Starting-Condition Dependence of Order Parameters Derived from Molecular Dynamics Simulations

Accuracy and Reproducibility of Lipari-Szabo Order Parameters From Molecular Dynamics

The Lipari-Szabo generalized order parameter probes the picosecond to nanosecond time scale motions of a protein and is useful for rationalizing a multitude of biological processes such as protein recognition and ligand binding. Although these fast motions are an important and intrinsic property of proteins, it remains unclear what simulation conditions are most suitable to reproduce methyl symmetry axis side chain order parameter data (Saxis2) from molecular dynamics simulations. In this study, we show that, while Saxis2 tends to converge within tens of nanoseconds, it is essential to run 10 to 20 replicas starting from configurations close to the experimental structure to obtain the best agreement with experimental Saxis2 values. Additionally, in a comparison of force fields, AMBER ff14SB outperforms CHARMM36m in accurately capturing these fast time scale motions, and we suggest that the origin of this performance gap is likely attributed to differences in side chain torsional parametrization and not due to differences in the global protein conformations sampled by the force fields. This study provides insight into obtaining accurate and reproducible Saxis2 values from molecular simulations and underscores the necessity of using replica simulations to compute equilibrium properties.

Interplay of halogen bonding and solvation in protein-ligand binding

Halogen bonding is increasingly utilized in efforts to achieve high affinity and selectivity of molecules designed to bind proteins, making it paramount to understand the relationship between structure, dynamics, and thermodynamic driving forces. We present a detailed analysis addressing this problem using a series of protein-ligand complexes involving single halogen substitutions - F, Cl, Br, and I - and nearly identical structures. Isothermal titration calorimetry reveals an increasingly favorable binding enthalpy from F to I that correlates with the halogen size and σ-hole electropositive character, but is partially counteracted by unfavorable entropy, which is constant from F to Cl and Br, but worse for I. Consequently, the binding free energy is roughly equal for Cl, Br, and I. QM and solvation-free-energy calculations reflect an intricate balance between halogen bonding, hydrogen bonds, and solvation. These advances have the potential to aid future drug design initiatives involving halogenated compounds.

Probing Methyl Group Dynamics in Proteins by NMR Cross-Correlated Dipolar Relaxation and Molecular Dynamics Simulations

Nuclear magnetic resonance (NMR) spin relaxation is the most informative approach to experimentally probe the internal dynamics of proteins on the picosecond to nanosecond time scale. At the same time, molecular dynamics (MD) simulations of biological macromolecules are steadily improving through better physical models, enhanced sampling methods, and increased computational power, and they provide exquisite information about flexibility and its role in protein stability and molecular interactions. Many examples have shown that MD is now adept in probing protein backbone motion, but improvements are still required toward a quantitative description of the dynamics of side chains, for example, probed by the dynamics of methyl groups. Thus far, the comparison of computation with experiment for side chain dynamics has primarily focused on the relaxation of ¹³C and ²H nuclei induced by autocorrelated variation of spin interactions. However, the cross-correlation of ¹³C-¹H dipolar interactions in methyl groups offers an attractive alternative. Here, we establish a computational framework to extract cross-correlation relaxation parameters of methyl groups in proteins from all-atom MD simulations. To demonstrate the utility of the approach, cross-correlation relaxation rates of ubiquitin are computed from MD simulations performed with the AMBER99SB*-ILDN and CHARMM36 force fields. Simulation results were found to agree well with those obtained by experiment. Moreover, the data obtained with the two force fields are highly consistent.

How Much Entropy Is Contained in NMR Relaxation Parameters?

Solution-state NMR relaxation experiments are the cornerstone to study internal protein dynamics at an atomic resolution on time scales that are faster than the overall rotational tumbling time τ_R. Since the motions described by NMR relaxation parameters are connected to thermodynamic quantities like conformational entropies, the question arises how much of the total entropy is contained within this tumbling time. Using all-atom molecular dynamics simulations of the T4 lysozyme, we found that entropy buildup is rather fast for the backbone, such that the majority of the entropy is indeed contained in the short-time dynamics. In contrast, the contribution of the slow dynamics of side chains on time scales beyond τ_R on the side-chain conformational entropy is significant and should be taken into account for the extraction of accurate thermodynamic properties.

