Mechanical unfolding and refolding pathways of ubiquitin

The mechanical cell - the role of force dependencies in synchronising protein interaction networks

The role of mechanical signals in the proper functioning of organisms is increasingly recognised, and every cell senses physical forces and responds to them. These forces are generated both from outside the cell or via the sophisticated force-generation machinery of the cell, the cytoskeleton. All regions of the cell are connected via mechanical linkages, enabling the whole cell to function as a mechanical system. In this Review, we define some of the key concepts of how this machinery functions, highlighting the critical requirement for mechanosensory proteins, and conceptualise the coupling of mechanical linkages to mechanochemical switches that enables forces to be converted into biological signals. These mechanical couplings provide a mechanism for how mechanical crosstalk might coordinate the entire cell, its neighbours, extending into whole collections of cells, in tissues and in organs, and ultimately in the coordination and operation of entire organisms. Consequently, many diseases manifest through defects in this machinery, which we map onto schematics of the mechanical linkages within a cell. This mapping approach paves the way for the identification of additional linkages between mechanosignalling pathways and so might identify treatments for diseases, where mechanical connections are affected by mutations or where individual force-regulated components are defective.

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

Emergence of effective temperatures in an out-of-equilibrium model of biopolymer folding

We investigate the possibility of extending the notion of temperature in a stochastic model for the RNA or protein folding driven out of equilibrium. We simulate the dynamics of a small RNA hairpin subject to an external pulling force, which is time-dependent. First, we consider a fluctuation-dissipation relation (FDR) whereby we verify that various effective temperatures can be obtained for different observables, only when the slowest intrinsic relaxation timescale of the system regulates the dynamics of the system. Then, we introduce a different nonequilibrium temperature, which is defined from the rate of heat exchanged with a weakly interacting thermal bath. Notably, this "kinetic" temperature can be defined for any frequency of the external switching force. We also discuss and compare the behavior of these two emerging parameters, by discriminating the time-delayed nature of the FDR temperature from the instantaneous character of the kinetic temperature. The validity of our numerics are corroborated by a simple four-state Markov model which describes the long-time behavior of the RNA molecule.

Native-state fingerprint on the ubiquitin translocation across a nanopore

We study the translocation of the ubiquitin molecule (Ubq) across a channel with a double section which constitutes a general feature of several transmembrane nanopores such as the α-hemolysin (αHL). Our purpose is to establish the structure-dependent character of the Ubq translocation pathway. This implies to find the correspondence, if any, between the translocational unfolding steps and the Ubq native state. For this reason, it is convenient to apply a coarse-grained computational approach, where the protein is described only by the backbone and the force field only exploits the information contained in the native state (in the spirit of Gō-like models, or native-centric models). The αHL-like pore is portrayed as two coaxial confining cylinders: a larger one for the vestibule and a narrower one for the barrel (or stem). Such simplified approach allows a large number of translocation events to be collected by limited computational resources. The co-translocational unfolding of Ubq is described via a few collective variables that characterize the translocation progress. We find two translocation intermediates (stalled conformations) that can be associated with specific unfolding stages. In particular, in the earliest step, the strand S5 unfolds and enters the pore. This step splits the native conformation into two structural clusters packing against each other in the Ubq fold. A second stall occurs when the hairpin of the N terminal engages the stem region.

