From Protein Structure to Function with Bioinformatics.pdf

238 M.B. Kubitzki et al.(Brooks and Karplus 1983; Austin et al. 1975). To perform a normal mode analysisone needs a set of coordinates, a force field describing the interactions betweenconstituent atoms, and software to perform the required calculations. The performanceof a normal mode analysis in Cartesian coordinate space requires three maincalculation steps.1. Minimization of the conformational potential energy as a function of the atomiccoordinates.2. The calculation of the so-called “Hessian” matrixVH = ∂ 2∂x∂xwhich is the matrix of second derivatives of the potential energy with respect to themass-weighted atomic coordinates.3. The diagonalization of the Hessian matrix. This final steps yields eigenvaluesand eigenvectors (the “normal modes”).Energy minimization can require quite a lot of CPU time. Furthermore, as theHessian matrix is a 3N × 3N matrix, where N is the number of atoms, the last stepcan be computationally demanding.ij9.4.2 Elastic Network ModelsElastic or Gaussian network models (Tirion 1996) (ENM) are basically a simplificationof normal mode analysis. Usually, instead of an all atom representation, onlyC αatoms are taken into account. This means a ten-fold reduction of the number ofparticles which decreases the computational effort dramatically. Moreover, as theinput coordinates are taken as representing the ground state, no energy minimizationis required.The potential energy is calculated according toγV = ∑ ( rij−rij)2 0rij< RCwhere γ denotes the spring constant and R Cthe cut-off distance. Regarding thedrastic assumptions inherent in normal mode analysis, these simplifications donot mean a severe loss of quality. This, together with the relatively low computationalcosts, explains the current popularity of elastic network models. ENM calculationsare also offered on web servers such as ElNemo (Suhre and Sanejouand2004a, b) (http://www.igs.cnrs-mrs.fr/elnemo/) and AD-ENM (Zheng andDoniach 2003; Zheng and Brooks 2005) (http://enm.lobos.nih.gov/).0 2