Direct evidence of the multidimensionality of the free-energy landscapes of proteins revealed by mechanical probes

Energy Landscape of Ubiquitin Is Weakly Multidimensional

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

Estimating the Directional Flexibility of Proteins from Equilibrium Thermal Fluctuations

The directional flexibility of proteins is an equilibrium molecular property which is accessible to both experiment and computation. Single molecule force spectroscopy (SMFS) experiments report effective directional spring constants to describe the collective anisotropic response of a protein structure to mechanical pulling forces applied along selected axes. On the other hand, computational methods have thus far employed either indirect force based nonequilibrium simulations or coarse-grained elastic network models (ENM) to predict protein directional spring constants. Here, we examine the ability of equilibrium atomistic Molecular Dynamics (MD) simulations to estimate the directional flexibility and mechanical anisotropy of proteins. MD-derived effective directional spring constants are found to correlate well with SMFS spring constants (ρ² = 0.97-0.99; Adj R² = 0.92-0.99) and unfolding forces (ρ² = 0.85-0.97; Adj R² = 0.63-0.91) for five different globular proteins. Specifically, the computed spring constants reproduce the mechanical anisotropy reported by SMFS along five different directions of green fluorescence protein (GFP) and six directions of the immunoglobulin-binding B1 domain of streptococcal protein G (GB1). Further, protein dynamics as captured in MD can be translated into spring constants which can distinguish the N-C directional flexibility of ubiquitin (Ub) from two structurally homologous small ubiquitin-like modifier (SUMO1 and SUMO2) isoforms. We apply our computational framework to study the mechanical anisotropy of Ub along the seven lysine-C-term directions which are functionally relevant. We show that Ub possesses two distinct flexibility scales along these directions which roughly differ by an order of magnitude. Further, our studies reveal that the mechanical anisotropy of Ub is modified in contrasting ways by the binding of two partner proteins (UBCH5A and UEV) which attach and recognize these biomolecular tag proteins. On the basis of equilibrium MD benchmarks for flexibility along 2485 bond vectors in Ub, we propose and validate a new covariance-propagation scheme to extract spring constants from ENM normal modes. We also critically examine the ability of ENM to predict directional flexibility of proteins and suggest modifications to improve these intuitive and scalable descriptions.

Non-Markov bond model for dynamic force spectroscopy

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

The hierarchical emergence of worm-like chain behaviour from globular domain polymer chains

Biological organisms make use of hierarchically organised structures to modulate mechanical behaviour across multiple lengthscales, allowing microscopic objects to generate macroscopic effects. Within these structural hierarchies, the resultant physical behaviour of the entire system is determined not only by the intrinsic mechanical properties of constituent subunits, but also by their organisation in three-dimensional space. When these subunits are polyproteins, colloidal chains or other globular domain polymers, the Kratky-Porod model is often assumed for the individual subunits. Hence, it is implicitly asserted that the polymeric object has an intrinsic parameter, the persistence length, that defines its flexibility. However, the persistence lengths extracted from experiment vary, and are often relatively small. Through a series of simulations on polymer chains formed of globular subunits, we show that the persistence length itself is a hierarchical structural property, related not only to the intrinsic mechanical properties of the underlying monomeric subunits, but emerging due to the organisation of inhomogenous geometry along the polymer contour.

Multiplexed protein force spectroscopy reveals equilibrium protein folding dynamics and the low-force response of von Willebrand factor

Single-molecule force spectroscopy has provided unprecedented insights into protein folding, force regulation, and function. So far, the field has relied primarily on atomic force microscope and optical tweezers assays that, while powerful, are limited in force resolution, throughput, and require feedback for constant force measurements. Here, we present a modular approach based on magnetic tweezers (MT) for highly multiplexed protein force spectroscopy. Our approach uses elastin-like polypeptide linkers for the specific attachment of proteins, requiring only short peptide tags on the protein of interest. The assay extends protein force spectroscopy into the low force (<1 pN) regime and enables parallel and ultra-stable measurements at constant forces. We present unfolding and refolding data for the small, single-domain protein ddFLN4, commonly used as a molecular fingerprint in force spectroscopy, and for the large, multidomain dimeric protein von Willebrand factor (VWF) that is critically involved in primary hemostasis. For both proteins, our measurements reveal exponential force dependencies of unfolding and refolding rates. We directly resolve the stabilization of the VWF A2 domain by Ca²⁺ and discover transitions in the VWF C domain stem at low forces that likely constitute the first steps of VWF's mechano-activation. Probing the force-dependent lifetime of biotin-streptavidin bonds, we find that monovalent streptavidin constructs with specific attachment geometry are significantly more force stable than commercial, multivalent streptavidin. We expect our modular approach to enable multiplexed force-spectroscopy measurements for a wide range of proteins, in particular in the physiologically relevant low-force regime.

