Protein Folding Kinetics Under Force from Molecular Simulation

Dissociation Rate Calculation via Constant-Force Steered Molecular Dynamics Simulation

Steered molecular dynamics (SMD) simulations are used to study molecular dissociation events by applying a harmonic force to the molecules and pulling them at a constant velocity. Instead of constant-velocity pulling, we use a constant force: the constant-force SMD (CF-SMD) simulation. The CF-SMD simulation employs a constant force to reduce the activation barrier of molecular dissociation, thereby enhancing the dissociation event. Here, we present the capability of the CF-SMD simulation to estimate the dissociation time at equilibrium. We performed all-atom CF-SMD simulations for NaCl and protein-ligand systems, producing dissociation time at various forces. We extrapolated these values to the dissociation rate without a constant force using Bell's model or the Dudko-Hummer-Szabo model. We demonstrate that the CF-SMD simulations with the models predicted the dissociation time in equilibrium. A CF-SMD simulation is a powerful tool for estimating the dissociation rate in a direct and computationally efficient manner.

Free energy and kinetic rate calculation via non-equilibrium molecular simulation: application to biomolecules

Non-equilibrium molecular dynamics (NEMD) simulation has been recognized as a powerful tool for examining biomolecules and provides fruitful insights into not only non-equilibrium but also equilibrium processes. We review recent advances in NEMD simulation and relevant, fundamental results of non-equilibrium statistical mechanics. We first introduce Crooks fluctuation theorem and Jarzynski equality that relate free energy difference to work done on a physical system during a non-equilibrium process. The theorems are beneficial for the analysis of NEMD trajectories. We then describe rate theory, a framework to calculate molecular kinetics from a non-equilibrium process; this theoretical framework enables us to calculate a reaction time-mean-first passage time-from NEMD trajectories. We, in turn, present recent NEMD techniques that apply an external force to a system to enhance molecular dissociation and introduce their application to biomolecules. Lastly, we show the current status of an appropriate selection of reaction coordinates for NEMD simulation.

Alginate Particles for Enzyme Immobilization Using Spray Drying

Enzymes are important catalysts for biological processes due to their high catalytic activity and selectivity. However, their low thermal stability limited their industrial applications. The present work demonstrates a simple and effective method for enzyme immobilization via spray drying. Alginate was used as a support material. Phytase, an important enzyme in the animal feed industry, was selected to study the effect of enzyme immobilization using alginate particles on its thermal stability. The physicochemical properties of alginate particles such as size, surface morphology, and heat resistance were studied. Successful immobilization of phytase was confirmed by confocal microscopy, and the immobilized phytase retained 58% of its original activity upon heating at 95 °C, compared to 4% when the alginate support material was absent. Phytase was released promptly in a simulated gastrointestinal tract with >95% of its original activity recovered. The spray drying method for phytase immobilization is scalable and applicable to other enzymes for various applications.

Reconstruction of mechanical unfolding and refolding pathways of proteins with atomic force spectroscopy and computer simulations

Most proteins in proteomes are large, typically consist of more than one domain and are structurally complex. This often makes studying their mechanical unfolding pathways challenging. Proteins composed of tandem repeat domains are a subgroup of multi-domain proteins that, when stretched, display a saw-tooth pattern in their mechanical unfolding force extension profiles due to their repetitive structure. However, the assignment of force peaks to specific repeats undergoing mechanical unraveling is complicated because all repeats are similar and they interact with their neighbors and form a contiguous tertiary structure. Here, we describe in detail a combination of experimental and computational single-molecule force spectroscopy methods that proved useful for examining the mechanical unfolding and refolding pathways of ankyrin repeat proteins. Specifically, we explain and delineate the use of atomic force microscope-based single molecule force spectroscopy (SMFS) to record the mechanical unfolding behavior of ankyrin repeat proteins and capture their unusually strong refolding propensity that is responsible for generating impressive refolding force peaks. We also describe Coarse Grain Steered Molecular Dynamic (CG-SMD) simulations which complement the experimental observations and provide insights in understanding the unfolding and refolding of these proteins. In addition, we advocate the use of novel coiled-coils-based mechanical polypeptide probes which we developed to demonstrate the vectorial character of folding and refolding of these repeat proteins. The combination of AFM-based SMFS on native and CC-equipped proteins with CG-SMD simulations is powerful not only for ankyrin repeat polypeptides, but also for other repeat proteins and more generally to various multidomain, non-repetitive proteins with complex topologies.

