Mechanical unfolding revisited through a simple but realistic model

Theory and Examples of Catch Bonds

We discuss slip bonds, catch bonds, and the tug-of-war mechanism using mathematical arguments. The aim is to explain the theoretical tool of molecular potential energy surfaces (PESs). For this, we propose simple 2-dimensional surface models to demonstrate how a molecule under an external force behaves. Examples are selectins. Catch bonds, in particular, are explained in more detail, and they are contrasted to slip bonds. We can support special two-dimensional molecular PESs for E- and L-selectin which allow the catch bond property. We demonstrate that Newton trajectories (NT) are powerful tools to describe these phenomena. NTs form the theoretical background of mechanochemistry.

Unfolding and Translocation of Knotted Proteins by Clp Biological Nanomachines: Synergistic Contribution of Primary Sequence and Topology Revealed by Molecular Dynamics Simulations

We use Langevin dynamics simulations to model, at an atomistic resolution, how various natively knotted proteins are unfolded in repeated allosteric translocating cycles of the ClpY ATPase. We consider proteins representative of different topologies, from the simplest knot (trefoil 3₁), to the three-twist 5₂ knot, to the most complex stevedore, 6₁, knot. We harness the atomistic detail of the simulations to address aspects that have so far remained largely unexplored, such as sequence-dependent effects on the ruggedness of the landscape traversed during knot sliding. Our simulations reveal the combined effect on translocation of the knotted protein structure, i.e., backbone topology and geometry, and primary sequence, i.e., side chain size and interactions, and show that the latter can dominate translocation hindrance. In addition, we observe that due to the interplay between the knotted topology and intramolecular contacts the transmission of tension along the polypeptide chain occurs very differently from that of homopolymers. Finally, by considering native and non-native interactions, we examine how the disruption or formation of such contacts can affect the translocation processivity and concomitantly create multiple unfolding pathways with very different activation barriers.

Comparative mechanical unfolding studies of spectrin domains R15, R16 and R17

Spectrins belong to repetitive three-helix bundle proteins that have vital functions in multicellular organisms and are of potential value in nanotechnology. To reveal the unique physical features of repeat proteins we have studied the structural and mechanical properties of three repeats of chicken brain α-spectrin (R15, R16 and R17) at the atomic level under stretching at constant velocities (0.01, 0.05 and 0.1 Å·ps^-1) and constant forces (700 and 900 pN) using molecular dynamics (MD) simulations at T = 300 K. 114 independent MD simulations were performed and their analysis has been done. Despite structural similarity of these domains we have found that R15 is less mechanically stable than R16, which is less stable than R17. This result is in agreement with the thermal unfolding rates. Moreover, we have observed the relationship between mechanical stability, flexibility of the domains and the number of aromatic residues involved in aromatic clusters.

Coarse-Grained Simulations of Topology-Dependent Mechanisms of Protein Unfolding and Translocation Mediated by ClpY ATPase Nanomachines

Clp ATPases are powerful ring shaped nanomachines which participate in the degradation pathway of the protein quality control system, coupling the energy from ATP hydrolysis to threading substrate proteins (SP) through their narrow central pore. Repetitive cycles of sequential intra-ring ATP hydrolysis events induce axial excursions of diaphragm-forming central pore loops that effect the application of mechanical forces onto SPs to promote unfolding and translocation. We perform Langevin dynamics simulations of a coarse-grained model of the ClpY ATPase-SP system to elucidate the molecular details of unfolding and translocation of an α/β model protein. We contrast this mechanism with our previous studies which used an all-α SP. We find conserved aspects of unfolding and translocation mechanisms by allosteric ClpY, including unfolding initiated at the tagged C-terminus and translocation via a power stroke mechanism. Topology-specific aspects include the time scales, the rate limiting steps in the degradation pathway, the effect of force directionality, and the translocase efficacy. Mechanisms of ClpY-assisted unfolding and translocation are distinct from those resulting from non-allosteric mechanical pulling. Bulk unfolding simulations, which mimic Atomic Force Microscopy-type pulling, reveal multiple unfolding pathways initiated at the C-terminus, N-terminus, or simultaneously from both termini. In a non-allosteric ClpY ATPase pore, mechanical pulling with constant velocity yields larger effective forces for SP unfolding, while pulling with constant force results in simultaneous unfolding and translocation.

