Smoluchowski dynamics of the vnd/NK-2 homeodomain from Drosophila melanogaster: second-order maximum correlation approximation

The mode coupling diffusion theory is applied to the derivation of local dynamics in proteins in solution. The rotational dynamics of the bonds along the protein sequence are calculated and compared to the experimentally measured nmr (15)N spin-lattice relaxation time T(1), at 36.5, 60.8, and 81.1 MHz of the vnd/NK-2 homeodomain from Drosophila melanogaster. The starting point for the calculations is the experimental three-dimensional solution structure of the homeodomain determined by multidimensional nmr spectroscopy. The higher order mode-coupling computations are compared also with the recently published first-order approximation calculations. The more accurate calculations improve substantially the first-order ORZLD calculations and show that the role of the strength of the hydrodynamic interactions becomes crucial to fix the order of magnitude of the rotational dynanics for these very compact molecules characterized by partial screening of the internal atoms to water. However, the relative mobility of the bonds along the sequence and the differential fluctuations depend only weakly on the hydrodynamic strength but strongly on the geometry of the three-dimensional structure and on the statistics incorporated into the theory. Both rigid and fluctuating dynamic models are examined, with fluctuations evaluated using molecular dynamics simulations. The comparison with nmr data shows that mode coupling diffusion accounts for the T(1) relaxation pattern at low frequency where the rotational tumbling dominates. An important contribution of internal motions in the nanosecond time scale is seen at high frequencies and is discussed in terms of diffusive concepts.     

Smoluchowski dynamics of the vnd/NK‐2 homeodomain from Drosophila melanogaster: First‐order mode‐coupling approximation

This work is the first in a series devoted to applying mode coupling diffusion theory to the derivation of local dynamics properties of proteins in solution. The first‐order mode‐coupling approximation, or optimized Rouse–Zimm local dynamics (ORZLD), is applied here to derive the rotational dynamics of the bonds and compare the calculated with the experimental nmr ¹⁵N spin–lattice relaxation time behavior of the vnd/NK‐2 homeodomain from Drosophila melanogaster. The starting point for the calculations is the experimental three‐dimensional structure of the homeodomain determined by multidimensional nmr spectroscopy. The results of the computations are compared with experimentally measured ¹⁵N spin–lattice relaxation times T₁, at 34.5 and 60.8 MHz, to check the first‐order approximation. To estimate the relative importance of internal and overall rotation, both rigid and fluctuating dynamic models are examined, with fluctuations evaluated using molecular dynamics (MD) simulations. The correlation times for the fundamental bond vector time correlation function and for the second‐order bond orientational TCF are obtained as a function of the residue number for vnd/NK‐2. The stability of the corresponding local dynamics pattern for the fluctuating structure as a function of the length of the MD trajectory is presented. Diffusive dynamics, which is essentially free of model parameters even at first order in the mode‐coupling diffusion approach, confirm that local dynamics of proteins can be described in terms of rotational diffusion of a fluctuating quasi‐rigid structure. The comparison with the nmr data shows that the first‐order mode coupling diffusion approximation accounts for the correct order of magnitude of the results and of important qualitative aspects of the data sensitive to conformational changes. Indications are obtained from this study to efficiently extend the theory to higher order in the mode‐coupling expansion. These results demonstrate the promise of the mode‐coupling approach, where the local dynamics of proteins is described in terms of rotational diffusion of a fluctuating quasi‐rigid structure, to analyze nmr spin–lattice relaxation behavior. © 1999 John Wiley & Sons, Inc. Biopoly 49: 235–254, 1999     

Local dynamics in biological macromolecules

The diffusive approach in the optimized Rouse-Zimm approximation to segment relaxation in the nanosecond time domain (ORZLD) is extended to consider chains of nonequivalent units as occurring in biological macromolecules. The correlation times for second-order time correlation functions of each virtual bond on the chain are calculated for some homopolypeptides, and random and regular copolypeptides. The expected correlation times for biological macromolecules organized in multiple domains are discussed via a simple model of the ORZLD hierarchy. Dynamic bond correlation times are compared with static local persistence lengths.     

