Small-amplitude protein conformational dynamics: Second-order analytic relation between cartesian coordinates and dihedral angles

An analytic expression for protein atomic displacements in Cartesian coordinate space (CCS) against small changes in dihedral angles is derived. To study time-dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necessary to convert small changes of internal coordinate variables to Cartesian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion is achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i.e., bond angles and bond lengths are fixed), and Cartesian coordinates of atoms are given analytically by a second-order Taylor expansion in terms of small deviations of variable dihedral angles. Coefficients of the first-order terms are collected in the K matrix obtained previously by Noguti and Go (1983) (see ref. 2). Coefficients of the second-order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulting quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihedral angle space (DAS) of the protein bovine pancreatic trypsin inhibitor (BPTI) has been performed to calculate shift of mean atomic positions and mean square fluctuations around the mean positions. The analysis shows that the second-order terms involving the L matrix have significant contributions to atomic fluctuations at room temperature. This indicates that NMA in CCS involves significant errors when applied for such large molecules as proteins. These errors can be avoided by carrying out NMA in DAS and by considering terms up to second order in the conversion of atomic motion from DAS to CCS. © 1995 by John Wiley & Sons, Inc.     

A high-performance matrix-assisted laser desorption/ionization orthogonal time-of-flight mass spectrometer with collisional cooling

A high-performance orthogonal time-of-flight (TOF) mass spectrometer was developed specifically for use in combination with a matrix-assisted laser desorption/ionization (MALDI) source. The MALDI source features an ionization region containing a buffer gas with variable pressure. The source is interfaced to the TOF section via a collisional focusing ion guide. The pressure in the source influences the rate of cooling and allows control of ion fragmentation. The instrument provides uniform resolution up to 18,000 FWHM (full width at half maximum). Mass accuracy routinely achieved with a single-point internal recalibration is below 2 ppm for protein digest samples. The instrument is also capable of recording spectra of samples containing compounds with a broad range of masses while using one set of experimental conditions and without compromising resolution or mass accuracy.     

Nuclear magnetic relaxation and molecular motion in solid lysozyme

Nuclear magnetic relaxation data for both proton and carbon-13 nuclei in solid lysozyme are analysed together to obtain information on local internal motions in protein. For this analysis the “model-free” approach is used. Three types of internal motion appear to determine the observed nuclear relaxation in protein. They may be attributed to local rotations of methyl groups around symmetry axes, the motion of main and side chain atoms like in rigid lattice, and large-amplitude motions of side groups (mainly, methylene groups). Conclusions on hydrated water influence on local dynamics of protein are made.     

Approximate values for force constant and wave number associated with a low-frequency concerted motion in proteins can be evaluated by a comparison of X-ray structures

Low-frequency internal motions in protein molecules play a key role in biological functions. A direct relationship between low-frequency motions and enzymatic activity has been suggested for bovine pancreatic ribonuclease (RNase A). The flexibility-function relationship in this enzyme has been attributed to a subtle and concerted breathing motion of the beta-sheet regions occurring upon substrate binding and release. Here, we calculate an approximate value for the force constant and the wave number of the low-frequency beta-sheet breathing motion of RNase A, by using the Boltzmann hypothesis on a set of data derived from a simple conventional structural superimposition of an unusual large number of X-ray structures available for the protein. The results agree with previous observations and with theoretical predictions on the basis of normal-mode analysis. To the best of our knowledge, this is the first example in which the wave number and the force constant of a low-frequency concerted motion in a protein are directly derived from X-ray structures.     

Structural and dynamic analysis of residual dipolar coupling data for proteins.

The measurement of residual dipolar couplings in weakly aligned proteins can potentially provide unique information on their structure and dynamics in the solution state. The challenge is to extract the information of interest from the measurements, which normally reflect a convolution of the structural and dynamic properties. We discuss here a formalism which allows a first order separation of their effects, and thus, a simultaneous extraction of structural and motional parameters from residual dipolar coupling data. We introduce some terminology, namely a generalized degree of order, which is necessary for a meaningful discussion of the effects of motion on residual dipolar coupling measurements. We also illustrate this new methodology using an extensive set of residual dipolar coupling measurements made on (15)N,(13)C-labeled human ubiquitin solvated in a dilute bicelle solution. Our results support a solution structure of ubiquitin which on average agrees well with the X-ray structure (Vijay-Kumar, et al., J. Mol. Biol. 1987, 194, 531--544) for the protein core. However, the data are also consistent with a dynamic model of ubiquitin, exhibiting variable amplitudes, and anisotropy, of internal motions. This work suggests the possibility of primary use of residual dipolar couplings in characterizing both structure and anisotropic internal motions of proteins in the solution state.     

