Thermal effects in stretching of Go-like models of titin and secondary structures

Steered molecular dynamics simulations reveal the role of Ca2+ in regulating mechanostability of cellulose-binding proteins

The conversion of cellulosic biomass into biofuels requires degradation of the biomass into fermentable sugars. The most efficient natural cellulase system for carrying out this conversion is an extracellular multi-enzymatic complex named the cellulosome. In addition to temperature and pH stability, mechanical stability is important for functioning of cellulosome domains, and experimental techniques such as Single Molecule Force Spectroscopy (SMFS) have been used to measure the mechanical strength of several cellulosomal proteins. Molecular dynamics computer simulations provide complementary atomic-resolution quantitative maps of domain mechanical stability for identification of experimental leads for protein stabilization. In this study, we used multi-scale steered molecular dynamics computer simulations, benchmarked against new SMFS measurements, to measure the intermolecular contacts that confer high mechanical stability to a family 3 Carbohydrate Binding Module protein (CBM3) derived from the archetypal Clostridium thermocellum cellulosome. Our data predicts that electrostatic interactions in the calcium binding pocket modulate the mechanostability of the cellulose-binding module, which provides an additional design rule for the rational re-engineering of designer cellulosomes for biotechnology. Our data offers new molecular insights into the origins of mechanostability in cellulose binding domains and gives leads for synthesis of more robust cellulose-binding protein modules. On the other hand, simulations predict that insertion of a flexible strand can promote alternative unfolding pathways and dramatically reduce the mechanostability of the carbohydrate binding module, which gives routes to rational design of tailormade fingerprint complexes for force spectroscopy experiments.

Dual binding mode in cohesin-dockerin complexes as assessed through stretching studies

A recent experimental study by Jobst et al. of stretching of a wild-type (WT) cohesin-dockerin complex has identified two kinds of the force-displacement patterns, with a single or double-peaked final rupture, which are termed "short" and "long" here. This duality has been interpreted as arising from the existence of two kinds of binding. Here, we analyze the separation of two cohesin-dockerin complexes of C. thermocellum theoretically. We use a coarse-grained structure-based model and the values of the pulling speeds are nearly experimental. In their native states, the two systems differ in the mutual binding orientations of the molecules in the complex. We demonstrate that the WT complex (PDB:1OHZ) unravels along two possible pathways that are qualitatively consistent with the presence of the short and long patterns observed experimentally. On the other hand, the mutated complex (PDB:2CCL) leads only to short trajectories. The short and long stretching pathways also appear in the cohesin-dockerin-Xmodule complex (PDB:4IU3, WT) of R. flavefaciens. Thus the duality in the stretching patterns need not be necessarily due to the duality in binding.

Connecting thermal and mechanical protein (un)folding landscapes

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

Periodic forces trigger knot untying during translocation of knotted proteins

Proteins need to be unfolded when translocated through the pores in mitochondrial and other cellular membranes. Knotted proteins, however, might get stuck during this process, jamming the pore, since the diameter of the pore is smaller than the size of maximally tightened knot. The jamming probability dramatically increases as the magnitude of the driving force exceeds a critical value, F_c. In this numerical study, we show that for deep knots F_c lies below the force range over which molecular import motors operate, which suggest that in these cases the knots will tighten and block the pores. Next, we show how such topological traps might be prevented by using a pulling protocol of a repetitive, on-off character. Such a repetitive pulling is biologically relevant, since the mitochondrial import motor, like other molecular motors transforms chemical energy into directed motions via nucleotide-hydrolysis-mediated conformational changes, which are cyclic in character.

Mechanostability of cohesin-dockerin complexes in a structure-based model: anisotropy and lack of universality in the force profiles

We use a structure-based coarse grained model to elucidate stretching of three cohesin-dockerin complexes that are found in the cellulosome. The average strength of mechanostability is comparable to that of the I27 domain of titin, but the force profiles depend on the pulling direction and anisotropy effects can be substantial. Even though the force profiles for individual cohesins and dockerins are similar, those for their complexes are visibly distinct for any pulling direction.

Proteins at air-water interfaces: a coarse-grained model

We present a coarse-grained model to describe the adsorption and deformation of proteins at an air-water interface. The interface is introduced empirically in the form of a localized field that couples to a hydropathy scale of amino acids. We consider three kinds of proteins: protein G, egg-white lysozyme, and hydrophobin. We characterize the nature of the deformation and the orientation of the proteins induced by their proximity to and association with the interface. We also study protein diffusion in the layer formed at the interface and show that the diffusion slows with increasing concentration in a manner similar to that for a colloidal suspension approaching the glass transition.

