Structure approximating inverse protein folding in 2D and 3D HPC models

The inverse protein folding (IPF) problem is that of designing an amino acid sequence which folds into a prescribed conformation/structure. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. Our goal here is to solve the structure approximating IPF problem in 2D and 3D in HP models. As for the 2D case, we consider a subclass of linear constructible structures designed by Gupta et. al 2004. These structures, called wave structures, are rich enough to approximate (although more coarsely) any given structure. We formally prove that protein sequence of any wave structure is stable under the HPC model. To prove the stability of wave structures we developed a computational tool, called 2DHPSolver, which we used to perform the large case analysis required for the proofs. 2DHPSolver can be used to prove the stability of any design in 2D square lattice. For the 3D case we introduce a robust class of protein structures, called tubular structures for 3D hexagonal prism lattice. These structures are capable of approximating target 3D shapes. Interestingly, the main building block of tubular structures, a tube, consists of six parallel ``alpha helix''-like structures. Similar designs appear in nature as a coiled coil structural motif in which 2--6 alpha-helices are coiled together. We show that the tubular structures are native for their proteins and we prove that a basic but infinite class of tubular structures consisting of a connector and three tubes of arbitrary length are structurally stable under the HPC model. Despite the tremendous amount of work on protein design for 2D lattices, to the best of our knowledge, this is the first general design of arbitrary long stable proteins for a 3D lattice.