Cooperativity and modularity in protein folding

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

Mapping the Topography of a Protein Energy Landscape

Protein energy landscapes are highly complex, yet the vast majority of states within them tend to be invisible to experimentalists. Here, using site-directed mutagenesis and exploiting the simplicity of tandem-repeat protein structures, we delineate a network of these states and the routes between them. We show that our target, gankyrin, a 226-residue 7-ankyrin-repeat protein, can access two alternative (un)folding pathways. We resolve intermediates as well as transition states, constituting a comprehensive series of snapshots that map early and late stages of the two pathways and show both to be polarized such that the repeat array progressively unravels from one end of the molecule or the other. Strikingly, we find that the protein folds via one pathway but unfolds via a different one. The origins of this behavior can be rationalized using the numerical results of a simple statistical mechanics model that allows us to visualize the equilibrium behavior as well as single-molecule folding/unfolding trajectories, thereby filling in the gaps that are not accessible to direct experimental observation. Our study highlights the complexity of repeat-protein folding arising from their symmetrical structures; at the same time, however, this structural simplicity enables us to dissect the complexity and thereby map the precise topography of the energy landscape in full breadth and remarkable detail. That we can recapitulate the key features of the folding mechanism by computational analysis of the native structure alone will help toward the ultimate goal of designed amino-acid sequences with made-to-measure folding mechanisms-the Holy Grail of protein folding.

The role of binding site on the mechanical unfolding mechanism of ubiquitin

We apply novel atomistic simulations based on potential energy surface exploration to investigate the constant force-induced unfolding of ubiquitin. At the experimentally-studied force clamping level of 100 pN, we find a new unfolding mechanism starting with the detachment between β₅ and β₃ involving the binding site of ubiquitin, the Ile44 residue. This new unfolding pathway leads to the discovery of new intermediate configurations, which correspond to the end-to-end extensions previously seen experimentally. More importantly, it demonstrates the novel finding that the binding site of ubiquitin can be responsible not only for its biological functions, but also its unfolding dynamics. We also report in contrast to previous single molecule constant force experiments that when the clamping force becomes smaller than about 300 pN, the number of intermediate configurations increases dramatically, where almost all unfolding events at 100 pN involve an intermediate configuration. By directly calculating the life times of the intermediate configurations from the height of the barriers that were crossed on the potential energy surface, we demonstrate that these intermediate states were likely not observed experimentally due to their lifetimes typically being about two orders of magnitude smaller than the experimental temporal resolution.

Cooperativity in thermal and force-induced protein unfolding: integration of crack propagation and network elasticity models

We investigate force-induced and temperature-induced unfolding of proteins using the combination of a gaussian network model and a crack propagation model based on "bond"-breaking independent events. We assume the existence of threshold values for the mean strain and strain fluctuations that dictate bond rupture. Surprisingly, we find that this stepwise process usually leads to a few cooperative, first-order-like, transitions in which several bonds break simultaneously, reminiscent of the "avalanches" seen in disordered networks.

Worm-like Ising model for protein mechanical unfolding under the effect of osmolytes

We show via single-molecule mechanical unfolding experiments that the osmolyte glycerol stabilizes the native state of the human cardiac I27 titin module against unfolding without shifting its unfolding transition state on the mechanical reaction coordinate. Taken together with similar findings on the immunoglobulin-binding domain of streptococcal protein G (GB1), these experimental results suggest that osmolytes act on proteins through a common mechanism that does not entail a shift of their unfolding transition state. We investigate the above common mechanism via an Ising-like model for protein mechanical unfolding that adds worm-like-chain behavior to a recent generalization of the Wako-Saitô-Muñoz-Eaton model with support for group-transfer free energies. The thermodynamics of the model are exactly solvable, while protein kinetics under mechanical tension can be simulated via Monte Carlo algorithms. Notably, our force-clamp and velocity-clamp simulations exhibit no shift in the position of the unfolding transition state of GB1 and I27 under the effect of various osmolytes. The excellent agreement between experiment and simulation strongly suggests that osmolytes do not assume a structural role at the mechanical unfolding transition state of proteins, acting instead by adjusting the solvent quality for the protein chain analyte.

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.

