Crystallographic evaluation of internal motion of human alpha-lactalbumin refined by full-matrix least-squares method

The low temperature form of human alpha-lactalbumin (HAL) was crystallized from a 2H2O solution and its structure was refined to the R value of 0.119 at 1.15 A resolution by the full-matrix least-squares method. Average estimated standard deviations of atomic parameters for non-hydrogen atoms were 0.038 A for coordinates and 0.044 A2 for anisotropic temperature factors (Uij). The magnitude of equivalent isotropic temperature factors (Ueqv) was highly correlated with the distance from the molecular centroid and fitted to a quadratic equation as a function of atomic coordinates. The atomic thermal motion was rather isotropic in the core region and the anisotropy increased towards the molecular surface. The statistical analysis revealed the out-of-plane motion of main-chain oxygen atoms, indicating that peptide groups are in rotational vibration around a Calpha.Calpha axis. The TLS model, which describes the rigid-body motion in terms of translation, libration, and screw motions, was adopted for the evaluation of the molecular motion and the TLS parameters were determined by the least-squares fit to Uij. The reproduced Ueqvcal from the TLS parameters was in fair agreement with observed Ueqv, but differences were found in regions of residues, 5-22, 44-48, 70-75, and 121-123, where Ueqv was larger than Ueqvcal because of large local motions. To evaluate the internal motion of HAL, the contribution of the rigid-body motion was determined to be 42.4 % of Ueqv in magnitude, which was the highest estimation to satisfy the condition that the Uijint tensors of the internal motion have positive eigen values. The internal motion represented with atomic thermal ellipsoids clearly showed local motions different from those observed in chicken-type lysozymes which have a backbone structure very similar to HAL. The result indicates that the internal motion is closely related to biological function of proteins.     

Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors

In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.     

Hydrodynamics of segmentally flexible macromolecules

Segmentally flexible macromolecules are composed of a few rigid subunits linked by joints which are more or less flexible. The dynamics in solution of this type of macromolecule present special aspects that are reviewed here. Three alternative approaches are described. One is the rigid-body treatment, which is shown to be valid for overall dynamic properties such as translational diffusion and intrinsic viscosity. Another approach is the Harvey-Wegener treatment, which is particularly suited for rotational diffusion. The simplest version of this treatment, which ignores hydrodynamic interaction (HI) effects, is found to be quite accurate when compared to a more rigorous version including HI. A third approach is the Brownian dynamics simulation that, albeit at some computational cost, might describe rigorously cases of arbitrary complexity. This technique has been used to test the approximations in the rigid-body and Harvey-Wegener treatments, thus allowing a better understanding of their validity. Brownian trajectories of simplified models such as the trumbbell and the broken rod have been simulated. The comparison of the decay rates of some correlation functions with the predictions of the two treatments leads to a general conclusion: the Harvey-Wegener treatment determines the initial rate, while the long-time behavior is dominated by the rigid-body relaxation time. As an example of application to a specific biological macromolecule, we present a simulation of an immunoglobulin molecule, showing how Brownian Dynamics can be used to predict rotational and internal dynamics. Another typical example is myosin. Literature data of hydrodynamic properties of whole myosin and the myosin rod are compared with predictions from the Harvey-Wegener and rigid-body treatments. The present situation of the problem on myosin flexibility is analyzed, and some indications are given for future experimental and simulation work.     

The crystal structure of the 1:1 inclusion complex of beta-cyclodextrin with squaric acid.

The crystal and molecular structure of the 1:1 inclusion complex of beta-cyclodextrin (cyclomaltoheptaose) with squaric acid (3,4-dihydroxycyclobutene-1,2-dione) was determined by X-ray diffraction. The complex crystallizes in the monoclinic P2(1) space group and belongs to the monomeric cage-type, characterized by a herringbone-like packing motif. Co-crystallized water molecules are present on seven sites, of which six are fully occupied. The guest molecule is placed inside the beta-cyclodextrin cavity, perpendicular to the plane defined by the glycosidic O-4n atoms, and held in place by direct and water-mediated hydrogen bonds mainly involving symmetry-related beta-cyclodextrin molecules. The accommodation of the planar guest molecule into the beta-cyclodextrin cavity determines a significant distortion of the latter from the sevenfold symmetry.     