Force probe simulations using an adaptive resolution scheme

Molecular simulations of the forced unfolding and refolding of biomolecules or molecular complexes allow to gain important kinetic, structural and thermodynamic information about the folding process and the underlying energy landscape. In force probe molecular dynamics (FPMD) simulations, one pulls one end of the molecule with a constant velocity in order to induce the relevant conformational transitions. Since the extended configuration of the system has to fit into the simulation box together with the solvent such simulations are very time consuming. Here, we apply a hybrid scheme in which the solute is treated with atomistic resolution and the solvent molecules far away from the solute are described in a coarse-grained manner. We use the adaptive resolution scheme (AdResS) that has very successfully been applied to various examples of equilibrium simulations. We perform FPMD simulations using AdResS on a well studied system, a dimer formed from mechanically interlocked calixarene capsules. The results of the multiscale simulations are compared to all-atom simulations of the identical system and we observe that the size of the region in which atomistic resolution is required depends on the pulling velocity, i.e. the particular non-equilibrium situation. For large pulling velocities a larger all atom region is required. Our results show that multiscale simulations can be applied also in the strong non-equilibrium situations that the system experiences in FPMD simulations.

Dynamic coarse-graining fills the gap between atomistic simulations and experimental investigations of mechanical unfolding

We present a dynamic coarse-graining technique that allows one to simulate the mechanical unfolding of biomolecules or molecular complexes on experimentally relevant time scales. It is based on Markov state models (MSMs), which we construct from molecular dynamics simulations using the pulling coordinate as an order parameter. We obtain a sequence of MSMs as a function of the discretized pulling coordinate, and the pulling process is modeled by switching among the MSMs according to the protocol applied to unfold the complex. This way we cover seven orders of magnitude in pulling speed. In the region of rapid pulling, we additionally perform steered molecular dynamics simulations and find excellent agreement between the results of the fully atomistic and the dynamically coarse-grained simulations. Our technique allows the determination of the rates of mechanical unfolding in a dynamical range from approximately 10^-8/ns to 1/ns thus reaching experimentally accessible time regimes without abandoning atomistic resolution.

Force probe simulations using a hybrid scheme with virtual sites

Hybrid simulations, in which a part of the system is treated with atomistic resolution and the remainder is represented on a coarse-grained level, allow for fast sampling while using the accuracy of atomistic force fields. We apply a hybrid scheme to study the mechanical unfolding and refolding of a molecular complex using force probe molecular dynamics (FPMD) simulations. The degrees of freedom of the solvent molecules are treated in a coarse-grained manner while atomistic resolution is retained for the solute. The coupling between the solvent and the solute is provided using virtual sites. We test two different common coarse-graining procedures, the iterative Boltzmann inversion method and the force matching procedure, and find that both methodologies give similar results. The results of the FPMD simulations are compared to all-atom simulations of the same system and we find that differences between these simulations and the ones using the hybrid scheme are in a similar range as the differences obtained when using different atomistic force fields. Thus, a hybrid scheme yields qualitatively correct results in the strong non-equilibrium situation the system is experiencing in FPMD simulations.

The influence of disulfide bonds on the mechanical stability of proteins is context dependent

Disulfide bonds play a crucial role in proteins, modulating their stability and constraining their conformational dynamics. A particularly important case is that of proteins that need to withstand forces arising from their normal biological function and that are often disulfide bonded. However, the influence of disulfides on the overall mechanical stability of proteins is poorly understood. Here, we used single-molecule force spectroscopy (smFS) to study the role of disulfide bonds in different mechanical proteins in terms of their unfolding forces. For this purpose, we chose the pilus protein FimG from Gram-negative bacteria and a disulfide-bonded variant of the I91 human cardiac titin polyprotein. Our results show that disulfide bonds can alter the mechanical stability of proteins in different ways depending on the properties of the system. Specifically, disulfide-bonded FimG undergoes a 30% increase in its mechanical stability compared with its reduced counterpart, whereas the unfolding force of I91 domains experiences a decrease of 15% relative to the WT form. Using a coarse-grained simulation model, we rationalized that the increase in mechanical stability of FimG is due to a shift in the mechanical unfolding pathway. The simple topology-based explanation suggests a neutral effect in the case of titin. In summary, our results indicate that disulfide bonds in proteins act in a context-dependent manner rather than simply as mechanical lockers, underscoring the importance of considering disulfide bonds both computationally and experimentally when studying the mechanical properties of proteins.

