Stretching lattice models of protein folding

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis

In a two-state molecular system, transition paths comprise the portions of trajectories during which the system transits from one stable state to the other. Because of their low population, it is essentially impossible to obtain information on transition paths from experiments on a large sample of molecules. However, single-molecule experiments such as laser optical tweezers or Förster resonance energy transfer (FRET) spectroscopy have allowed transition-path durations to be estimated. Here, we use molecular simulations to test the methodology for obtaining information on transition paths in single-molecule FRET by generating photon trajectories from the distance trajectories obtained in the simulation. Encouragingly, we find that this maximum likelihood analysis yields transition-path times within a factor of 2-4 of the values estimated using a good coordinate for folding, but tends to systematically underestimate them. The underestimation can be attributed partly to the fact that the large changes in the end-end distance occur mostly early in a folding trajectory. However, even if the transfer efficiency is a good reaction coordinate for folding, the assumption that the transition-path shape is a step function still leads to an underestimation of the transition-path time as defined here. We find that allowing more flexibility in the form of the transition path model allows more accurate transition-path times to be extracted and points the way toward further improvements in methods for estimating transition-path time and transition-path shape.

Molecular Simulations Suggest a Force-Dependent Mechanism of Vinculin Activation

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

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.

Reduction of All-Atom Protein Folding Dynamics to One-Dimensional Diffusion

Theoretical models have often modeled protein folding dynamics as diffusion on a low-dimensional free energy surface, a remarkable simplification. However, the accuracy of such an approximation and the number of dimensions required were not clear. For all-atom folding simulations of ten small proteins in explicit solvent we show that the folding dynamics can indeed be accurately described as diffusion on just a single coordinate, the fraction of native contacts (Q). The diffusion models reproduce both folding rates, and finer details such as transition-path durations and diffusive propagators. The Q-averaged diffusion coefficients decrease with chain length, as anticipated from energy landscape theory. Although the Q-diffusion model does not capture transition-path durations for the protein NuG2, we show that this can be accomplished by designing an improved coordinate Qopt. Overall, one-dimensional diffusion on a suitable coordinate turns out to be a remarkably faithful model for the dynamics of the proteins considered.

A hybrid MD-kMC algorithm for folding proteins in explicit solvent

We present a novel hybrid MD-kMC algorithm that is capable of efficiently folding proteins in explicit solvent. We apply this algorithm to the folding of a small protein, Trp-Cage. Different kMC move sets that capture different possible rate limiting steps are implemented. The first uses secondary structure formation as a relevant rate event (a combination of dihedral rotations and hydrogen-bonding formation and breakage). The second uses tertiary structure formation events through formation of contacts via translational moves. Both methods fold the protein, but via different mechanisms and with different folding kinetics. The first method leads to folding via a structured helical state, with kinetics fit by a single exponential. The second method leads to folding via a collapsed loop, with kinetics poorly fit by single or double exponentials. In both cases, folding times are faster than experimentally reported values, The secondary and tertiary move sets are integrated in a third MD-kMC implementation, which now leads to folding of the protein via both pathways, with single and double-exponential fits to the rates, and to folding rates in good agreement with experimental values. The competition between secondary and tertiary structure leads to a longer search for the helix-rich intermediate in the case of the first pathway, and to the emergence of a kinetically trapped long-lived molten-globule collapsed state in the case of the second pathway. The algorithm presented not only captures experimentally observed folding intermediates and kinetics, but yields insights into the relative roles of local and global interactions in determining folding mechanisms and rates.

Locating the barrier for folding of single molecules under an external force

Single-molecule pulling experiments on the folding of biomolecules are usually interpreted with one-dimensional models in which the dynamics occurs on the "pulling coordinate." Paradoxically, the free-energy profile along this coordinate may lack a refolding barrier, yet a barrier is known to exist for folding; thus, it has been argued that pulling experiments do not probe folding. Here, we show that transitions monitored in pulling experiments probe the true folding barrier but that the barrier may be hidden in the projection onto the pulling coordinate. However, one-dimensional theory using the pulling coordinate still yields physically meaningful energy landscape parameters.

