Dihedral Angle Space Research Articles

EncoderMap(II): Visualizing Important Molecular Motions with Improved Generation of Protein Conformations.

Dimensionality reduction can be used to project high-dimensional molecular data into a simplified, low-dimensional map. One feature of our recently introduced dimensionality reduction technique EncoderMap, which relies on the combination of an autoencoder with multidimensional scaling, is its ability to do the reverse. It is able to generate conformations for any selected points in the low-dimensional map. This transfers the simplified, low-dimensional map back into the high-dimensional conformational space. Although the output is again high-dimensional, certain aspects of the simplification are preserved. The generated conformations only mirror the most dominant conformational differences that determine the positions of conformational states in the low-dimensional map. This allows depicting such differences and-in consequence-visualizing molecular motions and gives a unique perspective on high-dimensional conformational data. In our previous work, protein conformations described in backbone dihedral angle space were used as the input for EncoderMap, and conformations were also generated in this space. For large proteins, however, the generation of conformations is inaccurate with this approach due to the local character of backbone dihedral angles. Here, we present an improved variant of EncoderMap which is able to generate large protein conformations that are accurate in short-range and long-range orders. This is achieved by differentiable reconstruction of Cartesian coordinates from the generated dihedrals, which allows adding a contribution to the cost function that monitors the accuracy of all pairwise distances between the Cα-atoms of the generated conformations. The improved capabilities to generate conformations of large, even multidomain, proteins are demonstrated for two examples: diubiquitin and a part of the Ssa1 Hsp70 yeast chaperone. We show that the improved variant of EncoderMap can nicely visualize motions of protein domains relative to each other but is also able to highlight important conformational changes within the individual domains.

Free-Energy Landscapes for Macromolecules that Form a Unique 3D Structure

Abstract—The entropic effects for the energy landscapes of macromolecules are considered. We use the ideas and methods of multidimensional geometry and topology, expansion into a multidimensional Fourier series for a potential-energy surface, and the Gaussian distribution function for the expansion coefficients to describe the interaction between conformational degrees of freedom to investigate the free energy surface in the space of dihedral (torsion) angles. We show that the free-energy surface for a macromolecule that forms a unique 3D structure in the framework of the proposed approach has the following features. The main properties of the free-energy surface can be represented in terms of a two-dimensional surface, which depends on two generalized variables. At low temperatures, the multidimensional space of torsion angles can be divided into region that belongs to the deepest central energy funnel and the region of satellite funnels. The funnels are separated by relatively high energy barriers, whose height decreases as the representative point moves up further away from the bottom of the funnel. The depths of satellite funnels decrease while the distances from the central funnel becomes larger. The funnels are surrounded by smooth entropic barriers. The height of entropic barrier is at a minimum for the central funnel, while the entropic barriers heights of satellite funnels increase as funnels move further away from the central funnel. As the temperature increases, the satellite funnels are destroyed, starting from the most distant central one. When the temperature is higher than the critical point (proportional to the specific gain in the potential energy during the formation of a unique 3D structure) the compact state becomes unstable and the global minimum of free energy disappears. As the number of interacting torsion angles (the length of the polymer chain) increases, the critical temperature becomes higher.

A Generative Angular Model of Protein Structure Evolution.

Recently described stochastic models of protein evolution have demonstrated that the inclusion of structural information in addition to amino acid sequences leads to a more reliable estimation of evolutionary parameters. We present a generative, evolutionary model of protein structure and sequence that is valid on a local length scale. The model concerns the local dependencies between sequence and structure evolution in a pair of homologous proteins. The evolutionary trajectory between the two structures in the protein pair is treated as a random walk in dihedral angle space, which is modeled using a novel angular diffusion process on the two-dimensional torus. Coupling sequence and structure evolution in our model allows for modeling both “smooth” conformational changes and “catastrophic” conformational jumps, conditioned on the amino acid changes. The model has interpretable parameters and is comparatively more realistic than previous stochastic models, providing new insights into the relationship between sequence and structure evolution. For example, using the trained model we were able to identify an apparent sequence–structure evolutionary motif present in a large number of homologous protein pairs. The generative nature of our model enables us to evaluate its validity and its ability to simulate aspects of protein evolution conditioned on an amino acid sequence, a related amino acid sequence, a related structure or any combination thereof.

