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PubMed2000

Efficient sampling in collective coordinate space.

Collective motions in biological macromolecules have been shown to be important for function. The most important collective motions occur on slow time scales, which poses a sampling problem in dynamic simulation of biomolecules. We present a novel method for efficient conformational sampling. The method combines the simulation of an ensemble of concurrent trajectories with restraints acting on the ensemble of structures as a whole. Two properties of the ensemble may be restrained: (i) the variance of the ensemble and (ii) the average position of the ensemble. Both properties are defined in a subspace of collective coordinate space spanned by an arbitrary number of modes. We show that weak restraints on the ensemble variance suffice for an increase in sampling efficiency along soft modes by two orders of magnitudes. The resulting trajectories exhibit virtually the same structural quality as trajectories generated by restraint-free-molecular dynamics simulation, as judged by standard structure validation tools. The method is used to probe the resistance of a structure against conformational changes along collective modes and clearly distinguishes soft from stiff modes. Further applications are discussed. Proteins 2000;39:82-88.

Computer Simulation↗

PubMed2005

Exploring the free-energy landscape of a short peptide using an average force.

The reversible folding of deca-alanine is chosen as a test case for characterizing a method that uses an adaptive biasing force (ABF) to escape from the minima and overcome the barriers of the free-energy landscape. This approach relies on the continuous estimation of a biasing force that yields a Hamiltonian in which no average force is exerted along the ordering parameter xi. Optimizing the parameters that control how the ABF is applied, the method is shown to be extremely effective when a nonequivocal ordering parameter can be defined to explore the folding pathway of the peptide. Starting from a beta-turn motif and restraining xi to a region of the conformational space that extends from the alpha-helical state to an ensemble of extended structures, the ABF scheme is successful in folding the peptide chain into a compact alpha helix. Sampling of this conformation is, however, marginal when the range of xi values embraces arrangements of greater compactness, hence demonstrating the inherent limitations of free-energy methods when ambiguous ordering parameters are utilized.

Alanine↗

PubMed2005

Dynamic receptor-based pharmacophore model development and its application in designing novel HIV-1 integrase inhibitors.

We present here a dynamic receptor-based pharmacophore model representing the complementary features of the active site region of HIV-1 integrase (IN), which was developed from a series of representative conformations of IN. Conformations of IN were sampled through a molecular dynamics study of the catalytic domain of an IN monomer, and an ensemble of representative IN structures were collected via a probability-based representative conformer sampling method that considers both the potential energy and the structural similarity of the protein conformations. The dynamic pharmacophore model was validated by a set of 128 known inhibitors, and the results showed that over 72% of the active inhibitors (IC(50) lower than 20 microM) could be successfully identified by the dynamic model. Therefore, we screened our in-house database of commercially available compounds against this model and successfully identified a set of structurally novel IN inhibitors. Compounds 7 and 18 with IC(50)s of 8 microM and 15 microM, respectively, against the strand transfer reaction were the most potent. Moreover, 7, 8 and 20 showed a 5-fold selectivity for the strand transfer reaction over 3'-processing.

Binding Sites↗

PubMed2005

On the influence of charged side chains on the folding-unfolding equilibrium of beta-peptides: a molecular dynamics simulation study.

The influence of charged side chains on the folding-unfolding equilibrium of beta-peptides was investigated by means of molecular dynamics simulations. Four different peptides containing only negatively charged side chains, positively charged side chains, both types of charged side chains (with the ability to form stabilizing salt bridges) or no charged side chains were studied under various conditions (different simulation temperatures, starting structures and solvent environment). The NMR solution structure in methanol of one of the peptides (A) has already been published; the synthesis and NMR analysis of another peptide (B) is described here. The other peptides (C and D) studied herein have hitherto not been synthesized. All four peptides A-D are expected to adopt a left-handed 3(14)-helix in solution as well as in the simulations. The resulting ensembles of structures were analyzed in terms of conformational space sampled by the peptides, folding behavior, structural properties such as hydrogen bonding, side chain-side chain and side chain-backbone interactions and in terms of the level of agreement with the NMR data available for two of the peptides. It was found that the presence of charged side chains significantly slows down the folding process in methanol solution due to the stabilization of intermediate conformers with side chain-backbone interactions. In water, where the solvent competes with the solute-solute polar interactions, the folding process to the 3(14)-helix is faster in the simulations.

