Collective NMR relaxation model applied to protein dynamics

Combining Molecular and Spin Dynamics Simulations with Solid-State NMR: A Case Study of Amphiphilic Lysine-Leucine Repeat Peptide Aggregates

Interpreting dynamics in solid-state molecular systems requires characterization of the potentially heterogeneous environmental contexts of molecules. In particular, the analysis of solid-state nuclear magnetic resonance (ssNMR) data to elucidate molecular dynamics (MD) involves modeling the restriction to overall tumbling by neighbors, as well as the concentrations of water and buffer. In this exploration of the factors that influence motion, we utilize atomistic MD trajectories of peptide aggregates with varying hydration to mimic an amorphous solid-state environment and predict ssNMR relaxation rates. We also account for spin diffusion in multiply spin-labeled (up to 19 nuclei) residues, with several models of dipolar-coupling networks. The framework serves as a general approach to determine essential spin couplings affecting relaxation, benchmark MD force fields, and reveal the hydration dependence of dynamics in a crowded environment. We demonstrate the methodology on a previously characterized amphiphilic 14-residue lysine-leucine repeat peptide, LKα14 (Ac-LKKLLKLLKKLLKL-c), which has an α-helical secondary structure and putatively forms leucine-burying tetramers in the solid state. We measure the R₁ relaxation rates of uniformly ¹³C-labeled and site-specific ²H-labeled leucines in the hydrophobic core of LKα14 at multiple hydration levels. Studies of 9 and 18 tetramer bundles reveal the following: (a) for the incoherent component of ¹³C relaxation, the nearest-neighbor spin interactions dominate, while the ¹H-¹H interactions have minimal impact; (b) the AMBER ff14SB dihedral barriers for the leucine C_γ-C_δ bond ("methyl rotation barriers") must be lowered by a factor of 0.7 to better match the ²H data; (c) proton-driven spin diffusion explains some of the discrepancy between experimental and simulated rates for the C_β and C_α nuclei; and (d) ¹³C relaxation rates are mostly underestimated in the MD simulations at all hydrations, and the discrepancies identify likely motions missing in the 50 ns MD trajectories.

Normal mode analysis and beyond

Normal mode analysis provides a powerful tool in biophysical computations. Particularly, we shed light on its application to protein properties because they directly lead to biological functions. As a result of normal mode analysis, the protein motion is represented as a linear combination of mutually independent normal mode vectors. It has been widely accepted that the large amplitude motions throughout the entire protein molecule can be well described with a few low-frequency normal modes. Furthermore, it is possible to represent the effect of external perturbations, e.g., ligand binding, hydrostatic pressure, as the shifts of normal mode variables. Making use of this advantage, we are able to explore mechanical properties of proteins such as Young's modulus and compressibility. Within thermally fluctuating protein molecules under physiological conditions, tightly packed amino acid residues interact with each other through heat and energy exchanges. Since the structure and dynamics of protein molecules are highly anisotropic, the flow of energy and heat should also be anisotropic. Based on the harmonic approximation of the heat current operator, it is possible to analyze the communication map of a protein molecule. By using this method, the energy transfer pathways of photoactive yellow protein were calculated. It turned out that these pathways are similar to those obtained via the Green-Kubo formalism with equilibrium molecular dynamics simulations, indicating that normal mode analysis captures the intrinsic nature of the transport properties of proteins.

Enhanced Sampling of Interdomain Motion Using Map-Restrained Langevin Dynamics and NMR: Application to Pin1

Many signaling proteins consist of globular domains connected by flexible linkers that allow for substantial domain motion. Because these domains often serve as complementary functional modules, the possibility of functionally important domain motions arises. To explore this possibility, we require knowledge of the ensemble of protein conformations sampled by interdomain motion. Measurements of NMR residual dipolar couplings (RDCs) of backbone HN bonds offer a per-residue characterization of interdomain dynamics, as the couplings are sensitive to domain orientation. A challenge in reaching this potential is the need to interpret the RDCs as averages over dynamic ensembles of domain conformations. Here, we address this challenge by introducing an efficient protocol for generating conformational ensembles appropriate for flexible, multi-domain proteins. The protocol uses map-restrained self-guided Langevin dynamics simulations to promote collective, interdomain motion while restraining the internal domain motion to near rigidity. Critically, the simulations retain an all-atom description for facile inclusion of site-specific NMR RDC restraints. The result is the rapid generation of conformational ensembles consistent with the RDC data. We illustrate this protocol on human Pin1, a two-domain peptidyl-prolyl isomerase relevant for cancer and Alzheimer's disease. The results include the ensemble of domain orientations sampled by Pin1, as well as those of a dysfunctional variant, I28A-Pin1. The differences between the ensembles corroborate our previous spin relaxation results that showed weakened interdomain contact in the I28A variant relative to wild type. Our protocol extends our abilities to explore the functional significance of protein domain motions.

