Mathematical Methods in Biophysics - MAT00095M

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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

Additional information

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

Module will run

Occurrence	Teaching period
A	Semester 1 2024-25

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.

• Diffusion. Fluctuation-dissipation theorem.

• 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

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

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.

Indicative reading

• 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)