Exploring the folding free energy landscape of a β-hairpin miniprotein, chignolin, using multiscale free energy landscape calculation method

The folding process for a β-hairpin miniprotein, chignolin, was investigated by free energy landscape (FEL) calculations using the recently proposed multiscale free energy landscape calculation method (MSFEL). First, coarse-grained molecular dynamics simulations searched a broad conformational space, then multiple independent, all-atom molecular dynamics simulations with explicit solvent determined the detailed local FEL using massively distributed computing. The combination of the two models enabled efficient calculation of the free energy landscapes. The MSFEL analysis showed that chignolin has an intermediate state as well as a misfolded state. The folding process is initiated by the formation of a β-hairpin turn, followed by the formation of contacts in the hydrophobic core between Tyr2 and Trp9. Furthermore, mutation of Tyr2 shifts the population to the misfolded conformation. The results indicate that the hydrophobic core plays an important role in stabilizing the native state of chignolin.

Probing static disorder in Arrhenius kinetics by single-molecule force spectroscopy

The widely used Arrhenius equation describes the kinetics of simple two-state reactions, with the implicit assumption of a single transition state with a well-defined activation energy barrier DeltaE, as the rate-limiting step. However, it has become increasingly clear that the saddle point of the free-energy surface in most reactions is populated by ensembles of conformations, leading to nonexponential kinetics. Here we present a theory that generalizes the Arrhenius equation to include static disorder of conformational degrees of freedom as a function of an external perturbation to fully account for a diverse set of transition states. The effect of a perturbation on static disorder is best examined at the single-molecule level. Here we use force-clamp spectroscopy to study the nonexponential kinetics of single ubiquitin proteins unfolding under force. We find that the measured variance in DeltaE shows both force-dependent and independent components, where the force-dependent component scales with F(2), in excellent agreement with our theory. Our study illustrates a novel adaptation of the classical Arrhenius equation that accounts for the microscopic origins of nonexponential kinetics, which are essential in understanding the rapidly growing body of single-molecule data.

Monte Carlo simulation of mechanical unfolding of proteins based on a simple two-state model

Single molecule methods are becoming routine biophysical techniques for studying biological macromolecules. In mechanical unfolding of proteins, an externally applied force is used to induce the unfolding of individual protein molecules. Such experiments have revealed novel information that has significantly enhanced our understanding of the function and folding mechanisms of several types of proteins. To obtain information on the unfolding kinetics and the free energy landscape of the protein molecule from mechanical unfolding data, a Monte Carlo simulation based on a simple two-state kinetic model is often used. In this paper, we provide a detailed description of the procedure to perform such simulations and discuss the approximations and assumptions involved. We show that the appearance of the force versus extension curves from mechanical unfolding of proteins is affected by a variety of experimental parameters, such as the length of the protein polymer and the force constant of the cantilever. We also analyze the errors associated with different methods of data pooling and present a quantitative measure of how well the simulation results fit experimental data. These findings will be helpful in experimental design, artifact identification, and data analysis for single molecule studies of various proteins using the mechanical unfolding method.

Design rules for biomolecular adhesion: lessons from force measurements

Cell adhesion to matrix, other cells, or pathogens plays a pivotal role in many processes in biomolecular engineering. Early macroscopic methods of quantifying adhesion led to the development of quantitative models of cell adhesion and migration. The more recent use of sensitive probes to quantify the forces that alter or manipulate adhesion proteins has revealed much greater functional diversity than was apparent from population average measurements of cell adhesion. This review highlights theoretical and experimental methods that identified force-dependent molecular properties that are central to the biological activity of adhesion proteins. Experimental and theoretical methods emphasized in this review include the surface force apparatus, atomic force microscopy, and vesicle-based probes. Specific examples given illustrate how these tools have revealed unique properties of adhesion proteins and their structural origins.

Jamming proteins with slipknots and their free energy landscape

Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their unfolding times when the stretching force crosses an intermediate threshold. This behavior arises as a consequence of the existence of alternative unfolding routes that are dominant at different force ranges. The existence of an intermediate, metastable configuration where the slipknot is jammed is responsible for longer unfolding times at higher forces. Simulations are performed with a coarse-grained model with further quantification using a refined description of the geometry of the slipknots. The simulation data are used to determine the free energy landscape of the protein, which supports recent analytical predictions.

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.

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.

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.