Discrete analyses of protein dynamics

Abstract

Protein structures are highly dynamic macromolecules. This dynamics is often analysed through experimental and/or computational methods only for an isolated or a limited number of proteins. Here, we explore large-scale protein dynamics simulation to observe dynamics of local protein conformations using different perspectives. We analysed molecular dynamics to investigate protein flexibility locally, using classical approaches such as RMSf, solvent accessibility, but also innovative approaches such as local entropy. First, we focussed on classical secondary structures and analysed specifically how β-strand, β–turns, and bends evolve during molecular simulations. We underlined interesting specific bias between β–turns and bends, which are considered as the same category, while their dynamics show differences. Second, we used a structural alphabet that is able to approximate every part of the protein structures conformations, namely protein blocks (PBs) to analyse (i) how each initial local protein conformations evolve during dynamics and (ii) if some exchange can exist among these PBs. Interestingly, the results are largely complex than simple regular/rigid and coil/flexible exchange. Abbreviations N_eq number of equivalent

PB Protein Blocks

PDB Protein DataBank

RMSf root mean square fluctuations

Communicated by Ramaswamy H. Sarma     

A comparison of the restrained molecular dynamics and distance geometry methods for determining three-dimensional structures of proteins on the basis of interproton distances

A direct comparison of the metric matrix distance geometry and restrained molecular dynamics methods for determining three-dimensional structures of proteins on the basis of interproton distances is presented using crambin as a model system. It is shown that both methods reproduce the overall features of the secondary and tertiary structure (shape and polypeptide fold). The region of conformational space sampled by the converged structures generated by the two methods is similar in size, and in both cases the converged structures are distributed about mean structures which are closer to the X-ray structure than any of the individual structures. The restrained molecular dynamics structures are superior to those obtained from distance geometry as regards local backbone conformation, side chain positions and non-bonding energies.     

Analyzing the flexibility of RNA structures by constraint counting

RNA requires conformational dynamics to undergo its diverse functional roles. Here, a new topological network representation of RNA structures is presented that allows analyzing RNA flexibility/rigidity based on constraint counting. The method extends the FIRST approach, which identifies flexible and rigid regions in atomic detail in a single, static, three-dimensional molecular framework. Initially, the network rigidity of a canonical A-form RNA is analyzed by counting on constraints of network elements of increasing size. These considerations demonstrate that it is the inclusion of hydrophobic contacts into the RNA topological network that is crucial for an accurate flexibility prediction. The counting also explains why a protein-based parameterization results in overly rigid RNA structures. The new network representation is then validated on a tRNA^ASP structure and all NMR-derived ensembles of RNA structures currently available in the Protein Data Bank (with chain length ≥40). The flexibility predictions demonstrate good agreement with experimental mobility data, and the results are superior compared to predictions based on two previously used network representations. Encouragingly, this holds for flexibility predictions as well as mobility predictions obtained by constrained geometric simulations on these networks. Potential applications of the approach to analyzing the flexibility of DNA and RNA/protein complexes are discussed.     

Conformational Particularities of the Tachykinin-Like Decapeptide Sialokinin I

This paper reports the study of the three-dimensional structure of the tachykinin-like decapeptide sialokinin I molecule using molecular mechanics and molecular dynamics. Fragment analysis was used to identify the stable spatial structures of sialokinin I, which may be present as a set of conformations that are characterized by the relatively labile N-terminal tripeptide and the conformationally rigid C-terminal heptapeptide. It has been demonstrated that the sialokinin I molecule tends to adopt nearly isoenergetic conformations with different structural types at the N-end of the peptide chain, which change into the alpha-helix turn at its C-end. Molecular dynamics was employed to model the sialokinin I molecule mobility in its stable conformations both in a vacuum and when surrounded by water molecules.

     

Protein Dynamics Governed by Interfaces of High Polarity and Low Packing Density

The folding pathway, three-dimensional structure and intrinsic dynamics of proteins are governed by their amino acid sequences. Internal protein surfaces with physicochemical properties appropriate to modulate conformational fluctuations could play important roles in folding and dynamics. We show here that proteins contain buried interfaces of high polarity and low packing density, coined as LIPs: Light Interfaces of high Polarity, whose physicochemical properties make them unstable. The structures of well-characterized equilibrium and kinetic folding intermediates indicate that the LIPs of the corresponding native proteins fold late and are involved in local unfolding events. Importantly, LIPs can be identified using very fast and uncomplicated computational analysis of protein three-dimensional structures, which provides an easy way to delineate the protein segments involved in dynamics. Since LIPs can be retained while the sequences of the interacting segments diverge significantly, proteins could in principle evolve new functional features reusing pre-existing encoded dynamics. Large-scale identification of LIPS may contribute to understanding evolutionary constraints of proteins and the way protein intrinsic dynamics are encoded.     

