## Mathematical Biology

 Unit code: MATH35032 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

This module aims to engage students with the applications of mathematical methods to current questions in biology.

#### Overview

The life sciences are arguably the greatest scientific adventure of the age. Over the last few decades a series of revolutions in experimental technique have made it possible to ask very detailed questions about how life works, ranging from the smallest, sub-cellular scales up through the organisation of tissues and the functioning of the brain and, on the very largest scales, the evolution of species and ecosystems. Mathematics has so far played a small, but honourable part in this development, especially by providing simple models designed to illuminate principles and test broad hypotheses.

The mathematics required for biology is not generally all that hard or deep (though there are exceptions: some of the most exciting recent work in phylogenetics requires tools from algebraic geometry), but as the sketches above suggest the range of tools is extremely broad. The point is that modern mathematical biology is genuinely applied maths: its techniques are chosen to suit the biological problems, not the traditional disciplinary subdivisions. Although some previous acquaintance with graph theory and probability would be helpful, this course is meant to be self-contained and will only assume knowledge of differential equations.

#### Learning outcomes

On successful completion of this course unit students will be able:

• Interpret differential equation models for populations, relating the expression 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, and relate the solutions of the associated PDEs to the processes of pattern-formation in developing organisms.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework: Homework, due on the last Friday before the Easter break, worth 20%
• End of semester examination: two hours weighting 80%

#### Syllabus

The course falls into two parts: for the first six weeks it is concerned mainly with standard ODE and PDE models and will rely strongly on the first volume of J. Murray's Mathematical Biology. The main topics will be:

Population models & questions of ecological and evolutionary stability

These topics are normally treated with ODEs or, when one wants to include spatial organisation, PDEs. This area is a good introduction to the illustrative model school of mathematical biology.

Models of chemical reaction networks

The ODE models used here are formally very similar to those used for interacting populations, but the emphasis on chemical reactions prepares the way for the more detailed models of cellular signalling.

Probabilistic simulation of chemical reaction networks

In the last part of the course we will be interested in models of genetic regulation, whihc naturally raise the question Is it sensible to use ODE-based models when there are only a very few reactants? Here we address this issue via a standard stochastic formulation, the Gillespie algorithm.

The latter part of the course is more directly connected to current questions in biology and the lectures will, in part, be designed to help the students read scientific papers, though some of the material is also covered in Uri Alon's book (see below).

Pattern selection and development of body plan

This topic forms a bridge between the textbook study and the research literature. We will begin by reading a famous old paper, Alan Turing's The Chemical Basis of Morphogenesis, and then look at the sorts of things that modern workboth experimental and theoreticalhas to say about related questions. The main tools here are, again, differential equations.

Analysis of regulatory networks

This topic follows naturally from Turing's work and begins to bring in some new mathematical methods and ideas, especially from graph theory and probability. This is mathematical biology at its closest to experimental data (see the online materials for a more detailed list of topics and links to articles).

I studied the following more or less mathematically-mindedbooks while preparing the course:

James D. Murray, Mathematical Biology I: An Introduction 3rd edition, (Springer, 2002). ISBN 0-387-95223-3

This book is available online through SpringerLink, a service to which the University subscribes.

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

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

For biological background one might also look at:

Bernhard . Palsson, Systems Biology: Properties of Reconstructed Networks (Cambridge University Press, 2006). ISBN 0-521-85903-4

Eric H. Davidson, Genomic Regulatory Systems (Academic Press, 2001). ISBN 0-12-205351-6

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

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Oliver Jensen - Unit coordinator