Metadynamics in essential coordinates: free energy simulation of conformational changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.     

Mapping the potential energy landscape of intrinsically disordered proteins at amino Acid resolution

Intrinsically disordered regions are predicted to exist in a significant fraction of proteins encoded in eukaryotic genomes. The high levels of conformational plasticity of this class of proteins endows them with unique capacities to act in functional modes not achievable by folded proteins, but also places their molecular characterization beyond the reach of classical structural biology. New techniques are therefore required to understand the relationship between primary sequence and biological function in this class of proteins. Although dependences of some NMR parameters such as chemical shifts (CSs) or residual dipolar couplings (RDCs) on structural propensity are known, so that sampling regimes are often inferred from experimental observation, there is currently no framework that allows for a statistical mapping of the available Ramachandran space of each amino acid in terms of conformational propensity. In this study we develop such an approach, combining highly efficient conformational sampling with ensemble selection to map the backbone conformational sampling of IDPs on a residue specific level. By systematically analyzing the ability of NMR data to map the conformational landscape of disordered proteins, we identify combinations of RDCs and CSs that can be used to raise conformational degeneracies inherent to different data types, and apply these approaches to characterize the conformational behavior of two intrinsically disordered proteins, the K18 domain from Tau protein and N(TAIL) from measles virus nucleoprotein. In both cases, we identify the enhanced populations of turn and helical regions in key regions of the proteins, as well as contiguous strands that show clear and enhanced polyproline II sampling.     

Hierarchical parallelization of divide-and-conquer density functional tight-binding molecular dynamics and metadynamics simulations

Massively parallel divide-and-conquer density functional tight-binding (DC-DFTB) molecular dynamics and metadynamics simulations are efficient approaches for describing various chemical reactions and dynamic processes of large complex systems via quantum mechanics. In this study, DC-DFTB simulations were combined with multi-replica techniques. Specifically, multiple walkers metadynamics, replica exchange molecular dynamics, and parallel tempering metadynamics methods were implemented hierarchically into the in-house Dcdftbmd program. Test simulations in an aqueous phase of the internal rotation of formamide and conformational changes of dialanine showed that the newly developed extensions increase the sampling efficiency and the exploration capabilities in DC-DFTB configuration space.     

Conformational analyses of peptides and proteins by vibrational Raman optical activity

Recent developments of vibrational Raman optical activity (ROA) spectroscopy enabled the detailed analyses of the backbone and side chain conformations of peptides and proteins in solution phases. ROA can be used as a powerful analytical technique for determining not only the structures of conformers, but also their populations even for systems in fast conformational equilibria where NMR spectroscopy is difficult to be applied. ROA enabled the monitoring of the secondary structures of denatured or unfolded proteins, such as an amyloid fibril and its prefibril intermediates.     

Application of time series analysis on molecular dynamics simulations of proteins: a study of different conformational spaces by principal component analysis

Time series analysis is applied on the collective coordinates obtained from principal component analysis of independent molecular dynamics simulations of alpha-amylase inhibitor tendamistat and immunity protein of colicin E7 based on the Calpha coordinates history. Even though the principal component directions obtained for each run are considerably different, the dynamics information obtained from these runs are surprisingly similar in terms of time series models and parameters. There are two main differences in the dynamics of the two proteins: the higher density of low frequencies and the larger step sizes for the interminima motions of colicin E7 than those of alpha-amylase inhibitor, which may be attributed to the higher number of residues of colicin E7 and/or the structural differences of the two proteins. The cumulative density function of the low frequencies in each run conforms to the expectations from the normal mode analysis. When different runs of alpha-amylase inhibitor are projected on the same set of eigenvectors, it is found that principal components obtained from a certain conformational region of a protein has a moderate explanation power in other conformational regions and the local minima are similar to a certain extent, while the height of the energy barriers in between the minima significantly change. As a final remark, time series analysis tools are further exploited in this study with the motive of explaining the equilibrium fluctuations of proteins.     

The free energy landscape of small peptides as obtained from metadynamics with umbrella sampling corrections

There is considerable interest in developing methodologies for the accurate evaluation of free energies, especially in the context of biomolecular simulations. Here, we report on a reexamination of the recently developed metadynamics method, which is explicitly designed to probe "rare events" and areas of phase space that are typically difficult to access with a molecular dynamics simulation. Specifically, we show that the accuracy of the free energy landscape calculated with the metadynamics method may be considerably improved when combined with umbrella sampling techniques. As test cases, we have studied the folding free energy landscape of two prototypical peptides: Ace-(Gly)(2)-Pro-(Gly)(3)-Nme in vacuo and trialanine solvated by both implicit and explicit water. The method has been implemented in the classical biomolecular code AMBER and is to be distributed in the next scheduled release of the code.     

From A to B in free energy space

The authors present a new method for searching low free energy paths in complex molecular systems at finite temperature. They introduce two variables that are able to describe the position of a point in configurational space relative to a preassigned path. With the help of these two variables the authors combine features of approaches such as metadynamics or umbrella sampling with those of path based methods. This allows global searches in the space of paths to be performed and a new variational principle for the determination of low free energy paths to be established. Contrary to metadynamics or umbrella sampling the path can be described by an arbitrary large number of variables, still the energy profile along the path can be calculated. The authors exemplify the method numerically by studying the conformational changes of alanine dipeptide.     

