Energy Minimization

The potential energy calculated by summing the energies of various interactions is a numerical value for a single conformation. This number can be used to evaluate a particular conformation, but it may not be a useful measure of a conformation because it can be dominated by a few bad interactions. For instance, a large molecule with an excellent conformation for nearly all atoms can have a large overall energy because of a single bad interaction, for instance two atoms too near each other in space and having a huge van der Waals repulsion energy. It is often preferable to carry out energy minimization on a conformation to find the best nearby conformation. Energy minimization is usually performed by gradient optimization: atoms are moved so as to reduce the net forces on them. The minimized structure has small forces on each atom and therefore serves as an excellent starting point for molecular dynamics simulations.

Energy minimization is usually performed in Cartesian coordinates, by optimizing along pathways in -dimensional space, where is the number of particles. This pathway can be the gradient, , where

In other words, each Cartesian component, , of the gradient equals the derivative of the potential energy with respect to that component. Only those interactions involving particle contribute to the gradients of the Cartesian coordinates of (). The components of constitute a path, P, in -dimensional space. Finding the minimum along this pathway typically involves an interpolation of two points in -space to find a new point where . Usually, however, at the new point, so a new path is chosen and minimization proceeds. It is possible to set at each new point, but it is more efficient to choose the new pathway to be orthogonal to all previous paths. This method of ``conjugate gradients'' is perhaps the most popular method of energy minimization. Details of this method can be found in Reference [16].

It is also possible to minimize the energy of a conformation by optimizing the dihedral angle degrees of freedom, rather than the Cartesian coordinates. The minimization occurs in -dimensional space, where is the number of dihedral angles. Torques, or derivatives of the forcefield with respect to dihedral angles, take the place of the gradient. We have found that ``torque minimization,'' when followed by Cartesian minimization, produces an overall lower-energy conformation than Cartesian minimization alone. Neither method, however, can guarantee that the lowest possible conformation (the global minimum) will be reached. The process of moving along pathways in conformational space usually ends at a ``local minimum'' - a well in the potential energy surface, where the energy is lower than for all other nearby conformations, but not necessarily lower than other local minima.