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| <!--[[Image:GNMproteinrepresentation.png|thumb|'''Figure 1:''' Three representations of a miniprotein structure. The protein is 20 residues long in amino acid sequence and (PDB ID: 1L2Y). Leftmost cartoon representation highlights secondary structure elements. Central chain connectivity representation shows alpha carbon positions at the kinks and connectivity between neighbor alpha carbons along the amino acid sequence. Rightmost Gaussian network representation shows connections between spatial neighbors of each residue within a cutoff distance of 7Å.|400px]]-->
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| The '''Gaussian network model (GNM)''' is a representation of a biological [[macromolecule]] as an elastic mass-and-[[spring (device)|spring]] network to study, understand, and characterize mechanical aspects of its long-scale [[dynamics (mechanics)|dynamics]]. The model has a wide range of applications from small proteins such as enzymes composed of a single [[protein domain|domain]], to large [[Macromolecular Assembly|macromolecular assemblies]] such as a [[ribosome]] or a viral [[capsid]].
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| The Gaussian network model is a minimalist, coarse-grained approach to study biological molecules. In the model, proteins are represented by nodes corresponding to alpha carbons of the amino acid residues. Similarly, DNA and RNA structures are represented with one to three nodes for each [[nucleotide]]. The model uses the harmonic approximation to model interactions, i.e. the spatial interactions between nodes (amino acids or nucleotides) are modeled with a uniform harmonic spring. This coarse-grained representation makes the calculations computationally inexpensive.
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| At molecular level, many biological phenomena, such as catalytic activity of an [[enzyme]], occur within the range of nano- to millisecond timescales. All atom simulation techniques, such as [[molecular dynamics]], rarely reach microsecond trajectory length, depending on the size of the system and accessible computational resources. Normal mode analysis in the context of GNM or elastic network (EN) models, in general, provides insights on the longer-scale functional behaviors of macromolecules. Here, the model captures native state functional motions of a biomolecule in the cost of atomic detail. The inference obtained from this model is complementary to atomic detail simulation techniques.
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| Another model for protein dynamics based on elastic mass-and-spring networks is the [[Anisotropic Network Model]].
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| == Gaussian network model theory ==
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| [[Image:MassSpringNetwork.jpg|thumb|'''Figure 2:''' Schematic representation of nodes in elastic network of GNM. Every node is connected to its spatial neighbors by uniform springs. Distance vector between two nodes, '''i''' and '''j''', is shown by an arrow and labeled '''R'''<sub>ij</sub>. Equilibrium positions of the '''i'''th and '''j'''th nodes, '''R'''<sup>0</sup><sub>i</sub> and '''R'''<sup>0</sup><sub>j</sub>, are shown in xyz coordinate system. '''R'''<sup>0</sup><sub>ij</sub> is the equilibrium distance between nodes '''i''' and '''j'''. Instantaneous fluctuation vectors, '''ΔR'''<sub>i</sub> and '''ΔR'''<sub>j</sub>, and instantaneous distance vector, '''R'''<sub>ij</sub>, are shown by the dashed arrows.|400px]]
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| The Gaussian network model was first proposed in 1996 by Tirion at the atomic level<ref name="Tirion, M.M 1905">Tirion, M.M. Large amplitude elastic motions in proteins from a single-parameter, atomic analysis, Phys. Rev. Lett., 77, 1905, 1996.</ref> and then one year later reconsidered at the amino-acid level by Bahar, Atilgan, Haliloglu and Erman.<ref name="I. Bahar, A. R 1997">Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential, I. Bahar, A. R. Atilgan, and B. Erman Folding & Design 2, 173-181, 1997.</ref><ref name="Haliloglu, T. Bahar 1997">Gaussian dynamics of folded proteins, Haliloglu, T. Bahar, I. & Erman, B. Phys. Rev. Lett. 79, 3090-3093, 1997.</ref> The model was influenced by work of PJ Flory on polymer networks <ref>Flory, P.J., Statistical thermodynamics of random networks, Proc. Roy. Soc. Lond. A, 351, 351, 1976.</ref> and other works that utilized normal mode analysis and simplified harmonic potentials to study dynamics of proteins.<ref>Go, N., Noguti, T. and Nishikawa, T. Dynamics of a small globular protein in terms of low-frequency vibrational modes, Proc. Natl. Acad. Sci. USA, 80, 3696, 1983.</ref>
