Algebra & Mathematical Biology

Algebra and Mathematical Biology

• Associate Professor Andrew R. Francis (School of Computing and Mathematics, Group Head)

Francis Group

Keywords: Hecke algebra, centre, Coxeter group, mathematical modelling, bacteria, evolution.

Collaborators (internal): none

Collaborators (external): Prof. L. Jones (Shippensburg, USA), Prof. W. Wang (Virginia, USA), Prof. H. Wynn (London School of Econ, UK), Dr J. Graham (Sydney), Dr M. Tanaka (UNSW), Dr S. Sisson (UNSW), Dr F. Luciani (UNSW).

Research: My interests fall into two fields with relatively unexplored intersection: algebra, and mathematical biology. My algebra work focuses on Hecke algebras and their centres. Hecke algebras are deformations of group algebras of finite Coxeter groups (reflection groups), that arise naturally in the representation theory of finite groups of Lie type. They also have connections to a wide range of mathematics and physics, for instance knot theory, quantum groups and statistical mechanics. My biology work involves attempting to understand the evolution of bacteria through the use of mathematical models. With my collaborators we have studied a number of questions relating to the spread of tuberculosis, modelling transmission together with mutation at the marker locus. This has allowed us to make some progress such as a method to detect emerging strains, and to make the best available estimates of key parameters to do with the development of drug resistance in tuberculosis. Hecke algebras In recent work with Lenny Jones (J. Algebra 2009) we used a connection between the centres of Hecke algebras and symmetric polynomials to find a new integral basis for the centre in type A (the symmetric group case). This might be important because an unknown at the moment is the multiplicative structure of the centre, and the only other known integral basis does not appear to have nice multiplicative properties. Drug resistance in tuberculosis In work with my biologist and statistician collaborators, we have just completed a lengthy project to use in vivo molecular data to estimate important epidemiological parameters of drug resistance in tuberculosis. We have been able to show, for instance, that the fitness of drug resistant strains may be as high as that of sensitive strains, and that it is likely that the vast majority of DR strains circulating are a result of transmission, rather than fresh evolution as a consequence of treatment failure. The methods here involved stochastic simulations and approximate Bayesian computation, and a model that allows evolution of the genotype as well as the drug resistance phenotype. This work is under review at the time of writing. Online tools for analysing tuberculosis enotypes Nanoscale Organisation and Dynamics Group Colloquium 11 June 2009 10 We have developed a number of tools that are now available to epidemiologists to aid in interpreting genotypic data from outbreaks of tuberculosis. In particular, we have developed method to detect emerging strains of TB (PNAS 2006), and a method to infer the most likely evolutionary history behind a collection of genotypes from a single outbreak. These methods are suited to genotypic data obtained using the spoligotyping marker and are available on the spolTools website (Bio-informatics 2008).

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