Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

Mechanical control of the directional stepping dynamics of the kinesin motor

Among the multiple steps constituting the kinesin mechanochemical cycle, one of the most interesting events is observed when kinesins move an 8-nm step from one microtubule (MT)-binding site to another. The stepping motion that occurs within a relatively short time scale ( approximately 100 mus) is, however, beyond the resolution of current experiments. Therefore, a basic understanding to the real-time dynamics within the 8-nm step is still lacking. For instance, the rate of power stroke (or conformational change) that leads to the undocked-to-docked transition of neck-linker is not known, and the existence of a substep during the 8-nm step still remains a controversial issue in the kinesin community. By using explicit structures of the kinesin dimer and the MT consisting of 13 protofilaments, we study the stepping dynamics with varying rates of power stroke (k(p)). We estimate that k(p)(-1) less, similar 20 micros to avoid a substep in an averaged time trace. For a slow power stroke with k(p)(-1) > 20 micros, the averaged time trace shows a substep that implies the existence of a transient intermediate, which is reminiscent of a recent single-molecule experiment at high resolution. We identify the intermediate as a conformation in which the tethered head is trapped in the sideway binding site of the neighboring protofilament. We also find a partial unfolding (cracking) of the binding motifs occurring at the transition state ensemble along the pathways before binding between the kinesin and MT.

Protein conformational transitions explored by mixed elastic network models

We develop a mixed elastic network model (MENM) to study large-scale conformational transitions of proteins between two (or more) known structures. Elastic network potentials for the beginning and end states of a transition are combined, in effect, by adding their respective partition functions. The resulting effective MENM energy function smoothly interpolates between the original surfaces, and retains the beginning and end structures as local minima. Saddle points, transition paths, potentials of mean force, and partition functions can be found efficiently by largely analytic methods. To characterize the protein motions during a conformational transition, we follow "transition paths" on the MENM surface that connect the beginning and end structures and are invariant to parameterizations of the model and the mathematical form of the mixing scheme. As illustrations of the general formalism, we study large-scale conformation changes of the motor proteins KIF1A kinesin and myosin II. We generate possible transition paths for these two proteins that reveal details of their conformational motions. The MENM formalism is computationally efficient and generally applicable even for large protein systems that undergo highly collective structural changes.

Docking of photosystem I subunit C using a constrained geometric simulation

The elucidation of assembly pathways of multi-subunit protein complexes is a problem of great interest in structural biology and biomolecular modeling. In this study, we use a new computer algorithm for the simulation of large-scale motion in proteins to dock the subunit PsaC onto Photosystem I. We find that a complicated docking pathway involving multiple conformational changes can be quickly simulated by actively targeting only a few residues at a time to their target positions. Simulations for two possible docking scenarios are explored, and experimental approaches to distinguish between them are discussed.

Coarse-grained free energy functions for studying protein conformational changes: a double-well network model

In this work, a double-well network model (DWNM) is presented for generating a coarse-grained free energy function that can be used to study the transition between reference conformational states of a protein molecule. Compared to earlier work that uses a single, multidimensional double-well potential to connect two conformational states, the DWNM uses a set of interconnected double-well potentials for this purpose. The DWNM free energy function has multiple intermediate states and saddle points, and is hence a "rough" free energy landscape. In this implementation of the DWNM, the free energy function is reduced to an elastic-network model representation near the two reference states. The effects of free energy function roughness on the reaction pathways of protein conformational change is demonstrated by applying the DWNM to the conformational changes of two protein systems: the coil-to-helix transition of the DB-loop in G-actin and the open-to-closed transition of adenylate kinase. In both systems, the rough free energy function of the DWNM leads to the identification of distinct minimum free energy paths connecting two conformational states. These results indicate that while the elastic-network model captures the low-frequency vibrational motions of a protein, the roughness in the free energy function introduced by the DWNM can be used to characterize the transition mechanism between protein conformations.

The rough energy landscape of superfolder GFP is linked to the chromophore

Many green fluorescent protein (GFP) variants have been developed for use as fluorescent tags, and recently a superfolder GFP (sfGFP) has been developed as a robust folding reporter. This new variant shows increased stability and improved folding kinetics, as well as 100% recovery of native protein after denaturation. Here, we characterize sfGFP, and find that this variant exhibits hysteresis as unfolding and refolding equilibrium titration curves are non-coincident even after equilibration for more than eight half-lives as estimated from kinetic unfolding and refolding studies. This hysteresis is attributed to trapping in a native-like intermediate state. Mutational studies directed towards inhibiting chromophore formation indicate that the novel backbone cyclization is responsible for the hysteresis observed in equilibrium titrations of sfGFP. Slow equilibration and the presence of intermediates imply a rough landscape. However, de novo folding in the absence of the chromophore is dominated by a smoother energy landscape than that sampled during unfolding and refolding of the post-translationally modified polypeptide.

Nanosecond-timescale conformational dynamics of the human alpha7 nicotinic acetylcholine receptor

We explore the conformational dynamics of a homology model of the human alpha7 nicotinic acetylcholine receptor using molecular dynamics simulation and analyses of root mean-square fluctuations, block partitioning of segmental motion, and principal component analysis. The results reveal flexible regions and concerted global motions of the subunits encompassing extracellular and transmembrane domains of the subunits. The most relevant motions comprise a bending, hinged at the beta10-M1 region, accompanied by concerted tilting of the M2 helices that widens the intracellular end of the channel. Despite the nanosecond timescale, the observations suggest that tilting of the M2 helices may initiate opening of the pore. The results also reveal direct coupling between a twisting motion of the extracellular domain and dynamic changes of M2. Covariance analysis of interresidue motions shows that this coupling arises through a network of residues within the Cys and M2-M3 loops where Phe135 is stabilized within a hydrophobic pocket formed by Leu270 and Ile271. The resulting concerted motion causes a downward shift of the M2 helices that disrupts a hydrophobic girdle formed by 9' and 13' residues.