Using free energy perturbation to predict effects of changing force field parameters on computed conformational equilibriums of peptides

We propose a method that may allow data about the conformational equilibriums of peptides to enter the parameter calibration phase in force field developments. The method combines free energy perturbation with techniques for extensive sampling in the conformational space. It predicts shifts in computed conformational equilibriums in response to separate or combined perturbations of force field parameters. As an example we considered a force field associated with an implicit solvent model. We considered two different approaches to define conformational states of four peptides. One is based on reaction coordinates and two-dimensional free energy surfaces. The other is based on the clustering analysis of sampled conformations. Effects of perturbing various model parameters on the equilibriums between nativelike states with other conformational states were considered. For one type of perturbation predicted to have consistent effects on different peptides, the predictions have been verified by actual simulations using a perturbed model.     

Implementation and application of helix-helix distance and crossing angle restraint potentials

Based on the definition of helix-helix distance and crossing angle introduced by Chothia et al. (J Mol Biol 1981, 145, 215), we have developed the restraint potentials by which the distance and crossing angle of two selected helices can be maintained around target values during molecular dynamics simulations. A series of assessments show that calculated restraint forces are numerically accurate. Since the restraint forces are only exerted on atoms which define the helical principal axes, each helix can rotate along its helical axis, depending on the helix-helix intermolecular interactions. Such a restraint potential enables us to characterize the helix-helix interactions at atomic details by sampling their conformational space around specific distance and crossing angle with (restraint) force-dependent fluctuations. Its efficacy is illustrated by calculating the potential of mean force as a function of helix-helix distance between two transmembrane helical peptides in an implicit membrane model.     

Quantum mechanical calculation of the effects of stiff and rigid constraints in the conformational equilibrium of the alanine dipeptide

If constraints are imposed on a macromolecule, two inequivalent classical models may be used: the stiff and the rigid one. This work studies the effects of such constraints on the conformational equilibrium distribution (CED) of the model dipeptide HCO-L-Ala-NH(2)without any simplifying assumption. We use ab initio quantum mechanics calculations including electron correlation at the MP2 level to describe the system, and we measure the conformational dependence of all the correcting terms to the naive CED based in the potential energy surface that appear when the constraints are considered. These terms are related to mass-metric tensors determinants and also occur in the Fixman's compensating potential. We show that some of the corrections are non-negligible if one is interested in the whole Ramachandran space. On the other hand, if only the energetically lower region, containing the principal secondary structure elements, is assumed to be relevant, then, all correcting terms may be neglected up to peptides of considerable length. This is the first time, as far as we know, that the analysis of the conformational dependence of these correcting terms is performed in a relevant biomolecule with a realistic potential energy function.     

The conformational free energy landscape of beta-D-glucopyranose. Implications for substrate preactivation in beta-glucoside hydrolases

Using ab initio metadynamics we have computed the conformational free energy landscape of beta-D-glucopyranose as a function of the puckering coordinates. We show that the correspondence between the free energy and the Stoddard's pseudorotational itinerary for the system is rather poor. The number of free energy minima (9) is smaller than the number of ideal structures (13). Moreover, only six minima correspond to a canonical conformation. The structural features, the electronic properties, and the relative stability of the predicted conformers permit the rationalization of the occurrence of distorted sugar conformations in all the available X-ray structures of beta-glucoside hydrolase Michaelis complexes. We show that these enzymes recognize the most stable distorted conformers of the isolated substrate and at the same time the ones better prepared for catalysis in terms of bond elongation/shrinking and charge distribution. This suggests that the factors governing the distortions present in these complexes are largely dictated by the intrinsic properties of a single glucose unit.     

Using Selectively Applied Accelerated Molecular Dynamics to Enhance Free Energy Calculations

Accelerated molecular dynamics (aMD) has been shown to enhance conformational space sampling relative to classical molecular dynamics; however, the exponential reweighting of aMD trajectories, which is necessary for the calculation of free energies relating to the classical system, is oftentimes problematic, especially for systems larger than small poly peptides. Here, we propose a method of accelerating only the degrees of freedom most pertinent to sampling, thereby reducing the total acceleration added to the system and improving the convergence of calculated ensemble averages, which we term selective aMD. Its application is highlighted in two biomolecular cases. First, the model system alanine dipeptide is simulated with classical MD, all-dihedral aMD, and selective aMD, and these results are compared to the infinite sampling limit as calculated with metadynamics. We show that both forms of aMD enhance the convergence of the underlying free energy landscape by 5-fold relative to classical MD; however, selective aMD can produce improved statistics over all-dihedral aMD due to the improved reweighting. Then we focus on the pharmaceutically relevant case of computing the free energy of the decoupling of oseltamivir in the active site of neuraminidase. Results show that selective aMD greatly reduces the cost of this alchemical free energy transformation, whereas all-dihedral aMD produces unreliable free energy estimates.     

