Mathematical Theory and Biological Applications of Diversity

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Diversity is the extent of variation in and between biological systems and encompasses variation from the scale of the molecule to the rainforest. Its influence and therefore assessment is critical to the life sciences, for example:- Genetic diversity is important for the health and productivity of crops and livestock- Immunological diversity is key to host protection from diverse and evolving pathogens- Pathogen diversity informs vaccine and drug development- Diversity in antimicrobial resistance is a serious clinical and drug development problem- Species diversity influences the health and sustainability of ecosystemsHowever, measuring diversity is hampered by the range of potential measures: species richness, Shannon entropy, expected heterozygosity, Gini-Simpson, and Berger-Parker, to name just a few. Often they exist to capture different aspects of diversity - species richness (where 'species' may be any unit e.g. antigenic phenotype, receptor transcript) counts the number of species ignoring abundance, while Berger-Parker assesses the proportion of the dominant species. But fundamental mathematical problems remain: different measures applied to the same aspect of diversity can give conflicting answers. A key test, which many popular measures fail, is whether a measure behaves intuitively. Suppose a meteorite wipes out 50% of the species on a continent of a million equally abundant species. The Shannon entropy drops by 5%, not the expected 50%, and the Gini-Simpson index by just 0.0001%. The measures that do behave intuitively and logically are called 'effective numbers'.Our ability to use effective numbers across diversity measurement crystallised during a BBSRC-funded workshop run and attended by the co-investigators. Using effective numbers, and a major theoretical advance that we have recently developed that allows us to include any kind of similarity between individuals (e.g. genetic, phylogenetic, functional etc.) in the same diversity framework, our long-term goal is to unify the measurement and interpretation of diversity across strategically important areas of the life sciences. During this FLIP award, we aim to set the groundwork for this by working with each other to understand the detail of how the theory and its applications connect. We will train each other in the mathematics of the diversity framework and in the science underpinning the BBSRC-funded biological applications that use diversity, respectively, bridging the gap across the mathematics - life sciences interface:- One of the developers of the theoretical framework (Leinster) will be taught about the role of the major histocompatibility complex in livestock disease resistance and about quantitative genetics and its application to animal breeding, and work to apply the diversity framework in this context.- The other mathematician (Cobbold) will learn about the measurement of genetic and phylogenetic diversity, and its application to measuring viral circulation for foot-and-mouth disease (FMD) epidemiology. Building on this, she will explore how antigenic diversity measurement can help in vaccine seed strain selection for FMD control.- The applied scientist (Reeve) will learn the pure mathematics that underpins the diversity framework, working on information theory, functional equations and category theory, and then identify how this work will apply to ongoing research on the ecology of antimicrobial resistance (AMR) on which he collaborates with Matthews. He will then work on identifying developments to the theory that will help in the study of the sources and spread of AMR.As well as enhancing the existing BBSRC-funded research of Reeve, Matthews and their collaborators, we will investigate the potential for this approach to unite research areas that have not previously been considered to be closely related, and to foster a powerful new, interdisciplinary research area in the field of diversity.