Spatial Structure and Diffusive Dynamics from Single-Particle Trajectories Using Spline Analysis

Single-particle tracking of biomolecular probes has provided a wealth of information about intracellular trafficking and the dynamics of proteins and lipids in the cell membrane. Conventional mean-square displacement (MSD) analysis of single-particle trajectories often assumes that probes are moving in a uniform environment. However, the observed two-dimensional motion of probe particles is influenced by the local three-dimensional geometry of the cell membrane and intracellular structures, which are rarely flat at the submicron scale. This complex geometry can lead to spatially confined trajectories that are difficult to analyze and interpret using conventional two-dimensional MSD analysis. Here we present two methods to analyze spatially confined trajectories: spline-curve dynamics analysis, which extends conventional MSD analysis to measure diffusive motion in confined trajectories; and spline-curve spatial analysis, which measures spatial structures smaller than the limits of optical resolution. We show, using simulated random walks and experimental trajectories of quantum dot probes, that differences in measured two-dimensional diffusion coefficients do not always reflect differences in underlying diffusive dynamics, but can instead be due to differences in confinement geometries of cellular structures.     

A local-optimization refinement algorithm in single particle analysis for macromolecular complex with multiple rigid modules

Single particle analysis, which can be regarded as an average of signals from thousands or even millions of particle projections, is an efficient method to study the three-dimensional structures of biological macromolecules. An intrinsic assumption in single particle analysis is that all the analyzed particles must have identical composition and conformation. Thus specimen heterogeneity in either composition or conformation has raised great challenges for high-resolution analysis. For particles with multiple conformations, inaccurate alignments and orientation parameters will yield an averaged map with diminished resolution and smeared density. Besides extensive classification approaches, here based on the assumption that the macromolecular complex is made up of multiple rigid modules whose relative orientations and positions are in slight fluctuation around equilibriums, we propose a new method called as local optimization refinement to address this conformational heterogeneity for an improved resolution. The key idea is to optimize the orientation and shift parameters of each rigid module and then reconstruct their three-dimensional structures individually. Using simulated data of 80S/70S ribosomes with relative fluctuations between the large (60S/50S) and the small (40S/30S) subunits, we tested this algorithm and found that the resolutions of both subunits are significantly improved. Our method provides a proof-of-principle solution for high-resolution single particle analysis of macromolecular complexes with dynamic conformations.     

Structure determination of a pseudotripeptide zinc complex with the COSMOS-NMR force field and DFT methods

A His-X-His pseudotripeptide zinc complex (X is a N-alkyl glycine derivative) similar to the catalytic center of the carbonic anhydrase was computer designed and experimentally synthesized. Using 2D-NMR techniques, all proton, carbon chemical shifts and nuclear overhauser effect signals were assigned. The three-dimensional structure of the complex was determined with the COSMOS (computer simulation of molecular structures) force field by applying ¹³C bond polarization theory chemical shift pseudo forces and restrictions for NOE distances. From molecular dynamics, simulated annealing simulations and geometry optimizations, the three best force field structures were taken for a final investigation by density functional theory calculations.     

