Mathematical Theory and Biological Applications of Diversity
Lead Research Organisation:
University of Glasgow
Department Name: College of Medical, Veterinary, Life Sci
Abstract
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.
- 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.
Planned Impact
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.
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.
Publications
Leinster Tom
(2019)
Entropy modulo a prime
in arXiv e-prints
Reeve Richard
(2014)
How to partition diversity
in arXiv e-prints
Sarker SK
(2019)
1980s-2010s: The world's largest mangrove ecosystem is becoming homogeneous.
in Biological conservation
Sarker S
(2019)
Modelling spatial biodiversity in the world's largest mangrove ecosystem-The Bangladesh Sundarbans: A baseline for conservation
in Diversity and Distributions
Allen L
(2019)
Are orchid bees useful indicators of the impacts of human disturbance?
in Ecological Indicators
Sarker S
(2021)
Solving the fourth-corner problem: forecasting ecosystem primary production from spatial multispecies trait-based models
in Ecological Monographs
Mittell E
(2020)
Feral populations of Brassica oleracea along Atlantic coasts in western Europe
in Ecology and Evolution
Peacock TP
(2018)
The molecular basis of antigenic variation among A(H9N2) avian influenza viruses.
in Emerging microbes & infections
McCulloch R
(2017)
Emerging challenges in understanding trypanosome antigenic variation.
in Emerging topics in life sciences
Davies V
(2019)
Improving the identification of antigenic sites in the H1N1 influenza virus through accounting for the experimental structure in a sparse hierarchical Bayesian model.
in Journal of the Royal Statistical Society. Series C, Applied statistics
Peacock TP
(2021)
Genetic determinants of receptor-binding preference and zoonotic potential of H9N2 avian influenza viruses.
in Journal of virology
Bush A
(2017)
Connecting Earth observation to high-throughput biodiversity data.
in Nature ecology & evolution
Ewing DA
(2019)
Uncovering mechanisms behind mosquito seasonality by integrating mathematical models and daily empirical population data: Culex pipiens in the UK.
in Parasites & vectors
Harvey WT
(2023)
A Bayesian approach to incorporate structural data into the mapping of genotype to antigenic phenotype of influenza A(H3N2) viruses.
in PLoS computational biology
Jayaraman S
(2019)
Application of long read sequencing to determine expressed antigen diversity in Trypanosoma brucei infections.
in PLoS neglected tropical diseases
Caudell MA
(2017)
Antimicrobial Use and Veterinary Care among Agro-Pastoralists in Northern Tanzania.
in PloS one
Buckley J
(2019)
Changing environments and genetic variation: natural variation in inbreeding does not compromise short-term physiological responses.
in Proceedings. Biological sciences
Hurford A
(2019)
Skewed temperature dependence affects range and abundance in a warming world.
in Proceedings. Biological sciences
Sarker SK
(2016)
Are we failing to protect threatened mangroves in the Sundarbans world heritage ecosystem?
in Scientific reports
Leinster
(2021)
Entropy and Diversity: The Axiomatic Approach
Leinster
(2021)
Entropy and Diversity: The Axiomatic Approach
Description | We have successfully deepened the linkages between the mathematical frameworks that underpin diversity and entropy on the one side, and the practical applications of it in measuring biodiversity of ecosystems and other forms of diversity. Several papers have already been published, with many more to come on this topic, and several new collaborations have been developed, notably with the Danish Natural History Museum and the Met Office in the UK. As a result of this, more rigorously founded appro |
Exploitation Route | An R and a Julia package are available for scientists to use in the research, and the ideas developed are already being considered for application in global assessments of biodiversity loss. |
Sectors | Environment Healthcare |
Description | We are now working with Peatland ACTION, a part of Scottish Natural Heritage to help guide their peatland restoration work and with Natural Resources Wales in the same area. This is very preliminary work at this stage, but has lead on from this initial project on understanding biodiversity assessment and change. |
First Year Of Impact | 2021 |
Sector | Environment |
Impact Types | Policy & public services |
Description | An integrated approach to tackling drug resistance in livestock trypanosomes. |
Amount | £343,273 (GBP) |
Funding ID | BB/S000143/1 |
Organisation | Biotechnology and Biological Sciences Research Council (BBSRC) |
Sector | Public |
Country | United Kingdom |
Start | 01/2019 |
End | 12/2020 |
Description | BBSRC FLIP award |
Amount | £100,592 (GBP) |
Funding ID | BB/P004202/1 |
Organisation | Biotechnology and Biological Sciences Research Council (BBSRC) |
Sector | Public |
Country | United Kingdom |
Start | 08/2016 |
End | 08/2018 |
Description | BBSRC LOLA |
Amount | £1,866,383 (GBP) |
