Gaussian network model

The Gaussian network model (GNM), one of many things named after Carl Gauss, is a representation of a biological macromolecule as an elastic mass-and-spring network to study, understand, and characterize mechanical aspects of its long-scale dynamics. The model has a wide range of applications from small proteins such as enzymes composed of a single domain, to large macromolecular assemblies such as a ribosome or a viral capsid.

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.

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.

Gaussian network model theory

The Gaussian network model was proposed by Bahar, Atilgan, Haliloglu and Erman in 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.] [Gaussian dynamics of folded proteins, Haliloglu, T. Bahar, I. & Erman, B. Phys. Rev. Lett. 79, 3090-3093, 1997.] . The model was influenced by work of PJ Flory on polymer networks [Flory, P.J., Statistical thermodynamics of random networks, Proc. Roy. Soc. Lond. A, 351, 351, 1976.] and other works that utilized normal mode analysis and simplified harmonic potentials to study dynamics of proteins [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.] [Tirion, M.M. Large amplitude elastic motions in proteins from a single-parameter, atomic analysis, Phys. Rev. Lett., 77, 1905, 1996.] .

The elastic network

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, R⁰_i and R⁰_j, equilibrium distance vector, R⁰_ij, instantaneous fluctuation vectors, ΔR_i and ΔR_j, and instantaneous distance vector, R_ij, are shown in Figure 2. Instantaneous position vectors of these nodes are defined by R_i and R_j. The difference between equilibrium position vector and instantaneous position vector of residue i gives the instantaneous fluctuation vector, ΔR_i = R_i - R⁰_i. Hence, the instantaneous fluctuation vector between nodes i and j is expressed as ΔR_ij = ΔR_j - ΔR_i = R_ij - R⁰_ij.

Potential of the Gaussian network

Using the harmonic potential approximation, potential energy of the network in terms of ΔR_i is

:

where γ is a force constant uniform for all springs and Γ_ij is the ijth element of the Kirchhoff (or connectivity) matrix of inter-residue contacts, Γ, defined by

:

"r"_c is a cutoff distance for spatial interactions and taken to be 7 Å for proteins.

"V"_GNM can be expressed in terms of ΔX_i, ΔY_i and ΔZ_i components of ΔR_i as follows

:$V\_\{GNM\}\; =\; frac\{gamma\}\{2\}left\; [\; sum\_\{i,j\}^\{N\}\; Gamma\_\{ij\}\; [(Delta\; X\_i-Delta\; X\_j)^2\; +\; (Delta\; Y\_i-Delta\; Y\_j)^2\; +\; (Delta\; Z\_i-Delta\; Z\_j)^2]\; ight]$

Expressing the X, Y and Z components of the fluctuation vectors ΔR_i as ΔX^T = [ΔX₁ ΔX₂ ..... ΔX_N] , ΔY^T = [ΔY₁ ΔY₂ ..... ΔY_N] , and ΔZ^T = [ΔZ₁ ΔZ₂ ..... ΔZ_N] , above equation simplifies to

:$V\_\{GNM\}\; =\; frac\{gamma\}\{2\}\; [Delta\; X^TGamma\; Delta\; X\; +\; Delta\; Y^TGamma\; Delta\; Y\; +\; Delta\; Z^TGamma\; Delta\; Z]$

Statistical mechanics foundations

In the GNM, the probability distribution of all fluctuations, "P"(ΔR) is "isotropic"

:$P(Delta\; R)=P(Delta\; X,Delta\; Y,Delta\; Z)=p(Delta\; X)p(Delta\; Y)p(Delta\; Z)$

and "Gaussian"

:$p(Delta\; X)propto\; expleft\{\; -frac\{gamma\}\{2\; k\_B\; T\}\; Delta\; X^TGamma\; Delta\; X\; ight\}=expleft\{\; -frac\{1\}\{2\}\; left(Delta\; X^Tleft(\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}\; ight)^\{-1\}\; Delta\; X\; ight)\; ight\}$

where "k"_"B" is the Boltzmann constant and "T" is the absolute temperature. "p"(ΔY) and "p"(ΔZ) are expressed similarly. N-dimensional Gaussian probability density function with random variable vector x, mean vector μ and covariance matrix Σ is

:$W(x,mu\; ,Sigma\; )\; =\; frac\{1\}\{sqrt\{(2pi)^N\; |Sigma|\; expleft\{\; -frac\{1\}\{2\}\; (x\; -\; mu)^T\; Sigma^\{-1\}\; (x\; -\; mu)\; ight\}$

$sqrt\{(2pi)^N\; |Sigma$ normalizes the distribution and |Σ| is the determinant of the covariance matrix.

