Langevin modes of macromolecules: applications to crambin and DNA hexamers

Langevin modes describe the behavior of atoms moving on a harmonic potential surface subject to viscous damping described by a classical Langevin equation. We present applications to the protein crambin and to the DNA duplex d(CpGpCpGpCpG)2 and its complex with ethidium. Our friction matrix is weighted according to surface area exposed to solvent, and results are reported for various values of the solvent viscosity and models for hydrodynamic interactions. Even for relatively small solvent friction (eta = 0.3 cp) a substantial number of modes are overdamped, and time correlation functions decay smoothly without the oscillations characteristic of gas-phase calculations. Perturbation theory starting from the gas-phase modes is accurate for many low-frequency modes (which are overdamped in the presence of solvent), but fails badly for higher modes. For correlation functions of interest to fluorescence depolarization or nmr relaxation, the plateau values are insensitive to solvent viscosity, but the relaxation times are not. The advantages and limitations of this analysis of macromolecular motions are discussed.     

Computer simulations of the flexibility of a series of synthetic cyclic peptide analogues

The flexibility of a series of cyclic peptides derived from the epitope of a snake toxin is investigated using computer simulation techniques. Molecular dynamics (MD) simulations and vibrational analyses are performed on chemically constrained peptides modeled on the parent protein loop. In the 50 ps MD simulations, large variations in the atomic fluctuations are seen between the peptides, and can be related to the nature of the chemical constraints present in the molecules. Normal mode analyses are performed on energy-minimized configurations derived from the dynamics trajectories. The atomic fluctuations calculated from the normal modes are about 30% of those of the molecular dynamics for the more flexible peptides and 70% for the more constrained molecules. The calculated differences in flexibility between the molecules are much less significant in the harmonic approximation. © 1993 John Wiley & Sons, Inc.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method.

This paper presents a general method for studying the harmonic dynamics of large biomolecules and molecular complexes. The performance and accuracy of the method applied to a number of molecules are also reported. The basic approach of the method is to divide a macromolecule into a number of smaller components. The local normal modes of the components are first calculated by treating individual components and the interactions between nearest neighboring components. The physical displacements of all atoms are then represented in the local normal mode space, in which a selected range of high-frequency local modes is neglected. The equation of motion of the molecule in the local normal mode space will then have a smaller dimension, and consequently the normal modes of the whole structure, particularly for large molecules, can be solved much more easily. The normal modes of two polypeptides--(Ala)6 and (Ala)12--and a double-helical DNA--d(ATATA).d(TATAT)--are analyzed with this method. Reductions on the dimensions of harmonic dynamic equations for these molecules have been made, with the fraction of the deleted high-frequency modes ranging from 1/2 to 5/6. The calculated low-frequency normal modes are found to be very accurate as compared to the exact solutions by standard procedure. The major advantage of the present approach on macromolecule harmonic dynamics is that the reduction on the dimensionality of the eigenvalue problems can be varied according to the size of molecules, so the method can be easily applied to large macromolecules with controlled accuracy.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method: Residue clustering and multiple-component approach

A study of the component synthesis method (CSM) for analyzing the normal mode dynamics of macromolecules is reported. The procedure involves a reduction of the dimensions of the normal mode problems for large molecular systems and the accurate extraction of the low-frequency modes. A macromolecule is divided into small components based on a hierarchical clustering of the residues in the structure. Interactions between coupled components are treated by the method of static correlation. The normal modes of the components are obtained first, and a fraction of the low-frequency normal modes of the components under mutual correlations are then used as a reduced basis for solving for the normal modes of the whole molecule. Multiple components are introduced for large macromolecules so that the dimensions of the eigenvalue problems at the component level are small. The method is applied to the protein crambin. In test calculations in which the dimensions of the eigenvalue equations are reduced to 1/6 of their natural size, the errors in the normal mode frequencies calculated by the CSM procedure are only about 1–2% when compared with the exact values. The rms fluctuations of all atoms in crambin calculated by the CSM procedure are basically identical to the exact results. The CSM procedure is shown to be accurate for calculating the normal modes of large macromolecules with a significant reduction of the size of the problem. © 1994 John Wiley & Sons, Inc.     

