Applications of Multidimensional Scaling to Molecular Conformation

Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing conngurations of points from information about interpoint distances. Such constructions arise in computational chemistry when one endeavors to infer the conformation (3-dimensional structure) of a molecule from information about its interatomic distances. For a number of reasons, this application of MDS poses computational challenges not encountered in more traditional applications. In this report we sketch the mathematical formulation of MDS for molecular conformation problems and describe two approaches that can be employed for their solution. 1 Molecular Conformation Consider a molecule with n atoms. We can represent its conformation, or 3-dimensional structure, by specifying the coordinates of each atom with respect to a Euclidean coordinate system for < 3. We store these coordinates in an n3 connguration matrix X. Given X, we can easily compute the matrix of interatomic distances, D(X). In this report we consider the inverse problem of inferring a mathematically plausible conformation from information about D(X). The mathematical problem of inferring a plausible conformation from information about interatomic distances is one of many problems that arise in computational chemistry under the heading of structure determination. Although it is beyond the scope of this report to document the importance of determining the structure of various complex molecules, we can conndently assert that the ability to do so has profound practical implications. Thus, mathematical techniques that potentially enhance this ability are of considerable interest. It may seem curious that one might have information about the interatomic distances of a molecule whose con-formation is not known. Indeed, complete information will not be available; however, incomplete information may be obtained from various sources: 1. Atoms bond at distances and angles that are approximately xed by nature. Assuming, as will typically be the case, that the chemical structure of the molecule is known a priori, fairly stringent upper and lower bounds on these distances and angles will be available. Bounds on bond distances transparently translate into bounds on distances between atoms that are bonded together (\1-2 pairs"); combined with bounds on bond angles, it is an elementary exercise in trigonometry to derive bounds on distances between atoms that are bonded to a common atom (\1-3 pairs"). 2. Some component structures within the molecule may be known (or guessed, by scrutinizing data banks of known structures) to be approximately rigid. For example, benzyl rings are approximately planar hexagonal structures. Approximate rigidity …