Course Materials for MATH35032, Mathematical Biology

New lecturer in 2018-19

The examples of lecture notes, problem sets and exams that appear below come from the period 2016-2018, but a new lecturer, Prof. Oliver Jensen, will be teaching Mathematical Biology starting in 2018-19 and he is likely to revise the course.

Lecture Notes and Articles

The first 5-6 weeks of the term will be devoted to what one might describe as mathematical biology in the classical style and the lectures will follow sections of Jim Murray's famous text, Mathematical Biology I: An Introduction. This book is available online, though only from within the University's network (e.g. in the campus clusters or in the halls), but if you want to access it from off campus, you can install software that will allow you to use all the Library's services via the University's excellent Virtual Private Network (VPN).

The notes below are examples of the kind of material the course has covered in recent years, but Prof. Jensen is likely to make changes of both content and emphasis.

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Opportunities for feedback

The main channel for formal, written feedback in this module is the coursework. It will be a problem set similar to the ones provided below, but devoted to a novel application that uses the ideas from the course. You'll prepare written solutions and I'll mark them over the Easter Break, 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

The problems below are intended as a guide for students considering taking the course in 2018-19 though, once again, Prof. Jensen may change the course.
Problems       Solutions

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Coursework & Exams

Coursework   Exams

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Intended Learning Outcomes

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

  • Interpret differential equation models for populations, relating the expressions appearing in the model to processes that affect the population.
  • 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.
  • 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.
  • 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.
  • 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 include diffusion.
  • Construct the ODEs associated with a system of chemical reactions subject to mass-action kinetics and analyse them to discover conserved quantities.
  • 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.
  • 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.
  • Explain the notion of a motif in a genetic regulatory network and, for small examples, compute the probability of seeing k instances of a given motif in a randomly-assembled network.

Reading Matter

A hyperlinked version of the lists below is available from Manchester University Library's Link2Lists system.

I studied the following—more or less mathematically-minded—books while preparing this course.

  • James D. Murray, Mathematical Biology I: An Introduction 3rd edition, (Springer, 2002). ISBN 0-387-95223-3
  • 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
  • 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
  • Uri Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC, 2007). ISBN 1-58488-642-0
  • Darren J. Wilkinson, Stochastic Modelling for Systems Biology (Chapman & Hall/CRC, 2006). ISBN 1-58488-540-8

I also looked at the books below for biological background.

  • Bernhard  Ø. Palsson, Systems Biology: Properties of Reconstructed Networks (Cambridge University Press, 2006). ISBN 0-521-85903-4
  • Eric  H. Davidson, The Regulatory Genome (Academic Press, 2006). ISBN 0-12-088563-8
  • Terry  A. Brown, Genomes 3 (Garland Science, 2007). ISBN 0-8153-4138-5
    The previous edition, Genomes 2, is available online from the National Center for Biotechnology Information (NCBI) Bookshelf, a service of the U.S.A's National Institutes of Health (NIH).
  • Bruce Alberts, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts and Peter Walter, Molecular Biology of the Cell 4th edition, (Garland Science, 2002). ISBN 0-8153-4072-9

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