How to detect internal motion by homonuclear NMR

NOESY and ROESY cross-peak intensities depend on internuclear distances and internal motion. Internal motion is usually ignored, and NOESY cross-peak intensities are interpreted in terms of internuclear distances only. Off-resonance ROESY experiments measure a weighted average of NOE and ROE. The weight can be described and experimentally set by an angle theta;. For large enough molecules, NOE and ROE have opposite signs. Therefore, each cross-peak intensity becomes zero for an angle theta;(0). For any sample, the maximum angle theta;(0) is determined by the overall motion of the molecule. Smaller theta;(0) values reflect the angular component of internal motions. Because individual cross-peaks are analyzed, the method evaluates internal motions of individual H,H vectors. The reduction of theta;(0) is largest for internal motions on a time scale of 100-300 ps. The sensitivity of theta;(0) for internal motions decreases with increasing molecular weight. We estimate that detecting internal motions will be practicable for molecules up to about 15 kDa. We describe a protocol to measure theta;(0) from a series of off-resonance ROESY spectra. For such a series, we describe the choice of experimental parameters, a procedure to extract theta;(0) from the raw data, and the interpretation of theta;(0) in terms of internal motions. In the small protein BPTI, we analyzed 75 cross-peaks. The precision of theta;(0) was 0.25 degrees, as compared to typical reductions of theta;(0) of 3 degrees. We found a well-defined maximum theta;(0) for cross-peaks in rigid parts of the molecule, which reflects the overall motion of the molecule. For BPTI, also many structurally important long-range cross-peaks appear rigid. The lower theta;(0) values of long-range contacts involving methyl groups are consistent with methyl rotation on the 25-ps time scale. The lower theta;(0) values of the flexible C-terminus and of flexible side chains translate into upper limits for the angular order parameter of 0.4 and 0.5-0.8, respectively. Off-resonance ROESY can monitor internal motions of H,H contacts that are used in a structure calculation. Because no isotope labeling is needed, off-resonance ROESY can be used to detect internal motions in a wide range of natural products.     

What does pressure decide to cook with orientationally disordered plastic phase of cubane: an orientational glass or crystal?

The effect of pressure on the structure and reorientational motion of molecules in orientationally disordered (OD) crystalline phase of cubane has been investigated in detail using variable shape molecular simulations in constant-pressure constant-temperature ensemble. Complete orientational ordering occurs at a pressure of 1.0 GPa and the OD phase transforms to an orientationally ordered phase at this pressure. The transition is associated with a kink in the variation of structural parameters such as cell parameters, unit-cell volume, and interaction energy. This transition is also associated with an anomaly in specific heat. Above this transition pressure, the structural quantities display only smaller changes with further increase in pressure. The structure of high-pressure orientationally ordered (HPOO) phase has been characterized using radial distribution functions and orientational distribution function. From detailed analysis of the structure of HPOO phase we conclude that it is isostructural with low-temperature orientationally ordered phase. The OD phase has four times larger compressibility than the HPOO phase.     

Conserving energy during molecular dynamics simulations of water,proteins, and proteins in water

Molecular dynamics simulations have been carried out for a series of systems of increasing complexity including: pure water, a model polypeptide (α-helical decaglycine) in vacuo, a protein (Pancreatic Trypsin Inhibitor, PTI) in vacuo, and a fully solvated protein (PTI in water). The equations of motion were integrated using Andersen's velocity version of the Verlet algorithm with internal contraints (the RATTLE algorithm). The accuracy with which the equations of motion are integrated has been analyzed for several different simulation conditions. The effects of various nonbonded interaction truncation schemes on the conservation of energy have been examined, including the use of atomic cutoffs, and (neutral group) residue cutoffs. The use of a smoothing function to eliminate the discontinuities in the potential at the cutoff leads to a significant improvement in the accuracy of the integration for each of the systems studied. The accuracy with which the equations of motion are integrated using the RATTLE algorithm for pure water and for the solvated protein are found to be comparable when the nonbonded interactions are tapered with a smoothing function at the cutoff.     