MinActionPath: maximum likelihood trajectory for large-scale structural transitions in a coarse-grained locally harmonic energy landscape

The non-linear problem of simulating the structural transition between two known forms of a macromolecule still remains a challenge in structural biology. The problem is usually addressed in an approximate way using 'morphing' techniques, which are linear interpolations of either the Cartesian or the internal coordinates between the initial and end states, followed by energy minimization. Here we describe a web tool that implements a new method to calculate the most probable trajectory that is exact for harmonic potentials; as an illustration of the method, the classical Calpha-based Elastic Network Model (ENM) is used both for the initial and the final states but other variants of the ENM are also possible. The Langevin equation under this potential is solved analytically using the Onsager and Machlup action minimization formalism on each side of the transition, thus replacing the original non-linear problem by a pair of linear differential equations joined by a non-linear boundary matching condition. The crossover between the two multidimensional energy curves around each state is found numerically using an iterative approach, producing the most probable trajectory and fully characterizing the transition state and its energy. Jobs calculating such trajectories can be submitted on-line at: http://lorentz.dynstr.pasteur.fr/joel/index.php.

Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines

Analyzing nonlinear conformational relaxation dynamics in elastic networks corresponding to two classical motor proteins, we find that they respond by well defined internal mechanical motions to various initial deformations and that these motions are robust against external perturbations. We show that this behavior is not characteristic for random elastic networks. However, special network architectures with such properties can be designed by evolutionary optimization methods. Using them, an example of an artificial elastic network, operating as a cyclic machine powered by ligand binding, is constructed.

Sampling small-scale and large-scale conformational changes in proteins and molecular complexes

Sampling of small-scale and large-scale motions is important in various computational tasks, such as protein-protein docking and ligand binding. Here, we report further development and applications of the activation-relaxation technique for internal coordinate space trajectories (ARTIST). This method generates conformational moves of any complexity and size by identifying and crossing well-defined saddle points connecting energy minima. Simulations on two all-atom proteins and three protein complexes containing between 70 and 300 amino acids indicate that ARTIST opens the door to the full treatment of all degrees of freedom in dense systems such as protein-protein complexes.

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.

Functional residues serve a dominant role in mediating the cooperativity of the protein ensemble

Conformational fluctuations in proteins have emerged as a potentially important aspect of biological function, although the precise relationship and the implications have yet to be fully explored. Numerous studies have reported that the binding of ligand can influence fluctuations. However, the role of the binding site in mediating these fluctuations is not known. Of particular interest is whether in addition to serving as structural scaffolds for recognition and catalysis, active-site residues may also play a role in modulating the cooperative network. To address this question, we employ an experimentally validated ensemble-based description of proteins to elucidate the extent to which perturbations at different sites can influence the cooperative network in the protein. Applying this method to a database of test proteins, it is found statistically that binding sites are located in regions most able to affect the cooperative network, even for cooperative interactions between residues distant to the binding sites. This indicates that the conformational manifold under native conditions is determined by the network of cooperative interactions within the protein and suggests that proteins have evolved to use these conformational fluctuations in carrying out their functions. Furthermore, because the energetic coupling pattern calculated for each protein is robust and relatively insensitive to sequence, these studies further suggest that binding sites evolved in regions of the protein that are inherently poised to take advantage of the fluctuations in the native structure.

Internal strain regulates the nucleotide binding site of the kinesin leading head

In the presence of ATP, kinesin proceeds along the protofilament of microtubule by alternated binding of two motor domains on the tubulin binding sites. Because the processivity of kinesin is much higher than other motor proteins, it has been speculated that there exists a mechanism for allosteric regulation between the two monomers. Recent experiments suggest that ATP binding to the leading head (L) domain in kinesin is regulated by the rearward strain built on the neck-linker. We test this hypothesis by explicitly modeling a Calpha-based kinesin structure whose motor domains are bound on the tubulin binding sites. The equilibrium structures of kinesin on the microtubule show disordered and ordered neck-linker configurations for the L and trailing head, respectively. The comparison of the structures between the two heads shows that several native contacts present at the nucleotide binding site in the L are less intact than those in the binding site of the rear head. The network of native contacts obtained from this comparison provides the internal tension propagation pathway, which leads to the disruption of the nucleotide binding site in the L. Also, using an argument based on polymer theory, we estimate the internal tension built on the neck-linker to be f approximately 12-15 pN. Both of these conclusions support the experimental hypothesis.

Harmonic Fourier beads method for studying rare events on rugged energy surfaces

We present a robust, distributable method for computing minimum free energy paths of large molecular systems with rugged energy landscapes. The method, which we call harmonic Fourier beads (HFB), exploits the Fourier representation of a path in an appropriate coordinate space and proceeds iteratively by evolving a discrete set of harmonically restrained path points-beads-to generate positions for the next path. The HFB method does not require explicit knowledge of the free energy to locate the path. To compute the free energy profile along the final path we employ an umbrella sampling method in two generalized dimensions. The proposed HFB method is anticipated to aid the study of rare events in biomolecular systems. Its utility is demonstrated with an application to conformational isomerization of the alanine dipeptide in gas phase.

