Inverse design of large molecules using linear diophantine equations

We have previously developed a method for the inverse design of small ligands. This method can be used to design novel compounds with optimized properties (such as drugs) and has been applied successfully to the design of small peptide antagonists to leukocyte functional antigen-1 (LFA-1) and its intercellular adhesion molecule (ICAM-1). A key step in our method involves computing the Hilbert basis of a system of linear Diophantine equations. In our previous application, the ligands considered were small peptide rings, so that the resulting system of Diophantine equations was relatively small and easy to solve. When considering larger molecules, however, the Diophantine system is larger and more difficult to solve. In this work we present a method for reducing the system of Diophantine equations before they are solved, allowing the inverse design of larger compounds. We present this reduction on our original LFA-1/ICAM-1 dataset, where we were able to reduce a system with 24 equations and 49 variables to an equivalent system with 11 equations and 34 variables, giving a 10 times speedup in performance. We also present the results of our reduction on two new datasets, neither of which we could solve previously: a set of 27 conazole fungicides and a set of 61 γ-secrerase inhibitors.