. This study not only provides a simple and high-throughput method to measure the trap stiffness of multiple particles inside a microfluidic device using DEP tweezers but also inspires the application of the trapped multiple particles to investigate the dynamics in molecular interactions.

Communication: Does force spectroscopy of biomolecules probe their intrinsic dynamic properties?

In single-molecule pulling experiments, the molecule of interest is attached to a much larger object such as an atomic force microscope tip or a micrometer sized bead. The measured dynamics of molecular transitions is therefore affected by the hydrodynamic drag on the pulling instrument itself. By considering the transitions within the combined system (the molecule and the instrument), it is shown here that two distinct physical regimes exist: when the intrinsic stiffness of the molecule is greater than that of the linker connecting the molecule to the pulling setup then the pulling experiment probes the intrinsic dynamics of the molecule with only relatively small (and quantifiable) corrections resulting from the pulling setup. In contrast, when the stiffness of the linker exceeds that of the molecule, the molecular transition in question involves concerted motion of the molecule and the pulling setup and the hydrodynamic drag on the pulling instrument becomes the dominant source of friction along the molecular reaction coordinate. An analytical formula interpolating between these two cases is further derived. These results explain recent conflicting observations where some single-molecule pulling measurements report anomalously low diffusion coefficients along molecular reaction coordinates while others do not.

Mapping the energy landscape for second-stage folding of a single membrane protein

Membrane proteins are designed to fold and function in a lipid membrane, yet folding experiments within a native membrane environment are challenging to design. Here we show that single-molecule forced unfolding experiments can be adapted to study helical membrane protein folding under native-like bicelle conditions. Applying force using magnetic tweezers, we find that a transmembrane helix protein, Escherichia coli rhomboid protease GlpG, unfolds in a highly cooperative manner, largely unraveling as one physical unit in response to mechanical tension above 25 pN. Considerable hysteresis is observed, with refolding occurring only at forces below 5 pN. Characterizing the energy landscape reveals only modest thermodynamic stability (ΔG = 6.5 kBT) but a large unfolding barrier (21.3 kBT) that can maintain the protein in a folded state for long periods of time (t1/2 ∼3.5 h). The observed energy landscape may have evolved to limit the existence of troublesome partially unfolded states and impart rigidity to the structure.

Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimum-energy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p(fold)(x). P(fold) encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p(fold) can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p(fold)(x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.

Spring-loaded unraveling of a single SNARE complex by NSF in one round of ATP turnover

During intracellular membrane trafficking, N-ethylmaleimide-sensitive factor (NSF) and alpha-soluble NSF attachment protein (α-SNAP) disassemble the soluble NSF attachment protein receptor (SNARE) complex for recycling of the SNARE proteins. The molecular mechanism by which NSF disassembles the SNARE complex is largely unknown. Using single-molecule fluorescence spectroscopy and magnetic tweezers, we found that NSF disassembled a single SNARE complex in only one round of adenosine triphosphate (ATP) turnover. Upon ATP cleavage, the NSF hexamer developed internal tension with dissociation of phosphate ions. After latent time measuring tens of seconds, NSF released the built-up tension in a burst within 20 milliseconds, resulting in disassembly followed by immediate release of the SNARE proteins. Thus, NSF appears to use a "spring-loaded" mechanism to couple ATP hydrolysis and unfolding of substrate proteins.

Transition-Path Probability as a Test of Reaction-Coordinate Quality Reveals DNA Hairpin Folding Is a One-Dimensional Diffusive Process

Chemical reactions are typically described in terms of progress along a reaction coordinate. However, the quality of reaction coordinates for describing reaction dynamics is seldom tested experimentally. We applied a framework for gauging reaction-coordinate quality based on transition-path analysis to experimental data for the first time, looking at folding trajectories of single DNA hairpin molecules measured under tension applied by optical tweezers. The conditional probability for being on a reactive transition path was compared with the probability expected for ideal diffusion over a 1D energy landscape based on the committor function. Analyzing measurements and simulations of hairpin folding where end-to-end extension is the reaction coordinate, after accounting for instrumental effects on the analysis, we found good agreement between transition-path and committor analyses for model two-state hairpins, demonstrating that folding is well-described by 1D diffusion. This work establishes transition-path analysis as a powerful new tool for testing experimental reaction-coordinate quality.