Molecular dynamics of apo-adenylate kinase: a distance replica exchange method for the free energy of conformational fluctuations

A large domain motion in adenylate kinase from E. coli (AKE) is studied with molecular dynamics. AKE undergoes a large-scale rearrangement of its lid and AMP-binding domains when the open form closes over its substrates, AMP, and Mg2+-ATP, whereby the AMP-binding and lid domains come closer to the core. The third domain, the core, is relatively stable during this motion. A reaction coordinate that monitors the distance between the AMP-binding and core domains is selected to be able to compare with the results of energy transfer experiments. Sampling along this reaction coordinate is carried out by using a distance replica exchange method (DREM), where systems that differ by a restraint potential enforcing different reaction coordinate values are independently simulated with periodic attempts at exchange of these systems. Several methods are used to study the efficiency and convergence properties of the DREM simulation and compared with an analogous non-DREM simulation. The DREM greatly accelerates the rate and extent of configurational sampling and leads to equilibrium sampling as measured by monitoring collective modes obtained from a principal coordinate analysis. The potential of mean force along the reaction coordinate reveals a rather flat region for distances from the open to a relatively closed AKE conformation. The potential of mean force for smaller distances has a distinct minimum that is quite close to that found in the closed form X-ray structure. In concert with a decrease in the reaction coordinate distance (AMP-binding-to-core distance) the lid-to-core distance of AKE also decreases. Therefore, apo AKE can fluctuate from its open form to conformations that are quite similar to its closed form X-ray structure, even in the absence of its substrates.

Pump-probe molecular dynamics as a tool for studying protein motion and long range coupling

A new method for analyzing the dynamics of proteins is developed and tested. The method, pump-probe molecular dynamics, excites selected atoms or residues with a set of oscillating forces, and the transmission of the impulse to other parts of the protein is probed using Fourier transform of the atomic motions. From this analysis, a coupling profile can be determined which quantifies the degree of interaction between pump and probe residues. Various physical properties of the method such as reciprocity and speed of transmission are examined to establish the soundness of the method. The coupling strength can be used to address questions such as the degree of interaction between different residues at the level of dynamics, and identify propagation of influence of one part of the protein on another via "pathways" through the protein. The method is illustrated by analysis of coupling between different secondary structure elements in the allosteric protein calmodulin, and by analysis of pathways of residue-residue interaction in the PDZ domain protein previously elucidated by genomics and mutational studies.

Dynamics of allosteric transitions in GroEL

The chaperonin GroEL-GroES, a machine that helps proteins to fold, cycles through a number of allosteric states, the T state, with high affinity for substrate proteins, the ATP-bound R state, and the R" (GroEL-ADP-GroES) complex. Here, we use a self-organized polymer model for the GroEL allosteric states and a general structure-based technique to simulate the dynamics of allosteric transitions in two subunits of GroEL and the heptamer. The T --> R transition, in which the apical domains undergo counterclockwise motion, is mediated by a multiple salt-bridge switch mechanism, in which a series of salt-bridges break and form. The initial event in the R -->R" transition, during which GroEL rotates clockwise, involves a spectacular outside-in movement of helices K and L that results in K80-D359 salt-bridge formation. In both the transitions there is considerable heterogeneity in the transition pathways. The transition state ensembles (TSEs) connecting the T, R, and R" states are broad with the TSE for the T --> R transition being more plastic than the R --> R" TSE.

Conformational transitions of adenylate kinase: switching by cracking

Conformational heterogeneity in proteins is known to often be the key to their function. We present a coarse grained model to explore the interplay between protein structure, folding and function which is applicable to allosteric or non-allosteric proteins. We employ the model to study the detailed mechanism of the reversible conformational transition of Adenylate Kinase (AKE) between the open to the closed conformation, a reaction that is crucial to the protein's catalytic function. We directly observe high strain energy which appears to be correlated with localized unfolding during the functional transition. This work also demonstrates that competing native interactions from the open and closed form can account for the large conformational transitions in AKE. We further characterize the conformational transitions with a new measure Phi(Func), and demonstrate that local unfolding may be due, in part, to competing intra-protein interactions.

Guanylate kinase, induced fit, and the allosteric spring probe

Since the introduction of the induced-fit theory by D. E. Koshland Jr., it has been established that conformational motion invariably accompanies the execution of protein function. The catalytic activity of kinases, specifically, is associated with large conformational changes ( approximately 1 nm amplitude). In the case of guanylate kinase, upon substrate binding, the LID and nucleotide-monophosphate-binding domains are brought together and toward the CORE with large concerted movements about the alpha3 (helix 3) axis. However, whether the change in conformation mostly affects the catalytic rate or mostly increases binding affinities for one or the other substrate is unclear. We investigate this question using a nanotechnology approach based on mechanical stress. Using an "allosteric spring probe", we bias conformational states in favor of the "open" (substrate-free) conformation of the enzyme; the result is that the binding constant for the substrate guanosine monophosphate (GMP) is reduced by up to a factor of 10, whereas the binding constant for adenosine triphosphate (ATP) and the catalytic rate are essentially unaffected. The results show that the GMP-induced conformational change, which promotes catalysis, does not promote ATP binding, consistent with previous mutagenesis studies. Furthermore, they show that this conformational change is of the induced-fit type with respect to GMP binding (but not ATP binding). We elaborate on this point by proposing a quantitative criterion for the classification of conformational changes with respect to the induced-fit theory. More generally, these results show that the allosteric spring probe can be used to affect enzymatic activity in a continuously controlled manner, and also to affect specific steps of the reaction mechanism while leaving others unaffected. It is presumed that this will enable informative comparisons with the results of future molecular dynamics or statistical mechanics computations.