Similar to Gaussian distribution, normalized distribution for ΔX^T = [ΔX₁ ΔX₂ ..... ΔX_N] around the equilibrium positions can be expressed as

:$p(Delta\; X\; )\; =\; frac\{1\}\{sqrt\{(2pi)^N\; frac\{k\_B\; T\}\{gamma\}\; |Gamma^\{-1\}|\; expleft\{\; -frac\{1\}\{2\}\; left(Delta\; X^Tleft(\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}\; ight)^\{-1\}\; Delta\; X\; ight)\; ight\}$

The normalization constant, also the partition function "Z"_X, is given by

:$Z\_X\; =\; int\_0^infty\; expleft\{\; -frac\{1\}\{2\}\; left(Delta\; X^Tleft(\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}\; ight)^\{-1\}\; Delta\; X\; ight)\; ight\}dDelta\; X$

where $frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}$ is the covariance matrix in this case. "Z"_Y and "Z"_Z 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 eigenvalue decomposition. Γ^-1 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

:$P(Delta\; R)\; =\; p(Delta\; X)\; p(Delta\; Y)\; p(Delta\; Z)=frac\{1\}\{sqrt\{(2pi)^\{3N\}\; |\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}|^3\; expleft\{\; -frac\{3\}\{2\}\; left(Delta\; X^Tleft(\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}\; ight)^\{-1\}\; Delta\; X\; ight)\; ight\}$

For this mass and spring system, the normalization constant in the preceding expression is the overall GNM partition function, "Z"_GNM,

:$Z\_\{GNM\}=Z\_X\; Z\_Y\; Z\_Z\; =\; frac\{1\}\{sqrt\{(2pi)^\{3N\}\; |\; frac\{k\_B\; T\}\{gamma\}\; Gamma^\{-1\}|^3$

Expectation values of fluctuations and correlations

Based on the statistical mechanics foundations of GNM, expectation values of residue fluctuations, <ΔR_i²> , and correlations, <ΔR_i ˑ ΔR_j> , can be calculated. Covariance matrix for ΔX is given by

:

Since,

:

<ΔR_i²> and <ΔR_i ˑ ΔR_j> follows

::$=\; frac\{k\_B\; T\}\{gamma\}(Gamma^\{-1\})\_\{ij\}$

Mode decomposition

The GNM normal modes are found by diagonalization of the Kirchhoff matrix, Γ = UΛU^"T". Here, U is a unitary matrix, U^"T" = U^-1, of the eigenvectors u_i of Γ and Λ is the diagonal matrix of eigenvalues λ_i. 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, λ₁, is zero and the corresponding eigenvector have all its elements equal to 1/√N. This shows that the network model is translation invariant.

Cross-correlations between residue fluctuations can be written as a sum over the N-1 nonzero modes as

:$=\; frac\{3\; k\_B\; T\}\{gamma\}\; [ULambda^\{-1\}U^T]\; \_\{ij\}=frac\{3\; k\_B\; T\}\{gamma\}sum\_\{i,j\}^\{N\}\; [lambda\_k^\{-1\}\; u\_k\; u\_k^T]\; \_\{ij\}$

It follows that, [ΔR_i ˑ ΔR_j] , the contribution of an individual mode is expressed as

:$[Delta\; R\_i\; cdot\; Delta\; R\_j]\; \_k\; =\; frac\{3\; k\_B\; T\}\{gamma\}lambda\_k^\{-1\}\; [u\_k]\; \_i\; [u\_k]\; \_j$

where [u_k] _i is the ith element of u_k.

Influence of local packing density

By definition, a diagonal element of the Kirchhoff matrix, Γ_ii, 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^-1 matrix. Γ can be written as a sum of two matrices, Γ = D + O, containing diagonal elements and off-diagonal elements of Γ.

:Γ = (D + O)^-1 = [ D (I + D^-1O) ] ^-1 = (I + D^-1O)^-1D^-1 = (I - D^-1O + ...)^-1D^-1 = D^-1 - D^-1O D^-1 + ...