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| === The elastic network ===
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| Figure 2 shows a schematic view of elastic network studied in GNM. Metal beads represent the nodes in this Gaussian network (residues of a protein) and springs represent the connections between the nodes of this network (covalent and non-covalent interactions between residues). For nodes '''i''' and '''j''', equilibrium position vectors, '''R'''<sup>0</sup><sub>i</sub> and '''R'''<sup>0</sup><sub>j</sub>, equilibrium distance vector, '''R'''<sup>0</sup><sub>ij</sub>, instantaneous fluctuation vectors, '''ΔR'''<sub>i</sub> and '''ΔR'''<sub>j</sub>, and instantaneous distance vector, '''R'''<sub>ij</sub>, are shown in Figure 2. Instantaneous position vectors of these nodes are defined by '''R'''<sub>i</sub> and '''R'''<sub>j</sub>. The difference between equilibrium position vector and instantaneous position vector of residue '''i''' gives the instantaneous fluctuation vector, '''ΔR'''<sub>i</sub> = '''R'''<sub>i</sub> - '''R'''<sup>0</sup><sub>i</sub>. Hence, the instantaneous fluctuation vector between nodes '''i''' and '''j''' is expressed as '''ΔR'''<sub>ij</sub> = '''ΔR'''<sub>j</sub> - '''ΔR'''<sub>i</sub> = '''R'''<sub>ij</sub> - '''R'''<sup>0</sup><sub>ij</sub>.
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| === Potential of the Gaussian network ===
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| Using the harmonic potential approximation, potential energy of the network in terms of '''ΔR'''<sub>i</sub> is
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| :<math>V_{GNM} = \frac{\gamma}{2}\left[ \sum_{i,j}^{N} (\Delta R_j-\Delta R_i)^2 \right]=
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| \frac{\gamma}{2}\left[ \sum_{i,j}^{N} \Delta R_i \Gamma_{ij} \Delta R_j\right]</math>
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| where '''γ''' is a force constant uniform for all springs and '''Γ'''<sub>ij</sub> is the '''ij'''th element of the [[Kirchhoff matrix|Kirchhoff]] (or connectivity) matrix of inter-residue contacts, '''Γ''', defined by | |
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| :<math>\Gamma_{ij} = \left\{\begin{matrix}
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| -1, & \mbox{if } i \ne j & \mbox{and }R_{ij} \le r_c \\
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| 0, & \mbox{if } i \ne j & \mbox{and }R_{ij} > r_c \\
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| -\sum_{j,j \ne i}^{N} \Gamma_{ij}, & \mbox{if } i = j \end{matrix}\right.</math>
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| ''r''<sub>c</sub> is a cutoff distance for spatial interactions and taken to be 7 Å for proteins.
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| Expressing the X, Y and Z components of the fluctuation vectors '''ΔR'''<sub>i</sub> as '''ΔX'''<sup>T</sup> = [ΔX<sub>1</sub> ΔX<sub>2</sub> ..... ΔX<sub>N</sub>], '''ΔY'''<sup>T</sup> = [ΔY<sub>1</sub> ΔY<sub>2</sub> ..... ΔY<sub>N</sub>], and '''ΔZ'''<sup>T</sup> = [ΔZ<sub>1</sub> ΔZ<sub>2</sub> ..... ΔZ<sub>N</sub>], above equation simplifies to
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| :<math>V_{GNM} = \frac{\gamma}{2} [\Delta X^T\Gamma \Delta X + \Delta Y^T\Gamma \Delta Y + \Delta Z^T\Gamma \Delta Z]</math>
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| === Statistical mechanics foundations ===
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| In the GNM, the probability distribution of all fluctuations, ''P''('''ΔR''') is ''isotropic''
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| :<math>P(\Delta R)=P(\Delta X,\Delta Y,\Delta Z)=p(\Delta X)p(\Delta Y)p(\Delta Z)</math>
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| and ''Gaussian''
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| :<math>p(\Delta X)\propto exp\left\{ -\frac{\gamma}{2 k_B T} \Delta X^T\Gamma \Delta X \right\}=exp\left\{ -\frac{1}{2} \left(\Delta X^T\left( \frac{k_B T}{\gamma} \Gamma^{-1} \right)^{-1} \Delta X \right) \right\}</math>
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| where ''k''<sub>''B''</sub> is the Boltzmann constant and ''T'' is the absolute temperature. ''p''('''ΔY''') and ''p''('''ΔZ''') are expressed similarly.
