Chapter 4 - Vector calculus

Abstract

Biomolecular structures correspond to the positions of the minima on a potential energy surface (PES) and functional mode analysis (FMA) identifies the collective motions of atoms in a protein related to a specific function. Protein conformations are represented by vectors and the PES is described as a function of the vectors. Evidently, to optimize (minimize) the energy on a PES or describe the equations of motion in FMA, derivatives of such functions must be computed. The inherent complexity of the system requires approximation of the potential energy by Taylor series expansion about a “starting geometry.” When computation of second- and higher-order derivates is too much taxing, the analysis proceeds by linear approximation (linearization). In general, the minima of a function are determined by a “second derivative test,” whereas the equations of motion are written in terms of the second-order derivatives.