C5.9 Mathematical Mechanical Biology - Material for the year 2020-2021

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Prof. Derek Moulton
General Prerequisites: 

The Part C courses Solid Mechanics and Elasticity/Plasticity are recommended. Familiarity with common techniques of applied mathematics will be assumed, in particular asymptotic methods and calculus of variations. Mathematical biology or physiology is desirable but not necessary as the material for a particular biological system will be part of the course.

Course Term: 
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 

Assessment type:

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, Gauss–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

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