Full-matrix least-squares refinement of lysozymes and analysis of anisotropic thermal motion

Crystal structures of turkey egg lysozyme (TEL) and human lysozyme (HL) were refined by full-matrix least-squares method using anisotropic temperature factors. The refinement converged at the conventional R-values of 0.104 (TEL) and 0.115 (HL) for reflections with F_o > 0 to the resolution of 1.12 Å and 1.15 Å, respectively. The estimated r.m.s. coordinate errors for protein atoms were 0.031 Å (TEL) and 0.034 Å (HL). The introduction of anisotropic temperature factors markedly reduced the R-value but did not significantly affect the main chain coordinates. The degree of anisotropy of atomic thermal motion has strong positive correlation with the square of distance from the molecular centroid. The ratio of the radial component of thermal ellipsoid to the r.m.s. magnitude of three principal components has negative correlation with the distance from the molecular centroid, suggesting the domination of libration rather than breathing motion. The TLS model was applied to elucidate the characteristics of the rigid-body motion. The TLS tensors were determined by the least-squares fit to observed temperature factors. The profile of the magnitude of reproduced temperature factors by the TLS method well fitted to that of observed B_eqv. However, considerable disagreement was observed in the shape and orientation of thermal ellipsoid for atoms with large temperature factors, indicating the large contribution of local motion. The upper estimate of the external motion, 67% (TEL) and 61% (HL) of B_eqv, was deduced from the plot of the magnitude of TLS tensors determined for main chain atoms which were grouped into shells according to the distance from the center of libration. In the external motion, the translational portion is predominant and the contribution of libration and screw motion is relatively small. The internal motion, estimated by subtracting the upper estimate of the external motion from the observed temperature factor, is very similar between TEL and HL in spite of the difference in 54 of 130 amino acid residues and in crystal packing, being suggested to reflect the intrinsic internal motion of chicken-type lysozymes. Proteins 30:232–243, 1998. © 1998 Wiley-Liss, Inc.     

The crystal structure of the 1:1 complex of beta-cyclodextrin with trans-cinnamic acid

The crystal structure of the 1:1 complex of beta-cyclodextrin (cyclomaltoheptaose) with trans-cinnamic acid was studied by X-ray diffraction. Two beta-cyclodextrin molecules related by a twofold crystal axis form dimers in the hydrophobic cavity of which, two guest molecules are entirely buried. The complex crystallizes in the monoclinic C2 space group with channel-type molecular packing. The oxygen atoms of the carboxylate group of the trans-cinnamic acid molecule form strong hydrogen bonds with two water molecules lying in the interdimeric space of the hydrophobic channel.     

The transmission of translational seat vibration to the head--II. Horizontal seat vibration

The second part of this study of the six axes of head motion caused by translational seat vibration is concerned with the effect of fore-and-aft (x-axis) and lateral (y-axis) seat vibration. Seat-to-head transmissibilities have been determined at frequencies up to 16 Hz for each of the three translational and three rotational axes of the head during exposure to random vibration of the seat. Repeatability measures within a single subject and studies of the variability across a group of twelve subjects have been conducted with two seating conditions: a rigid seat with a backrest, and the same seat with no backrest. Fore-and-aft seat motion mainly resulted in head motion within the mid-sagittal plane (x-z plane). Without the backrest, transmissibilities for the fore-and-aft, vertical and pitch axes of the head were greatest at about 2 Hz. The backrest greatly increased head vibration at frequencies above 4 Hz and caused a second peak in the transmissibility curves at about 6 to 8 Hz. Lateral seat motion mainly caused lateral head motion with a maximum transmissibility at about 2 Hz. The backrest had little effect on the transmission of lateral vibration to the head. For both axes of excitation inter-subject variability was much greater than intra-subject variability.     

Structure of the complex of beta-cyclodextrin with beta-naphthyloxyacetic acid in the solid state and in aqueous solution

The structure of the complex of beta-cyclodextrin (cyclomaltoheptaose) with beta-naphthyloxyacetic acid was studied in solid state by X-ray diffraction and in aqueous solution by 1H NMR spectroscopy. The complex crystallizes in the channel mode, space group C2, with a stoichiometry of 2:1; two beta-cyclodextrin molecules related by a twofold crystal axis form dimers, in the cavity of which one guest molecule is found on average. The above stoichiometry indicates one guest per beta-CD dimer statistically oriented over two positions or two guest molecules in pi-pi interactions in half of the beta-CD dimers and the rest of the beta-CD dimers empty. In addition, occupancy of 0.5 for the guest per every beta-CD dimer is in accord with the occupancy of the two disordered primary hydroxyls. These two hydroxyl groups, to which the carboxylic oxygen atoms of the guest are hydrogen bonded, point towards the interior of the beta-CD cavity. In aqueous solution, the 1H NMR spectroscopic study indicated that there is a mixture of complexes with host-guest stoichiometries both 1:1 and 2:1.     