Effects of confinement on thermal stability and folding kinetics in a simple Ising-like model

In a cellular environment, confinement and macromolecular crowding play an important role in thermal stability and folding kinetics of a protein. We have resorted to a generalized version of the Wako-Saitô-Muñoz-Eaton model for protein folding to study the behavior of six different protein structures confined between two walls. Changing the distance 2R between the walls, we found, in accordance with previous studies, two confinement regimes: starting from large R and decreasing R, confinement first enhances the stability of the folded state as long as this is compact and until a given value of R; then a further decrease of R leads to a decrease of folding temperature and folding rate. We found that in the low confinement regime, both unfolding temperatures and logarithm of folding rates scale as R(-γ) where γ values lie in between 1.42 and 2.35.

Direction-dependent mechanical unfolding and green fluorescent protein as a force sensor

An Ising-like model of proteins is used to investigate the mechanical unfolding of the green fluorescent protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.

Analysis of the equilibrium and kinetics of the ankyrin repeat protein myotrophin

We apply the Wako-Saito-Muñoz-Eaton model to the study of myotrophin, a small ankyrin repeat protein, whose folding equilibrium and kinetics have been recently characterized experimentally. The model, which is a native-centric with binary variables, provides a finer microscopic detail than the Ising model that has been recently applied to some different repeat proteins, while being still amenable for an exact solution. In partial agreement with the experiments, our results reveal a weakly three-state equilibrium and a two-state-like kinetics of the wild-type protein despite the presence of a nontrivial free-energy profile. These features appear to be related to a careful "design" of the free-energy landscape, so that mutations can alter this picture, stabilizing some intermediates and changing the position of the rate-limiting step. Also, the experimental findings of two alternative pathways, an N-terminal and a C-terminal one, are qualitatively confirmed, even if the variations in the rates upon the experimental mutations cannot be quantitatively reproduced. Interestingly, the folding and unfolding pathways appear to be different, even if closely related: a property that is not generally considered in the phenomenological interpretation of the experimental data.

Quantitative prediction of protein folding behaviors from a simple statistical model

The statistical nature of the protein folding process requires the use of equally detailed yet simple models that lend themselves to characterize experiments. One such model is the Wako-Saitô-Muñoz-Eaton model, that we extend here to include solvation effects (WSME-S), introduced via empirical terms. We employ the novel version to analyze the folding of two proteins, gpW and SH3, that have similar size and thermodynamic stability but with the former folding 3 orders of magnitude faster than SH3. A quantitative analysis reveals that gpW presents at most marginal barriers, in contrast to SH3 that folds following a simple two-state approximation. We reproduce the observed experimental differences in melting temperature in gpW as seen by different experimental spectroscopic probes and the shape of the rate-temperature plot. In parallel, we predict the folding complexity expected in gpW from the analysis of both the residue-level thermodynamics and kinetics. SH3 serves as a stringent control with neither folding complexity nor dispersion in melting temperatures being observed. The extended model presented here serves as an ideal tool not only to characterize folding data but also to make experimentally testable predictions.

Simplified exactly solvable model for β-amyloid aggregation

We propose an exactly solvable simplified statistical mechanical model for the thermodynamics of β-amyloid aggregation, generalizing a well-studied model for protein folding. The monomer concentration is explicitly taken into account as well as a nontrivial dependence on the microscopic degrees of freedom of the single peptide chain, both in the α-helix folded isolated state and in the fibrillar one. The phase diagram of the model is studied and compared to the outcome of fibril formation experiments which is qualitatively reproduced.

Molecular basis for the structural stability of an enclosed β-barrel loop

We present molecular dynamics simulation studies of the structural stability of an enclosed loop in the β domain of the Escherichia coli O157:H7 autotransporter EspP. Our investigation revealed that, in addition to its excellent resistance to thermal perturbations, EspP loop 5 (L5) also has remarkable mechanical stability against pulling forces along the membrane norm. These findings are consistent with the experimental report that EspP L5 helps to maintain the permeability barrier in the outer membrane. In contrast to the major secondary structure elements of globular proteins such as ubiquitin, whose resistance to thermal and mechanical perturbations depends mainly on backbone hydrogen bonds and hydrophobic interactions, the structural stability of EspP L5 can be attributed mainly to geometric constraints and side-chain interactions dominated by hydrogen bonds. Examination of B-factors from available high-resolution structures of membrane-embedded β barrels indicates that most of the enclosed loops have stable structures. This finding suggests that loops stabilized by geometric constraints and side-chain interactions might be used more generally to restrict β-barrel channels for various functional purposes.