Hydrogen Bonding↗

PubMed2000

pH dependence of stability of staphylococcal nuclease: evidence of substantial electrostatic interactions in the denatured state.

The pH dependence of stability of staphylococcal nuclease was studied with two independent equilibrium thermodynamic approaches. First, by measurement of stability in the pH range 9 to 3.5 by fluorescence-monitored denaturation with urea (Delta), GdnHCl (Delta), and heat (Delta). Second, by numerical integration of H(+) titration curves (Delta) measured potentiometrically under native (100 mM KCl) and unfolding (6.0 M GdnHCl) conditions. The pH dependence of stability described by Delta, Delta, and Delta was comparable but significantly different from the one described by Delta. The decrease in Delta between pH 9 and pH 4 was 4 kcal/mol greater than the decrease in Delta, Delta, and Delta in the same pH range. In 6 M GdnHCl, all the ionizable groups titrated with the pK(a) values of model compounds. Therefore, Delta represents the free energy difference between the native state (N) and an ensemble of unstructured, or expanded, and highly screened conformations. In contrast, the shallower pH dependence of stability described by Delta and by Delta between pH 9 and 5 was consistent with the titration of histidines with depressed, nativelike pK(a) values in the denatured state (D). These depressed pK(a) values likely reflect long-range electrostatic interactions with the other 29 basic groups and are a consequence of the compact character of the D state. The steep change in Delta and Delta at pH < 5 suggests that near pH 5 the structural and thermodynamic character of the D state shifts toward a state in which acidic residues titrate with normal pK(a) values, presumably because the electrostatic interactions with basic residues are lost, maybe as a consequence of an expansion.

Acids↗

PubMed2005

Reversible dimerization of acid-denatured ACBP controlled by helix A4.

The peptide segment corresponding to helix A4 in acyl-coenzyme-A-binding protein (ACBP) is an exceptionally stable helix in the denatured state of the protein as well as in its isolated form. Circular dichroism spectroscopy showed an alpha-helix content in the helix A4 peptide (HA4) of 45%, and under denaturing conditions at pH 2.3, helix conformations are still populated in 24% of the ensemble of molecules. The structure of HA4 at atomic resolution was assessed using nuclear magnetic resonance (NMR) spectroscopy. Long-range NOEs between remote residues at opposite peptide ends suggested the formation of an antiparallel homodimer, and the resulting structure was treated as the minimum higher-order structure. The dimerization property of helix A4 is maintained in the full-length protein under denaturing conditions. NMR diffusion studies and concentration-dependent experiments on ACBP at low pH proved the formation of dimers and revealed a cooperative stabilization of helix A4 in this process. This emphasizes its special role in the structure formation in the denatured state of ACBP. No dimers are formed in the presence of guanidine hydrochloride, which underlines the fundamental difference between the nature of these two denatured states.

Acids↗

PubMed2006

Single molecule Michaelis-Menten equation beyond quasistatic disorder.

The classic Michaelis-Menten equation describes the catalytic activities for ensembles of enzyme molecules very well. But recent single-molecule experiments showed that the waiting time distribution and other properties of single enzyme molecules were not consistent with the prediction based on the ensemble viewpoint. They have contributed to the slow conformational changes of a single enzyme in the catalytic processes. In this work, we study the general dynamics of single enzymes in the presence of dynamic disorder. We find that, within the time separation regimes, i.e., the slow reaction and nondiffusion limits, the Michaelis-Menten equation holds exactly. In particular, by employing the decoupling approximation we demonstrate analytically that the classic Michaelis-Menten equation is still an excellent approximation in the presence of general dynamic disorder.

Catalysis↗

PubMed2001

Brain function, nonlinear coupling, and neuronal transients.

