Spinor product computations for protein conformations

Spinor operators in geometric algebra (GA) can efficiently describe conformational changes of proteins by ordered products that act on individual bonds and represent their net rotations. Backward propagation through the protein backbone yields all rotational spinor axes in advance allowing the efficient computation of atomic coordinates from internal coordinates. The introduced mathematical framework enables to efficiently manipulate and generate protein conformations to any arbitrary degree. Moreover, several new formulations in the context of rigid body motions are added. Emphasis is placed on the intimate relationship between spinors and quaternions, which can be recovered from within the GA approach. The spinor methodology is implemented and tested versus the state of the art algorithms for both protein construction and coordinate updating. Spinor calculations have a smaller computational cost and turn out to be slightly faster than current alternatives. © 2012 Wiley Periodicals, Inc.     

Internal coordinate density of state from molecular dynamics simulation

The vibrational density of states (DoS), calculated from the Fourier transform of the velocity autocorrelation function, provides profound information regarding the structure and dynamic behavior of a system. However, it is often difficult to identify the exact vibrational mode associated with a specific frequency if the DoS is determined based on velocities in Cartesian coordinates. Here, the DoS is determined based on velocities in internal coordinates, calculated from Cartesian atomic velocities using a generalized Wilson's B ‐matrix. The DoS in internal coordinates allows for the correct detection of free dihedral rotations that may be mistaken as hindered rotation in Cartesian DoS. Furthermore, the pronounced enhancement of low frequency modes in Cartesian DoS for macromolecules should be attributed to the coupling of dihedral and angle motions. The internal DoS, thus deconvolutes the internal motions and provides fruitful insights to the dynamic behaviors of a system. © 2015 Wiley Periodicals, Inc.     

Ferreting out correlations from trajectory data

Thermally driven materials characterized by complex energy landscapes, such as proteins, exhibit motions on a broad range of space and time scales. Principal component analysis (PCA) is often used to extract modes of motion from protein trajectory data that correspond to coherent, functional motions. In this work, two other methods, maximum covariance analysis (MCA) and canonical correlation analysis (CCA) are formulated in a way appropriate to analyze protein trajectory data. Both methods partition the coordinates used to describe the system into two sets (two measurement domains) and inquire as to the correlations that may exist between them. MCA and CCA provide rotations of the original coordinate system that successively maximize the covariance (MCA) or correlation (CCA) between modes of each measurement domain under suitable constraint conditions. We provide a common framework based on the singular value decomposition of appropriate matrices to derive MCA and CCA. The differences between and strengths and weaknesses of MCA and CCA are discussed and illustrated. The application presented here examines the correlation between the backbone and side chain of the peptide met-enkephalin as it fluctuates between open conformations, found in solution, to closed conformations appropriate to when it is bound to its receptor. Difficulties with PCA carried out in Cartesian coordinates are found and motivate a formulation in terms of dihedral angles for the backbone atoms and selected atom distances for the side chains. These internal coordinates are a more reliable basis for all the methods explored here. MCA uncovers a correlation between combinations of several backbone dihedral angles and selected side chain atom distances of met-enkephalin. It could be used to suggest residues and dihedral angles to focus on to favor specific side chain conformers. These methods could be applied to proteins with domains that, when they rearrange upon ligand binding, may have correlated functional motions or, for multi-subunit proteins, may exhibit correlated subunit motions.     

Open science grid study of the coupling between conformation and water content in the interior of a protein

Computational grids are a promising resource for modeling complex biochemical processes such as protein folding, penetration of gases or water into proteins, or protein structural rearrangements coupled to ligand binding. We have enabled the molecular dynamics program CHARMM to run on the Open Science Grid. The implementation is general, flexible, easily modifiable for use with other molecular dynamics programs and other grids and automated in terms of job submission, monitoring, and resubmission. The usefulness of grid computing was demonstrated through the study of hydration of the Glu-66 side chain in the interior of protein staphylococcal nuclease. Multiple simulations started with and without two internal water molecules shown crystallographically to be associated with the side chain of Glu-66 yielded two distinct populations of rotameric states of Glu-66 that differed by as much as 20%. This illustrates how internal water molecules can bias protein conformations. Furthermore, there appeared to be a temporal correlation between dehydration of the side chain and conformational transitions of Glu-66. This example demonstrated how difficult it is to get convergence even in the relatively simple case of a side chain oscillating between two conformations. With grid computing, we also benchmarked the self-guided Langevin dynamics method against the Langevin dynamics method traditionally used for temperature control in molecular dynamics simulations and showed that the two methods yield comparable results.     