Reconciling Intermediates in Mechanical Unfolding Experiments with Two-State Protein Folding in Bulk

Most experimentally well-characterized single domain proteins of less than 100 residues have been found to be two-state folders. That is, only two distinct populations can explain both equilibrium and kinetic measurements. Results from single molecule force spectroscopy, where a protein is unfolded by applying a mechanical pulling force to its ends, have largely confirmed this description for proteins found to be two-state in ensemble experiments. Recently, however, stable intermediates have been reported in mechanical unfolding experiments on a cold-shock protein previously found to be a prototypical two-state folder. Here, we tackle this discrepancy using free energy landscapes and Markov state models derived from coarse-grained molecular simulations. We show that protein folding intermediates can be selectively stabilized by the pulling force and that the populations of these intermediates vary in a force-dependent manner. Our model qualitatively captures the experimental results and suggests a possible origin of the apparent discrepancy.

Structural Determinants of Misfolding in Multidomain Proteins

Recent single molecule experiments, using either atomic force microscopy (AFM) or Förster resonance energy transfer (FRET) have shown that multidomain proteins containing tandem repeats may form stable misfolded structures. Topology-based simulation models have been used successfully to generate models for these structures with domain-swapped features, fully consistent with the available data. However, it is also known that some multidomain protein folds exhibit no evidence for misfolding, even when adjacent domains have identical sequences. Here we pose the question: what factors influence the propensity of a given fold to undergo domain-swapped misfolding? Using a coarse-grained simulation model, we can reproduce the known propensities of multidomain proteins to form domain-swapped misfolds, where data is available. Contrary to what might be naively expected based on the previously described misfolding mechanism, we find that the extent of misfolding is not determined by the relative folding rates or barrier heights for forming the domains present in the initial intermediates leading to folded or misfolded structures. Instead, it appears that the propensity is more closely related to the relative stability of the domains present in folded and misfolded intermediates. We show that these findings can be rationalized if the folded and misfolded domains are part of the same folding funnel, with commitment to one structure or the other occurring only at a relatively late stage of folding. Nonetheless, the results are still fully consistent with the kinetic models previously proposed to explain misfolding, with a specific interpretation of the observed rate coefficients. Finally, we investigate the relation between interdomain linker length and misfolding, and propose a simple alchemical model to predict the propensity for domain-swapped misfolding of multidomain proteins.

On artifacts in single-molecule force spectroscopy

In typical force spectroscopy experiments, a small biomolecule is attached to a soft polymer linker that is pulled with a relatively large bead or cantilever. At constant force, the total extension stochastically changes between two (or more) values, indicating that the biomolecule undergoes transitions between two (or several) conformational states. In this paper, we consider the influence of the dynamics of the linker and mesoscopic pulling device on the force-dependent rate of the conformational transition extracted from the time dependence of the total extension, and the distribution of rupture forces in force-clamp and force-ramp experiments, respectively. For these different experiments, we derive analytic expressions for the observables that account for the mechanical response and dynamics of the pulling device and linker. Possible artifacts arise when the characteristic times of the pulling device and linker become comparable to, or slower than, the lifetimes of the metastable conformational states, and when the highly anharmonic regime of stretched linkers is probed at high forces. We also revisit the problem of relating force-clamp and force-ramp experiments, and identify a linker and loading rate-dependent correction to the rates extracted from the latter. The theory provides a framework for both the design and the quantitative analysis of force spectroscopy experiments by highlighting, and correcting for, factors that complicate their interpretation.

Effect of confinement on DNA, solvent and counterion dynamics in a model biological nanopore

The application of recent advances in nanopore technology to high-throughput DNA sequencing requires a more detailed understanding of solvent, ion and DNA interactions occurring within these pores. Here we present a combination of atomistic and coarse-grained modeling studies of the dynamics of short single-stranded DNA (ssDNA) homopolymers within the alpha-hemolysin pore, for the two single-stranded homopolymers poly(dA)40 and poly(dC)40. Analysis of atomistic simulations along with the per-residue decomposition of protein-DNA interactions in these simulations gives new insight into the very complex issues that have yet to be fully addressed with detailed MD simulations. We discuss a modification of the solvent properties and ion distribution around DNA within nanopore confinement and put it into the general framework of counterion condensation theory. There is a reasonable agreement in computed properties from our all-atom simulations and the resulting predictions from analytical theories with experimental data, and our equilibrium results here support the conclusions from our previous non-equilibrium Brownian dynamics studies with a recently developed BROMOC protocol that cations are the primary charge carriers through alpha-hemolysin nanopores under an applied voltage in the presence of ssDNA. Clustering analysis led to an identification of distinct conformational states of captured polymer and depth of the current blockade. Therefore, our data suggest that confined polymer may act as a flickering gate, thus contributing to excess noise phenomena. We also discuss the extent of water structuring due to nanopore confinement and the relationship between the conformational dynamics of a captured polymer and the distribution of blocked current.

