computer modeling in molecular biology.pdf

222 Oliver S. Smart(8-2)and applying some optimization technique. However, as Eq. (8-2) involves the maximumfunction it is impossible to find its derivatives. This would preclude the useof efficient optimization methods which require the derivative vector of the objectivefunction to be known.This problem can be avoided by noting that for any set of n positive numbers[kl, k2, c3 ... 5,) the expression:MvGi=l 9(8-3)tends towards the maximum of It1, k2, k3 . . , 5,) as M tends to infinity. This expressionis used to construct the PEM objective function:where N, is the number of sample conformations to be taken in each line section.This function is continuous, differentiable and tends towards the peak energy thehigher the number Mis set. This allows the use of conventional optimization techniquessuch as the Polak-Ribere conjugate gradients minimization [46-481 as used inthis study. Alternatively simulated annealing [49] could be employed in order toavoid some of the problems of the dependence of the final result on the initial setof moving conformations. Importantly, it can be shown that, by the application ofthe chain rule, the derivative vector VS can be calculated analytically from themolecular force and potential energy functions. Note that all the potential energiesin Eq. (8-4) are relative to the energy of fixed position 0:E'(X) = E(X) - E(Xo), (8-5)which is assumed to be the lowest energy position for the molecule - so that E'(X)is always greater than zero.If it is necessary to exactly locate the transition state conformation an adaptionof the PEM procedure (focusing down) can be used [52].The PEM objective function is dominated by the high areas of the energy profileof the path (provided the power M is set to a reasonably high value). The procedurewill concentrate on lowering these regions, resulting in a path which will ultimatelyfollow the optimal vector close to a saddle point (see Figure 8-1). However, as the