Normal modes and phase transition of the protein chain based on the Hamiltonian formalism

We use the torsional angles of the protein chain as generalized coordinates in the canonical formalism, derive canonical equations of motion, and investigate the coordinate dependence of the kinetic energy expressed in terms of the canonical momenta. We use the formalism to compute the normal-frequency distributions of the α helix and the β sheet, under the assumption that they are stabilized purely through hydrogen bonding. In addition, we obtain the free-energy relations of the α helix, the β sheet, and the random coil of a 15-residue polyalanine. Interestingly, our results predict a phase transition from an α helix to a β sheet at a critical temperature.

An experimentally guided umbrella sampling protocol for biomolecules

We present a simple method for utilizing experimental data to improve the efficiency of numerical calculations of free energy profiles from molecular dynamics simulations. The method involves umbrella sampling simulations with restraining potentials based on a known approximate estimate of the free energy profile derived solely from experimental data. The use of the experimental data results in optimal restraining potentials, guides the simulation along relevant pathways, and decreases overall computational time. In demonstration of the method, two systems are showcased. First, guided, unguided (regular) umbrella sampling simulations and exhaustive sampling simulations are compared to each other in the calculation of the free energy profile for the distance between the ends of a pentapeptide. The guided simulation use restraints based on a simulated "experimental" potential of mean force of the end-to-end distance that would be measured by fluorescence resonance energy transfer (obtained from exhaustive sampling). Statistical analysis shows a dramatic improvement in efficiency for a 5 window guided umbrella sampling over 5 and 17 window unguided umbrella sampling simulations. Moreover, the form of the potential of mean force for the guided simulations evolves, as one approaches convergence, along the same milestones as the extensive simulations, but exponentially faster. Second, the method is further validated by replicating the forced unfolding pathway of the titin I27 domain using guiding umbrella sampling potentials determined from actual single molecule pulling data. Comparison with unguided umbrella sampling reveals that the use of guided sampling encourages unfolding simulations to converge faster to a forced unfolding pathway that agrees with previous results and produces a more accurate potential of mean force.

Algorithmic dimensionality reduction for molecular structure analysis

Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation--a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation.

Second derivatives in generalized Born theory

Generalized Born solvation models offer a popular method of including electrostatic aspects of solvation free energies within an analytical model that depends only upon atomic coordinates, charges, and dielectric radii. Here, we describe how second derivatives with respect to Cartesian coordinates can be computed in an efficient manner that can be distributed over multiple processors. This approach makes possible a variety of new methods of analysis for these implicit solvation models. We illustrate three of these methods here: the use of Newton-Raphson optimization to obtain precise minima in solution; normal mode analysis to compute solvation effects on the mechanical properties of DNA; and the calculation of configurational entropies in the MM/GBSA model. An implementation of these ideas, using the Amber generalized Born model, is available in the nucleic acid builder (NAB) code, and we present examples for proteins with up to 45,000 atoms. The code has been implemented for parallel computers using both the OpenMP and MPI environments, and good parallel scaling is seen with as many as 144 OpenMP processing threads or MPI processing tasks.

An overview of recent developments in the interpretation and prediction of fast internal protein dynamics

During the past decades, NMR spectroscopy has emerged as a unique tool for the study of protein dynamics. Indeed, relaxation studies on isotopically labeled proteins can provide information on the overall motions as well as the internal fast, sub-nanosecond, dynamics. Therefore, the interpretation and the prediction of spin relaxation rates in proteins are important issues that have motivated numerous theoretical and methodological developments, including the description of overall dynamics and its possible coupling to internal mobility, the introduction of models of internal dynamics, the determination of correlation functions from experimental data, and the relationship between relaxation and thermodynamical quantities. A brief account of recent developments that have proven useful in this domain are presented.

Amplitudes and directions of internal protein motions from a JAM analysis of 15N relaxation data