Direct Observation of the Intrinsic Backbone Torsional Mobility of Disordered Proteins

The fundamental backbone dynamics of unfolded proteins arising due to intrinsic ϕ-ψ dihedral angle fluctuations dictate the course of protein folding, binding, assembly, and function. These internal fluctuations are also critical for protein misfolding associated with a range of human diseases. However, direct observation and unambiguous assignment of this inherent dynamics in chemically denatured proteins is extremely challenging due to various experimental limitations. To directly map the backbone torsional mobility in the ϕ-ψ dihedral angle space, we used a model intrinsically disordered protein, namely, α-synuclein, that adopts an expanded state under native conditions. We took advantage of nonoccurrence of tryptophan in α-synuclein and created a number of single-tryptophan variants encompassing the entire polypeptide chain. We then utilized highly sensitive picosecond time-resolved fluorescence depolarization measurements that allowed us to discern the site-specific torsional relaxation at a low protein concentration under physiological conditions. For all the locations, the depolarization kinetics exhibited two well-separated rotational-correlation-time components. The shorter, subnanosecond component arises due to the local mobility of the indole side chain, whereas the longer rotational-correlation-time component (1.37 ± 0.15 ns), independent of global tumbling, represents a characteristic timescale for short-range conformational exchange in the ϕ-ψ dihedral space. This correlation time represents an intrinsic timescale for torsional relaxation and is independent of position, which is expected for an extended polypeptide chain having little or no propensity to form persistent structures. We were also able to capture this intrinsic timescale at the N-terminal unstructured domain of the prion protein. Our estimated timescale of the segmental mobility is similar to that of unfolded proteins studied by nuclear magnetic resonance in conjunction with molecular dynamics simulations. Our results have broader implications for a diverse range of functionally and pathologically important intrinsically disordered proteins and disordered regions.

Optimized Markov state models for metastable systems.

A method is proposed to identify target states that optimize a metastability index amongst a set of trial states and use these target states as milestones (or core sets) to build Markov State Models (MSMs). If the optimized metastability index is small, this automatically guarantees the accuracy of the MSM, in the sense that the transitions between the target milestones is indeed approximately Markovian. The method is simple to implement and use, it does not require that the dynamics on the trial milestones be Markovian, and it also offers the possibility to partition the system's state-space by assigning every trial milestone to the target milestones it is most likely to visit next and to identify transition state regions. Here the method is tested on the Gly-Ala-Gly peptide, where it is shown to correctly identify the expected metastable states in the dihedral angle space of the molecule without a priori information about these states. It is also applied to analyze the folding landscape of the Beta3s mini-protein, where it is shown to identify the folded basin as a connecting hub between an helix-rich region, which is entropically stabilized, and a beta-rich region, which is energetically stabilized and acts as a kinetic trap.

Computing Free Energy Differences of Configurational Basins.

A simulation-based approach is proposed to estimate free energy differences between configurational states A and B, defined in terms of collective coordinates of the molecular system. The computational protocol is organized into three stages that can be carried on simultaneously. Two of them consist of independent simulations aimed at sampling, in turn, A and B states. In order to limit the evolution of the system around A and B, biased sampling simulations such as umbrella sampling can be employed. These simulations allow us to estimate local configuration integrals associated with A and B, which can be viewed as vibrational contributions to the free energy. Free energy evaluation is completed by the linking-path stage, in which the potential of mean force difference is estimated between two arbitrary points of the configurational surface, located the first around A and the second around B. The linking path in the space of the collective coordinates is arbitrary and can be computed with any method, starting from adaptive biasing potential/force approaches to nonequilibrium techniques. As an illustrative example, we present the calculation of free energy differences between conformational states of the alanine dipeptide in the space of backbone dihedral angles. The basic advantage of this method, that we term "path-linked domains" scheme, is to prevent accurate calculation of the whole free energy hypersurface in the space of the collective coordinates, thus limiting the statistical sampling to a minimum. Path-linked domains schemes can be applied to a variety of biochemical processes, such as protein-ligand complexation or folding-unfolding interconversion.