Nanoindentation of 35 virus capsids in a molecular model: relating mechanical properties to structure

A coarse-grained model is used to study the mechanical response of 35 virus capsids of symmetries T = 1, T = 2, T = 3, pseudo T = 3, T = 4, and T = 7. The model is based on the native structure of the proteins that constitute the capsids and is described in terms of the C atoms associated with each amino acid. The number of these atoms ranges between 8 460 (for SPMV – satellite panicum mosaic virus) and 135 780 (for NBV – nudaureli virus). Nanoindentation by a broad AFM tip is modeled as compression between two planes: either both flat or one flat and one curved. Plots of the compressive force versus plate separation show a variety of behaviors, but in each case there is an elastic region which extends to a characteristic force . Crossing results in a drop in the force and irreversible damage. Across the 35 capsids studied, both and the elastic stiffness are observed to vary by a factor of 20. The changes in mechanical properties do not correlate simply with virus size or symmetry. There is a strong connection to the mean coordination number , defined as the mean number of interactions to neighboring amino acids. The Young's modulus for thin shell capsids rises roughly quadratically with , where 6 is the minimum coordination for elastic stability in three dimensions.

Topological features in stretching of proteins

In the present article, we highlight the diversity of mechanical clamps, some of them topological in nature, that have been found by making surveys of mechanostability of approximately 18000 proteins within structure-based models. The existence of superstable proteins (with the characteristic unfolding force in the region of 1000 pN) is predicted.

Formation of cystine slipknots in dimeric proteins

We consider mechanical stability of dimeric and monomeric proteins with the cystine knot motif. A structure based dynamical model is used to demonstrate that all dimeric and some monomeric proteins of this kind should have considerable resistance to stretching that is significantly larger than that of titin. The mechanisms of the large mechanostability are elucidated. In most cases, it originates from the induced formation of one or two cystine slipknots. Since there are four termini in a dimer, there are several ways of selecting two of them to pull by. We show that in the cystine knot systems, there is strong anisotropy in mechanostability and force patterns related to the selection. We show that the thermodynamic stability of the dimers is enhanced compared to the constituting monomers whereas machanostability is either lower or higher.

Denaturation of proteins near polar surfaces

All-atom molecular dynamics simulations for proteins placed near a model mica surface indicate existence of two types of evolution. One type leads to the surface-induced unfolding and the other just to a deformation. The two behaviors are characterized by distinct properties of the radius of gyration and of a novel distortion parameter that distinguishes between elongated, globular, and planar shapes. They also differ in the nature of their single site diffusion and two-site distance fluctuations. The four proteins chosen for the studies, the tryptophan cage, protein G, hydrophobin and lyzozyme, are small to allow for a fair determination of the forces generated by the surface as the effects of finite cutoffs in the Coulombic interactions are thus minimized. When the net charge on the surface is set to zero artificially, infliction of deformation is seen to persists but no unfolding takes place. Unfolding may also be prevented by a cluster of disulfide bonds, as we observe in simulations of hydrophobin.

Molecular jamming--the cystine slipknot mechanical clamp in all-atom simulations

A recent survey of 17 134 proteins has identified a new class of proteins which are expected to yield stretching induced force peaks in the range of 1 nN. Such high force peaks should be due to forcing of a slip-loop through a cystine ring, i.e., by generating a cystine slipknot. The survey has been performed in a simple coarse grained model. Here, we perform all-atom steered molecular dynamics simulations on 15 cystine knot proteins and determine their resistance to stretching. In agreement with previous studies within a coarse grained structure based model, the level of resistance is found to be substantially higher than in proteins in which the mechanical clamp operates through shear. The large stretching forces arise through formation of the cystine slipknot mechanical clamp and the resulting steric jamming. We elucidate the workings of such a clamp in an atomic detail. We also study the behavior of five top strength proteins with the shear-based mechanostability in which no jamming is involved. We show that in the atomic model, the jamming state is relieved by moving one amino acid at a time and there is a choice in the selection of the amino acid that advances the first. In contrast, the coarse grained model also allows for a simultaneous passage of two amino acids.