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| N-dimensional Gaussian probability density function with random variable vector '''x''', mean vector '''μ''' and covariance matrix '''Σ''' is
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| :<math>W(x,\mu ,\Sigma ) = \frac{1}{\sqrt{(2\pi)^N |\Sigma|}} exp\left\{ -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right\}</math>
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| <math>\sqrt{(2\pi)^N |\Sigma|}</math> normalizes the distribution and '''|Σ|''' is the determinant of the covariance matrix.
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| Similar to Gaussian distribution, normalized distribution for '''ΔX'''<sup>T</sup> = [ΔX<sub>1</sub> ΔX<sub>2</sub> ..... ΔX<sub>N</sub>] around the equilibrium positions can be expressed as
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| :<math>p(\Delta X ) = \frac{1}{\sqrt{(2\pi)^N \frac{k_B T}{\gamma} |\Gamma^{-1}|}} exp\left\{ -\frac{1}{2} \left(\Delta X^T\left( \frac{k_B T}{\gamma} \Gamma^{-1} \right)^{-1} \Delta X \right) \right\}</math>
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| The normalization constant, also the partition function ''Z''<sub>X</sub>, is given by
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| :<math>Z_X = \int_0^\infty exp\left\{ -\frac{1}{2} \left(\Delta X^T\left( \frac{k_B T}{\gamma} \Gamma^{-1} \right)^{-1} \Delta X \right) \right\}d\Delta X</math>
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| where <math>\frac{k_B T}{\gamma} \Gamma^{-1}</math> is the covariance matrix in this case. ''Z''<sub>Y</sub> and ''Z''<sub>Z</sub> are expressed similarly. This formulation requires inversion of the Kirchhoff matrix. In the GNM, the determinant of the Kirchhoff matrix is zero, hence calculation of its inverse requires [[Spectral theorem|eigenvalue decomposition]]. '''Γ'''<sup>−1</sup> is constructed using the N-1 non-zero eigenvalues and associated eigenvectors. Expressions for ''p''('''ΔY''') and ''p''('''ΔZ''') are similar to that of ''p''('''ΔX'''). The probability distribution of all fluctuations in GNM becomes
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| :<math>P(\Delta R) = p(\Delta X) p(\Delta Y) p(\Delta Z)=\frac{1}{\sqrt{(2\pi)^{3N} | \frac{k_B T}{\gamma} \Gamma^{-1}|^3}} exp\left\{ -\frac{3}{2} \left(\Delta X^T\left( \frac{k_B T}{\gamma} \Gamma^{-1} \right)^{-1} \Delta X \right) \right\}</math>
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| For this mass and spring system, the normalization constant in the preceding expression is the overall GNM partition function, ''Z''<sub>GNM</sub>,
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| :<math>Z_{GNM}=Z_X Z_Y Z_Z = \frac{1}{\sqrt{(2\pi)^{3N} | \frac{k_B T}{\gamma} \Gamma^{-1}|^3}}</math>
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| === Expectation values of fluctuations and correlations ===
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| Based on the statistical mechanics foundations of GNM, expectation values of residue fluctuations, <'''ΔR'''<sub>i</sub><sup>2</sup>> , and correlations, <'''ΔR'''<sub>i</sub> · '''ΔR'''<sub>j</sub>> , can be calculated. Covariance matrix for '''ΔX''' is given by
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| :<math><\Delta X \cdot \Delta X^T > = \int \Delta X \cdot \Delta X^T p(\Delta X)d\Delta X=\frac{k_B T}{\gamma}\Gamma^{-1} </math>
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| Since,
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| :<math><\Delta X \cdot \Delta X^T > = <\Delta Y \cdot \Delta Y^T > = <\Delta Z \cdot \Delta Z^T > =\frac{1}{3} <\Delta R \cdot \Delta R^T ></math>