Normal mode refinement: crystallographic refinement of protein dynamic structure. I. Theory and test by simulated diffraction data.

A dynamic structure refinement method for X-ray crystallography, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the low-frequency normal modes whose amplitudes and eigenvectors are experimentally optimized in the process of the crystallographic refinement. In this model, the atomic fluctuations are treated as anisotropic and concerted. The normal modes of the external motion (TLS model) are also introduced to cover the factors other than the internal fluctuations, such as the lattice disorder and diffusion. A program for the normal mode refinement (NM-REF) has been developed. The method has first been tested against simulated diffraction data for human lysozyme calculated by a Monte Carlo simulation. Applications of the method have demonstrated that the normal mode refinement has: (1) improved the fitting to the diffraction data, even with fewer adjustable parameters; (2) distinguished internal fluctuations from external ones; (3) determined anisotropic thermal factors; and (4) identified concerted fluctuations in the protein molecule.     

Normal mode refinement: crystallographic refinement of protein dynamic structure applied to human lysozyme.

A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.     

Coarse-grained modeling of the actin filament derived from atomistic-scale simulations

A coarse-grained (CG) procedure that incorporates the information obtained from all-atom molecular dynamics (MD) simulations is presented and applied to actin filaments (F-actin). This procedure matches the averaged values and fluctuations of the effective internal coordinates that are used to define a CG model to the values extracted from atomistic MD simulations. The fluctuations of effective internal coordinates in a CG model are computed via normal-mode analysis (NMA), and the computed fluctuations are matched with the atomistic MD results in a self-consistent manner. Each actin monomer (G-actin) is coarse-grained into four sites, and each site corresponds to one of the subdomains of G-actin. The potential energy of a CG G-actin contains three bonds, two angles, and one dihedral angle; effective harmonic bonds are used to describe the intermonomer interactions in a CG F-actin. The persistence length of a CG F-actin was found to be sensitive to the cut-off distance of assigning intermonomer bonds. Effective harmonic bonds for a monomer with its third nearest neighboring monomers are found to be necessary to reproduce the values of persistence length obtained from all-atom MD simulations. Compared to the elastic network model, incorporating the information of internal coordinate fluctuations enhances the accuracy and robustness for a CG model to describe the shapes of low-frequency vibrational modes. Combining the fluctuation-matching CG procedure and NMA, the achievable time- and length scales of modeling actin filaments can be greatly enhanced. In particular, a method is described to compute the force-extension curve using the CG model developed in this work and NMA. It was found that F-actin is easily buckled under compressive deformation, and a writhing mode is developed as a result. In addition to the bending and twisting modes, this novel writhing mode of F-actin could also play important roles in the interactions of F-actin with actin-binding proteins and in the force-generation process via polymerization.     

Comparison of full-atomic and coarse-grained models to examine the molecular fluctuations of c-AMP dependent protein kinase

Molecular fluctuations of the native conformation of c-AMP dependent protein kinase (cAPK) have been investigated with three different approaches. The first approach is the full atomic normal mode analysis (NMA) with empirical force fields. The second and third approaches are based on a coarse-grained model with a single single-parameter- harmonic potential between close residues in the crystal structure of the molecule without any residue specificity. The second method calculates only the magnitude of fluctuations whereas the third method is developed to find the directionality of the fluctuations which are essential to understand the functional importance of biological molecules. The aim, in this study, is to determine whether using such coarse-grained models are appropriate for elucidating the global dynamic characteristics of large proteins which reduces the size of the system at least by a factor of ten. The mean-square fluctuations of C(alpha) atoms and the residue cross-correlations are obtained by three approaches. These results are then compared to test the results of coarse grained models on the overall collective motions. All three of the approaches show that highly flexible regions correspond to the activation and solvent exposed loops, whereas the conserved residues (especially in substrate binding regions) exhibit almost no flexibility, adding stability to the structure. The anti-correlated motions of the two lobes of the catalytic core provide flexibility to the molecule. High similarities among the results of these methods indicate that the slowest modes governing the most global motions are preserved in the coarse grained models for proteins. This finding may suggest that the general shapes of the structures are representative of their dynamic characteristics and the dominant motions of protein structures are robust at coarse-grained levels.     

