Using fractal to solve the multiple minima problem in molecular mechanics calculation

Abstract

ABSTRACT:This article presents an approach using fractal to solve the multiple minima problem. We use the Newton-Raphson method of the MM3 molecular mechanics program to scan the conformational spaces of a model molecule and a real molecule. The results show each energy minimum, maximum point, and saddle point has a basin of initial points converging to it in conformational spaces. Points converging to different extrema are mixed, and form fractal structures around basin boundaries. Singular points seem to involve in the formation of fractal. When searching within a small region of fractal basin boundaries, the self-similarity of fractal makes it possible to find all energy minima, maxima, and saddle points from which global minimum may be extracted. Compared with other methods, this approach is efficient, accurate, conceptually simple, and easy to implement.

Numerical Errors in Minimization Based Binding Energy Calculations

“…37 The plots in Figures 10 and 11 clearly show that input regions that optimize to a given cluster of MM-GBSA solutions are not contiguous, and boundaries between regions appear to have fractal properties in places. The plots resemble both plots of fractal basin boundaries reported for simple MM minimizations 38 as well as plots of docking output as a function of input torsions angles we reported in a recent publication. 22 The plot in Figure 10 shows that while most input structures optimize to the Cluster1 minimum (region 1), a significant portion optimize to the Cluster2 minimum (region 2).…”

Numerical Errors and Chaotic Behavior in Docking Simulations

“…Well-known examples of fractal basin boundaries include the logistic difference equation and Newton’s method applied to solving complex roots . Fractal boundaries have already been reported in MM minimizations of small molecules, so their presence in docking searches is not inconceivable.…”