Mechanical unfolding of spectrin reveals a super-exponential dependence of unfolding rate on force

We investigated the mechanical unfolding of single spectrin molecules over a broad range of loading rates and thus unfolding forces by combining magnetic tweezers with atomic force microscopy. We find that the mean unfolding force increases logarithmically with loading rate at low loading rates, but the increase slows at loading rates above 1pN/s. This behavior indicates an unfolding rate that increases exponentially with the applied force at low forces, as expected on the basis of one-dimensional models of protein unfolding. At higher forces, however, the increase of the unfolding rate with the force becomes faster than exponential, which may indicate anti-Hammond behavior where the structures of the folded and transition states become more different as their free energies become more similar. Such behavior is rarely observed and can be explained by either a change in the unfolding pathway or as a reflection of a multidimensional energy landscape of proteins under force.

Mechanical Softening of a Small Ubiquitin-Related Modifier Protein Due to Temperature Induced Flexibility at the Core

Despite the growing interest in the thermal softening of proteins, the mechanistic details of it are far from understood. β-Grasp proteins have globular shape with compact structure and they are mechanically resilient. The β-clamp or mechanical clamp in them formed by the interactions between the terminal β-strands is generally associated with the protein mechanical resistance. Although previous studies showed that temperature can perturb the protein mechanical stability, the structural changes leading to the lowered mechanical resistance are not known. Here, we investigated the temperature dependent mechanical stability of small ubiquitin-related modifier 2 (SUMO2) using single-molecule force spectroscopy (SMFS) and the corresponding conformational changes using ensemble experiments. SMFS studies on the polyprotein of SUMO2 estimate a decrease in the spring constant of the protein from 4.50 to 1.35 N/m upon increasing the temperature from 5 to 45 °C. Interestingly, near-UV circular dichroism spectroscopy reveals a decrease in tertiary structure content while the overall secondary structure of the protein remains unchanged. Steady-state fluorescence and quenching studies on SUMO2 with a tryptophan mutation at the core (F60W) show that the nonpolar environment of the tryptophan is unchanged and the protein core is inaccessible to the bulk solvent, in the same temperature range. We attribute the thermal softening observed in atomic force microscopy (AFM) experiments to the reduction in tertiary structure of SUMO2. Our results provide evidence for the importance of the intramolecular interactions at the protein core along with the β-clamp or mechanical clamp in providing the mechanical resistance as well as in modulating the protein stiffness.

Molecular Simulations Suggest a Force-Dependent Mechanism of Vinculin Activation

Focal adhesions are dynamic constructs at the leading edge of migrating cells, linking them to the extracellular matrix and enabling force sensing and transmission. The lifecycle of a focal adhesion is a highly coordinated process involving spatial and temporal variations of protein composition, interaction, and cellular tension. The assembly of focal adhesions requires the recruitment and activation of vinculin. Vinculin is present in the cytoplasm in an autoinhibited conformation in which its tail is held pincerlike by its head domains, further stabilized by two high-affinity head-tail interfaces. Vinculin has binding sites for talin and F-actin, but effective binding requires vinculin activation to release its head-tail associations. In migrating cells, it has been shown that the locations of vinculin activation coincide with areas of high cellular tension, and that the highest recorded tensions across vinculin are associated with adhesion assembly. Here, we use a structure-based model to investigate vinculin activation by talin modulated by tensile force generated by transient associations with F-actin. We show that vinculin activation may proceed from an intermediate state stabilized by partial talin-vinculin association. There is a low-force regime and a high-force regime where vinculin activation is dominated by two different pathways with distinct responses to force. Specifically, at zero or low forces, vinculin activation requires substantial destabilization of the main head-tail interface, which is rigid and undergoes very limited fluctuations, despite the other being relatively flexible. This pathway is not significantly affected by force; instead, higher forces favor an alternative pathway, which seeks to release the vinculin tail from its pincerlike head domains before destabilizing the head-tail interfaces. This pathway has a force-sensitive activation barrier and is significantly accelerated by force. Experimental data of vinculin during various stages of the focal adhesion lifecycle are consistent with the proposed force-regulated activation pathway.