Organisation | Biotechnology and Biological Sciences Research Council (BBSRC) |
Sector | Public |
Country | United Kingdom |
Start | 01/2015 |
End | 09/2019 |
Description | BBSRC ZELS |
Amount | £1,667,136 (GBP) |
Funding ID | BB/L018926/1 |
Organisation | Biotechnology and Biological Sciences Research Council (BBSRC) |
Sector | Public |
Country | United Kingdom |
Start | 09/2014 |
End | 09/2018 |
Description | JPIAMR |
Amount | £1,000,000 (GBP) |
Organisation | Medical Research Council (MRC) |
Sector | Public |
Country | United Kingdom |
Start | 03/2017 |
End | 03/2020 |
Description | Newton Fund swine and poultry research initiative |
Amount | £622,000 (GBP) |
Funding ID | BB/R012679/1 |
Organisation | Biotechnology and Biological Sciences Research Council (BBSRC) |
Sector | Public |
Country | United Kingdom |
Start | 01/2018 |
End | 12/2020 |
Description | Supporting the National Action Plan for Antimicrobial Resistance (SNAP-AMR) in Tanzania |
Amount | £3,189,370 (GBP) |
Funding ID | MR/S004815/1 |
Organisation | Medical Research Council (MRC) |
Sector | Public |
Country | United Kingdom |
Start | 04/2018 |
End | 04/2023 |
Description | The interplay of land-use, climate and plant biodiversity on the UK stage |
Amount | £50,330 (GBP) |
Funding ID | NE/T004193/1 |
Organisation | Natural Environment Research Council |
Sector | Public |
Country | United Kingdom |
Start | 09/2019 |
End | 09/2020 |
Title | Are orchid bees useful indicators of the impacts of human disturbance? |
Description | |
Type Of Material | Database/Collection of data |
Year Produced | 2018 |
Provided To Others? | Yes |
Description | Collaboration with the Natural History Museum London. |
Organisation | Natural History Museum |
Department | Life Sciences Department |
Country | United Kingdom |
Sector | Public |
PI Contribution | Using data provided by our collaborators at the Natural History museum, we are developing general ecosystem models that we will use to test the current suite of biodiversity metrics and their use and efficacy from a monitoring point of view. To this end, we hope to use the information to answer questions surrounding current biodiversity patterns of endangered plant species and how we might better predict/measure these. |
Collaborator Contribution | Our Collaborator Neil Brummitt provides expertise on the biological aspect of this work, including data and information on plant species occurrences, traits and current conservation policy (including the process involved in the IUCN red list). |
Impact | This is a multi-disciplinary collaboration spanning computer science, ecology and mathematics. |
Start Year | 2016 |
Description | Diversity |
Organisation | Natural History Museum |
Department | Lepidoptera Collection |
Country | United Kingdom |
Sector | Charity/Non Profit |
PI Contribution | We have provided an understanding of the links between the fundamental mathematics studied by the mathematicians with whom we collaborate and the applied biodiversity problems that the ecologists we work with wish to understand. |
Collaborator Contribution | Tom Leinster at the University of Edinburgh has been providing mathematical expertise to understand the fundamental properties of diversity measures. Jill Thompson at CEH and Neil Brummitt at NHM have been providing practical assistance in understanding the underlying biodiversity that we are studying. Michael Krabbe Borregaard at NHMD has been working on development of Julia code to help with analyses. |
Impact | How to partition diversity (arXiv paper) multidisciplinary - maths, ecology, biodiversity BB/P004202/1 Mathematical Theory and Biological Applications of Diversity (further funding) multidisciplinary - maths, ecology, biodiversity, evolutionary biology |
Start Year | 2012 |
Description | Diversity |
Organisation | UK Centre for Ecology & Hydrology |
Country | United Kingdom |
Sector | Public |
PI Contribution | We have provided an understanding of the links between the fundamental mathematics studied by the mathematicians with whom we collaborate and the applied biodiversity problems that the ecologists we work with wish to understand. |
Collaborator Contribution | Tom Leinster at the University of Edinburgh has been providing mathematical expertise to understand the fundamental properties of diversity measures. Jill Thompson at CEH and Neil Brummitt at NHM have been providing practical assistance in understanding the underlying biodiversity that we are studying. Michael Krabbe Borregaard at NHMD has been working on development of Julia code to help with analyses. |
Impact | How to partition diversity (arXiv paper) multidisciplinary - maths, ecology, biodiversity BB/P004202/1 Mathematical Theory and Biological Applications of Diversity (further funding) multidisciplinary - maths, ecology, biodiversity, evolutionary biology |
Start Year | 2012 |
Description | Diversity |
Organisation | University of Copenhagen |
Department | Natural History Museum of Denmark |
Country | Denmark |
Sector | Public |
