Molecular alignment as a penalized permutation Procrustes problem
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Molecular alignment as a penalized permutation Procrustes problem Farnaz Heidar Zadeh · Paul W. Ayers
Received: 3 September 2012 / Accepted: 19 November 2012 / Published online: 18 December 2012 © Springer Science+Business Media New York 2012
Abstract Molecular alignment is viewed as a permutation Procrustes problem, where the goal is to find the best assignment of points (or functional groups) in one molecule to the points in another molecule. A penalty function ensures that the optimal alignment respects the underlying connectivity between atoms/points. This method helps reveal why molecular alignment suffers from the curse of dimension. Keywords Molecular similarity · 3D-QSAR · Quantum QSAR · Molecular alignment · Permutation Procrustes problem · k-nearest neighbor alignment
1 Motivation Three-dimensional quantitative structure activity relationships (3D-QSAR) are built by computing the similarity between different molecules, then using the precept that similar molecules have similar properties to make predictions. [1–8] In assessing the similarity of two molecules, however, one must first choose the appropriate relative position, orientation, and (sometimes) conformation of the molecules: a molecule will not even be similar to itself if it is misaligned. This leads to the problem of finding the optimal alignment between molecules. There are many approaches in the literature (see Ref. [9]) and references cited therein). We are primarily interested in the quantum QSAR alignment problem, where the molecules are aligned based on quantum mechanical properties. Cf. Refs. [10–16]. However, our approach is valid for any sort of molecular alignment and, indeed, the general problem of structural alignment.
F. H. Zadeh · P. W. Ayers (B) Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON, Canada e-mail: [email protected]
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J Math Chem (2013) 51:927–936
One problem in molecular alignment is that there is no way to quantify what indicates a “good alignment” method. Consider two ambiphilic molecules. (A molecule is ambiphilic if it has both electrophilic and nucleophilic sites.) The activity of these molecules with respect to an electrophile is determined by their nucleophilic sites, so assessing the similarity of the molecules’ nucleophilic activity requires aligning their nucleophilic regions. Similarly, for assessing the activity of the molecules to a nucleophilic reagent, their electrophilic regions should be aligned. For another property (e.g., their solubility), an entirely different alignment protocol might be preferable. It is important, then, to have a flexible alignment method that can align molecules based on many different criteria. In this paper we present a flexible alignment method of this type. (M) (M) (M) Npts . We assume that each molecule, M, is described by a set of Npts points, qi i=1 (M) Npts (M) (M) (M) . At each point one has a vector of Nprop properties, p1 , p2 , . . . , p Nprop i=1 The similarity of the molecules is the written as (M) (M) min
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