# C5.9 Mathematical Mechanical Biology - Material for the year 2019-2020

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Fluid Mechanics: Part A Waves and Fluids and Part B Viscous flow are recommended. Solid mechanics: The Part C courses Solid Mechanics and Elasticity/Plasticity are recommended. Mathematical biology or physiology is desirable but not necessary as the material for a particular biological system will be part of the course.

16 lectures

### Assessment type:

- Written Examination

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.

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.

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 the cell, plants and microbes.

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

The following module will not be taught in 2019-20:

3. 3D Biological Mechanics.

(a) Low Reynolds number flows: Scallop theorem, Cell Motility, Ciliary Pumping.

(b) Introduction to nonlinear 3D elasticity for soft tissues

(c) Coupling low Reynolds fluids and non-linear elastic solids, with application to poro-elastic tissue

- Physical Cell Biology, second ed. Rob Phillips et al. Garland Science.
- Cardiovascular solid mechanics. Cells, tissues, and organs, Humpherey, 2002, Springer.
- Nonlinear Solid Mechanics: A Continuum Approach for Enineering: A Continuum Approach for Engineering, G. Holzapfel, 200, Wiley.
- The Mathematics and Mechanics of Biological Growth, A. Goriely, 2017, Springer.