Conformational Entropy of Intrinsically Disordered Proteins from Amino Acid Triads.

This work quantitatively characterizes intrinsic disorder in proteins in terms of sequence composition and backbone conformational entropy. Analysis of the normalized relative composition of the amino acid triads highlights a distinct boundary between globular and disordered proteins. The conformational entropy is calculated from the dihedral angles of the middle amino acid in the amino acid triad for the conformational ensemble of the globular, partially and completely disordered proteins relative to the non-redundant database. Both Monte Carlo (MC) and Molecular Dynamics (MD) simulations are used to characterize the conformational ensemble of the representative proteins of each group. The results show that the globular proteins span approximately half of the allowed conformational states in the Ramachandran space, while the amino acid triads in disordered proteins sample the entire range of the allowed dihedral angle space following Flory’s isolated-pair hypothesis. Therefore, only the sequence information in terms of the relative amino acid triad composition may be sufficient to predict protein disorder and the backbone conformational entropy, even in the absence of well-defined structure. The predicted entropies are found to agree with those calculated using mutual information expansion and the histogram method.

Inhibition of GPR18 through Docking of known Antagonists using a Homology Model

The Class A G-protein coupled receptor, GPR18, is a newly de-orphanized cannabinoid receptor that binds the endogenous ligand, N-arachidonoyl glycine (NAGly). As part of a larger project to model the interaction of ligands with GPR18, we undertook the conformational analysis of NAGly. Because NAGLY is an arachidonic acid derivative and arachidonic acid has been shown to be highly flexible, we began by investigating the conformational flexibility of this endogenous ligand. To this end, the Conformational Memories (CM) method was employed. This method combines Monte Carlo exploration of the dihedral angle and bond angle space with simulated annealing (MC/SA) to determine the range of values that each dihedral angle and bond angle is capable of exploring in a broad temperature range (Whitnell J Comp Chem 2007). Similar to previously reported CM results for arachidonic acid (Barnett-Norris J Med Chem 1998), NAGly's conformers could be divided into four groups: linear, U-shape, J-shape, and helical. These results will be used to explore the binding site interactions of NAGly at GPR18.

Protein secondary structure classification revisited: processing DSSP information with PSSC.

A first step toward three-dimensional protein structure description is the characterization of secondary structure. The most widely used program for secondary structure assignment remains DSSP, introduced in 1983, with currently more than 400 citations per year. DSSP output is in a one-letter representation, where much of the information on DSSP's internal description is lost. Recently it became evident that DSSP overlooks most π-helical structures, which are more prevalent and important than anticipated before. We introduce an alternative concept, representing the internal structure characterization of DSSP as an eight-character string that is human-interpretable and easy to parse by software. We demonstrate how our protein secondary structure characterization (PSSC) code allows for inspection of complicated structural features. It recognizes ten times more π-helical residues than does the standard DSSP. The plausibility of introduced changes in interpreting DSSP information is demonstrated by better clustering of secondary structures in (φ, ψ) dihedral angle space. With a sliding sequence window (SSW), helical assignments with PSSC remain invariant compared with an assignment based on the complete structure. In contrast, assignment with DSSP can be changed by residues in the neighborhood that are in fact not interacting with the residue under consideration. We demonstrate how one can easily define new secondary structure classification schemes with PSSC and perform the classifications. Our approach works without changing the DSSP source code and allows for more detailed protein characterization.

The Principle of Stationary Action in Biophysics: Stability in Protein Folding

We conceptualize protein folding as motion in a large dimensional dihedral angle space. We use Lagrangian mechanics and introduce an unspecified Lagrangian to study the motion. The fact that we have reliable folding leads us to conjecture the totality of paths forms caustics that can be recognized by the vanishing of the second variation of the action. There are two types of folding processes: stable against modest perturbations and unstable. We also conjecture that natural selection has picked out stable folds. More importantly, the presence of caustics leads naturally to the application of ideas from catastrophe theory and allows us to consider the question of stability for the folding process from that perspective. Powerful stability theorems from mathematics are then applicable to impose more order on the totality of motions. This leads to an immediate explanation for both the insensitivity of folding to solution perturbations and the fact that folding occurs using very little free energy. The theory of folding, based on the above conjectures, can also be used to explain the behavior of energy landscapes, the speed of folding similar to transition state theory, and the fact that random proteins do not fold.