Picosecond timescale rigid-helix and side-chain motions in deoxymyoglobin

The contribution of rigidbody motions to the atomic trajectories in a 100 ps molecular dynamics simulation of deoxymyoglobin is examined. Two typesof rigid-body motions are considered: one in which the helices are rigid units and one in which the side-chains are rigid units. Using a quaternionbased algorithm, fits of the rigid reference structures are made to each time frame of the simulation to derive trajectories of the rigid-body motions. The fitted trajectories are analysed in terms of atomic position fluctuations, mean-square displacements as a function of time, velocity autocorrelation functions and densities of states. The results are compared with the corresponding quantities calculated from the full trajectory. The relative contribution of the rigid helix motions to the helix atom dynamics depends on which quantity is examined and on which subset of atoms is chosen: rigid-helix motions contribute 86% of the rms helix backbone atomic position fluctuations, but 30% of the helix,: atom (backbone and side-chain) mean square displacements and only 1.1% of total kinetic energy. Only very low-frequency motions contribute to the rigid-helix dynamics; the rigid-body analysis allows characteristic rigid-helix vibrations to be identified and described. Treating the side-chains as rigid bodies is foundto be an excellent approximation to both their diffusive and vibrationalmean-square displacements: 96% of side-chain atom mean-square displacements originate from rigid side-Chain motions. However, the errors in theside-chain atomic positional fits are not always small. An analysis is madeof factors contributing to the positional error for different types of side-chain. © Wiley-Liss, Inc.     

The impact of protein flexibility on protein-protein docking

Accounting for protein flexibility in protein-protein docking algorithms is challenging, and most algorithms therefore treat proteins as rigid bodies or permit side-chain motion only. While the consequences are obvious when there are large conformational changes upon binding, the situation is less clear for the modest conformational changes that occur upon formation of most protein-protein complexes. We have therefore studied the impact of local protein flexibility on protein-protein association by means of rigid body and torsion angle dynamics simulation. The binding of barnase and barstar was chosen as a model system for this study, because the complexation of these 2 proteins is well-characterized experimentally, and the conformational changes accompanying binding are modest. On the side-chain level, we show that the orientation of particular residues at the interface (so-called hotspot residues) have a crucial influence on the way contacts are established during docking from short protein separations of approximately 5 A. However, side-chain torsion angle dynamics simulations did not result in satisfactory docking of the proteins when using the unbound protein structures. This can be explained by our observations that, on the backbone level, even small (2 A) local loop deformations affect the dynamics of contact formation upon docking. Complementary shape-based docking calculations confirm this result, which indicates that both side-chain and backbone levels of flexibility influence short-range protein-protein association and should be treated simultaneously for atomic-detail computational docking of proteins.     

Relationship of protein flexibility to thermostability

Thermostability of proteins arises from the simultaneous effect of several forces, which in fact lead to decreased flexibility of the polypeptide chain. This is verified by flexibility indices, which are derived from normalized B-values of individual amino acids in several refined three-dimensional structures. Flexibility indices show that overall flexibility is reduced when thermostability is increased. Protein molecules require both flexibility and rigidity to function, but the higher the temperature optimum and stability the more rigid is the structure needed to compensate for increased thermal fluctuations. Flexibilities of proteins performing the same catalytic activity seem to be about the same at their temperature optima, but the more rigid thermostable proteins reach the flexibility of thermolabile proteins at higher temperatures. In several proteins such as allosteric enzymes, some local sites of flexibility are highly conserved. The relevance of reduced flexibility to overall stability of proteins is also discussed. Flexibility indices and profiles can be used in the design of more stable proteins by site-directed mutagenesis.     

Localized Modeling of Biochemical and Flow Interactions during Cancer Cell Adhesion

This work focuses on one component of a larger research effort to develop a simulation tool to model populations of flowing cells. Specifically, in this study a local model of the biochemical interactions between circulating melanoma tumor cells (TC) and substrate adherent polymorphonuclear neutrophils (PMN) is developed. This model provides realistic three-dimensional distributions of bond formation and attendant attraction and repulsion forces that are consistent with the time dependent Computational Fluid Dynamics (CFD) framework of the full system model which accounts local pressure, shear and repulsion forces. The resulting full dynamics model enables exploration of TC adhesion to adherent PMNs, which is a known participating mechanism in melanoma cell metastasis. The model defines the adhesion molecules present on the TC and PMN cell surfaces, and calculates their interactions as the melanoma cell flows past the PMN. Biochemical rates of reactions between individual molecules are determined based on their local properties. The melanoma cell in the model expresses ICAM-1 molecules on its surface, and the PMN expresses the β-2 integrins LFA-1 and Mac-1. In this work the PMN is fixed to the substrate and is assumed fully rigid and of a prescribed shear-rate dependent shape obtained from micro-PIV experiments. The melanoma cell is transported with full six-degrees-of-freedom dynamics. Adhesion models, which represent the ability of molecules to bond and adhere the cells to each other, and repulsion models, which represent the various physical mechanisms of cellular repulsion, are incorporated with the CFD solver. All models are general enough to allow for future extensions, including arbitrary adhesion molecule types, and the ability to redefine the values of parameters to represent various cell types. The model presented in this study will be part of a clinical tool for development of personalized medical treatment programs.     

