Diffuse scattering in protein crystallography

Molecular-dynamics simulations of macromolecular diffraction, part II: Analysis of protein crystal simulations

Molecular-dynamics (MD) simulations have contributed substantially to our understanding of protein structure and dynamics, yielding insights into many biological processes including protein folding, drug binding, and mechanisms of protein-protein interactions. Much of what is known about protein structure comes from macromolecular crystallography (MX) experiments. MD simulations of protein crystals are useful in the study of MX because the simulations can be analyzed to calculate almost any crystallographic observable of interest, from atomic coordinates to structure factors and densities, B-factors, multiple conformations and their populations/occupancies, and diffuse scattering intensities. Computing resources and software to support crystalline MD simulations are now readily available to many researchers studying protein structure and dynamics and who may be interested in advanced interpretation of MX data, including diffuse scattering. In this work, we outline methods of analyzing MD simulations of protein crystals and provide accompanying Jupyter notebooks as practical resources for researchers wishing to perform similar analyses on their own systems of interest.

Molecular Dynamics Simulations of Macromolecular Crystals

The structures of biological macromolecules would not be known to their present extent without X-ray crystallography. Most simulations of globular proteins in solution begin by surrounding the crystal structure of the monomer in a bath of water molecules, but the standard simulation employing periodic boundary conditions is already close to a crystal lattice environment. With simple protocols, the same software and molecular models can perform simulations of the crystal lattice, including all asymmetric units and solvent to fill the box. Throughout the history of molecular dynamics, studies of crystal lattices have served to investigate the quality of the underlying force fields, correlate the simulated ensembles to experimental structure factors, and extrapolate the behavior in lattices to behavior in solution. Powerful new computers are enabling molecular simulations with greater realism and statistical convergence. Meanwhile, the advent of exciting new methods in crystallography, including femtosecond free-electron lasers and image reconstruction for time-resolved crystallography on slurries of small crystals, is expanding the range of structures accessible to X-ray diffraction. We review past fusions of simulations and crystallography, then look ahead to the ways that simulations of crystal structures will enhance structural biology in the future.

Intermolecular correlations are necessary to explain diffuse scattering from protein crystals

Conformational changes drive protein function, including catalysis, allostery and signaling. X-ray diffuse scattering from protein crystals has frequently been cited as a probe of these correlated motions, with significant potential to advance our understanding of biological dynamics. However, recent work has challenged this prevailing view, suggesting instead that diffuse scattering primarily originates from rigid-body motions and could therefore be applied to improve structure determination. To investigate the nature of the disorder giving rise to diffuse scattering, and thus the potential applications of this signal, a diverse repertoire of disorder models was assessed for its ability to reproduce the diffuse signal reconstructed from three protein crystals. This comparison revealed that multiple models of intramolecular conformational dynamics, including ensemble models inferred from the Bragg data, could not explain the signal. Models of rigid-body or short-range liquid-like motions, in which dynamics are confined to the biological unit, showed modest agreement with the diffuse maps, but were unable to reproduce experimental features indicative of long-range correlations. Extending a model of liquid-like motions to include disorder across neighboring proteins in the crystal significantly improved agreement with all three systems and highlighted the contribution of intermolecular correlations to the observed signal. These findings anticipate a need to account for intermolecular disorder in order to advance the interpretation of diffuse scattering to either extract biological motions or aid structural inference.

X-ray Scattering Studies of Protein Structural Dynamics

X-ray scattering is uniquely suited to the study of disordered systems and thus has the potential to provide insight into dynamic processes where diffraction methods fail. In particular, while X-ray crystallography has been a staple of structural biology for more than half a century and will continue to remain so, a major limitation of this technique has been the lack of dynamic information. Solution X-ray scattering has become an invaluable tool in structural and mechanistic studies of biological macromolecules where large conformational changes are involved. Such systems include allosteric enzymes that play key roles in directing metabolic fluxes of biochemical pathways, as well as large, assembly-line type enzymes that synthesize secondary metabolites with pharmaceutical applications. Furthermore, crystallography has the potential to provide information on protein dynamics via the diffuse scattering patterns that are overlaid with Bragg diffraction. Historically, these patterns have been very difficult to interpret, but recent advances in X-ray detection have led to a renewed interest in diffuse scattering analysis as a way to probe correlated motions. Here, we will review X-ray scattering theory and highlight recent advances in scattering-based investigations of protein solutions and crystals, with a particular focus on complex enzymes.

