computer modeling in molecular biology.pdf

218 Oliver S. Smartvolving large molecules as the evaluation, storage and manipulation of the large Hessiansbecomes difficult.A technique commonly used in both fields is the reaction coordinate or adiabaticmapping method [18, 281. A variable or function of the variables judged to be importantin the transition of interest is controlled and energy minimization is appliedchanging all other variables. By judicious control of the “reaction coordinate” it ishoped that the transition of interest can be provoked. Often two coordinates are usedand a two dimensional contour map is built up. The results are referred to as theadiabatic surface as the method gives a reasonable approximation to the potentialof mean force (free energy profile) along the reaction coordinate, provided the timescale for the motion along the coordinate is much larger than for the other variables[MI. The method is widely used [18, 29-34] despite being prone to failure: even inthe case of a two dimensional model function [35]. It is important to note that theroute obtained by adiabatic mapping may be quite distinct from the steepest descentspath even if the same transition state conformation is found [36].A set of procedures which are related to the method proposed by Sinclair andFletcher [37] are more applicable to large systems. In their original method an initialsearch was made to find the maximum energy position along the line connecting thetwo given end points. The system at this point was then subjected to energyminimization along directions conjugate to the original search direction. This ismeant to ensure that the gradient norm drops to zero while the energy profile alongthe original search direction is a maximum, thus locating a transition state. The problemwith the method is that on a complex energy hypersurface, as soon as a singlestep is taken form the original point, the energy profile along the original searchdirection ceases to be a maximum [38]. In similar methods proposed by Halgren andLipscomb [39] and Bell and Crighton [28] this problem is avoided by making thesearch for a maximum along the parabola through the current point joining the endpoints. If the end points are energy minima there is guaranteed to be at least onemaximum along the parabola. The problem with this procedure on complicatedenergy hypersurfaces is when the case can arise where the energy profile displays twomaxima [38]. Whatever choice is made as to which of the maxima is accepted theroutine eventually ends up taking a large displacement along the parabola and doesnot converge [38]. To avoid this problem an adapted procedure has been proposed[38]. In common with the other methods an initial search is made for a maximumalong the line section joining the end points. This is followed by energy minimizationalong all directions orthogonal to the original search direction. A further search formaxima is then made along the lines joining the resulting point to the end points.In practice, the procedure was found to produce a reasonable result, but was cumbersomeand each minimization had to be restrained to avoid jumping to remote partsof conformational space. The methods set out by Fischer and Karplus [40] andLiotard [41] are based on broadly similar ideas but have proved to be more successful.