Atomic resolution structure of prokaryotic phospholipase A2: analysis of internal motion and implication for a catalytic mechanism

We have found a secreted phospholipase A(2) (PLA(2), EC 3.1.1.4) from Streptomyces violaceoruber A-2688, which is the first PLA(2) identified in prokaryote, and determined its tertiary structure by NMR and X-ray analyses. In this study, we collected the X-ray diffraction data of the bacterial PLA(2) at room temperature (297 K) using conventional MoK(alpha) radiation and refined the structure at a 1.05 A resolution. The atomic resolution analysis led us to introduce disordered conformations and hydrogen atoms into a full anisotropic model. The molecular motion, which is expressed as the sum of rigid-body motion and internal motion of protein, is roughly estimated as the thermal motion when the X-ray diffraction data are collected at room temperature. In this study, we applied a TLS (rigid-body motion in terms of translation, libration, and screw motions) model to analyze the rigid-body motion of the bacterial PLA(2) and calculated the internal motion by subtracting the estimate of the rigid-body motion from the observed anisotropic temperature factor. We also subjected the TLS model to estimate the internal motion of the bovine pancreatic PLA(2) using the anisotropic temperature factor deposited in the Protein Data Bank. Both results indicate that the localization of regions exhibiting larger internal motion in the bacterial PLA(2) is almost the same as that in the bovine pancreatic PLA(2), suggesting that although the tertiary structure of the bacterial PLA(2) is strikingly different from that of the bovine pancreatic PLA(2), the internal motion, which is associated with the calcium(II) ion-binding, phospholipid-binding, and allosteric interfacial activation, is commonly observed in both PLA(2)s.     

Hydrodynamic study of flexibility in immunoglobulin IgG1 using Brownian dynamics and the Monte Carlo simulations of a simple model

A simple bead model is proposed for the antibody molecule immunoglobulin IgG1. The partial flexibility of the hinge is represented by a quadratic potential associated to the angles between arms. Conformational and hydrodynamic properties are calculated using Monte Carlo (rigid-body) and Brownian dynamics simulations. Comparison of experimental and calculated values for some overall properties allows the assignment of dimensions and other model parameters. The Brownian dynamics technique is used next to simulate a rotational correlation function that is comparable with the decay of fluorescence emission anisotropy. This is done with varying flexibility at the hinge. The longest relaxation time shows a threefold decrease when going from the rigid Y-shaped conformation to the completely flexible case. The calculations are in good agreement with the decay times observed for IgG1. A flexibility analysis of the latter indicates that a variability of +/- 55 degrees (standard deviation) in the angle between the Fab arms.     

Manifestation of chiral recognition of camphor enantiomers by alpha-cyclodextrin in longitudinal and transverse relaxation rates of the corresponding 1:2 complexes and determination of the orientation of the guest inside the host capsule

The 1:2 complexes of camphor enantiomers with alpha-cyclodextrin in (2)H(2)O manifest differences in longitudinal and transverse relaxation rates of camphor methyl protons owing to chiral recognition. The relaxation data obtained at two magnetic fields were quantitatively analyzed using the model of anisotropic overall tumbling with internal motion. In experimental conditions (guest-to-host ratio = 1:20, T = 300.6K), all camphor molecules are complexed. The complexes are not rigid but the rotational diffusion of camphor enantiomers embedded inside the capsules formed by two alpha-cyclodextrin hosts is well outside the extreme narrowing region. Both differences in the anisotropic overall tumbling and internal rotation of all methyl groups participate in enantiomeric differentiation of the relaxation rates. Anisotropic tumbling of camphor molecules provides information on the orientation of the guest in the host capsule that for the complex under study could not be obtained by other methods.     