Mechanical stability of multidomain proteins and novel mechanical clamps

We estimate the size of mechanostability for 318 multidomain proteins which are single-chain and contain up to 1021 amino acids. We predict existence of novel types of mechanical clamps in which interdomain contacts play an essential role. Mechanical clamps are structural regions which are the primary source of a protein's resistance to pulling. Among these clamps there is one that opposes tensile stress due to two domains swinging apart. This movement strains and then ruptures the contacts that hold the two domains together. Another clamp also involves tensile stress but it originates from an immobilization of a structural region by a surrounding knot-loop (without involving any disulfide bonds). Still another mechanism involves shear between helical regions belonging to two domains. We also consider the amyloid-prone cystatin C which provides an example of a two-chain 3D domain-swapped protein. We predict that this protein should withstand remarkably large stress, perhaps of order 800 pN, when inducing a shearing strain. The survey is generated through molecular dynamics simulations performed within a structure-based coarse grained model.

The influence of hydrodynamic interactions on protein dynamics in confined and crowded spaces-assessment in simple models

We consider several systems that are confined within a softly repulsive sphere. The first one is a model protein, crambin, which is described by a structure-based coarse grained model. We demonstrate that the folding process is accelerated by the hydrodynamic interactions (HI) in a way that depends on the radius of the sphere. The tighter the encompassing sphere, the smaller the effect, independent of the nature of the starting conformations. The second system is a protein surrounded by protein-like softly repulsive spheres that make the confined space crowded. In this case, the HI shorten the folding times in a way which depends on the degree of crowdedness only weakly. The third system is a collection of spheres that are meant to represent molecules. We show that confinement increases association times. We also observe that the HI either facilitate or obstruct association of two spheres depending on the crowding conditions. The dependence of the association time on crowdedness in the confining sphere is qualitatively distinct from that derived by Wieczorek and Zielenkiewicz for a cube with the periodic boundary conditions.

BSDB: the biomolecule stretching database

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

Nanoindentation of virus capsids in a molecular model

A molecular-level model is used to study the mechanical response of empty cowpea chlorotic mottle virus (CCMV) and cowpea mosaic virus (CPMV) capsids. The model is based on the native structure of the proteins that constitute the capsids and is described in terms of the C(alpha) atoms. Nanoindentation by a large tip is modeled as compression between parallel plates. Plots of the compressive force versus plate separation for CCMV are qualitatively consistent with continuum models and experiments, showing an elastic region followed by an irreversible drop in force. The mechanical response of CPMV has not been studied, but the molecular model predicts an order of magnitude higher stiffness and a much shorter elastic region than for CCMV. These large changes result from small structural changes that increase the number of bonds by only 30% and would be difficult to capture in continuum models. Direct comparison of local deformations in continuum and molecular models of CCMV shows that the molecular model undergoes a gradual symmetry breaking rotation and accommodates more strain near the walls than the continuum model. The irreversible drop in force at small separations is associated with rupturing nearly all of the bonds between capsid proteins in the molecular model, while a buckling transition is observed in continuum models.

Stretching and twisting of the DNA duplexes in coarse-grained dynamical models

Three coarse-grained molecular dynamics models of the double-stranded DNA are proposed and compared in the context of single molecule mechanical manipulation such as twisting and various schemes of stretching-unzipping, shearing, two-strand stretching and stretching of only one strand. The models differ in the number of effective beads (between two and five) representing each nucleotide. They all show similar behaviour, but the bigger the resolution, the more details in the force patterns. The models incorporate the effective Lennard-Jones potentials in the couplings between two strands and harmonic potentials to describe the structure of a single strand. The force patterns are shown to depend on the sequence studied. In particular, both shearing and unzipping for an all-AT sequence lead to lower forces than for an all-CG sequence. The unzipping patterns and the corresponding scenario diagrams for the contact rupture events are found to reflect the sequential information if the temperature is moderate and initial transients are discarded. The derived torque-force phase diagram is found to be qualitatively consistent with experiments and all-atom simulations.

Temperature and chemical denaturant dependence of forced unfolding of titin I27

Single-molecule force measurement opens a new door for investigating detailed biomolecular interactions and their thermodynamic properties by pulling molecules apart while monitoring the force exerted on them. Recent advances in the nonequilibrium work theorem allows one to determine the free-energy landscapes of these events. Such information is valuable for understanding processes such as protein and RNA folding and receptor-ligand binding. Here, we used force as a physical parameter under the traditional chemical and temperature denaturing environment to alter the protein folding energy landscape and compared the change in the unfolding free-energy barrier of the I27 domain of human cardiac titin. We found that the trends in protein unfolding free-energy barriers are consistent for single-molecule force measurements and bulk chemical and temperature studies. The results suggest that the information from single-molecule pulling experiments are meaningful and useful for understanding the mechanism of folding of titin I27.