Re-evaluation of the model-free analysis of fast internal motion in proteins using NMR relaxation

NMR spin relaxation retains a central role in the characterization of the fast internal motion of proteins and their complexes. Knowledge of the distribution and amplitude of the motion of amino acid side chains is critical for the interpretation of the dynamical proxy for the residual conformational entropy of proteins, which can potentially significantly contribute to the entropy of protein function. A popular treatment of NMR relaxation phenomena in macromolecules dissolved in liquids is the so-called model-free approach of Lipari and Szabo. The robustness of the mode-free approach has recently been strongly criticized and the remarkable range and structural context of the internal motion of proteins, characterized by such NMR relaxation techniques, attributed to artifacts arising from the model-free treatment, particularly with respect to the symmetry of the underlying motion. We develop an objective quantification of both spatial and temporal asymmetry of motion and re-examine the foundation of the model-free treatment. Concerns regarding the robustness of the model-free approach to asymmetric motion appear to be generally unwarranted. The generalized order parameter is robustly recovered. The sensitivity of the model-free treatment to asymmetric motion is restricted to the effective correlation time, which is by definition a normalized quantity and not a true time constant and therefore of much less interest in this context. With renewed confidence in the model-free approach, we then examine the microscopic distribution of side chain motion in the complex between calcium-saturated calmodulin and the calmodulin-binding domain of the endothelial nitric oxide synthase. Deuterium relaxation is used to characterize the motion of methyl groups in the complex. A remarkable range of Lipari-Szabo model-free generalized order parameters are seen with little correlation with basic structural parameters such as the depth of burial. These results are contrasted with the homologous complex with the neuronal nitric oxide synthase calmodulin-binding domain, which has distinctly different thermodynamic origins for high affinity binding.     

Protein motions promote catalysis

A relationship between molecular dynamics motions of noncatalytic residues and enzyme activity has recently been proposed. We present examples where mutations either near or distal from the active site residues modify internal enzyme motion with resulting modification of catalysis. A better understanding of internal protein motions correlated to catalysis will lead to a greater insight into enzyme function.     

Effect of Molecular Crowding on the Temperature–Pressure Stability Diagram of Ribonuclease A

FT‐IR spectroscopic and thermodynamic measurements were designed to explore the effect of a macromolecular crowder, dextran, on the temperature and pressure‐dependent phase diagram of the protein Ribonuclease A (RNase A), and we compare the experimental data with approximate theoretical predictions based on configuration entropy. Exploring the crowding effect on the pressure‐induced unfolding of proteins provides insight in protein stability and folding under cell‐like dense conditions, since pressure is a fundamental thermodynamic variable linked to molecular volume. Moreover, these studies are of relevance for understanding protein stability in deep‐sea organisms, which have to cope with pressures in the kbar range. We found that not only temperature‐induced equilibrium unfolding of RNase A, but also unfolding induced by pressure is markedly prohibited in the crowded dextran solutions, suggesting that crowded environments such as those found intracellularly, will also oppress high‐pressure protein unfolding. The FT‐IR spectroscopic measurements revealed a marked increase in unfolding pressure of 2 kbar in the presence of 30 wt % dextran. Whereas the structural changes upon thermal unfolding of the protein are not significantly influenced in the presence of the crowding agent, through stabilization by dextran the pressure‐unfolded state of the protein retains more ordered secondary structure elements, which seems to be a manifestation of the entropic destabilization of the unfolded state by crowding.     

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

A structural mode-coupling approach to 15N NMR relaxation in proteins.

The two-body Slowly Relaxing Local Structure (SRLS) model was applied to (15)N NMR spin relaxation in proteins and compared with the commonly used original and extended model-free (MF) approaches. In MF, the dynamic modes are assumed to be decoupled, local ordering at the N-H sites is represented by generalized order parameters, and internal motions are described by effective correlation times. SRLS accounts for dynamical coupling between the global diffusion of the protein and the internal motion of the N-H bond vector. The local ordering associated with the coupling potential and the internal N-H diffusion are tensors with orientations that may be tilted relative to the global diffusion and magnetic frames. SRLS generates spectral density functions that differ from the MF formulas. The MF spectral densities can be regarded as limiting cases of the SRLS spectral density. SRLS-based model-fitting and model-selection schemes similar to the currently used MF-based ones were devised, and a correspondence between analogous SRLS and model-free parameters was established. It was found that experimental NMR data are sensitive to the presence of mixed modes. Our results showed that MF can significantly overestimate order parameters and underestimate local motion correlation times in proteins. The extent of these digressions in the derived microdynamic parameters is estimated in the various parameter ranges, and correlated with the time scale separation between local and global motions. The SRLS-based analysis was tested extensively on (15)N relaxation data from several isotropically tumbling proteins. The results of SRLS-based fitting are illustrated with RNase H from E. coli, a protein extensively studied previously with MF.     