Protein folding and unfolding by all-atom molecular dynamics simulations

Computational protein folding can be classified into pathway and sampling approaches. Here, we use the AMBER simulation package as an example to illustrate the protocols for all-atom molecular simulations of protein folding, including system setup, simulation, and analysis. We introduced two traditional pathway approaches: ab inito folding and high-temperature unfolding. The popular replica exchange method was chosen to represent sampling approaches. Our emphasis is placed on the analysis of the simulation trajectories, and some in-depth discussions are provided for commonly encountered problems.

Sampling of states for estimating the folding funnel entropy and energy landscape of a model alpha-helical hairpin peptide

Protein folding times are many orders of magnitude shorter than would occur if the peptide chain randomly sampled possible configurations, which implies that protein folding is a directed process. The detailed shape of protein's energy landscape determines the rate and reliability of folding to the native state, but the large number of structural degrees of freedom generates an energy landscape that is hard to visualize because of its high dimensionality. A commonly used picture is that of an energy funnel leading from high energy random coil state down to the low energy native state. As lattice computer models of protein dynamics become more realistic, the number of possible configurations becomes too large to count directly. Statistical mechanic and thermodynamic approaches allow us to count states in an approximate manner to quantify the entropy and energy of the energy landscape within a folding funnel for an alpha-helical protein. We also discuss the problems that arise in attempting to count the huge number of individual states of the random coil at the top of the funnel.

Impact of salt bridges on the equilibrium binding and adhesion of human CD2 and CD58

This study describes quantitative investigations of the impact of single charge mutations on equilibrium binding, kinetics, and the adhesion strength of the CD2-CD58 interaction. Previously steered molecular dynamics simulations guided the selection of the charge mutants investigated, which include the CD2 mutants D31A, K41A, K51A, and K91A. This set includes mutations in which the previous cell aggregation and binding data either agreed or disagreed with the steered molecular dynamics predictions. Surface plasmon resonance measurements quantified the solution binding properties. Adhesion was quantified with the surface force apparatus, which was used previously to study the closely related CD2-CD48 interaction. The results reveal roles that these salt bridges play in equilibrium binding and adhesion. We discuss both the molecular basis of this behavior and its implications for cell adhesion.

Multiple-basin energy landscapes for large-amplitude conformational motions of proteins: Structure-based molecular dynamics simulations

Biomolecules often undergo large-amplitude motions when they bind or release other molecules. Unlike macroscopic machines, these biomolecular machines can partially disassemble (unfold) and then reassemble (fold) during such transitions. Here we put forward a minimal structure-based model, the "multiple-basin model," that can directly be used for molecular dynamics simulation of even very large biomolecular systems so long as the endpoints of the conformational change are known. We investigate the model by simulating large-scale motions of four proteins: glutamine-binding protein, S100A6, dihydrofolate reductase, and HIV-1 protease. The mechanisms of conformational transition depend on the protein basin topologies and change with temperature near the folding transition. The conformational transition rate varies linearly with driving force over a fairly large range. This linearity appears to be a consequence of partial unfolding during the conformational transition.

Elasticity of peptide omega bonds

We calculated the changes of the free energy profile of the peptidyl-prolyl torsional angle of the dipeptide valine-proline under pulling forces by simulations. Using a dynamic model built on the equilibrium properties of this system and previously studied dynamic properties of cis-trans isomerization of other dipeptides, we calculated the dynamic viscoelasticity of this degree of freedom. The results show significant differences between how thermal and mechanical forces alter the equilibrium and the dynamics of the isomerization transition. The former does not change the barrier heights but changes the prefactor of the kinetics owing to temperature effects, while the latter changes minima and thus the population. The force that is required to "excite" this degree of freedom is small. Compared to other systems, we found that this degree of freedom becomes already quite rigid at several hertz, which is a much lower value due to the high barrier of the cis-trans isomerization. We also found that the tensile elastic modulus of densely packed omega bonds is at the order of GPa, which is comparable to that of polymer materials. These results give mechanical properties of polyproline elasticity of a local nature and provide guidance for future experimental designs.

Numerical simulation of gel electrophoresis of DNA knots in weak and strong electric fields

Gel electrophoresis allows one to separate knotted DNA (nicked circular) of equal length according to the knot type. At low electric fields, complex knots, being more compact, drift faster than simpler knots. Recent experiments have shown that the drift velocity dependence on the knot type is inverted when changing from low to high electric fields. We present a computer simulation on a lattice of a closed, knotted, charged DNA chain drifting in an external electric field in a topologically restricted medium. Using a Monte Carlo algorithm, the dependence of the electrophoretic migration of the DNA molecules on the knot type and on the electric field intensity is investigated. The results are in qualitative and quantitative agreement with electrophoretic experiments done under conditions of low and high electric fields.

