Protein dynamics and distance determination by NOE measurements

Present analysis procedures for NMR structure determination of macromolecules presuppose fixed internuclear distances. Improvement of the precision of the requisite NOE information has stimulated the use of more quantitative distance constraints thus necessitating examination as to whether the assumption of a rigid model systematically biases the distance estimates. Analysis using the simple (r-6) dependence of NOE buildup rates seriously underestimates the correct distance for spatially proximal proton pairs having fluctuations comparable to those observed in X-ray temperature factor analysis. However, by calculating the proper generalized order parameter it is shown that for nuclei undergoing rapid isotropic uncorrelated fluctuations the effective distance is identical to the distance between the mean positions of the nuclei. Similar analysis of molecular dynamics simulation data from bovine pancreatic trypsin inhibitor indicates that the distance obtained from the generalized order parameter predicts the distance between the mean positions to within a few percent regardless of the degree of correlation of the pairwise motion for virtually all main chain and dynamically constrained side chain protons.     

Simulation of protein conformational freedom as a function of pH: constant-pH molecular dynamics using implicit titration

Solution pH is a determinant parameter on protein function and stability, and its inclusion in molecular dynamics simulations is attractive for studies at the molecular level. Current molecular dynamics simulations can consider pH only in a very limited way, through a somewhat arbitrary choice of a set of fixed charges on the titrable sites. Conversely, continuum electrostatic methods that explicitly treat pH effects assume a single protein conformation whose choice is not clearly defined. In this paper we describe a general method that combines both titration and conformational freedom. The method is based on a potential of mean force for implicit titration and combines both usual molecular dynamics and pH-dependent calculations based on continuum methods. A simple implementation of the method, using a mean field approximation, is presented and applied to the bovine pancreatic trypsin inhibitor. We believe that this constant-pH molecular dynamics method, by correctly sampling both charges and conformation, can become a valuable help in the understanding of the dependence of protein function and stability on pH. © 1997 Wiley-Liss Inc.     

Rigidity loss of protein macromolecule induced by force—Effective field theory

In the framework of the effective field theory for the order parameter, which characterizes the degree of deviating the protein globule structure from its native state, the phase transition of the protein macromolecule from the elastic state into the plastic one under its mechanical stretching is considered. Elastic properties of a protein are studied as a function of the applied force, temperature, and the mean coordination number of the protein “network.” Proteins 2014; 82:966–974. © 2013 Wiley Periodicals, Inc.     

Insights into the local residual entropy of proteins provided by NMR relaxation.

A simple model is used to illustrate the relationship between the dynamics measured by NMR relaxation methods and the local residual entropy of proteins. The expected local dynamic behavior of well-packed extended amino acid side chains are described by employing a one-dimensional vibrator that encapsulates both the spatial and temporal character of the motion. This model is then related to entropy and to the generalized order parameter of the popular "model-free" treatment often used in the analysis of NMR relaxation data. Simulations indicate that order parameters observed for the methyl symmetry axes in, for example, human ubiquitin correspond to significant local entropies. These observations have obvious significance for the issue of the physical basis of protein structure, dynamics, and stability.     

Influence of the conservative force on transport coefficients in the DPD method

The transport coefficients of a dissipative particle dynamics system are investigated numerically taking into account the conservative force. The influence of the conservative force parameter on kinetic and dissipative viscosity is considered and compared with theoretical predictions. The analytical solution of the potential term that arises from the conservative force is generally very complicated; therefore, this term has been ignored in most previous work and deemed negligible. In the present work, an expression for the effect of the conservative force on potential viscosity is semi-empirically obtained revealing the dependence of the latter on various parameters such as conservative force strength, density number and temperature. This relation offers a proper approximation of conservative force impact on potential viscosity.     