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| <'''ΔR'''<sub>i</sub><sup>2</sup>> and <'''ΔR'''<sub>i</sub> · '''ΔR'''<sub>j</sub>> follows
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| :<math><\Delta R_i^2 > = \frac{3 k_B T}{\gamma}(\Gamma^{-1})_{ii}</math>
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| :<math><\Delta R_i \cdot \Delta R_j > = \frac{3 k_B T}{\gamma}(\Gamma^{-1})_{ij}</math>
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| === Mode decomposition ===
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| The GNM normal modes are found by diagonalization of the Kirchhoff matrix, '''Γ''' = '''UΛU'''<sup>''T''</sup>. Here, '''U''' is a unitary matrix, '''U'''<sup>''T''</sup> = '''U'''<sup>−1</sup>, of the eigenvectors '''u'''<sub>i</sub> of '''Γ''' and '''Λ''' is the diagonal matrix of eigenvalues '''λ'''<sub>i</sub>. The frequency and shape of a mode is represented by its eigenvalue and eigenvector, respectively. Since the Kirchhoff matrix is positive semi-definite, the first eigenvalue, '''λ'''<sub>1</sub>, is zero and the corresponding eigenvector have all its elements equal to 1/√N. This shows that the network model is translation invariant.
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| Cross-correlations between residue fluctuations can be written as a sum over the N-1 nonzero modes as
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| :<math><\Delta R_i \cdot \Delta R_j> = \frac{3 k_B T}{\gamma}[U\Lambda^{-1}U^T]_{ij}=\frac{3 k_B T}{\gamma}\sum_{k=1}^{N-1}\lambda_k^{-1} [u_k u_k^T]_{ij}</math>
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| It follows that, ['''ΔR'''<sub>i</sub> · '''ΔR'''<sub>j</sub>], the contribution of an individual mode is expressed as
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| :<math>[\Delta R_i \cdot \Delta R_j]_k = \frac{3 k_B T}{\gamma}\lambda_k^{-1} [u_k]_i [u_k]_j</math>
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| where ['''u'''<sub>k</sub>]<sub>i</sub> is the '''i'''th element of '''u'''<sub>k</sub>.
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| === Influence of local packing density ===
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| By definition, a diagonal element of the Kirchhoff matrix, '''Γ'''<sub>ii</sub>, is equal to the degree of a node in GNM that represents the corresponding residue’s coordination number. This number is a measure of the local packing density around a given residue. The influence of local packing density can be assessed by series expansion of '''Γ'''<sup>−1</sup> matrix. '''Γ''' can be written as a sum of two matrices, '''Γ''' = '''D''' + '''O''', containing diagonal elements and off-diagonal elements of '''Γ'''.
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| :'''Γ'''<sup>-1</sup> = ('''D''' + '''O''')<sup>-1</sup> = [ '''D''' ('''I''' + '''D'''<sup>-1</sup>'''O''') ]<sup>-1</sup> = ('''I''' + '''D'''<sup>-1</sup>'''O''')<sup>-1</sup>'''D'''<sup>-1</sup> = ('''I''' - '''D'''<sup>-1</sup>'''O''' + ...)<sup>-1</sup>'''D'''<sup>-1</sup> = '''D'''<sup>-1</sup> - '''D'''<sup>-1</sup>'''O''' '''D'''<sup>-1</sup> + ...
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| This expression shows that local packing density makes a significant contribution to expected fluctuations of residues.<ref>Halle, B. Flexibility and packing in proteins, Proc. Natl. Acad. Sci. USA, 99, 1274, 2002.</ref> The terms that follow inverse of the diagonal matrix, are contributions of positional correlations to expected fluctuations.
