Efficient sampling in collective coordinate space

Collective motions in biological macromolecules have been shown to be important for function. The most important collective motions occur on slow time scales, which poses a sampling problem in dynamic simulation of biomolecules. We present a novel method for efficient conformational sampling. The method combines the simulation of an ensemble of concurrent trajectories with restraints acting on the ensemble of structures as a whole. Two properties of the ensemble may be restrained: (i) the variance of the ensemble and (ii) the average position of the ensemble. Both properties are defined in a subspace of collective coordinate space spanned by an arbitrary number of modes. We show that weak restraints on the ensemble variance suffice for an increase in sampling efficiency along soft modes by two orders of magnitudes. The resulting trajectories exhibit virtually the same structural quality as trajectories generated by restraint-free-molecular dynamics simulation, as judged by standard structure validation tools. The method is used to probe the resistance of a structure against conformational changes along collective modes and clearly distinguishes soft from stiff modes. Further applications are discussed. Proteins 2000;39:82-88.     

Normal mode analysis of macromolecular motions in a database framework: developing mode concentration as a useful classifying statistic

We investigated protein motions using normal modes within a database framework, determining on a large sample the degree to which normal modes anticipate the direction of the observed motion and were useful for motions classification. As a starting point for our analysis, we identified a large number of examples of protein flexibility from a comprehensive set of structural alignments of the proteins in the PDB. Each example consisted of a pair of proteins that were considerably different in structure given their sequence similarity. On each pair, we performed geometric comparisons and adiabatic-mapping interpolations in a high-throughput pipeline, arriving at a final list of 3,814 putative motions and standardized statistics for each. We then computed the normal modes of each motion in this list, determining the linear combination of modes that best approximated the direction of the observed motion. We integrated our new motions and normal mode calculations in the Macromolecular Motions Database, through a new ranking interface at http://molmovdb.org. Based on the normal mode calculations and the interpolations, we identified a new statistic, mode concentration, related to the mathematical concept of information content, which describes the degree to which the direction of the observed motion can be summarized by a few modes. Using this statistic, we were able to determine the fraction of the 3,814 motions where one could anticipate the direction of the actual motion from only a few modes. We also investigated mode concentration in comparison to related statistics on combinations of normal modes and correlated it with quantities characterizing protein flexibility (e.g., maximum backbone displacement or number of mobile atoms). Finally, we evaluated the ability of mode concentration to automatically classify motions into a variety of simple categories (e.g., whether or not they are "fragment-like"), in comparison to motion statistics. This involved the application of decision trees and feature selection (particular machine-learning techniques) to training and testing sets derived from merging the "list" of motions with manually classified ones.     

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.     

Energy landscape of a native protein: Jumping-among-minima model

We have investigated energy landscape of human lysozyme in its native state by using principal component analysis and a model, jumping-among-minima (JAM) model. These analyses are applied to 1 nsec molecular dynamics trajectory of the protein in water. An assumption embodied in the JAM model allows us to divide protein motions into intra-substate and inter-substate motions. By examining intra-substate motions, it is shown that energy surfaces of individual conformational substates are nearly harmonic and mutually similar. As a result of principal component analysis and JAM model analysis, protein motions are shown to consist of three types of collective modes, multiply hierarchical modes, singly hierarchical modes, and harmonic modes. Multiply hierarchical modes, the number of which accounts only for 0.5% of all modes, dominate contributions to total mean-square atomic fluctuation. Inter-substate motions are observed only in a small-dimensional subspace spanned by the axes of multiplyhierarchical and singly hierarchical modes. Inter-substate motions have two notable time components: faster component seen within 200 psec and slower component. The former involves transitions among the conformational substates of the low-level hierarchy, whereas the latter involves transitions of the higher level substates observed along the first four multiply hierarchical modes. We also discuss dependence of the subspace, which contains conformational substates, on time duration of simulation. Proteins 33:496–517, 1998. © 1998 Wiley-Liss, Inc.     

Diagonalization in a mixed basis: A method to compute low-frequency normal modes for large macromolecules