Perspective: Mechanochemistry of biological and synthetic molecules

Coupling of mechanical forces and chemical transformations is central to the biophysics of molecular machines, polymer chemistry, fracture mechanics, tribology, and other disciplines. As a consequence, the same physical principles and theoretical models should be applicable in all of those fields; in fact, similar models have been invoked (and often repeatedly reinvented) to describe, for example, cell adhesion, dry and wet friction, propagation of cracks, and action of molecular motors. This perspective offers a unified view of these phenomena, described in terms of chemical kinetics with rates of elementary steps that are force dependent. The central question is then to describe how the rate of a chemical transformation (and its other measurable properties such as the transition path) depends on the applied force. I will describe physical models used to answer this question and compare them with experimental measurements, which employ single-molecule force spectroscopy and which become increasingly common. Multidimensionality of the underlying molecular energy landscapes and the ensuing frequent misalignment between chemical and mechanical coordinates result in a number of distinct scenarios, each showing a nontrivial force dependence of the reaction rate. I will discuss these scenarios, their commonness (or its lack), and the prospects for their experimental validation. Finally, I will discuss open issues in the field.

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.

Effect of external pulling forces on the length distribution of peptides

BACKGROUND

The distribution of the length of a polypeptide, or that of the distance between any two of its atoms, is an important property as it can be analytically or numerically estimated for a number of polymer models. Importantly, it is directly measurable through a number of different experimental techniques. Length distributions can be straightforwardly assessed from molecular dynamics simulation; however, true convergence through full accurate coverage of the length range is difficult to achieve.

METHODS

The application of external constant force combined with the weighted-histogram analysis method (WHAM) is used to enhance sampling of unlikely 'long' or 'short' conformations and obtain the potential of mean force, while also collecting dynamic properties of the chain under variable tension.

RESULTS

We demonstrate the utility of constant force to enhance the sampling efficiency and obtain experimentally measurable quantities on a series of short peptides, including charge-rich sequences that are known to be highly helical but whose properties are distinct from those of helical peptides undergoing helix-coil transitions.

CONCLUSIONS

Force-enhanced sampling enhances the range and accuracy of the length-based potential of mean force of the peptide, in particular those sequences that contain increased numbers of charged residues.

GENERAL SIGNIFICANCE

This approach allows users to simultaneously probe the force-dependent behaviour of peptides directly, enhance the range and accuracy of the length-based PMF of the peptide and also test the convergence of simulations by comparing the overlap of PMF profiles from different constant forces. This article is part of a special issue entitled Recent developments of molecular dynamics.

A structure-based model fails to probe the mechanical unfolding pathways of the titin I27 domain

We discuss the use of a structure based Cα-Go model and Langevin dynamics to study in detail the mechanical properties and unfolding pathway of the titin I27 domain. We show that a simple Go-model does detect correctly the origin of the mechanical stability of this domain. The unfolding free energy landscape parameters x(u) and ΔG(‡), extracted from dependencies of unfolding forces on pulling speeds, are found to agree reasonably well with experiments. We predict that above v = 10(4) nm/s the additional force-induced intermediate state is populated at an end-to-end extension of about 75 Å. The force-induced switch in the unfolding pathway occurs at the critical pulling speed v(crit) ≈ 10(6)-10(7) nm/s. We argue that this critical pulling speed is an upper limit of the interval where Bell's theory works. However, our results suggest that the Go-model fails to reproduce the experimentally observed mechanical unfolding pathway properly, yielding an incomplete picture of the free energy landscape. Surprisingly, the experimentally observed intermediate state with the A strand detached is not populated in Go-model simulations over a wide range of pulling speeds. The discrepancy between simulation and experiment is clearly seen from the early stage of the unfolding process which shows the limitation of the Go model in reproducing unfolding pathways and deciphering the complete picture of the free energy landscape.

Model studies of force-dependent kinetics of multi-barrier reactions

According to transition state theory, the rate of a reaction that traverses multiple energy barriers is determined by the least stable (rate-determining) transition state. The preceding ('inner') energy barriers are kinetically 'invisible' but mechanistically significant. Here we show experimentally and computationally that the reduction rate of organic disulphides by phosphines in water, which in the absence of force proceeds by an equilibrium formation of a thiophosphonium intermediate, measured as a function of force applied on the disulphide moiety yields a usefully accurate estimate of the height of the inner barrier. We apply varying stretching force to the disulphide by incorporating it into a series of increasingly strained macrocycles. This force accelerates the reduction, even though the strain-free rate-determining step is orthogonal to the pulling direction. The observed rate-force correlation is consistent with the simplest model of force-dependent kinetics of a multi-barrier reaction.

