A normal mode analysis of 20 proteins in 'open' or 'closed' forms was performed using simple potential and protein models. The quality of the results was found to depend upon the form of the protein studied, normal modes obtained with the open form of a given protein comparing better with the conformational change than those obtained with the closed form. Moreover, when the motion of the protein is a highly collective one, then, in all cases considered, there is a single low-frequency normal mode whose direction compares well with the conformational change. When it is not, in most cases there is still a single low-frequency normal mode giving a good description of the pattern of the atomic displacements, as they are observed experimentally during the conformational change. Hence a lot of information on the nature of the conformational change of a protein is often found in a single low-frequency normal mode of its open form. Since this information can be obtained through the normal mode analysis of a model as simple as that used in the present study, it is likely that the property captured by such an analysis is for the most part a property of the shape of the protein itself. One of the points that has to be clarified now is whether or not amino acid sequences have been selected in order to allow proteins to follow a single normal mode direction, as least at the very beginning of their conformational change.

Normal mode analysis (NMA) is a powerful tool for predicting the possible movements of a given macromolecule. It has been shown recently that half of the known protein movements can be modelled by using at most two low-frequency normal modes. Applications of NMA cover wide areas of structural biology, such as the study of protein conformational changes upon ligand binding, membrane channel opening and closure, potential movements of the ribosome, and viral capsid maturation. Another, newly emerging field of NMA is related to protein structure determination by X-ray crystallography, where normal mode perturbed models are used as templates for diffraction data phasing through molecular replacement (MR). Here we present ElNémo, a web interface to the Elastic Network Model that provides a fast and simple tool to compute, visualize and analyse low-frequency normal modes of large macro-molecules and to generate a large number of different starting models for use in MR. Due to the 'rotation-translation-block' (RTB) approximation implemented in ElNémo, there is virtually no upper limit to the size of the proteins that can be treated. Upon input of a protein structure in Protein Data Bank (PDB) format, ElNémo computes its 100 lowest-frequency modes and produces a comprehensive set of descriptive parameters and visualizations, such as the degree of collectivity of movement, residue mean square displacements, distance fluctuation maps, and the correlation between observed and normal-mode-derived atomic displacement parameters (B-factors). Any number of normal mode perturbed models for MR can be generated for download. If two conformations of the same (or a homologous) protein are available, ElNémo identifies the normal modes that contribute most to the corresponding protein movement. The web server can be freely accessed at http://igs-server.cnrs-mrs.fr/elnemo/index.html.

Normal mode analysis of proteins of various sizes, ranging from 46 (crambin) up to 858 residues (dimeric citrate synthase) were performed, by using standard approaches, as well as a recently proposed method that rests on the hypothesis that low‐frequency normal modes of proteins can be described as pure rigid‐body motions of blocks of consecutive amino‐acid residues. Such a hypothesis is strongly supported by our results, because we show that the latter method, named RTB, yields very accurate approximations for the low‐frequency normal modes of all proteins considered. Moreover, the quality of the normal modes thus obtained depends very little on the way the polypeptidic chain is split into blocks. Noteworthy, with six amino‐acids per block, the normal modes are almost as accurate as with a single amino‐acid per block. In this case, for a protein of n residues and N atoms, the RTB method requires the diagonalization of an n × n matrix, whereas standard procedures require the diagonalization of a 3N × 3N matrix. Being a fast method, our approach can be useful for normal mode analyses of large systems, paving the way for further developments and applications in contexts for which the normal modes are needed frequently, as for example during molecular dynamics calculations. Proteins 2000;41:1–7. © 2000 Wiley‐Liss, Inc.

The Elastic Network Model is used to investigate the open/closed transition in all DNA-dependent polymerases whose structure is known in both forms. For each structure the model accounts well for experimental crystallographic B-factors. It is found in all cases that the transition can be well described with just a handful of the normal modes. Usually, only the lowest and/or the second lowest frequency normal modes deduced from the open form give rise to calculated displacement vectors that have a correlation coefficient larger than 0.50 with the observed difference vectors between the two forms. This is true for every structural class of DNA-dependent polymerases where a direct comparison with experimental structural data is available. In cases where only one form has been observed by X-ray crystallography, it is possible to make predictions concerning the possible existence of another form in solution by carefully examining the vector displacements predicted for the lowest frequency normal modes. This simple model, which has the advantage to be computationally inexpensive, could be used to design novel kind of drugs directed against polymerases, namely drugs preventing the open/closed transition from occurring in bacterial or viral DNA-dependent polymerases.

A normal mode analysis of the closed form of dimeric citrate synthase has been performed. The largest-amplitude collective motion predicted by this method compares well with the crystallographically observed hinge-bending motion. Such a result supports those obtained previously in the case of hinge-bending motions of smaller systems, such as lysozyme or hexokinase. Taken together, all these results suggest that low-frequency normal modes may become useful for determining a first approximation of the conformational path between the closed and open forms of these proteins.

