Protein Native Dynamics

The dynamics of biological polymers, including proteins, RNA, and DNA, occur in very high-dimensional spaces. Many naturally occurring polymers can navigate a vast phase space and rapidly find their lowest free energy (folded) state. Thus, although the search process is stochastic, it is not completely random. Instead, it is best described in terms of diffusion along a downhill energy landscape. In this context, there have been many efforts to use simplified representations of the energetics, for which the potential energy is chosen to be a relatively smooth function with a global minimum that corresponds to the folded state. That is, instead of including every type of physical interaction, the broad characteristics of the landscape are encoded in approximate energy functions. We describe a particular class of models, called structure-based models, that can be used to explore the diffusive properties of biomolecular folding and conformational rearrangements. These energy functions m...

Proteins have evolved to use water to help guide folding. A physically motivated, nonpairwise-additive model of water-mediated interactions added to a protein structure prediction Hamiltonian yields marked improvement in the quality of structure prediction for larger proteins. Free energy profile analysis suggests that long-range water-mediated potentials guide folding and smooth the underlying folding funnel. Analyzing simulation trajectories gives direct evidence that water-mediated interactions facilitate native-like packing of supersecondary structural elements. Long-range pairing of hydrophilic groups is an integral part of protein architecture. Specific water-mediated interactions are a universal feature of biomolecular recognition landscapes in both folding and binding.

Allostery can be viewed as an effect of binding at one site of the protein to a second, often significantly distant functional site, enabling regulation of the protein function. In majority of cases, the molecular mechanisms and networks mediating distant communications that give rise to allostery are poorly understood. We have developed a rigidity-transmission allostery (RTA) algorithm, a computational method based on mathematical rigidity theory. Starting with an X-ray crystal structure or an ensemble of three dimensional snapshots, we model the protein as a constraint network (graph) consisting of vertices (atoms) and edges (i.e., constraints including covalent bonds, electrostatic bonds, hydrogen bonds, and hydrophobic contacts).

We report a fully general technique addressing a long standing challenge of calculating conformational free energy differences between various states of a polymer chain from simulations using explicit solvent force fields. The main feature of our method is a special mapping variable, a path coordinate, which continuously connects two conformations. The path variable has been designed to preserve locality in the phase space near the path endpoints. We avoid the problem of sampling the unfolded states by creating an artificial confinement ''tube" in the phase space that prevents the molecule from unfolding without affecting the calculation of the desired free energy difference. We applied our technique to compute the free energy difference between two native-like conformations of the small protein Trp-cage using the CHARMM force field with explicit solvent. We verified this result by comparing it with an independent, significantly more expensive calculation. Overall, the present study suggests that the new method of computing free energy differences between polymer chain conformations is accurate and highly computationally efficient.

Many biologically interesting functions such as allosteric switching or protein-ligand binding are determined by the kinetics and mechanisms of transitions between various conformational substates of the native basin of globular proteins. To advance our understanding of these processes, we constructed a two-dimensional free energy surface (FES) of the native basin of a small globular protein, Trp-cage. The corresponding order parameters were defined using two native substructures of Trp-cage. These calculations were based on extensive explicit water all-atom molecular dynamics simulations. Using the obtained two-dimensional FES, we studied the transition kinetics between two Trp-cage conformations, finding that switching process shows a borderline behavior between diffusive and weakly-activated dynamics. The transition is well-characterized kinetically as a biexponential process. We also introduced a new one-dimensional reaction coordinate for the conformational transition, finding reasonable qualitative agreement with the two-dimensional kinetics results. We investigated the distribution of all the 38 native nuclear magnetic resonance structures on the obtained FES, analyzing interactions that stabilize specific lowenergy conformations. Finally, we constructed a FES for the same system but with simple dielectric model of water instead of explicit water, finding that the results were surprisingly similar in a small region centered on the native conformations. The dissimilarities between the explicit and implicit model on the larger-scale point to the important role of water in mediating interactions between amino acid residues.