Model of the toxic complex of anthrax: responsive conformational changes in both the lethal factor and the protective antigen heptamer

The toxic complex of anthrax is formed when the monomeric protective antigen (PA) (83 kDa), while bound to its cell-surface receptor, is first converted to PA63 heptamers (PA63h) following N-terminal proteolytic cleavage, and then lethal (LF) (90 kDa) or edema factor (EF) binds to the heptamer. We report a "pseudoatomic" model for the complex of PA63h and full-length LF determined by applying the normal-mode flexible fitting procedure to a approximately 18 A cryo-electron microscopy (EM) density map of the complex. The model describes the interacting surface that buries a total area of approximately 10,140 A2 comprising approximately 40% charged, and approximately 30% each of polar and hydrophobic residues. For the heptamer, the buried surface, composed of approximately 110 residues, involves primarily three monomers and includes for two, similar stretches of the polypeptide chain from domain 1. For LF, the interface again involves approximately 110 residues, mostly from the N-terminal domain I (LF(N)), and the structurally homologous C-terminal domain IV. Most interestingly, bound LF displays a marked conformational change resulting from a "collapse" of domains I, III, and IV on domain II, with the largest movement of approximately 9 A noted for domain I. On the other hand, primarily, rigid-body movements, larger than approximately 10 A for three PA63 monomers, cause the hourglass-shaped heptamer lumen to enlarge by as much as approximately 50% near the middle of the molecule. Such concerted structural rearrangements in LF and the heptamer can facilitate ingress of the ligand into the heptamer lumen prior to unfolding and release through the PA63h channel formed in the acidic late endosomal membrane.

A novel disulfide bond in the SH2 Domain of the C-terminal Src kinase controls catalytic activity

The SH2 domain of the C-terminal Src kinase [Csk] contains a unique disulfide bond that is not present in other known SH2 domains. To investigate whether this unusual disulfide bond serves a novel function, the effects of disulfide bond formation on catalytic activity of the full-length protein and on the structure of the SH2 domain were investigated. The kinase activity of full-length Csk decreases by an order of magnitude upon formation of the disulfide bond in the distal SH2 domain. NMR spectra of the fully oxidized and fully reduced SH2 domains exhibit similar chemical shift patterns and are indicative of similar, well-defined tertiary structures. The solvent-accessible disulfide bond in the isolated SH2 domain is highly stable and far from the small lobe of the kinase domain. However, reduction of this bond results in chemical shift changes of resonances that map to a cluster of residues that extend from the disulfide bond across the molecule to a surface that is in direct contact with the small lobe of the kinase domain in the intact molecule. Normal mode analyses and molecular dynamics calculations suggest that disulfide bond formation has large effects on residues within the kinase domain, most notably within the active-site cleft. Overall, the data indicate that reversible cross-linking of two cysteine residues in the SH2 domain greatly impacts catalytic function and interdomain communication in Csk.

The open state gating mechanism of gramicidin a requires relative opposed monomer rotation and simultaneous lateral displacement

The gating mechanism of the open state of the gramicidin A (gA) channel is studied by using a new Monte Carlo Normal Mode Following (MC-NMF) technique, one applicable even without a target structure. The results demonstrate that the lowest-frequency normal mode (NM) at approximately 6.5 cm(-1) is the crucial mode that initiates dissociation. Perturbing the gA dimer in either direction along this NM leads to opposed, nearly rigid-body rotations of the gA monomers around the central pore axis. Tracking this NM by using the eigenvector-following technique reveals the channel's gating mechanism: dissociation via relative opposed monomer rotation and simultaneous lateral displacement. System evolution along the lowest-frequency eigenvector shows that the large-amplitude motions required for gating (dissociation) are not simple relative rigid-body motions of the monomers. Gating involves coupling intermonomer hydrogen bond breaking, backbone realignment, and relative monomer tilt with complex side chain reorganization at the intermonomer junction.

Exploring biomolecular machines: energy landscape control of biological reactions

For almost 15 years, our Pathway model has been the most powerful model in terms of predicting the tunnelling mechanism for electron transfer (ET) in biological systems, particularly proteins. Going beyond the conventional Pathway models, we have generalized our method to understand how protein dynamics modulate not only the Franck-Condon factor, but also the tunnelling matrix element. We have demonstrated that when interference among pathways modulates the electron tunnelling interactions in proteins (particularly destructive interference), dynamical effects are of critical importance. Tunnelling can be controlled by protein conformations that lie far from equilibrium-those that minimize the effect of destructive interference during tunnelling, for example. In the opposite regime, electron tunnelling is mediated by one (or a few) constructively interfering pathway tubes and dynamical effects are modest. This new mechanism for dynamical modulation of the ET rate has been able to explain and/or predict several rates that were later confirmed by experiment. However, thermal fluctuations can also affect these molecular machines in many other ways. For example, we show how global transformations, which control protein functions such as allostery, may involve large-scale motion and possibly partial unfolding during the reaction event.

Protein complex formation by acetylcholinesterase and the neurotoxin fasciculin-2 appears to involve an induced-fit mechanism

Specific, rapid association of protein complexes is essential for all forms of cellular existence. The initial association of two molecules in diffusion-controlled reactions is often influenced by the electrostatic potential. Yet, the detailed binding mechanisms of proteins highly depend on the particular system. A complete protein complex formation pathway has been delineated by using structural information sampled over the course of the transformation reaction. The pathway begins at an encounter complex that is formed by one of the apo forms of neurotoxin fasciculin-2 (FAS2) and its high-affinity binding protein, acetylcholinesterase (AChE), followed by rapid conformational rearrangements into an intermediate complex that subsequently converts to the final complex as observed in crystal structures. Formation of the intermediate complex has also been independently captured in a separate 20-ns molecular dynamics simulation of the encounter complex. Conformational transitions between the apo and liganded states of FAS2 in the presence and absence of AChE are described in terms of their relative free energy profiles that link these two states. The transitions of FAS2 after binding to AChE are significantly faster than in the absence of AChE; the energy barrier between the two conformational states is reduced by half. Conformational rearrangements of FAS2 to the final liganded form not only bring the FAS2/AChE complex to lower energy states, but by controlling transient motions that lead to opening or closing one of the alternative passages to the active site of the enzyme also maximize the ligand's inhibition of the enzyme.