1H magic angle rotation NMR in polymer studies

The principles of the method of NMR line narrowing by measurement with spinning of the sample about the magic axis (MAR-NMR) are introduced, with particular emphasis on the effects of internal motion upon the possibilities and limitations of the method. The applications of the method in ¹H-NMR studies of polymer structure and dynamics are then reviewed. Due to both theoretical and experimental limitations, narrowing of dipolar broadened NMR lines by MAR can be observed in ¹H NMR spectra only in those cases where internal motion is anisotropic, or in heterogeneous systems where line width is limited by differences of magnetic susceptibility. In polymers, both solid and liquid, the method makes possible differentiation between isotropic and anisotropic internal motion. In systems with anisotropic internal motion, MAR-NMR makes possible a characterization of motional codes which normally are obscured by residual dipolar interactions, as well as of geometrical restrictions upon these motions.     

Combining High-Pressure Perturbation with NMR Spectroscopy for a Structural and Dynamical Characterization of Protein Folding Pathways

High-hydrostatic pressure is an alternative perturbation method that can be used to destabilize globular proteins. Generally perfectly reversible, pressure exerts local effects on regions or domains of a protein containing internal voids, contrary to heat or chemical denaturant that destabilize protein structures uniformly. When combined with NMR spectroscopy, high pressure (HP) allows one to monitor at a residue-level resolution the structural transitions occurring upon unfolding and to determine the kinetic properties of the process. The use of HP-NMR has long been hampered by technical difficulties. Owing to the recent development of commercially available high-pressure sample cells, HP-NMR experiments can now be routinely performed. This review summarizes recent advances of HP-NMR techniques for the characterization at a quasi-atomic resolution of the protein folding energy landscape.     

Mechanical simulation of the pressure and the relaxation to thermal equilibrium of a hot and dense rare gas cluster

A cold atomic cluster can be very rapidly heated and compressed by a hypersonic impact at a hard surface. The impact can be simulated by computing a classical trajectory for the motion of the atoms. By suddenly confining the hot and dense cluster within a rigid container, it is possible to monitor the time evolution of the force acting on the faces of the container. It is found that the pressure computed this way very rapidly decays to a time-independent value. After a somewhat longer time, this value reproduces the value for the pressure computed as the sum of the kinetic and internal pressures. This agreement is expected for a system in equilibrium. These observations support the conclusion that there is a fast relaxation to thermal equilibrium in these essentially hard-sphere systems. The deviation from equilibrium is primarily due to the propagation of shock waves within the cluster. The equilibrium pressure can reach up to the megabar range.     

Capillary Stokes drift: a new driving mechanism for mixing in AC-electrowetting

We studied the flow fields generated inside sessile drops that oscillate periodically between states of high and low contact angle under the influence of alternating electric fields of variable frequency and amplitude. Following the motion of dye patches, we show that the number of oscillation cycles required to achieve mixing scales logarithmically with the Péclet number as expected for chaotic mixing. High speed movies reveal an asymmetry of the drop shape between the spreading and receding phase of the oscillations. This results in net internal flow fields that we characterize by tracing the motion of colloidal seed particles. The strength and frequency dependence of the flow are explained in terms of Stokes drift driven by capillary waves that emanate from the oscillating contact line.     

Molecular dynamics in the isothermal-isobaric ensemble: the requirement of a "shell" molecule. II. Simulation results

The results of a series of constant pressure and temperature molecular-dynamics (MD) simulation studies based on the rigorous shell particle formulation of the isothermal-isobaric (NpT) ensemble are presented. These MD simulations validate the newly proposed constant pressure equations of motion in which a "shell" particle is used to define uniquely the volume of the system [M. J. Uline and D. S. Corti, J. Chem. Phys. (to be published), preceding paper]. Ensemble averages obtained with the new MD NpT algorithm match the ensemble averages obtained using the previously derived shell particle Monte Carlo NpT method [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. In addition, we also verify that the Hoover NpT MD algorithm [W. G. Hoover, Phys. Rev. A 31, 1695 (1985); 34, 2499 (1986)] generates the correct ensemble averages, though only when periodic boundary conditions are employed. The extension of the shell particle MD algorithm to multicomponent systems is also discussed, in which we show for equilibrium properties that the identity of the shell particle is completely arbitrary when periodic boundary conditions are applied. Self-diffusion coefficients determined with the shell particle equations of motion are also identical to those obtained in other ensembles. Finally, since the mass of the shell particle is known, the system itself, and not a piston of arbitrary mass, controls the time scales for internal pressure and volume fluctuations. We therefore consider the effects of the shell particle on the dynamics of the system. Overall, the shell particle MD algorithm is an effective simulation method for studying systems exposed to a constant external pressure and may provide an advantage over other existing constant pressure approaches when developing nonequilibrium MD methods.