Direct design of an energy landscape with bistable DNA origami mechanisms

Structural DNA nanotechnology provides a feasible technique for the design and fabrication of complex geometries even exhibiting controllable dynamic behavior. Recently we have demonstrated the possibility of implementing macroscopic engineering design approaches to construct DNA origami mechanisms (DOM) with programmable motion and tunable flexibility. Here, we implement the design of compliant DNA origami mechanisms to extend from prescribing motion to prescribing an energy landscape. Compliant mechanisms facilitate motion via deformation of components with tunable stiffness resulting in well-defined mechanical energy stored in the structure. We design, fabricate, and characterize a DNA origami nanostructure with an energy landscape defined by two stable states (local energy minima) separated by a designed energy barrier. This nanostructure is a four-bar bistable mechanism with two undeformed states. Traversing between those states requires deformation, and hence mechanical energy storage, in a compliant arm of the linkage. The energy barrier for switching between two states was obtained from the conformational distribution based on a Boltzmann probability function and closely follows a predictive mechanical model. Furthermore, we demonstrated the ability to actuate the mechanism into one stable state via additional DNA inputs and then release the actuation via DNA strand displacement. This controllable multistate system establishes a foundation for direct design of energy landscapes that regulate conformational dynamics similar to biomolecular complexes.

Reconstructing folding energy landscape profiles from nonequilibrium pulling curves with an inverse Weierstrass integral transform

The energy landscapes that drive structure formation in biopolymers are difficult to measure. Here we validate experimentally a novel method to reconstruct landscape profiles from single-molecule pulling curves using an inverse Weierstrass transform (IWT) of the Jarzysnki free-energy integral. The method was applied to unfolding measurements of a DNA hairpin, replicating the results found by the more-established weighted histogram (WHAM) and inverse Boltzmann methods. Applying both WHAM and IWT methods to reconstruct the folding landscape for a RNA pseudoknot having a stiff energy barrier, we found that landscape features with sharper curvature than the force probe stiffness could not be recovered with the IWT method. The IWT method is thus best for analyzing data from stiff force probes such as atomic force microscopes.

Mechano-transduction: from molecules to tissues

Biological mechano-transduction and force-dependent changes scale from protein conformation (â„« to nm) to cell organization and multi-cell function (mm to cm) to affect cell organization, fate, and homeostasis.

External forces play complex roles in cell organization, fate, and homeostasis. Changes in these forces, or how cells respond to them, can result in abnormal embryonic development and diseases in adults. How cells sense and respond to these mechanical stimuli requires an understanding of the biophysical principles that underlie changes in protein conformation and result in alterations in the organization and function of cells and tissues. Here, we discuss mechano-transduction as it applies to protein conformation, cellular organization, and multi-cell (tissue) function.

Direct single-molecule observation of calcium-dependent misfolding in human neuronal calcium sensor-1

Neurodegenerative disorders are strongly linked to protein misfolding, and crucial to their explication is a detailed understanding of the underlying structural rearrangements and pathways that govern the formation of misfolded states. Here we use single-molecule optical tweezers to monitor misfolding reactions of the human neuronal calcium sensor-1, a multispecific EF-hand protein involved in neurotransmitter release and linked to severe neurological diseases. We directly observed two misfolding trajectories leading to distinct kinetically trapped misfolded conformations. Both trajectories originate from an on-pathway intermediate state and compete with native folding in a calcium-dependent manner. The relative probability of the different trajectories could be affected by modulating the relaxation rate of applied force, demonstrating an unprecedented real-time control over the free-energy landscape of a protein. Constant-force experiments in combination with hidden Markov analysis revealed the free-energy landscape of the misfolding transitions under both physiological and pathological calcium concentrations. Remarkably for a calcium sensor, we found that higher calcium concentrations increased the lifetimes of the misfolded conformations, slowing productive folding to the native state. We propose a rugged, multidimensional energy landscape for neuronal calcium sensor-1 and speculate on a direct link between protein misfolding and calcium dysregulation that could play a role in neurodegeneration.

Reconstructing folding energy landscapes by single-molecule force spectroscopy

Folding may be described conceptually in terms of trajectories over a landscape of free energies corresponding to different molecular configurations. In practice, energy landscapes can be difficult to measure. Single-molecule force spectroscopy (SMFS), whereby structural changes are monitored in molecules subjected to controlled forces, has emerged as a powerful tool for probing energy landscapes. We summarize methods for reconstructing landscapes from force spectroscopy measurements under both equilibrium and nonequilibrium conditions. Other complementary, but technically less demanding, methods provide a model-dependent characterization of key features of the landscape. Once reconstructed, energy landscapes can be used to study critical folding parameters, such as the characteristic transition times required for structural changes and the effective diffusion coefficient setting the timescale for motions over the landscape. We also discuss issues that complicate measurement and interpretation, including the possibility of multiple states or pathways and the effects of projecting multiple dimensions onto a single coordinate.

