Time-averaged order parameter restraints in molecular dynamics simulations

A Method to Derive Structural Information on Molecules from Residual Dipolar Coupling NMR Data

A method for structure refinement of molecules based on residual dipolar coupling (RDC) data is proposed. It calculates RDC values using rotational and molecule-internal configurational sampling instead of the common refinement procedure that is based on the approximation of the nonuniform rotational distribution of the molecule by a single alignment tensor representing the average nonuniformity of this distribution. Using rotational sampling, as is occurring in the experiment leading to observable RDCs, the method stays close to the experiment. It avoids the use of an alignment tensor and thus the assumption that the overall rotation of the molecule is decoupled from its internal motions and that the molecule be rigid. Two simple molecules, two-united-atomic ethane and a cyclooctane molecule with eight side chains, containing 24 united atoms, serve as the so-called "toy model" test systems. The method demonstrates the influence of molecular flexibility and force-field deficiencies on the outcome of structure refinement based on RDCs. For a molecule of a given size (number of atoms N_at), there must be a sufficiently large number N_RDC of measured RDC values available to allow the restraining forces to bias the overall orientation distribution of the molecule. If the ratio N_RDC/N_at gets too small, the RDC-restraining forces will either not be strong enough to change the overall rotational direction of the molecule such that the target RDC values are approximated well or will be so strong that they induce a local deformation of the molecule. In the latter case, the size or inertia of the molecule hinders a restraining-induced overall rotation and the internal structure of the molecule is not strong enough to avoid local deformation due to the restraining forces.

Molecular dynamics simulation or structure refinement of proteins: are solvent molecules required? A case study using hen lysozyme

In protein simulation or structure refinement based on values of observable quantities measured in (aqueous) solution, solvent (water) molecules may be explicitly treated, omitted, or represented by a potential of mean-solvation-force term, depending on protein coordinates only, in the force field used. These three approaches are compared for hen egg white lysozyme (HEWL). This 129-residue non-spherical protein contains a variety of secondary-structure elements, and ample experimental data are available: 1630 atom–atom Nuclear Overhauser Enhancement (NOE) upper distance bounds, 213 ³ J-couplings and 200 S² order parameters. These data are used to compare the performance of the three approaches. It is found that a molecular dynamics (MD) simulation in explicit water approximates the experimental data much better than stochastic dynamics (SD) simulation in vacuo without or with a solvent-accessible-surface-area (SASA) implicit-solvation term added to the force field. This is due to the missing energetic and entropic contributions and hydrogen-bonding capacities of the water molecules and the missing dielectric screening effect of this high-permittivity solvent. Omission of explicit water molecules leads to compaction of the protein, an increased internal strain, distortion of exposed loop and turn regions and excessive intra-protein hydrogen bonding. As a consequence, the conformation and dynamics of groups on the surface of the protein, which may play a key role in protein–protein interactions or ligand or substrate binding, may be incorrectly modelled. It is thus recommended to include water molecules explicitly in structure refinement of proteins in aqueous solution based on nuclear magnetic resonance (NMR) or other experimentally measured data.

Model-Free or Not?

Relaxation in nuclear magnetic resonance is a powerful method for obtaining spatially resolved, timescale-specific dynamics information about molecular systems. However, dynamics in biomolecular systems are generally too complex to be fully characterized based on NMR data alone. This is a familiar problem, addressed by the Lipari-Szabo model-free analysis, a method that captures the full information content of NMR relaxation data in case all internal motion of a molecule in solution is sufficiently fast. We investigate model-free analysis, as well as several other approaches, and find that model-free, spectral density mapping, LeMaster's approach, and our detector analysis form a class of analysis methods, for which behavior of the fitted parameters has a well-defined relationship to the distribution of correlation times of motion, independent of the specific form of that distribution. In a sense, they are all "model-free." Of these methods, only detectors are generally applicable to solid-state NMR relaxation data. We further discuss how detectors may be used for comparison of experimental data to data extracted from molecular dynamics simulation, and how simulation may be used to extract details of the dynamics that are not accessible via NMR, where detector analysis can be used to connect those details to experiments. We expect that combined methodology can eventually provide enough insight into complex dynamics to provide highly accurate models of motion, thus lending deeper insight into the nature of biomolecular dynamics.

On the use of 3J-coupling NMR data to derive structural information on proteins

Values of ³J-couplings as obtained from NMR experiments on proteins cannot easily be used to determine protein structure due to the difficulty of accounting for the high sensitivity of intermediate ³J-coupling values (4-8 Hz) to the averaging period that must cover the conformational variability of the torsional angle related to the ³J-coupling, and due to the difficulty of handling the multiple-valued character of the inverse Karplus relation between torsional angle and ³J-coupling. Both problems can be solved by using ³J-coupling time-averaging local-elevation restraining MD simulation. Application to the protein hen egg white lysozyme using 213 backbone and side-chain ³J-coupling restraints shows that a conformational ensemble compatible with the experimental data can be obtained using this technique, and that accounting for averaging and the ability of the algorithm to escape from local minima for the torsional angle induced by the Karplus relation, are essential for a comprehensive use of ³J-coupling data in protein structure determination.

