Molecular structure determination by convex global underestimation of local energy minima

Abstract

The determination of a stable molecular structure can often be formulated in terms of calculating the global (or approximate global) minimum of a potential energy function. Computing the global minimum of this function is very difficult because it typically has a very large number of local minima which may grow exponentially with molecule size. The optimization method presented involves collecting a large number of conformers, each attained by finding a local minimum of the potential energy function from a random starting point. The information from these conformers is then used to form a convex quadratic global underestimating function for the potential energy of all known conformers. This underestimator is an L 1 approximation to all known local minima, and is obtained by a linear programming formulation and solution. The minimum of this underestimator is used to predict the global minimum for the function, allowing a localized conformer search to be performed based on the predicted minimum. The new set of conformers generated by the localized search serves as the basis for another quadratic underestimation step in an iterative algorithm. This algorithm has been used to determine the structures of homopolymers of length n ≤ 30 with no sidechains. While it is estimated that there are O(3 n) local minima for a chain of length n, this method requires O(n 4) computing time on average. It is also shown that the global minimum potential energy values lie on a concave quadratic curve for n ≤ 30. This important property permits estimation of the minimum energy for larger molecules, and also can be used to accelerate the global minimization algorithm.

A Branch‐and‐Prune algorithm for the Molecular Distance Geometry Problem

“…Furthermore, all the instances in the Lavor "large set" are solved by dgsol to a solution accuracy of order 10 −1 : given that in BP pruning often occurs for feasibility differences of order 10 −1 and even 10 −2 , such a slack solution accuracy may mean that dgsol is not actually finding the correct solution. Table 2 reports the averages of the same parameters as in Table 1 taken over 10 Lavor instances in a sample of the "small set" and over 4 Lavor instances in a sample of the "large set". It appears clear from these data that BP's strong points are indeed speed and accuracy.…”

“…1. In most molecular conformation calculations, all covalent bond lengths and bond angles are assumed to be known a priori [10]. Thus, the first three atoms in the sequence can be fixed and the fourth atom is determined by the torsion angle ω 4 .…”

“…. , ω n of a molecule with n atoms, the Cartesian coordinates x i = (x i1 , x i2 , x i3 ) for each atom i in the molecule can be obtained using the following formulae [10]:…”

“…These instances, described in [7], are based on the model proposed by [10], whereby a molecule is represented as a linear chain of atoms. Bond lengths and angles are kept fixed, and a set of likely torsion angles is generated randomly.…”

Double variable neighbourhood search with smoothing for the molecular distance geometry problem

“…The "Lavor" instances, described in [7], are based on the model proposed by [16], whereby a molecule is represented as a linear chain of atoms. Bond lengths and angles are kept fixed, and a set of likely torsion angles is generated randomly based on an energy function including some Lennard-Jones potential terms.…”