Low-frequency collective motions in proteins are generally very important for their biological functions. To study such motions, harmonic dynamics proved most useful since it is a straightforward method; it consists of the diagonalization of the Hessian matrix of the potential energy, yielding the vibrational spectrum and the directions of internal motions. Unfortunately, the diagonalization of this matrix requires a large computer memory, which is a limiting factor when the protein contains several thousand atoms. To circumvent this limitation we have developed three methods that enable us to diagonalize large matrices using much less computer memory than the usual harmonic dynamics. The first method is approximate; it consists of diagonalizing small blocks of the Hessian matrix, followed by the coupling of the low-frequency modes obtained for each block. It yields the low-frequency vibrational spectrum with a maximum error of 20%. The second method consists, after diagonalizing small blocks, of coupling the high- and low-frequency modes using an iterative procedure. It yields the exact low-frequency normal modes, but requires a long computational time with convergence problems. The third method, DIMB (Diagonalization in a Mixed Basis), which has the best performance, consists of coupling the approximate low-frequency modes with the mass-weighted cartesian coordinates, also using an iterative procedure. It reduces significantly the required computer memory and converges rapidly. The eigenvalues and eigenvectors obtained by this method are without significant error in the chosen frequency range. Moreover, it is a general method applicable to any problem of diagonalization of a large matrix. We report the application of these methods to a deca-alanine helix, trypsin inhibitor, a neurotoxin, and lysozyme. © 1993 John Wiley & Sons, Inc.     

Continuous Interdomain Orientation Distributions Reveal Components of Binding Thermodynamics

The flexibility of biological macromolecules is an important structural determinant of function. Unfortunately, the correlations between different motional modes are poorly captured by discrete ensemble representations. Here, we present new ways to both represent and visualize correlated interdomain motions. Interdomain motions are determined directly from residual dipolar couplings, represented as a continuous conformational distribution, and visualized using the disk-on-sphere representation. Using the disk-on-sphere representation, features of interdomain motions, including correlations, are intuitively visualized. The representation works especially well for multidomain systems with broad conformational distributions.This analysis also can be extended to multiple probability density modes, using a Bingham mixture model. We use this new paradigm to study the interdomain motions of staphylococcal protein A, which is a key virulence factor contributing to the pathogenicity of Staphylococcus aureus. We capture the smooth transitions between important states and demonstrate the utility of continuous distribution functions for computing the reorientational components of binding thermodynamics. Such insights allow for the dissection of the dynamic structural components of functionally important intermolecular interactions.     

An analysis of the influence of protein intrinsic dynamical properties on its thermal unfolding behavior

The influence of the protein topology-encoded dynamical properties on its thermal unfolding motions was studied in the present work. The intrinsic dynamics of protein topology was obtained by the anisotropic network model (ANM). The ANM has been largely used to investigate protein collective functional motions, but it is not well elucidated if this model can also reveal the preferred large-scale motions during protein unfolding. A small protein barnase is used as a typical case study to explore the relationship between protein topology-encoded dynamics and its unfolding motions. Three thermal unfolding simulations at 500 K were performed for barnase and the entire unfolding trajectories were sampled and partitioned into several windows. For each window, the preferred unfolding motions were investigated by essential dynamics analysis, and then associated with the intrinsic dynamical properties of the starting conformation in this window, which is detected by ANM. The results show that only a few slow normal modes imposed by protein structure are sufficient to give a significant overlap with the preferred unfolding motions. Especially, the large amplitude unfolding movements, which imply that the protein jumps out of a local energy basin, can be well described by a single or several ANM slow modes. Besides the global motions, it is also found that the local residual fluctuations encoded in protein structure are highly correlated with those in the protein unfolding process. Furthermore, we also investigated the relationship between protein intrinsic flexibility and its unfolding events. The results show that the intrinsic flexible regions tend to unfold early. Several early unfolding events can be predicted by analysis of protein structural flexibility. These results imply that protein structure-encoded dynamical properties have significant influences on protein unfolding motions.     

Close correspondence between the motions from principal component analysis of multiple HIV-1 protease structures and elastic network modes

The large number of available HIV-1 protease structures provides a remarkable sampling of conformations of the different conformational states, which can be viewed as direct structural information about the dynamics of the HIV-1 protease. After structure matching, we apply principal component analysis (PCA) to obtain the important apparent motions for both bound and unbound structures. There are significant similarities between the first few key motions and the first few low-frequency normal modes calculated from a static representative structure with an elastic network model (ENM), strongly suggesting that the variations among the observed structures and the corresponding conformational changes are facilitated by the low-frequency, global motions intrinsic to the structure. Similarities are also found when the approach is applied to an NMR ensemble, as well as to molecular dynamics (MD) trajectories. Thus, a sufficiently large number of experimental structures can directly provide important information about protein dynamics, but ENM can also provide similar sampling of conformations.     

Tuning Triangular Prism Dimer into Fano Resonance for Plasmonic Sensor

We theoretically investigate the plasmonic Fano resonance in a triangular nanoprism dimer. By adjusting the geometry parameters, we have observed a Fano line shape in the scattering spectra, which is induced by the competence of bonding and antibonding modes in the triangular nanoprism dimer. The Fano line shape can be well described by a theoretical model of two harmonic oscillators. A figure of merit value as high as 16.1 is achieved in the triangular nanoprism dimer, which is caused by the Fano resonance. The electric field at the corner of the triangular prisms is the highest among the circular cylinder dimer and square rod dimmers, which shows that the triangular prism dimer is more suitable for the detection of biomolecules. The triangular prism dimer may also used in plasmonic circuits.     

