Filtering molecular dynamics trajectories to reveal low-frequency collective motions: phospholipase A2

A novel method for analysing molecular dynamics trajectories has been developed, which filters out high frequencies using digital signal processing techniques and facilitates focusing on the low-frequency collective motions of proteins. These motions involve low energy slow motions, which lead to important biological phenomena such as domain closure and allosteric effects in enzymes. The filtering method treats each of the atomic trajectories obtained from the molecular dynamics simulation as a "signal". The trajectories of each of the atoms in the system (or any subset of interest) are Fourier transformed to the frequency domain, a filtering function is applied and then an inverse transformation back to the time domain yields the filtered trajectory. The filtering method has been used to study the dynamics of the enzyme phospholipase A2. In the filtered trajectory, all the high frequency bond and valence angle vibrations were eliminated, leaving only low-frequency motion, mainly fluctuations in torsions and conformational transitions. Analysis of this trajectory revealed interesting motions of the protein, including concerted movements of helices, and changes in shape of the active site cavity. Unlike normal mode analysis, which has been used to study the motion of proteins, this method does not require converged minimizations or diagonalization of a matrix of second derivatives. In addition, anharmonicity, multiple minima and conformational transitions are treated explicitly. Thus, the filtering method avoids most of the approximations implicit in other investigations of the dynamic behaviour of large systems.     

Essential domain motions in barnase revealed by MD simulations

The wealth of data accumulated on the bacterial ribonuclease barnase is complemented by molecular dynamics trajectories starting from four different experimental structures and covering a total of >10 ns. Using principal component analysis, the simulations are interpreted in view of dynamic domains and hinges promoting relative motions of these domains. Two domains with residues 7-22 and 52-108 for the first domain and residues 25-51 for the second domain were consistently observed. Hinge regions consist primarily of Tyr24, Ser50, Ile51, and Gly52. Earlier mutation studies have demonstrated that the residues of the hinge regions play essential roles for the stability and activity of barnase. The domain motions are correlated to inter-domain interactions involving functionally important active site residues, such as Lys27 and Glu73. A model is presented that combines the observation of dynamic domains and their motions with the extensive mutation data from the literature. Enthalpic energy contributions originating from specific inter-domain interactions as well as entropic energy contributions due to the domain motions are discussed in the frame of this model and compared with destabilization energies measured for corresponding mutants.     

Comprehensive analysis of motions in molecular dynamics trajectories of the actin capping protein and its inhibitor complexes

The actin capping protein (CP) binds to actin filaments to block further elongation. The capping activity is inhibited by proteins V‐1 and CARMIL interacting with CP via steric and allosteric mechanisms, respectively. The crystal structures of free CP, CP/V‐1, and CP/CARMIL complexes suggest that the binding of CARMIL alters the flexibility of CP rather than the overall structure of CP, and this is an allosteric inhibition mechanism. Here, we performed molecular dynamics (MD) simulations of CP in the free form, and in complex with CARMIL or V‐1. The resulting trajectories were analyzed exhaustively using Motion Tree, which identifies various rigid‐body motions ranging from small local motions to large domain motions. After enumerating all the motions, CP flexibilities with different ligands were characterized by a list of frequencies for 20 dominant rigid‐body motions, some of which were not identified in previous studies. The comparative analysis highlights the influence of the binding of the CARMIL peptide to CP flexibility. In free CP and the CP/V‐1 complex, domain motions around a large crevice between the N‐stalk and the CP‐S domain occur frequently. The CARMIL peptide binds the crevice and suppresses the motions effectively. In addition, the binding of the CARMIL peptide enhances and alters local motions around the pocket that participates in V‐1 binding. These newly identified motions are likely to suppress the binding of V‐1 to CP. The observed changes in CP motion provide insights that describe the mechanism of allosteric regulation by CARMIL through modulating CP flexibility. Proteins 2016; 84:948–956. © 2016 Wiley Periodicals, Inc.     

Projection of Monte Carlo and molecular dynamics trajectories onto the normal mode axes: human lysozyme

A method is presented to describe the internal motions of proteins obtained from molecular dynamics or Monte Carlo simulations as motions of normal mode variables. This method calculates normal mode variables by projecting trajectories of these simulations onto the axes of normal modes and expresses the trajectories as a linear combination of normal mode variables. This method is applied to the result of the molecular dynamics and the Monte Carlo simulations of human lysozyme. The motion of the lowest frequency mode extracted from the simulations represents the hinge bending motion very faithfully. Analysis of the obtained motions of the normal mode variables provides an explanation of the anharmonic aspects of protein dynamics as due first to the anharmonicity of the actual potential energy surface near a minimum and second to trans-minimum conformational changes.     