The effects of Ca 2C binding on the side-chain methyl dynamics of calbindin D 9k have been characterized by 2 H NMR relaxation rate measurements. Longitudinal, transverse in-phase, quadrupolar order, transverse antiphase and double quantum relaxation rates are reported for both the apo and Ca 2C-loaded states of the protein at two magnetic field strengths. The relatively large size of the data set allows for a detailed analysis of the underlying conformational dynamics by spectral density mapping and model-free fitting procedures. The results reveal a correlation between a methyl group's distance from the Ca 2C binding sites and its conformational dynamics. Several methyl groups segregate into two limiting classes, one proximal and the other distal to the binding sites. Methyl groups in these two classes respond differently to Ca 2C binding, both in terms of the timescale and amplitude of their fluctuations. Ca 2C binding elicits a partial immobilization among methyl groups in the proximal class, which is consistent with previous studies of calbindin's backbone dynamics. The distal class, however, exhibits a trend that could not be inferred from the backbone data in that its mobility actually increases with Ca 2C binding. We have introduced the term polar dynamics to describe this type of organization across the molecule. The trend may represent an important mechanism by which calbindin D 9k achieves high affinity binding while minimizing the corresponding loss of conformational entropy.

The long-held views on lock-and-key versus induced fit in binding arose from the notion that a protein exists in a single, most stable conformation, dictated by its sequence. However, in solution proteins exist in a range of conformations, which may be described by statistical mechanical laws and their populations follow statistical distributions. Upon binding, the equilibrium will shift in favor of the bound conformation from the ensemble of conformations around the bottom of the folding funnel. Hence here we extend the implications and the usefulness of the folding funnel concept to explain fundamental binding mechanisms.

proton channels from influenza A and B viruses by solution NMR spectroscopy. The channel structures reveal pore features that are important for proton gating and proton relay. Structural details of the anti-influenza drug, rimantadine, bound to the channel suggests an unexpected allosteric mechanism of drug inhibition and drug resistance, which has been verified by thorough functional and mutagenesis experiments. The work is supported by NIH grant AI067438.

Structural excursions of a protein at equilibrium are key to biomolecular recognition and function modulation. Protein modeling research is driven by the need to aid wet laboratories in characterizing equilibrium protein dynamics. In principle, structural excursions of a protein can be directly observed via simulation of its dynamics, but the disparate temporal scales involved in such excursions make this approach computationally impractical. On the other hand, an informative representation of the structure space available to a protein at equilibrium can be obtained efficiently via stochastic optimization, but this approach does not directly yield information on equilibrium dynamics. We present here a novel methodology that first builds a multi-dimensional map of the energy landscape that underlies the structure space of a given protein and then queries the computed map for energetically-feasible excursions between structures of interest. An evolutionary algorithm builds such maps w...

Predicting conformational changes of proteins is needed in order to fully comprehend functional mechanisms. With the large number of available structures in sets of related proteins, it is now possible to directly visualize the clusters of conformations and their conformational transitions through the use of principal component analysis. The most striking observation about the distributions of the structures along the principal components is their highly non-uniform distributions. In this work, we use principal component analysis of experimental structures of 50 diverse proteins to extract the most important directions of their motions, sample structures along these directions, and estimate their free energy landscapes by combining knowledge-based potentials and entropy computed from elastic network models. When these resulting motions are visualized upon their coarse-grained free energy landscapes, the basis for conformational pathways becomes readily apparent. Using three well-stu...

The large number of available HIV-1 protease structures provides a remarkable sampling of conformations of the different conformational states, which can be viewed as direct structural information about the dynamics of the HIV-1 protease. After structure matching, we apply principal component analysis (PCA) to obtain the important apparent motions for both bound and unbound structures. There are significant similarities between the first few key motions and the first few low-frequency normal modes calculated from a static representative structure with an elastic network model (ENM), strongly suggesting that the variations among the observed structures and the corresponding conformational changes are facilitated by the low-frequency, global motions intrinsic to the structure. Similarities are also found when the approach is applied to an NMR ensemble, as well as to molecular dynamics (MD) trajectories. Thus, a sufficiently large number of experimental structures can directly provide important information about protein dynamics, but ENM can also provide similar sampling of conformations.

