computer modeling in molecular biology.pdf

224 Oliver S. SmartThe repulsion term ensures that moving conformations do not aggregate in lowenergy region :As the SPW procedure imposes restraints on the distances between moving positionsit is necessary to ensure that “rigid body” motions do not effect the result. In thisstudy this is accomplished by imposing a penalty function [44, 501where rj is the Cartesian vector giving the position of atom j and ($7j=1,Natom isthe position of a reference set of atoms which is set to the arithmetic average of theend positions. Czerminski and Elber [44] introduced an alternative to this penaltyfunction in a gradient projection constraint technique which is much more efficient(but more difficult to implement).The SPW method differs fundamentally from PEM in that it favors routes withthe lowest average energy rather than the lowest peak energy. It is easy to imaginethe case of a large molecular system with many distinct routes between two conformationswhere the route with the lowest average energy may not be the mostfavorable [50]. As noted by Fischer and Karplus [40] the SPW method results inroutes with molecular conformations which lie as low as possible on each side of thesaddle point. As shown below, the PEM formalism encourages exactly the oppositebehavior in that if moving a point from a low energy region to the high energy partof a profile leads to improved sampling and a reduction of peak energy then the objectivefunction (Eq. (8-4)) will favor this. The PEM approach will avoid problemsof abrupt shifts in atomic position sometimes found by the SPW procedure [45] asif such a shift produced a path with a high energy intermediate sample position this,would automatically be disfavored by an increase in the objective function.The methods also differ in the respect that SPW requires the concept of distanceto be applicable to the coordinate system under consideration. The PEM methoddoes not rely on any such notion and is immediately applicable to a non-Cartesiancoordinate representation, such as internal coordinates. In particular, it would be difficultto apply the SPW formalism to case in which the conformational transitionof interest also involved a rigid body displacement and rotation under the influenceof a mild external field, such as that imposed by a crystalline environment.