The brain can be regarded as an ensemble of connected dynamical systems and as such conforms to some simple principles relating the inputs and outputs of its constituent parts. The ensuing implications, for the way we think about, and measure, neuronal interactions, can be quite profound. These range from 1) implications for which aspects of neuronal activity are important to measure and how to characterize coupling among neuronal populations; 2) implication for understanding the emergence of dynamic receptive fields and functionally specialized brain architectures; and 3) teleological implications pertaining to the genesis of dynamic instability and complexity, which is necessary for adaptive self-organization. This review focuses on the first set of implications by looking at neuronal interactions, coupling, and implicit neuronal codes from a dynamical perspective. By considering the brain in this light, one can show that a sufficient description of neuronal activity must comprise activity at the current time and its recent history. This history constitutes a neuronal transient. Such transients represent an essential metric of neuronal interactions and, implicitly, a code employed in the functional integration of brain systems. The nature of transients, expressed conjointly in different neuronal populations, reflects the underlying coupling among brain systems. A complete description of this coupling, or effective connectivity, can be expressed in terms of generalized convolution kernels (Volterra kernels) that embody high-order or nonlinear interactions. This coupling may be synchronous, and possibly oscillatory, or asynchronous. A critical distinction between synchronous and asynchronous coupling is that the former is essentially linear and the latter is nonlinear. The nonlinear nature of asynchronous coupling enables the rich, context-sensitive interactions that characterize real brain dynamics, suggesting that it plays an important role in functional integration.

Animals↗

PubMed1980

Correlations between internal mobility and stability of globular proteins.

The recent work is surveyed which leads to the suggestions that the conformation of globular proteins in solution corresponds to a dynamic ensemble of rapidly interconverting spatial structures, that clusters of hydrophobic amino acid side chains have an important role in the architecture of protein molecules, and that mechanistic aspects of protein denaturation can be correlated with internal mobility seen in the native conformation. These conclusions resulted originally from high resolution 1H nuclear magnetic resonance (NMR) studies of aromatic ring mobility, exchange of interior amide protons and thermal denaturation of the basic pancreatic trypsin inhibitor and a group of related proteins. Various new approaches to further characterize proteins in solution have now been taken and preliminary data are presented. These include computer graphics to outline hydrophobic clusters in globular protein structures, high resolution 1H-NMR experiments at variable hydrostatic pressure and 13C-NMR relaxation measurements. At the present early stage of these new investigations it appears that the hydrophobic cluster model for globular proteins is compatible with the data obtained.

Amino Acids↗

PubMed1998

Limited internal friction in the rate-limiting step of a two-state protein folding reaction.

Small, single-domain proteins typically fold via a compact transition-state ensemble in a process well fitted by a simple, two-state model. To characterize the rate-limiting conformational changes that underlie two-state folding, we have investigated experimentally the effects of changing solvent viscosity on the refolding of the IgG binding domain of protein L. In conjunction with numerical simulations, our results indicate that the rate-limiting conformational changes of the folding of this domain are strongly coupled to solvent viscosity and lack any significant "internal friction" arising from intrachain collisions. When compared with the previously determined solvent viscosity dependencies of other, more restricted conformational changes, our results suggest that the rate-limiting folding transition involves conformational fluctuations that displace considerable amounts of solvent. Reconciling evidence that the folding transition state ensemble is comprised of highly collapsed species with these and similar, previously reported results should provide a significant constraint for theoretical models of the folding process.

Bacterial Proteins↗

PubMed2002

Thermodynamics and folding kinetics analysis of the SH3 domain form discrete molecular dynamics.

We perform a detailed analysis of the thermodynamics and folding kinetics of the SH3 domain fold with discrete molecular dynamic simulations. We propose a protein model that reproduces some of the experimentally observed thermodynamic and folding kinetic properties of proteins. Specifically, we use our model to study the transition state ensemble of the SH3 fold family of proteins, a set of unstable conformations that fold to the protein native state with probability 1/2. We analyze the participation of each secondary structure element formed at the transition state ensemble. We also identify the folding nucleus of the SH3 fold and test extensively its importance for folding kinetics. We predict that a set of amino acid contacts between the RT-loop and the distal hairpin are the critical folding nucleus of the SH3 fold and propose a hypothesis that explains this result.

Amino Acid Sequence↗

PubMed2004

Mesoscopic analysis of conformational and entropic contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations.

Dynamic Monte Carlo simulations of short linear HP-type copolymers exhibiting proteinlike characteristics are used to investigate both chain dynamics and changes in chain conformational entropy and their contributions to the energetics of adsorption onto a solid-liquid interface. The dMC results show that the conformations and energies of adsorbed chains are highly degenerate. The ensemble-averaged energy of the adsorbed state is dependent on temperature, chain sequence, native-state stability, and sorbent surface geometry and hydrophobicity. Mesoscopic thermodynamic analyses reveal that, although increased chain conformational entropy contributes to the driving force for adsorption in certain cases, many conditions exist where the change in conformational entropy is either negligible or unfavorable due to constraints imposed by the need to form a large and specific number of favorable intra- and intermolecular contacts and by the impenetrable nature of the sorbent surface. Step-number-averaged energy trajectories, based on sampling of a large number of energy trajectories and thus conformational states at each step number, suggest that the search for a global energy minimum is gradual, so that adsorption is first reversible but becomes apparently irreversible with longer exposure to the sorbent. These results appear to be connected to the conformational adaptability of the chain both on the surface and in solution, and an adsorption model taking chain conformational dynamics into account is proposed.