Effective stochastic dynamics on a protein folding energy landscape

We present an approach to protein folding kinetics using stochastic reaction-coordinate dynamics, in which the effective drift velocities and diffusion coefficients are determined from microscopic simulation data. The resultant Langevin equation can then be used to directly simulate the folding process. Here, we test this approach by applying it to a toy two-state dynamical system and to a funnellike structure-based (Go-type) model. The folding time predictions agree very well with full simulation results. Therefore, we have in hand a fast numerical tool for calculating the folding kinetic properties, even when full simulations are not feasible. In addition, the local drift and diffusion coefficients provide an alternative way to compute the free energy profile in cases where only local sampling can be achieved.

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.

Reconciling the "old" and "new" views of protein allostery: a molecular simulation study of chemotaxis Y protein (CheY)

A combination of thirty-two 10-ns-scale molecular dynamics simulations were used to explore the coupling between conformational transition and phosphorylation in the bacteria chemotaxis Y protein (CheY), as a simple but representative example of protein allostery. Results from these simulations support an activation mechanism in which the beta4-alpha4 loop, at least partially, gates the isomerization of Tyr106. The roles of phosphorylation and the conserved Thr87 are deemed indirect in that they stabilize the active configuration of the beta4-alpha4 loop. The indirect role of the activation event (phosphorylation) and/or conserved residues in stabilizing, rather than causing, specific conformational transition is likely a feature in many signaling systems. The current analysis of CheY also helps to make clear that neither the "old" (induced fit) nor the "new" (population shift) views for protein allostery are complete, because they emphasize the kinetic (mechanistic) and thermodynamic aspects of allosteric transitions, respectively. In this regard, an issue that warrants further analysis concerns the interplay of concerted collective motion and sequential local structural changes in modulating cooperativity between distant sites in biomolecules.

SYMMETRY, FORM, AND SHAPE: Guiding Principles for Robustness in Macromolecular Machines

Computational studies of large macromolecular assemblages have come a long way during the past 10 years. With the explosion of computer power and parallel computing, timescales of molecular dynamics simulations have been extended far beyond the hundreds of picoseconds timescale. However, limitations remain for studies of large-scale conformational changes occurring on timescales beyond nanoseconds, especially for large macromolecules. In this review, we describe recent methods based on normal mode analysis that have enabled us to study dynamics on the microsecond timescale for large macromolecules using different levels of coarse graining, from atomically detailed models to those employing only low-resolution structural information. Emerging from such studies is a control principle for robustness in Nature's machines. We discuss this idea in the context of large-scale functional reorganization of the ribosome, virus particles, and the muscle protein myosin.

Ensemble-based signatures of energy propagation in proteins: a new view of an old phenomenon

The ability of a protein to transmit the energetic effects of binding from one site to another constitutes the underlying basis for allosterism and signal transduction. Despite clear experimental evidence indicating the ability of proteins to transmit the effects of binding, the means by which this propagation is facilitated is not well understood. Using our previously developed ensemble-based description of the equilibrium, we investigated the physical basis of energy propagation and identified several fundamental and general aspects of energetic coupling between residues in a protein. First, partitioning of a conformational ensemble into four distinct sub-ensembles allows for explanation of the range of experimentally observed coupling behaviors (i.e., positive, neutral, and negative coupling between various regions of the protein structure). Second, the relative thermodynamic properties of these four sub-ensembles define the energetic coupling between residues as either positive, neutral, or negative. Third, analysis of the structural and thermodynamic features of the states within each sub-ensemble reveals significant variability. This third result suggests that a quantitative description of energy propagation in proteins requires an understanding of the structural and energetic features of more than just one or a few low-energy states, but also of many high-energy states. Such findings illuminate the difficulty in interpreting energy propagation in proteins in terms of a structural pathway that physically links coupled sites.

A New Method for Coarse-Grained Elastic Normal-Mode Analysis

In this paper, we report a new method for coarse-grained elastic normal-mode analysis. The purpose is to overcome a long-standing problem in the conventional analysis called the tip effect that makes the motional patterns (eigenvectors) of some low-frequency modes irrational. The new method retains the merits of a conventional method such as not requiring lengthy initial energy minimization, which always distorts structures, and also delivers substantially more accurate low-frequency modes with no tip effect for proteins of any size. This improvement of modes is crucial for certain types of applications such as structural refinement or normal-mode-based sampling.

Folding-based molecular simulations reveal mechanisms of the rotary motor F1-ATPase

Biomolecular machines fulfill their function through large conformational changes that typically occur on the millisecond time scale or longer. Conventional atomistic simulations can only reach microseconds at the moment. Here, extending the minimalist model developed for protein folding, we propose the "switching Gō model" and use it to simulate the rotary motion of ATP-driven molecular motor F(1)-ATPase. The simulation recovers the unidirectional 120 degrees rotation of the gamma-subunit, the rotor. The rotation was induced solely by steric repulsion from the alpha(3)beta(3) subunits, the stator, which undergoes conformation changes during ATP hydrolysis. In silico alanine mutagenesis further elucidated which residues play specific roles in the rotation. Finally, regarding the mechanochemical coupling scheme, we found that the tri-site model does not lead to successful rotation but that the always-bi-site model produces approximately 30 degrees and approximately 90 degrees substeps, perfectly in accord with experiments. In the always-bi-site model, the number of sites occupied by nucleotides is always two during the hydrolysis cycle. This study opens up an avenue of simulating functional dynamics of huge biomolecules that occur on the millisecond time scales involving large-amplitude conformational change.

