Collaborative reference program for paper :

Minimization of empirical energy functions in proteins including hydrophobic surface area effects

“…After the update, the coaugate gradient minimizer is restarted. 31 To avoid singularities in the ECEPP/ 2 energy, the distance r between atoms that get closer than 2.0 A was replaced by a distance T, 1.0 A < F < 2.0 A, such that the smoothed ECEPP/2 energy function and its fiist and second derivatives are continuous for all r > 0 A. The minimization of the conformational energy was stopped if the absolute value of the gradient fell below 0.4 kcal/mol/ deg; if the energy change between subsequent steps was below lOV% or at least below lo-" kcal/mol during the previous five iterations; or latest after 1000 energy function and gradient evaluations.…”

Direct solution of continuous densities given in the Fourier magnitudes

“…Fast scalar and vector machines are common, and this section describes an algorithm which may be efficiently implemented on one of these. The conjugate gradients algorithm has already been described in detail in the literature (Powell, 1977;McCormick, 1983), so it is appropriate to concentrate on the specific details of a practical implementation of this algorithm to Patterson solution. The search direction is allowed to be an arbitrary vector, in this case the steepest descent direction, and the search directions are forced to be conjugate with a recursive formula.…”

“…The second major problem in a general shift algorithm is the breakdown of conjugacy in the directions (Powell, 1977). The recursive formula used to generate a set of conjugate directions is prone to fail if the problem being minimized is not well conditioned, which can happen when some reflections are missing or much smaller than average, or if the search along each conjugate direction is not exact.…”

“…While almost any of the iterative formulae for enforcing conjugate directions (McCormick, 1983) are better than steepest descents, in general the Polak-Ribeire formula works best (Powell, 1977). In this procedure each search direction is the sum of the current steepest descent direction and a constant multiple of the last search direction.…”

“…where di is the search direction and F~ is the function value. This formula fails in two major ways (Powell, 1977). It can force the search direction to be a nondescent, and will produce a fl which gives zero search directions when the gradients are nearly parallel.…”