Banding 2of NMR‐derived methyl order parameters: Implications for protein dynamics

Our understanding of protein folding, stability, and function has begun to more explicitly incorporate dynamical aspects. Nuclear magnetic resonance has emerged as a powerful experimental method for obtaining comprehensive site‐resolved insight into protein motion. It has been observed that methyl‐group motion tends to cluster into three “classes” when expressed in terms of the popular Lipari‐Szabo model‐free squared generalized order parameter. Here the origins of the three classes or bands in the distribution of order parameters are examined. As a first step, a Bayesian based approach, which makes no a priori assumption about the existence or number of bands, is developed to detect the banding of values derived either from NMR experiments or molecular dynamics simulations. The analysis is applied to seven proteins with extensive molecular dynamics simulations of these proteins in explicit water to examine the relationship between O² and fine details of the motion of methyl bearing side chains. All of the proteins studied display banding, with some subtle differences. We propose a very simple yet plausible physical mechanism for banding. Finally, our Bayesian method is used to analyze the measured distributions of methyl group motions in the catabolite activating protein and several of its mutants in various liganded states and discuss the functional implications of the observed banding to protein dynamics and function. Proteins 2014; 82:2106–2117. © 2014 Wiley Periodicals, Inc.     

Effective potentials between gold nano crystals – functional dependence on temperature

A method is presented that allows to combine the effective potential between two nano crystals (NC), the potential of mean force (PMF), as obtained from all-atomistic molecular dynamics simulations with perturbation theory. In this way, a functional dependence of the PMF on temperature is derived, which enables the prediction of the PMF in a wide temperature range. We applied the method to systems of capped gold NCs of different size. They show very good agreement with data from atomistic simulations.     

On the ability of molecular dynamics force fields to recapitulate NMR derived protein side chain order parameters

Molecular dynamics (MD) simulations have become a central tool for investigating various biophysical questions with atomistic detail. While many different proxies are used to qualify MD force fields, most are based on largely structural parameters such as the root mean square deviation from experimental coordinates or nuclear magnetic resonance (NMR) chemical shifts and residual dipolar couplings. NMR derived Lipari–Szabo squared generalized order parameter (O²) values of amide N? H bond vectors of the polypeptide chain were also often employed for refinement and validation. However, with a few exceptions, side chain methyl symmetry axis order parameters have not been incorporated into experimental reference sets. Using a test set of five diverse proteins, the performance of several force fields implemented in the NAMDD simulation package was examined. It was found that simulations employing explicit water implemented using the TIP3 model generally performed significantly better than those using implicit water in reproducing experimental methyl symmetry axis O² values. Overall the CHARMM27 force field performs nominally better than two implementations of the Amber force field. It appeared that recent quantum mechanics modifications to side chain torsional angles of leucine and isoleucine in the Amber force field have significantly hindered proper motional modeling for these residues. There remained significant room for improvement as even the best correlations of experimental and simulated methyl group Lipari–Szabo generalized order parameters fall below an R² of 0.8.     

Dynamic Light Scattering Application to Study Protein Interactions in Electrolyte Solutions

The concentration dependence of the diffusion coefficient of particles suspended in solution depends primarily on the occupied volume fraction and on repulsive and attractive forces. This dependency is expressed by the interaction parameter, which can be assessed experimentally by light scattering measurements and have been determined for the diffusion coefficient of BSA under different salt concentration conditions in the present work. The result shows that the diffusion coefficient of protein grows up with increasing protein concentration, and when the ionic strength turns up gradually the diffusion coefficient decreases with protein concentrations increasing. The concentration dependence of BSA diffusion coefficients is interpreted in the context of a two-body potential of mean force, which includes repulsive hard-sphere and Coulombic interactions and attractive dispersion. With the increase of ionic strength, Debye screening decreases, protein interaction changes from repulsion to attraction, and protein begins to aggregate. By means of the concentration dependence of BSA diffusion coefficients, one can obtain the parameters of protein interactions and can find that protein bears a net effective charge of –9.0 e and has a Hamaker constant of 2.8k_BT. This work demonstrates that DLS is an effective technique of studying protein interactions.     