Generalized correlation for biomolecular dynamics

Correlated motions in biomolecules are often essential for their function, e.g., allosteric signal transduction or mechanical/thermodynamic energy transport. Because correlated motions in biomolecules remain difficult to access experimentally, molecular dynamics (MD) simulations are particular useful for their analysis. The established method to quantify correlations from MD simulations via calculation of the covariance matrix, however, is restricted to linear correlations and therefore misses part of the correlations in the atomic fluctuations. Herein, we propose a general statistical mechanics approach to detect and quantify any correlated motion from MD trajectories. This generalized correlation measure is contrasted with correlations obtained from covariance matrices for the B1 domain of protein G and T4 lysozyme. The new method successfully quantifies correlations and provides a valuable global overview over the functionally relevant collective motions of lysozyme. In particular, correlated motions of helix 1 together with the two main lobes of lysozyme are detected, which are not seen by the conventional covariance matrix. Overall, the established method misses more than 50% of the correlation. This failure is attributed to both, an interfering and unnecessary dependence on mutual orientations of the atomic fluctuations and, to a lesser extent, attributed to nonlinear correlations. Our generalized correlation measure overcomes these problems and, moreover, allows for an improved understanding of the conformational dynamics by separating linear and nonlinear contributions of the correlation.     

Effects of Disease Causing Mutations on the Essential Motions in Proteins

Abstract

This study embodies a detailed comparative analysis of the essential motions of the Wild type and the eight different disease mutant forms of the Human CYPlbl. The mutations considered in this study have been implicated in Primary Congenital Glaucoma, an in-born, genetic disorder associated with eye-abnormality. The principal component analysis for Wild type and the Mutants was carried out using the stabilized molecular dynamics trajectories, which ranged from 35 to 45 nanoseconds. Investigations revealed the nature of the collective motions that characterize functionally relevant ‘essential motions’. The essential motions in Wild type are characterized by the collective motions of the Substrate Access Channel including the β-rich domain and the loops in the region of p450-reductase interaction. Comparative analysis of the essential motions of the Wild type and Mutants, especially those involving the functionally important regions indicated distinct differences in their magnitudes as well as the residue-wise distribution. The Mutants in general are associated with higher root mean square fluctuations, and involve some of the relatively intact core regions of the protein, in large collective motions. This study sheds light on the possible effects of disease causing mutations on the large functionally important collective motions in proteins.     

A comparison of structural and dynamic properties of different simulation methods applied to SH3.

The dynamic and static properties of molecular dynamics simulations using various methods for treating solvent were compared. The SH3 protein domain was chosen as a test case because of its small size and high surface-to-volume ratio. The simulations were analyzed in structural terms by examining crystal packing, distribution of polar residues, and conservation of secondary structure. In addition, the "essential dynamics" method was applied to compare each of the molecular dynamics trajectories with a full solvent simulation. This method proved to be a powerful tool for the comparison of large concerted atomic motions in SH3. It identified methods of simulation that yielded significantly different dynamic properties compared to the full solvent simulation. Simulating SH3 using the stochastic dynamics algorithm with a vacuum (reduced charge) force field produced properties close to those of the full solvent simulation. The application of a recently described solvation term did not improve the dynamic properties. The large concerted atomic motions in the full solvent simulation as revealed by the essential dynamics method were analyzed for possible biological implications. Two loops, which have been shown to be involved in ligand binding, were seen to move in concert to open and close the ligand-binding site.     

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.     