Exploring ligand dynamics in protein crystal structures with ensemble refinement

Understanding the dynamics of ligands bound to proteins is an important task in medicinal chemistry and drug design. However, the dominant technique for determining protein-ligand structures, X-ray crystallography, does not fully account for dynamics and cannot accurately describe the movements of ligands in protein binding sites. In this article, an alternative method, ensemble refinement, is used on six protein-ligand complexes with the aim of understanding the conformational diversity of ligands in protein crystal structures. The results show that ensemble refinement sometimes indicates that the flexibility of parts of the ligand and some protein side chains is larger than that which can be described by a single conformation and atomic displacement parameters. However, since the electron-density maps are comparable and Rfree values are slightly increased, the original crystal structure is still a better model from a statistical point of view. On the other hand, it is shown that molecular-dynamics simulations and automatic generation of alternative conformations in crystallographic refinement confirm that the flexibility of these groups is larger than is observed in standard refinement. Moreover, the flexible groups in ensemble refinement coincide with groups that give high atomic displacement parameters or non-unity occupancy if optimized in standard refinement. Therefore, the conformational diversity indicated by ensemble refinement seems to be qualitatively correct, indicating that ensemble refinement can be an important complement to standard crystallographic refinement as a tool to discover which parts of crystal structures may show extensive flexibility and therefore are poorly described by a single conformation. However, the diversity of the ensembles is often exaggerated (probably partly owing to the rather poor force field employed) and the ensembles should not be trusted in detail.

Entropy-Entropy Compensation between the Protein, Ligand, and Solvent Degrees of Freedom Fine-Tunes Affinity in Ligand Binding to Galectin-3C

Molecular recognition is fundamental to biological signaling. A central question is how individual interactions between molecular moieties affect the thermodynamics of ligand binding to proteins and how these effects might propagate beyond the immediate neighborhood of the binding site. Here, we investigate this question by introducing minor changes in ligand structure and characterizing the effects of these on ligand affinity to the carbohydrate recognition domain of galectin-3, using a combination of isothermal titration calorimetry, X-ray crystallography, NMR relaxation, and computational approaches including molecular dynamics (MD) simulations and grid inhomogeneous solvation theory (GIST). We studied a congeneric series of ligands with a fluorophenyl-triazole moiety, where the fluorine substituent varies between the ortho, meta, and para positions (denoted O, M, and P). The M and P ligands have similar affinities, whereas the O ligand has 3-fold lower affinity, reflecting differences in binding enthalpy and entropy. The results reveal surprising differences in conformational and solvation entropy among the three complexes. NMR backbone order parameters show that the O-bound protein has reduced conformational entropy compared to the M and P complexes. By contrast, the bound ligand is more flexible in the O complex, as determined by ¹⁹F NMR relaxation, ensemble-refined X-ray diffraction data, and MD simulations. Furthermore, GIST calculations indicate that the O-bound complex has less unfavorable solvation entropy compared to the other two complexes. Thus, the results indicate compensatory effects from ligand conformational entropy and water entropy, on the one hand, and protein conformational entropy, on the other hand. Taken together, these different contributions amount to entropy-entropy compensation among the system components involved in ligand binding to a target protein.

