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Mathematical Mechanical Biology (Autumn 2018)

General informations

Lecturer

Thomas Lessinnes

Assistant

Thomas Zwahlen

Schedule

Lectures: Wednesday from 08:15 to 10:00, room MAA112
Exercices: Wednesday from 10h15 to 12:00, room MAA112

Objectives

The aim of the course is to provide students with adequate technique to build and analyse biomechanical models.

Requirements

1st and 2nd year courses in math, physics, or engineering (or with teacher's permission).

Recommended

Ordinary Differential Equations, BA Math (MATH-301).

Analysis I-IV.

Contents

  • 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.

Lecture notes

Will be provided as the semester unfolds.

Week-by-week correspondence

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)
Week 13(12.12)

Exercices

Exercice sheet Answers

Bibliography