Analysis of transient behavior in complex trajectories: application to secretory vesicle dynamics

Analysis of trajectories of dynamical biological objects, such as breeding ants or cell organelles, is essential to reveal the interactions they develop with their environments. Many previous works used a global characterization based on parameters calculated for entire trajectories. In cases where transient behavior was detected, this usually concerned only a particular type, such as confinement or directed motion. However, these approaches are not appropriate in situations in which the tracked objects may display many different types of transient motion. We have developed a method to exhaustively analyze different kinds of transient behavior that the tracked objects may exhibit. The method discriminates stalled periods, constrained and directed motions from random dynamics by evaluating the diffusion coefficient, the mean-square displacement curvature, and the trajectory asymmetry along individual trajectories. To detect transient motions of various durations, these parameters are calculated along trajectories using a rolling analysis window whose width is variable. The method was applied to the study of secretory vesicle dynamics in the subplasmalemmal region of human carcinoid BON cells. Analysis of transitions between transient motion periods, combined with plausible assumptions about the origin of each motion type, leads to a model of dynamical subplasmalemmal organization.     

Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular dynamics and normal mode simulations.

A method is described for identifying collective motions in proteins from molecular dynamics trajectories or normal mode simulations. The method makes use of the covariances of atomic positional fluctuations. It is illustrated by an analysis of the bovine pancreatic trypsin inhibitor. Comparison of the covariance and cross-correlation matrices shows that the relative motions have many similar features in the different simulations. Many regions of the protein, especially regions of secondary structure, move in a correlated manner. Anharmonic effects, which are included in the molecular dynamics simulations but not in the normal analysis, are of some importance in determining the larger scale collective motions, but not the more local fluctuations. Comparisons of molecular dynamics simulations in the present and absence of solvent indicate that the environment is of significance for the long-range motions.     

Evidence of protein collective motions on the picosecond timescale

We investigate the presence of structural collective motions on a picosecond timescale for the heme protein, cytochrome c, as a function of oxidation and hydration, using terahertz (THz) time domain spectroscopy and molecular dynamics simulations. The THz response dramatically increases with oxidation, with the largest increase for lowest hydrations, and highest frequencies. For both oxidation states the THz response rapidly increases with hydration saturating above ∼25% (g H₂O/g protein). Quasiharmonic vibrational modes and dipole-dipole correlation functions were calculated from molecular dynamics trajectories. The collective mode density of states alone reproduces the measured hydration dependence, providing strong evidence of the existence of these motions. The large oxidation dependence is reproduced only by the dipole-dipole correlation function, indicating the contrast arises from diffusive motions consistent with structural changes occurring in the vicinity of buried internal water molecules. This source for the observed oxidation dependence is consistent with the lack of an oxidation dependence in nuclear resonant vibrational spectroscopy measurements.     

Conformational dynamics of a multidomain protein by neutron scattering and computational analysis

The flexible conformations of a multidomain protein are responsible for its biological functions. Although MurD, a 47-kDa protein that consists of three domains, sequentially changes its domain conformation from an open form to a closed form through a semiclosed form in its enzymatic reaction, the domain dynamics in each conformation remains unclear. In this study, we verify the conformational dynamics of MurD in the corresponding three states (apo and ATP- and inhibitor-bound states) with a combination of small-angle x-ray and neutron scattering (SAXS and SANS), dynamic light scattering (DLS), neutron backscattering (NBS), neutron spin echo (NSE) spectroscopy, and molecular dynamics (MD) simulations. Applying principal component analysis of the MD trajectories, twisting and open-closed domain modes are identified as the major collective coordinates. The deviations of the experimental SAXS profiles from the theoretical calculations based on the known crystal structures become smaller in the ATP-bound state than in the apo state, and a further decrease is evident upon inhibitor binding. These results suggest that domain motions of the protein are suppressed step by step of each ligand binding. The DLS and NBS data yield collective and self-translational diffusion constants, respectively, and we used them to extract collective domain motions in nanometer and nanosecond scales from the NSE data. In the apo state, MurD shows both twisting and open-closed domain modes, whereas an ATP binding suppresses twisting domain motions, and a further reduction of open-closed mode is seen in the inhibitor-binding state. These observations are consistent with the structure modifications measured by the small-angle scattering as well as the MD simulations. Such changes in the domain dynamics associated with the sequential enzymatic reactions should be related to the affinity and reaction efficiency with a ligand that binds specifically to each reaction state.     