Coarse-grained normal mode analysis in structural biology

The realization that experimentally observed functional motions of proteins can be predicted by coarse-grained normal mode analysis has renewed interest in applications to structural biology. Notable applications include the prediction of biologically relevant motions of proteins and supramolecular structures driven by their structure-encoded collective dynamics; the refinement of low-resolution structures, including those determined by cryo-electron microscopy; and the identification of conserved dynamic patterns and mechanically key regions within protein families. Additionally, hybrid methods that couple atomic simulations with deformations derived from coarse-grained normal mode analysis are able to sample collective motions beyond the range of conventional molecular dynamics simulations. Such applications have provided great insight into the underlying principles linking protein structures to their dynamics and their dynamics to their functions.

Slow protein conformational dynamics from multiple experimental structures: the helix/sheet transition of arc repressor

Conformational transitions underlie the function of many biomolecular systems. Resolving intermediate structural changes, however, is challenging for both experiments and all-atom simulations because the duration of transitions is short relative to the lifetime of the stable species. Simplified descriptions based on a single experimental structure, such as elastic network models or Gō models, are not immediately applicable. Here, we develop a general method that combines multiple coarse-grained models to capture slow conformational transitions. Individually, each model describes one of the experimental structures; together, they approximate the complete energy surface. We demonstrate the method for the helix-to-sheet transition in Arc repressor N11L. We find that the transition involves the partial unfolding of the switch region, and rapid refolding into the alternate structure. Transient local unfolding is consistent with the low hydrogen exchange protection factors of the switch region. Also in agreement with experiment, the isomerization occurs independently of the global folding/dimerization transition.

Roles of static and dynamic domains in stability and catalysis of adenylate kinase

Protein dynamics, including conformational switching, are recognized to be crucial for the function of many systems. These motions are more challenging to study than simple static structures. Here, we present evidence suggesting that in the enzyme adenylate kinase large "hinge bending" motions closely related to catalysis are regulated by intrinsic properties of the moving domains and not by their hinges, by anchoring domains, or by remote allosteric-like regions. From a pair of highly homologous mesophilic and thermophilic adenylate kinases, we generated a series of chimeric enzymes using a previously undescribed method with synthetic genes. Subsequent analysis of the chimeras has revealed unexpected spatial separation of stability and activity control. Our results highlight specific contributions of dynamics to catalysis in adenylate kinase. Furthermore, the overall strategy and the specific mutagenesis method used in this study can be generally applied.

Coupled protein domain motion in Taq polymerase revealed by neutron spin-echo spectroscopy

Long-range conformational changes in proteins are ubiquitous in biology for the transmission and amplification of signals; such conformational changes can be triggered by small-amplitude, nanosecond protein domain motion. Understanding how conformational changes are initiated requires the characterization of protein domain motion on these timescales and on length scales comparable to protein dimensions. Using neutron spin-echo spectroscopy (NSE), normal mode analysis, and a statistical-mechanical framework, we reveal overdamped, coupled domain motion within DNA polymerase I from Thermus aquaticus (Taq polymerase). This protein utilizes correlated domain dynamics over 70 angstroms to coordinate nucleotide synthesis and cleavage during DNA synthesis and repair. We show that NSE spectroscopy can determine the domain mobility tensor, which determines the degree of dynamical coupling between domains. The mobility tensor defines the domain velocity response to a force applied to it or to another domain, just as the sails of a sailboat determine its velocity given the applied wind force. The NSE results provide insights into the nature of protein domain motion that are not appreciated by conventional biophysical techniques.

Channel opening motion of alpha7 nicotinic acetylcholine receptor as suggested by normal mode analysis

The gating motion of the human nicotinic acetylcholine receptor (nAChR) alpha7 was investigated with normal mode analysis (NMA) of two homology models. The first model, referred to as model I, was built from both the Lymnaea stagnalis acetylcholine binding protein (AChBP) and the transmembrane (TM) domain of the Torpedo marmorata nAChR. The second model, referred to as model C, was based solely on the recent electron microscopy structure of the T. marmorata nAChR. Despite structural differences, both models exhibit nearly identical patterns of flexibility and correlated motions. In addition, both models show a similar global twisting motion that may represent channel gating. The similar results obtained for the two models indicate that NMA is most sensitive to the contact topology of the structure rather than its finer detail. The major difference between the low-frequency motions sampled for the two models is that a symmetrical pore-breathing motion, favoring channel opening, is present as the second most dominant motion in model I, whilst largely absent from model C. The absence of this mode in model C can be attributed to its less symmetrical architecture. Finally, as a further goal of the present study, an approximate open channel model, consistent with many experimental findings, has been produced.

Large amplitude conformational change in proteins explored with a plastic network model: adenylate kinase

The plastic network model (PNM) is used to generate a conformational change pathway for Escherichia coli adenylate kinase based on two crystal structures, namely that of an open and a closed conformer. In this model, the energy basins corresponding to known conformers are connected at their lowest common energies. The results are used to evaluate and analyze the minimal energy pathways between these basins. The open to closed transition analysis provides an identification of hinges that is in agreement with the existing definitions based on the available X-ray structures. The elastic energy distribution and the C(alpha) pseudo-dihedral variation provide similar information on these hinges. The ensemble of the 45 published structures for this protein and closely related proteins is shown to always be within 3.0 A of the pathway, which corresponds to a conformational change between two end structures that differ by a C(alpha)-atom root-mean-squared deviation of 7.1A.

