Gaussian network model


The Gaussian network model is a representation of a biological macromolecule as an elastic mass-and-spring network to study, understand, and characterize the mechanical aspects of its long-time large-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. Protein domain dynamics plays key roles in a multitude of molecular recognition and cell signalling processes.
Protein domains, connected by intrinsically disordered flexible linker domains, induce long-range allostery via.
The resultant dynamic modes cannot be generally predicted from static structures of either the entire protein or individual domains.
The Gaussian network model is a minimalist, coarse-grained approach to study biological molecules. In the model, proteins are represented by nodes corresponding to α-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. This coarse-grained representation makes the calculations computationally inexpensive.
At the 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 simulations, 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 models in general, provides insights on the longer-scale functional dynamic behaviors of macromolecules. Here, the model captures native state functional motions of a biomolecule at the cost of atomic detail. The inference obtained from this model is complementary to atomic detail simulation techniques.
Another model for protein dynamics based on elastic mass-and-spring networks is the Anisotropic Network Model.

Gaussian network model theory

The Gaussian network model was proposed by Bahar, Atilgan, Haliloglu and Erman in 1997. The GNM is often analyzed using normal mode analysis, which offers an analytical formulation and unique solution for each structure. The GNM normal mode analysis differs from other normal mode analyses in that it is exclusively based on inter-residue contact topology, influenced by the theory of elasticity of Flory and the Rouse model and does not take the three-dimensional directionality of motions into account.

Representation of structure as an elastic network

Figure 2 shows a schematic view of elastic network studied in GNM. Metal beads represent the nodes in this Gaussian network and springs represent the connections between the nodes. For nodes i and j, equilibrium position vectors, R0i and R0j, equilibrium distance vector, R0ij, instantaneous fluctuation vectors, ΔRi and ΔRj, and instantaneous distance vector, Rij, are shown in Figure 2. Instantaneous position vectors of these nodes are defined by Ri and Rj. The difference between equilibrium position vector and instantaneous position vector of residue i gives the instantaneous fluctuation vector, ΔRi = Ri - R0i. Hence, the instantaneous fluctuation vector between nodes i and j is expressed as ΔRij = ΔRj - ΔRi = Rij - R0ij.

Potential of the Gaussian network

The potential energy of the network in terms of ΔRi is
where γ is a force constant uniform for all springs and Γij is the ijth element of the Kirchhoff matrix of inter-residue contacts, Γ, defined by
rc is a cutoff distance for spatial interactions and taken to be 7 Å for amino acid pairs.
Expressing the X, Y and Z components of the fluctuation vectors ΔRi as ΔXT = , ΔYT = , and ΔZT = , above equation simplifies to

Statistical mechanics foundations

In the GNM, the probability distribution of all fluctuations, P is isotropic
and Gaussian
where kB is the Boltzmann constant and T is the absolute temperature. p and p are expressed similarly.
N-dimensional Gaussian probability density function with random variable vector x, mean vector μ and covariance matrix Σ is
normalizes the distribution and |Σ| is the determinant of the covariance matrix.
Similar to Gaussian distribution, normalized distribution for ΔXT = around the equilibrium positions can be expressed as
The normalization constant, also the partition function ZX, is given by
where is the covariance matrix in this case. ZY and ZZ 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 and p are similar to that of p. The probability distribution of all fluctuations in GNM becomes
For this mass and spring system, the normalization constant in the preceding expression is the overall GNM partition function, ZGNM,

Expectation values of fluctuations and correlations

The expectation values of residue fluctuations, <ΔRi2>, and their cross-correlations, <ΔRi · ΔRj> can be organized as the diagonal and off-diagonal terms, respectively, of a covariance matrix. Based on statistical mechanics, the covariance matrix for ΔX is given by
The last equality is obtained by inserting the above p and taking the integral. Since,
<ΔRi2> and <ΔRi · ΔRj> follows

Mode decomposition

The GNM normal modes are found by diagonalization of the Kirchhoff matrix, Γ = UΛUT. Here, U is a unitary matrix, UT = U−1, of the eigenvectors ui of Γ and Λ is the diagonal matrix of eigenvalues λi. 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/. This shows that the network model translationally invariant.
Cross-correlations between residue fluctuations can be written as a sum over the N-1 nonzero modes as
It follows that, , the contribution of an individual mode is expressed as
where i is the ith element of uk.

Influence of local packing density

By definition, a diagonal element of the Kirchhoff matrix, Γii, 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.
This expression shows that local packing density makes a significant contribution to expected fluctuations of residues. 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 the B-factor of each atom is a measure of its mean-square fluctuation near its equilibrium position in the native structure. In NMR experiments, this measure can be obtained by calculating root-mean-square differences between different models.
In many applications and publications, including the original articles, it has been shown that expected residue fluctuations obtained by the GNM are in good agreement with the experimentally measured native state fluctuations. The relation between B-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-specificity phosphatase.

Physical meanings of slow and fast modes

Diagonalization of the Kirchhoff matrix decomposes the conformational motions into a spectrum of collective modes. 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 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. Fast modes, on the other hand, describe uncorrelated motions not inducing notable changes in the structure. GNM-based methods do not provide real dynamics but only an approximation based on the combination and interpolation of normal modes. Their applicability strongly depends on how collective the motion is.

Other specific applications

There are several major areas in which the Gaussian network model and other elastic network models have proved to be useful. These include:
In practice, two kinds of calculations can be performed.
The first kind makes use of the Kirchhoff matrix. The second kind makes use of the Hessian matrix associated to the corresponding set of harmonic springs. Both kinds of models can be used online, using the following servers.

GNM servers