TITLE:
A Novel Design Method for Protein-Like Molecules from the Perspective of Sheaf Theory
AUTHORS:
Naoto Morikawa
KEYWORDS:
Discrete Differential Geometry, Protein Design, Sheaf Theory, Protein Structure
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.13 No.3,
June
13,
2023
ABSTRACT: Proteins
perform a variety of functions in living organisms and their functions are
largely determined by their shape. In this paper, we propose a novel
mathematical method for designing protein-like molecules of a given shape. In
the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles
and 3-simplices are tetrahedra). We design a new molecule of a given shape by
patching together a set of smaller molecules that cover the shape. The covering
set of small molecules is defined using a binary relation between sets of
molecules. A new molecule is then obtained as a sum of the smaller molecules,
where addition of molecules is defined using transformations acting on a set of
(n + 1)-dimensional
cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered.
No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is
required. The author hopes that this paper will encourage further collaboration
between Mathematics and Protein Science.