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.     

A mezoscopic model of nucleic acids. Part 2. An effective potential energy function for DNA

In this study we present an effective Potential of Mean Force (PMF) designed for Lagrangian and Quaternion Molecular Dynamics (LQMD) of DNA. The DNA model is built from pseudoatoms as well as rigid and pseudo-elastic bodies described by a limited number of selected Cartesian and internal degrees of freedom. Phosphate groups, deoxyribose rings and nucleic acid bases are represented by pseudoparticles, some of them with internal degrees of freedom. PMF is defined as the sum of effective bonded and long-range potentials. The potentials were fitted to numerical free energy surfaces. Over 50 free energy surfaces, each depending on a conformational variable (pseudobond length, angle or dihedral angle) and the pseudorotation phase of the nearest neighbour deoxribose ring, were computed. The numerical free energy surfaces were obtained from probability distributions derived from a 1.5 ns conventional, microscopic MD simulation of the d(GpC)9 double helical DNA molecule. An umbrella sampling method was used to simulate transitions between the A and B DNA forms, and PMF reproduces these transitions.     

Molecular dynamics simulations and rigid body (TLS) analysis of aspartate carbamoyltransferase: evidence for an uncoupled R state.

In the R form of ATCase complexed with the bisubstrate analogue, N-(phosphonacetyl)-L-aspartate, large temperature factors are reported for the allosteric domains of the regulatory chains. We studied the conformational flexibility of the holoenzyme with molecular dynamics simulations and rigid body (TLS) analysis. The results of the molecular dynamics simulations suggest that, although local atomic fluctuations account for the temperature factors of the catalytic and zinc domains, they do not account for the large temperature factors of the allosteric regions. However, the temperature factors of the allosteric domains can be satisfactorily analyzed using a rigid body model. The simulations and rigid body analysis support the idea that the allosteric regions are mechanically uncoupled from the rest of the enzyme in the PALA structure. Implications of this uncoupling for allosteric regulation are discussed.     

Virtual rigid body dynamics

An important direction in biological simulations is the development of methods that permit the study of larger systems and/or longer simulation time scales than is currently feasible by molecular dynamics. One such method designed with this objective in mind is stochastic boundary molecular dynamics (SBMD). SBMD was developed for the characterization of spatially localized processes in proteins, and has been shown to successfully reproduce structural and dynamical properties of these macromolecules, as compared to a molecular dynamics control simulation, when concerted or global motions are not present. The virtual rigid body dynamics method presented in this work extends the range of applicability of the SBMD method, by providing a framework to include these important long time scale conformational transitions. In this paper we describe the two-step implementation of the virtual rigid body model: first, the reduction of the full atomic representation to a reduced particle (virtual bond) model, and second, the propagation of the dynamics of flexibly connected rigid bodies containing virtual atom sites.     

Dipolar NMR relaxation of nonprotonated aromatic carbons in proteins. Structural and dynamical effects.

The crystal structure and a 96-ps molecular dynamics simulation used to analyze structural and motional contributions to spin-lattice (T1) relaxation times of phenylalanine and tyrosine C gamma carbons of the pancreatic trypsin inhibitor. The H beta and H delta protons geminal to C gamma are calculated to account for approximately 80% of the dipolar relaxation for each residue. Experimental T1 values for the phenylalanine residues obtained at 25 MHz are observed to be 15-25% longer than estimates based on the rigid crystal structure. It is shown how an increase in T1 can be related to order parameters for the picosecond motional averaging of the important C,H dipolar interactions, and how these order parameters can be calculated from a protein molecular dynamics trajectory.     

Experimentally tractable,pseudo-elastic constitutive law for biomembranes: I. Theory

Although visco-elastic in general, the stress-strain relation of biomembranes is one-to-one or pseudo-elastic when being loaded after preconditioning. This pseudo-elastic relation is hypoelastic (i.e., it is not hyperelastic), yet much of the stress response can be characterized by a scalar function omega that represents the work done (per unit reference volume) on the specimen during loading. (Since a pseudo-strain-energy function W is optimized to fit the test data and not the work done, omega is not equal to W in general.) The remaining part tR of the stress response does no work during loading. With biaxial testing, omega can be definitively determined from data. Moreover, for tests with the stretch directions coaxial to the axes of anisotropy, tR can be accurately characterized by a scalar function omega that depends on the strain. This paper is part 1 of 2 with "I. Theory" and "II. Application."     