Anisotropy and anharmonicity of atomic fluctuations in proteins: implications for X-ray analysis

The effects of anisotropy and anharmonicity of the atomic fluctuations on the results of crystallographic refinement of proteins are examined. Atomic distribution functions from a molecular dynamics simulation for lysozyme are introduced into a real-space (electron density) refinement procedure for individual atoms. Several models for the atomic probability distributions are examined. When isotropic, harmonic motion is assumed, the largest discrepancies between the true first moments (means) and second moments (B factors) of the positions calculated from the dynamics and the fitted values occur for probability densities with multiple peaks. The refined mean is at the center of the largest peak, and the refined B factor is slightly larger than that of the largest peak, unless the distance between the peaks is small compared to the peak width. The resulting values are often significantly different from the true first and second moments of the distribution. To improve the results, alternate conformations, rather than anharmonic corrections, should be included.     

Stability of triple-helical poly(dT)-poly(dA)-poly(dT) DNA with counterions.

Structural conformation of triple-helical poly(dT)-poly(dA)-poly(dT) has been a very controversial issue recently. Earlier investigations, based on fiber diffraction data and molecular modeling, indicated an A-form conformation with C'3-endo sugar pucker. On the other hand, Raman, solution infrared spectral, and NMR studies show a B-form structure with C'2-endo sugars. In accordance with these experimental results, a theoretical model with B-form, C'2-endo sugars was proposed in 1993. In the present work we investigate the dynamics and stability of the two conformations within the effective local field approach applied to the normal mode calculations for the system. The presence of counterions was explicitly taken into account. Stable equilibrium positions for the counterions were calculated by analyzing the normal mode dynamics and free energy of the system. The breathing modes of the triple helix are shifted to higher frequencies over those of the double helix by 4-16 cm-1. The characteristic marker band for the B conformation at 835 cm-1 is split up into two marker bands at 830 and 835 cm-1. A detailed comparison of the normal modes and the free energies indicates that the B-form structure, with C'2-endo sugar pucker, is more stable than the A-form structure. The normal modes and the corresponding dipole moments are found to be in close agreement with recent spectroscopic findings.     

A molecular dynamics study of the C-terminal fragment of the L7/L12 ribosomal protein. Secondary structure motion in a 150 picosecond trajectory

A 150 picosecond molecular dynamics computer simulation of the C-terminal fragment of the L7/L12 ribosomal protein from Escherichia coli is reported. The molecular dynamics results are compared with the available high-resolution X-ray data in terms of atomic positions, distances and positional fluctuations. Good agreement is found between the molecular dynamics results and the X-ray data. The form and parameters of the interaction potential energy function and the procedures for deriving it are discussed. Some current misunderstandings concerning the ways of evaluating the efficiency of molecular dynamics algorithms and of application of bond-length constraints in protein simulations are cleared up. The 150 picosecond trajectory has been scanned in a search for correlated motions within and between secondary structure elements. The beta-strands have diffusional stretching modes, and uncorrelated transversal displacements. The dynamic analysis of alpha-helices shows a variety of features. The atomic fluctuations differ between the helix ends; this effect reflects long time-scale motions. Two alpha-helices, alpha A and alpha C, show diffusive longitudinal stretching modes. The third helix, alpha B, has a correlated asymmetric longitudinal stretching; the N-terminal part dominates this behaviour. Furthermore, alpha B presents a librational motion with respect to the other parts of the molecule with a frequency of approximately 5 cm-1. This motion is coupled to helix stretching. Interestingly, the regions of highly conserved residues contain the most mobile parts of the molecule.     