Dynamic neutron scattering from conformational dynamics. II. Application using molecular dynamics simulation and Markov modeling

Neutron scattering experiments directly probe the dynamics of complex molecules on the sub pico- to microsecond time scales. However, the assignment of the relaxations seen experimentally to specific structural rearrangements is difficult, since many of the underlying dynamical processes may exist on similar timescales. In an accompanying article, we present a theoretical approach to the analysis of molecular dynamics simulations with a Markov State Model (MSM) that permits the direct identification of structural transitions leading to each contributing relaxation process. Here, we demonstrate the use of the method by applying it to the configurational dynamics of the well-characterized alanine dipeptide. A practical procedure for deriving the MSM from an MD is introduced. The result is a 9-state MSM in the space of the backbone dihedral angles and the side-chain methyl group. The agreement between the quasielastic spectrum calculated directly from the atomic trajectories and that derived from the Markov state model is excellent. The dependence on the wavevector of the individual Markov processes is described. The procedure means that it is now practicable to interpret quasielastic scattering spectra in terms of well-defined intramolecular transitions with minimal a priori assumptions as to the nature of the dynamics taking place.

Normal mode analysis based on an elastic network model for biomolecules in the Protein Data Bank, which uses dihedral angles as independent variables

We have developed a computer program, named PDBETA, that performs normal mode analysis (NMA) based on an elastic network model that uses dihedral angles as independent variables. Taking advantage of the relatively small number of degrees of freedom required to describe a molecular structure in dihedral angle space and a simple potential-energy function independent of atom types, we aimed to develop a program applicable to a full-atom system of any molecule in the Protein Data Bank (PDB). The algorithm for NMA used in PDBETA is the same as the computer program FEDER/2, developed previously. Therefore, the main challenge in developing PDBETA was to find a method that can automatically convert PDB data into molecular structure information in dihedral angle space. Here, we illustrate the performance of PDBETA with a protein–DNA complex, a protein–tRNA complex, and some non-protein small molecules, and show that the atomic fluctuations calculated by PDBETA reproduce the temperature factor data of these molecules in the PDB. A comparison was also made with elastic-network-model based NMA in a Cartesian-coordinate system.

Dynamic and Thermodynamic Signatures of Native and Non-Native Protein States with Application to the Improvement of Protein Structures.

Accurate knowledge of the 3D structural ensemble of proteins is important for understanding of their biological function. We report here the application of microsecond all-atom molecular dynamics (MD) simulations in explicit solvent for the improvement of the quality of low-resolution structures obtained by protein structure prediction (decoys). Seventy MD simulations of ∼1 μs average duration were performed on 13 different protein systems starting from X-ray crystal structures and decoys. Their behavior can be divided into three groups: 22 trajectories converged toward the native state, 27 trajectories displayed a quasi-equilibrium by populating mainly a single non-native free energy basin, and 21 trajectories drifted away from their initial decoy structure transiently visiting multiple free energy minima. To determine whether the native structure can be identified among non-native ensembles, the free energy was determined for each basin by the MM/GBSA method together with the von Mises entropy estimator in dihedral angle space. For the proteins studied here, it is found that the ensembles belonging to free energy basins with the lowest free energies and the longest residence times are most native-like. The results demonstrate that explicit solvent microsecond MD simulations using the latest generation of protein force fields and free energy metrics are sufficiently accurate to permit positive identification of native state ensembles against low-resolution structural models and decoys. The approach can be applied to the direct refinement of predicted or experimental low-resolution protein structures.

Analysis and elimination of a bias in targeted molecular dynamics simulations of conformational transitions: application to calmodulin.