A principal component analysis of the dynamics of subdomains and binding sites in human serum albumin

The conformational dynamics of human serum albumin (HSA) was investigated by principal component analysis (PCA) applied to three molecular dynamics trajectories of 200 ns each. The overlap of the essential subspaces spanned by the first 10 principal components (PC) of different trajectories was about 0.3 showing that the PCA based on a trajectory length of 200 ns is not completely convergent for this protein. The contributions of the relative motion of subdomains and of the subdomains (internal) distortion to the first 10 PCs were found to be comparable. Based on the distribution of the first 3 PC, 10 protein conformers are identified showing relative root mean square deviations (RMSD) between 2.3 and 4.6 Å. The main PCs are found to be delocalized over the whole protein structure indicating that the motions of different protein subdomains are coupled. This coupling is considered as being related to the allosteric effects observed upon ligand binding to HSA. On the other hand, the first PC of one of the three trajectories describes a conformational transition of the protein domain I that is close to that experimentally observed upon myristate binding. This is a theoretical support for the older hypothesis stating that changes of the protein onformation favorable to binding can precede the ligand complexation. A detailed all atoms PCA performed on the primary Sites 1 and 2 confirms the multiconformational character of the HSA binding sites as well as the significant coupling of their motions. © 2013 Wiley Periodicals, Inc. Biopolymers 101: 561–572, 2014.     

Characteristic domain motion in the ribosome recycling factor revealed by 15N NMR relaxation experiments and molecular dynamics simulations

The backbone dynamics of ribosome recycling factor (RRF) from Escherichia coli in water were characterized by (15)N NMR relaxation analysis and molecular dynamics (MD) simulation. RRF is composed of two domains connected by a joint region that consists of two peptide chains, such that the overall structure seems to mimic that of tRNA. MD trajectories indicated that the relative orientation of domains varies on the nanosecond time scale. We analyzed the observed (15)N T(1), T(2), and NOE using an extended model-free spectral density function in which the domain motions with a nanosecond time scale were considered. At 30 degrees C, the order parameters of slow motion () were determined to be approximately 0.9 for domain I and 0.7 for domain II, respectively. These values indicate that domain I is nearly fixed on the molecular diffusion frame, and domain II is wobbling in a cone for which the semi-angle is about 30 degrees.     

Two-arm trajectory planning in a manipulation task

This paper addresses the trajectory planning problem for a task which requires positioning and orienting an object firmly grasped by two hands at a visually specified goal configuration in the horizontal plane. The motor task involves three degrees of freedom (two translational and one rotational), and the motions of the arms are constrained by the physical coupling through the held object. Experimentally measured trajectories of two arms in the coordinated positioning/orienting task are presented. The hypothesis that the rotational and translational components of motions are decoupled and independently planned is tested. Two explicit mathematical models to account for the kinematic features of the two-arm motions are formulated, and the predictions of the models are compared with the experimental data. Both models extend the minimum-jerk model to the two-arm coordinated motions case. The trajectories predicted by the models were found to be in qualitative agreement with the experimental data. However, neither model could account for the observed configuration dependence of the motions, nor for some of the properties of the measured velocity components of the motions. Our findings support the idea that the rotational and translational components of two-arm motions in the positioning/orienting task are independently planned in extra-personal space, and are further combined in a hierarchical fashion to produce the observed motions. The tested models may serve as a basis for further investigations of issues pertinent to the generation of two-arm trajectories. Received: 27 March 1995 / Accepted in revised form: 17 September 1996     

Functionally relevant motions of haloalkane dehalogenases occur in the specificity-modulating cap domains

One-nanosecond molecular dynamics trajectories of three haloalkane dehalogenases (DhlA, LinB, and DhaA) are compared. The main domain was rigid in all three dehalogenases, whereas the substrate specificity-modulating cap domains showed considerably higher mobility. The functionally relevant motions were spread over the entire cap domain in DhlA, whereas they were more localized in LinB and DhaA. The highest amplitude of essential motions of DhlA was noted in the alpha4'-helix-loop-alpha4-helix region, formerly proposed to participate in the large conformation change needed for product release. The highest amplitude of essential motions of LinB and DhaA was observed in the random coil before helix 4, linking two domains of these proteins. This flexibility is the consequence of the modular composition of haloalkane dehalogenases. Two members of the catalytic triad, that is, the nucleophile and the base, showed a very high level of rigidity in all three dehalogenases. This rigidity is essential for their function. One of the halide-stabilizing residues, important for the catalysis, shows significantly higher flexibility in DhlA compared with LinB and DhaA. Enhanced flexibility may be required for destabilization of the electrostatic interactions during the release of the halide ion from the deeply buried active site of DhlA. The exchange of water molecules between the enzyme active site and bulk solvent was very different among the three dehalogenases. The differences could be related to the flexibility of the cap domains and to the number of entrance tunnels.     

