Not Running 2016/17
Lecturer: Nigel Burroughs
Term: Term 2
Status for Mathematics students: List C
Commitment: 30 lectures and weekly assignments
Assessment: 3 hour examination (100%)
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) is advised; students from Mathematics or Physics with backgrounds covering some of these areas should find the course accessible. MA235 Introduction to Mathematical Biology, MA256 Introduction to Systems Biology or MA390 Topics in Mathematical Biology provide some useful background in modelling, while Probability A/B (ST111/2), MA253 Probability and Discrete Mathematics or ST202 Stochastic Processes provide some background for the probabilistic aspects of the course. Programming: A small number of the examples will involve a programming component, so MatLab, or another high-level language would be useful.
1. Basic Techniques. Dynamical Systems/stability, modelling of small circuits.
2. Spatial systems and organisation principles. Cell adhesion, protein patterns and diffusion driven instability.
3. Molecule diffusion and search times. Diffusion along DNA (1D), in membranes (2D). Molecule tracking.
5. Polymerisation underpinning motion. Work and catastrophes.
6. Cell movement. Actin gels and pushing beads.
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 architecture governing dynamic processes such as division. We will also 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; including dynamical systems approaches (20%), probabilistic modelling (60%) and mechanics (20%); indicated percentages are approximate and may vary from year to year. Note that just because we are applying mathematics to biology (and biological is in the title), it does not imply the models, or the mathematics, will be simple. The models can very quickly become very complex and we will draw on a wide variety of techniques to best address the issues.
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
Determine dynamic capabilities 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)
There are currently no specialized text books in this area available. But all the standard textbooks related to the prerequisite modules indicated are relevant.