Topological Analysis of Enzymatic Actions on DNA Polyhedral Links

Abstract

Current synthetic biology has witnessed a revolution that natural DNA molecule steps onto a broad scientific area by assembling a large variety of three-dimensional structures with the connectivity of polyhedra. A mathematical model of these biomolecules is crucial to clarify the biological self-assembly principle, and unravel a first-step understanding of biological regulation and controlling mechanisms. In this paper, mechanisms of two different enzymatic actions on DNA polyhedra are elucidated through theoretical models of polyhedral links: (1) topoisomerase that untangles DNA polyhedral links produces separated single-stranded DNA circles through the crossing change operation; (2) recombinase generates a class of polyhedral circular paths or polyhedral knots by applying the crossing smoothing operation. Furthermore, we also discuss the possibility of applying two theoretical operations in molecular design of DNA polyhedra. Thus, our research provides a new sight of how geometry and topology of DNA polyhedra can be manipulated and controlled by enzymes, as well as has implications for molecular design and structural analysis of structural genome organization.

A New Euler's Formula for DNA Polyhedra

“…As an example, we illustrate its use to characterize the recombinase regulation and controlling mechanisms for DNA polyhedra [33]. Recombinase is a site-specific enzyme, which, by cutting two segments and interchanging the ends of DNA, can result in the inversion or the deletion or insertion of a DNA segment.…”

“…By analogy with the classic Euler's formula, we expect that such a simple and elegant relation will greatly contribute to an understanding of the topologically complex structures of polyhedral links, as well as some potential biological processes. As an example, we illustrate its use to characterize the recombinase regulation and controlling mechanisms for DNA polyhedra [33] . Recombinase is a site-specific enzyme, which, by cutting two segments and interchanging the ends of DNA, can result in the inversion or the deletion or insertion of a DNA segment.…”

Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid

“…e crossing number of a polyhedral link c is the least number of crossings that occur in any projection of the polyhedral link. e component number of a polyhedral link μ is the number of loops (rings) knotted with each other [15].…”

“…Benson et al proposed a general method to construct scaffolded DNA nanostructures from flat sheet meshes with the computer-aided technique [13]. Qiu and his colleagues took a mathematical approach to answer the complex question of what models are suitable to describe the DNA polyhedron structure [14][15][16][17][18][19]. Guo et al provided a topological approach to assemble DNA tetrahedra and DNA triangular prism [20,21].…”