Force unfolding single RNAs

We develop a continue time Monte Carlo algorithm to simulate single RNAs unfolded by a time-dependent external force on the secondary structure level. Two recent unfolding RNA experiments carried out by Bustamante group are mainly investigated. We find that, in contrast to popular two-state assumption about the RNAs free energy landscape along the molecular extension, the molecules used in the experiments do not present apparent energy barriers. The strong cooperative folding and unfolding transitions of the RNAs observed in the experiments and in our simulations arise from the interaction of the molecules and the light trap. In addition, we also investigate the properties of Jarzynski's remarkable equality, whose experimental test has received considerable attention.

Dissecting the mechanical unfolding of ubiquitin

The unfolding behavior of ubiquitin under the influence of a stretching force recently was investigated experimentally by single-molecule constant-force methods. Many observed unfolding traces had a simple two-state character, whereas others showed clear evidence of intermediate states. Here, we use Monte Carlo simulations to investigate the force-induced unfolding of ubiquitin at the atomic level. In agreement with experimental data, we find that the unfolding process can occur either in a single step or through intermediate states. In addition to this randomness, we find that many quantities, such as the frequency of occurrence of intermediates, show a clear systematic dependence on the strength of the applied force. Despite this diversity, one common feature can be identified in the simulated unfolding events, which is the order in which the secondary-structure elements break. This order is the same in two- and three-state events and at the different forces studied. The observed order remains to be verified experimentally but appears physically reasonable.

Energy landscape distortions and the mechanical unfolding of proteins

Molecular simulations and an energy landscape analysis are used to examine the stretching of a model protein. A mapping of the energy landscape shows that stretching the protein causes energy minima and energy barriers to flatten out and disappear, and new energy minima to be created. The implications of these landscape distortions depend on the timescale regime under which the protein is stretched. When the timescale for thermally activated processes is longer than the timescale of stretching, the disappearances of energy barriers provide the mechanism for protein unfolding. When the timescale for thermally activated processes is shorter than the timescale of stretching, the landscape distortions influence the stretching process by changing the number and types of energy minima in which the system can exist.

Unfolding single RNA molecules by mechanical force: a stochastic kinetic method

Using simple polymer elastic theory and known RNA free energies, we study the single RNA folding and unfolding on the secondary structure level under mechanical constant force by stochastic kinetic simulation. As a primary application, this method is used to simulate the experiment performed by Science 292, 733 (2001)]. The extension-force curves in equilibrium and kinetic reaction rate constants for folding and unfolding are calculated. Our results show that the agreement between simulation and experimental measurements is satisfactory.

Unfolding of titin domains studied by molecular dynamics simulations

Titin, a approximately 1 micron long protein found in striated muscle myofibrils, possesses unique elastic properties. The extensible behavior of titin has been demonstrated in atomic force microscopy and optical tweezer experiments to involve the reversible unfolding of individual immunoglobulin-like (Ig) domains. We have used steered molecular dynamics (SMD), a novel computer simulation method, to investigate the mechanical response of single titin Ig domains upon stress. Simulations of stretching Ig domains I1 and I27 have been performed in a solvent of explicit water molecules. The SMD approach provides a detailed structural and dynamic description of how Ig domains react to external forces. Validation of SMD results includes both qualitative and quantitative agreement with AFM recordings. Furthermore, combining SMD with single molecule experimental data leads to a comprehensive understanding of Ig domains' mechanical properties. A set of backbone hydrogen bonds that link the domains' terminal beta-strands play a key role in the mechanical resistance to external forces. Slight differences in architecture permit a mechanical unfolding intermediate for I27, but not for I1. Refolding simulations of I27 demonstrate a locking mechanism.