Refinement of the NMR structures for acyl carrier protein with scalar coupling data

Structure determination of small proteins using NMR data is most commonly pursued by combining NOE derived distance constraints with inherent constraints based on chemical bonding. Ideally, one would make use of a variety of experimental observations, not just distance constraints. Here, coupling constant constraints have been added to molecular mechanics and molecular dynamics protocols for structure determination in the form of a psuedoenergy function that is minimized in a search for an optimum molecular conformation. Application is made to refinement of a structure for a 77 amino acid protein involved in fatty acid synthesis, Escherichia coli acyl carrier protein (ACP). 54 3JHN alpha coupling constants, 12 coupling constants for stereospecifically assigned side chain protons, and 450 NOE distance constraints were used to calculate the 3-D structure of ACP. A three-step protocol for a molecular dynamics calculation is described, in analogy to the protocol previously used in molecular mechanics calculations. The structures calculated with the molecular mechanics approach and the molecular dynamics approach using a rigid model for the protein show similar molecular energies and similar agreement with experimental distance and coupling constant constraints. The molecular dynamics approach shows some advantage in overcoming local minimum problems, but only when a two-state averaging model for the protein was used, did molecular energies drop significantly.     

Ion-specific diffusion rates through transmembrane protein channels. A molecular dynamics study

The migration of different alkali metal cations through a transmembrane model channel is simulated by means of the molecular dynamics technique. The parameters of the model are chosen in close relation to the gramicidin A channel. Coulomb- and van der Waals-type potentials between the ions and flexible carbonyl groups of the pore-forming molecule are used to describe the ion channel interaction. The diffusion properties of the ions are obtained from three-dimensional trajectory calculations. The diffusion rates for the different ions Li+, Na+, K+ and Rb+ are affected not only by the mass of the particles but also very strongly by their size. The latter effect is more pronounced for rigid channels, i.e., for binding vibrational frequencies of the CO groups with v greater than 400 cm-1. In this range the selectivity sequence for the diffusion rates is the inverse of that expected from normal rate theory but agrees with that found in experiments for gramicidin A.     

Coarse-Grained Description of Protein Internal Dynamics: An Optimal Strategy for Decomposing Proteins in Rigid Subunits

The possibility of accurately describing the internal dynamics of proteins, in terms of movements of a few approximately-rigid subparts, is an appealing biophysical problem with important implications for the analysis and interpretation of data from experiments or numerical simulations. The problem is tackled here by means of a novel variational approach that exploits information about equilibrium fluctuations of interresidues distances, provided, e.g., by atomistic molecular dynamics simulations or coarse-grained models. No contiguity in primary sequence or in space is enforced a priori for amino acids grouped in the same rigid unit. The identification of the rigid protein moduli, or dynamical domains, provides valuable insight into functionally oriented aspects of protein internal dynamics. To illustrate this point, we first discuss the decomposition of adenylate kinase and HIV-1 protease and then extend the investigation to several representatives of the hydrolase enzymatic class. The known catalytic site of these enzymes is found to be preferentially located close to the boundary separating the two primary dynamical subdomains.     

Dynamic and structural analysis of isotropically distributed molecular ensembles.

An efficient new method is presented for the characterization of motional correlations derived from a set of protein structures without requiring the separation of overall and internal motion. In this method, termed isotropically distributed ensemble (IDE) analysis, each structure is represented by an ensemble of isotropically distributed replicas corresponding to the situation found in an isotropic protein solution. This leads to a covariance matrix of the cartesian atomic positions with elements proportional to the ensemble average of scalar products of the position vectors with respect to the center of mass. Diagonalization of the covariance matrix yields eigenmodes and amplitudes that describe concerted motions of atoms, including overall rotational and intramolecular dynamics. It is demonstrated that this covariance matrix naturally distinguishes between "rigid" and "mobile" parts without necessitating a priori selection of a reference structure and an atom set for the orientational alignment process. The method was applied to the analysis of a 5-ns molecular dynamics trajectory of native ubiquitin and a 40-ns trajectory of a partially folded state of ubiquitin. The results were compared with essential dynamics analysis. By taking advantage of the spherical symmetry of the IDE covariance matrix, more than a 10-fold speed up is achieved for the computation of eigenmodes and mode amplitudes. IDE analysis is particularly suitable for studying the correlated dynamics of flexible and large molecules.     