Anisotropy and anharmonicity of atomic fluctuations in proteins: analysis of a molecular dynamics simulation

Positional probability density functions (pdf) for the atomic fluctuations are determined from a molecular dynamics simulation for hen egg-white lysozyme. Most atoms are found to have motions that are highly anisotropic but only slightly anharmonic. The largest deviations from harmonic motion are in the direction of the largest rms fluctuations in the local principal axis frame. Backbone atoms tend to be more nearly harmonic than sidechain atoms. The atoms with the largest anharmonicities tend to have pdfs with multiple peaks, each of which is close to harmonic. Several model pdfs are evaluated on the basis of how well they fit probability densities from the dynamics simulations when parameterized in terms of the moments of the distribution. Gram-Charlier and Edgeworth perturbation expansions, which have been successful in describing the motions of small molecules in crystals, are shown to be inadequate for the distributions found in the dynamics of proteins. Multipeaked distribution functions are found to be more appropriate.     

Vibrational normal modes and dynamical stability of DNA triplex poly(dA). 2poly(dT): S-type structure is more stable and in better agreement with observations in solution.

A normal-mode and statistical mechanical calculation was carried out to determine the vibrational normal modes, contribution of internal fluctuations to the free energy, and hydrogen bond disruption of DNA triplex poly(dA).2poly(dT). The calculation was performed on both the x-ray fiber diffraction model with a N-type sugar conformation, and a newly proposed model with a S-type sugar conformation. Our calculated normal modes for the S-type structure are in better agreement with observed IR spectra for samples in D2O solution. We also find that the contribution of internal fluctuations to free energy, premelting hydrogen bond disruption probability, and hydrogen bond melting temperatures for the Hoogsteen and Watson-Crick hydrogen bonds all show that the S-type structure is dynamically more stable than the N-type structure in a nominal solution environment. Therefore our calculation supports experimental findings that the triplex d(T)n.d(A)nd(T)n most likely adopts a S-type sugar conformation in solution or at high humidity. Our calculations, however, do not preclude the possibility of an N-type conformation at lower humidities.     

A combined 2D-NMR and molecular dynamics analysis of the structure of the actinomycin D: d(ATGCAT)2 complex

We present a comparative analysis of an NMR experiment and molecular and harmonic dynamics simulations of an actinomycin D: d(ATGCAT)2 complex. A comparison of NOE measurements and 1/R6 weighted proton-proton distances confirm the general correctness of the Actinomycin D-DNA model proposed by Sobell. There are, however, some substantial differences between the proton-proton distances inferred from the NOE results and the molecular and harmonic dynamics simulations. The remaining discrepancies could either come from contributions of other conformations to the average properties of the complex or from uncertainties in the NMR distance analysis. An analysis of the molecular dynamics helix properties, sugar puckers, hydrogen bonding, rms fluctuations and torsional properties are qualitatively consistent with those from previous simulations, but the presence of an intercalated drug leads to some new structural and dynamical features.     

Filtering molecular dynamics trajectories to reveal low-frequency collective motions: phospholipase A2

A novel method for analysing molecular dynamics trajectories has been developed, which filters out high frequencies using digital signal processing techniques and facilitates focusing on the low-frequency collective motions of proteins. These motions involve low energy slow motions, which lead to important biological phenomena such as domain closure and allosteric effects in enzymes. The filtering method treats each of the atomic trajectories obtained from the molecular dynamics simulation as a "signal". The trajectories of each of the atoms in the system (or any subset of interest) are Fourier transformed to the frequency domain, a filtering function is applied and then an inverse transformation back to the time domain yields the filtered trajectory. The filtering method has been used to study the dynamics of the enzyme phospholipase A2. In the filtered trajectory, all the high frequency bond and valence angle vibrations were eliminated, leaving only low-frequency motion, mainly fluctuations in torsions and conformational transitions. Analysis of this trajectory revealed interesting motions of the protein, including concerted movements of helices, and changes in shape of the active site cavity. Unlike normal mode analysis, which has been used to study the motion of proteins, this method does not require converged minimizations or diagonalization of a matrix of second derivatives. In addition, anharmonicity, multiple minima and conformational transitions are treated explicitly. Thus, the filtering method avoids most of the approximations implicit in other investigations of the dynamic behaviour of large systems.     

