Course Materials for MATH35032 Mathematical Biology, 2019-20

Lecture: Prof. Oliver Jensen

If you have any questions about this course, feel free to contact me by email or find me in ATB 1.205a during my office hour (12-1pm Fridays).


Intended Learning Outcomes

Once you've successfully completed this module you should be able to:

  1. Interpret differential equation models for populations, relating the expressions appearing in the model to processes that affect the population.
  2. Formulate and analyse ordinary differential equation (ODE) models for the population of a single species, finding equilibrium populations and determining how their stability depends on parameters.
  3. Analyse delay-differential equation (DDE) models for the population of a single species and use linear stability analysis to determine which values of the parameters induce oscillatory instabilities.
  4. Analyse ODE models for the populations of two interacting species, finding equilibria and using information about their linear stability to characterise the long-term behaviour of the system.
  5. Define a conserved quantity for a system of ODEs and, where possible, use such quantities to determine the long-term behaviour of both two-species ODE models and single-species models population models that include diffusion.
  6. Construct the ODEs associated with a system of chemical reactions subject to mass-action kinetics and analyse them to discover conserved quantities.
  7. Construct the Markov process associated with a system of chemical reactions and, for small numbers of reactions, analyse it to determine the long-term behaviour of the system.
  8. Analyse two key models, Wolpert's Frech flag model and Turing's reaction-diffusion model, relating the solutions of the associated PDEs to the processes of pattern-formation in developing organisms.

Lecture Notes

The first half of the course will cover classical topics in mathematical biology, following sections of Jim Murray's famous text, Mathematical Biology I: An Introduction. This book is available via Reading Lists Online.

Printed notes are posted below. They outline topics that will be discussed in more depth in lectures.

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The main channel for formal, written feedback in this module is the coursework. It will be a problem set similar to the ones provided on this page, but devoted to a specific application that uses the ideas from the course. Written solutions will be marked and scripts returned, providing both written comments and a mark. In addition, the weekly examples classes provide further opportunities for verbal feedback and, for students who bring written solutions to the exercises, on-the-spot marking and written feedback as well.

Problem Sets & Solutions

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The coursework will be due at 1:00 PM on Friday, 20 March 2020. Please hand it in at Reception. The coursework contributes 20% to your final mark for the module and will be marked out of 20.
The 2020 Coursework problem concerns the SIR model (and variants) for the spread of a disease in a population. Scripts and marks will be returned at the start of Week 10.

Coursework & Exams

Previous exams include short questions on gene network motifs; these are not part of the 2020 version of the course and can be ignored. However the exam format will remain the same as in previous years.

Exam Solutions:

... will be posted here

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Background materials

  • Books for key mathematical ideas and techniques (see also Reading Lists Online via the Library Catalogue):
    • Nick Britton, Essential Mathematical Biology, Springer ISBN 978-85233-536-6
    • James D. Murray, Mathematical Biology I: An Introduction 3rd edition, (Springer, 2002) ISBN 0-387-95223-3. The first half of the course covers classical topics described in this book.
    • James D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications 3rd edition, (Springer, 2002, ISBN 0-387-95228-4
    • Lee A. Segel, Modeling dynamic phenomena in molecular and cellular biology (Cambridge University Press, 1984). ISBN 0-521-27477-X An text from the early modern era, using deterministic models and ideas from dynamical systems.
    • Darren J. Wilkinson, Stochastic Modelling for Systems Biology (Chapman & Hall/CRC, 2006). ISBN 1-58488-540-8 An up-to-date account of a developing field.
  • Books for biological background:
    • Bruce Alberts, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts and Peter Walter (2002) Molecular Biology of the Cell. 4th edition, Garland Science. ISBN 0-8153-4072-9 This is detailed, comprehensive and big!
    • Uri Alon (2008) An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC, 2007). ISBN 1-58488-642-0
    • Evelyn Fox Keller (2002) Making Sense of Life, Harvard University Press. ISBN 0-674-01250-X A historical and philosophical view of mathematical biology.
    • James Keener, James Sneyd (2009) Mathematical Physiology, Springer-Verkag New York, ISBN 978-0-387-75847-3
    • Edda Klipp, Wolfram Liebermeister, Christoph Wierling, Axel Kowald, Hans Lehrach, Ralf Herwig (2009) Systems Biology: A Textbook. Wiley-Blackwell. ISBN 978-3-527-31874-2
    • Bernhard Ø. Palsson (2006) Systems Biology: Properties of Reconstructed Networks. Cambridge University Press. ISBN 0-521-85903-4
    • Ron Milo, Rob Philips (2016) Cell Biology by the Numbers. Taylor & Francis. ISBN 978-0-8153-4537-4. A great resource for practical calculations where quantification is needed.
  • A tool (created by Chris Johnson) demonstrating reaction-diffusion patterns can be found here. The equations solved relate to those on page 65 of Turing's 1952 Phil. Trans. paper: α scales diffusion coefficients, setting the lengthscale of the pattern; β sets the ratio of diffusion coefficients, leading to stripes or spots; γ sets the ratio of reaction terms.
  • 20 equations that changed biology

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