On the Use of Side-Chain NMR Relaxation Data to Derive Structural and Dynamical Information on Proteins: A Case Study Using Hen Lysozyme

Values of S2CH and S2NH order parameters derived from NMR relaxation measurements on proteins cannot be used straightforwardly to determine protein structure because they cannot be related to a single protein structure, but are defined in terms of an average over a conformational ensemble. Molecular dynamics simulation can generate a conformational ensemble and thus can be used to restrain S2CH and S2NH order parameters towards experimentally derived target values S2CH (exp) and S2NH (exp). Application of S2CH and S2NH order‐parameter restraining MD simulation to bond vectors in 63 side chains of the protein hen egg white lysozyme using 51 S2CH (exp) target values and 28 S2NH (exp) target values shows that a conformational ensemble compatible with the experimentally derived data can be obtained by using this technique. It is observed that S2CH order‐parameter restraining of C−H bonds in methyl groups is less reliable than S2NH order‐parameter restraining because of the possibly less valid assumptions and approximations used to derive experimental S2CH (exp) values from NMR relaxation measurements and the necessity to adopt the assumption of uniform rotational motion of methyl C−H bonds around their symmetry axis and of the independence of these motions from each other. The restrained simulations demonstrate that side chains on the protein surface are highly dynamic. Any hydrogen bonds they form and that appear in any of four different crystal structures, are fluctuating with short lifetimes in solution.

Orientational and Folding Thermodynamics via Electric Dipole Moment Restraining

The projection of molecular processes onto a small set of relevant descriptors, the so-called reaction coordinates or collective variables (CVs), is a technique nowadays routinely employed by the biomolecular simulation community. In this work, we implemented two CVs to manipulate the orientation (i.e., angle) (μ⃗_a) and magnitude (|μ⃗|) of the electric dipole moment. In doing so, we studied the thermodynamics of water orientation under the application of external voltages and the folding of two polypeptides at zero-field conditions. The projection of the free-energy [potential of mean force (PMF)] along water orientation defined an upper limit of around 0.3 V for irrelevant thermodynamic effects. On the other hand, sufficiently strong μ⃗_a restraints applied on 12-alanine (Ala₁₂) triggered structural effects because of the alignment of local dipoles; for lower restraints, a full-body rotation is achieved. The manipulation of |μ⃗| produced strong perturbations on the secondary structure of Ala₁₂, promoting an enhanced sampling to its configurational space. Rigorous free-energy calculations in the form of 2-D PMFs for deca-alanine showed the utility of |μ⃗| as a reaction coordinate to study folding in small α helices. As a whole, we propose that the manipulation of both components of the dipole moment, μ⃗_a and |μ⃗|, provides thermodynamics insights into the structural conformation and stability of biomolecules. These new CVs are implemented in the Colvars module, available for NAMD and LAMMPS.

A Hybrid Hamiltonian for the Accelerated Sampling along Experimental Restraints

In this article, we present an enhanced sampling method based on a hybrid Hamiltonian which combines experimental distance restraints with a bias dependent from multiple path-dependent variables. This simulation method determines the bias-coordinates on the fly and does not require a priori knowledge about reaction coordinates. The hybrid Hamiltonian accelerates the sampling of proteins, and, combined with experimental distance information, the technique considers the restraints adaptively and in dependency of the system's intrinsic dynamics. We validate the methodology on the dipole relaxation of two water models and the conformational landscape of dialanine. Using experimental NMR-restraint data, we explore the folding landscape of the TrpCage mini-protein and in a second example apply distance restraints from chemical crosslinking/mass spectrometry experiments for the sampling of the conformation space of the Killer Cell Lectin-like Receptor Subfamily B Member 1A (NKR-P1A). The new methodology has the potential to adaptively introduce experimental restraints without affecting the conformational space of the system along an ergodic trajectory. Since only a limited number of input- and no-order parameters are required for the setup of the simulation, the method is broadly applicable and has the potential to be combined with coarse-graining methods.

Using Complementary NMR Data Sets To Detect Inconsistencies and Model Flaws in the Structure Determination of Human Interleukin-4

The derivation of protein structure from values of observable quantities measured in NMR experiments is a rather nontrivial task due to (i) the limited number of data compared to degrees of freedom of a protein, (ii) the uncertainty inherent to the function connecting an observable quantity to molecular structure, (iii) the finite quality of biomolecular models and force fields used in structure refinement, and (iv) the conformational freedom of a protein in aqueous solution, which requires extensive conformational sampling and appropriate conformational averaging when calculating or restraining to sets of NMR data. The protein interleukin-4 (IL-4) has been taken as a test case using NOE distances, S² order parameters, and ³J-couplings as test data and the former two types of data as restraints. It is shown that, by combining sets of different, complementary NMR data as restraints in MD simulations, inconsistencies in the data or flaws in the model and procedures used to derive protein structure from NMR data can be detected. This leads to an improved structural interpretation of such data particularly in more mobile loop regions.