Leapfrog Rotational Algorithms

Abstract

A new implicit rotational integrator for the orientation of rigid molecules is introduced, and compared with an existing explicit integrator. Both algorithms are categorised as leapfrogs since the quantities saved between time steps are on the on-step orientation and the mid-step angular momentum. Orientations may be expressed in terms of principal axis vectors or, as in the implementations used here, quaternions. Thermostatted versions of the algorithms as well as conventional energy-conserving versions are described. The algorithms are extensively tested in simulations of liquid water, the aim being to study the effect of increased time steps on a range of measured properties. The implicit algorithm is superior to the explicit algorithm, and can be used with time steps up to 3 fs with energy-conserving dynamics. When thermostatted, it may be used with time steps up to at least 6 fs.     

Co-operative dynamics in organelles

Some organelles produce elementary life phenomena which are characterized by the spontaneous formation and/or maintenance of ordered macroscopic dynamics like e.g. the shortening of sarcomeres in striated muscle and the transmission of electrical impulses in an axon. It has been widely accepted that such organelles are organized molecular systems where molecular elements work independently under constraint of a more or less rigid and regular structure of the system. On the other hand, such organelles should be regarded as self-organizing systems if the ordered macroscopic dynamics are self-organized. As the macroscopic dynamics gradually emerge, the microscopic dynamics of its elements become linked to each other through a feedback loop. It is crucial for the feedback loop to operate that the macroscopic dynamics are "free" in their behavior. In the present paper, it is pointed out that the traditional view of independent molecular elements has been obtained from experiments in which, by means of external constraint, the macroscopic dynamics is "clamped". Under such conditions, the self-organizing system may behave as an organized one. Based on synergetics we propose criterions for proving self-organizing systems, and, by applying the criterions, we conclude that skeletal muscle actomysin is a co-operative element in the sense of self-organization.     

Dynamical properties of fasciculin-2.

Fasciculin-2 (FAS2) is a potent protein inhibitor of the hydrolytic enzyme acetylcholinesterase. A 2-ns isobaric-isothermal ensemble molecular dynamics simulation of this toxin was performed to examine the dynamic structural properties which may play a role in this inhibition. Conformational fluctuations of the FAS2 protein were examined by a variety of techniques to identify flexible residues and determine their characteristic motion. The tips of the toxin "finger" loops and the turn connecting loops I and II were found to fluctuate, while the rest of the protein remained fairly rigid throughout the simulation. Finally, the structural fluctuations were compared to NMR data of fluctuations on a similar timescale in a related three-finger toxin. The molecular dynamics results were in good qualitative agreement with the experimental measurements. Proteins 1999;36:447-453.     

Functional flexibility of human cyclin-dependent kinase-2 and its evolutionary conservation

Cyclin-dependent kinase 2 (CDK2) is the most thoroughly studied of the cyclin-dependent kinases that regulate essential cellular processes, including the cell cycle, and it has become a model for studies of regulatory mechanisms at the molecular level. This contribution identifies flexible and rigid regions of CDK2 based on temperature B-factors acquired from both X-ray data and molecular dynamics simulations. In addition, the biological relevance of the identified flexible regions and their motions is explored using information from the essential dynamics analysis related to conformational changes of CDK2 and knowledge of its biological function(s). The conserved regions of CMGC protein kinases' primary sequences are located in the most rigid regions identified in our analyses, with the sole exception of the absolutely conserved G13 in the tip of the glycine-rich loop. The conserved rigid regions are important for nucleotide binding, catalysis, and substrate recognition. In contrast, the most flexible regions correlate with those where large conformational changes occur during CDK2 regulation processes. The rigid regions flank and form a rigid skeleton for the flexible regions, which appear to provide the plasticity required for CDK2 regulation. Unlike the rigid regions (which as mentioned are highly conserved) no evidence of evolutionary conservation was found for the flexible regions.     