Viral capsid equilibrium dynamics reveals nonuniform elastic properties

The long wavelength, low-frequency modes of motion are the relevant motions for understanding the continuum mechanical properties of biomolecules. By examining these low-frequency modes, in the context of a spherical harmonic basis set, we identify four elastic moduli that are required to describe the two-dimensional elastic behavior of capsids. This is in contrast to previous modeling and theoretical studies on elastic shells, which use only the two-dimensional Young's modulus (Y) and the bending modulus (κ) to describe the system. Presumably, the heterogeneity of the structure and the anisotropy of the biomolecular interactions lead to a deviation from the homogeneous, isotropic, linear elastic shell theory. We assign functional relevance of the various moduli governing different deformation modes, including a mode primarily sensed in atomic force microscopy nanoindentation experiments. We have performed our analysis on the T = 3 cowpea chlorotic mottle virus and our estimate for the nanoindentation modulus is in accord with experimental measurements.     

Picosecond dynamics of tyrosine side chains in proteins.

To probe the details of small amplitude motions in proteins, a dynamical analysis of the orientation fluctuations of two tyrosine side chains in the bovine pancreatic trypsin inhibitor is presented. Detailed results are given for the time history and correlation functions obtained for the ring motion from a molecular dynamics simulation of the entire protein. It is shown that even on a picosecond time scale orientational fluctuations of +/-30 degrees from the average position occur for the tyrosine rings in the interior of the protein. It is found that the Langevin equation is applicable to the ring torsional motion, which corresponds to that of an angular harmonic oscillator with near-critical damping. Two possible microscopic models for the observed damping effects are outlined. One of these, analogous to liquid behavior, is based on kinetic theory and takes account of the collisions which occur between atoms of the protein; the other, more analogous to solid behavior, involves the coupling among a large number of harmonic oscillators. The collisional model with parameters obtained from theoretical estimates leads to good agreement with the correlation functions from the dynamic simulation. However, the dephasing of harmonic oscillations can yield similar short-time results so that a distinction between the two models is difficult. The importance of damping effects on the motions involved in conformational transitions and enzymatic reactions is discussed.     

Comparison of tRNA motions in the free and ribosomal bound structures

A general method is presented that allows the separation of the rigid body motions from the nonrigid body motions of structural subunits when bound in a complex. The application presented considers the motions of the tRNAs: free, bound to the ribosome and to a synthase. We observe that both the rigid body and nonrigid body motions of the structural subunits are highly controlled by the large ribosomal assembly and are important for the functional motions of the assembly. For the intact ribosome, its major parts, the 30S and the 50S subunits, are found to have counterrotational motions in the first few slowest modes, which are consistent with the experimentally observed ratchet motion. The tRNAs are found to have on average approximately 72-75% rigid body motions and principally translational motions within the first 100 slow modes of the complex. Although the three tRNAs exhibit different apparent total motions, after the rigid body motions are removed, the remaining internal motions of all three tRNAs are essentially the same. The direction of the translational motions of the tRNAs are in the same direction as the requisite translocation step, especially in the first slowest mode. Surprisingly the small intrinsically flexible mRNA has all of its internal motions completely inhibited and shows mainly a rigid-body translation in the slow modes of the ribosome complex. On the other hand, the required nonrigid body motions of the tRNA during translocation reveal that the anticodon-stem-loop, as well as the acceptor arm, of the tRNA enjoy a large mobility but act as rigid structural units. In summary, the ribosome exerts its control by enforcing rigidity in the functional parts of the tRNAs as well as in the mRNA.     

Studying biomolecular complexes with pulsed electron-electron double resonance spectroscopy

The function of biomolecules is intrinsically linked to their structure and the complexes they form during function. Techniques for the determination of structures and dynamics of these nanometre assemblies are therefore important for an understanding on the molecular level. PELDOR (pulsed electron-electron double resonance) is a pulsed EPR method that can be used to reliably and precisely measure distances in the range 1.5-8?nm, to unravel orientations and to determine the number of monomers in complexes. In conjunction with site-directed spin labelling, it can be applied to biomolecules of all sizes in aqueous solutions or membranes. PELDOR is therefore complementary to the methods of X-ray crystallography, NMR and FRET (fluorescence resonance energy transfer) and is becoming a powerful method for structural determination of biomolecules. In the present review, the methods of PELDOR are discussed and examples where PELDOR has been used to obtain structural information on biomolecules are summarized.     