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| == GNM applications ==
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| [[Image:EQFluct.jpg|thumb|'''Figure 3:''' Example of theoretical prediction of expected residue fluctuations for the catalytic domain of the protein Cdc25B, a cell division cycle dual-specifity phosphatase. '''A.''' Comparison of β-factors from X-ray structure (yellow) and theoretical calculations (red). '''B.''' Structure of catalytic domain of Cdc25B colored according to theoretical motility of regions. Light blue regions, e.g. topmost alpha-helix next to the catalytic site of this protein, are expected to be more mobile than the rest of the domain. '''C.''' Cross-correlation map i.e. normalized <'''ΔR'''<sub>i</sub>·'''ΔR'''<sub>j</sub>> values. Red-colored regions correspond to collective residue motions and blue-colored regions correspond to uncorrelated motions. The results are retrieved iGNM server. PDB ID of Cdc25B is 1QB0.|400px]]
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| === Equilibrium fluctuations ===
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| Equilibrium fluctuations of biological molecules can be experimentally measured. In [[X-ray crystallography]] β-factor (or temperature factor) of each atom is a measure of mean-squared fluctuation of the native structure. In NMR experiments, this measure can be obtained by calculating root-mean-squared differences between different models.
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| In many applications and publications, including the original articles, it has been shown that expected residue fluctuations obtained from GNM is in good agreement with the experimentally measured native state fluctuations.<ref>Correlation between native state hydrogen exchange and cooperative residue fluctuations from a simple model, I. Bahar, A. Wallqvist, D. G. Covell, & R.L. Jernigan Biochemistry 37, 1067-1075, 1998.</ref><ref>Vibrational dynamics of proteins: Significance of slow and fast modes in relation to function and stability, I. Bahar, A. R. Atilgan, M. C. Demirel, & B. Erman, Phys. Rev. Lett. 80, 2733-2736, 1998.</ref> The relation between b-factors, for example, and expected residue fluctuations obtained from GNM is as follows
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| :<math>B_i = \frac{8\pi^2}{3}< \Delta R_{i} \cdot \Delta R_{i} > = \frac{8\pi^2 k_B T}{\gamma}(\Gamma^{-1})_{ii}</math>
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| Figure 3 shows an example of GNM calculation for the catalytic domain of the protein Cdc25B, a [[cell division cycle]] dual-specifity phosphatase.
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| [[Image:SlowestMode.jpg|thumb|'''Figure 4:''' Slow modes obtained from GNM calculations are depicted on Cdc2B catalytic domain. '''A.''' Plot of the slowest mode. '''B.''' Mapping of the amplitude of motion in the slowest mode onto protein structure. The alpha-helix nearby the catalytic site of this domain is the most mobile region of the protein along the slowest mode. Expected values of fluctuations were also highest at this region, as shown in Figure 3. The results are retrieved iGNM server. PDB ID of Cdc25B is 1QB0.|400px]]
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| === Physical meanings of slow and fast modes ===
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| Diagonalization of the Kirchhoff matrix decomposes the normal modes of collective motions of the Gaussian network model of a biomolecule. The expected values of fluctuations and cross-correlations are obtained from linear combinations of fluctuations along these normal modes. The contribution of each mode is scaled with the inverse of that modes frequency. Hence, slow (low frequency) modes contribute most to the expected fluctuations. Along the few slowest modes, motions are shown to be collective and global and potentially relevant to functionality of the biomolecules [9,13,15-18]. Fast (high frequency) modes, on the other hand, describe uncorrelated motions not inducing notable changes in the structure.