Tcb2 is a calcium-binding protein that localizes to the membrane-associated skeleton of the ciliated protozoan Tetrahymena thermophila with hypothesized roles in ciliary movement, cell cortex signaling, and pronuclear exchange. Tcb2 has also been implicated in a unique calcium-triggered, ATP-independent type of contractility exhibited by filamentous networks isolated from the Tetrahymena cytoskeleton. To gain insight into Tcb2's structure-function relationship and contractile properties, we determined solution NMR structures of its C-terminal domain in the calcium-free and calcium-bound states. The overall architecture is similar to other calcium-binding proteins, with paired EF-hand calcium-binding motifs. Comparison of the two structures reveals that Tcb2-C's calcium-induced conformational transition differs from the prototypical calcium sensor calmodulin, suggesting that the two proteins play distinct functional roles in Tetrahymena and likely have different mechanisms of target recognition. Future studies of the full-length protein and the identification of Tcb2 cellular targets will help establish the molecular basis of Tcb2 function and its unique contractile properties.

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δ E , the energetic roughness Δ E , and the scale of landscape measured by the entropy S . We show that the dimensionless ratio between the gap, roughness, and entropy of the system accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated wi...

Protein folding processes are generally described statistically with the help of multidimensional free energy landscape, typically reduced to a 1-D free energy profile along good reaction co-ordinate. There are many physical parameters which are responsible for protein molecule to hop between the native and unfolded states. The transition path region across the barrier is the region corresponding to the minimum fluctuation. The extent to which this transition region can extend beyond the obvious 1/2 kBT region has been a question of interest for a long time. We propose a new method to locate this transition path region and to study its dependence on the asymmetry of the transition state for a given free energy landscape. We have performed Brownian dynamics simulations with Gaussian white noise and Monte-Carlo simulation by sampling ten thousand successful transitions across the barrier for three different energy landscape having fixed barrier height with asymmetry in their curvature...

Protein motion simulation is still a troublesome problem yet to be solved, especially due to its high computational requirements. Accurate methodologies such as molecular dynamics have too much computational cost in order to simulate protein motion. On the other hand, less accurate procedures, such as interpolation methods, do not obtain realistic paths but mere estimations, usually resulting in kinematically impossible paths. The procedure presented in this paper tries to merge both approaches as it obtains protein motion by adaptive dihedral angle increment application, without making use of energy minimization processes. The presented procedure is an evolution of our previous works, as it provides the protein with more movement capacities, being able to search a wider configurational space. Also, structures are normalized in order to minimize inaccuracies introduced by experimental methods, providing a more efficient but still accurate structure for motion simulation.

Protein folding occurs in a high dimensional phase space, and the representation of the associated energy landscape is non-trivial. A widely applied approach to studying folding landscapes is to describe the dynamics along a small number of reaction coordinates. However, other strategies involve more elaborate analysis of the complex phase space. There have been many attempts to obtain a more detailed representation of all available conformations for a given system. In this work, we address this problem using a metric based on internal distances between amino acids to describe the differences between any two conformations. Using an effective projection method, we are able to go beyond the usual one-dimensional representation and thus visualize landscapes in two dimensions. We refer to this method as Energy Landscape Visualization Method (ELViM). We have applied this methodology using Cα structure-based model (SBM) to study the folding of two well-known proteins: SH3 domain and Prote...

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δ E , the energetic roughness Δ E , and the scale of landscape measured by the entropy S . We show that the dimensionless ratio between the gap, roughness, and entropy of the system accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated wi...

We propose a new theoretical approach to study the kinetics of the electron transfer (ET) under the dynamical influence of the complex environments with the first passage times (FPT) of the reaction events. By measuring the mean and high order moments of FPT and their ratios, the full kinetics of ET, especially the dynamical transitions across different temperature zones, is revealed. The potential applications of the current results to single molecule electron transfer are discussed.

There is considerable interest in the dynamic aspect of allosteric action, and in a growing list of proteins allostery has been characterized as being mediated predominantly by a change in dynamics, not a transition in conformation. For considering conformational dynamics, a protein molecule can be simplified into a number of relatively rigid microdomains connected by joints, corresponding to, e.g., communities and edges from a community network analysis. Binding of an allosteric activator strengthens intermicrodomain coupling, thereby quenching fast (e.g., picosecond to nanosecond) local motions but initiating slow (e.g., microsecond to millisecond), cross-microdomain correlated motions that are potentially of functional importance. This scenario explains allosteric effects observed in many unrelated proteins.

