Mathematical Mechanical Biology (Autumn 2018)
The aim of the course is to provide students with adequate technique to build and analyse biomechanical models.
1st and 2nd year courses in math, physics, or engineering (or with teacher's permission).
Ordinary Differential Equations, BA Math (MATH-301).
- statistical mechanics of different chains of growing complexity.
- classical rod mechanics (Kirchhoff and Cosserat).
- Geometry of surfaces and its application to mechanics.
- Fluid bio-membranes.
- Axisymmetric Membranes and Shells in linear and nonlinear elasticity.
- Growth of rods.
- A brief introduction to classical nonlinear elasticity.
- Volumetric growth.
Will be provided as the semester unfolds.
|Week 1 (19.9)||Module 1: Bio-filaments Chapter 1: Chain models The freely jointed chain without and with external force. Boltzmann distribution in a thermal bath. Gyration radius and averaged end-to-end distance. Force-extension relation. An elastic behaviour that comes purely from the entropic regime. Appendix on asymptotic expansions.|
|Week 2 (26.9)||The Worm-like chain. Computation of averaged end-to-end distance for WLC. The case of stiff chains. Definition of persistence length. The continuous limit. Averaged end-to-end distance in the continuous limit: the persistence length comes up again. Discussion of order of magnitudes for different macro-molecules. Force-extension behaviour. Chapter 2: Continuous filaments Dynamical space curves and Frenet frames. Curvature and torsion. Notion of the reference configuration. Reference arclength and current arclength.|
|Week 3 (3.10)|