Acoustic vibrations contribute to the diffuse scatter produced by ribosome crystals

The diffuse scattering pattern produced by frozen crystals of the 70S ribosome from Thermus thermophilus is as highly structured as it would be if it resulted entirely from domain-scale motions within these particles. However, the qualitative properties of the scattering pattern suggest that acoustic displacements of the crystal lattice make a major contribution to it.

On the relationship between diffraction patterns and motions in macromolecular crystals

The quality of many macromolecular crystal structures published recently has been enhanced through the use of new methods for treating the effects of molecular motion and disorder on diffraction patterns, among them a technique called translation, libration, screw-axis (TLS) parameterization. TLS parameterization rationalizes those effects in terms of domain-scale, rigid-body motions and, interestingly, the models for molecular motion that emerge when macromolecular diffraction data are analyzed this way often make sense biochemically. Here it is pointed out that all such models should be treated with caution until it is shown that they are consistent with the diffuse scatter produced by the crystals that provided the diffraction data from which they derive.

Application of elastic network models to proteins in the crystalline state

Normal mode analysis using elastic network models has grown popular for probing the low-frequency collective dynamics of proteins and other biomolecular assemblies. In most previous studies, these models were validated by comparing calculated atomic fluctuations for isolated proteins with experimental temperature factors determined in the crystalline state, although there were also hints that including crystal contacts in the calculations has an impact on the comparison. In this study, a set of 83 ultra-high resolution crystal structures with experimentally determined anisotropic displacement parameters is used to evaluate several C(alpha)-based elastic network models that either ignore or treat the crystal environment in different ways; the latter include using periodic boundary conditions defined with respect to the asymmetric unit or the primitive unit cell as well as using the Born-von Kármán boundary condition that accounts for lattice vibrations. For all elastic network models, treating the crystal environment leads to better agreement with experimental anisotropic displacement parameters with the Born-von Kármán boundary condition giving the best agreement. Atomic correlations over the entire protein are clearly affected by the presence of the crystal contacts and fairly sensitive to the way that the crystal environment is treated. These observations highlight the importance of properly treating the protein system in an environment consistent with experiment when either evaluating approximate protein models or using approximate dynamic models in structural refinement application types. Finally, investigation of the scaling behaviors of the cumulative density of states and the heat capacity indicates that there are still gaps between simplified elastic models and all-atom models for proteins.

Protein dynamics from X-ray crystallography: Anisotropic, global motion in diffuse scattering patterns

Understanding X-ray crystallographic diffuse scattering is likely to improve our comprehension of equilibrium collective protein dynamics. Here, using molecular dynamics (MD) simulation, a detailed analysis is performed of the origins of diffuse scattering in crystalline Staphylococcal nuclease, for which the complete diffuse scattering pattern has been determined experimentally. The hydrogen-atom contribution and the scattering range over which the scattering can be considered to be a sum of solvent and protein scattering are determined. Two models of correlated protein motion are investigated by calculating the model-derived diffuse scattering and comparing with the scattering calculated directly from MD trajectories. In one model, previously used in diffuse scattering interpretation, the atomic displacement correlations decay isotropically with increasing separation. Model correlation lengths are obtained by refining the model scattering against the simulation-derived scattering pattern, and are found to be significantly different from those correlation lengths derived directly from the MD trajectories. Furthermore, the convergence between the model-derived and MD-derived scattering is poor. The second model, in which the displacement correlations are calculated from the principal components of the MD trajectories, is capable of fully reproducing the MD-derived diffuse scattering if the approximately 50% lowest-frequency modes are included. However, a small number ( approximately 10) of lowest-frequency and largest-amplitude modes dominates the diffuse scattering and thus the correlated protein motions. A detailed analysis of the principal components is performed. In particular, the effective free energy profile associated with each principle mode is analyzed and the eigenfrequency and damping coefficient computed using a model of Brownian dynamics. Those collective modes with effective frequencies below approximately 0.5 THz, including those that determine the diffuse scattering, are overdamped.