Open-Boundary Molecular Dynamics of a DNA Molecule in a Hybrid Explicit/Implicit Salt Solution

The composition and electrolyte concentration of the aqueous bathing environment have important consequences for many biological processes and can profoundly affect the behavior of biomolecules. Nevertheless, because of computational limitations, many molecular simulations of biophysical systems can be performed only at specific ionic conditions: either at nominally zero salt concentration, i.e., including only counterions enforcing the system’s electroneutrality, or at excessive salt concentrations. Here, we introduce an efficient molecular dynamics simulation approach for an atomistic DNA molecule at realistic physiological ionic conditions. The simulations are performed by employing the open-boundary molecular dynamics method that allows for simulation of open systems that can exchange mass and linear momentum with the environment. In our open-boundary molecular dynamics approach, the computational burden is drastically alleviated by embedding the DNA molecule in a mixed explicit/implicit salt-bathing solution. In the explicit domain, the water molecules and ions are both overtly present in the system, whereas in the implicit water domain, only the ions are explicitly present and the water is described as a continuous dielectric medium. Water molecules are inserted and deleted into/from the system in the intermediate buffer domain that acts as a water reservoir to the explicit domain, with both water molecules and ions free to enter or leave the explicit domain. Our approach is general and allows for efficient molecular simulations of biomolecules solvated in bathing salt solutions at any ionic strength condition.     

Lees–Edwards boundary condition for simulation of polymer suspension with dissipative particle dynamics method

The Lees–Edwards boundary condition (LEbc) is widely used in particle-based simulation for producing shear flow. Application of traditional LEbc in dissipative particle dynamics (DPD) method may encounter certain problems, e.g. it will destroy the momentum conservation law at the near boundary region, and the coordinate system gives an incorrect end-to-end vector for polymer beads. Special treatments of the implementation of LEbc in DPD method are introduced in this paper. A single side ghost layer is used to keep the momentum conservation, and the global coordinate system is employed to obtain a correct calculation of the spring force between polymer beads. The simulation results give a good prediction of velocity profile and system temperature, and the elastic dumbbell model for current method can well represent the Oldroyd-B fluid.     

The effects of rigid motions on elastic network model force constants

Elastic network models provide an efficient way to quickly calculate protein global dynamics from experimentally determined structures. The model's single parameter, its force constant, determines the physical extent of equilibrium fluctuations. The values of force constants can be calculated by fitting to experimental data, but the results depend on the type of experimental data used. Here, we investigate the differences between calculated values of force constants and data from NMR and X-ray structures. We find that X-ray B factors carry the signature of rigid-body motions, to the extent that B factors can be almost entirely accounted for by rigid motions alone. When fitting to more refined anisotropic temperature factors, the contributions of rigid motions are significantly reduced, indicating that the large contribution of rigid motions to B factors is a result of over-fitting. No correlation is found between force constants fit to NMR data and those fit to X-ray data, possibly due to the inability of NMR data to accurately capture protein dynamics.     

Characterization of Stratum Corneum Molecular Dynamics by Natural-Abundance 13C Solid-State NMR

Despite the enormous potential for pharmaceutical applications, there is still a lack of understanding of the molecular details that can contribute to increased permeability of the stratum corneum (SC). To investigate the influence of hydration and heating on the SC, we record the natural-abundance ¹³C signal of SC using polarization transfer solid-state NMR methods. Resonance lines from all major SC components are assigned. Comparison of the signal intensities obtained with the INEPT and CP pulse sequences gives information on the molecular dynamics of SC components. The majority of the lipids are rigid at 32°C, and those lipids co-exist with a small pool of mobile lipids. The ratio between mobile and rigid lipids increases with hydration. An abrupt change of keratin filament dynamics occurs at RH = 80–85%, from completely rigid to a structure with rigid backbone and mobile protruding terminals. Heating has a strong effect on the lipid mobility, but only a weak influence on the keratin filaments. The results provide novel molecular insight into how the SC constituents are affected by hydration and heating, and improve the understanding of enhanced SC permeability, which is associated with elevated temperatures and SC hydration.