Interpretation of Seemingly Contradictory Data: Low NMR S2 Order Parameters Observed in Helices and High NMR S2 Order Parameters in Disordered Loops of the Protein hGH at Low pH

At low pH, human growth hormone (hGH) adopts a partially folded state, in which the native helices are maintained, but the long loop regions and side-chain packing become disordered. Some of the S² order parameters for backbone N-H vectors derived from NMR relaxation measurements on hGH at low pH initially seem contradictory. Three isolated residues (15, 20, and 171) in helices A and D exhibit low order parameter values (<0.5) indicating flexibility, whereas residue 143 in the centre of a long flexible loop region has a high order parameter (0.82). Using S² order parameter restraining MD simulations, this paradox has been resolved. Low S² values in helices are due to the presence of a mixture of 3₁₀ -helical and α-helical hydrogen bonds. High S² values in relatively disordered parts of a protein may be due to fluctuating networks of hydrogen bonds between the backbone and the side chains, which restrict the motion of N-H bond vectors.

Deriving Structural Information from Experimentally Measured Data on Biomolecules

During the past half century, the number and accuracy of experimental techniques that can deliver values of observables for biomolecular systems have been steadily increasing. The conversion of a measured value Q^exp of an observable quantity Q into structural information is, however, a task beset with theoretical and practical problems: 1) insufficient or inaccurate values of Q^exp , 2) inaccuracies in the function Q(r→) used to relate the quantity Q to structure r→ , 3) how to account for the averaging inherent in the measurement of Q^exp , 4) how to handle the possible multiple-valuedness of the inverse r→(Q) of the function Q(r→) , to mention a few. These apply to a variety of observable quantities Q and measurement techniques such as X-ray and neutron diffraction, small-angle and wide-angle X-ray scattering, free-electron laser imaging, cryo-electron microscopy, nuclear magnetic resonance, electron paramagnetic resonance, infrared and Raman spectroscopy, circular dichroism, Förster resonance energy transfer, atomic force microscopy and ion-mobility mass spectrometry. The process of deriving structural information from measured data is reviewed with an eye to non-experts and newcomers in the field using examples from the literature of the effect of the various choices and approximations involved in the process. A list of choices to be avoided is provided.

Bayesian ensemble refinement by replica simulations and reweighting

We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy "ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraints should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the "Bayesian inference of ensembles" method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations.

On the use of time-averaging restraints when deriving biomolecular structure from ³J -coupling values obtained from NMR experiments

Deriving molecular structure from [Formula: see text]-couplings obtained from NMR experiments is a challenge due to (1) the uncertainty in the Karplus relation [Formula: see text] connecting a [Formula: see text]-coupling value to a torsional angle [Formula: see text], (2) the need to account for the averaging inherent to the measurement of [Formula: see text]-couplings, and (3) the sampling road blocks that may emerge due to the multiple-valuedness of the inverse function [Formula: see text] of the function [Formula: see text]. Ways to properly handle these issues in structure refinement of biomolecules are discussed and illustrated using the protein hen egg white lysozyme as example.

Synergy between NMR measurements and MD simulations of protein/RNA complexes: application to the RRMs, the most common RNA recognition motifs

RNA recognition motif (RRM) proteins represent an abundant class of proteins playing key roles in RNA biology. We present a joint atomistic molecular dynamics (MD) and experimental study of two RRM-containing proteins bound with their single-stranded target RNAs, namely the Fox-1 and SRSF1 complexes. The simulations are used in conjunction with NMR spectroscopy to interpret and expand the available structural data. We accumulate more than 50 μs of simulations and show that the MD method is robust enough to reliably describe the structural dynamics of the RRM-RNA complexes. The simulations predict unanticipated specific participation of Arg142 at the protein-RNA interface of the SRFS1 complex, which is subsequently confirmed by NMR and ITC measurements. Several segments of the protein-RNA interface may involve competition between dynamical local substates rather than firmly formed interactions, which is indirectly consistent with the primary NMR data. We demonstrate that the simulations can be used to interpret the NMR atomistic models and can provide qualified predictions. Finally, we propose a protocol for 'MD-adapted structure ensemble' as a way to integrate the simulation predictions and expand upon the deposited NMR structures. Unbiased μs-scale atomistic MD could become a technique routinely complementing the NMR measurements of protein-RNA complexes.

Investigation of the structural preference and flexibility of the loop residues in amyloid fibrils of the HET-s prion

The structural variability of HET-s(218–289) loops is restricted by the β-sheet core.