Title: A diffusion model on toric varieties with application to protein loop modeling
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Protein loop, Toric varieties, Jacobian, Tangent space, MHC, Nanobody

Abstract: The conformation spaces of loop regions in proteins as well as closed kinematic linkages in robotics can be described by systems of polynomial equations, forming Toric varieties. These are real algebraic varieties, formulated as the zero sets of polynomial equations constraining the rotor angles in a linkage or macromolecular chain. These spaces are essentially stitched manifolds and contain singularities. Diffusion models have achieved spectacular success in applications in Cartesian space and smooth manifolds but have not been extended to varieties. Here we develop a diffusion model on the underlying variety by utilizing an appropriate Jacobian, whose loss of rank indicates singularities. This allows our method to explore the variety, without encountering singular or infeasible states. We demonstrated the approach on two important protein structure prediction problems: one is prediction of Major Histocompatibility Complex (MHC) peptide interactions, a critical part in the design of neoantigen vaccines, and the other is loop prediction for nanobodies, an important class of drugs. In both, we improve upon the state of the art open source AlphaFold.