Computational Design of Enantiocomplementary Epoxide Hydrolases for Asymmetric Synthesis of Aliphatic and Aromatic Diols

The use of enzymes in preparative biocatalysis often requires tailoring enzyme selectivity by protein engineering. Herein we explore the use of computational library design and molecular dynamics simulations to create variants of limonene epoxide hydrolase that produce enantiomeric diols from meso‐epoxides. Three substrates of different sizes were targeted: cis‐2,3‐butene oxide, cyclopentene oxide, and cis‐stilbene oxide. Most of the 28 designs tested were active and showed the predicted enantioselectivity. Excellent enantioselectivities were obtained for the bulky substrate cis‐stilbene oxide, and enantiocomplementary mutants produced (S,S)‐ and (R,R)‐stilbene diol with >97 % enantiomeric excess. An (R,R)‐selective mutant was used to prepare (R,R)‐stilbene diol with high enantiopurity (98 % conversion into diol, >99 % ee). Some variants displayed higher catalytic rates (k_cat) than the original enzyme, but in most cases K_M values increased as well. The results demonstrate the feasibility of computational design and screening to engineer enantioselective epoxide hydrolase variants with very limited laboratory screening.

Interplay between Conformational Entropy and Solvation Entropy in Protein-Ligand Binding

Understanding the driving forces underlying molecular recognition is of fundamental importance in chemistry and biology. The challenge is to unravel the binding thermodynamics into separate contributions and to interpret these in molecular terms. Entropic contributions to the free energy of binding are particularly difficult to assess in this regard. Here we pinpoint the molecular determinants underlying differences in ligand affinity to the carbohydrate recognition domain of galectin-3, using a combination of isothermal titration calorimetry, X-ray crystallography, NMR relaxation, and molecular dynamics simulations followed by conformational entropy and grid inhomogeneous solvation theory (GIST) analyses. Using a pair of diastereomeric ligands that have essentially identical chemical potential in the unbound state, we reduced the problem of dissecting the thermodynamics to a comparison of the two protein-ligand complexes. While the free energies of binding are nearly equal for the R and S diastereomers, greater differences are observed for the enthalpy and entropy, which consequently exhibit compensatory behavior, ΔΔ H°(R - S) = -5 ± 1 kJ/mol and - TΔΔ S°(R - S) = 3 ± 1 kJ/mol. NMR relaxation experiments and molecular dynamics simulations indicate that the protein in complex with the S-stereoisomer has greater conformational entropy than in the R-complex. GIST calculations reveal additional, but smaller, contributions from solvation entropy, again in favor of the S-complex. Thus, conformational entropy apparently dominates over solvation entropy in dictating the difference in the overall entropy of binding. This case highlights an interplay between conformational entropy and solvation entropy, pointing to both opportunities and challenges in drug design.

Assessing the stability of free-energy perturbation calculations by performing variations in the method

We have calculated relative binding affinities for eight tetrafluorophenyl-triazole-thiogalactoside inhibitors of galectin-3 with the alchemical free-energy perturbation approach. We obtain a mean absolute deviation from experimental estimates of only 2-3 kJ/mol and a correlation coefficient (R²) of 0.5-0.8 for seven relative affinities spanning a range of up to 11 kJ/mol. We also studied the effect of using different methods to calculate the charges of the inhibitor and different sizes of the perturbed group (the atoms that are described by soft-core potentials and are allowed to have differing coordinates). However, the various approaches gave rather similar results and it is not possible to point out one approach as consistently and significantly better than the others. Instead, we suggest that such small and reasonable variations in the computational method can be used to check how stable the calculated results are and to obtain a more accurate estimate of the uncertainty than if performing only one calculation with a single computational setup.

Molecular Basis of S100A1 Activation at Saturating and Subsaturating Calcium Concentrations

The S100A1 protein mediates a wide variety of physiological processes through its binding of calcium (Ca(2+)) and endogenous target proteins. S100A1 presents two Ca(2+)-binding domains: a high-affinity "canonical" EF (cEF) hand and a low-affinity "pseudo" EF (pEF) hand. Accumulating evidence suggests that both Ca(2+)-binding sites must be saturated to stabilize an open state conducive to peptide recognition, yet the pEF hand's low affinity limits Ca(2+) binding at normal physiological concentrations. To understand the molecular basis of Ca(2+) binding and open-state stabilization, we performed 100 ns molecular dynamics simulations of S100A1 in the apo/holo (Ca(2+)-free/bound) states and a half-saturated state, for which only the cEF sites are Ca(2+)-bound. Our simulations indicate that the pattern of oxygen coordination about Ca(2+) in the cEF relative to the pEF site contributes to the former's higher affinity, whereas Ca(2+) binding strongly reshapes the protein's conformational dynamics by disrupting β-sheet coupling between EF hands. Moreover, modeling of the half-saturated configuration suggests that the open state is unstable and reverts toward a closed state in the absence of the pEF Ca(2+) ion. These findings indicate that Ca(2+) binding at the cEF site alone is insufficient to stabilize opening; thus, posttranslational modification of the protein may be required for target peptide binding at subsaturating intracellular Ca(2+) levels.

