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WIREs Comput Mol Sci
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Cooperative dynamics of proteins unraveled by network models

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Abstract Recent years have seen a significant increase in the number of computational studies that adopted network models for investigating biomolecular systems dynamics and interactions. In particular, elastic network models have proven useful in elucidating the dynamics and allosteric signaling mechanisms of proteins and their complexes. Here we present an overview of two most widely used elastic network models, the Gaussian Network Model (GNM) and Anisotropic Network Model (ANM). We illustrate their use in (i) explaining the anisotropic response of proteins observed in external pulling experiments, (ii) identifying residues that possess high allosteric potentials, and demonstrating in this context the propensity of catalytic sites and metal‐binding sites for enabling efficient signal transduction, and (iii) assisting in structure refinement, molecular replacement and comparative modeling of ligand‐bound forms via efficient sampling of energetically favored conformers. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 426–439 DOI: 10.1002/wcms.44 This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods

Schematic representation of equilibrium fluctuations. A portion of the protein backbone is displayed by the dotted curve. Filled dots refer to interaction sites (e.g., Cα‐atoms) that are adopted as the network nodes. Ri0 and Rj0 designate the equilibrium positions of residues i and j; Ri and Rj are their instantaneous position vectors. The original and instantaneous separations are indicated by the solid line and dashed line, respectively. The fluctuations in the position vectors are given by ΔRj and ΔRi. Rij0 and Rij designate the equilibrium and instantaneous distance vectors between residues i and j. The change in the interresidue distance vector is related to the fluctuations in residue positions as ΔRij = RijRij0 = ΔRj − ΔRi, and may be expressed as a weighted sum over the contributions ΔRij(k) of individual modes.

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Structural refinement of the bacteria ferric ion‐binding protein by deforming a template protein along its normal modes. (a) Superposition of the target and template proteins [Protein Data Bank codes 1xvy (red) and 1qvs (blue), respectively]. The proteins share 37% sequence identity. The root‐mean‐square deviation (RMSD) between the structures is 3.21 Å. (b) Reduction of the RMSD by reconfigurations of the template along the first 30 normal modes accessible to the template structure, calculated using the anisotropic network model. The red dotted line shows the RMSD between the two structures prior to refinement.

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Correlation between signaling efficiency and evolutionary conservation. Panels (a; 1a0c) and (a′; 1a42) show the ribbon diagrams color coded by the distance of residues from the origin in Figure 4 (c) and (c′). The regions colored blue refer to the fastest and most precise communication sites. Panels (b) and (b′) are color coded by conservation score of each residue as obtained from the Consurf server, the most conserved sites being colored purple. The metal‐binding and catalytic sites are shown in space‐filling representation.

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Efficient signal propagation properties of active sites illustrated for a metal‐binding protein (a–c) and an enzyme (a′–c′). Hit time maps for cobalt‐binding xylose isomerase [Protein Data Bank (PDB) code: 1a0c] and for human carbonic anhydrase II (PDB code: 1a42) are displayed in the respective panels (a) and (a′). Average hit time profiles as a function of residue number are shown in panels (b) and (b′) with metal‐binding and catalytic residues labeled by the red dots in the two respective cases. Panels (c) and (c′) display the average hit time versus standard deviation for each residue for the two cases. The residues that participate in metal binding (b) and catalysis (b′) (shown in red markers) exhibit small average hit time and small‐to‐moderate standard deviation, indicative of their fast and precise communication properties.

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(a) Comparison of mechanical stress (uniaxial tension) experiments with anisotropic network model (ANM) predictions for green fluorescent protein (GFP). The plots on the left display the normalized contributions dij(k) of ANM modes k = 1, 3N‐6 (abscissa) to the deformation of the molecule in response to uniaxial tension exerted at residues (i, j) = (K3, N212) (top) and (D117,Y182) (bottom). When pulling apart the residues Lys3–Asn212 (upper diagram), the deformation is largely accommodated one soft mode (note the peak at mode 1). In the case of Asp117–Tyr182 (lower diagram), the response is distributed over a broader range of modes, including high‐frequency modes that entail high energy cost. The force required to unfold the molecule by stretching in the Lys3–Asn112 direction was reported46 to be five times weaker than that needed in the Asp117–Tyr182 direction, consistent with the mode distributions. (b) Complete mechanical resistance map generated for GFP by ANM. The entries in the map represent the effective force constants <κij> (Eq. 15) in response to uniaxial tensions exerted at each of pair of residues. The secondary structure of the protein is shown along the upper abscissa and on the right (blue arrows, β‐strand; zigzag line, α‐helix). The profile at the lower part of the map displays the mean resistance per residue, averaged over all values in a given column. See our study12 for more details.

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Schematic description of the evaluation of affinity matrix elements. Two residues within interaction range are displayed, composed of Ni = 7 and Nj = 11 atoms. The affinity is evaluated based on atom–atom contacts that are closer than a cutoff separation, e.g., 4.0 Å. In the present case, there is only one pair of atom within this interaction rage, such that Nij = 1, and the affinity between this pair becomes aij = Nij/√(NiNj) = 1/(77)½ (see Markovian Propagation of Allosteric Signals section).

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