The energy landscape of proteins is thought to have an intricate, corrugated structure. Such roughness should have important consequences on the folding and binding kinetics of proteins, as well as on their equilibrium fluctuations. So far, no direct measurement of protein energy landscape roughness has been made. Here, we combined a recent theory with single-molecule dynamic force spectroscopy experiments to extract the overall energy scale of roughness e for a complex consisting of the small GTPase Ran and the nuclear transport receptor importin-b. The results gave e45k B T, indicating a bumpy energy surface, which is consistent with the ability of importin-b to accommodate multiple conformations and to interact with different, structurally distinct ligands.

In protein databases there is a substantial number of proteins structurally determined but without function annotation. Understanding the relationship between function and structure can be useful to predict function on a large scale. We have analyzed the similarities in global physicochemical parameters for a set of enzymes which were classified according to the four Enzyme Commission (EC) hierarchical levels. Using relevance theory we introduced a distance between proteins in the space of physicochemical characteristics. This was done by minimizing a cost function of the metric tensor built to reflect the EC classification system. Using an unsupervised clustering method on a set of 1025 enzymes, we obtained no relevant clustering formation compatible with EC classification. The distance distributions between enzymes from the same EC group and from different EC groups were compared by histograms. Such analysis was also performed using sequence alignment similarity as a distance. Our results suggest that global structure parameters are not sufficient to segregate enzymes according to EC hierarchy. This indicates that features essential for function are rather local than global. Consequently, methods for predicting function based on global attributes should not obtain high accuracy in main EC classes prediction without relying on similarities between enzymes from training and validation datasets. Furthermore, these results are consistent with a substantial number of studies suggesting that function evolves fundamentally by recruitment, i.e., a same protein motif or fold can be used to perform different enzymatic functions and a few specific amino acids (AAs) are actually responsible for enzyme activity. These essential amino acids should belong to active sites and an effective method for predicting function should be able to recognize them.

Dynamical connectivity graphs, which describe dynamical transition rates between local energy minima of a system, can be displayed against the background of a disconnectivity graph which represents the energy landscape of the system. The resulting supergraph describes both dynamics and statics of the system in a unified coarse-grained sense. We give examples of the supergraphs for several two dimensional spin and protein-related systems. We demonstrate that disordered ferromagnets have supergraphs akin to those of model proteins whereas spin glasses behave like random sequences of aminoacids which fold badly.

We introduce a nuclear magnetic resonance method for quantifying the shape of axially symmetric microscopic diffusion tensors in terms of a new diffusion anisotropy metric, DΔ, which has unique values for oblate, spherical, and prolate tensor shapes. The pulse sequence includes a series of equal-amplitude magnetic field gradient pulse pairs, the directions of which are tailored to give an axially symmetric diffusion-encoding tensor b with variable anisotropy bΔ. Averaging of data acquired for a range of orientations of the symmetry axis of the tensor b renders the method insensitive to the orientation distribution function of the microscopic diffusion tensors. Proof-of-principle experiments are performed on water in polydomain lyotropic liquid crystals with geometries that give rise to microscopic diffusion tensors with oblate, spherical, and prolate shapes. The method could be useful for characterizing the geometry of fluid-filled compartments in porous solids, soft matter, and bio...

associated concepts such as the thermodynamics, kinetics, free energy, enthalpy, entropy, and frustration/ruggedness of the landscape are also introduced. This chapter will facilitate not only the protein engineering studies aimed at modifying protein structure and function but also the rational drug design and understanding of life in the post-genomic era. 2. The problems of protein folding Protein folding is the process of transformation of one-dimensional linear information encoded in the amino acid sequence into a functional 3D structure. This is essentially a physical chemistry process for which the mechanism remains elusive due to the following three incompletely resolved problems (Dill et al., 2007): i) the thermodynamic question of how a native structure results from the inter-atomic forces acting on an amino acid sequence  the folding code; ii) the kinetic problem of how a native structure can fold so fast  the folding rate; iii) the computational problem of how to predict the native structure of a protein from its amino acid sequence  the protein structure prediction.

Proteins have evolved to use water to help guide folding. A physically motivated, nonpairwise-additive model of water-mediated interactions added to a protein structure prediction Hamiltonian yields marked improvement in the quality of structure prediction for larger proteins. Free energy profile analysis suggests that long-range water-mediated potentials guide folding and smooth the underlying folding funnel. Analyzing simulation trajectories gives direct evidence that water-mediated interactions facilitate native-like packing of supersecondary structural elements. Long-range pairing of hydrophilic groups is an integral part of protein architecture. Specific water-mediated interactions are a universal feature of biomolecular recognition landscapes in both folding and binding.