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| === Other specific applications ===
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| There are several major areas in which the Gaussian network model and other elastic network models are applied and found to be useful.<ref>Chennubhotla C, Rader AJ, Yang LW, Bahar I (2005). Elastic network models for understanding biomolecular machinery: from enzymes to supramolecular assemblies. Phys. Biol. 2:S173-S180 PMID 16280623</ref> These include:
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| * Decomposition of flexible/rigid regions and domains of proteins <ref>Analysis of domain motions by approximate normal mode calculations, Hinsen, K, Proteins 33(3), 417-429, 1999</ref><ref>Identification of core amino acids stabilizing rhodopsin, Rader, AJ., G. Anderson, B. Isin, H. G. Khorana, I. Bahar, & J. Klein-Seetharaman. Proc. Natl. Acad Sci USA 101: 7246-7251, 2004.</ref><ref>Automatic domain decomposition of proteins by a Gaussian Network Model, Kundu, S., Sorensen, D.C., Phillips, G.N. Jr., Proteins 57(4), 725-733, 2004.</ref>
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| * Characterization of functional motions and functionally important sites/residues of proteins, enzymes and large macromolecular assemblies <ref>Keskin, O. et al. Relating molecular flexibility to function: a case study of tubulin, Biophys. J., 83, 663, 2002.</ref><ref>Inhibitor binding alters the directions of domain motions in HIV-1 reverse transcriptase, Temiz NA & Bahar I, Proteins: Structure, Function and Genetics 49, 61-70, 2002.</ref><ref>Xu, C., Tobi, D. and Bahar, I. Allosteric changes in protein structure computed by a simple mechanical model: hemoglobin T<-> R2 transition, J. Mol. Biol., 333, 153, 2003.</ref><ref>Structural Changes Involved in Protein Binding Correlate with intrinsic Motions of Proteins in the Unbound State, Dror Tobi & Ivet Bahar. Proc Natl Acad Sci (USA) 102, 18908-18913, 2005.</ref><ref>Common Mechanism of Pore Opening Shared by Five Different Potassium Channels, Indira H. Shrivastava & Ivet Bahar. Biophys J 90, 3929-3940, 2006.</ref><ref>Yang LW, Bahar I (2005). Coupling between Catalytic Site and Collective Dynamics: A requirement for Mechanochemical Activity of Enzymes. Structure 13:893-904 {{DOI|10.1016/j.str.2005.03.015}} PMID 15939021</ref><ref>Markov Methods for Hierarchical Coarse-Graining of Large Protein Dynamics, Chakra Chennubhotla & Ivet Bahar. Lecture Notes in Computer Science 3909, 379-393, 2006.</ref><ref>Global Ribosome Motions Revealed with Elastic Network Model, Wang, Y. Rader, AJ, Bahar, I. & Jernigan, RL. , J. Struct Biol 147: 302-314, 2004.</ref><ref>Maturation Dynamics of Bacteriophage HK97 Capsid, AJ Rader, Daniel Vlad & Ivet Bahar. Structure (Camb) 13:413-21, 2005.</ref><ref>Hamacher, K., Trylska, J., McCammon, J.A. Dependency Map of Proteins in the Small Ribosomal Subunit. PLoS Comput. Biol. 2(2): e10, 2006</ref>
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| * Refinement and dynamics of low-resolution structural data, e.g. [[Cryo-electron microscopy]] <ref>Ming, D. et al. How to describe protein motion without amino acid sequence and atomic coordinates, Proc. Natl. Acad. Sci. USA, 99, 8620, 2002.</ref><ref>Tama, F., Wriggers, W. and Brooks III, C.L. Exploring global distortions of biological macromolecules and assemblies from low-resolution structural information and elastic network theory, J. Mol. Biol., 321, 297, 2002.</ref><ref>Delarue, M. and Dumas, P. On the use of low-frequency normal modes to enforce collective movements in refining macromolecular structural models, Proc. Natl. Acad. Sci. USA, 101, 6957, 2004.</ref><ref>Micheletti, C., Carloni, P. and Maritan, A. Accurate and efficient description of protein vibrational dynamics: comparing molecular dynamics and gaussian models, Proteins, 55, 635, 2004.</ref>
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| * [[Molecular replacement]] for solving [[X-ray structure]]s, when a [[conformational change]] occurred, with respect to a known structure<ref>Suhre, K. and Sanejouand, Y.H. On the potential of normal mode analysis for solving difficult molecular replacement problems, Acta Cryst. D 60, 796, 2004.
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| </ref>
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| * Integration with atomistic models and simulations <ref>Zhang, Z.Y., Shi, Y.Y. and Liu, H.Y. Molecular dynamics simulations of peptides and proteins with amplified collective motions, Bipohys. J., 84, 3583, 2003.</ref><ref>Micheletti, C., Lattanzi, G. and Maritan, A. Elastic properties of proteins: insight on the folding process and evolutionary selection of native structures, J. Mol. Biol., 321, 909, 2002.</ref>
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| * Investigation of folding/unfolding pathways and kinetics.<ref>Micheletti, C. et al. Crucial stages of protein folding through a solvable model: predicting target sites for enzyme-inhibiting drugs, Prot. Sci., 11, 1878, 2002.</ref><ref>Portman, J.J., Takada, S. and Wolynes, P.G. Microscopic theory of protein folding rates. I. fine structure of the free energy profile and folding routes from a variational approach, J. Chem. Phys., 114, 5069, 2001.</ref>
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| * Annotation of functional implication in molecular evolution <ref>Hamacher, K. Relating Sequence Evolution of HIV1-Protease to Its Underlying Molecular Mechanics, Gene, 422, 30-36, 2008.</ref><ref>Hamacher, K., McCammon J.A. Computing the amino acid specificity of fluctuations in biomolecular systems, J.Chem.Theo.Comp. 2:873, 2006.</ref>
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| == Web servers ==
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| In practice, two kinds of calculations can be performed.