The role of backbone conformational heat capacity in protein stability: temperature dependent dynamics of the B1 domain of Streptococcal protein G

The contributions of backbone NH group dynamics to the conformational heat capacity of the B1 domain of Streptococcal protein G have been estimated from the temperature dependence of 15N NMR-derived order parameters. Longitudinal (R1) and transverse (R2) relaxation rates, transverse cross-relaxation rates (eta(xy)), and steady state [1H]-15N nuclear Overhauser effects were measured at temperatures of 0, 10, 20, 30, 40, and 50 degrees C for 89-100% of the backbone secondary amide nitrogen nuclei in the B1 domain. The ratio R2/eta(xy) was used to identify nuclei for which conformational exchange makes a significant contribution to R2. Relaxation data were fit to the extended model-free dynamics formalism, incorporating an axially symmetric molecular rotational diffusion tensor. The temperature dependence of the order parameter (S2) was used to calculate the contribution of each NH group to conformational heat capacity (Cp) and a characteristic temperature (T*), representing the density of conformational energy states accessible to each NH group. The heat capacities of the secondary structure regions of the B1 domain are significantly higher than those of comparable regions of other proteins, whereas the heat capacities of less structured regions are similar to those in other proteins. The higher local heat capacities are estimated to contribute up to approximately 0.8 kJ/mol K to the total heat capacity of the B1 domain, without which the denaturation temperature would be approximately 9 degrees C lower (78 degrees C rather than 87 degrees C). Thus, variation of backbone conformational heat capacity of native proteins may be a novel mechanism that contributes to high temperature stabilization of proteins.     

Temperature dependence of the internal dynamics of a calmodulin-peptide complex

The temperature dependence of the fast internal dynamics of calcium-saturated calmodulin in complex with a peptide corresponding to the calmodulin-binding domain of the smooth muscle myosin light chain kinase is examined using 15N and 2H NMR relaxation methods. NMR relaxation studies of the complex were carried out at 13 temperatures that span 288-346 K. The dynamics of the backbone and over four dozen methyl-bearing side chains, distributed throughout the calmodulin molecule, were probed. The side chains show a much more variable and often considerably larger response to temperature than the backbone. A significant variation in the temperature dependence of the amplitude of motion of individual side chains is seen. The amplitude of motion of some side chains is essentially temperature-independent while many show a simple roughly linear temperature dependence. In a few cases, angular order increases with temperature, which is interpreted as arising from interactions with neighboring residues. In addition, a number of side chains display a nonlinear temperature dependence. The significance of these and other results is illuminated by several simple interpretative models. Importantly, analysis of these models indicates that changes in generalized order parameters can be robustly related to corresponding changes in residual entropy. A simple cluster model that incorporates features of cooperative or conditional motion reproduces many of the unusual features of the experimentally observed temperature dependence and illustrates that side chain interactions result in a dynamically changing environment that significantly influences the motion of internal side chains. This model also suggests that the intrinsic entropy of interacting clusters of side chains is only modestly reduced from that of independent side chain motion. Finally, estimates of protein heat capacity support the view that the major contribution to the heat capacity of protein solutions largely arises from local bond vibrations and solvent interactions and not from torsional oscillations of side chains.     

Distance dependent centroid to centroid force fields using high resolution decoys

Simplified force fields play an important role in protein structure prediction and de novo protein design by requiring less computational effort than detailed atomistic potentials. A side chain centroid based, distance dependent pairwise interaction potential has been developed. A linear programming based formulation was used in which non-native "decoy" conformers are forced to take a higher energy compared with the corresponding native structure. This model was trained on an enhanced and diverse protein set. High quality decoy structures were generated for approximately 1400 nonhomologous proteins using torsion angle dynamics along with restricted variations of the hydrophobic cores of the native structure. The resulting decoy set was used to train the model yielding two different side chain centroid based force fields that differ in the way distance dependence has been used to calculate energy parameters. These force fields were tested on an independent set of 148 test proteins with 500 decoy structures for each protein. The side chain centroid force fields were successful in correctly identifying approximately 86% native structures. The Z-scores produced by the proposed centroid-centroid distance dependent force fields improved compared with other distance dependent C(alpha)-C(alpha) or side chain based force fields.     

Chain-length dependence of lipid bilayer properties near the liquid crystal to gel phase transition.