The energy landscape picture of protein folding and binding is employed to optimize a number of pair potentials for direct and water-mediated interactions in protein complex interfaces. We find that water-mediated interactions greatly complement direct interactions in discriminating against various types of trap interactions that model those present in the cell. We highlight the context dependent nature of knowledge-based binding potentials, as contrasted with the situation for autonomous folding. By performing a Principal Component Analysis (PCA) of the corresponding interaction matrixes, we rationalize the strength of the recognition signal for each combination of the contact type and reference trap states using the differential in the idealized "canonical" amino acid compositions of native and trap layers. The comparison of direct and water-mediated contact potential matrixes emphasizes the importance of partial solvation in stabilizing charged groups in the protein interfaces. Specific water-mediated interresidue interactions are expected to influence significantly the kinetics as well as thermodynamics of protein association.

While the energy landscape theory of protein folding is now a widely accepted view for understanding how relatively weak molecular interactions lead to rapid and cooperative protein folding, such a framework must be extended to describe the large-scale functional motions observed in molecular machines. In this review, we discuss (1) the development of the energy landscape theory of biomolecular folding, (2) recent advances toward establishing a consistent understanding of folding and function and (3) emerging themes in the functional motions of enzymes, biomolecular motors and other biomolecular machines. Recent theoretical, computational and experimental lines of investigation have provided a very dynamic picture of biomolecular motion. In contrast to earlier ideas, where molecular machines were thought to function similarly to macroscopic machines, with rigid components that move along a few degrees of freedom in a deterministic fashion, biomolecular complexes are only marginally stable. Since the stabilizing contribution of each atomic interaction is on the order of the thermal fluctuations in solution, the rigid body description of molecular function must be revisited. An emerging theme is that functional motions encompass order-disorder transitions and structural flexibility provides significant contributions to the free energy. In this review, we describe the biological importance of order-disorder transitions and discuss the statistical-mechanical foundation of theoretical approaches that can characterize such transitions.

The energy landscape of proteins is thought to have an intricate, corrugated structure. Such roughness should have important consequences on the folding and binding kinetics of proteins, as well as on their equilibrium fluctuations. So far, no direct measurement of protein energy landscape roughness has been made. Here, we combined a recent theory with single-molecule dynamic force spectroscopy experiments to extract the overall energy scale of roughness e for a complex consisting of the small GTPase Ran and the nuclear transport receptor importin-b. The results gave e45k B T, indicating a bumpy energy surface, which is consistent with the ability of importin-b to accommodate multiple conformations and to interact with different, structurally distinct ligands.

NMR spectroscopic analysis of the C-terminal Kunitz domain fragment ~a3~VI!! from the human a3-chain of type VI collagen has revealed that the side chain of Trp21 exists in two unequally populated conformations. The major conformation ~M! is identical to the conformation observed in the X-ray crystallographic structure, while the minor conformation ~m! cannot structurally be resolved in detail by NMR due to insufficient NOE data. In the present study, we have applied: ~1! rigid and adiabatic mapping, ~2! free energy simulations, and ~3! molecular dynamic simulations to elucidate the structure of the m conformer and to provide a possible pathway of the Trp21 side chain between the two conformers. Adiabatic energy mapping of conformations of the Trp21 side chain obtained by energy minimization identified two energy minima: One corresponding to the conformation of Trp21 observed in the X-ray crystallographic structure and solution structure of a3~VI! ~the M conformation! and the second corresponding to the m conformation predicted by NMR spectroscopy. A transition pathway between the M and m conformation is suggested. The free-energy difference between the two conformers obtained by the thermodynamic integration method is calculated to 1.77 6 0.7 kcal0mol in favor of the M form, which is in good agreement with NMR results. Structural and dynamic properties of the major and minor conformers of the a3~VI! molecule were investigated by molecular dynamic. Essential dynamics analysis of the two resulting 800 ps trajectories reveals that when going from the M to the m conformation only small, localized changes in the protein structure are induced. However, notable differences are observed in the mobility of the binding loop ~residues Thr13-Ile18!, which is more flexible in the m conformation than in the M conformation. This suggests that the reorientation of Trp21 might influence the inhibitory activity against trypsin, despite the relative large distance between the binding loop and Trp21.

