Sample PhD Project in Mathematical Biology

General Interests:

Excitation waves in the heart; population dynamics; epidemiology; evolution; systems biology; biological fluid dynamics.

Professor Rachel Bearon

Migration of cancer cells

This project will make use of 3D imaging data from cancer ‘spheroids’; experimental model systems that help us understand cancer biology. We shall develop individual-based stochastic simulations and population-level models to describe the movement of the cells and resultant spatial organisation of cells. This will help us better understand the efficacy of treatments such as chemotherapy. 

Richards, R. and Mason, D. and Levy, R. and Bearon, R. and See, V. (Preprint). 4D imaging and analysis of multicellular tumour spheroid cell migration and invasion.

Biofilm formation and emergence of anti-microbial resistance

This project will look at how bacteria move in flows and on surfaces, so as to develop a model which allows us to investigate the colonisation rate of surfaces. We will apply this model to the early-stage formation of biofilms in catheters, in particular seeking to better understand the emergence of anti-microbial resistant strains of bacteria.

Bearon, R. N., & Hazel, A. L. (2015). The trapping in high-shear regions of slender bacteria undergoing chemotaxis in a channel. J. Fluid Mech.

Strategies for bacterial chemotaxis in shear

Many bacteria are able to respond to chemical cues; preferentially moving towards attractants and away from harmful substances. In this project we develop individual-level & population-level models to describe bacterial chemotaxis, and investigate how different strategies impact aggregation & dispersion and what are the optimal strategies depending on the flow environment.

Bearon, R., & Durham, W. (2019). A model of strongly-biased chemotaxis reveals the tradeoffs of different bacterial migration strategies. Mathematical Medicine and Biology: A Journal of the IMA.

Drug efficacy & toxicity: Connecting lab experiments to patient predictions

3D micro-tissue experimental models are increasingly being recognised as being a useful tool for testing drug efficacy and toxicity & providing a link between standard test-tube experiments and in-vivo tests. Mathematical models provide a key tool for testing and refining such models, and providing predictions as to the spatial-temporal dynamics of drugs within the experimental system. The area of systems pharmacology allows for the extrapolation to patient predictions through techniques such as Physiologically Based PharmacoKinetic (PKPB) models. A rich area for modelling with many problems to tackle!

Leedale, J. A., Kyffin, J., Harding, A., Colley, H., Murdoch, C., Sharma, P., Williams, D., Webb, S. and Bearon, R. (Accepted). Multiscale modelling of drug transport and metabolism in liver spheroids. Interface Focus.

Vertical transport of phytoplankton in flow

Phytoplankton are photosynthetic micro-organisms that live in aquatic environments. They are critically important in oceanic food webs; forming the base and sustaining all the world’s commercially important fisheries, and playing a critical part in the carbon cycle where they act as a biological pump transporting carbon from the atmosphere to the deep ocean. Understanding how their behaviour (e.g. ability to form chains or to swim) interacts with their flow environment to modify the rate at which vertical migration occurs is therefore important, yet there are still several fundamental unsolved problems.   

Bearon, R., Clifton, W., & Bees, M. (2018). Enhanced sedimentation of elongated plankton in simple flows. IMA Journal of Applied Mathematics.

Dr Mirela Domijan

Circadian rhythms

There are several possible projects within this theme.

Relevant starting literature: Gould, Domijan et al. (2018) Coordination of robust single cell rhythms in the Arabidopsis circadian clock via spatial waves of gene expression, eLife.

A possible project focusses on the integration of light inputs in the circadian clock circuitry. Techniques from dynamical systems theory and perturbation theory are relevant here. Some starting point literature: Hajdu et al. (2018) ELONGATED HYPOCOTYL 5 mediates blue light signalling to the Arabidopsis circadian clock, The Plant Journal.

Modelling the impact of insulin pulses on the cell cycle

Muscle development and maintenance is strongly linked to insulin, yet how insulin exactly affects cell proliferation and energy production is not well known. Various pathways have all been identified as facilitators of this interaction, but their integration and linkage to the cell cycle remain unclear. Since these pathways and their interactions are complex, a greater understanding can only be gained using mathematical models. This work is done in collaboration with Prof. Francesco Falciani at the Institute of Integrative Biology in Liverpool.