NMR Order Parameter Determination from Long Molecular Dynamics Trajectories for Objective Comparison with Experiment

Functional protein motions covering a wide range of time scales can be studied, among other techniques, by NMR and by molecular dynamics (MD) computer simulations. MD simulations of proteins now routinely extend into the hundreds of nanoseconds time scale range exceeding the overall tumbling correlation times of proteins in solution by several orders of magnitude. This provides a unique opportunity to rigorously validate these simulations by quantitative comparison with model-free order parameters derived from NMR relaxation experiments. However, presently there is no consensus on how such a comparison is best done. We address here how this can be accomplished in a way that is both efficient and objective. For this purpose, we analyze (15)N R1 and R2 and heteronuclear {(1)H}-(15)N NOE NMR relaxation parameters computed from 500 ns MD trajectories of 10 different protein systems using the model-free analysis. The resulting model-free S(2) order parameters are then used as targets for S(2) values computed directly from the trajectories by the iRED method by either averaging over blocks of variable lengths or by using exponentially weighted snapshots (wiRED). We find that the iRED results are capable of reproducing the target S(2) values with high accuracy provided that the averaging window is chosen 5 times the length of the overall tumbling correlation time. These results provide useful guidelines for the derivation of NMR order parameters from MD for a meaningful comparison with their experimental counterparts.

Computationally efficient and accurate enantioselectivity modeling by clusters of molecular dynamics simulations

Computational approaches could decrease the need for the laborious high-throughput experimental screening that is often required to improve enzymes by mutagenesis. Here, we report that using multiple short molecular dynamics (MD) simulations makes it possible to accurately model enantioselectivity for large numbers of enzyme-substrate combinations at low computational costs. We chose four different haloalkane dehalogenases as model systems because of the availability of a large set of experimental data on the enantioselective conversion of 45 different substrates. To model the enantioselectivity, we quantified the frequency of occurrence of catalytically productive conformations (near attack conformations) for pairs of enantiomers during MD simulations. We found that the angle of nucleophilic attack that leads to carbon-halogen bond cleavage was a critical variable that limited the occurrence of productive conformations; enantiomers for which this angle reached values close to 180° were preferentially converted. A cluster of 20-40 very short (10 ps) MD simulations allowed adequate conformational sampling and resulted in much better agreement to experimental enantioselectivities than single long MD simulations (22 ns), while the computational costs were 50-100 fold lower. With single long MD simulations, the dynamics of enzyme-substrate complexes remained confined to a conformational subspace that rarely changed significantly, whereas with multiple short MD simulations a larger diversity of conformations of enzyme-substrate complexes was observed.

Polarized Protein-Specific Charges from Atoms-in-Molecule Electron Density Partitioning

Atomic partial charges for use in traditional force fields for biomolecular simulation are often fit to the electrostatic potentials of small molecules and, hence, neglect large-scale electronic polarization. On the other hand, recent advances in atoms-in-molecule charge derivation schemes show promise for use in flexible force fields but are limited in size by the underlying quantum mechanical calculation of the electron density. Here, we implement the density derived electrostatic and chemical charges method in the linear-scaling density functional theory code ONETEP. Our implementation allows the straightforward derivation of partial atomic charges for systems comprising thousands of atoms, including entire proteins. We demonstrate that the derived charges are chemically intuitive, reproduce ab initio electrostatic potentials of proteins and are transferable between closely related systems. Simulated NMR data derived from molecular dynamics of three proteins using force fields based on the ONETEP charges are in good agreement with experiment.