Journal Article↗

PubMed2006

Understanding ensemble protein folding at atomic detail.

It has long been known that a protein's amino acid sequence dictates its native structure. However, despite significant recent advances, an ensemble description of how a protein achieves its native conformation from random coil under physiologically relevant conditions remains incomplete. Here we present a detailed all-atom model with a transferable potential that is capable of ab initio folding of entire protein domains using only sequence information. The computational efficiency of this model allows us to perform thousands of microsecond-time scale-folding simulations of the engrailed homeodomain and to observe thousands of complete independent folding events. We apply a graph-theoretic analysis to this massive data set to elucidate which intermediates and intermediary states are common to many trajectories and thus important for the folding process. This method provides an atomically detailed and complete picture of a folding pathway at the ensemble level. The approach that we describe is quite general and could be used to study the folding of proteins on time scales orders of magnitude longer than currently possible.

Algorithms↗

PubMed2006

Network and graph analyses of folding free energy surfaces.

Protein folding is governed by a complex free energy surface whose entropic contributions are relevant because of the large number of degrees of freedom involved. Such complexity, in particular the conformational heterogeneity of the denatured state, is hidden in projections onto one or two order parameters (e.g. fraction of native contacts and/or radius of gyration), which usually results in relatively smooth surfaces. Recent approaches borrowed from network and graph theory have yielded quantitative unprojected representations of the free energy surfaces of a beta-hairpin and a three-stranded beta-sheet peptide using equilibrium folding-unfolding molecular dynamics simulations. Interestingly, the network and graph analyses of these structured peptides have revealed a very heterogeneous denatured state ensemble. It includes high-enthalpy, high-entropy conformations with fluctuating non-native secondary structure, as well as low-enthalpy, low-entropy traps.

Computational Biology↗

PubMed2007

Ensemble docking of multiple protein structures: considering protein structural variations in molecular docking.

One approach to incorporate protein flexibility in molecular docking is the use of an ensemble consisting of multiple protein structures. Sequentially docking each ligand into a large number of protein structures is computationally too expensive to allow large-scale database screening. It is challenging to achieve a good balance between docking accuracy and computational efficiency. In this work, we have developed a fast, novel docking algorithm utilizing multiple protein structures, referred to as ensemble docking, to account for protein structural variations. The algorithm can simultaneously dock a ligand into an ensemble of protein structures and automatically select an optimal protein structure that best fits the ligand by optimizing both ligand coordinates and the conformational variable m, where m represents the m-th structure in the protein ensemble. The docking algorithm was validated on 10 protein ensembles containing 105 crystal structures and 87 ligands in terms of binding mode and energy score predictions. A success rate of 93% was obtained with the criterion of root-mean-square deviation <2.5 A if the top five orientations for each ligand were considered, comparable to that of sequential docking in which scores for individual docking are merged into one list by re-ranking, and significantly better than that of single rigid-receptor docking (75% on average). Similar trends were also observed in binding score predictions and enrichment tests of virtual database screening. The ensemble docking algorithm is computationally efficient, with a computational time comparable to that for docking a ligand into a single protein structure. In contrast, the computational time for the sequential docking method increases linearly with the number of protein structures in the ensemble. The algorithm was further evaluated using a more realistic ensemble in which the corresponding bound protein structures of inhibitors were excluded. The results show that ensemble docking successfully predicts the binding modes of the inhibitors, and discriminates the inhibitors from a set of noninhibitors with similar chemical properties. Although multiple experimental structures were used in the present work, our algorithm can be easily applied to multiple protein conformations generated by computational methods, and helps improve the efficiency of other existing multiple protein structure(MPS)-based methods to accommodate protein flexibility.

Algorithms↗

PubMed1996

Kinetics and thermodynamics of folding of a de novo designed four-helix bundle protein.