Conformational dynamics of ATP∕Mg:ATP in motor proteins via data mining and molecular simulation

The conformational diversity of ATP∕Mg:ATP in motor proteins was investigated using molecular dynamics and data mining. Adenosine triphosphate (ATP) conformations were found to be constrained mostly by inter cavity motifs in the motor proteins. It is demonstrated that ATP favors extended conformations in the tight pockets of motor proteins such as F(1)-ATPase and actin whereas compact structures are favored in motor proteins such as RNA polymerase and DNA helicase. The incorporation of Mg(2+) leads to increased flexibility of ATP molecules. The differences in the conformational dynamics of ATP∕Mg:ATP in various motor proteins was quantified by the radius of gyration. The relationship between the simulation results and those obtained by data mining of motor proteins available in the protein data bank is analyzed. The data mining analysis of motor proteins supports the conformational diversity of the phosphate group of ATP obtained computationally.     

Dissociation of carbonic acid: gas phase energetics and mechanism from ab initio metadynamics simulations

A comprehensive metadynamics study of the energetics, stability, conformational changes, and mechanism of dissociation of gas phase carbonic acid, H2CO3, yields significant new insight into these reactions. The equilibrium geometries, vibrational frequencies, and conformer energies calculated using the density functional theory are in good agreement with the previous theoretical predictions. At 315 K, the cis-cis conformer has a very short life time and transforms easily to the cis-trans conformer through a change in the O=C-O-H dihedral angle. The energy difference between the trans-trans and cis-trans conformers is very small (approximately 1 kcal/mol), but the trans-trans conformer is resistant to dissociation to carbon dioxide and water. The cis-trans conformer has a relatively short path for one of its hydroxyl groups to accept the proton from the other end of the molecule, resulting in a lower activation barrier for dissociation. Comparison of the free and potential energies of dissociation shows that the entropic contribution to the dissociation energy is less than 10%. The potential energy barrier for dissociation of H2CO3 to CO2 and H2O from the metadynamics calculations is 5-6 kcal/mol lower than in previous 0 K studies, possibly due to a combination of a finite temperature and more efficient sampling of the energy landscape in the metadynamics calculations. Gas phase carbonic acid dissociation is triggered by the dehydroxylation of one of the hydroxyl groups, which reorients as it approaches the proton on the other end of the molecule, thus facilitating a favorable H-O-H angle for the formation of a product H2O molecule. The major atomic reorganization of the other part of the molecule is a gradual straightening of the O=C=O bond. The metadynamics results provide a basis for future simulation of the more challenging carbonic acid-water system.     

Dihedral angle principal component analysis of molecular dynamics simulations

It has recently been suggested by Mu et al. [Proteins 58, 45 (2005)] to use backbone dihedral angles instead of Cartesian coordinates in a principal component analysis of molecular dynamics simulations. Dihedral angles may be advantageous because internal coordinates naturally provide a correct separation of internal and overall motion, which was found to be essential for the construction and interpretation of the free energy landscape of a biomolecule undergoing large structural rearrangements. To account for the circular statistics of angular variables, a transformation from the space of dihedral angles {phi(n)} to the metric coordinate space {x(n)=cos phi(n),y(n)=sin phi(n)} was employed. To study the validity and the applicability of the approach, in this work the theoretical foundations underlying the dihedral angle principal component analysis (dPCA) are discussed. It is shown that the dPCA amounts to a one-to-one representation of the original angle distribution and that its principal components can readily be characterized by the corresponding conformational changes of the peptide. Furthermore, a complex version of the dPCA is introduced, in which N angular variables naturally lead to N eigenvalues and eigenvectors. Applying the methodology to the construction of the free energy landscape of decaalanine from a 300 ns molecular dynamics simulation, a critical comparison of the various methods is given.     

Using intrinsic X-ray absorption spectral differences to identify and map peptides and proteins

The intrinsic variation in the near-edge X-ray absorption fine structure (NEXAFS) spectra of peptides and proteins provide an opportunity to identify and map them in various biological environments, without additional labeling. In principle, with sufficiently accurate spectra, peptides (<50 amino acids) or proteins with unusual sequences (e.g., cysteine- or methionine-rich) should be differentiable from other proteins, since the NEXAFS spectrum of each amino acid is distinct. To evaluate the potential for this approach, we have developed X-SpecSim, a tool for quantitatively predicting the C, N, and O 1s NEXAFS spectra of peptides and proteins from their sequences. Here we present the methodology for predicting such spectra, along with tests of its precision using comparisons to the spectra of various proteins and peptides. The C 1s, N 1s, and O 1s spectra of two novel antimicrobial peptides, Indolicidin (ILPWKWPWWPWRR-NH2) and Sub6 (RWWKIWVIRWWR-NH2), as well as human serum albumin and fibrinogen are reported and interpreted. The ability to identify, differentiate, and quantitatively map an antimicrobial peptide against a background of protein is demonstrated by a scanning transmission X-ray microscopy study of a mixture of albumin and sub6.