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| The first kind (the GNM per se) makes use of the [[Kirchhoff matrix]].<ref name="I. Bahar, A. R 1997"/><ref name="Haliloglu, T. Bahar 1997"/> The second kind (more specifically called either the Elastic Network Model or the Anisotropic Network Model) makes use of the [[Hessian matrix]] associated to the corresponding set of harmonic springs.<ref name="Tirion, M.M 1905"/> Both kinds of models can be used online, using the following servers.
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| === GNM servers ===
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| * iGNM: A database of protein functional motions based on GNM http://ignm.ccbb.pitt.edu/Index.htm
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| * oGNM: Online calculation of structural dynamics using GNM http://ignm.ccbb.pitt.edu/GNM_Online_Calculation.htm
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| * GNM server http://gor.bb.iastate.edu/gnm/gnm.htm
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| === ENM/ANM servers ===
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| * [[Anisotropic Network Model]] web server http://www.ccbb.pitt.edu/anm <ref>Anisotropy of fluctuation dynamics of proteins with an elastic network model, Atilgan, AR, Durrell, SR, Jernigan, RL, Demirel, MC, Keskin, O. & Bahar, I. Biophys. J. 80, 505-515, 2001.</ref>
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| * [[Anisotropic Network Model|ANM]] server http://gor.bb.iastate.edu/anm/anm.htm
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| * elNemo: Web-interface to The Elastic Network Model http://www.igs.cnrs-mrs.fr/elnemo/
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| * AD-ENM: Analysis of Dynamics of an Elastic Network Model http://enm.lobos.nih.gov/
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| * WEBnm@: Web-server for Normal Mode Analysis of proteins http://apps.cbu.uib.no/webnma/home
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| === Other relevant servers ===
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| * ProMode: Database of normal mode analysis of proteins http://cube.socs.waseda.ac.jp/pages/jsp/index.jsp
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| * HingeProt: An algorithm for protein hinge prediction using elastic network models http://www.prc.boun.edu.tr/appserv/prc/hingeprot/, or http://bioinfo3d.cs.tau.ac.il/HingeProt/hingeprot.html
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| * DNABindProt: A Server for Determination of Potential DNA Binding Sites of Proteins http://www.prc.boun.edu.tr/appserv/prc/dnabindprot/
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| * MolMovDB: A database of macromolecular motions: http://www.molmovdb.org/
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| * The [[Protein Data Bank]] (PDB) http://www.pdb.org/
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| * A comprephensive elastic network model server: http://omega.psi.iastate.edu
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| == See also ==
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| * [[Normal distribution|Gaussian distribution]]
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| * [[Harmonic oscillator]]
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| * [[Hooke's law]]
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| * [[Molecular dynamics]]
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| * [[Normal mode]]
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| * [[Principal component analysis]]
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| * [[Protein dynamics]]
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| * [[Rubber elasticity]]
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| * [[Statistical mechanics]]
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| == References ==
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| === Primary sources ===
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| <div class="references-small">
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| * Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential, I. Bahar, A. R. Atilgan, and B. Erman Folding & Design 2, 173-181, 1997.
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| * Gaussian dynamics of folded proteins, Haliloglu, T. Bahar, I. & Erman, B. Phys. Rev. Lett. 79, 3090-3093, 1997.
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| * Cui Q, Bahar I, (2006). Normal Mode Analysis: Theory and applications to biological and chemical systems, Chapman & Hall/CRC, London, UK
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| </div>
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| | |
| === Specific citations ===
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| {{reflist}}
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| {{DEFAULTSORT:Gaussian Network Model}}
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| [[Category:Molecular modelling]]
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