NMR spectroscopic analysis of the C-terminal Kunitz domain fragment ~a3~VI!! from the human a3-chain of type VI collagen has revealed that the side chain of Trp21 exists in two unequally populated conformations. The major conformation ~M! is identical to the conformation observed in the X-ray crystallographic structure, while the minor conformation ~m! cannot structurally be resolved in detail by NMR due to insufficient NOE data. In the present study, we have applied: ~1! rigid and adiabatic mapping, ~2! free energy simulations, and ~3! molecular dynamic simulations to elucidate the structure of the m conformer and to provide a possible pathway of the Trp21 side chain between the two conformers. Adiabatic energy mapping of conformations of the Trp21 side chain obtained by energy minimization identified two energy minima: One corresponding to the conformation of Trp21 observed in the X-ray crystallographic structure and solution structure of a3~VI! ~the M conformation! and the second corresponding to the m conformation predicted by NMR spectroscopy. A transition pathway between the M and m conformation is suggested. The free-energy difference between the two conformers obtained by the thermodynamic integration method is calculated to 1.77 6 0.7 kcal0mol in favor of the M form, which is in good agreement with NMR results. Structural and dynamic properties of the major and minor conformers of the a3~VI! molecule were investigated by molecular dynamic. Essential dynamics analysis of the two resulting 800 ps trajectories reveals that when going from the M to the m conformation only small, localized changes in the protein structure are induced. However, notable differences are observed in the mobility of the binding loop ~residues Thr13-Ile18!, which is more flexible in the m conformation than in the M conformation. This suggests that the reorientation of Trp21 might influence the inhibitory activity against trypsin, despite the relative large distance between the binding loop and Trp21.

Protein function is the result of a complex yet precise relationship between protein structure and dynamics. The ability of a protein to assume different structural states is key to biomolecular recognition and function modulation. Protein modeling research is driven by the need to complement experimental techniques in obtaining a comprehensive and detailed characterization of protein equilibrium dynamics. This is a non-trivial task, as it requires mapping the structure space (and underlying energy landscape) available to a protein under physiological conditions. Existing algorithms invariably adopt a stochastic optimization approach to explore the non-linear and multimodal protein energy landscapes. At the present, such algorithms suffer from limited sampling, particularly in high-dimensional and non-linear variable spaces rich in local minima. In this paper, we equip a recently published evolutionary algorithm with novel evolutionary search strategies to enhance the sampling capability for mapping multibasin protein energy landscapes. We investigate initialization strategies to delay premature convergence and techniques to maintain and update on-the-fly a sample-based representation that serves as a map of the energy landscape. Applications on three proteins central to human disease show that the novel strategies are effective at locating basins in complex energy landscapes with a practical computational budget.

A lattice model of protein folding is developed to distinguish between amino acid sequences that do and do not fold into unique conformations. Although Monte Carlo simulations provide insights into the long-time processes involved in protein folding, these simulations cannot systematically chart the conformational energy surface that enables folding. By assuming that protein folding occurs after chain collapse, a kinetic map of important pathways on this surface is constructed through the use of an analytical theory of probability flow. Convergent kinetic pathways, or "folding funnels," guide folding to a unique, stable, native conformation. Solution of the probability flow equations is facilitated by limiting treatment to diffusion between geometrically similar collapsed conformers. Similarity is measured in terms of a reconfigurational distance. Two specific amino acid sequences are deemed foldable and nonfoldable because one gives rise to a single, large folding funnel leading to a native conformation and the other has multiple pathways leading to several stable conformers. Monte Carlo simulations demonstrate that folding funnel calculations accurately predict the fact of and the pathways involved in folding-specific sequences. The existence of folding funnels for specific sequences suggests that geometrically related families of stable, collapsed conformers fulfill kinetic and thermodynamic requirements of protein folding.

The catalytic conversion ATP + AMP → 2ADP by the enzyme adenylate kinase (ADK) involves the binding of one ATP molecule to the LID domain and one AMP molecule to the NMP domain. The latter is followed by a phosphate transfer and then the release of two ADP ...

