### C5.9 Mathematical Mechanical Biology (2023-24)

General Prerequisites:
There are no strict prerequisites for this course, though familiarity with common techniques of applied mathematics will be assumed, in particular asymptotic methods and calculus of variations. Mathematical biology, Elasticity&Plasticity, and/or Physiology make nice companion courses but are not necessary as the material for a particular system, both in terms of mechanics and biology, will be presented as part of the course.
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
The course will be motivated by outstanding problems in physiology and biology but the emphasis is on the mathematical tools needed to answer some biologically relevant problems. The course is divided into modules and three modules will be given during a term but these modules can change from one year to the next.
Learning Outcomes:
The goal of this course is to learn the physical background and mathematical methods behind many problems arising in mechanical biology from the cellular level and upwards. Students will familiarise themselves with key notions used in modern research in bio-physics and mechano-biology.
Course Synopsis:
1. 1D Biological Mechanics. Bio-Filaments (2 1/2 weeks)
(a) Introduction: bio-molecules (actin, microtubules, DNA,...)
(b) Randomly fluctuating chains (statistical mechanics)
(c) Continuous filaments (neurons, stems, roots, plants)
(d) Differential geometry of curves: Kirchhoff rod theory and beam theory

2. 2D Biological Mechanics. Bio-Membranes (2 1/2 weeks)
(a) Introduction: lipid bilayer, cell membranes
(b) Differential geometry of surfaces: curvatures, Gaussâ€“Bonnet theorem
(c) Fluid membranes: shape equation, fluctuating membranes
(d) Solid membranes, shells and their application to biological membranes.

3. Bio-solids and growth (3 weeks)
(a) Introduction: nonlinear elasticity for soft tissues
(b) one-dimensional growth theory
(c) Application to mechanical pattern formation
(d) volumetric growth: multiplicative decomposition