A method has been developed for characterizing dynamic structures of proteins in solution by using nuclear magnetic resonance (NMR) restraints and 15N relaxation data. This method is based on the concept of the jumping-among-minima (JAM) model. In this model we assume that protein dynamics can be described on the basis of conformational substates, and involves intra- and inter-substate motion. A set of substates is created by picking energy-minimized conformations from the conformational space consistent with the geometric NMR restraints. Intra-substate motions, which occur on the timescale of approximately 10 ps, are simulated with molecular dynamics (MD) calculations with force-field energy terms. Statistical weights of the conformational substates are determined to reproduce the NMR relaxation parameters. The refinement procedure consists of four stages: (i) determination of the ensemble of structures that satisfy NMR restraints, (ii) determination of intra-substate fluctuation, (iii) determination of statistical weights of conformational substates to reproduce model-free relaxation parameters, and (iv) analysis of the resulting dynamic structure to determine amplitudes and directions of internal protein motions. This method was employed to investigate structure and dynamics of the adhesion domain of human CD2 (hCD2) in solution. Two major collective modes, whose contributions to atomic mean-square fluctuations are 77.1% in total, are identified by the refinement. The first mode is interpreted as a rigid-body motion of a protein segment consisting of a part of the B--C loop, a part of the F strand, and the F--G loop. Another type of smaller-amplitude mode is indicated for the C'--C'' loop. The motions affect primarily the curvature of the slightly concave counterreceptor-binding site and represent transitions between a concave (closed) and flat (open) binding face. By comparing the ensemble of structures in solution to the complex structure with counterreceptor CD58, we found that these two types of motions resemble the change upon counterreceptor binding.

SYMMETRY, FORM, AND SHAPE: Guiding Principles for Robustness in Macromolecular Machines

Computational studies of large macromolecular assemblages have come a long way during the past 10 years. With the explosion of computer power and parallel computing, timescales of molecular dynamics simulations have been extended far beyond the hundreds of picoseconds timescale. However, limitations remain for studies of large-scale conformational changes occurring on timescales beyond nanoseconds, especially for large macromolecules. In this review, we describe recent methods based on normal mode analysis that have enabled us to study dynamics on the microsecond timescale for large macromolecules using different levels of coarse graining, from atomically detailed models to those employing only low-resolution structural information. Emerging from such studies is a control principle for robustness in Nature's machines. We discuss this idea in the context of large-scale functional reorganization of the ribosome, virus particles, and the muscle protein myosin.

Reorientational contact-weighted elastic network model for the prediction of protein dynamics: comparison with NMR relaxation

A new model for the prediction of protein backbone motions is presented. The model, termed reorientational contact-weighted elastic network model, is based on a multidimensional reorientational harmonic potential of the backbone amide bond vector orientations and it is applied to the interpretation of dynamics parameters obtained from NMR relaxation data. The individual energy terms are weighted as a function of the intervector distances and by the contact strengths of each bond vector with respect to its local environment. Correlated reorientational motional properties of the bond vectors are obtained by means of normal mode analysis. Application to a set of proteins with known three-dimensional structures yields good to excellent agreement between predicted and experimental NMR order parameters presenting an improvement over the local contact model. The reorientational eigenmodes of the reorientational contact-weighted elastic network model method provide direct information on the collective nature of protein backbone motions. The dominant eigenmodes have a notably low collectivity, which is consistent with the behavior found for reorientational eigenmodes from molecular dynamics simulations.

Solvent Interactions and Protein Dynamics in Spin-labeled T4 Lysozyme

Aspects of T4 lysozyme dynamics and solvent interaction are investigated using atomically detailed Molecular Dynamics (MD) simulations. Two spin-labeled mutants of T4 lysozyme are analyzed (T4L-N40C and T4L-K48C), which have been found from electronic paramagnetic resonance (EPR) experiments to exhibit different mobilities at the site of spin probe attachment (N- and C-terminus of helix B, respectively). Similarities and differences in solvent distribution and diffusion around the spin label, as well as around exposed and buried residues within the protein, are discussed. The purpose is to capture possible strong interactions between the spin label (ring) and solvent molecules, which may affect EPR lineshapes. The effect of backbone motions on the water density profiles is also investigated. The focus is on the domain closure associated with the T4 lysozyme hinge-bending motion, which is analyzed by Essential Dynamics (ED). The N-terminus of helix B is found to be a "hinge" residue, which explains the high degree of flexibility and motional freedom at this site.

A coarse-grained normal mode approach for macromolecules: an efficient implementation and application to Ca(2+)-ATPase

A block normal mode (BNM) algorithm, originally proposed by Tama et al., (Proteins Struct. Func. Genet. 41:1-7, 2000) was implemented into the simulation program CHARMM. The BNM approach projects the hessian matrix into local translation/rotation basis vectors and, therefore, dramatically reduces the size of the matrix involved in diagonalization. In the current work, by constructing the atomic hessian elements required in the projection operation on the fly, the memory requirement for the BNM approach has been significantly reduced from that of standard normal mode analysis and previous implementation of BNM. As a result, low frequency modes, which are of interest in large-scale conformational changes of large proteins or protein-nucleic acid complexes, can be readily obtained. Comparison of the BNM results with standard normal mode analysis for a number of small proteins and nucleic acids indicates that many properties dominated by low frequency motions are well reproduced by BNM; these include atomic fluctuations, the displacement covariance matrix, vibrational entropies, and involvement coefficients for conformational transitions. Preliminary application to a fairly large system, Ca(2+)-ATPase (994 residues), is described as an example. The structural flexibility of the cytoplasmic domains (especially domain N), correlated motions among residues on domain interfaces and displacement patterns for the transmembrane helices observed in the BNM results are discussed in relation to the function of Ca(2+)-ATPase. The current implementation of the BNM approach has paved the way for developing efficient sampling algorithms with molecular dynamics or Monte Carlo for studying long-time scale dynamics of macromolecules.

