MFBM Subgroup Contributed Talks

Gregory Szep

King's College London

"Parameter Inference with Bifurcation Diagrams"

Estimation of parameters in differential equation models can be achieved by applying learning algorithms to quantitative time-series data. However, sometimes it is only possible to measure qualitative changes of a system in response to a controlled condition. In dynamical systems theory, such change points are known as bifurcations and lie on a function of the controlled condition called the bifurcation diagram. In this work, we propose a gradient-based semi-supervised approach for inferring the parameters of differential equations that produce a user-specified bifurcation diagram. The cost function contains a supervised error term that is minimal when the model bifurcations match the specified targets and an unsupervised bifurcation measure which has gradients that push optimisers towards bifurcating parameter regimes. The gradients can be computed without the need to differentiate through the operations of the solver that was used to compute the diagram. We demonstrate parameter inference with minimal models which explore the space of saddle-node and pitchfork diagrams and the genetic toggle switch from synthetic biology. Furthermore, the cost landscape allows us to organise models in terms of topological and geometric equivalence.

Sta Léa

University of Leeds, United Kingdom

" IL-7R mathematical modelling: algebraic expressions for amplitude and EC50"

Effector T cells rely on the cytokine IL-7 to receive receptor-mediated signalling for their survival. The IL-7 receptor (IL-7R), composed of the common gamma chain and the specific alpha chain, is also associated with the kinase JAK3, which triggers its signalling pathway. Recently, study of cell-to-cell variability and flow cytometry data yielded a seemingly paradoxical observation: increased expression of gamma chains reduces the IL-7 response. We introduce a mathematical model of cytokine IL-7 and IL-7R signalling that provides an explanation for this empirical observation. Our results show the formation of dummy complexes (those receptors that are bound to ligand but not to the JAK3 kinase, and are thus, unable to signal) and indicate that the balance between the number of IL-7R subunits in one cell is crucial for optimal signalling. We make use of a method in algebraic geometry, the Groebner basis, to compute exact analytical expressions for the maximum IL-7 response (or amplitude) and for the half-maximal effective concentration of ligand (EC50) of our mathematical models of cytokine-receptor signalling. While predicted amplitudes agree with the experimental data, measurements of EC50 exhibit more complicated behaviour than we have managed to capture with a variation of our IL-7R model.

Johannes Borgqvist

Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford

"Symmetry methods for model-construction and analysis in the context of collective cell migration"

Mathematical modelling is a vital tool in coping with complexity on numerous spatial and temporal scales, and a key goal of modelling is to be able to predict future outcomes using model analysis and simulation. The challenge for this dream scenario is the difficulty of validating a particular model, and it is often achieved by attempting to fit the model to observed data. However, often there are multiple candidate models available which renders the task of knowing which description is “correct” very difficult. In order to encode physical properties of the studied system in the construction phase of a model, a novel mathematical technique called symmetry methods can be used. The method of symmetries originates from mathematical physics, and they are transformations that encode physical properties, often formulated as conservation laws. Symmetries have been used with huge success in theoretical physics, but are relatively unexplored in a biological context. Here, the application of symmetries for finding analytical solutions to partial differential equation models of cell migration is showcased, as well as a methodology for model selection. Finally, the difficulties of finding symmetries of large biological models in an automated fashion are discussed.

Remus Stana

University of Leeds

"Diffusion in a domain with inclusion"

Many cells of the immune system have molecules which are produced in the nucleus and these move under the influence of diffusion until they reach the outer membrane of the cell. Depending on the type of molecule, they might also diffuse on the surface of the cell until either a certain period of time has passed or the molecule forms a complex. After either of these events the molecules re-enter the cytoplasm and diffuse until they are absorbed by the nucleus. We are interested in the first passage properties of the molecules. For this purpose, we derive an analytic expression for the Green's function of the Laplace equation for a domain bounded by non-concentric surfaces in two dimensions and three dimensions subject to mixed boundary conditions. Utilizing the Green's function we derive an exact expression for the mean time for a Brownian molecule to return to the nuclear surface given that it hit the cellular surface and compare with previous results in the literature. Furthermore, using the Green's function we calculate an exact formula for the hitting density of molecules on the cellular surface and compare it with numerical results.