PI Contribution | We have provided an understanding of the links between the fundamental mathematics studied by the mathematicians with whom we collaborate and the applied biodiversity problems that the ecologists we work with wish to understand. |
Collaborator Contribution | Tom Leinster at the University of Edinburgh has been providing mathematical expertise to understand the fundamental properties of diversity measures. Jill Thompson at CEH and Neil Brummitt at NHM have been providing practical assistance in understanding the underlying biodiversity that we are studying. Michael Krabbe Borregaard at NHMD has been working on development of Julia code to help with analyses. |
Impact | How to partition diversity (arXiv paper) multidisciplinary - maths, ecology, biodiversity BB/P004202/1 Mathematical Theory and Biological Applications of Diversity (further funding) multidisciplinary - maths, ecology, biodiversity, evolutionary biology |
Start Year | 2012 |
Description | Diversity |
Organisation | University of Edinburgh |
Department | Centre for Integrative Physiology |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | We have provided an understanding of the links between the fundamental mathematics studied by the mathematicians with whom we collaborate and the applied biodiversity problems that the ecologists we work with wish to understand. |
Collaborator Contribution | Tom Leinster at the University of Edinburgh has been providing mathematical expertise to understand the fundamental properties of diversity measures. Jill Thompson at CEH and Neil Brummitt at NHM have been providing practical assistance in understanding the underlying biodiversity that we are studying. Michael Krabbe Borregaard at NHMD has been working on development of Julia code to help with analyses. |
Impact | How to partition diversity (arXiv paper) multidisciplinary - maths, ecology, biodiversity BB/P004202/1 Mathematical Theory and Biological Applications of Diversity (further funding) multidisciplinary - maths, ecology, biodiversity, evolutionary biology |
Start Year | 2012 |
Description | Peatland ACTION |
Organisation | NatureScot |
Country | United Kingdom |
Sector | Public |
PI Contribution | We are developing our diversity framework and biodiversity simulation environment to target peatland environments to help Peatland ACTION (a part of SNH) better target their peatland restoration work. |
Collaborator Contribution | Peatland ACTION have provided us with expert advice and guidance on modelling of the peatland environment as well as a detailed understanding of how they operate, and data that they have collected in Scotland. |
Impact | One grant at the moment - NE/T010355/1 - which is multidisciplinary, involving botany, ecology, biodiversity assessment, mathematics, computer science (high performance computing) and policy. |
Start Year | 2019 |
Title | Diversity.jl package in Julia |
Description | Diversity.jl is a package for Julia based around a framework of similarity-sensitive diversity measures. It calculates the diversity of a population and its constituent subcommunities inclusive of similarity (taxonomic, phenotypic, genetic, phylogenetic, functional, and so on) between individuals. |
Type Of Technology | Software |
Year Produced | 2017 |
Open Source License? | Yes |
Impact | New collaboration with Danish Natural History Museum. |
URL | https://github.com/richardreeve/Diversity.jl |
Title | rdiversity R package |
Description | rdiversity is a package for R based around a framework of similarity-sensitive diversity measures. It calculates the diversity of a population and its constituent subcommunities inclusive of similarity (taxonomic, phenotypic, genetic, phylogenetic, functional, and so on) between individuals. |
Type Of Technology | Software |
Year Produced | 2016 |
Open Source License? | Yes |
Impact | Used in research by internal and external colleagues. |
URL | https://github.com/boydorr/rdiversity |
Description | BES meetings |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | The British Ecological Society meeting is a huge annual meeting that offers an opportunity to communicate with other researchers, students, policymakers and NGOs. Several people contacted us during the meeting to discuss adapting our work to their needs. |
Year(s) Of Engagement Activity | 2016 |
Description | Invited talk at Danish Natural History Museum |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | 30 members of the Danish Natural History Museum in Copenhagen attended an invited talk, sparking questions and discussion, ultimately leading to decisions to work together on further development of our work. |
Year(s) Of Engagement Activity | 2018 |
Description | Shiny app |
Form Of Engagement Activity | Engagement focused website, blog or social media channel |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Public/other audiences |
Results and Impact | We have developed a shiny app that enables us to graphically illustrate the (suitably anonymised) socio-economic survey and resistance data from our study sites in Tanzania) |
Year(s) Of Engagement Activity | 2017 |