Discrete differential geometry of triangle tiles and algebra of closed trajectories

This paper proposes a new mathematical framework that can be applied to biological problems such as analysis of the structures of proteins and protein complexes. In particular, it gives a new method for encoding the three-dimensional structure of a protein into a binary sequence, where proteins are approximated by a folded tetrahedron sequence. The feature of the method is the correspondence between protein folding and “integration”. And the binary code of a protein is obtained as the “second derivative” of the native folded structure of the protein. With this method at hand, it becomes possible to describe the structure of a protein without any subjective hierarchical classification. It also gives a new algebraic framework for describing molecular complexes and their interactions. First we describe the biological background briefly.