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.

Low folding cooperativity of HP35 revealed by single-molecule force spectroscopy and molecular dynamics simulation

Some small proteins, such as HP35, fold at submicrosecond timescale with low folding cooperativity. Although these proteins have been extensively investigated, still relatively little is known about their folding mechanism. Here, using single-molecule force spectroscopy and steered molecule dynamics simulation, we study the unfolding of HP35 under external force. Our results show that HP35 unfolds at extremely low forces without a well-defined unfolding transition state. Subsequently, we probe the structure of unfolded HP35 using the persistence length obtained in the force spectroscopy. We found that the persistence length of unfolded HP35 is around 0.72 nm, >40% longer than typical unstructured proteins, suggesting that there are a significant amount of residual secondary structures in the unfolded HP35. Molecular dynamics simulation further confirmed this finding and revealed that many native contacts are preserved in HP35, even its two ends have been extended up to 8 nm. Our results therefore suggest that retaining a significant amount of secondary structures in the unfolded state of HP35 may be an efficient way to reduce the entropic cost for the formation of tertiary structure and increase the folding speed, although the folding cooperativity is compromised. Moreover, we anticipate that the methods we used in this work can be extended to the study of other proteins with complex folding behaviors and even intrinsically disordered ones.

Direct observation of a force-induced switch in the anisotropic mechanical unfolding pathway of a protein

Many biological processes generate force, and proteins have evolved to resist and respond to tension along different force axes. Single-molecule force spectroscopy allows for molecular insight into the behavior of proteins under force and the mechanism of protein folding in general. Here, we have used src SH3 to investigate the effect of different pulling axes under the low-force regime afforded by an optical trap. We find that this small cooperatively folded protein shows an anisotropic response to force; the protein is more mechanically resistant to force applied along a longitudinal axis compared to force applied perpendicular to the terminal β strand. In the longitudinal axis, we observe an unusual biphasic behavior revealing a force-induced switch in the unfolding mechanism suggesting the existence of two parallel unfolding pathways. A site-specific variant can selectively affect one of these pathways. Thus, even this simple two-state protein demonstrates a complex mechanical unfolding trajectory, accessing multiple unfolding pathways under the low-force regime of the optical trap; the specific unfolding pathway depends on the perturbation axis and the applied force.

Peculiar reduction of graphene oxide into graphene after diffusion in exponentially growing polyelectrolyte multilayers

In the present work, in situ reduction of graphene oxide (GO) into graphene was preformed, after diffusion in exponentially growing polyelectrolyte multilayers, using sodium citrate as the reducing agent. First, the graphene oxide was obtained by treating a commercial grade of Expanded Graphite (EG). Based on XRD and Raman spectroscopy results, a complete exfoliation of graphene nanopellets down to one layer was achieved during the oxidation process. Secondly, the diffusion of GO was carried out in an exponentially growing polyelectrolyte multilayer film made from poly(diallyldimethylammonium chloride) as the polycation and from poly(acrylic acid) as the polyanion. Electrical conductivity of the GO based films was measured during the reduction process as a function of time. The conductivity reached values of the order of 10(-4) S cm(-1), whereas the pristine polyelectrolyte multilayer was highly insulating (∼10(-8) S cm(-1)). The conductivity also reached a maximal value after about 24 h of reduction and decreased for longer reduction duration. Some tentative explanations for this peculiar finding will be given.