Molecular dynamics simulations of a protein model in uniform and elongational flows

We present a molecular dynamics study of the conformational deformation of a minimalist beta-barrel protein model in two different types of hydrodynamic flows: uniform and elongational. We investigate the characteristics of protein stretching, paying special attention to the unfolding intermediate states and their relationship to the protein folding/unfolding problem. In the uniform flow simulations, one end of the modeled protein was tethered to a fixed point in space and the forced unfolding process was observed. The unfolding takes place via a few stages involving one or two intermediate states, depending on which end is tethered. The calculated force-extension curves show plateau regimes and hysteresis as the protein is stretched and refolded, in qualitative agreement with the experimental measurements. The physical behavior observed in our numerical simulations of the forced unfolding in an elongational flow is very different from that in uniform flow. The protein unfolds abruptly from the globular state to a fully stretched state without going through any observable intermediate states. From these observation, we stress that protein unfolding pathways under the influence of an external force are highly dependent on the mechanism of the exerted force.

Unfolding proteins in an external field: can we always observe the intermediate states?

A protein molecule under the stress of an external denaturing force acting on a terminal end or on the entire molecule is expected to unfold, possibly through a few intermediate stages depending on the magnitude of the denaturing force. We have investigated two protein minimal models under various types of denaturing force fields using the collision molecular-dynamics simulation, in order to critically examine the relationship between the folding pathways observed in different protein denaturing experiments.

Protein folding: bringing theory and experiment closer together

The ability to perform enzymatic function by harnessing random molecular motion into self-organized protein structures is one of the most fascinating results of evolution. A close interplay between theory and experiment is driving the progress in understanding the principles that determine the behaviour of proteins. New techniques that significantly increase the amount of information obtainable from experimental data have been recently proposed; it is now becoming possible to describe at atomic resolution the events that take place during the folding process. Successful predictions of these events are being reported at an increasing rate and general principles are being outlined.

Thermal folding and mechanical unfolding pathways of protein secondary structures

Mechanical stretching of secondary structures is studied through molecular dynamics simulations of a Go-like model. Force versus displacement curves are studied as a function of the stiffness and velocity of the pulling device. The succession of stretching events, as measured by the order in which contacts are ruptured, is compared to the sequencing of events during thermal folding and unfolding. Opposite cross-correlations are found for an alpha-helix and a beta-hairpin structure. In a tandem of two alpha-helices, the two constituent helices unravel nearly simultaneously. A simple condition for simultaneous versus sequential unraveling of repeat units is presented.

Folding and stretching in a Go-like model of titin

Mechanical stretching of the I27 domain of titin and of its double and triple repeats are studied through molecular dynamics simulations of a Go-like model with Lennard-Jones contact interactions. We provide a thorough characterization of the system and correlate the sequencing of the folding and unraveling events with each other and with the contact order. The roles of cantilever stiffness and pulling rate are studied. Unraveling of tandem titin structures has a serial nature. The force-displacement curves in this coarse-grained model are similar to those obtained through all atom calculations.

Unfolding proteins under external forces: a solvable model under the self-consistent pair contact probability approximation

We extend a model of Micheletti et al. [Phys. Rev. Lett. 87, 088102 (2001)]] used to study protein con-formations to the case in which there is an external force field. Under the self-consistent pair contact probability approximation, this residue-level resolution model can still be solved under pulling forces. We implement the algorithm using heterogeneous parameters and study the force-induced unfolding of a helical segment from the protein transformylase and of the beta-stranded domains from the protein titin. The results are qualitatively consistent with the results from more expensive, atomistic dynamics simulation. Despite the mean-field-like approach, we observed a sharp and cooperative unfolding transition.

Steered molecular dynamics investigations of protein function

Molecular recognition and mechanical properties of proteins govern molecular processes in the cell that can cause disease and can be targeted for drug design. Single molecule measurement techniques have greatly advanced knowledge but cannot resolve enough detail to be interpreted in terms of protein structure. We seek to complement the observations through so-called Steered Molecular Dynamics (SMD) simulations that link directly to experiments and provide atomic-level descriptions of the underlying events. Such a research program has been initiated in our group and has involved, for example, studies of elastic properties of immunoglobulin and fibronectin domains as well as the binding of biotin and avidin. In this article we explain the SMD method and suggest how it can be applied to the function of three systems that are the focus of modern molecular biology research: force transduction by the muscle protein titin and extracellular matrix protein fibronectin, recognition of antibody-antigene pairs, and ion selective conductivity of the K+ channel.