Local chain dynamics of a model polycarbonate near glass transition temperature: A molecular dynamics simulation

Constant pressure constant temperature molecular dynamics method is employed to investigate the atomistic scale dynamics of a model Bisphenol A polycarbonate in the vicinity of its glass transition temperature. First, the glass transition temperature and the thermal expansion coefficients of the polymer are predicted by performing simulations at different temperatures. To explore the significance of different modes of motion, various types of time correlation functions are utilized in analyzing the trajectories. In these nanosecond scale simulations, the motion of the chain segments is found to be highly localized with little reorientation of the vectors representing these segments. Detailed analysis of trajectories and the correlation functions of the backbone dihedrals and side methyl groups indicates that they exhibit numerous conformational transitions. The activation energies of the conformational transitions obtained from the simulation are generally larger than the potential barriers for the rotations of these dihedrals, however, both show the same trend. We also have estimated the phenylene ring flip activation energy as 12.6 kcal/mol and the flip frequency as 0.77 MHz at 300 K. These values fall either fall within the range determined by various NMR spectroscopy experiments or slightly out of the range. The study shows that the conformational transitions between the adjacent dihedrals are strongly correlated. Three basic cooperative modes are identified from the simulation. They are: a positive synchronous rotation of two phenylene rings, a negative synchronous rotation of two phenylene rings, and a carbonate group rotation. Above the glass transition temperature, the large scale cooperative motions become much more significant.     

The Lagrange multiplier method for manipulating geometries. Implementation and applications in molecular mechanics

It is shown that a Lagrange multiplier method to constrain one or several internal coordinates, or averages and combinations of these, is easily implemented in a molecular mechanics computer program that uses Newton–Raphson (NR ) minimization. Results are given for constraints on nonbonded distances and torsion angles. When a potential energy surface is to be explored, it is much better to constrain the average of three torsion angles around a bond than to constrain a single torsion angle. Certain conversions can only be achieved when averages of torsion angles around different bonds are constrained. Combinations of constraints have been applied to evaluate differences between calculated and observed geometries and to obtain transition states for relatively large molecules from results for smaller molecules at relatively low costs. The efficieny of the combination of the Lagrange multiplier method and NR minimization in terms of computing time can be rated as good.     

Rovibrational molecular hamiltonian in mixed bond-angle and umbrella-like coordinates

A new exact quantum mechanical rovibrational Hamiltonian operator for molecules exhibiting large amplitude inversion and torsion motions is derived. The derivation is based on a division of a molecule into two parts: a frame and a top. The nuclei of the frame only are used to construct a molecular system of axes. The inversion motion of the frame is described in the umbrella-like coordinates, whereas the torsion motion of the top is described by the nonstandard torsion angle defined in terms of the nuclear vectors and one of the molecular axes. The internal coordinates chosen take into account the properties of the inversion and torsion motions. Vibrational s and rotational Omega vectors obtained for the introduced internal coordinates determine the rovibrational tensor G defined by simple scalar products of these vectors. The Jacobian of the transformation from the Cartesian to the internal coordinates considered and the G tensor specify the rovibrational Hamiltonian. As a result, the Hamiltonian for penta-atomic molecules like NH2OH with one inverter is presented and a complete set of the formulas necessary to write down the Hamiltonian of more complex molecules, like NH2NH2 with two inverters, is reported. The approach considered is essentially general and sufficiently simple, as demonstrated by derivation of a polyatomic molecule Hamiltonian in polyspherical coordinates, obtained by other methods with much greater efforts.     