Free-energy inference from partial work measurements in small systems

Fluctuation relations (FRs) are among the few existing general results in nonequilibrium systems. Their verification requires the measurement of the total work performed on a system. Nevertheless in many cases only a partial measurement of the work is possible. Here we consider FRs in dual-trap optical tweezers where two different forces (one per trap) are measured. With this setup we perform pulling experiments on single molecules by moving one trap relative to the other. We demonstrate that work should be measured using the force exerted by the trap that is moved. The force that is measured in the trap at rest fails to provide the full dissipation in the system, leading to a (incorrect) work definition that does not satisfy the FR. The implications to single-molecule experiments and free-energy measurements are discussed. In the case of symmetric setups a second work definition, based on differential force measurements, is introduced. This definition is best suited to measure free energies as it shows faster convergence of estimators. We discuss measurements using the (incorrect) work definition as an example of partial work measurement. We show how to infer the full work distribution from the partial one via the FR. The inference process does also yield quantitative information, e.g., the hydrodynamic drag on the dumbbell. Results are also obtained for asymmetric dual-trap setups. We suggest that this kind of inference could represent a previously unidentified and general application of FRs to extract information about irreversible processes in small systems.

Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has, therefore, been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase-space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs/PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, a 1 ms Anton MD simulation of the bovine pancreatic trypsin inhibitor protein, and an optical tweezer force probe trajectory of an RNA hairpin.

Sequence-resolved free energy profiles of stress-bearing vimentin intermediate filaments

Intermediate filaments (IFs) are key to the mechanical strength of metazoan cells. Their basic building blocks are dimeric coiled coils mediating hierarchical assembly of the full-length filaments. Here we use single-molecule force spectroscopy by optical tweezers to assess the folding and stability of coil 2B of the model IF protein vimentin. The coiled coil was unzipped from its N and C termini. When pulling from the C terminus, we observed that the coiled coil was resistant to force owing to the high stability of the C-terminal region. Pulling from the N terminus revealed that the N-terminal half is considerably less stable. The mechanical pulling assay is a unique tool to study and control seed formation and structure propagation of the coiled coil. We then used rigorous theory-based deconvolution for a model-free extraction of the energy landscape and local stability profiles. The data obtained from the two distinct pulling directions complement each other and reveal a tripartite stability of the coiled coil: a labile N-terminal half, followed by a medium stability section and a highly stable region at the far C-terminal end. The different stability regions provide important insight into the mechanics of IF assembly.

Kinetics and energetics of biomolecular folding and binding

The ability of biomolecules to fold and to bind to other molecules is fundamental to virtually every living process. Advanced experimental techniques can now reveal how single biomolecules fold or bind against mechanical force, with the force serving as both the regulator and the probe of folding and binding transitions. Here, we present analytical expressions suitable for fitting the major experimental outputs from such experiments to enable their analysis and interpretation. The fit yields the key determinants of the folding and binding processes: the intrinsic on-rate and the location and height of the activation barrier.

Single-molecule studies of riboswitch folding

The folding dynamics of riboswitches are central to their ability to modulate gene expression in response to environmental cues. In most cases, a structural competition between the formation of a ligand-binding aptamer and an expression platform (or some other competing off-state) determines the regulatory outcome. Here, we review single-molecule studies of riboswitch folding and function, predominantly carried out using single-molecule FRET or optical trapping approaches. Recent results have supplied new insights into riboswitch folding energy landscapes, the mechanisms of ligand binding, the roles played by divalent ions, the applicability of hierarchical folding models, and kinetic vs. thermodynamic control schemes. We anticipate that future work, based on improved data sets and potentially combining multiple experimental techniques, will enable the development of more complete models for complex RNA folding processes. This article is part of a Special Issue entitled: Riboswitches.

Interrogating biology with force: single molecule high-resolution measurements with optical tweezers

Single molecule force spectroscopy methods, such as optical and magnetic tweezers and atomic force microscopy, have opened up the possibility to study biological processes regulated by force, dynamics of structural conformations of proteins and nucleic acids, and load-dependent kinetics of molecular interactions. Among the various tools available today, optical tweezers have recently seen great progress in terms of spatial resolution, which now allows the measurement of atomic-scale conformational changes, and temporal resolution, which has reached the limit of the microsecond-scale relaxation times of biological molecules bound to a force probe. Here, we review different strategies and experimental configurations recently developed to apply and measure force using optical tweezers. We present the latest progress that has pushed optical tweezers' spatial and temporal resolution down to today's values, discussing the experimental variables and constraints that are influencing measurement resolution and how these can be optimized depending on the biological molecule under study.

Diverse metastable structures formed by small oligomers of α-synuclein probed by force spectroscopy

Oligomeric aggregates are widely suspected as toxic agents in diseases caused by protein aggregation, yet they remain poorly characterized, partly because they are challenging to isolate from a heterogeneous mixture of species. We developed an assay for characterizing structure, stability, and kinetics of individual oligomers at high resolution and sensitivity using single-molecule force spectroscopy, and applied it to observe the formation of transient structured aggregates within single oligomers of α-synuclein, an intrinsically-disordered protein linked to Parkinson's disease. Measurements of the molecular extension as the proteins unfolded under tension in optical tweezers revealed that even small oligomers could form numerous metastable structures, with a surprisingly broad range of sizes. Comparing the structures formed in monomers, dimers and tetramers, we found that the average mechanical stability increased with oligomer size. Most structures formed within a minute, with size-dependent rates. These results provide a new window onto the complex α-synuclein aggregation landscape, characterizing the microscopic structural heterogeneity and kinetics of different pathways.