Cell survival is critically dependent on tightly regulated protein quality control, which includes chaperone-mediated folding and degradation. In the degradation pathway, AAA+ nanomachines, such as bacterial Clp proteases, use ATP-driven mechanisms to mechanically unfold, translocate, and destroy excess or defective proteins. Understanding these remodeling mechanisms is of central importance for deciphering the details of essential cellular processes. We perform coarse-grained computer simulations to extensively probe the effect of substrate protein topology on unfolding and translocation actions of the ClpY ATPase nanomachine. We find that, independent of SP topology, unfolding proceeds from the tagged C-terminus, which is engaged by the ATPase, and translocation involves coordinated steps. Topology-specific aspects include more complex unfolding and translocation pathways of the α/β SP compared with the all-α SP due to high stability of β-hairpins and interplay of tertiary contacts. In addition, directionality of the mechanical force applied by the Clp ATPase gives rise to distinct unfolding pathways.

Single-molecule chemo-mechanical unfolding reveals multiple transition state barriers in a small single-domain protein

A fundamental question in protein folding is whether proteins fold through one or multiple trajectories. While most experiments indicate a single pathway, simulations suggest proteins can fold through many parallel pathways. Here, we use a combination of chemical denaturant, mechanical force and site-directed mutations to demonstrate the presence of multiple unfolding pathways in a simple, two-state folding protein. We show that these multiple pathways have structurally different transition states, and that seemingly small changes in protein sequence and environment can strongly modulate the flux between the pathways. These results suggest that in vivo, the crowded cellular environment could strongly influence the mechanisms of protein folding and unfolding. Our study resolves the apparent dichotomy between experimental and theoretical studies, and highlights the advantage of using a multipronged approach to reveal the complexities of a protein's free-energy landscape.

Although most protein folding experiments can be explained by a single pathway, theoretical evidence suggests the presence of multiple pathways. Here, the authors resolve this using a combination of force, chemical denaturation and mutagenesis to modulate the flux between parallel pathways.

Mechanical stability analysis of the protein L immunoglobulin-binding domain by full alanine screening using molecular dynamics simulations

This article is the first to study the mechanical properties of the immunoglobulin-binding domain of protein L (referred to as protein L) and its mutants at the atomic level. In the structure of protein L, each amino acid residue (except for alanines and glycines) was replaced sequentially by alanine. Thus, 49 mutants of protein L were obtained. The proteins were stretched at their termini at constant velocity using molecular dynamics simulations in water, i.e. by forced unfolding. 19 out of 49 mutations resulted in a large decrease of mechanical protein stability. These amino acids were affecting either the secondary structure (11 mutations) or loop structures (8 mutations) of protein L. Analysis of mechanical unfolding of the generated protein that has the same topology as protein L but consists of only alanines and glycines allows us to suggest that the mechanical stability of proteins, and specifically protein L, is determined by interactions between certain amino acid residues, although the unfolding pathway depends on the protein topology. This insight can now be used to modulate the mechanical properties of proteins and their unfolding pathways in the desired direction for using them in various biochips, biosensors and biomaterials for medicine, industry, and household purposes.

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.

Nanomechanics of β-rich proteins related to neuronal disorders studied by AFM, all-atom and coarse-grained MD methods

Computer simulations of protein unfolding substantially help to interpret force-extension curves measured in single-molecule atomic force microscope (AFM) experiments. Standard all-atom (AA) molecular dynamics simulations (MD) give a good qualitative mechanical unfolding picture but predict values too large for the maximum AFM forces with the common pulling speeds adopted here. Fine tuned coarse-grain MD computations (CG MD) offer quantitative agreement with experimental forces. In this paper we address an important methodological aspect of MD modeling, namely the impact of numerical noise generated by random assignments of bead velocities on maximum forces (F_max) calculated within the CG MD approach. Distributions of CG forces from 2000 MD runs for several model proteins rich in β structures and having folds with increasing complexity are presented. It is shown that F_max have nearly Gaussian distributions and that values of F_max for each of those β-structures may vary from 93.2 ± 28.9 pN (neurexin) to 198.3 ± 25.2 pN (fibronectin). The CG unfolding spectra are compared with AA steered MD data and with results of our AFM experiments for modules present in contactin, fibronectin and neurexin. The stability of these proteins is critical for the proper functioning of neuronal synaptic clefts. Our results confirm that CG modeling of a single molecule unfolding is a good auxiliary tool in nanomechanics but large sets of data have to be collected before reliable comparisons of protein mechanical stabilities are made.