How to detect internal motion by homonuclear NMR

NOESY and ROESY cross-peak intensities depend on internuclear distances and internal motion. Internal motion is usually ignored, and NOESY cross-peak intensities are interpreted in terms of internuclear distances only. Off-resonance ROESY experiments measure a weighted average of NOE and ROE. The weight can be described and experimentally set by an angle theta;. For large enough molecules, NOE and ROE have opposite signs. Therefore, each cross-peak intensity becomes zero for an angle theta;(0). For any sample, the maximum angle theta;(0) is determined by the overall motion of the molecule. Smaller theta;(0) values reflect the angular component of internal motions. Because individual cross-peaks are analyzed, the method evaluates internal motions of individual H,H vectors. The reduction of theta;(0) is largest for internal motions on a time scale of 100-300 ps. The sensitivity of theta;(0) for internal motions decreases with increasing molecular weight. We estimate that detecting internal motions will be practicable for molecules up to about 15 kDa. We describe a protocol to measure theta;(0) from a series of off-resonance ROESY spectra. For such a series, we describe the choice of experimental parameters, a procedure to extract theta;(0) from the raw data, and the interpretation of theta;(0) in terms of internal motions. In the small protein BPTI, we analyzed 75 cross-peaks. The precision of theta;(0) was 0.25 degrees, as compared to typical reductions of theta;(0) of 3 degrees. We found a well-defined maximum theta;(0) for cross-peaks in rigid parts of the molecule, which reflects the overall motion of the molecule. For BPTI, also many structurally important long-range cross-peaks appear rigid. The lower theta;(0) values of long-range contacts involving methyl groups are consistent with methyl rotation on the 25-ps time scale. The lower theta;(0) values of the flexible C-terminus and of flexible side chains translate into upper limits for the angular order parameter of 0.4 and 0.5-0.8, respectively. Off-resonance ROESY can monitor internal motions of H,H contacts that are used in a structure calculation. Because no isotope labeling is needed, off-resonance ROESY can be used to detect internal motions in a wide range of natural products.     

Metadynamics in the conformational space nonlinearly dimensionally reduced by Isomap

Atomic motions in molecules are not linear. This infers that nonlinear dimensionality reduction methods can outperform linear ones in analysis of collective atomic motions. In addition, nonlinear collective motions can be used as potentially efficient guides for biased simulation techniques. Here we present a simulation with a bias potential acting in the directions of collective motions determined by a nonlinear dimensionality reduction method. Ad hoc generated conformations of trans,trans-1,2,4-trifluorocyclooctane were analyzed by Isomap method to map these 72-dimensional coordinates to three dimensions, as described by Brown and co-workers [J. Chem. Phys. 129, 064118 (2008)]. Metadynamics employing the three-dimensional embeddings as collective variables was applied to explore all relevant conformations of the studied system and to calculate its conformational free energy surface. The method sampled all relevant conformations (boat, boat-chair, and crown) and corresponding transition structures inaccessible by an unbiased simulation. This scheme allows to use essentially any parameter of the system as a collective variable in biased simulations. Moreover, the scheme we used for mapping out-of-sample conformations from the 72D to 3D space can be used as a general purpose mapping for dimensionality reduction, beyond the context of molecular modeling.     

A normal mode-based geometric simulation approach for exploring biologically relevant conformational transitions in proteins

A three-step approach for multiscale modeling of protein conformational changes is presented that incorporates information about preferred directions of protein motions into a geometric simulation algorithm. The first two steps are based on a rigid cluster normal-mode analysis (RCNMA). Low-frequency normal modes are used in the third step (NMSim) to extend the recently introduced idea of constrained geometric simulations of diffusive motions in proteins by biasing backbone motions of the protein, whereas side-chain motions are biased toward favorable rotamer states. The generated structures are iteratively corrected regarding steric clashes and stereochemical constraint violations. The approach allows performing three simulation types: unbiased exploration of conformational space; pathway generation by a targeted simulation; and radius of gyration-guided simulation. When applied to a data set of proteins with experimentally observed conformational changes, conformational variabilities are reproduced very well for 4 out of 5 proteins that show domain motions, with correlation coefficients r > 0.70 and as high as r = 0.92 in the case of adenylate kinase. In 7 out of 8 cases, NMSim simulations starting from unbound structures are able to sample conformations that are similar (root-mean-square deviation = 1.0-3.1 ?) to ligand bound conformations. An NMSim generated pathway of conformational change of adenylate kinase correctly describes the sequence of domain closing. The NMSim approach is a computationally efficient alternative to molecular dynamics simulations for conformational sampling of proteins. The generated conformations and pathways of conformational transitions can serve as input to docking approaches or as starting points for more sophisticated sampling techniques.     