Single molecule measurements of titin elasticity

Titin, with a massive single chain of 3--4MDa and multiple modular motifs, spans the half-sarcomere of skeletal and cardiac muscles and serves important, multifaceted functions. In recent years, titin has become a favored subject of single molecule observations by atomic force microscopy (AFM) and laser optical trap (LOT). Here we review these single titin molecule extension studies with an emphasis on understanding their relevance to titin elasticity in muscle function. Some fundamental aspects of the methods for single titin molecule investigations, including the application of dynamic force, the elasticity models for filamentous titin motifs, the technical foundations and calibrations of AFM and LOT, and titin sample preparations are provided. A chronological review of major publications on recent single titin extension observations is presented. This is followed by summary evaluations of titin domain folding/unfolding results and of elastic properties of filamentous titin motifs. Implications of these single titin measurements for muscle physiology/pathology are discussed and forthcoming advances in single titin studies are anticipated.

Can non-mechanical proteins withstand force? Stretching barnase by atomic force microscopy and molecular dynamics simulation

Atomic force microscopy (AFM) experiments have provided intriguing insights into the mechanical unfolding of proteins such as titin I27 from muscle, but will the same be possible for proteins that are not physiologically required to resist force? We report the results of AFM experiments on the forced unfolding of barnase in a chimeric construct with I27. Both modules are independently folded and stable in this construct and have the same thermodynamic and kinetic properties as the isolated proteins. I27 can be identified in the AFM traces based on its previous characterization, and distinct, irregular low-force peaks are observed for barnase. Molecular dynamics simulations of barnase unfolding also show that it unfolds at lower forces than proteins with mechanical function. The unfolding pathway involves the unraveling of the protein from the termini, with much more native-like secondary and tertiary structure being retained in the transition state than is observed in simulations of thermal unfolding or experimentally, using chemical denaturant. Our results suggest that proteins that are not selected for tensile strength may not resist force in the same way as those that are, and that proteins with similar unfolding rates in solution need not have comparable unfolding properties under force.

Computer modeling of force-induced titin domain unfolding

Titin, a 1 micron long protein found in striated muscle myofibrils, possesses unique elastic and extensibility properties, and is largely composed of a PEVK region and beta-sandwich immunoglobulin (Ig) and fibronectin type III (FnIII) domains. The extensibility behavior of titin has been shown in atomic force microscope and optical tweezer experiments to partially depend on the reversible unfolding of individual Ig and FnIII domains. We performed steered molecular dynamics simulations to stretch single titin Ig domains in solution with pulling speeds of 0.1-1.0 A/ps, and FnIII domains with a pulling speed of 0.5 A/ps. Resulting force-extension profiles exhibit a single dominant peak for each domain unfolding, consistent with the experimentally observed sequential, as opposed to concerted, unfolding of Ig and FnIII domains under external stretching forces. The force peaks can be attributed to an initial burst of a set of backbone hydrogen bonds connected to the domains' terminal beta-strands. Constant force stretching simulations, applying 500-1000 pN of force, were performed on Ig domains. The resulting domain extensions are halted at an initial extension of 10 A until the set of all six hydrogen bonds connecting terminal beta-strands break simultaneously. This behavior is accounted for by a barrier separating folded and unfolded states, the shape of which is consistent with AFM and chemical denaturation data.

Stress-induced structural transitions in DNA and proteins

The ability to manipulate, stretch and twist biomolecules opens the way to an understanding of their structural transitions. We review some of the recently discovered stress-induced structural transitions in DNA as well as the application of single molecule manipulation techniques to DNA unzipping and to the study of protein folding/unfolding transitions.

Mechanical unfolding and refolding of proteins: an off-lattice model study

Using an off-lattice model, we investigate the response of a protein molecule to external stretching and release. In particular we study the passive force in a protein as a function of the extension of the protein. These force-extension curves exhibit hysteresis loops, whose areas increase with the pulling rate and decrease with thermal noise. Most of these results seem to be appropriately described by a cusp catastrophe.

Force spectroscopy with single bio-molecules

For many biological molecules, force is an important functional and structural parameter. With the rapidly growing knowledge about the relation between structure, function, and force, single-molecule force spectroscopy has become a versatile analytical tool for the structural and functional investigation of single bio-molecules in their native environments. Within the past year, detailed insights into binding potentials of receptor ligand pairs, protein folding pathways, molecular motors, DNA mechanics and the functioning of DNA-binding agents (such as proteins and drugs), as well as the function of molecular motors, have been obtained.