# MA6J6 Mathematics and Biophysics of Cell Dynamics

**Lecturer: **Nigel Burroughs** **

**Term:** Term 2

**Commitment:** 30 lectures and weekly assignments

**Assessment:** Oral Exam

**Prerequisites:**

Previous experience with at least a couple of Dynamical Systems, PDEs, probability theory/stochastic processes, continuum mechanics and physical principles such as elasticity and energy (thermodynamics) would be beneficial; students from Mathematics or Physics with backgrounds covering some of these areas should also find the course accessible. Given the diversity of techniques used in the course, do not worry if you haven’t got them all. MA256 Introduction to Systems Biology or MA390 Topics in Mathematical Biology provide some useful background in modelling, Probability A/B (ST111/2), or ST202 Stochastic Processes provide background for the probabilistic aspects of the course, and MA250 Introduction to partial differential equations provides some background in PDEs. Programming: A small number of the examples will involve a programming component, so MatLab, or another high-level language would be useful.

**Content:
**

1. Spatial systems and organisation principles. Cell adhesion, protein (Turing) patterns and diffusion driven instability.

2. Molecule diffusion and search times. Diffusion along DNA (1D), in membranes (2D) and in 3D.

3. Polymerisation underpinning motion and work. Microtubule dynamics (dynamic instability and catastrophes) and actin.

4. Molecular motors. the flashing ratchet.

5. Cell movement. Actin gels and pushing beads.

6. Cell division if time permits. Chromosome self-organisation processes.

**Aims:**

How cells manage to do seeming intelligent things and respond appropriately to stimulus has generated scientific and philosophical debate for centuries given that they are just a ‘bag’ of chemicals. This course will attempt to offer some answers using state-of-the-art mathematical/physics models of fundamental cell behaviour from both bacteria and mammals. A number of key models have emerged over the last decade dealing with spatial-temporal dynamics in cells, in particular cell movement, but also in developing crucial understanding of the basic cellular architecture governing dynamic processes. We will thus explore a number of biological phenomena to illustrate fundamental biological principles and mechanisms, including for example molecular polymerisation to perform work. This course will take a mathematical modelling viewpoint, developing both modelling techniques but also essentials of model analysis. We will draw on a large body of mathematical areas from both deterministic- dynamical systems methods (20%), (continuum) mechanics (20%), and probabilistic arenas - Fokker Planck equations, diffusions (60%); indicated percentages are approximate and may vary from year to year. We will thus draw on a wide variety of techniques to best address the issues, techniques that will be covered in the course.

**
Objectives:
**By the end of the module the student should be able to:

Develop spatial-temporal models of biological phenomena from basic principles

Understand the basic organisation and physical principles governing cell dynamics and structure

Characterise the dynamics of simple (stochastic) models of biological polymers (actin, tubulin)

Construct and solve optimisation problems in biological systems, e.g. for a diffusing protein to find a target binding site

Reproduce models and fundamental results for a number of cell behaviours (division, actin gels)

**Books:
**There are currently no specialised text books in this area available.