Mathematical modelling applications to plant science projects other than circadian rhythms

Aside from the circadian rhythms, there are numerous examples in plant sciences where dynamical systems theory has been successfully applied to study gene regulatory networks and hormonal pathways. Interested student should get in touch to discuss further options.

Some relevant literature:

  1. Vimont et al. (2018), Hormonal balance finely tunes dormancy status in sweet cherry flower buds, DOI:10.1101/423871, BiorXiv.
  2. Jung, Domijan et al. (2016), Phytochomes function as thermosensors in Arabidopsis, Science.

Dr David Lewis

The influence of turbulence on plankton population dynamics

The objective of this project would be to try and develop new population models, which, for the first time, will seek to include the effects of the local turbulent environment. To a certain extent this is made easier by the very small size of a typical plankton, which ensures that on the scale of an individual plankton the turbulence can be assumed to be homogeneous and isotropic. However, one objective would be to examine how much larger scale features of the turbulent flow influence the development of a large population of an individual species. This would be done by building upon existing plankton population models in the literature, combined with a working model of the turbulent flow field.

The influence of prey distribution on plankton predation

The role of wave induced Coriolis-Stokes forcing on the wind driven mixed layer

Dr Kieran Sharkey

Theoretical epidemiology/ population dynamics

Developing a better understanding of the mathematical ideas underpinning the modelling of epidemics and how these ideas relate to each other.

Applied epidemiology/ population dynamics

Mathematical modelling to gain insights into the spread of epidemics in real systems such as livestock industries and humans.

Mathematics of networks

Developing mathematical approaches to understand and control the dynamics on and of networks.

Systems biology

Mathematical modelling of biochemical networks to understand biological function.

Dr Bakhti Vasiev

Mathematical modelling of embryogenesis

This PhD project is devoted to mathematical study of the mechanisms of differentiation and migration of cells taking place during embryogenesis. The main paradigm here is that there are biochemical substances (morphogens), which are produced/decayed at different rates in different cells, form concentration gradients. The concentration of morphogens in turn cause differentiation and migration of cells and results to formation of different tissues. The evolution of the entire system is driven by feedback mechanisms and this project is about identifying feedback loops involved in development of embryonic tissues. The project involves development of models, represented by partial differential equations, analysis of these equations and programming in C++/Matlab for numerical integration of model equations. 

Pattern formation in excitable systems with applications to developmental biology

Excitable media are characterized by homogeneous steady state, where over-threshold disturbances can result to formation of propagating waves or localized structures.  Both types of solutions are important for mathematical studies in developmental biology. This project is devoted to the analysis of solutions, their types and bifurcations in models describing excitable dynamics. This research has various applications in developmental biology and can, for example, be used for a general explanation of developmental processes in populations of unicellular organisms. 

Spatio-temporal dynamics in heterogeneous populations of interacting bacteria

This project is focused on mathematical studies of interference competition between growing bacterial populations. The outcome of such competition is commonly manifested by spatial distribution of bacteria and reflect such phenomena as prevention of colonisation, displacement of the competitor or coexistence in spatially separated patches. The project involves development of models, discrete (represented by cellular automata) as well as continuous (represented by systems of partial differential equations). Continuous model are used to reveal general conditions which should be satisfied for one of populations to take over or for their coexistence. Discrete model will be used for the analysis of cellular events in smaller populations (i.e. up to 105 bacteria) where behaviour of each bacterial species and its interactions will be modelled by phenomenological rules. The project aims a design of medical treatment based on competition between toxic/nontoxic bacteria as well as understanding biophysical mechanisms of biofilm formation.

Mathematical modelling of the dynamics of aging and mortality in human populations

This project is focused on mathematical analysis of mortality data for human populations and development of mechanistic models of aging reproducing these data. In this project we consider human population as consisting of subpopulations with different aging and mortality characteristics and the latter being affected by stochastic factors. The role of heterogeneity and stochasticity on the net rate of mortality is studied by means of mathematical modelling and analysis. Mechanistic models of aging will be represented by differential equations with stochastic terms. These models will be studied analytically and integrated numerically using C++/Matlab programming. The project aims an impact to the improvement of policies and medical services for an increase of lifespan in human populations.


Mathematical Biology

EPSRC Liverpool Centre for Mathematics in Healthcare