Force as a single molecule probe of multidimensional protein energy landscapes

Force spectroscopy has developed into an indispensable tool for studying folding and binding of proteins on a single molecule level in real time. Design of the pulling geometry allows tuning the reaction coordinate in a very precise manner. Many recent experiments have taken advantage of this possibility and have provided detailed insight the folding pathways on the complex high dimensional energy landscape. Beyond its potential to provide control over the reaction coordinate, force is also an important physiological parameter that affects protein conformation under in vivo conditions. Single molecule force spectroscopy studies have started to unravel the response and adaptation of force bearing protein structures to mechanical loads.

The molten globule state is unusually deformable under mechanical force

Recently, the role of force in cellular processes has become more evident, and now with advances in force spectroscopy, the response of proteins to force can be directly studied. Such studies have found that native proteins are brittle, and thus not very deformable. Here, we examine the mechanical properties of a class of intermediates referred to as the molten globule state. Using optical trap force spectroscopy, we investigated the response to force of the native and molten globule states of apomyoglobin along different pulling axes. Unlike natively folded proteins, the molten globule state of apomyoglobin is compliant (large distance to the transition state); this large compliance means that the molten globule is more deformable and the unfolding rate is more sensitive to force (the application of force or tension will have a more dramatic effect on the unfolding rate). Our studies suggest that these are general properties of molten globules and could have important implications for mechanical processes in the cell.

Effect of loading conditions on the dissociation behaviour of catch bond clusters

Under increasing tensile load, the lifetime of a single catch bond counterintuitively increases up to a maximum and then decreases exponentially like a slip bond. So far, the characteristics of single catch bond dissociation have been extensively studied. However, it remains unclear how a cluster of catch bonds behaves under tensile load. We perform computational analysis on the following models to examine the characteristics of clustered catch bonds: (i) clusters of catch bonds with equal load sharing, (ii) clusters of catch bonds with linear load sharing, and (iii) clusters of catch bonds in micropipette-manipulated cell detachment. We focus on the differences between the slip and catch bond clusters, identifying the critical factors for exhibiting the characteristics of catch bond mechanism for the multiple-bond system. Our computation reveals that for a multiple-bond cluster, the catch bond behaviour could only manifest itself under relatively uniform loading conditions and at certain stages of decohesion, explaining the difficulties in observing the catch bond mechanism under real biological conditions.

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

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

Mechanically unfolding protein L using a laser-feedback-controlled cantilever

Force spectroscopy using the atomic force microscope (AFM) can yield important information on the strength and lifetimes of the folded states of single proteins and their complexes when they are loaded with force. For example, by mechanically unfolding concatenated proteins at different velocities, a dynamic force spectrum can be built up that allows reconstruction of the energy landscape that the protein traverses during unfolding. To characterize fully the unfolding landscape, however, it is necessary both to explore the entire force spectrum and to characterize each species populated during unfolding. In the conventional AFM apparatus, force is applied to the protein construct through a compliant cantilever. This limits the dynamic range of the force spectrum that can be probed, and the cantilever recoil after unfolding may mask the presence of metastable intermediates. Here, we describe to our knowledge a new technique-constant-deflection AFM-in which the compliance of the AFM cantilever is removed. Using this technique, we show that protein L exhibits a more complex unfolding energy landscape than previously detected using the conventional technique. This technique is also able to detect the presence of a refolding intermediate whose formation is otherwise prevented by cantilever recoil.

Complex unfolding kinetics of single-domain proteins in the presence of force

Single-molecule force spectroscopy is providing unique, and sometimes unexpected, insights into the free-energy landscapes of proteins. Despite the complexity of the free-energy landscapes revealed by mechanical probes, forced unfolding experiments are often analyzed using one-dimensional models that predict a logarithmic dependence of the unfolding force on the pulling velocity. We previously found that the unfolding force of the protein filamin at low pulling speed did not decrease logarithmically with the pulling speed. Here we present results from a large number of unfolding simulations of a coarse-grain model of the protein filamin under a broad range of constant forces. These show that a two-path model is physically plausible and produces a deviation from the behavior predicted by one-dimensional models analogous to that observed experimentally. We also show that the analysis of the distributions of unfolding forces (p[F]) contains crucial and exploitable information, and that a proper description of the unfolding of single-domain proteins needs to account for the intrinsic multidimensionality of the underlying free-energy landscape, especially when the applied perturbation is small.