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Experimental and theoretical studies of mechanical unfolding of different proteins

Mechanical properties of proteins are important for a wide range of biological processes including cell adhesion, muscle contraction, and protein translocation across biological membranes. It is necessary to reveal how proteins achieve their required mechanical stability under natural conditions in order to understand the biological processes and also to use the knowledge for constructing new biomaterials for medical and industrial purposes. In this connection, it is important to know how a protein will behave in response to various impacts. Theoretical and experimental works on mechanical unfolding of globular proteins will be considered in detail in this review.

Direct sampling of multiple single-molecular rupture dominant pathways involving a multistep transition

We report a novel single-molecular rupture mechanism revealed by direct sampling of the dominant pathway using a self-optimized path sampling method.

Right- and left-handed three-helix proteins. II. Similarity and differences in mechanical unfolding of proteins

Here, we study mechanical properties of eight 3-helix proteins (four right-handed and four left-handed ones), which are similar in size under stretching at a constant speed and at a constant force on the atomic level using molecular dynamics simulations. The analysis of 256 trajectories from molecular dynamics simulations with explicit water showed that the right-handed three-helix domains are more mechanically resistant than the left-handed domains. Such results are observed at different extension velocities studied (192 trajectories obtained at the following conditions: v = 0.1, 0.05, and 0.01 Å ps(-1) , T = 300 K) and under constant stretching force (64 trajectories, F = 800 pN, T = 300 K). We can explain this by the fact, at least in part, that the right-handed domains have a larger number of contacts per residue and the radius of cross section than the left-handed domains.

Early events in helix unfolding under external forces: a milestoning analysis

Initial events of helix breakage as a function of load are considered using molecular dynamics simulations and milestoning analysis. A helix length of ∼100 amino acids is considered as a model for typical helices found in molecular machines and as a model that minimizes end effects for early events of unfolding. Transitions of individual amino acids (averaged over the helix's interior residues) are examined and its surrounding hydrogen bonds are considered. Dense kinetic networks are constructed that, with milestoning analysis, provide the overall kinetics of early breakage events. Network analysis and selection of MaxFlux pathways illustrate that load impacts unfolding mechanisms in addition to time scales. At relatively high (100 pN) load levels, the principal intermediate is the 3(10)-helix, while at relatively low (10 pN) levels the π-helix is significantly populated, albeit not as an unfolding intermediate. Coarse variables are examined at different levels of resolution; the rate of unfolding illustrates remarkable stability under changes in the coarsening. Consistent prediction of about ∼5 ns for the time of a single amino-acid unfolding event are obtained. Hydrogen bonds are much faster coarse variables (by about 2 orders of magnitude) compared to backbone torsional transition, which gates unfolding and thereby provides the appropriate coarse variable for the initiation of unfolding. Results provide an atomic description of "catch-bond" behavior, based on alternative pathways, in which unfolding of a simple protein structural element occurs over longer timescales for intermediate (10 pN) loads than for zero (0 pN) or large (100 pN) loads.

Evolution of the potential energy landscape with static pulling force for two model proteins

The energy landscape is analyzed for off-lattice bead models of protein L and protein G as a function of a static pulling force. Two different pairs of attachment points (pulling directions) are compared in each case, namely, residues 1/56 and 10/32. For the terminal residue pulling direction 1/56, the distinct global minimum structures are all extended, aside from the compact geometry that correlates with zero force. The helical turns finally disappear at the highest pulling forces considered. For the 10/32 pulling direction, the changes are more complicated, with a variety of competing arrangements for beads outside the region where the force is directly applied. These alternatives produce frustrated energy landscapes, with low-lying minima separated by high barriers. The calculated folding pathways in the absence of force are in good agreement with previous work. The N-terminal hairpin folds first for protein L and the C-terminal hairpin for protein G, which exhibits an intermediate. However, for a relatively low static force, where the global minimum retains its structure, the folding mechanisms change, sometimes dramatically, depending on the protein and the attachment points. The scaling relations predicted by catastrophe theory are found to hold in the limit of short path lengths.

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.

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.