Comparison of Full-atomic and Coarse-grained Models to Examine the Molecular Fluctuations of c-AMP Dependent Protein Kinase

Abstract

Molecular fluctuations of the native conformation of c-AMP dependent protein kinase (cAPK) have been investigated with three different approaches. The first approach is the full atomic normal mode analysis (NMA) with empirical force fields. The second and third approaches are based on a coarse-grained model with a single single-parameter- harmonic potential between close residues in the crystal structure of the molecule without any residue specificity. The second method calculates only the magnitude of fluctuations whereas the third method is developed to find the directionality of the fluctuations which are essential to understand the functional importance of biological molecules. The aim, in this study, is to determine whether using such coarse-grained models are appropriate for elucidating the global dynamic characteristics of large proteins which reduces the size of the system at least by a factor of ten. The mean-square fluctuations of C^α atoms and the residue cross-correlations are obtained by three approaches. These results are then compared to test the results of coarse grained models on the overall collective motions. AH three of the approaches show that highly flexible regions correspond to the activation and solvent exposed loops, whereas the conserved residues (especially in substrate binding regions) exhibit almost no flexibility, adding stability to the structure. The anti-correlated motions of the two lobes of the catalytic core provide flexibility to the molecule. High similarities among the results of these methods indicate that the slowest modes governing the most global motions are preserved in the coarse grained models for proteins. This finding may suggest that the general shapes of the structures are representative of their dynamic characteristics and the dominant motions of protein structures are robust at coarse-grained levels.     

A kinetic model for the internal motions of proteins: diffusion between multiple harmonic wells.

The dynamics of collective protein motions derived from Molecular Dynamics simulations have been studied for two small model proteins: initiation factor I and the B1 domain of Protein G. First, we compared the structural fluctuations, obtained by local harmonic approximations in different energy minima, with the ones revealed by large scale molecular dynamics (MD) simulations. It was found that a limited set of harmonic wells can be used to approximate the configurational fluctuations of these proteins, although any single harmonic approximation cannot properly describe their dynamics. Subsequently, the kinetics of the main (essential) collective protein motions were characterized. A dual-diffusion behavior was observed in which a fast type of diffusion switches to a much slower type in a typical time of about 1-3 ps. From these results, the large backbone conformational fluctuations of a protein may be considered as "hopping" between multiple harmonic wells on a basically flat free energy surface.     

Why are large conformational changes well described by harmonic normal modes?

Low-frequency normal modes generated by elastic network models tend to correlate strongly with large conformational changes of proteins, despite their reliance on the harmonic approximation, which is only valid in close proximity of the native structure. We consider 12 variants of the torsional network model (TNM), an elastic network model in torsion angle space, that adopt different sets of torsion angles as degrees of freedom and reproduce with similar quality the thermal fluctuations of proteins but present drastic differences in their agreement with conformational changes. We show that these differences are related to the extent of the deviations from the harmonic approximation, assessed through an anharmonic energy function whose harmonic approximation coincides with the TNM. Our results indicate that mode anharmonicity is more strongly related to its collectivity, i.e., the number of atoms displaced by the mode, than to its amplitude; low-frequency modes can remain harmonic even at large amplitudes, provided they are sufficiently collective. Finally, we assess the potential benefits of different strategies to minimize the impact of anharmonicity. The reduction of the number of degrees of freedom or their regularization by a torsional harmonic potential significantly improves the collectivity and harmonicity of normal modes and the agreement with conformational changes. In contrast, the correction of normal mode frequencies to partially account for anharmonicity does not yield substantial benefits. The TNM program is freely available at https://github.com/ugobas/tnm.