Coupled motions in the SH2 and kinase domains of Csk control Src phosphorylation

The C-terminal Src kinase (Csk) phosphorylates and down-regulates Src family tyrosine kinases. The Csk-binding protein (Cbp) localizes Csk close to its substrates at the plasma membrane, and increases the specific activity of the kinase. To investigate this long-range catalytic effect, the phosphorylation of Src and the conformation of Csk were investigated in the presence of a high-affinity phosphopeptide derived from Cbp. This peptide binds tightly to the SH2 domain and enhances Src recognition (lowers K(m)) by increasing the apparent phosphoryl transfer rate in the Csk active site, a phenomenon detected in rapid quench flow experiments. Previous studies demonstrated that the regulation of Csk activity is linked to conformational changes in the enzyme that can be probed with hydrogen-deuterium exchange methods. We show that the Cbp peptide impacts deuterium incorporation into its binding partner (the SH2 domain), and into the SH2-kinase linker and several sequences in the kinase domain, including the glycine-rich loop in the active site. These findings, along with computational data from normal mode analyses, suggest that the SH2 domain moves in a cantilever fashion with respect to the small lobe of the kinase domain, ordering the active site for catalysis. The binding of a small Cbp-derived peptide to the SH2 domain of Csk modifies these motions, enhancing Src recognition.

Usefulness and limitations of normal mode analysis in modeling dynamics of biomolecular complexes

Various types of large-amplitude molecular deformation are ubiquitously involved in the functions of biological macromolecules, especially supramolecular complexes. They can be very effectively analyzed by normal mode analysis with well-established procedures. However, despite its enormous success in numerous applications, certain issues related to the applications of normal mode analysis require further discussion. In this review, the author addresses some common issues so as to raise the awareness of the usefulness and limitations of the method in the general community of structural biology.

Plasticity of acetylcholine receptor gating motions via rate-energy relationships

Like other protein conformational changes, ion channel gating requires the protein to achieve a high-energy transition-state structure. It is not known whether ion channel gating takes place on a broad energy landscape on which many alternative transition state structures are accessible, or on a narrow energy landscape where only a few transition-state structures are possible. To address this question, we measured how rate-equilibrium free energy relationships (REFERs) for di-liganded and unliganded acetylcholine receptor gating vary as a function of the gating equilibrium constant. A large slope for the REFER plot indicates an openlike transition state, whereas a small slope indicates a closedlike transition state. Due to this relationship between REFERs and transition-state structure, the sensitivity of the REFER slope to mutation-induced energetic perturbations allows estimation of the breadth of the energy landscape underlying gating. The relatively large sensitivity of di-liganded REFER slopes to energetic perturbations suggests that the motions underlying di-liganded gating take place on a broad, shallow energy landscape where many alternative transition-state structures are accessible.

Thermodynamic and dynamic factors involved in the stability of native protein structure in amorphous solids in relation to levels of hydration

The internal, dynamical fluctuations of protein molecules exhibit many of the features typical of polymeric and bulk small molecule glass forming systems. The response of a protein's internal molecular mobility to temperature changes is similar to that of other amorphous systems, in that different types of motions freeze out at different temperatures, suggesting they exhibit the alpha-beta-modes of motion typical of polymeric glass formers. These modes of motion are attributed to the dynamic regimes that afford proteins the flexibility for function but that also develop into the large-scale collective motions that lead to unfolding. The protein dynamical transition, T(d), which has the same meaning as the T(g) value of other amorphous systems, is attributed to the temperature where protein activity is lost and the unfolding process is inhibited. This review describes how modulation of T(d) by hydration and lyoprotectants can determine the stability of protein molecules that have been processed as bulk, amorphous materials. It also examines the thermodynamic, dynamic, and molecular factors involved in stabilizing folded proteins, and the effects typical pharmaceutical processes can have on native protein structure in going from the solution state to the solid state.

The role of shape in determining molecular motions

We examined the role of molecular shape in determining the patterns of low-frequency deformational motions of biological macromolecules. The low-frequency subspace of eigenvectors in normal mode analysis was found to be robustly similar upon randomization of the Hessian matrix elements as long as the structure of the matrix is maintained, which indicates that the global shape of molecules plays a more dominant role in determining the highly anisotropic low-frequency motions than the absolute values of stiffness and directionality of local interactions. The results provided a quantitative foundation for the validity of elastic normal mode analysis.

Rigid-cluster models of conformational transitions in macromolecular machines and assemblies

We present a rigid-body-based technique (called rigid-cluster elastic network interpolation) to generate feasible transition pathways between two distinct conformations of a macromolecular assembly. Many biological molecules and assemblies consist of domains which act more or less as rigid bodies during large conformational changes. These collective motions are thought to be strongly related with the functions of a system. This fact encourages us to simply model a macromolecule or assembly as a set of rigid bodies which are interconnected with distance constraints. In previous articles, we developed coarse-grained elastic network interpolation (ENI) in which, for example, only Calpha atoms are selected as representatives in each residue of a protein. We interpolate distance differences of two conformations in ENI by using a simple quadratic cost function, and the feasible conformations are generated without steric conflicts. Rigid-cluster interpolation is an extension of the ENI method with rigid-clusters replacing point masses. Now the intermediate conformations in an anharmonic pathway can be determined by the translational and rotational displacements of large clusters in such a way that distance constraints are observed. We present the derivation of the rigid-cluster model and apply it to a variety of macromolecular assemblies. Rigid-cluster ENI is then modified for a hybrid model represented by a mixture of rigid clusters and point masses. Simulation results show that both rigid-cluster and hybrid ENI methods generate sterically feasible pathways of large systems in a very short time. For example, the HK97 virus capsid is an icosahedral symmetric assembly composed of 60 identical asymmetric units. Its original Hessian matrix size for a Calpha coarse-grained model is >(300,000)(2). However, it reduces to (84)(2) when we apply the rigid-cluster model with icosahedral symmetry constraints. The computational cost of the interpolation no longer scales heavily with the size of structures; instead, it depends strongly on the minimal number of rigid clusters into which the system can be decomposed.

