# Modern Mathematical Biology 3

## Unit aims

To provide students with the mathematical tools used to study and solve a variety of problems in biology at different scales. Examples will be taken from problems at different length and timescales - from the scale of the cell, tissue to organisms.

## Unit description

Mathematical Biology is one of the most rapidly growing and exciting areas of Applied Mathematics. This is because recently developed experimental techniques in the biological sciences are generating an unprecedented amount of quantitative data. This new 'quantitative revolution' is changing the way biology is done - requiring methods of generating hypotheses and then testing them that rely heavily on sophisticated mathematical analyses. Biological systems are complex systems and the modern process of studying them requires an iterative process of communication between mathematicians (modellers) and biologists. This starts with making quantitative measurements; second, this biological data is used to develop mathematical models; third, approximate solutions of the models are obtained; and fourth, these solutions are used to make new predictions which can be further tested by new measurements - thus starting the cycle anew.

Professionals in the biomedical sector are increasingly using technology that is reliant on sophisticated mathematics. Examples include ECG readings of the heart, MRI brain scans, blood flow through arteries, tumor invasion, drug design and immunology. Therefore research in this area has the promise of quickly finding real world applications with a positive impact on society. Mathematical Biology also encompasses other interesting phenomena observed in nature, such as the swimming of microorganisms, spread of infectious diseases, and the emergence of patterns in the development and growth.

In this unit we shall use a number of fundamental biological problems as the motivation and starting point for developing mathematical models, explore methods for solving these models and discuss the implications of the predictions that can be made based on them.

## Relation to other units

This is a Level H/6 version of Modern Mathematical Biology 34, sharing the lectures but with differentiated problems and exam.

## Learning objectives

By the end of the unit the students will be familiar with (1) the applications of ODE models in a variety of biological systems, (2) Reaction-Diffusion equations and their applications in biology, (3) the use of linear and nonlinear stability analysis to study the dynamics of complex systems, (4) the dynamical systems approach to describing excitable media.

## Transferable skills

Clear, logical thinking and an ability to comprehend and solve problems of quantitative biology.

## Syllabus

- The role of numbers in biology in the 21st century (particularly at the cellular and tissue scale)
- Ordinary differential equations for modelling biochemical networks (examples from metabolic, signalling and gene regulation networks)
- Stochasticity and randomness in cellular biology
- Spatial dynamics of biochemical networks: reaction-diffusion and all that.

There may be minor changes to this syllabus, or to the order of presentation.

## Reading and References

- R. Philips, J. Kondev , J. Theriot and H. Garcia,
*Physical Biology of the Cell*, Garland Science, ISBN 0815341636 - J.D. Murray,
*Mathematical Biology I. An Introduction*, Springer, ISBN 0387952233 - J.D. Murray,
*Mathematical Biology II. Spatial models and biomedical applications*, Springer, ISBN 0387952284 - L.A. Segel,
*Modelling dynamic phenomena in molecular and cellular biology*, Cambridge University Press, ISBN 052127477X - P. Nelson,
*Biological Physics: Energy, Information, Life*, W. H. Freeman, ISBN 0716798972 - U Alon,
*An Introduction to Systems Biology: Design Principles of Biological Circuits*, CRC press, ISBN: 978-1584886426

**Unit code:** MATH30004

**Level of study:** H/6

**Credit points:** 10

**Teaching block (weeks):** 1 (1-6)

**Lecturer:** Professor Tannie Liverpool

## Pre-requisites

MATH11300 Probability 1, MATH11009 Mechanics 1, MATH11007 Calculus 1, MATH20901 Multivariable Calculus and MATH20001 Methods of Complex Functions

## Co-requisites

None

## Methods of teaching

Lectures, with problems classes or informal discussion to meet the needs of individual students. Regular homework problems set and marked. Homework will include simple numerical exercises using Maple and MATLAB.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 1 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.