The popular targeted molecular dynamics (TMD) method for generating transition paths in complex biomolecular systems is revisited. In a typical TMD transition path, the large-scale changes occur early and the small-scale changes tend to occur later. As a result, the order of events in the computed paths depends on the direction in which the simulations are performed. To identify the origin of this bias, and to propose a method in which the bias is absent, variants of TMD in the restraint formulation are introduced and applied to the complex open ↔ closed transition in the protein calmodulin. Due to the global best-fit rotation that is typically part of the TMD method, the simulated system is guided implicitly along the lowest-frequency normal modes, until the large spatial scales associated with these modes are near the target conformation. The remaining portion of the transition is described progressively by higher-frequency modes, which correspond to smaller-scale rearrangements. A straightforward modification of TMD that avoids the global best-fit rotation is the locally restrained TMD (LRTMD) method, in which the biasing potential is constructed from a number of TMD potentials, each acting on a small connected portion of the protein sequence. With a uniform distribution of these elements, transition paths that lack the length-scale bias are obtained. Trajectories generated by steered MD in dihedral angle space (DSMD), a method that avoids best-fit rotations altogether, also lack the length-scale bias. To examine the importance of the paths generated by TMD, LRTMD, and DSMD in the actual transition, we use the finite-temperature string method to compute the free energy profile associated with a transition tube around a path generated by each algorithm. The free energy barriers associated with the paths are comparable, suggesting that transitions can occur along each route with similar probabilities. This result indicates that a broad ensemble of paths needs to be calculated to obtain a full description of conformational changes in biomolecules. The breadth of the contributing ensemble suggests that energetic barriers for conformational transitions in proteins are offset by entropic contributions that arise from a large number of possible paths.

Fast Flexible Modeling of RNA Structure Using Internal Coordinates

Modeling the structure and dynamics of large macromolecules remains a critical challenge. Molecular dynamics (MD) simulations are expensive because they model every atom independently, and are difficult to combine with experimentally derived knowledge. Assembly of molecules using fragments from libraries relies on the database of known structures and thus may not work for novel motifs. Coarse-grained modeling methods have yielded good results on large molecules but can suffer from difficulties in creating more detailed full atomic realizations. There is therefore a need for molecular modeling algorithms that remain chemically accurate and economical for large molecules, do not rely on fragment libraries, and can incorporate experimental information. RNABuilder works in the internal coordinate space of dihedral angles and thus has time requirements proportional to the number of moving parts rather than the number of atoms. It provides accurate physics-based response to applied forces, but also allows user-specified forces for incorporating experimental information. A particular strength of RNABuilder is that all Leontis-Westhof basepairs can be specified as primitives by the user to be satisfied during model construction. We apply RNABuilder to predict the structure of an RNA molecule with 160 bases from its secondary structure, as well as experimental information. Our model matches the known structure to 10.2 Angstroms RMSD and has low computational expense.

Multicanonical molecular dynamics simulations combined with Metadynamics for the free energy landscape of a biomolecular system with high energy barriers

Multicanonical molecular-dynamics (McMD) simulation and Metadynamics (MetaD) are useful for obtaining the free-energies, and can be mutually complementary. We combined McMD with MetaD, and applied it to the conformational free energy calculations of a proline dipeptide. First, MetaD was performed along the dihedral angle at the prolyl bond and we obtained a coarse biasing potential. After adding the biasing potential to the dihedral angle potential energy, we conducted McMD with the modified potential energy. Enhanced sampling was achieved for all degrees-of-freedom, and the sampling of the dihedral angle space was facilitated. After reweighting, we obtained an accurate free energy landscape.

Entropy Localization in Proteins

The configurational entropy of a protein is under physiological conditions a major contributor to the free energy. Its quantitative characterization is therefore an important step toward the understanding of protein function. The configurational entropy of the oncoprotein MDM2, whose determination is a challenge by experiment alone, is studied here by means of 0.4 μs molecular dynamics computer simulations in both the presence and absence of the p53-peptide ligand. By characterizing protein motions in dihedral angle space, it is found that the motional amplitudes change considerably upon ligand binding while correlations between dihedral angle motions are remarkably well conserved. This applies for backbone and side-chain dihedral angle pairs at both short- and long-range distance to the binding site. As a direct consequence, the change of the configurational entropy can be decomposed into a sum of local contributions. This significantly facilitates the understanding of the relationship between protein dynamics and thermodynamics, which is important, for example, in the context of protein-ligand and protein-protein interactions. The findings also have implications for the direct derivation of entropy changes from site-specific dynamics measurements as afforded by NMR spectroscopy.