This expression shows that local packing density makes a significant contribution to expected fluctuations of residues [Halle, B. Flexibility and packing in proteins, Proc. Natl. Acad. Sci. USA, 99, 1274, 2002.] . The terms that follow inverse of the diagonal matrix, are contributions of positional correlations to expected fluctuations.

GNM applications

Equilibrium fluctuations

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.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 [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.] [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.] . The relation between -factors, for example, and expected residue fluctuations obtained from GNM is as follows

:

Figure 3 shows an example of GNM calculation for the catalytic domain of the protein Cdc25B, a cell division cycle dual-specifity phosphatase.

Physical meanings of slow and fast modes

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.

Other specific applications

There are several major areas in which the Gaussian network model and other elastic network models are applied and found to be useful [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] . These include
* decomposition of flexible/rigid regions and domains of proteins [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.] [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.]
* characterization of functional motions and functionally important sites/residues of proteins, enzymes and large macromolecular assemblies [Keskin, O. et al. Relating molecular flexibility to function: a case study of tubulin, Biophys. J., 83, 663, 2002.] [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.] [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.] [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.] [Common Mechanism of Pore Opening Shared by Five Different Potassium Channels, Indira H. Shrivastava & Ivet Bahar. Biophys J 90, 3929-3940, 2006.] [Yang LW, Bahar I (2005). Coupling between Catalytic Site and Collective Dynamics: A requirement for Mechanochemical Activity of Enzymes. Structure 13:893-904 PMID 15939021] [Markov Methods for Hierarchical Coarse-Graining of Large Protein Dynamics, Chakra Chennubhotla & Ivet Bahar. Lecture Notes in Computer Science 3909, 379-393, 2006.] [Global Ribosome Motions Revealed with Elastic Network Model, Wang, Y. Rader, AJ, Bahar, I. & Jernigan, RL. , J. Struct Biol 147: 302-314, 2004.] [Maturation Dynamics of Bacteriophage HK97 Capsid, AJ Rader, Daniel Vlad & Ivet Bahar. Structure (Camb) 13:413-21, 2005.]
* refinement and dynamics of low-resolution structural data, e.g. Cryo-electron microscopy [Ming, D. et al. How to describe protein motion without amino acid sequence and atomic coordinates, Proc. Natl. Acad. Sci. USA, 99, 8620, 2002.] [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.] [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.] [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.]
* integration with atomistic models and simulations [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.] [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.]
* investigation of folding/unfolding pathways and kinetics [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.] [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.] .

Web servers

GNM servers

* iGNM: A database of protein functional motions based on GNM http://ignm.ccbb.pitt.edu/Index.htm
* oGNM: Online calculation of structural dynamics using GNM http://ignm.ccbb.pitt.edu/GNM_Online_Calculation.htm
* GNM server http://gor.bb.iastate.edu/gnm/gnm.htm

ANM servers

* Anisotropic Network Model web server http://www.ccbb.pitt.edu/anm [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.]
* ANM server http://gor.bb.iastate.edu/anm/anm.htm

ENM servers

* elNemo: Web-interface to The Elastic Network Model http://www.igs.cnrs-mrs.fr/elnemo/
* AD-ENM: Analysis of Dynamics of Elastic Network Model http://enm.lobos.nih.gov/

Other relevant servers

* ProMode: Database of normal mode analysis of proteins http://cube.socs.waseda.ac.jp/pages/jsp/index.jsp
* HingeProt: An algorithm for protein hinge prediction using elastic network models http://www.prc.boun.edu.tr/appserv/prc/hingeprot/http://bioinfo3d.cs.tau.ac.il/HingeProt/hingeprot.html
* MolMovDB: A database of macromolecular motions http://www.molmovdb.org/
* Protein Data Bank (PDB) http://www.pdb.org/

See also

* Gaussian distribution
* Harmonic oscillator
* Hooke's law
* Molecular dynamics
* Normal mode
* Principal component analysis
* Rubber elasticity
* Statistical mechanics

References

Primary sources

* 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.
* Gaussian dynamics of folded proteins, Haliloglu, T. Bahar, I. & Erman, B. Phys. Rev. Lett. 79, 3090-3093, 1997.
* Cui Q, Bahar I, (2006). Normal Mode Analysis: Theory and applications to biological and chemical systems, Chapman & Hall/CRC, London, UK

Specific citations