Protein high-force pulling simulations yield low-force results

All-atom explicit-solvent molecular dynamics simulations are used to pull with extremely large constant force (750–3000 pN) on three small proteins. The introduction of a nondimensional timescale permits direct comparison of unfolding across all forces. A crossover force of approximately 1100 pN divides unfolding dynamics into two regimes. At higher forces, residues sequentially unfold from the pulling end while maintaining the remainder of the protein force-free. Measurements of hydrodynamic viscous stresses are made easy by the high speeds of unfolding. Using an exact low-Reynolds-number scaling, these measurements can be extrapolated to provide, for the first time, an estimate of the hydrodynamic force on low-force unfolding. Below 1100 pN, but surprisingly still at extremely large applied force, intermediate states and cooperative unfoldings as seen at much lower forces are observed. The force-insensitive persistence of these structures indicates that decomposition into unfolded fragments requires a large fluctuation. This finding suggests how proteins are constructed to resist transient high force. The progression of helix and sheet unfolding is also found to be insensitive to force. The force-insensitivity of key aspects of unfolding opens the possibility that numerical simulations can be accelerated by high applied force while still maintaining critical features of unfolding.

Free energy profiles from single-molecule pulling experiments

Nonequilibrium pulling experiments provide detailed information about the thermodynamic and kinetic properties of molecules. We show that unperturbed free energy profiles as a function of molecular extension can be obtained rigorously from such experiments without using work-weighted position histograms. An inverse Weierstrass transform is used to relate the system free energy obtained from the Jarzynski equality directly to the underlying molecular free energy surface. An accurate approximation for the free energy surface is obtained by using the method of steepest descent to evaluate the inverse transform. The formalism is applied to simulated data obtained from a kinetic model of RNA folding, in which the dynamics consists of jumping between linker-dominated folded and unfolded free energy surfaces.

BSDB: the biomolecule stretching database

We describe the Biomolecule Stretching Data Base that has been recently set up at http://www.ifpan.edu.pl/BSDB/. It provides information about mechanostability of proteins. Its core is based on simulations of stretching of 17 134 proteins within a structure-based model. The primary information is about the heights of the maximal force peaks, the force-displacement patterns, and the sequencing of the contact-rupturing events. We also summarize the possible types of the mechanical clamps, i.e. the motifs which are responsible for a protein's resistance to stretching.

Collapse dynamics of single proteins extended by force

Single-molecule force spectroscopy has opened up new approaches to the study of protein dynamics. For example, an extended protein folding after an abrupt quench in the pulling force was shown to follow variable collapse trajectories marked by well-defined stages that departed from the expected two-state folding behavior that is commonly observed in bulk. Here, we explain these observations by developing a simple approach that models the free energy of a mechanically extended protein as a combination of an entropic elasticity term and a short-range potential representing enthalpic hydrophobic interactions. The resulting free energy of the molecule shows a force-dependent energy barrier of magnitude, DeltaE =epsilon(F - F(c))(3/2), separating the enthalpic and entropic minima that vanishes at a critical force F(c). By solving the Langevin equation under conditions of a force quench, we generate folding trajectories corresponding to the diffusional collapse of an extended polypeptide. The predicted trajectories reproduce the different stages of collapse, as well as the magnitude and time course of the collapse trajectories observed experimentally in ubiquitin and I27 protein monomers. Our observations validate the force-clamp technique as a powerful approach to determining the free-energy landscape of proteins collapsing and folding from extended states.

Designing the folding mechanics of coiled coils

Naturally occurring coiled coils are often not homogeneous throughout their entire structure but rather interrupted by sequence discontinuities and non-coiled-coil-forming subsegments. We apply atomic force microscopy to locally probe the mechanical folding/unfolding process of a well-understood model coiled coil when unstructured subsegments with different sizes are added. We find that the refolding force decreases from 7.8 pN with increasing size of the added unstructured subsegment, while the unfolding properties of the model coiled coil remain unchanged. We show that this behavior results from the increased size of the nucleation seed which has to form before further coiled-coil folding can proceed. Since the nucleation seed size is linked to the width of the energetic folding barrier, we are able to directly measure the dependence of folding forces on the barrier width. Our results allow the design of coiled coils with designated refolding forces by simply adjusting the nucleation seed size.

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

Kinetic bulk and single molecule folding experiments characterize barrier properties but the shape of folding landscapes between barrier top and native state is difficult to access. Here, we directly extract the full free energy landscape of a single molecule of the GCN4 leucine zipper using dual beam optical tweezers. To this end, we use deconvolution force spectroscopy to follow an individual molecule's trajectory with high temporal and spatial resolution. We find a heterogeneous energy landscape of the GCN4 leucine zipper domain. The energy profile is divided into two stable C-terminal heptad repeats and two less stable repeats at the N-terminus. Energies and transition barrier positions were confirmed by single molecule kinetic analysis. We anticipate that deconvolution sampling is a powerful tool for the model-free investigation of protein energy landscapes.