The temperature dependence of the mean orientational order parameter in the vicinity of the liquid crystal to gel phase transition is obtained from the first moment M1 of deuterium nuclear magnetic resonance spectra for bilayers of chain perdeuterated phosphatidylcholines with acyl chains of 12, 14, 16, and 18 carbons. The data clearly show an increasing temperature dependence of the orientational order parameter in the vicinity of the transition, with the effect becoming more pronounced with decreasing chain length. Assuming a linear relationship between the mean orientational order parameter and the extension of the acyl chain, estimates of the change in area of the membrane at the transition are shown to be consistent with those obtained from other measurements. It is shown that the transition may be modeled in terms of a Landau expansion of the free energy involving a small number of phenomenological parameters. From this it is shown that the behavior of these systems in the temperature range of interest is, in large part, controlled by the close proximity of a spinodal to the transition temperature.     

Concordance of residual dipolar couplings, backbone order parameters and crystallographic B-factors for a small alpha/beta protein: a unified picture of high probability, fast atomic motions in proteins

Using ensemble refinement of the third immunoglobulin binding domain (GB3) of streptococcal protein G (a small alpha/beta protein of 56 residues), we demonstrate that backbone (N-H, N-C', Calpha-Halpha, Calpha-C') residual dipolar coupling data in five independent alignment media, generalized order parameters from 15N relaxation data, and B-factors from a high-resolution (1.1A), room temperature crystal structure are entirely consistent with one another within experimental error. The optimal ensemble size representation is between four and eight, as assessed by complete cross-validation of the residual dipolar couplings. Thus, in the case of GB3, all three observables reflect the same low-amplitude anisotropic motions arising from fluctuations in backbone phi/psi torsion angles in the picosecond to nanosecond regime in both solution and crystalline environments, yielding a unified picture of fast, high-probability atomic motions in proteins. An understanding of these motions is crucial for understanding the impact of protein dynamics on protein function, since they provide part of the driving force for triggered conformational changes that occur, for example, upon ligand binding, signal transduction and enzyme catalysis.     

Thermal coefficients of the methyl groups within ubiquitin

Physiological processes such as protein folding and molecular recognition are intricately linked to their dynamic signature, which is reflected in their thermal coefficient. In addition, the local conformational entropy is directly related to the degrees of freedom, which each residue possesses within its conformational space. Therefore, the temperature dependence of the local conformational entropy may provide insight into understanding how local dynamics may affect the stability of proteins. Here, we analyze the temperature dependence of internal methyl group dynamics derived from the cross-correlated relaxation between dipolar couplings of two CH bonds within ubiquitin. Spanning a temperature range from 275 to 308 K, internal methyl group dynamics tend to increase with increasing temperature, which translates to a general increase in local conformational entropy. With this data measured over multiple temperatures, the thermal coefficient of the methyl group order parameter, the characteristic thermal coefficient, and the local heat capacity were obtained. By analyzing the distribution of methyl group thermal coefficients within ubiquitin, we found that the N-terminal region has relatively high thermostability. These results indicate that methyl groups contribute quite appreciably to the total heat capacity of ubiquitin through the regulation of local conformational entropy.     

Protein molecular dynamics with electrostatic force entirely determined by a single Poisson-Boltzmann calculation

The electrostatic force including the intramolecular Coulombic interactions and the electrostatic contribution of solvation effect were entirely calculated by using the finite difference Poisson-Boltzmann method (FDPB), which was incorporated into the GROMOS96 force field to complete a new finite difference stochastic dynamics procedure (FDSD). Simulations were performed on an insulin dimer. Different relative dielectric constants were successively assigned to the protein interior; a value of 17 was selected as optimal for our system. The simulation data were analyzed and compared with those obtained from 500-ps molecular dynamics (MD) simulation with explicit water and a 500-ps conventional stochastic dynamics (SD) simulation without the mean solvent force. The results indicate that the FDSD method with GROMOS96 force field is suitable to study the dynamics and structure of proteins in solution if used with the optimal protein dielectric constant.     

