The inverse protein folding problem is that of designing an amino acid sequence which has a prescribed native protein fold. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. The input to the inverse protein folding problem is a shape and the goal is to design a protein sequence with a unique native fold that closely approximates the input shape. Gupta et al.1 introduced a design in the 2D HP model of Dill that can be used to approximate any given (2D) shape. They conjectured that the protein sequences of their design are stable but only proved the stability for an infinite class of very basic structures. The HP model divides amino acids to two groups: hydrophobic (H) and polar (P), and considers only hydrophobic interactions between neighboring H amino in the energy formula. Another significant force acting during the protein folding are sulfide (SS) bridges between two cysteine amino acids. In this paper, we will enrich the HP model by adding cysteines as the third group of amino acids. A cysteine monomer acts as an H amino acid, but in addition two neighboring cysteines can form a bridge to further reduce the energy of the fold. We call our model the HPC model. We consider a subclass of linear structures designed in Gupta et al.1 which is rich enough to approximate (although more coarsely) any given structure. We refine the structures for the HPC model by setting approximately a half of H amino acids to cysteine ones. We conjecture that these structures are stable under the HPC model and prove it under an additional assumption that non-cysteine amino acids act as cysteine ones, i.e., they tend to form their own bridges to reduce the energy. In the proof we will make an efficient use of a computational tool 2DHPSolver which significantly speeds up the progress in the technical part of the proof. This is a preliminary work, and we believe that the same techniques can be used to prove this result without the artificial assumption about non-cysteine H monomers.