Mechanical unzipping and rezipping of a single SNARE complex reveals hysteresis as a force-generating mechanism

Formation of the soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) complex provides mechanical thrust for membrane fusion, but its molecular mechanism is still unclear. Here using magnetic tweezers, we observe mechanical responses of a single neuronal SNARE complex under constant pulling force. Single SNARE complexes may be unzipped with 34 pN force. When rezipping is induced by lowering the force to 11 pN, only a partially assembled state results, with the C-terminal half of the SNARE complex remaining disassembled. Reassembly of the C-terminal half occurs only when the force is further lowered below 11 pN. Thus, mechanical hysteresis, characterized by the unzipping and rezipping cycle of a single SNARE complex, produces the partially assembled state. In this metastable state, unzipping toward the N-terminus is suppressed while zippering toward the C-terminus is initiated as a steep function of force. This ensures the directionality of SNARE-complex formation, making the SNARE complex a robust force-generating machine.

Interactions between (SNARE) proteins on vesicle and target membranes provide the force necessary to drive membrane fusion. By applying piconewton forces to single SNARE complexes, the authors identify a partially assembled intermediate state that reveals how force is generated in a consistent direction.

Kinetics of allosteric transitions in S-adenosylmethionine riboswitch are accurately predicted from the folding landscape

Riboswitches are RNA elements that allosterically regulate gene expression by binding cellular metabolites. The SAM-III riboswitch, one of several classes that binds S-adenosylmethionine (SAM), represses translation upon binding SAM (OFF state) by encrypting the ribosome binding sequence. We have carried out simulations of the RNA by applying mechanical force (f) to the ends of SAM-III, with and without SAM, to get quantitative insights into the f-dependent structural changes. Force-extension (z) curves (FECs) for the apo (ON) state, obtained in simulations in which f is increased at a constant loading rate, show three intermediates, with the first one being the rupture of SAM binding region, which is greatly stabilized in the OFF state. Force-dependent free energy profiles, G(z,f), as a function of z, obtained in equilibrium constant force simulations, reveal the intermediates observed in FECs. The predicted stability difference between the ON and OFF states using G(z,f) is in excellent agreement with experiments. Remarkably, using G(z,f)s and estimate of an effective diffusion constant at a single value of f allows us to predict the f-dependent transition rates using theory of first passage times for both the apo and holo states. To resolve the kinetics of assembly of SAM-III riboswitch in structural terms, we use force stretch-quench pulse sequences in which the force on RNA is maintained at a low (fq) value starting from a high value for a time period tq. Variation of tq over a wide range results in resolution of elusive states involved in the SAM binding pocket and leads to accurate determination of folding times down to fq = 0. Quantitative measure of the folding kinetics, obtained from the folding landscape, allows us to propose that, in contrast to riboswitches regulating transcription, SAM-III functions under thermodynamic control provided the basal concentration of SAM exceeds a small critical value. All of the predictions are amenable to tests in single molecule pulling experiments.

Single-chain mechanics of poly(N,N-diethylacrylamide) and poly(N-isopropylacrylamide): comparative study reveals the effect of hydrogen bond donors

The single-chain mechanics of two similar thermosensitive polymers, poly(N,N-diethylacrylamide) (PDEAM) and poly(N-isopropylacrylamide) (PNIPAM), have been studied by atomic force microscopy-based single-molecule force spectroscopy (SMFS). In a typical nonpolar organic solvent, octane, both of the polymers show the same inherent elasticity, although they have different substitutional groups. However, the mechanics of the two polymers presents large differences in water. The energies needed for the rearrangement of the bound water during elongation at room temperature are estimated by the SMFS method at the single-chain level, which is ~1.13 ± 0.10 and ~5.19 ± 0.10 kJ/mol for PDEAM and PNIPAM, respectively. In addition, PNIPAM shows a temperature-dependent single-chain mechanics when the temperature is increased across the lower critical solution temperature (LCST), while PDEAM does not. These differences observed in aqueous solution originate from the different structures of the two polymers. With a hydrogen bond donor in the amide group, PNIPAM will be more hydrated when T < LCST. When T > LCST, PNIPAM will have larger changes in both conformation and hydration. These findings also suggest that PNIPAM is a good candidate for a thermo-driven single-molecule motor, while PDEAM is not.