Structure of inclusion complexes of cyclomaltoheptaose (cycloheptaamylose): crystal structure of the 1-adamantanemethanol adduct

Cyclomaltoheptaose (cycloheptaamylose) has been crystallized with 1-adamantanemethanol as the guest molecule. The complex crystallized in space group C222(1), with unit-cell dimensions a = 19.162 (13), b = 23.965 (17), and c = 32.597 (27) A. The structure was solved by rotation-translation search-methods. The cyclomaltoheptaose exists as a dimer in the crystal by means of extensive hydrogen-bonding across the secondary hydroxyl ends of two cyclomaltoheptaose molecules. The two halves of the dimer are related by a crystallographic two-fold axis. The primary hydroxyl ends of two adjacent cyclomaltoheptaose molecules are also related by a crystallographic two-fold axis, but do not directly hydrogen bond to one another. Instead, they are held in place by a strong hydrogen bond from the hydroxyl group of the 1-adamantanemethanol to a primary hydroxyl group on an adjacent cyclomaltoheptaose molecule. Other stabilizing hydrogen bonds are formed via three water molecules which are situated at the primary hydroxyl interface, and others that form parallel columns stabilizing the crystal structure. A unique feature of this complex is the presence of trapped water in the cavity at the secondary hydroxyl interface. This water is distributed over 3 disordered sites. Its presence blocks one possible site for the 1-adamantanemethanol, which, instead, binds near the primary hydroxyl end, with its hydroxyl group and part of the adamantane moiety protruding from the cyclomaltoheptaose.     

Crystallographic dissection of the thermal motion of protein-sugar complex

The crystal structure of turkey egg lysozyme (TEL) complexed with di-N-acetylchitobiose (NAG2) was refined at 1.19 A resolution by the full-matrix least-squares method with anisotropic temperature factors, and its thermal motion was evaluated by the TLS method. The average ESDs of atomic parameters of nonhydrogen atoms were 0.030 A for coordinates and 0.025 A(2) for anisotropic temperature factors. The active site cleft of TEL binds the alpha-anomer of NAG2 in a nonproductive binding mode with its pyranose rings parallel to a beta-sheet. The TEL structure was compared with the re-refined 1.12 A structure of native TEL. The RMS difference for equivalent Calpha atoms was 0.103 A and a relatively large difference was observed in the region of residues 104-125 rather than in the beta-sheet region where NAG2 was bound. In contrast, the temperature factor of the beta-sheet region was significantly decreased by the NAG2 binding. The TLS model that describes the rigid body motion in translation, libration, and screw motion was adopted for the evaluation of the molecular motion of TEL and NAG2, and the TLS parameters were determined by the least-squares fit to U(ij). The contribution of the external motion of TEL was estimated to be 55.8% of the observed temperature factor for the native structure and 45.9% for the NAG2 complex. The internal motion of TEL represented with atomic thermal ellipsoids was very similar between the native and complex structures except the NAG2 binding region. In the structure of NAG2, the rigid body motion dominates the thermal motion. The center of rotation of NAG2, 4.45A far from the center of gravity, is on the nitrogen atom of the acetylamino group that is hydrogen bonded to the main-chain peptide groups of Asn49 and Ala107. The rigid body motion of NAG2 indicates that the acetylamino group is most strongly bound to the active site, and the recognition of this group is a crucial step of the substrate binding.     

The dimeric complex of cyclomaltoheptaose with 1,14-tetradecanedioic acid. Comparison with related complexes

The structure of the complex of cyclomaltoheptaose (beta-cyclodextrin, betaCD) with 1,14-tetradecanedioic acid has been determined and refined to a final R=0.0693 based on 9824 observed reflections. Each diacid molecule threads through two betaCD monomers arranged in dimers thus, forming a [3]pseudorotaxane. The end carboxylic groups of adjacent dimers, far apart and fully hydrated, are associated indirectly through water molecules. The positioning of the carboxylic groups with respect to the betaCD dimer and the H-bonds with water molecules are very similar to these of the corresponding complexes of the diacids with 12 and 13 carbon atoms. The bending in the middle of the aliphatic chain is more prominent, compared to that of the corresponding guests with less carbon atoms, thus the end carboxylic groups stay in the same height of the primary faces of the betaCD dimeric complex. As a consequence of the present structure, more close contacts are observed between calculated H-atoms of the guest and O-atoms of the host inside the cavity. This bending is allowed by the width of the betaCD dimer cavity at the secondary interface region.     