Molecular dynamics and NMR spin relaxation in proteins

Molecular dynamics simulations often play a central role in the analysis of biomolecular NMR data. The focus here is on NMR spin-relaxation, which can provide unique insights into the time-dependence of conformational fluctuations, especially on picosecond to nanosecond time scales which can be directly probed by simulations. A great deal has been learned from such simulations about the general nature of such motions and their impact on NMR observables. In principle, relaxation measurements should also provide valuable benchmarks for judging the quantitative accuracy of simulations, but there are a variety of experimental and computational obstacles to making useful direct comparisons. It seems likely that simulations on time scales that are just now becoming generally feasible may provide important new information on internal motions, overall rotational diffusion, and the coupling between internal and rotational motion. Such information could provide a sound foundation for a new generation of detailed interpretation of NMR spin-relaxation results.

Nmr probes of molecular dynamics: overview and comparison with other techniques

NMR spin relaxation spectroscopy is a powerful approach for characterizing intramolecular and overall rotational motions in proteins. This review describes experimental methods for measuring laboratory frame spin relaxation rate constants by high-resolution solution-state NMR spectroscopy, together with theoretical approaches for interpreting spin relaxation data in order to quantify protein conformational dynamics on picosecond-nanosecond time scales. Recent applications of these techniques to proteins are surveyed, and investigations of the contribution of conformational chain entropy to protein function are highlighted. Insights into the dynamical properties of proteins obtained from NMR spin relaxation spectroscopy are compared with results derived from other experimental and theoretical techniques.

Reorientational eigenmode dynamics: a combined MD/NMR relaxation analysis method for flexible parts in globular proteins

An approach is presented for the interpretation of heteronuclear NMR spin relaxation data in mobile protein parts in terms of reorientational eigenmode dynamics. The method is based on the covariance matrix of the spatial functions of the nuclear spin interactions that cause relaxation expressed as spherical harmonics of rank 2. The approach was applied to characterize the dynamics of a loop region of ubiquitin. The covariance matrix was determined from a conformational ensemble generated by a 5 ns molecular dynamics simulation. It was found that the time correlation functions of the dominant eigenmodes decay in good approximation with a single correlation time. From the reorientational eigenmodes, their eigenvalues, and correlation times, NMR relaxation data were calculated in accordance with Bloch-Wangsness-Redfield relaxation theory and directly compared with experimental (15)N relaxation parameters. Using a fitting procedure, agreement between calculated and experimental data was improved significantly by adjusting eigenvalues and correlation times of the dominant modes. The presented procedure provides detailed information on correlated reorientational dynamics of flexible parts in globular proteins. The covariance matrix was linked to the covariance matrix of backbone dihedral angle fluctuations, allowing one to study the motional behavior of these degrees of freedom on nano- and subnanosecond time scales.

Characterization of collective and anisotropic reorientational protein dynamics

Intramolecular reorientational dynamics of proteins are described in terms of reorientational quasiharmonic modes. These modes provide important insight into anisotropic and collective axial fluctuations of distinct molecular fragments, and they represent a highly compact description of intramolecular protein motions that are spectroscopically observable via nuclear spin relaxation. The method is applied to a molecular dynamics computer simulation of the protein ubiquitin.

A space-time structure determination of human CD2 reveals the CD58-binding mode

We describe a procedure for a space-time description of protein structures. The method is capable of determining populations of conformational substates, and amplitudes and directions of internal protein motions. This is achieved by fitting static and dynamic NMR data. The approach is based on the jumping-among-minima concept. First, a wide conformational space compatible with structural NMR data is sampled to find a large set of substates. Subsequently, intrasubstate motions are sampled by using molecular dynamics calculations with force field energy terms. Next, the populations of substates are fitted to NMR relaxation data. By diagonalizing a second moment matrix, directions and amplitudes of motions are identified. The method was applied to the adhesion domain of human CD2. We found that very few substates can account for most of the experimental data. Furthermore, only two types of collective motions have high amplitudes. They represent transitions between a concave (closed) and flat (open) binding face and resemble the change upon counter-receptor (CD58) binding.

Investigating protein dynamics in collective coordinate space

Currently, collective coordinates are commonly employed in order to examine protein dynamics. In recent studies, they have been successfully applied to finding functionally relevant motions, to investigating the physical nature of protein dynamics, to sampling of the conformational space and to the analysis of experimental data. Collective coordinates also have other possible applications.