NMR spectroscopy and computer simulations were used to examine changes in chemical shifts and in dynamics of the ribonuclease barnase that result upon binding to its natural inhibitor barstar. Although the spatial structures of free and bound barnase are very similar, binding results in changes of the dynamics of both fast side-chains, as revealed by 2 H relaxation measurements, and NMR chemical shifts in an extended β-sheet that is located far from the binding interface. Both side-chain dynamics and chemical shifts are sensitive to variations in the ensemble populations of the inter-converting molecular states, which can escape direct structural observation. Molecular dynamics simulations of free barnase and barnase in complex with barstar, as well as a normal mode analysis of barnase using a Gaussian network model, reveal relatively rigid domains that are separated by the extended β-sheet mentioned above. The observed changes in NMR parameters upon ligation can thus be rationalized in terms of changes in inter-domain dynamics and in populations of exchanging states, without measurable structural changes. This provides an alternative model for the propagation of a molecular response to ligand binding across a protein that is based exclusively on changes in dynamics.

The dynamics of biological polymers, including proteins, RNA, and DNA, occur in very highdimensional spaces. Many naturally-occurring polymers can navigate a vast phase space and rapidly find their lowest free energy (folded) state. Thus, although the search process is stochastic, it is not completely random. Instead, it is best described in terms of diffusion along a downhill free energy landscape. In this context, there have been many efforts to use simplified representations of the energetics, for which the potential energy is chosen to be a relatively smooth function with a global minima that corresponds to the folded state. That is, instead of including every type of physical interaction, the broad characteristics of the landscape are encoded in approximate energy functions. We describe a particular class of models, called structure-based models, that can be used to explore the diffusive properties of biomolecular folding and conformational rearrangements. These energy function...

Hemoglobin exhibits allosteric structural changes upon ligand binding due to the dynamic interactions between the ligand binding sites, the amino acids residues and some other solutes present under physiological conditions. In the present study, the dynamical and quaternary structural changes occurring in two unligated (deoxy-) T structures, and two fully ligated (oxy-) R, R2 structures of adult human hemoglobin were investigated with molecular dynamics. It is shown that, in the sub-microsecond time scale, there is no marked difference in the global dynamics of the amino acids residues in both the oxy-and the deoxy-forms of the individual structures. In addition, the R, R2 are relatively stable and do not present quaternary conformational changes within the time scale of our simulations while the T structure is dynamically more flexible and exhibited the T→R quaternary conformational transition, which is propagated by the relative rotation of the residues at the α 1 β 2 and α 2 β 1 interface.

The long-held views on lock-and-key versus induced fit in binding arose from the notion that a protein exists in a single, most stable conformation, dictated by its sequence. However, in solution proteins exist in a range of conformations, which may be described by statistical mechanical laws and their populations follow statistical distributions. Upon binding, the equilibrium will shift in favor of the bound conformation from the ensemble of conformations around the bottom of the folding funnel. Hence here we extend the implications and the usefulness of the folding funnel concept to explain fundamental binding mechanisms.

The contributions of conformational dynamics to substrate specificity have been examined by the application of principal component analysis to molecular dynamics trajectories of ␣-lytic protease. The wild-type ␣-lytic protease is highly specific for substrates with small hydrophobic side chains at the specificity pocket, while the Met190→Ala binding pocket mutant has a much broader specificity, actively hydrolyzing substrates ranging from Ala to Phe. Based on a combination of multiconformation analysis of cryo-X-ray crystallographic data, solution nuclear magnetic resonance (NMR), and normal mode calculations, we had hypothesized that the large alteration in specificity of the mutant enzyme is mainly attributable to changes in the dynamic movement of the two walls of the specificity pocket. To test this hypothesis, we performed a principal component analysis using 1-nanosecond molecular dynamics simulations using either a global or local solvent boundary condition. The results of this analysis strongly support our hypothesis and verify the results previously obtained by in vacuo normal mode analysis. We found that the walls of the wild-type substrate binding pocket move in tandem with one another, causing the pocket size to remain fixed so that only small substrates are recognized. In contrast, the M190A mutant shows uncoupled movement of the binding pocket walls, allowing the pocket to sample both smaller and larger sizes, which appears to be the cause of the observed broad specificity. The results suggest that the protein dynamics of ␣-lytic protease may play a significant role in defining the patterns of substrate specificity. As shown here, concerted local movements within proteins can be efficiently analyzed through a combination of principal component analysis and molecular dynamics trajectories using a local solvent boundary condition to reduce computational time and matrix size.