Correlated motions and residual frustration in thrombin

Thrombin is the central protease in the cascade of blood coagulation proteases. The structure of thrombin consists of a double β-barrel core surrounded by connecting loops and helices. Compared to chymotrypsin, thrombin has more extended loops that are thought to have arisen from insertions in the serine protease that evolved to impart greater specificity. Previous experiments showed thermodynamic coupling between ligand binding at the active site and distal exosites. We present a combined approach of molecular dynamics (MD), accelerated molecular dynamics (AMD), and analysis of the residual local frustration of apo-thrombin and active-site-bound (PPACK-thrombin). Community analysis of the MD ensembles identified changes upon active site occupation in groups of residues linked through correlated motions and physical contacts. AMD simulations, calibrated on measured residual dipolar couplings, reveal that upon active site ligation, correlated loop motions are quenched, but new ones connecting the active site with distal sites where allosteric regulators bind emerge. Residual local frustration analysis reveals a striking correlation between frustrated contacts and regions undergoing slow time scale dynamics. The results elucidate a motional network that probably evolved through retention of frustrated contacts to provide facile conversion between ensembles of states.

Starting-structure dependence of nanosecond timescale intersubstate transitions and reproducibility of MD-derived order parameters

Factors affecting the accuracy of molecular dynamics (MD) simulations are investigated by comparing generalized order parameters for backbone NH vectors of the B3 immunoglobulin-binding domain of streptococcal protein G (GB3) derived from simulations with values obtained from NMR spin relaxation (Yao L, Grishaev A, Cornilescu G, Bax A, J Am Chem Soc 2010;132:4295-4309.). Choices for many parameters of the simulations, such as buffer volume, water model, or salt concentration, have only minor influences on the resulting order parameters. In contrast, seemingly minor conformational differences in starting structures, such as orientations of sidechain hydroxyl groups, resulting from applying different protonation algorithms to the same structure, have major effects on backbone dynamics. Some, but not all, of these effects are mitigated by increased sampling in simulations. Most discrepancies between simulated and experimental results occur for residues located at the ends of secondary structures and involve large amplitude nanosecond timescale transitions between distinct conformational substates. These transitions result in autocorrelation functions for bond vector reorientation that do not converge when calculated over individual simulation blocks, typically of length similar to the overall rotational diffusion time. A test for convergence before averaging the order parameters from different blocks results in better agreement between order parameters calculated from different sets of simulations and with NMR-derived order parameters. Thus, MD-derived order parameters are more strongly affected by transitions between conformational substates than by fluctuations within individual substates themselves, while conformational differences in the starting structures affect the frequency and scale of such transitions.

The dynamic structure of thrombin in solution

The backbone dynamics of human α-thrombin inhibited at the active site serine were analyzed using R(1), R(2), and heteronuclear NOE experiments, variable temperature TROSY 2D [(1)H-(15)N] correlation spectra, and R(ex) measurements. The N-terminus of the heavy chain, which is formed upon zymogen activation and inserts into the protein core, is highly ordered, as is much of the double beta-barrel core. Some of the surface loops, by contrast, remain very dynamic with order parameters as low as 0.5 indicating significant motions on the ps-ns timescale. Regions of the protein that were thought to be dynamic in the zymogen and to become rigid upon activation, in particular the γ-loop, the 180s loop, and the Na(+) binding site have order parameters below 0.8. Significant R(ex) was observed in most of the γ-loop, in regions proximal to the light chain, and in the β-sheet core. Accelerated molecular dynamics simulations yielded a molecular ensemble consistent with measured residual dipolar couplings that revealed dynamic motions up to milliseconds. Several regions, including the light chain and two proximal loops, did not appear highly dynamic on the ps-ns timescale, but had significant motions on slower timescales.