Application of geometric algebra for the description of polymer conformations

In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.     

Importance of correlated motions on the low barrier rotational potentials of crystalline molecular gyroscopes

The energetic and structural changes taking place upon rotation of the central phenylene of 1,4-bis(3,3,3-triphenylpropynyl)benzene in the solid state were computed using molecular mechanics calculations. Pseudopolymorphic crystals of a benzene clathrate (1A) and a desolvated form (1B) were analyzed with models that account for varying degrees of freedom within the corresponding lattices. The calculated rotational barriers in a rigid lattice approximation, 78 kcal/mol for 1A and 72 kcal/mol for 1B, are about 5 times greater than those previously measured by variable-temperature 13C CPMAS NMR and quadrupolar echo 2H NMR line-shape analysis: 12.8 kcal/mol for 1A and 14.6 kcal/mol for 1B. The potential energy barriers calculated with a model that restricts whole body rotation and translational motions but allows for internal rotations give results that are near the experimental free-energy barriers. The calculated barriers for 1A and 1B are 15.5 and 16.2 kcal/mol, respectively. The differences between the rigid and partially relaxed models are attributed to the effect of correlated motions of the lattice and the rotating group, which are evident from the structural analysis of the atomic position data as a function of the dihedral angle of the rotator. The displacements of neighboring molecules near the rotary transition states for 1A and 1B can be as large as 2.7 and 1.1 A, respectively. The displacement and oscillation (C2) of interpenetrating phenyl rings from neighboring rotors proximal to the event are significant for both 1A and 1B. In addition, 6-fold (C6) benzene rotations in clathrate 1A were found to be directly correlated to the rotation of the phenylene rotator.     

Two-dimensional NMR methods for studying solid polymers

A survey is given about two-dimensional (2D) NMR experiments on solid polymers involving ²H- and ¹³C-NMR. 2D exchange NMR spectra of static samples directly reflect the distribution of rotational angles resulting from ultraslow molecular motions. Typical examples are the chain motion above the glass transition or rotations around a helix axis in semi-crystalline polymers. 2D-Magic angle spinning not only allows the detection of molecular order and motion. By combining rotor synchronized MAS with rotations in spin space the correlation of order and mobility can be studied.     

Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis

A systematic approach to construct a low-dimensional free energy landscape from a classical molecular dynamics (MD) simulation is presented. The approach is based on the recently proposed dihedral angle principal component analysis (dPCA), which avoids artifacts due to the mixing of internal and overall motions in Cartesian coordinates and circumvents problems associated with the circularity of angular variables. Requiring that the energy landscape reproduces the correct number, energy, and location of the system's metastable states and barriers, the dimensionality of the free energy landscape (i.e., the number of essential components) is obtained. This dimensionality can be determined from the distribution and autocorrelation of the principal components. By performing an 800 ns MD simulation of the folding of hepta-alanine in explicit water and using geometric and kinetic clustering techniques, it is shown that a five-dimensional dPCA energy landscape is a suitable and accurate representation of the full-dimensional landscape. In the second step, the dPCA energy landscape can be employed (e.g., in a Langevin simulation) to facilitate a detailed investigation of biomolecular dynamics in low dimensions. Finally, several ways to visualize the multidimensional energy landscape are discussed.     