The catch bond mechanism between von Willebrand factor and platelet surface receptors investigated by molecular dynamics simulations

The multi-domain protein von Willebrand factor is crucial in the blood coagulation process at high shear. The A1 domain binds to the platelet surface receptor glycoprotein Ibalpha (GpIb alpha) and this interaction is known to be strengthened by tensile force. The molecular mechanism behind this observation was investigated here by molecular dynamics simulations. The results suggest that the proteins unbind through two distinct pathways depending whether a high-tensile force is applied or whether unbinding happens through thermal fluctuations. In the high-force unbinding pathway the A1 domain was observed to rotate away from the C-terminus of GpIb alpha. In contrast, during thermal unbinding the A1 domain rotated in the opposite direction as in the high-force pathway and the distance between the terminii of A1 and the GpIb alpha C-terminus shortened. This shortening was reduced and the lifetime of the bond extended if a moderate tensile force was applied across the complex. This suggests that the thermal unbinding pathway is inhibited by a moderate tensile force which is in agreement with the catch bond property shown previously in single molecule experiments. A designed mutant of GpIb alpha is suggested here in order to test in vitro the thermal unbinding pathway observed in silico.

Hexameric helicase deconstructed: interplay of conformational changes and substrate coupling

Hexameric helicases are molecular motor proteins that utilize energy obtained from ATP hydrolysis to translocate along and/or unwind nucleic acids. In this study, we investigate the dynamic behavior of the Simian Virus 40 hexameric helicase bound to DNA by performing molecular dynamics simulations employing a coarse-grained model. Our results elucidate the two most important molecular features of the helicase motion. First, the attractive interactions between the DNA-binding domain of the helicase and the DNA backbone are essential for the helicase to exhibit a unidirectional motion along the DNA strand. Second, the sequence of ATP binding at multiple binding pockets affects the helicase motion. Specifically, concerted ATP binding does not generate a unidirectional motion of the helicase. It is only when the binding of ATP occurs sequentially from one pocket to the next that the helicase moves unidirectionally along the DNA. Interestingly, in the reverse order of sequential ATP binding, the helicase also moves unidirectionally but in the opposite direction. These observations suggest that in nature ATP molecules must distinguish between different available ATP binding pockets of the hexameric helicase in order to function efficiently. To this end, simulations reveal that the binding of ATP in one pocket induces an opening of the next ATP-binding pocket and such an asymmetric deformation may coordinate the sequential ATP binding in a unidirectional manner. Overall, these findings may provide clues toward understanding the mechanism of substrate translocation in other motor proteins.

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

The study of mechanical unfolding, through the combined efforts of atomic force microscopy and simulation, is yielding fresh insights into the free-energy landscapes of proteins. Thus far, experiments have been mostly analyzed with one-dimensional models of the free-energy landscape. We show that as the two ends of a protein, filamin, are pulled apart at a speed tending to zero, the measured mechanical strength plateaus at approximately 30 pN instead of going toward zero, deviating from the Bell model. The deviation can only be explained by a switch between parallel pathways. Insightful analysis of mechanical unfolding kinetics needs to account for the multidimensionality of the free-energy landscapes of proteins, which are crucial for understanding the behavior of proteins under the small forces experienced in vivo.

Stress Analysis at the Molecular Level: A Forced Cucurbituril-Guest Dissociation Pathway

Changes in mechanical stresses in a tight-binding host-guest system were computed and visualized as the cationic was computationally pulled out of the cucurbituril host in a series of steps. A sharp conformational transition was observed as one of the guest's ammonium groups jumped through the center of the host to the opposite portal. The conformation immediately prior to this transition was found to possess high levels of Lennard-Jones and electrostatic stress. This observation, along with the specific distribution of Lennard-Jones stress around the portals, suggested that the conformational transition resulted from steric constriction, which had been expected, and electrostatics, which was not expected. An important role for electrostatics, at least at the level of these calculations, was confirmed by a comparative computational pulling study of another guest molecule lacking the critical ammonium group. These calculations suggest that the binding kinetics of diammonium guests that position an ammonium at each cucurbituril portal will be found to be slower than the kinetics of monoammonium guests. More generally, the results suggest that computational stress analysis can provide mechanistic insight into supramolecular systems. It will be of considerable interest to extend such applications to biomolecules, for which the mechanisms of conformational change are of great scientific and practical interest.

Single-molecule rupture dynamics on multidimensional landscapes

We explore emergent effects of multidimensionality of the free energy landscape on single-molecule kinetics under constant force. The proposed minimal model reveals the existence of a spectrum of unusual scenarios for the force-dependent lifetime, all of which are shown to occur on a free energy landscape with a single transition state. We present an analytical solution that governs single-molecule responses to a constant force and relates them to microscopic parameters of the system.

