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# Mathematical Methods in Biophysics - MAT00095M

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• Department: Mathematics
• Module co-ordinator: Prof. Reidun Twarock
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

In this module, students will learn how mathematics can be applied to study a variety of topics in soft matter in biological systems, such as viral capsid shells, liquid thin films, protein-ligand interactions, biological polymers, and membranes. Through the use of statistical mechanics, random walks, group graph and tiling theory and representation theory, the student will explore topics in biological soft matter ranging from protein buckling to the geometry and dynamics of protein shells of viruses.

## Related modules

• None

### Prohibited combinations

• None

Physics students can use Thermodynamics and Statistical Mechanics (PHY00013H) as a pre-requisite if necessary

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

In this module, students will learn how mathematics can be applied to study a variety of topics in soft matter in biological systems, such as viral capsid shells, liquid thin films, protein-ligand interactions, biological polymers, and membranes. Through the use of statistical mechanics, random walks, group graph and tiling theory and representation theory, the student will explore topics in biological soft matter ranging from protein buckling to the geometry and dynamics of protein shells of viruses.

## Module learning outcomes

By the end of the module, students will be able to:

1. Apply the methods of Classical and Statistical Mechanics to describe a wide variety of biophysical systems

2. Explain viral geometry using groups, graphs, and tilings, including applications to vaccine design

3. Use the symmetry of an object to understand its biophysical properties, such as its dynamic behaviour

4. Use problem-solving skills and abilities developed in the module to treat problems using a set of different but complementary approaches

5. Apply mathematical skills and techniques in interdisciplinary contexts

## Module content

The following topics will be covered:

• Soft matter and biological materials as subjects of statistical mechanics (SM).

• Mechanical equilibrium as energy minimisation problem. Protein buckling as a bifurcation.

• Liquid films and membranes.

• Fundamentals of Thermodynamics. Entropy. Free energy. Ideal gas.

• Fundamentals of Statistical Mechanics.

• Biological systems: optical trap, mechanosensitive ion channel, and ligand-receptor binding.

• Polymers. Random walk and the entropic origin of elasticity.

• Liquid solutions. Mixing cocktails: phase transitions. The mean field theory.

• Cayley’s theorem and symmetric group descriptions of viral geometry

• Caspar-Klug tiling theory for viral shells, generalisation of viral tilings using Archimedean lattices, and applications to vaccine design

• Coxeter theory, root systems and their applications to multi-shell models in viruses and chemistry applications.

• Application to the study of biophysical properties of symmetric objects

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Mathematical Methods in Biophysics
3 hours 100

None

None

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

• R. Phillips et al. The physical biology of the cell. Garland science (2013).

• S. J. Blundell and K.M. Blundell. Concepts in thermal physics. OUP (2010).

• M. Doi. Soft matter physics. OUP (2013).

• D.-G. De Gennes et al. Capillarity and Wetting Phenomena. Springer (2012).

• Coxeter, Regular polytopes, Dover (1973)

• Humphries, Reflection Groups and Coxeter Groups, Cambridge University Press (1990)

• Senechal, Quasicrystals and Geometry, Cambridge University Press (2009)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.