Correlated dynamics determining x-ray diffuse scattering from a crystalline protein revealed by molecular dynamics simulation

The dynamical origin of the x-ray diffuse scattering by crystals of a protein, Staphylococcal nuclease, is determined using molecular dynamics simulation. A smooth, nearly isotropic scattering shell at originates from equal contributions from correlations in nearest-neighbor water molecule dynamics and from internal protein motions, the latter consisting of -helix pitch and inter--strand fluctuations. Superposed on the shell are intense, three-dimensional scattering features that originate from a very small number of slowly varying (>10 ns) collective motions. The individual three-dimensional features are assigned to specific collective motions in the protein, and some of these explicitly involve potentially functional active-site deformations.

Fluctuations and correlations in crystalline protein dynamics: a simulation analysis of staphylococcal nuclease

Understanding collective motions in protein crystals is likely to furnish insight into functional protein dynamics and will improve models for refinement against diffraction data. Here, four 10 ns molecular dynamics simulations of crystalline Staphylococcal nuclease are reported and analyzed in terms of fluctuations and correlations in atomic motion. The simulation-derived fluctuations strongly correlate with, but are slightly higher than, the values derived from the experimental B-factors. Approximately 70% of the atomic fluctuations are due to internal protein motion. For 65% of the protein atoms the internal fluctuations converge on the nanosecond timescale. Convergence is much slower for the elements of the interatomic displacement correlation matrix--of these, >80% converge within 1 ns for interatomic distances less, approximately <6 A, but only 10% for separations approximately =12 A. Those collective motions that converged on the nanosecond timescale involve mostly correlations within the beta-barrel or between alpha-helices of the protein. The R-factor with the experimental x-ray diffuse scattering for the crystal, which is determined by the displacement variance-covariance matrix, decreases to 8% after 10 ns simulation. Both the number of converged correlation matrix elements and the R-factor depend logarithmically on time, consistent with a model in which the number of energy minima sampled depends exponentially on the maximum energy barrier crossed. The logarithmic dependence is also extrapolated to predict a convergence time for the whole variance-covariance matrix of approximately 1 micros.

Molecular dynamics simulations of a double unit cell in a protein crystal: volume relaxation at constant pressure and correlation of motions between the two unit cells

Eight molecular dynamics simulations of a double crystal unit cell of ubiquitin were performed to investigate the effects of simulating at constant pressure and of simulating two unit cells compared to a single unit cell. To examine the influence of different simulation conditions, the constant-pressure and constant-volume simulations were each performed with and without counterions and using two different treatments of the long-range electrostatic interactions (lattice-sum and reaction-field methods). The constant-pressure simulations were analyzed in terms of unit cell deformation and accompanying protein deformations. Energetic and structural properties of the proteins in the simulations of the double unit cell were compared to the results of previously reported one-unit-cell simulations. Correlation between the two unit cells was also investigated based on relative translational and rotational movements of the proteins and on dipole fluctuations. The box in the constant-pressure simulations is found to deform slowly to reach convergence only after 5-10 ns. This deformation does not result from a distortion in the structure of the proteins but rather from changes in protein packing within the unit cell. The results of the double-unit-cell simulations are closely similar to the results of the single-unit-cell simulations, and little motional correlation is found between the two unit cells.