Harmonic analysis of DNA dynamics in a viscous medium

The harmonic dynamics of normal modes of double-stranded DNA in a viscous fluid are investigated. The model DNA consists of two backbone-supported DNA strands coiling around a common helix axis with base stacking, sugar puckering, interstrand hydrogen bonding, and intrastrand sugar-base interactions assigned values based on published data. Assuming that the DNA bases are shielded from direct bombardment by the solvent, analytical solutions are obtained. The dissipation and fluctuation of the normal modes of the bases moving along the spirals display the effect of the medium indirectly through interactions with the backbone. The dynamics of the backbone are found to be overdamped with the characteristic damping times extending to the picosecond region for disturbance in position and to the sub-picosecond region for disturbance in velocity. In addition to the dynamic mode of a rigid rod, the motions of the bases are coupled to the motions of the backbone with comparable amplitudes for disturbance in position. For disturbance in velocity, however, the bases are effectively at rest, not being able to follow the motions of the backbone. The angular frequencies of the underdamped vibrational modes, identified as the ringing modes of the bases with the backbone effectively at rest, are insensitive to the viscosity and lie in the low frequency region of the Raman spectrum. These findings indicate that the backbone of DNA plays a significant role in modulating the dynamics of double-stranded DNA in an overdamping environment. This modulation of the dynamics of the motions of the bases in DNA by environmental impediments to molecular motion is briefly discussed in connection with protein- and drug- DNA interactions as well as gene regulation.     

Normal modes of vibration in bovine pancreatic trypsin inhibitor and its mechanical property

The normal mode analysis of conformational fluctuation is carried out for a small globular protein, bovine pancreatic trypsin inhibitor. Results are analyzed mainly to reveal the mechanical construction of the protein molecule. We take dihedral angles, including peptide omega angles, as independent variables for the normal mode analysis. There are 306 such angles in this molecule. Motions in modes with frequencies lower than 120 cm-1 are shown to involve atoms in the whole protein molecule, and spatial change of displacement vectors is continuous, i.e., those of atoms near in space are similar. To quantitate the observation of the continuity, a correlation function of direction vectors of atomic displacements is calculated. From this function we define a quantity that is interpreted as the wave length of an equivalent elastic plane wave. From this quantity we deduce effective Young's modulus for each mode. For the mode with the lowest frequency 4.4 cm-1, it turned out to be 0.8 x 10(9) dyn cm-2, the value two orders of magnitude softer than, for instance, alpha-helices. Prompted by this observation, the four lowest frequency modes and also the harmonic motions in the thermal equilibrium are analyzed further mainly to detect relatively rigid structural elements in the molecule. From this analysis emerges a mechanical picture of the protein molecule that is made up of relatively rigid elements held together by very soft parts.     

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.     

Nonlinear optical studies of heme protein dynamics: implications for proteins as hybrid states of matter

Protein structure is fundamentally related to function. However, static structures alone are insufficient to understand how a protein works. Dynamics play an equally important role. Given that proteins are highly associated aperiodic systems, it may be expected that protein dynamics would follow glass-like dynamics. However, protein functions occur on time scales orders of magnitude faster than the time scales typically associated with glassy systems. It is becoming clear that the reaction forces driving functions do not sample entirely the large number of configurations available to a protein but are highly directed along an optimized pathway. Could there be any correlation between specific topological features in protein structures and dynamics that leads to strongly correlated atomic displacements in the dynamical response to a perturbation? This review will try to provide an answer by focusing upon recent nonlinear optical studies with the aim of directly observing functionally important protein motions over the entire dynamic range of the protein response function. The specific system chosen is photoinduced dynamics of ligand dissociation at the active site in heme proteins, with myoglobin serving as the simplest model system. The energetics and nuclear motions from the very earliest events involved in bond breaking on the femtosecond time scale all the way out to ligand escape and bimolecular rebinding on the microsecond and millisecond time scale have been mapped out. The picture that is emerging is that the system consists of strongly coupled motions from the very instant the bond breaks at the active site that cascade into low frequency collective modes specific to the protein structure. It is this coupling that imparts the ability of a protein to function on time scales more commensurate with liquids while simultaneously conserving structural integrity akin to solids.     

Normal modes as refinement parameters for the F-actin model.

The slow normal modes of G-actin were used as structural parameters to refine the F-actin model against 8-A resolution x-ray fiber diffraction data. The slowest frequency normal modes of G-actin pertain to collective rearrangements of domains, motions that are characterized by correlation lengths on the order of the resolution of the fiber diffraction data. Using a small number of normal mode degrees of freedom (< or = 12) improved the fit to the data significantly. The refined model of F-actin shows that the nucleotide binding cleft has narrowed and that the DNase I binding loop has twisted to a lower radius, consistent with other refinement techniques and electron microscopy data. The methodology of a normal mode refinement is described, and the results, as applied to actin, are detailed.