Normal mode based flexible fitting of high-resolution structure into low-resolution experimental data from cryo-EM

A new method for the flexible fitting of high-resolution structures into low-resolution maps of macromolecular complexes from electron microscopy has been recently described in applications to simulated electron density maps. This method uses a linear combination of low-frequency normal modes in an iterative manner to deform the structure optimally to conform to the low-resolution electron density map. Gradient-following techniques in the coordinate space of collective normal modes are used to optimize the overall correlation coefficient between computed and measured electron densities. With this approach, multi-scale flexible fitting can be performed using all-atoms or Calpha atoms. In this paper, illustrative studies of normal mode based flexible fitting to experimental cryo-EM maps are presented for three different systems. Large, functionally relevant conformational changes for elongation factor G bound to the ribosome, Escherichia coli RNA polymerase and cowpea chlorotic mottle virus are elucidated as the result of the application of NMFF from high-resolution structures to cryo-electron microscopy maps.

Energy landscapes and solved protein-folding problems

Energy-landscape theory has led to much progress in protein folding kinetics, protein structure prediction and protein design. Funnel landscapes describe protein folding and binding and explain how protein topology determines kinetics. Landscape-optimized energy functions based on bioinformatic input have been used to correctly predict low-resolution protein structures and also to design novel proteins automatically.

The requirement for mechanical coupling between head and S2 domains in smooth muscle myosin ATPase regulation and its implications for dimeric motor function

A combination of experimental structural data, homology modelling and elastic network normal mode analysis is used to explore how coupled motions between the two myosin heads and the dimerization domain (S2) in smooth muscle myosin II determine the domain movements required to achieve the inhibited state of this ATP-dependent molecular motor. These physical models rationalize the empirical requirement for at least two heptads of non-coiled alpha-helix at the junction between the myosin heads and S2, and the dependence of regulation on S2 length. The results correlate well with biochemical data regarding altered conformational-dependent solubility and stability. Structural models of the conformational transition between putative active states and the inhibited state show that torsional flexibility of the S2 alpha-helices is a key mechanical requirement for myosin II regulation. These torsional motions of the myosin heads about their coiled coil alpha-helices affect the S2 domain structure, which reciprocally affects the motions of the myosin heads. This inter-relationship may explain a large body of data on function of molecular motors that form dimers through a coiled-coil domain.

Simple Energy Landscape Model for the Kinetics of Functional Transitions in Proteins

It is evident that protein conformational transitions play important roles in biological machinery; however, detailed pictures of these transition processes capable of making kinetic prediction are not yet available. For a full description of these transitions, we first need to describe kinematically movements between stable states. Then, more importantly, a free energy profile associated with the conformational change needs to be obtained. Recently, a new model to describe the energy landscape of protein conformational changes was applied to the conformational transition of adenylate kinase. In this model, the conformational change coupled to the ligand binding is described as a switching between two energy surfaces that correspond to ligand bound and unbound states. The nonlinearity of the protein conformational changes is described through an iterative usage of normal mode calculations. In addition, another kind of nonlinearity enters the dynamics of the conformational transitions due to cracking, or partial unfolding, which may occur during the conformational transitions. The consequences of this theoretical model are explored in greater detail. An improved model for the cracking that includes the cooperativity of the partial unfolding in analogy to nucleation is introduced.

Diversity and Identity of Mechanical Properties of Icosahedral Viral Capsids Studied with Elastic Network Normal Mode Analysis

We analyze the mechanical properties and putative dynamical fluctuations of a variety of viral capsids comprising different sizes and quasi-equivalent symmetries by performing normal mode analysis using the elastic network model. The expansion of the capsid to a swollen state is studied using normal modes and is compared with the experimentally observed conformational change for three of the viruses for which experimental data exist. We show that a combination of one or two normal modes captures remarkably well the overall translation that dominates the motion between the two conformational states, and reproduces the overall conformational change. We observe for all of the viral capsids that the nature of the modes is different. In particular for the T=7 virus, HK97, for which the shape of the capsid changes from spherical to faceted polyhedra, two modes are necessary to accomplish the conformational transition. In addition, we extend our study to viral capsids with other T numbers, and discuss the similarities and differences in the features of virus capsid conformational dynamics. We note that the pentamers generally have higher flexibility and propensity to move freely from the other capsomers, which facilitates the shape adaptation that may be important in the viral life cycle.

Protein Unfolding and Refolding Under Force: Methodologies for Nanomechanics

An increasing number of inter- and intramolecular interactions can nowadays be probed using single-molecule manipulation techniques. Protein unfolding and refolding is the most representative--though complex--of these interactions. Herein, we review the main modes of performing a force unfolding experiment: the velocity clamp and the new force clamp mode. We also compare some of the physical aspects behind the two most frequently used single-molecule manipulation instrumentations: optical tweezers and atomic force microscopes.