Dynamics and calcium association to the N-terminal regulatory domain of human cardiac troponin C: a multiscale computational study

Troponin C (TnC) is an important regulatory molecule in cardiomyocytes. Calcium binding to site II in TnC initiates a series of molecular events that result in muscle contraction. The most direct change upon Ca(2+) binding is an opening motion of the molecule that exposes a hydrophobic patch on the surface allowing for Troponin I to bind. Molecular dynamics simulations were used to elucidate the dynamics of this crucial protein in three different states: apo, Ca(2+)-bound, and Ca(2+)-TnI-bound. Dynamics between the states are compared, and the Ca(2+)-bound system is investigated for opening motions. On the basis of the simulations, NMR chemical shifts and order parameters are calculated and compared with experimental observables. Agreement indicates that the simulations sample the relevant dynamics of the system. Brownian dynamics simulations are used to investigate the calcium association of TnC. We find that calcium binding gives rise to correlative motions involving the EF hand and collective motions conducive of formation of the TnI-binding interface. We furthermore indicate the essential role of electrostatic steering in facilitating diffusion-limited binding of Ca(2+).

Will molecular dynamics simulations of proteins ever reach equilibrium?

We show that conformational entropies calculated for five proteins and protein-ligand complexes with dihedral-distribution histogramming, the von Mises approach, or quasi-harmonic analysis do not converge to any useful precision even if molecular dynamics (MD) simulations of 380-500 ns length are employed (the uncertainty is 12-89 kJ mol(-1)). To explain this, we suggest a simple protein model involving dihedrals with effective barriers forming a uniform distribution and show that for such a model, the entropy increases logarithmically with time until all significantly populated dihedral states have been sampled, in agreement with the simulations (during the simulations, 52-70% of the available dihedral phase space has been visited). This is also confirmed by the analysis of the trajectories of a 1 ms simulation of bovine pancreatic trypsin inhibitor (31 kJ mol(-1) difference in the entropy between the first and second part of the simulation). Strictly speaking, this means that it is practically impossible to equilibrate MD simulations of proteins. We discuss the implications of such a lack of strict equilibration of protein MD simulations and show that ligand-binding free energies estimated with the MM/GBSA method (molecular mechanics with generalised Born and surface-area solvation) vary by 3-15 kJ mol(-1) during a 500 ns simulation (the higher estimate is caused by rare conformational changes), although they involve a questionable but well-converged normal-mode entropy estimate, whereas free energies estimated by free-energy perturbation vary by less than 0.6 kJ mol(-1) for the same simulation.

Synergistic applications of MD and NMR for the study of biological systems

Modern biological sciences are becoming more and more multidisciplinary. At the same time, theoretical and computational approaches gain in reliability and their field of application widens. In this short paper, we discuss recent advances in the areas of solution nuclear magnetic resonance (NMR) spectroscopy and molecular dynamics (MD) simulations that were made possible by the combination of both methods, that is, through their synergistic use. We present the main NMR observables and parameters that can be computed from simulations, and how they are used in a variety of complementary applications, including dynamics studies, model-free analysis, force field validation, and structural studies.

NMR order parameters calculated in an expanding reference frame: identifying sites of short- and long-range motion

NMR order parameters are calculated from molecular dynamics computer simulations of ubiquitin and the apo (Ca(2+)-free) state of calbindin D(9k). Calculations are performed in an expanding reference frame so as to discriminate between the effects of short- and long-range motions. This approach reveals that the dominant contributions to the order parameters are short-range. Longer-range contributions are limited to specific sites, many of which have been recognized in previous studies of correlated motions. These sites are identified on the basis of an effective reorientational number, n ( eff ). Not only does this parameter identify sites of short- and long-range motion, it also provides a way of evaluating the separability condition that is key to the Lipari-Szabo model-free method. When analyzed in conjunction with the Prompers-Brüschweiler separability index, the n ( eff ) values indicate that longer-range motions play a more prominent role in apo calbindin than they do in ubiquitin.