A simple continuum model of a de novo designed model of a four-helix bundle is presented. The thermodynamics and kinetics of the model are studied using Langevin simulations. We use a three-letter minimal off-lattice representation of a de novo designed four-helix bundle protein. The native state of the model, which can be thought of as an alpha-carbon representation of the peptide chain, is a caricature of the sequence designed by Ho and Degrado and shows several characteristics found in the naturally occurring four-helix bundles. These include the structural aspects and the relative stability of the native conformation. The model four-helix bundle shows two characteristic temperatures T theta and Tf. The former is the temperature above which the structure resembles that of the random coil. Below the first-order folding transition temperature Tf the chain adopts the native conformation corresponding to the four-helix bundle. It is shown that in order to obtain a unique native structure a proper free energy balance between secondary and tertiary interactions is needed. The thermal denaturation starting from the unique native conformation indicates that at least a three-state analysis is required. The intermediates in the equilibrium thermal denaturation are all found to be native-like. The kinetics of refolding starting from an ensemble of denatured states shows that the acquisition of the native conformation takes place via a kinetic partitioning mechanism. A fraction of molecules, phi, reaches the native state by a topology inducing nucleation collapse mechanism, while the remainder (1-phi) follows a complex three-stage multipathway process. We suggest, in accord with our earlier studies, that phi is essentially determined by the intrinsic temperature scales T theta and Tf. Our studies indicate that better design of proteins can be achieved by making T theta as close to Tf as possible. Experimental implications for de novo design of proteins are briefly discussed.

Kinetics↗

PubMed2000

A conformational study of the xyloglucan oligomer, XXXG, by NMR spectroscopy and molecular modeling.

A structural study of the XXXG xyloglucan heptasaccharide (X = alpha-D-Xylp(1 --> 6)-beta-D-Glcp and G = beta-D-Glcp) isolated from apple fruit has been undertaken with nmr and molecular mechanics methods. Quantitative 400 MHz nmr data including nuclear Overhauser effect spectroscopy (NOESY) volumes were recorded at both 6 and 20 degrees C. In spite of severe overlapping of resonances, it was possible to estimate summed NOEs for the majority of the anomeric and glucosyl methylene protons. An ensemble-average population of preferred geometries has been established with the CICADA conformational searching algorithm associated with the MM3 force field. Comparison of the theoretical data obtained by back-calculation of the NOESY volumes from the ensemble-average distance matrix program and motional models based on the Stokes-Einstein-Debye relation satisfactorily reproduce the experimental data. Conformational averaging about the mainchain glycosidic linkages includes both the syn and anti conformers and a minor gauche-gauche population is highly probable. The theoretical data overestimate the syn preference of the Glc(c) --> Glc(b) linkage as well as the Glc(c) GT rotamer population. Finally, both the motional models and the conformational search indicate a fairly rigid backbone and greater flexiblity for the xylose side chains.

Carbohydrate Conformation↗

PubMed1996

Hippocampal place fields: relationship between degree of field overlap and cross-correlations within ensembles of hippocampal neurons.

The capacity to record from multiple neurons in awake freely moving animals provides a means for characterizing organizational principles of place field encoding within ensembles of hippocampal neurons. In this study, cross-correlations between pairs of hippocampal place cells and degree of overlap between their respective place fields were analyzed during behavioral performance of delayed matching (DMS) or non-matching sample (DNMS) tasks, or while the same rats chased pellets in a different environment. The relationship between field overlap and cross-correlations of neural spike activity within ensembles was shown to be a positive, exponentially increasing, function. Place fields from the same neurons were markedly "remapped" between the Delay and Pellet-chasing tasks, with respect to physical location and size of fields. However individual pairs of place cells within each ensemble retained nearly the same degree of overlap and cross-correlation even though the spatial environment and the tasks differed markedly. This suggested that place cells were organized in functional "clusters" which exhibited the same inter-relations with respect to place field overlap and cross-correlations, irrespective of actual field of location. When cross-correlations between place cells were compared to placement of the array recording electrodes within the hippocampus, the strongest correlations were found along previously defined posterior-projecting fiber gradients between CA3 and CA1 subfields (Ishizuka et al. [1990], J Comp Neurol 295:580-623; Li et al. [1994] (J Comp Neurol 339:181-208). These findings suggest that the functional organization of place fields conforms to anatomical principles suspected to operate within hippocampal ensembles.

Animals↗