Insight into the dynamic properties of a-lytic protease (aLP) has been obtained through the use of low-temperature X-ray crystallography and multiple-conformation refinement. Previous studies of a L P have shown that the residues around the active site are able to move significantly to accommodate substrates of different sizes. Here we show a link between the ability to accommodate ligands and the dynamics of the binding pocket. Although the structure of aLP at 120 K has E-factors with a uniformly low value of 4.8 A* for the main chain, four regions stand out as having significantly higher E-factors. Because thermal motion should be suppressed at cryogenic temperatures, the high E-factors are interpreted as the result of trapped conformational substates. The active site residues that are perturbed during accommodation of different substrates are precisely those showing conformational substates, implying that substrate binding selects a subset of conformations from the ensemble of accessible states. To better characterize the precise nature of these substates, a protein model consisting of 16 structures has been refined and evaluated. The model reveals a number of features that could not be well-described by conventional B-factors: for example, 40% of the main-chain residue conformations are distributed asymmetrically or in discrete clusters. Furthermore, these data demonstrate an unexpected correlation between motions on either side of the binding pocket that we suggest is a consequence of "dynamic close packing." These results provide strong evidence for the role of protein dynamics in substrate binding and are consistent with the results of dynamic studies of ligand binding in myoglobin and ribonuclease A.

The minimum frustration principle is a computational approach which states that, in the long timescales of evolution, proteins’ freeenergy decreases more than expected by thermodynamical contraints as their aminoacids assume conformations progressively closer to the lowest energetic state. Here we show that this general principle, borrowed from protein folding dynamics, can be fruitfully applied to nervous function too. Highligting the foremost role of energetic requirements, macromolecular dynamics, and, above all, intertwined timescales in brain activity, the minimum frustration principle elucidates a wide range of mental processes, from sensations to memory retrieval. Brain functions are compared to trajectories which, in long nervous timescales, are attracted towards the lowenergy bottom of funnel-like structures characterized both by robustness and plasticity. We discuss how the principle, as derived explicitly from evolution and selection of a funneling structure from microdyn...

Protein function is the result of a complex yet precise relationship between protein structure and dynamics. The ability of a protein to assume different structural states is key to biomolecular recognition and function modulation. Protein modeling research is driven by the need to complement experimental techniques in obtaining a comprehensive and detailed characterization of protein equilibrium dynamics. This is a non-trivial task, as it requires mapping the structure space (and underlying energy landscape) available to a protein under physiological conditions. Existing algorithms invariably adopt a stochastic optimization approach to explore the non-linear and multimodal protein energy landscapes. At the present, such algorithms suffer from limited sampling, particularly in high-dimensional and non-linear variable spaces rich in local minima. In this paper, we equip a recently published evolutionary algorithm with novel evolutionary search strategies to enhance the sampling capability for mapping multibasin protein energy landscapes. We investigate initialization strategies to delay premature convergence and techniques to maintain and update on-the-fly a sample-based representation that serves as a map of the energy landscape. Applications on three proteins central to human disease show that the novel strategies are effective at locating basins in complex energy landscapes with a practical computational budget.

Allostery is a regulatory mechanism in proteins where an effector molecule binds distal from an active site to modulate its activity. Allosteric signaling may occur via a continuous path of residues linking the active and allosteric sites, which has been suggested by large conformational changes evident in crystal structures. An alternate possibility is that the signal occurs in the realm of ensemble dynamics via an energy landscape change. While the latter was first proposed on theoretical grounds, increasing evidence suggests that such a control mechanism is plausible. A major difficulty for testing the two methods is the ability to definitively determine that a residue is directly involved in allosteric signal transduction. Statistical Coupling Analysis (SCA) is a method that has been successful at predicting pathways, and experimental tests involving mutagenesis or domain substitution provide the best available evidence of signaling pathways. However, ascertaining energetic pathways which need not be contiguous is far more difficult. To date, simple estimates of the statistical significance of a pathway in a protein remain to be established. The focus of this work is to estimate such benchmarks for the statistical significance of contiguous pathways for the null model of selecting residues at random. We found that when 20% of residues in proteins are randomly selected, contiguous pathways at the 6 Å cutoff level were found with success rates of 51% in PDZ, 30% in p53, and 3% in MutS. The results suggest that the significance of pathways may have system specific factors involved. Furthermore, the possible existence of false positives for contiguous pathways implies that signaling could be occurring via alternate routes including those consistent with the energetic landscape model.