University of Oxford

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of quantitative data now being generated. This sets a new challenge for the field – to develop, calibrate and analyse new, biologically realistic models to interpret these data. In this talk I will showcase how quantitative comparisons between models and data can help tease apart subtle details of biological mechanisms, as well as present some steps we have taken to tackle the mathematical challenges in developing models that are both identifiable and can be efficiently calibrated to quantitative data.

Amy Kent

Multiscale Mathematical Models for Tendon Tissue Engineering

Tendon tissue engineering aims to grow functional tendon in vitro. In bioreactor chambers, cells growing on a solid scaffold are fed with nutrient-rich media and stimulated by mechanical loads. The Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences is developing a Humanoid Robotic Bioreactor, where cells grow on a flexible fibrous scaffold actuated by a robotic shoulder. Tendon cells modulate their behaviour in response to shear stresses - experimentally, it is desirable to design robotic loading regimes that mimic physiological loads. The shear stresses are generated by flowing cell media; this flow induces deformation of the scaffold which in turn modulates the flow. Here, we capture this fluid-structure interaction using a homogenised model of fluid flow and scaffold deformation in a simplified bioreactor geometry. The homogenised model admits analytical solutions for a broad class of forces representing robotic loading. Given the solution to the microscale problem, we can determine microscale shear stresses at any point in the domain. In this presentation, we will outline the model derivation and discuss the experimental implications of model predictions.

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Michael Negus

High-Speed Droplet Impact Onto Deformable Substrates: Analysis And Simulations

The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

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Edwina Yeo

Modelling of Magnetically Targeted Stem Cell Delivery

Targeting delivery of stem cells to the site of an injury is a key challenge in regenerative medicine. One possible approach is to inject cells implanted withmagnetic nanoparticles into the blood stream. Cells can then be targeted to the delivery site by an external magnetic field. At the injury site, it is of criticalimportance that the cells do not form an aggregate which could significantly occlude the vessel.We develop a model for the transport of magnetically tagged cells in blood under the action of an external magnetic field. We consider a system of blood and stem cells in a single vessel.  We exploit the small aspect ratio of the vessel to examine the system asymptotically. We consider the system for a range of magnetic field strengths and varying strengths of the diffusion coefficient of the stem cells. We explore the different regimes of the model and determine the optimal conditions for the effective delivery of stem cells while minimising vessel occlusion.

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Helen Zha

Mathematical model of a valve-controlled, gravity driven bioreactor for platelet production

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into~\textit{in vitro}~production of platelets. We model a novel platelet bioreactor described in Shepherd et al [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of unsteady Stokes flow coupled to Darcy. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of varying pressure head and valve dynamics on shear stress.

[1] Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., 2018. Structurally graduated collagen scaffolds applied to the ex vivo generation of platelets from human pluripotent stem cell-derived megakaryocytes: enhancing production and purity. Biomaterials.

We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives we show agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, we derive a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.

One of the fundamental examples of geometric representation theory is the Springer correspondence which parameterises the irreducible representations of the Weyl group of a lie algebra in terms of nilpotent orbits of the lie algebra and irreducible representations of the equivariant fundamental group of said nilpotent orbits. If you don’t like geometry this may sound entirely mysterious. In this talk I will hopefully offer a gentle introduction to the subject and present a preprint by Lusztig (2020) which gives an entirely algebraic description of the springer correspondence.

A Real representation of a $C_2$-graded group $H < G$ ($H$ an index two subgroup) is a complex representation of $H$ with an action of the other coset $G \backslash H$ (“odd" elements) satisfying appropriate algebraic coherence conditions. In this talk I will present three such Real representation theories. In these, each odd element acts as an antilinear operator, a bilinear form or a sesquilinear form (equivalently a linear map to $V$ from the conjugate, the dual, or the conjugate dual of $V$) respectively. I will describe how these theories are related, how representations in each are classified, and how the first generalises the classical representation theory of $H$ over the real numbers - retaining much of its beauty and subtlety.

Let $k$ be a field and $A$ a $k$-algebra. The classical Quillen's Lemma states that if $A$ if is equipped with an exhaustive filtration such that the associated graded ring is commutative and finitely generated $k$-algebra then for any finitely generated $A$-module $M$, every element of the endomorphism ring of $M$ is algebraic over $k$. In particular, Quillen's Lemma may be applied to the enveloping algebra of a finite dimensional Lie algebra. I aim to present an affinoid version of Quillen's Lemma which strengthness a theorem proved by Ardakov and Wadsley. Depending on time, I will show how this leads to an (almost) classification of the primitive spectrum of the affinoid enveloping algebra of a semisimple Lie algebra.

Polynomial rings $R[X]$ are a fundamental construction in commutative algebra, under which Hilbert's basis theorem controls a finiteness property: being Noetherian. We will describe the picture for the non-commutative world; this leads us towards other interesting finiteness conditions.

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will explore various ways in which this result can and can't be extended.

Empirical networks often exhibit different meso-scale structures, such as community and core-periphery structure. Core-periphery typically consists of a well-connected core, and a periphery that is well-connected to the core but sparsely connected internally. Most core-periphery studies focus on undirected networks. In this talk we discuss  a generalisation of core-periphery to directed networks which  yields a family of core-periphery blockmodel formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. Then we shall  focus on a particular structure consisting of two core sets and two periphery sets, and we introduce  two measures to assess the statistical significance and quality of this  structure in empirical data, where one often has no ground truth. The idea will be illustrated on three empirical networks --  faculty hiring, a world trade data-set, and political blogs.

This is based on joint work with Andrew Elliott, Angus Chiu, Marya Bazzi and Mihai Cucuringu, available at https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0783