Mechanical strength of 17,134 model proteins and cysteine slipknots

A new theoretical survey of proteins' resistance to constant speed stretching is performed for a set of 17 134 proteins as described by a structure-based model. The proteins selected have no gaps in their structure determination and consist of no more than 250 amino acids. Our previous studies have dealt with 7510 proteins of no more than 150 amino acids. The proteins are ranked according to the strength of the resistance. Most of the predicted top-strength proteins have not yet been studied experimentally. Architectures and folds which are likely to yield large forces are identified. New types of potent force clamps are discovered. They involve disulphide bridges and, in particular, cysteine slipknots. An effective energy parameter of the model is estimated by comparing the theoretical data on characteristic forces to the corresponding experimental values combined with an extrapolation of the theoretical data to the experimental pulling speeds. These studies provide guidance for future experiments on single molecule manipulation and should lead to selection of proteins for applications. A new class of proteins, involving cystein slipknots, is identified as one that is expected to lead to the strongest force clamps known. This class is characterized through molecular dynamics simulations.

The advances in nanotechnology have allowed for manipulation of single biomolecules and determination of their elastic properties. Titin was among the first proteins studied in this way. Its unravelling by stretching requires a 204 pN force. The resistance to stretching comes mostly from a localized region known as a force clamp. In titin, the force clamp is simple as it is formed by two parallel β-strands that are sheared on pulling. Studies of a set of under a hundred proteins accomplished in the last decade have revealed a variety of the force clamps that lead to forces ranging from under 20 pN to about 500 pN. This set comprises only a tiny fraction of proteins known. Thus one needs guidance as to what proteins should be considered for specific mechanical properties. Such a guidance is provided here through simulations within simplified coarse-grained models on 17 134 proteins that are stretched at constant speed. We correlate their unravelling forces with two structure classification schemes. We identify proteins with large resistance to unravelling and characterize their force clamps. Quite a few top strength proteins owe their sturdiness to a new type of the force clamp: the cystein slipknot in which the force peak is due to dragging of a piece of the backbone through a closed ring formed by two other pieces of the backbone and two connecting disulphide bonds.

Experimental and computational characterization of biological liquid crystals: a review of single-molecule bioassays

Quantitative understanding of the mechanical behavior of biological liquid crystals such as proteins is essential for gaining insight into their biological functions, since some proteins perform notable mechanical functions. Recently, single-molecule experiments have allowed not only the quantitative characterization of the mechanical behavior of proteins such as protein unfolding mechanics, but also the exploration of the free energy landscape for protein folding. In this work, we have reviewed the current state-of-art in single-molecule bioassays that enable quantitative studies on protein unfolding mechanics and/or various molecular interactions. Specifically, single-molecule pulling experiments based on atomic force microscopy (AFM) have been overviewed. In addition, the computational simulations on single-molecule pulling experiments have been reviewed. We have also reviewed the AFM cantilever-based bioassay that provides insight into various molecular interactions. Our review highlights the AFM-based single-molecule bioassay for quantitative characterization of biological liquid crystals such as proteins.

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.

Dodging the crisis of folding proteins with knots

Proteins with nontrivial topology, containing knots and slipknots, have the ability to fold to their native states without any additional external forces invoked. A mechanism is suggested for folding of these proteins, such as YibK and YbeA, that involves an intermediate configuration with a slipknot. It elucidates the role of topological barriers and backtracking during the folding event. It also illustrates that native contacts are sufficient to guarantee folding in approximately 1-2% of the simulations, and how slipknot intermediates are needed to reduce the topological bottlenecks. As expected, simulations of proteins with similar structure but with knot removed fold much more efficiently, clearly demonstrating the origin of these topological barriers. Although these studies are based on a simple coarse-grained model, they are already able to extract some of the underlying principles governing folding in such complex topologies.

Stabilizing effect of knots on proteins

Molecular dynamics studies within a coarse-grained, structure-based model were used on two similar proteins belonging to the transcarbamylase family to probe the effects of the knot in the native structure of a protein. The first protein, N-acetylornithine transcarbamylase, contains no knot, whereas human ormithine transcarbamylase contains a trefoil knot located deep within the sequence. In addition, we also analyzed a modified transferase with the knot removed by the appropriate change of a knot-making crossing of the protein chain. The studies of thermally and mechanically induced unfolding processes suggest a larger intrinsic stability of the protein with the knot.

