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-dimens...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 (n=2,3,4...), 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 n+1 -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of n=2 using the discrete differential geometry of triangles (i.e., 2-simplices).展开更多
Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the na...Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the nanoscale, researchers are mimicking the self-assembly strategy for engineering of complex nanostructures. However, the general principles underlying the design of self-assembled molecules have not yet been identified. The question is “How to obtain a well-defined shape with desired properties by folding a chain of subunits (such as amino acids and nucleic acids)”, where properties are determined by the precise spatial arrangement of the subunits on the surface. In this paper, we consider the question from the viewpoint of the discrete differential geometry of n-simplices. Self-assembling molecules are then represented as a union of trajectories of 3-simplices (i.e., tetrahedrons), and the question is rephrased as a “boundary value problem” for flows on a space of tetrahedrons. Also considered is a characterization of two types of surface flows of n-simplices. It is a rough classification of surface flows, but may be essential in characterizing important properties of biomolecules such as allosteric regulation. The author believes this paper not only provides a new perspective for the engineering of self-assembling molecules, but also promotes further collaboration between mathematics and other disciplines in life science.展开更多
文摘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 (n=2,3,4...), 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 n+1 -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of n=2 using the discrete differential geometry of triangles (i.e., 2-simplices).
文摘Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the nanoscale, researchers are mimicking the self-assembly strategy for engineering of complex nanostructures. However, the general principles underlying the design of self-assembled molecules have not yet been identified. The question is “How to obtain a well-defined shape with desired properties by folding a chain of subunits (such as amino acids and nucleic acids)”, where properties are determined by the precise spatial arrangement of the subunits on the surface. In this paper, we consider the question from the viewpoint of the discrete differential geometry of n-simplices. Self-assembling molecules are then represented as a union of trajectories of 3-simplices (i.e., tetrahedrons), and the question is rephrased as a “boundary value problem” for flows on a space of tetrahedrons. Also considered is a characterization of two types of surface flows of n-simplices. It is a rough classification of surface flows, but may be essential in characterizing important properties of biomolecules such as allosteric regulation. The author believes this paper not only provides a new perspective for the engineering of self-assembling molecules, but also promotes further collaboration between mathematics and other disciplines in life science.