N-terminal strands of filamin Ig domains act as a conformational switch under biological forces

Conformational changes of filamin A under stress have been postulated to play crucial roles in signaling pathways of cell responses. Direct observation of conformational changes under stress is beyond the resolution of current experimental techniques. On the other hand, computational studies are mainly limited to either traditional molecular dynamics simulations of short durations and high forces or simulations of simplified models. Here we perform all-atom discrete molecular dynamics (DMD) simulations to study thermally and force-induced unfolding of filamin A. The high conformational sampling efficiency of DMD allows us to observe force-induced unfolding of filamin A Ig domains under physiological forces. The computationally identified critical unfolding forces agree well with experimental measurements. Despite a large heterogeneity in the population of force-induced intermediate states, we find a common initial unfolding intermediate in all the Ig domains of filamin, where the N-terminal strand unfolds. We also study the thermal unfolding of several filamin Ig-like domains. We find that thermally induced unfolding features an early-stage intermediate state similar to the one observed in force-induced unfolding and characterized by the N-terminal strand being unfurled. We propose that the N-terminal strand may act as a conformational switch that unfolds under physiological forces leading to exposure of cryptic binding sites, removal of native binding sites, and modulating the quaternary structure of domains.

Mechanical characterization of protein L in the low-force regime by electromagnetic tweezers/evanescent nanometry

Mechanical manipulation at the single molecule level of proteins exhibiting mechanical stability poses a technical challenge that has been almost exclusively approached by atomic force microscopy (AFM) techniques. However, due to mechanical drift limitations, AFM techniques are restricted to experimental recordings that last less than a minute in the high-force regime. Here we demonstrate a novel combination of electromagnetic tweezers and evanescent nanometry that readily captures the forced unfolding trajectories of protein L at pulling forces as low as 10-15 pN. Using this approach, we monitor unfolding and refolding cycles of the same polyprotein for a period of time longer than 30 min. From such long-lasting recordings, we obtain ensemble averages of unfolding step sizes and rates that are consistent with single-molecule AFM data obtained at higher stretching forces. The unfolding kinetics of protein L at low stretching forces confirms and extends the observations that the mechanical unfolding rate is exponentially dependent on the pulling force within a wide range of stretching forces spanning from 13 pN up to 120 pN. Our experiments demonstrate a novel approach for the mechanical manipulation of single proteins for extended periods of time in the low-force regime.

A multi-timescale strength model of alpha-helical protein domains

Here we report a constitutive model that characterizes the strength of an alpha-helical protein domain subjected to tensile deformation, covering more than ten orders of magnitude in timescales. The model elucidates multiple physical mechanisms of failure in dependence on the deformation rate, quantitatively linking atomistic simulation results with experimental strength measurements of alpha-helical protein domains. The model provides a description of the strength of alpha-helices based on fundamental physical parameters such as the H-bond energy and the polypeptide's persistence length, showing that strength is controlled by energetic, nonequilibrium processes at high rates and by thermodynamical, equilibrium processes at low rates. Our model provides a novel perspective on the strength of protein domains at ultra-slow pulling speeds relevant under physiologic and experimental conditions.

Selection of optimal variants of Gō-like models of proteins through studies of stretching

The Gō-like models of proteins are constructed based on the knowledge of the native conformation. However, there are many possible choices of a Hamiltonian for which the ground state coincides with the native state. Here, we propose to use experimental data on protein stretching to determine what choices are most adequate physically. This criterion is motivated by the fact that stretching processes usually start with the native structure, in the vicinity of which the Gō-like models should work the best. Our selection procedure is applied to 62 different versions of the Gō model and is based on 28 proteins. We consider different potentials, contact maps, local stiffness energies, and energy scales--uniform and nonuniform. In the latter case, the strength of the nonuniformity was governed either by specificity or by properties related to positioning of the side groups. Among them is the simplest variant: uniform couplings with no i, i + 2 contacts. This choice also leads to good folding properties in most cases. We elucidate relationship between the local stiffness described by a potential which involves local chirality and the one which involves dihedral and bond angles. The latter stiffness improves folding but there is little difference between them when it comes to stretching.

Elastic bond network model for protein unfolding mechanics

Recent advances in single molecule mechanics have made it possible to investigate the mechanical anisotropy of protein stability in great detail. A quantitative prediction of protein unfolding forces at experimental time scales has so far been difficult. Here, we present an elastically bonded network model to describe the mechanical unfolding forces of green fluorescent protein in eight different pulling directions. The combination of an elastic network and irreversible bond fracture kinetics offers a new concept to understand the determinants of mechanical protein stability.