Effects of confinement and crowding on folding of model proteins

We perform molecular dynamics simulations for a simple coarse-grained model of crambin placed inside of a softly repulsive sphere of radius R. The confinement makes folding at the optimal temperature slower and affects the folding scenarios, but both effects are not dramatic. The influence of crowding on folding are studied by placing several identical proteins within the sphere, denaturing them, and then by monitoring refolding. If the interactions between the proteins are dominated by the excluded volume effects, the net folding times are essentially like for a single protein. An introduction of inter-proteinic attractive contacts hinders folding when the strength of the attraction exceeds about a half of the value of the strength of the single protein contacts. The bigger the strength of the attraction, the more likely is the occurrence of aggregation and misfolding.

The effect of temperature on mechanical resistance of the native and intermediate states of I27

We investigated the effect of temperature on the mechanical unfolding of I27 from human cardiac titin, employing a custom-built temperature control device for single-molecule atomic force microscopy measurement. A sawtooth pattern was observed in the force curves where each force peak reports on the unfolding of an I27 domain. In early unfolding events, we observed a "hump-like" deviation due to the detachment of beta-strand A from each I27 domain. The force at which the humps appear was approximately 130 pN and showed no temperature dependence, at least in the temperature range of 2 degrees C-30 degrees C. The hump structure was successfully analyzed with a two-state worm-like chain model, and the Gibbs free energy difference of the detachment reaction was estimated to be 11.6 +/- 0.58 kcal/mol and found to be temperature independent. By contrast, upon lowering the temperature, the mean unfolding force from the partly unfolded intermediate state was found to markedly increase and the unfolding force distribution to broaden significantly, suggesting that the distance (x(u)) between the folded and transition states in the energy landscape along the pulling direction is decreased. These results suggest that the local structure of beta-strand A are stabilized by topologically simple local hydrogen-bond network and that the temperature does not affect the detachment reaction thermodynamically and kinetically, whereas the interaction between the beta-strands A' and G, which is a critical region for its mechanical stability, is strongly dependent on the temperature.

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.

Predicting the order in which contacts are broken during single molecule protein stretching experiments

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force-displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments.

Tightening of knots in proteins

We perform theoretical studies of stretching of 20 proteins with knots within a coarse-grained model. The knot's ends are found to jump to well defined sequential locations that are associated with sharp turns, whereas in homopolymers they diffuse around and eventually slide off. The waiting times of the jumps are increasingly stochastic as the temperature is raised. Knots typically do not return to their native locations when a protein is released after stretching.

Proteins in a shear flow

The conformational dynamics of a single protein molecule in a shear flow is investigated using Brownian dynamics simulations. A structure-based coarse grained model of a protein is used. We consider two proteins, ubiquitin and integrin, and find that at moderate shear rates they unfold through a sequence of metastable states-a pattern which is distinct from a smooth unraveling found in homopolymers. Full unfolding occurs only at very large shear rates. Furthermore, the hydrodynamic interactions between the amino acids are shown to hinder the shear flow unfolding. The characteristics of the unfolding process depend on whether a protein is anchored or not, and if it is, on the choice of an anchoring point.

Secondary and tertiary structure elasticity of titin Z1Z2 and a titin chain model

The giant protein titin, which is responsible for passive elasticity in muscle fibers, is built from approximately 300 regular immunoglobulin-like (Ig) domains and FN-III repeats. While the soft elasticity derived from its entropic regions, as well as the stiff mechanical resistance derived from the unfolding of the secondary structure elements of Ig- and FN-III domains have been studied extensively, less is known about the mechanical elasticity stemming from the orientation of neighboring domains relative to each other. Here we address the dynamics and energetics of interdomain arrangement of two adjacent Ig-domains of titin, Z1, and Z2, using molecular dynamics (MD) simulations. The simulations reveal conformational flexibility, due to the domain-domain geometry, that lends an intermediate force elasticity to titin. We employ adaptive biasing force MD simulations to calculate the energy required to bend the Z1Z2 tandem open to identify energetically feasible interdomain arrangements of the Z1 and Z2 domains. The finding is cast into a stochastic model for Z1Z2 interdomain elasticity that is generalized to a multiple domain chain replicating many Z1Z2-like units and representing a long titin segment. The elastic properties of this chain suggest that titin derives so-called tertiary structure elasticity from bending and twisting of its domains. Finally, we employ steered molecular dynamics simulations to stretch individual Z1 and Z2 domains and characterize the so-called secondary structure elasticity of the two domains. Our study suggests that titin's overall elastic response at weak force stems from a soft entropic spring behavior (not described here), from tertiary structure elasticity with an elastic spring constant of approximately 0.001-1 pN/A and, at strong forces, from secondary structure elasticity.