Interpreting protein structural dynamics from NMR chemical shifts

In this investigation, semiempirical NMR chemical shift prediction methods are used to evaluate the dynamically averaged values of backbone chemical shifts obtained from unbiased molecular dynamics (MD) simulations of proteins. MD-averaged chemical shift predictions generally improve agreement with experimental values when compared to predictions made from static X-ray structures. Improved chemical shift predictions result from population-weighted sampling of multiple conformational states and from sampling smaller fluctuations within conformational basins. Improved chemical shift predictions also result from discrete changes to conformations observed in X-ray structures, which may result from crystal contacts, and are not always reflective of conformational dynamics in solution. Chemical shifts are sensitive reporters of fluctuations in backbone and side chain torsional angles, and averaged (1)H chemical shifts are particularly sensitive reporters of fluctuations in aromatic ring positions and geometries of hydrogen bonds. In addition, poor predictions of MD-averaged chemical shifts can identify spurious conformations and motions observed in MD simulations that may result from force field deficiencies or insufficient sampling and can also suggest subsets of conformational space that are more consistent with experimental data. These results suggest that the analysis of dynamically averaged NMR chemical shifts from MD simulations can serve as a powerful approach for characterizing protein motions in atomistic detail.     

Least constraint approach to the extraction of internal motions from molecular dynamics trajectories of flexible macromolecules

We propose a rigorous method for removing rigid-body motions from a given molecular dynamics trajectory of a flexible macromolecule. The method becomes exact in the limit of an infinitesimally small sampling step for the input trajectory. In a recent paper [G. Kneller, J. Chem. Phys. 128, 194101 (2008)], one of us showed that virtual internal atomic displacements for small time increments can be derived from Gauss' principle of least constraint, which leads to a rotational superposition problem for the atomic coordinates in two consecutive time frames of the input trajectory. Here, we demonstrate that the accumulation of these displacements in a molecular-fixed frame, which evolves in time according to the virtual rigid-body motions, leads to the desired trajectory for internal motions. The atomic coordinates in the input and output trajectory are related by a roto-translation, which guarantees that the internal energy of the molecule is left invariant. We present a convenient implementation of our method, in which the accumulation of the internal displacements is performed implicitly. Two numerical examples illustrate the difference to the classical approach for removing macromolecular rigid-body motions, which consists of aligning its configurations in the input trajectory with a fixed reference structure.     

Static atomistic modelling of the structure and ring dynamics of bulk amorphous polystyrene

A detailed static atomistic model of dense, glassy polystyrene is simulated using a well established technique that previously proved successful for simple vinyl polymers. Initial chain conformations that are generated using a Monte Carlo technique including periodic continuation conditions are “relaxed” by potential energy minimization. In total 24 microstructures at densities of 1,07 g/cm³ were obtained with a cube-edge length of 18,65 Å. Detailed analysis of the minimized structures indicates that intermolecular packing influences create a large variety of chain conformations different from the purely intramolecular ground states. The systems are amorphous, exhibiting random coil behavior. The described structures have been used for a quasistatic simulation of localized motions. These motions include stepwise rotation and oscillation of the phenyl groups. The frequency distribution for the simulated ring motions covers many orders of magnitude. It is very rare that an energy barrier with a reorientation angle indicating a ring “flip” is overcome. Motions with small reorientation of the phenyl rings, and therefore not leading to a ring “flip”, dominate with an average reorientation angle of 16° (±12°). The intermolecular effects of the analyzed processes were found very important and far-reaching, widely influencing the cooperative motions of molecular groups.     

CAMERRA: An analysis tool for the computation of conformational dynamics by evaluating residue–residue associations

A computational method which extracts the dominant motions from an ensemble of biomolecular conformations via a correlation analysis of residue–residue contacts is presented. The algorithm first renders the structural information into contact matrices, then constructs the collective modes based on the correlated dynamics of a selected set of dynamic contacts. Associated programs can bridge the results for further visualization using graphics software. The aim of this method is to provide an analysis of conformations of biopolymers from the contact viewpoint. It may assist a systematical uncovering of conformational switching mechanisms existing in proteins and biopolymer systems in general by statistical analysis of simulation snapshots. In contrast to conventional correlation analyses of Cartesian coordinates (such as distance covariance analysis and Cartesian principal component analysis), this program also provides an alternative way to locate essential collective motions in general. Herein, we detail the algorithm in a stepwise manner and comment on the importance of the method as applied to decoding allosteric mechanisms. © 2018 Wiley Periodicals, Inc.