Discrete Differential Geometry and the Structural Study of Protein Complexes

This paper proposes a novel four-dimensional approach to the structural study of protein complexes. In the approach, the surface of a protein molecule is to be described using the intersection of a pair of four-dimensional triangular cones (with multiple top vertexes). As a mathematical toy model of protein complexes, we consider complexes of closed trajectories of n-simplices ( 2,3,4, n =  ), where the design problem of protein complexes corresponds to an extended version of the Hamiltonian cycle problem. The problem is to find “a set of” closed trajectories of n-simplices which fills the n -dimensional region defined by a given pair of 1 n + -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of 2 n = using the discrete differential geometry of triangles (i.e., 2-simplices).