Stretching to understand proteins - a survey of the protein data bank

We make a survey of resistance of 7510 proteins to mechanical stretching at constant speed as studied within a coarse-grained molecular dynamics model. We correlate the maximum force of resistance with the native structure, predict proteins which should be especially strong, and identify the nature of their force clamps.

Stretching of proteins in a uniform flow

Stretching of a protein by a fluid flow is compared to that in a force-clamp apparatus. The comparison is made within a simple topology-based dynamical model of a protein in which the effects of the flow are implemented using Langevin dynamics. We demonstrate that unfolding induced by a uniform flow shows a richer behavior than that in the force clamp. The dynamics of unfolding is found to depend strongly on the selection of the amino acid, usually one of the termini, which is anchored. These features offer potentially wider diagnostic tools to investigate structure of proteins compared to experiments based on the atomic force microscopy.

Mechanical unfolding of ubiquitin molecules

Mechanical stretching of ubiquitin and of its several repeats are studied through molecular-dynamics simulations. A Go-type model [H. Abe and N. Go, Biopolymers 20, 1013 (1981)] with a realistic contact map and with Lennard-Jones contact interactions is used. The model qualitatively reproduces the experimentally observed differences between force-extension patterns obtained on polyubiquitins stretched by various linkages. The terminal-to-terminal stretching of polyubiquitin results in peak forces similar to those measured for titin-based polyproteins and of a magnitude that matches measurements. Consistent with the experimental measurements, the simulated peak forces depend on the pulling speed logarithmically when thermal fluctuations are explicitly introduced. These results validate the application of topology-based models in the study of the mechanical stretching of proteins.

Mechanical unfolding revisited through a simple but realistic model

Single-molecule experiments and their application to probe the mechanical resistance and related properties of proteins provide a new dimension in our knowledge of these important and complex biological molecules. Single-molecule techniques may not have yet overridden solution experiments as a method of choice to characterize biophysical and biological properties of proteins, but have stimulated a debate and contributed considerably to bridge theory and experiment. Here we demonstrate this latter contribution by illustrating the reach of some theoretical findings using a solvable but nontrivial molecular model whose properties are analogous to those of the corresponding experimental systems. In particular, we show the relationship between the thermodynamic and the mechanical properties of a protein. The simulations presented here also illustrate how forced and spontaneous unfolding occur through different pathways and that folding and unfolding rates at equilibrium cannot in general be obtained from forced unfolding experiments or simulations. We also study the relationship between the energy surface and the mechanical resistance of a protein and show how a simple analysis of the native state can predict much of the mechanical properties of a protein.

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.

Mechanical properties of the domains of titin in a Go-like model

Comparison of properties of three domains of titin, I1, I27, and I28, in a simple geometry-based model shows that despite a high structural homology between their native states different domains show similar but distinguishable mechanical properties. Folding properties of the separate domains are predicted to be diversified which reflects sensitivity of the kinetics to the details of native structures. The Go-like model corresponding to the experimentally resolved native structure of the I1 domain is found to provide the biggest thermodynamic and mechanical stability compared to the other domains studied here. We analyze elastic, thermodynamic, and kinetic properties of several structures corresponding to the I28 domain as obtained through homology-based modeling. We discuss the ability of the models of the I28 domain to reproduce experimental results qualitatively. A strengthening of contacts that involve hydrophobic amino acids does not affect theoretical comparisons of the domains. Tandem linkages of up to five identical or different domains unravel in a serial fashion at low temperatures. We study the nature of the intermediate state that arises in the early stages of the serial unraveling and find it to qualitatively agree with the results of Marszalek et al.

Pulling single bacteriorhodopsin out of a membrane: Comparison of simulation and experiment

Mechanical unfolding of single bacteriorhodopsins from a membrane bilayer is studied using molecular dynamics simulations. The initial conformation of the lipid membrane is determined through all-atom simulations and then its coarse-grained representation is used in the studies of stretching. A Go-like model with a realistic contact map and with Lennard-Jones contact interactions is applied to model the protein-membrane system. The model qualitatively reproduces the experimentally observed differences between force-extension patterns obtained on bacteriorhodopsin at different temperatures and predicts a lack of symmetry in the choice of the terminus to pull by. It also illustrates the decisive role of the interactions of the protein with the membrane in determining the force pattern and thus the stability of transmembrane proteins.