Coordinate-dependent diffusion in protein folding

Diffusion on a low-dimensional free-energy surface is a remarkably successful model for the folding dynamics of small single-domain proteins. Complicating the interpretation of both simulations and experiments is the expectation that the effective diffusion coefficient D will in general depend on the position along the folding coordinate, and this dependence may vary for different coordinates. Here we explore the position dependence of D, its connection to protein internal friction, and the consequences for the interpretation of single-molecule experiments. We find a large decrease in D from unfolded to folded, for reaction coordinates that directly measure fluctuations in Cartesian configuration space, including those probed in single-molecule experiments. In contrast, D is almost independent of Q, the fraction of native amino acid contacts: Near the folded state, small fluctuations in position cause large fluctuations in Q, and vice versa for the unfolded state. In general, position-dependent free energies and diffusion coefficients for any two good reaction coordinates that separate reactant, product, and transition states, are related by a simple transformation, as we demonstrate. With this transformation, we obtain reaction coordinates with position-invariant D. The corresponding free-energy surfaces allow us to justify the assumptions used in estimating the speed limit for protein folding from experimental measurements of the reconfiguration time in the unfolded state, and also reveal intermediates hidden in the original free-energy projection. Lastly, we comment on the design of future single-molecule experiments that probe the position dependence of D directly.

Mechanics of forced unfolding of proteins

We describe and solve a two-state kinetic model for the forced unfolding of proteins. The protein oligomer is modeled as a heterogeneous, freely jointed chain with two possible values of Kuhn length and contour length representing its folded and unfolded configurations. We obtain analytical solutions for the force-extension response of the protein oligomer for different types of loading conditions. We fit the analytical solutions for constant-velocity pulling to the force-extension data for ubiquitin and fibrinogen and obtain model parameters, such as Kuhn lengths and kinetic coefficients, for both proteins. We then predict their response under a linearly increasing force and find that our solutions for ubiquitin are consistent with a different set of experiments. Our calculations suggest that the refolding rate of proteins at low forces is several orders larger than the unfolding rate, and neglecting it can lead to lower predictions for the unfolding force, especially at high stretching velocities. By accounting for the refolding of proteins we obtain a critical force below which equilibrium is biased in favor of the folded state. Our calculations also suggest new methods to determine the distance of the transition state from the energy wells representing the folded and unfolded states of a protein.

Ligand-dependent equilibrium fluctuations of single calmodulin molecules

Single-molecule force spectroscopy allows superb mechanical control of protein conformation. We used a custom-built low-drift atomic force microscope to observe mechanically induced conformational equilibrium fluctuations of single molecules of the eukaryotic calcium-dependent signal transducer calmodulin (CaM). From this data, the ligand dependence of the full energy landscape can be reconstructed. We find that calcium ions affect the folding kinetics of the individual CaM domains, whereas target peptides stabilize the already folded structure. Single-molecule data of full length CaM reveal that a wasp venom peptide binds noncooperatively to CaM with 2:1 stoichiometry, whereas a target enzyme peptide binds cooperatively with 1:1 stoichiometry. If mechanical load is applied directly to the target peptide, real-time binding/unbinding transitions can be observed.

Theory, analysis, and interpretation of single-molecule force spectroscopy experiments

Dynamic force spectroscopy probes the kinetic and thermodynamic properties of single molecules and molecular assemblies. Here, we propose a simple procedure to extract kinetic information from such experiments. The cornerstone of our method is a transformation of the rupture-force histograms obtained at different force-loading rates into the force-dependent lifetimes measurable in constant-force experiments. To interpret the force-dependent lifetimes, we derive a generalization of Bell's formula that is formally exact within the framework of Kramers theory. This result complements the analytical expression for the lifetime that we derived previously for a class of model potentials. We illustrate our procedure by analyzing the nanopore unzipping of DNA hairpins and the unfolding of a protein attached by flexible linkers to an atomic force microscope. Our procedure to transform rupture-force histograms into the force-dependent lifetimes remains valid even when the molecular extension is a poor reaction coordinate and higher-dimensional free-energy surfaces must be considered. In this case the microscopic interpretation of the lifetimes becomes more challenging because the lifetimes can reveal richer, and even nonmonotonic, dependence on the force.