Molecular dynamics (MD) simulations for the prediction of chiral discrimination of N-acetylphenylalanine enantiomers by cyclomaltoheptaose (beta-cyclodextrin, beta-CD) based on the MM-PBSA (molecular mechanics-Poisson-Boltzmann surface area) approach

Molecular dynamics (MD) simulations were performed for the prediction of chiral discrimination of N-acetylphenylalanine enantiomers by cyclomaltoheptaose (beta-cyclodextrin, beta-CD). Binding free energies and various conformational properties were obtained using by the MM-PBSA (molecular mechanics Poisson-Boltzmann/surface area) approach. The calculated relative difference (DeltaDeltabinding) of binding free energy was in fine agreement with the experimentally determined value. The difference of rotameric distributions of guest N-acetylphenylalanine enantiomers complexed with the host, beta-CD, was observed after the conformational analyses, suggesting that the conformational changes of guest captured within host cavity would be a decisive factor for enantiodifferentiation at a molecular level.     

Transport properties of rigid bent-rod macromolecules and of semiflexible broken rods in the rigid-body treatment. Analysis of the flexibility of myosin rod.

The translational diffusion coefficients, rotational relaxation times and intrinsic viscosities of rigid bent rods, composed by two rodlike arms joined rigidly at an angle alpha, have been evaluated for varying conformation using the latest advances in hydrodynamic theory. We have considered semiflexible rods in which the joint is an elastic hinge or swivel, with a potential V(alpha) = 1/2Q alpha 2 with constant Q. Accepting the rigid-body treatment, we calculate properties of broken rods by averaging alpha-dependent values for rigid rods. The results are finally used to interpret literature values of the properties of myosin rod. Q is regarded as an adjustable parameter, and the value fitted is such that the average bending angle of myosin rod is approximately 60 degrees.     

Protein motions at zero-total angular momentum: the importance of long-range correlations

A constant-energy molecular dynamics simulation is used to monitor protein motion at zero-total angular momentum. With a simple protein model, it is shown that overall rotation is possible at zero-total angular momentum as a result of flexibility. Since the rotational motion is negligible on a time scale of 1000 reduced time units, the essentially rotation-free portion of the trajectory provides an unbiased test of the common approximate methods for separating overall rotation from internal motions by optimal superposition. Removing rotation by minimizing the root-mean-square deviation (RMSD) for the entire system is found to be more appropriate than using the RMSD for only the more rigid part of the system. The results verify the existence of positive cross-correlation in the motions of atoms separated by large distances.     

Vibrot,a Simple Device for the Conversion of Vibration into Rotation Mediated by Friction: Preliminary Evaluation

While “vibrational noise” induced by rotating components of machinery is a common problem constantly faced by engineers, the controlled conversion of translational into rotational motion or vice-versa is a desirable goal in many scenarios ranging from internal combustion engines to ultrasonic motors. In this work, we describe the underlying physics after isolating a single degree of freedom, focusing on devices that convert a vibration along the vertical axis into a rotation around this axis. A typical Vibrot (as we label these devices) consists of a rigid body with three or more cantilevered elastic legs attached to its bottom at an angle. We show that these legs are capable of transforming vibration into rotation by a “ratchet effect”, which is caused by the anisotropic stick-slip-flight motion of the leg tips against the ground. Drawing an analogy with the Froude number used to classify the locomotion dynamics of legged animals, we discuss the walking regime of these robots. We are able to control the rotation frequency of the Vibrot by manipulating the shaking amplitude, frequency or waveform. Furthermore, we have been able to excite Vibrots with acoustic waves, which allows speculating about the possibility of reducing the size of the devices so they can perform tasks into the human body, excited by ultrasound waves from the outside.     

The transmission of translational seat vibration to the head--I. Vertical seat vibration

Vibration in the three translational (fore-and-aft, lateral and vertical) and the three rotational (roll, pitch and yaw) axes of the head has been measured during exposure to whole-body random vibration. Using an instrumented bar gripped between the teeth, the influence of variations in bite grip and bite-bar mass on movements of the head were found to be small up to a mass of 375 g. The repeatability of measures of seat-to-head transmissibility within a single subject and the variability in transmissibility across a group of twelve subjects have been determined with two seating conditions: a rigid seat with a backrest and the same seat with no backrest. Seat-to-head transmissibilities associated with vertical seat vibration are presented at frequencies up to 25 Hz for all six axes of head vibration both with and without a backrest. Head motion occurred principally in the fore-and-aft, vertical and pitch axes of the head. The backrest increased the magnitude of head vibration in most cases. Intra-subject variability was generally small compared to inter-subject variability.