Protein conformations explored by difference high-angle solution X-ray scattering: oxidation state and temperature dependent changes in cytochrome C

We demonstrate the use of high-angle X-ray scattering to explore protein conformational states in solution by resolving oxidation state- and temperature-dependent changes in the conformation of horse heart cytochrome c. Several detailed models exist for oxidation-dependent changes in mitochondrial class I c cytochromes determined by X-ray crystallography and solution NMR techniques. These models differ in the magnitude and locations of structural change. Our scattering measurements show that high-angle X-ray scattering can discriminate between these models, and that the experimental scattering data for horse cytochrome c can be best reconciled with selected NMR models for the same protein. These results demonstrate the ability to use high-angle X-ray scattering to resolve conformational states of proteins in solution, and to relate these measurements to detailed structural models. Furthermore, temperature-dependent changes are found in the high angle scattering patterns for horse cytochrome c, illustrating the sensitivity of these measurements to dynamic aspects of protein structure. These results demonstrate the ability to use difference high angle scattering as a quantitative monitor of reaction-linked changes in protein conformation and structural dynamics. Synchrotron-based high-angle scattering holds promise as a widely applicable, high throughput technique for exploring conformational states linked to physiological protein function, for resolving configurational differences between protein structures in solution and crystalline states, and for bridging the gap between solution NMR and crystallographic structure techniques.

X-ray diffuse scattering and rigid-body motion in crystalline lysozyme probed by molecular dynamics simulation

Rigid-body motions are determined from a 1 ns molecular dynamics simulation of the unit cell of orthorhombic hen egg-white lysozyme and their contribution to X-ray diffuse scattering intensities are examined. Using a dynamical cluster technique, groups of backbone atoms that move as approximately rigid bodies are derived from the intramolecular interatomic fluctuation matrix. These groups tend to be local in the sequence or connected by disulphide bonds, and contain on average five residues each, X-ray diffuse scattering patterns, which are sensitive to collective motions, are calculated from the full simulation trajectory (including all the protein degrees of freedom). The results reproduce the main features of the experimental scattering. Diffuse scattering is also calculated from fitted trajectories of the rigid bodies. The full simulation diffuse scattering and atomic displacements are found to be well reproduced by a model in which the backbone atoms form the rigid groups determined using the dynamical cluster technique and the individual side-chains behave as separate rigid bodies: the resulting R-factor with the full simulation scattering is 5%. Quantitatively poorer agreement is obtained from trajectories in which the secondary structural elements of the protein are considered rigid. Rigid whole-molecule and domain motions make only minor contributions to the protein atom displacements. Finally, correlations in the interatomic fluctuations are examined directly using a canonical method.

Motions of calmodulin characterized using both Bragg and diffuse X-ray scattering

BACKGROUND

Calmodulin is a calcium-activated regulatory protein which can bind to many different targets. The protein resembles a highly flexible dumbbell, and bends in the middle as it binds. This and other motions must be understood to formulate a realistic model of calmodulin function.

RESULTS

Using the Bragg reflections from X-ray crystallography, a multiple-conformer refinement of a calmodulin-peptide complex shows anisotropic displacements, with high variations of dihedral angles in several nonhelical domains: the flexible linker; three of the four calcium-binding sites (including both of the N-terminal sites); and a turn connecting the C-terminal EF-hand calcium-binding domains. Three-dimensional maps of the large scale diffuse X-ray scattering data show isotropic liquid-like motions with an unusually small correlation length. Three-dimensional maps of the small scale diffuse streaks show highly coupled, anisotropic motions along the head-to-tail molecular packing direction in the unit cell. There is also weak coupling perpendicular to the head-to-tail packing direction, particularly across a cavity occupied by the disordered linker domain of the molecule.

CONCLUSIONS

Together, the Bragg and diffuse scattering present a self-consistent description of the motions in the flexible linker of calmodulin. The other mobile regions of the protein are also of great interest. In particular, the high variations in the calcium-binding sites are likely to influence how strongly they bind ions. This is especially important in the N-terminal sites, which regulate the activity of the molecule.