A differential geometric model of supercoiled DNA: A study of the geometry, energy, and electrostatics

The familiar picture of DNA is the classical Watson-Crick linear double helix. However, in the cell, DNA is not a linear structure, but exists in compact folded forms. Superimposed upon the right-handed coiling of the familiar DNA double helix is a higher order of coiling of the helix itself, called supercoiling. The unique structural features of supercoiled DNA (as related to the study of curves) poses an interesting mathematical problem. There are two branches of mathematics that can address this problem: topology and differential geometry.;Previous models of supercoiled DNA, while accounting for macroscopic properties, have ignored the structural details of the molecule. These models treat the DNA as an ideal symmetric elastic rod or rubber hose. An alternative differential geometric procedure to obtain detailed realistic models of DNA folding has been developed. This approach is an extension of the methods currently used to describe topological and geometric parameters of a space curve.;Chapter 1 introduces the topological and geometric concepts of space curves and summarizes the previous literature relating to the structure of supercoiled DNA. In Chapter 2, the differential geometric method used to model the DNA is introduced. Chapter 3 presents some results of the effects of the supercoiling upon the local helical structure of the DNA in terms of base-base geometry, non-bonded stacking energy, and phosphate-phosphate interactions. In Chapter 4 structural consequences of bending the DNA are analyzed in terms of local backbone conformations and groove widths. The structures are minimized using the molecular mechanics package AMBER to relieve strain on the backbone bond lengths, bond angles, and torsion angles introduced by the differential geometric model building procedure.;The geometrical and conformational properties of nucleic acids play an important role in their biochemical behavior. This behavior is also influenced by the electrostatic characteristics of the macromolecule. Background for electrostatic concepts as well as a literature overview are found in Chapter 5. Chapters 6 and 7 discuss the development of a tool to study the effects of DNA bending and sequence upon the electrostatic potential surface, field, and gradient of the molecule. DNA fragments used for these studies are generated using the differential geometric procedure developed in this thesis.