- Contact au sein des structures élancées: sur-enroulement de l’ADN et noeuds élastiques (Self-contact in elastic rods: DNA supercoiling and elastic knots)
- Thesis defended on December 19th 2008 and awarded summa cum laude, Université Pierre & Marie Curie Paris VI (now Sorbonne Université).
- Doctoral work performed at the Institut D’Alembert under the supervision of Dr. Basile Audoly and Dr. Sébastien Neukirch.
This thesis concerns the mechanics of elastic rods in case of configurations with self-contacts. In this context we present two different studies: the first one presents a mechanical model for DNA supercoiling in single molecule experiments and the second study deals with the elasticity of knotted rods. The first part of the thesis presents the elastic rod theory based on the Kirchhoff equations, which addresses the mechanical equilibrium of elastic rods considered as one dimensional bodies. These equations are completed with constitutive relations for an isotropic and inextensible rod in the case of a hookean material. Single molecule experiments allow to exert mechanical stresses onto DNA molecules and we focus on extension-rotation measurements. In such experiments the DNA molecule supercoils and forms plectonemes. We present an analytical model based on a variational formulation which takes into account DNA-DNA interactions and thermal fluctuations. Our model allows to calculate the mechanical response of the DNA molecule and also to predict the main experimental results. We compare our predictions with experimental data and find a good agreement. The last part presents an analytical model which addresses the mechanical response of a knotted rod when subjected to both a tensile force and a torsional moment. Our model uses a matched asymptotic expansions method and takes into account the impenetrability constraint. We then provide a general method for building a solution of the Kirchhoff equations in the case of a knotted rod. Our main result is the prediction of the contact set without formulating hypothesis on its structure. We also predict an instability related to the applied torsional moment.
Full manuscript of my PhD thesis (French)