Mathematical Theory and Biological Applications of Diversity

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 ecosystems

However, 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.

Assessing diversity is fundamental to research across the life sciences. The project will strengthen interdisciplinary links across the mathematics-life sciences interface, enhance existing BBSRC-funded research using diversity and, if we succeed in demonstrating the power and generality of the approach, it will offer long-term impact for researchers across the life sciences and to many beneficiaries beyond the academic arena.

1. The researchers involved in this proposal will directly benefit from their enhanced understanding of the mathematical theory and biological applications of diversity, will develop their leadership skills through their joint pursuit of a currently unrecognised unifying theme across well established BBSRC research areas, and will develop their interdisciplinary skills by working across traditional academic boundaries. These promise to be long-term benefits and connections thanks to the universities' joint commitment to give permanent affiliate / visiting researcher status to each of the interchangers.

2. The Boyd Orr Centre for Population and Ecosystem Health will benefit directly from expanding its interdisciplinary remit beyond its current focus on external impact. The centre won the Queen's Anniversary Prize in 2013 for its excellence in applied interdisciplinary research, achieving a diverse range of development and policy impacts across the world. This project allows us to integrate pure mathematicians into the centre and provide them with a route for academic impact, as well as strengthening our increasingly interdisciplinary and inter-institutional foundation. It creates a new way of working that cuts across the life sciences and offers a direct link into the physical sciences, expanding the reach of the centre into more areas of basic science.

3. The development of such a new cross-cutting theme across the life sciences would also allow research areas to cross-fertilise each other more easily, with advances in analytical techniques in one area, such as quantitative genetics and animal breeding, being able to be translated to and applied in other areas, such as assessment of biodiversity impacts or landscape epidemiology.

4. Correctly identifying the links between these fields will also allow us to generate theoretically sound and widely applicable tools for analysis and provide a consistent approach to new research areas as we integrate them into this framework, leading to higher quality proposals for future research. The symposia we will run at the end of the project will specifically address this goal: discussing the conclusions of our work with our collaborators in the life sciences will enable us to utilise these developments in future applications.

5. Our use of branches of pure mathematics not previously associated with applications has already begun to draw pure mathematicians (e.g. analysts and topologists) into the study of diversity, establishing new connections between scientific disciplines and creating a rich new pathway for knowledge transfer from mathematics into the life sciences.

6. Our longer term aim is to forge close links with biodiversity research, where key beneficiaries include biodiversity and conservation organisations. By providing a coherent and powerful framework that allows the quantification of diversity and its partitioning across space and time, we will provide better and more powerful tools for assessing biodiversity loss and the impact of human activity, environment, and climate change on diversity. Policy makers responsible for decisions on biodiversity management will benefit from a more powerful and systematic scientific basis for assessing biodiversity loss. We anticipate longer-term success in global biodiversity management through impact on the newly emerging Essential Biodiversity Variables.