Stability and rigidity/flexibility—Two sides of the same coin?

Protein molecules require both flexibility and rigidity for functioning. The fast and accurate prediction of protein rigidity/flexibility is one of the important problems in protein science. We have determined flexible regions for four homologous pairs from thermophilic and mesophilic organisms by two methods: the fast FoldUnfold which uses amino acid sequence and the time consuming MDFirst which uses three-dimensional structures. We demonstrate that both methods allow determining flexible regions in protein structure. For three of the four thermophile–mesophile pairs of proteins, FoldUnfold predicts practically the same flexible regions which have been found by the MD/First method. As expected, molecular dynamics simulations show that thermophilic proteins are more rigid in comparison to their mesophilic homologues. Analysis of rigid clusters and their decomposition provides new insights into protein stability. It has been found that the local networks of salt bridges and hydrogen bonds in thermophiles render their structure more stable with respect to fluctuations of individual contacts. Such network includes salt bridge triads Agr-Glu-Lys and Arg-Glu-Arg, or salt bridges (such as Arg-Glu) connected with hydrogen bonds. This ionic network connects alpha helices and rigidifies the structure. Mesophiles can be characterized by stand alone salt bridges and hydrogen bonds or small ionic clusters. Such difference in the network of salt bridges results in different flexibility of homologous proteins. Combining both approaches allows characterizing structural features in atomic detail that determine the rigidity/flexibility of a protein structure. This article is a part of a Special Issue entitled: The emerging dynamic view of proteins: Protein plasticity in allostery, evolution and self-assembly.     

Ergoline Derivatives as a Probe for Featuring the 5-HT1A Receptor Pharmacophore

A 5-HT_1A pharmacophore has been obtained employing a set of rigid templates encompassing the 5-HT1A structure. The use of rigid templates allowed us to overcome the discrepancy found when flexible structures where the energy of the active conformers are sometimes higher than the global minimum energy are used. On the basis of the results herein reported the three-dimensional requirements necessary for the binding interaction have been defined within this set of molecules. In this study forbidden zones of the receptor have been characterised. The pharmacophore model derived places some agonist/antagonist pharmacophore models appeared in the literature.     

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.     

Site-Specific Structural Variations Accompanying Tubular Assembly of the HIV-1 Capsid Protein

The 231-residue capsid (CA) protein of human immunodeficiency virus type 1 (HIV-1) spontaneously self-assembles into tubes with a hexagonal lattice that is believed to mimic the surface lattice of conical capsid cores within intact virions. We report the results of solid-state nuclear magnetic resonance (NMR) measurements on HIV-1 CA tubes that provide new information regarding changes in molecular structure that accompany CA self-assembly, local dynamics within CA tubes, and possible mechanisms for the generation of lattice curvature. This information is contained in site-specific assignments of signals in two- and three-dimensional solid-state NMR spectra, conformation-dependent ¹⁵N and ¹³C NMR chemical shifts, detection of highly dynamic residues under solution NMR conditions, measurements of local variations in transverse spin relaxation rates of amide ¹H nuclei, and quantitative measurements of site-specific ¹⁵N–¹⁵N dipole–dipole couplings. Our data show that most of the CA sequence is conformationally ordered and relatively rigid in tubular assemblies and that structures of the N-terminal domain (NTD) and the C-terminal domain (CTD) observed in solution are largely retained. However, specific segments, including the N-terminal β-hairpin, the cyclophilin A binding loop, the inter-domain linker, segments involved in intermolecular NTD–CTD interactions, and the C-terminal tail, have substantial static or dynamical disorder in tubular assemblies. Other segments, including the 3₁₀-helical segment in CTD, undergo clear conformational changes. Structural variations associated with curvature of the CA lattice appear to be localized in the inter-domain linker and intermolecular NTD–CTD interface, while structural variations within NTD hexamers, around local 3-fold symmetry axes, and in CTD–CTD dimerization interfaces are less significant.