Picosecond timescale rigid-helix and side-chain motions in deoxymyoglobin

The contribution of rigidbody motions to the atomic trajectories in a 100 ps molecular dynamics simulation of deoxymyoglobin is examined. Two typesof rigid-body motions are considered: one in which the helices are rigid units and one in which the side-chains are rigid units. Using a quaternionbased algorithm, fits of the rigid reference structures are made to each time frame of the simulation to derive trajectories of the rigid-body motions. The fitted trajectories are analysed in terms of atomic position fluctuations, mean-square displacements as a function of time, velocity autocorrelation functions and densities of states. The results are compared with the corresponding quantities calculated from the full trajectory. The relative contribution of the rigid helix motions to the helix atom dynamics depends on which quantity is examined and on which subset of atoms is chosen: rigid-helix motions contribute 86% of the rms helix backbone atomic position fluctuations, but 30% of the helix,: atom (backbone and side-chain) mean square displacements and only 1.1% of total kinetic energy. Only very low-frequency motions contribute to the rigid-helix dynamics; the rigid-body analysis allows characteristic rigid-helix vibrations to be identified and described. Treating the side-chains as rigid bodies is foundto be an excellent approximation to both their diffusive and vibrationalmean-square displacements: 96% of side-chain atom mean-square displacements originate from rigid side-Chain motions. However, the errors in theside-chain atomic positional fits are not always small. An analysis is madeof factors contributing to the positional error for different types of side-chain. © Wiley-Liss, Inc.     

Principal component analysis for protein folding dynamics

Protein folding is considered here by studying the dynamics of the folding of the triple β-strand WW domain from the Formin-binding protein 28. Starting from the unfolded state and ending either in the native or nonnative conformational states, trajectories are generated with the coarse-grained united residue (UNRES) force field. The effectiveness of principal components analysis (PCA), an already established mathematical technique for finding global, correlated motions in atomic simulations of proteins, is evaluated here for coarse-grained trajectories. The problems related to PCA and their solutions are discussed. The folding and nonfolding of proteins are examined with free-energy landscapes. Detailed analyses of many folding and nonfolding trajectories at different temperatures show that PCA is very efficient for characterizing the general folding and nonfolding features of proteins. It is shown that the first principal component captures and describes in detail the dynamics of a system. Anomalous diffusion in the folding/nonfolding dynamics is examined by the mean-square displacement (MSD) and the fractional diffusion and fractional kinetic equations. The collisionless (or ballistic) behavior of a polypeptide undergoing Brownian motion along the first few principal components is accounted for.     

Essential spaces defined by NMR structure ensembles and molecular dynamics simulation show significant overlap

Large concerted motions of proteins which span its “essential space,” are an important component of protein dynamics. We investigate to what extent structure ensembles generated with standard structure calculation techniques such as simulated annealing can capture these motions by comparing them to long-time molecular dynamics (MD) trajectories. The motions are analyzed by principal component analysis and compared using inner products of eigenvectors of the respective covariance matrices. Two very different systems are studied, the β-spectrin PH domain and the single-stranded DNA binding protein (ssDBP) from the filamentous phage Pf3. A comparison of the ensembles from NMR and MD shows significant overlap of the essential spaces, which in the case of ssDBP is extraordinarily high. The influence of variations in the specifications of distance restraints is investigated. We also study the influence of the selection criterion for the final structure ensemble on the definition of mobility. The results suggest a modified criterion that improves conformational sampling in terms of amplitudes of correlated motion. Proteins 31:370–382, 1998. © 1998 Wiley-Liss, Inc.     

The STATIS method: Characterization of conformational states of flexible molecules from molecular dynamics simulations in solution

STATIS, a data analysis method used when data can be expressed as matrices, seems particularly well suited to characterize the internal molecular motions and conformational states extracted from MD trajectories. We first outline this method and the “adapted STATIS” method. Applications are presented for 18-crown-6 (simulated for 2 nsec in acetonitrile solution) and for the (L30)₂Cu⁺ catenate (stimulated for 150 psec in chloroform). STATIS should be valuable for the classification of molecular conformations and simplified visualization of MD trajectories.