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Generation of Ligand Based Structural Hypothesis: The Use of Markov Random Fields

For many therapeutically established drug targets the 3D structure is not known.  There is however often experimental information on ligands active against the target or target class.  A key goal in generalizing this information and in generating predictive models is the elucidation of pharmacophore hypotheses common to subsets of these ligands.

Here we specifically address the problem of three-dimensional superposition of small molecules using the technique of Markov Random Fields (MRF). This as well as many other problems in computational chemistry and biology can be reduced to finding a match of chemical features and  of distances between them in the form of three dimensional graphs.

We have developed a general purpose computational engine for such graph-matching problems that makes use of a Markov Random Fields representation.  Convergent solutions of  MRF's is achieved using one of several methods such as Beliefs Propagation  to minimize a free energy function associated with MRF.  Such partial weighted graph matching solutions are ideally suited to the difficult problem of small-molecule superposition. The formulation of the  MRF problem and of resulting MRF solutions is probabilistic and multiple solutions can be obtained.  A related but distinct application of this computational engine to the problem of docking is described in Pharmacophore Match for Docking-Based Virtual Screen.

At the start every molecule is broken into constituent parts called fragments, and every fragment is represented by a reduced set of chemical vectors and points that represent hydrogen bonds, hydrophobic centers, formal charges and etc.

For every ligand a set of conformers is generated, and represented in a factorized form. This factorized representation can be envisaged as a tree with a scaffold (largest fragment) at the root of this tree. Conformations are generated using fixed-angle rotations whose number depends on bond hybridization. The resulting factorized representation is very compact and grows linearly with the number of ligand atoms.  The MRF construct uses this factorized representation of flexible molecules, yielding an advantage in both speed and completeness over alignment packages that use a  fixed small number of conformers only.

The problem of finding a structural hypothesis for a set of molecules is first reduced to the problem of finding the best alignment of a flexible ligand to a fixed ligand. Subsequently, the best aligned conformations of pair wise alignments are compared and alignments common to a ligand set are found. From common alignments one can then easily infer structural pharmacophoric hypothesis.

In many cases a partial pharmacophoric hypothesis is known.  MRF models, by boosting prior beliefs associated with this partial pharmacophore, can be restricted to solutions that incorporate this previous partial pharmacophoric hypothesis.