Nearly symmetrical proteins: folding pathways and transition states

The folding pathways of the B domain of protein A have been the subject of many experimental and computational studies. Based on a statistical mechanical model, it has been suggested that the native state symmetry leads to multiple pathways, highly dependent on temperature and denaturant concentration. Experiments, however, have not confirmed this scenario. By considering four nearly symmetrical proteins, one of them being the above molecule, here we show that, if contact energies are properly taken into account, a different picture emerges from kinetic simulations of the above-mentioned model. This is characterized by a dominant folding pathway, which is consistent with the most recent experimental results. Given the simplicity of the model, we also report on a direct sampling of the transition state.

Equilibrium properties and force-driven unfolding pathways of RNA molecules

The mechanical unfolding of a simple RNA hairpin and of a 236-base portion of the Tetrahymena thermophila ribozyme is studied by means of an Ising-like model. Phase diagrams and free energy landscapes are computed exactly and suggest a simple two-state behavior for the hairpin and the presence of intermediate states for the ribozyme. Nonequilibrium simulations give the possible unfolding pathways for the ribozyme, and the dominant pathway corresponds to the experimentally observed one.

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.

Periodic forces trigger a complex mechanical response in ubiquitin

Mechanical forces govern physiological processes in all living organisms. Many cellular forces, for example, those generated in cyclic conformational changes of biological machines, have repetitive components. In apparent contrast, little is known about how dynamic protein structures respond to periodic mechanical information. Ubiquitin is a small protein found in all eukaryotes. We developed molecular dynamics simulations to unfold single and multimeric ubiquitins with periodic forces. By using a coarse-grained representation, we were able to model forces with periods about 2 orders of magnitude longer than the protein's relaxation time. We found that even a moderate periodic force weakened the protein and shifted its unfolding pathways in a frequency- and amplitude-dependent manner. A complex dynamic response with secondary structure refolding and an increasing importance of local interactions was revealed. Importantly, repetitive forces with broadly distributed frequencies elicited very similar molecular responses compared to fixed-frequency forces. When testing the influence of pulling geometry on ubiquitin's mechanical stability, it was found that the linkage involved in the mechanical degradation of cellular proteins renders the protein remarkably insensitive to periodic forces. We also devised a complementary kinetic energy landscape model that traces these observations and explains periodic-force, single-molecule measurements. In turn, this analytical model is capable of predicting dynamic protein responses. These results provide new insights into ubiquitin mechanics and a potential mechanical role during protein degradation, as well as first frameworks for dynamic protein stability and the modeling of repetitive mechanical processes.

Changing the mechanical unfolding pathway of FnIII10 by tuning the pulling strength

We investigate the mechanical unfolding of the tenth type III domain from fibronectin (FnIII(10)) both at constant force and at constant pulling velocity, by all-atom Monte Carlo simulations. We observe both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. All three major intermediates lack two of seven native beta-strands, and share a quite similar extension. The unfolding behavior is found to depend strongly on the pulling conditions. In particular, we observe large variations in the relative frequencies of occurrence for the intermediates. At low constant force or low constant velocity, all three major intermediates occur with a significant frequency. At high constant force or high constant velocity, one of them, with the N- and C-terminal beta-strands detached, dominates over the other two. Using the extended Jarzynski equality, we also estimate the equilibrium free-energy landscape, calculated as a function of chain extension. The application of a constant pulling force leads to a free-energy profile with three major local minima. Two of these correspond to the native and fully unfolded states, respectively, whereas the third one can be associated with the major unfolding intermediates.