Temperature-dependent dynamics of the villin headpiece helical subdomain,an unusually small thermostable protein

(15)N spin relaxation experiments were used to measure the temperature-dependence of protein backbone conformational fluctuations in the thermostable helical subdomain, HP36, of the F-actin-binding headpiece domain of chicken villin. HP36 is the smallest domain of a naturally occurring protein that folds cooperatively to a compact native state. Spin-lattice, spin-spin, and heteronuclear nuclear Overhauser effect relaxation data for backbone amide (15)N spins were collected at five temperatures in the range of 275-305 K. The data were analyzed using a model-free formalism to determine generalized order parameters, S, that describe the distribution of N-H bond vector orientations in a molecular reference frame. A novel parameter, Lambda=dln(1-S)/dln T is introduced to characterize the temperature-dependence of S. An average value of Lambda=4.5 is obtained for residues in helical conformations in HP36. This value of Lambda is not reproduced by model potential energy functions commonly used to parameterize S. The maximum entropy principle was used to derive a new model potential function that reproduces both S and Lambda. Contributions to the entropy, S(r), and heat capacity, C(r)(p), from reorientational conformational fluctuations were analyzed using this potential energy function. Values of S(r) show a qualitative dependence on S similar to that obtained for the diffusion-in-a-cone model; however, quantitative differences of up to 0.5k, in which k is the Boltzmann constant, are observed. Values of C(r)(p) approach zero for small values of S and approach k for large values of S; the largest values of C(r)(p) are predicted to occur for intermediate values of S. The results suggest that backbone dynamics, as probed by relaxation measurements, make very little contribution to the heat capacity difference between folded and unfolded states for HP36.     

Red fluorescent protein DsRed: Parametrization of its chromophore as an amino acid residue for computer modeling in the OPLS-AA force field

Topology of the neutral form of the DsRed fluorescent protein chromophore as a residue of [(4-cis)-2-[(1-cis)-4-amino-4-oxobutanimidoyl]-4-(4-hydroxybenzylidene)-5-oxo-4,5-dihydro-1H-imidazol-1-yl]acetic acid was calculated with OPLS-AA force field. Use of this topology and molecular dynamics simulation allows calculating the parameters of proteins that contain such residue in their polypeptide chains. The chromophore parameters were obtained by ab initio (RHF/6-31G**) quantum chemical calculations applying density functional theory (B3LYP). Using this chromophore, we have calculated the molecular dynamics trajectory of tetrameric fluorescent protein DsRed in solution at 300 K (4 nsec). Correctness of the chromophore parametrization was revealed by comparison of quantitative characteristics of the chromophore structure obtained from the molecular dynamic simulations of DsRed protein with the quantitative characteristics of the chromophore based on the crystallographic X-ray data of fluorescent protein DsRed (PDB ID: 1ZGO, 1G7K, and 1GGX), and also with the quantitative characteristics of the chromophore obtained by quantum chemical calculations. Inclusion of the neutral form of DsRed protein chromophore topology into the OPLS-AA force field yielded the extended force field OPLS-AA/DsRed. This force field can be used for molecular dynamics calculations of proteins containing the DsRed chromophore. The parameter set presented in this study can be applied for similar extension in any other force fields.     

Temperature dependence of protein dynamics: computer simulation analysis of neutron scattering properties

The temperature dependence of the internal dynamics of an isolated protein, bovine pancreatic trypsin inhibitor, is examined using normal mode analysis and molecular dynamics (MD) simulation. It is found that the protein exhibits marked anharmonic dynamics at temperatures of approximately 100-120 K, as evidenced by departure of the MD-derived average mean square displacement from that of the harmonic model. This activation of anharmonic dynamics is at lower temperatures than previously detected in proteins and is found in the absence of solvent molecules. The simulation data are also used to investigate neutron scattering properties. The effects are determined of instrumental energy resolution and of approximations commonly used to extract mean square displacement data from elastic scattering experiments. Both the presence of a distribution of mean square displacements in the protein and the use of the Gaussian approximation to the dynamic structure factor lead to quantified underestimation of the mean square displacement obtained.