Pulling direction as a reaction coordinate for the mechanical unfolding of single molecules

The folding and unfolding kinetics of single molecules, such as proteins or nucleic acids, can be explored by mechanical pulling experiments. Determining intrinsic kinetic information, at zero stretching force, usually requires an extrapolation by fitting a theoretical model. Here, we apply a recent theoretical approach describing molecular rupture in the presence of force to unfolding kinetic data obtained from coarse-grained simulations of ubiquitin. Unfolding rates calculated from simulations over a broad range of stretching forces, for different pulling directions, reveal a remarkable "turnover" from a force-independent process at low force to a force-dependent process at high force, akin to the "roll-over" in unfolding rates sometimes seen in studies using chemical denaturant. While such a turnover in rates is unexpected in one dimension, we demonstrate that it can occur for dynamics in just two dimensions. We relate the turnover to the quality of the pulling direction as a reaction coordinate for the intrinsic folding mechanism. A novel pulling direction, designed to be the most relevant to the intrinsic folding pathway, results in the smallest turnover. Our results are in accord with protein engineering experiments and simulations which indicate that the unfolding mechanism at high force can differ from the intrinsic mechanism. The apparent similarity between extrapolated and intrinsic rates in experiments, unexpected for different unfolding barriers, can be explained if the turnover occurs at low forces.

A structural, kinetic model of soft tissue thermomechanics

A structure-based kinetic model was developed to predict the thermomechanical response of collagenous soft tissues. The collagen fibril was represented as an ensemble of molecular arrays with cross-links connecting the collagen molecules within the same array. A two-state kinetic model for protein folding was employed to represent the native and the denatured states of the collagen molecule. The Monte Carlo method was used to determine the state of the collagen molecule when subjected to thermal and mechanical loads. The model predictions were compared to existing experimental data for New Zealand white rabbit patellar tendons. The model predictions for one-dimensional tissue shrinkage and the corresponding mechanical property degradation agreed well with the experimental data, showing that the gross tissue behavior is dictated by molecular-level phenomena.

Secondary structure, mechanical stability, and location of transition state of proteins

It is well known that the unfolding times of proteins, tauu, scales with the external mechanical force f as tauu=tauu0exp(-fxu/kBT), where xu is the location of the average transition state along the reaction coordinate given by the end-to-end distance. Using the off-lattice Go-like models, we have shown that in terms of xu, proteins may be divided into two classes. The first class, which includes beta- and beta/alpha-proteins, has xu approximately 2-5 A whereas the second class of alpha-proteins has xu about three times larger than that of the first class, xu approximately 7-15 A. These results are in good agreement with the experimental data. The secondary structure is found to play the key role in determining the shape of the free energy landscape. Namely, the distance between the native state and the transition state depends on the helix content linearly. It is shown that xu has a strong correlation with mechanical stability of proteins. Defining the unfolding force, fu, from the constant velocity pulling measurements as a measure of the mechanical stability, we predict that xu decays with fu by a power law, xu approximately fu(-mu), where the exponent mu is approximately 0.4. We have demonstrated that the unfolding force correlates with the helix content of a protein. The contact order, which is a measure of fraction of local contacts, was found to strongly correlate with the mechanical stability and the distance between the transition state and native state. Our study reveals that xu and fu might be estimated using either the helicity or the contact order.

The mechanical unfolding of ubiquitin through all-atom Monte Carlo simulation with a Go-type potential

The mechanical unfolding of proteins under a stretching force has an important role in living systems and is a logical extension of the more general protein folding problem. Recent advances in experimental methodology have allowed the stretching of single molecules, thus rendering this process ripe for computational study. We use all-atom Monte Carlo simulation with a Gō-type potential to study the mechanical unfolding pathway of ubiquitin. A detailed, robust, well-defined pathway is found, confirming existing results in this vein though using a different model. Additionally, we identify the protein's fundamental stabilizing secondary structure interactions in the presence of a stretching force and show that this fundamental stabilizing role does not persist in the absence of mechanical stress. The apparent success of simulation methods in studying ubiquitin's mechanical unfolding pathway indicates their potential usefulness for future study of the stretching of other proteins and the relationship between protein structure and the response to mechanical deformation.

Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality

The equilibrium free energy difference between two long-lived molecular species or "conformational states" of a protein (or any other molecule) can in principle be estimated by measuring the work needed to shuttle the system between them, independent of the irreversibility of the process. This is the meaning of the Jarzynski equality (JE), which we test in this paper by performing simulations that unfold a protein by pulling two atoms apart. Pulling is performed fast relative to the relaxation time of the molecule and is thus far from equilibrium. Choosing a simple protein model for which we can independently compute its equilibrium properties, we show that the free energy can be exactly and effectively estimated from nonequilibrium simulations. To do so, one must carefully and correctly determine the ensemble of states that are pulled, which is more important the farther from equilibrium one performs simulations; this highlights a potential problem in using the JE to extract the free energy from forced unfolding experiments. The results presented here also demonstrate that the free energy difference between the native and denatured states of a protein measured in solution is not always equal to the free energy profile that can be estimated from forced unfolding simulations (or experiments) using the JE.

Internal protein dynamics shifts the distance to the mechanical transition state

Mechanical unfolding of polyproteins by force spectroscopy provides valuable insight into their free energy landscapes. Most experiments of the unfolding process have been fit to two-state and/or one dimensional models, with the details of the protein and its dynamics often subsumed into a zero-force unfolding rate and a distance x{u}{1D} to the transition state. We consider the entire phase space of a model protein under a constant force, and show that x{u}{1D} contains a sizeable contribution from exploring the full multidimensional energy landscape. This effect is greater for proteins with many degrees of freedom that are affected by force; and surprisingly, we predict that externally attached flexible linkers also contribute to the measured unfolding characteristics.

Stretching the immunoglobulin 27 domain of the titin protein: the dynamic energy landscape

Molecular simulations are carried out on the Immunoglobulin 27 domain of the titin protein. The energy landscape is mapped out using an implicit solvent model, and molecular dynamics simulations are run with the solvent explicitly modeled. Stretching a protein is shown to produce a dynamic energy landscape in which the energy minima move in configuration space, change in depth, and are created and destroyed. The connections of these landscape changes to the mechanical unfolding of the Immunoglobulin 27 domain are addressed. Hydrogen bonds break upon stretching by either intrabasin processes associated with the movement of energy minima, or interbasin processes associated with transitions between energy minima. Intrabasin changes are reversible and dominate for flexible interactions, whereas interbasin changes are irreversible and dominate for stiff interactions. The most flexible interactions are Glu-Lys salt bridges, which can act like tethers to bind strands even after all backbone interactions between the strands have been broken. As the protein is stretched, different types of structures become the lowest energy structures, including structures that incorporate nonnative hydrogen bonds. Structures that have flat energy versus elongation profiles become the lowest energy structures at elongations of several Angstroms, and are associated with the unfolding intermediate state observed experimentally.

Sequential unfolding of individual helices of bacterioopsin observed in molecular dynamics simulations of extraction from the purple membrane

Multiple molecular dynamics simulations of bacterioopsin pulling from its C-terminus show that its alpha-helices unfold individually. In the first metastable state observed in the simulations, helix G is unfolded at its C-terminal segment while the rest of helix G (residues 200-216) is folded and opposes resistance because of a salt-bridge network consisting of Asp-212 and Lys-216 on helix G and Arg-82 and Asp-85 on helix C. Helix G unfolds inside the bundle because the external force is applied to its C-terminal end in a direction perpendicular to the surface of the membrane. Inversely, helix F has to flip by 180 degrees to exit from the membrane because the applied force and the helical N-C axis point in opposite directions. At the highest peak of the force, which cannot be interpreted in single-molecule force spectroscopy experiments, helix F has a pronounced kink at Pro-186. Mutation of Pro-186 and/or the charged side chains mentioned above, which are involved in very favorable electrostatic interactions in the low-dielectric region of the membrane, are expected to reduce the highest peak of the force. Helices E and D unfold in a similar way to helices G and F, respectively. Hence, the force-distance profile and sequence of events during forced unfolding of bacterioopsin are influenced by the up-and-down topology of the seven-helix bundle. The sequential extraction of individual helices from the membrane suggests that the spontaneous (un)folding of bacterioopsin proceeds through metastable bundles of fewer than seven helices. The metastable states observed in the simulations provide atomic level evidence that corroborates the interpretation of very recent force spectroscopy experiments of bacteriorhodopsin refolding.