Team:TU Darmstadt/Labjournal/Simulation/GNM

Gaussian network model

Theory

Nearly all biologically important processes such as enzyme catalysis,ligand binding and allosteric regulation occur on a large time-scale (micro- to millisecond). A Gaussian network model (GNM) is a coarse-grained representation of a protein as an network consisting of balls and springs. In our approach, proteins are represented by balls corresponding to the CA –atom of each residue[1]⁠ . While Molecular Dynamics (MD) simulations are computational expensive, a GNM calculation only needs a few seconds.

Computation

The dynamics of the structure in the GNM is described by the topology of contacts within the Kirchhoff matrix G. Thus in this network of N interacting sites, the elements of G are computed as:

<math> \Gamma_{ij} = \left\{\begin{matrix} -1, & \mbox{if } i \ne j & \mbox{and }R_{ij} \le r_c \\ 0, & \mbox{if } i \ne j & \mbox{and }R_{ij} > r_c \\ -\sum_{j,j \ne i}^{N} \Gamma_{ij}, & \mbox{if } i = j \end{matrix}\right. <math>

where Rij is the distance between point i and j. We used Gamma as the intra CA-contact matrix. The inverse of it describes correlations between fluctuations within the proteins native state. The diagonal of the matrix is replaced by the sum of contacts of one CA-atom within the whole protein. After a singular value decomposition (SVD) we have calculated the normal modes of the protein. Slow modes describe functionally relevant residues within a biomolecule[2]⁠. The opposite, Fast modes, represent an uncorrelated motion without significant changes in the structure.

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A recent examination of the X-ray crystallographic B-factors of over 100 proteins showed that the GNM closely reproduces the experimental data [3]⁠.

Application to our Proteins

We computed the GNM in R [4]⁠ by using the BioPhysConnectoR [5]⁠ library.

• pnB-Esterase
• Fusarium solani cutinase

References

[1] A. R. Atilgan, S. R. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin, and I. Bahar, “Anisotropy of fluctuation dynamics of proteins with an elastic network model.,” Biophys J, vol. 80, no. 1, pp. 505–515, Jan. 2001.

[2] C. Chennubhotla, A. J. Rader, L.-W. Yang, and I. Bahar, “Elastic network models for understanding biomolecular machinery: from enzymes to supramolecular assemblies.,” Physical Biology, vol. 2, no. 4, pp. S173–S180, 2005.

[3] I. Bahar and A. J. Rader, “Coarse-grained normal mode analysis in structural biology.,” Current Opinion in Structural Biology, vol. 15, no. 5, pp. 586–592, 2005.

[4] R. D. C. Team, “R: A Language and Environment for Statistical Computing.” Vienna, Austria, 2008.

[5] F. Hoffgaard, P. Weil, and K. Hamacher, “BioPhysConnectoR: Connecting sequence information and biophysical models.,” BMC Bioinformatics, vol. 11, p. 199, 2010.