SYNOPSISA new method for calculating a set of low-frequency normal modes in macromolecules is proposed and applied to the case of proteins. In a first step, the protein chain is partitioned into blocks of one or more residues and the low-frequency modes are evaluated at a lowresolution level by combining the local translations and rotations of each block. In a second step, these low-resolution modes are perturbed by high-frequency modes explicitly calculated in each block, thus leading to the exact low-frequency modes. The procedure is tested for three cases-decaalanine, icosaleucin, and crambin-using a perturbation-iteration scheme in the second step. Convergence properties and numerical accuracy are assessed and tested for various partitions. The low-resolution modes obtained in the first step are always found to be good starting approximations. Potential advantages of the method include a central processing unit time roughly N 2 dependent on the size of the problem ( N being the number of degrees of freedom), the possibility of using parallel processing, the nonrequirement for loading the complete mass-weighted second-derivative input matrix into central memory, and the possibility of introducing in the procedure further structural hierarchy, such as secondary structures or motifs. In addition, any improvement or refinement of the algorithm benefits from the efficient formalism of the effective Hamiltonian theory. 0 1994 John Wiley & Sons. Inc.

We introduce a topology-based nonlinear network model of protein dynamics with the aim of investigating the interplay of spatial disorder and nonlinearity. We show that spontaneous localization of energy occurs generically and is a site-dependent process. Localized modes of nonlinear origin form spontaneously in the stiffest parts of the structure and display site-dependent activation energies. Our results provide a straightforward way for understanding the recently discovered link between protein local stiffness and enzymatic activity. They strongly suggest that nonlinear phenomena may play an important role in enzyme function, allowing for energy storage during the catalytic process. The predictions of elastic network models (ENMs) of proteins [1,2,3,4] have proven useful in quantitatively describing amino-acid fluctuations at room temperature [1], often in good agreement with isotropic [2], as well as anisotropic measurements [5,6]. Moreover, it has been shown that a few low-frequency normal modes can provide fair insight on the large amplitude motions of proteins upon ligand binding [7,8,9], as previously noticed when more detailed models were considered [10,11,12], also by virtue of the robust character of the collective functional motions [13].However, low-frequency modes of proteins are known to be highly anharmonic [14,15], a property which has to be taken into account in order to understand energy storage and transfer within their structure as a consequence of ligand binding, chemical reaction, etc [16,17]. Indeed, there is growing experimental evidence that long-lived modes of nonlinear origin may exist in proteins [18,19]. Likewise, many theoretical studies have appeared suggesting that localized vibrations may play an active role in, e.g., enzyme catalysis [20]. These include topological excitations such as solitons [21] as well as discrete breathers (DBs) [22,23].The latter are nonlinear modes that emerge in many contexts as a result of both nonlinearity and discreteness [24]. Although their existence and stability properties are well understood in systems with translational invariance, much less is known of the subtle effects arising from the interplay of spatial disorder and anharmonicity [25,26,27]. For this purpose, in the present work we introduce the nonlinear network model (NNM). Our aim is to extend the simple scheme of ENMs, known to capture the topology-based features of protein dynamics [1,2,3], by adding anharmonic terms. Within the NNM framework, we show that spontaneous localization of energy can occur in protein-like systems and that its properties may be intuitively rationalized in the context of specific biological functions. In our model, the potential energy of a protein, E p , has the following form:where d ij is the distance between atoms i and j, d0 ij their distance in the structure under examination (as e.g. solved through X-ray crystallography) and R c is a cutoff that specifies the interacting pairs. As done in numerous studies, only C α atoms are taken into account [4] and...

Recently, using a numerical surface cooling approach, we have shown that highly energetic discrete breathers (DBs) can form in the stiffest parts of nonlinear network models of large protein structures. In the present study, using an analytical approach, we extend our previous results to low-energy discrete breathers as well as to smaller proteins. We confirm and further scrutinize the striking site selectiveness of energy localization in the presence of spatial disorder. In particular, we find that, as a sheer consequence of disorder, a non-zero energy gap for exciting a DB at a given site either exists or not. Remarkably, in the former case, the gaps arise as a result of the impossibility of exciting small-amplitude modes in the first place. In contrast, in the latter case, a small subset of linear edge modes acts as accumulation points, whereby DBs can be continued to arbitrary small energies, while unavoidably approaching one of such normal modes. In particular, the case of the edge mode seems peculiar, its dispersion relation being simple and little system dependent. Concerning the structure-dynamics relationship, we find that the regions of protein structures where DBs form easily (zero or small gaps) are unfailingly the most highly connected ones, also characterized by weak local clustering. Remarkably, a systematic analysis on a large database of enzyme structures reveals that amino-acid residues involved in catalysis tend to be located in such regions. This finding reinforces the idea that localized modes of nonlinear origin may play an important biological role, e.g., by providing a ready channel for energy storage and/or contributing to lower energy barriers of chemical reactions.