Using the Chemical Shift Anisotropy Tensor of Carbonyl Backbone Nuclei As a Probe of Secondary Structure in Proteins

This study seeks to establish that the chemical shift anisotropy (CSA) tensor of the backbone carbonyl ((13)C') nucleus is a useful indicator of secondary structure elements in proteins. The CSA tensors of protein backbone nuclei in different secondary structures were computed for experimentally determined dihedral angles using ab initio methods and by calculating the CSA tensor for a model peptide over the entire dihedral angle space. It is shown that 2D and 3D cluster plots of CSA tensor parameters for (13)C' nuclei are able to distinguish between different secondary structure elements with little to no overlap. As evidenced by multinuclear 2D plots, the CSA of the (13)C' nucleus when correlated with different CSA parameters of the other backbone nuclei (such as C(alpha) or (1)H(alpha)) is also useful in secondary structure identification. The differentiation of alpha-helix versus beta-sheet motifs (the most populated regions of the Ramachandran map) for experimentally determined values of the carbonyl CSA tensor for proteins ubiquitin and binase (obtained from the literature) agrees well with the quantum chemical predictions.

Latent dynamics of a protein molecule observed in dihedral angle space

Dihedral angles are alternative set of variables to Cartesian coordinates for representing protein dynamics. The two sets of variables exhibit extremely different behavior. Motions in dihedral angle space are characterized by latent dynamics, in which motion induced in each dihedral angle is always compensated for by motions of many other dihedral angles, in order to maintain a rigid globular shape. Using molecular dynamics simulations, we propose a molecular mechanism for the latent dynamics in dihedral angle space. It was found that, due to the unique structure of dihedral principal components originating in the globular shape of the protein, the dihedral principal components with large (small) amplitudes are highly correlated with the eigenvectors of the metric matrix with small (large) eigenvalues. Such an anticorrelation in the eigenmode structures minimizes the mean square displacement of Cartesian coordinates upon rotation of dihedral angles. In contrast, a short peptide, deca-alanine in this study, does not show such behavior of the latent dynamics in the dihedral principal components, but shows similar behaviors to those of the Cartesian principal components, due to the absence of constraints to maintain a rigid globular shape.

Electrostatic Persistence Length

The persistence length is calculated for polyelectrolyte chains with fixed bond lengths and bond angles (pi-theta), and a potential energy consisting of the screened Coulomb interaction between beads, potential wells alpha phi(i)2 for the dihedral angles phi(i), and coupling terms beta phi(i) phi(i+/-1). This model defines a librating chain that reduces in appropriate limits to the freely rotating or wormlike chains, it can accommodate local crumpling or extreme stiffness, and it is easy to simulate. A planar-quadratic (pq), analytic approximation is based on an expansion of the electrostatic energy in eigenfunctions of the quadratic form that describes the backbone energy, and on the assumption that the quadratic form not only is positive but also adequately confines the chain in an infinite phase space of dihedral angles to the physically unique part with all |phi(i)| < pi. The pq approximation is available under these weak constraints, but the simulations confirm its quantitative accuracy only under the expected condition that alpha is large, that is, for very stiff chains. Stiff chains can also be simulated with small alpha and small theta and compared to an OSF approximation suitably generalized to chains with finite rather than vanishing theta, and increasing agreement with OSF is found the smaller is theta. The two approximations, one becoming exact as alpha --> infinity with fixed theta, the other as theta --> 0 with fixed alpha, are quantitatively similar in behavior, both giving a persistence length P = P0 + aD2 for stiff chains, where D is the Debye length. However, the coefficient apq is about twice the value of aOSF. Under other conditions the simulations show that P may or not be linear in D2 at small or moderate D, depending on the magnitudes of alpha, beta, theta, and the charge density but always becomes linear at large D. Even at a moderately low charge density, corresponding to fewer than 20% of the beads being charged, and with strong crumpling induced by large beta, increasing D dissolves blobs and recovers a linear dependence of P on D2, although a lower power of D gives an adequate fit at moderate D. For the class of models considered, it is concluded that the only universal feature is the asymptotic linearity of P in D2, regardless of flexibility or stiffness.