L4

Multicomponent fluids are mixtures of distinct chemical species (i.e. components) that interact through complex physical processes such as cross-diffusion and chemical reactions. Additional physical phenomena often must be accounted for when modelling these fluids; examples include momentum transport, thermality and (for charged species) electrical effects. Despite the ubiquity of chemical mixtures in nature and engineering, multicomponent fluids have received almost no attention from the finite element community, with many important applications remaining out of reach from numerical methods currently available in the literature. This is in spite of the fact that, in engineering applications, these fluids often reside in complicated spatial regions -- a situation where finite elements are extremely useful! In this talk, we present a novel class of high-order finite element methods for simulating cross-diffusion and momentum transport (i.e. convection) in multicomponent fluids. Our model can also incorporate local electroneutrality when the species carry electrical charge, making the numerical methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We discuss challenges that arise when discretising the partial differential equations of multicomponent flow, as well as some salient theoretical properties of our numerical schemes. Finally, we present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the transient evolution of a lithium-ion battery electrolyte in a Hull cell electrode.

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.

The motion of a particle suspended in a fluid flow is governed by hydrodynamic interactions. In this talk, I will present the rich nonlinear dynamics that arise from particle-fluid interactions for two different setups: (i) passive particles in 3D channel flows where fluid inertia is important, and (ii) active particles in 3D channel flows in the Stokes regime (i.e. without fluid inertia).

For setup (i), the particle-fluid interactions result in focusing of particles in the channel cross section, which has been exploited in biomedical microfluidic technologies to separate particles by size. I will offer insights on how dynamical system features of bifurcations and tipping phenomena might be exploited to efficiently separate particles of different sizes. For setup (ii), microswimmers routinely experience unidirectional flows in confined environment such as sperm cells swimming in fallopian tubes, pathogens moving through blood vessels, and microrobots programed for targeted drug delivery applications. I will show that our minimal model of the system exhibits rich nonlinear and chaotic dynamics resulting in a diverse set of active particle trajectories.

We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where one or more bulk diffusing species are coupled to nonlinear intracellular
reactions that are confined within a disjoint collection of small compartments. The bulk species are coupled to the spatially segregated
intracellular reactions through Robin conditions across the cell boundaries. For this compartmental-reaction diffusion system, we show that
symmetry-breaking bifurcations leading to stable asymmetric steady-state patterns, as regulated by a membrane binding rate ratio, occur even when
two bulk species have equal bulk diffusivities. This result is in distinct contrast to the usual, and often biologically unrealistic, large
differential diffusivity ratio requirement for Turing pattern formation from a spatially uniform state. Secondly, for the case of one-bulk
diffusing species in R^2, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular
oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the
``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various
spatial arrangements of cells using the Kuramoto order parameter. This theoretical modeling framework, relevant when spatially localized nonlinear
oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on
networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).

Our visual perception of the world heavily relies on sophisticated and delicate biological mechanisms, and any disruption to these mechanisms negatively impacts our lives. Age-related macular degeneration (AMD) affects the central field of vision and has become increasingly common in our society, thereby generating a surge of academic and clinical interest. I will present some recent developments in the mathematical modeling of the retinal pigment epithelium (RPE) in the retina in the context of AMD; the RPE cell layer supports photoreceptor survival by providing nutrients and participating in the visual cycle and “cellular maintenance". Our objectives include modeling the aging and degeneration of the RPE with a mechanistic approach, as well as predicting the progression of atrophic lesions in the epithelial tissue. This is a joint work with the research team of Prof. M. Paques at Hôpital National des Quinze-Vingts.

The rheological (deformation and flow) properties of biological tissues  are important in processes such as embryo development, wound healing and 
tumour invasion. Indeed, processes such as these spontaneously generate  stresses within living tissue via active process at the single cell level. 
Tissues are also continually subject to external stresses and deformations  from surrounding tissues and organs. The success of numerous physiological 
functions relies on the ability of cells to withstand stress under some conditions, yet to flow collectively under others. Biological tissue is 
furthermore inherently viscoelastic, with a slow time-dependent mechanics.  Despite this rich phenomenology, the mechanisms that govern the 
transmission of stress within biological tissue, and its response to bulk deformation, remain poorly understood to date.

This talk will describe three recent research projects in modelling the rheology of biological tissue. The first predicts a strain-induced 
stiffening transition in a sheared tissue [1]. The second elucidates the interplay of external deformations applied to a tissue as a whole with 
internal active stresses that arise locally at the cellular level, and shows how this interplay leads to a host of fascinating rheological 
phenomena such as yielding, shear thinning, and continuous or discontinuous shear thickening [2]. The third concerns the formulation of 
a continuum constitutive model that captures several of these linear and nonlinear rheological phenomena [3].

[1] J. Huang, J. O. Cochran, S. M. Fielding, M. C. Marchetti and D. Bi, 
Physical Review Letters 128 (2022) 178001

[2] M. J. Hertaeg, S. M. Fielding and D. Bi, Physical Review X 14 (2024) 
011017.

[3] S. M. Fielding, J. O. Cochran, J. Huang, D. Bi, M. C. Marchetti, 
Physical Review E (Letter) 108 (2023) L042602.

Circadian clocks generate autonomous daily rhythms of gene expression in anticipation of daily sunlight and temperature cycles in a variety of organisms. The simples and best characterised of all circadian clocks in nature is the cyanobacterial clock, the core of which consists of just 3 proteins - KaiA, KaiB and KaiC - locked in a 24-h phosphorylation-dephosphorylation loop. Substantial progress has been made in understanding how cells generate and sustain this rhythm, but important questions remain: how does the clock maintain resilience in the face of internal and external fluctuations, how is the clock coupled to other cellular processes and what dynamics arise from this coupling? We address these questions using an interdisciplinary approach combining time-lapse microscopy and modelling. In this talk, I will first characterise the clock's free-running robustness and explore how the clock buffers environmental noise and genetic mutations. Our stochastic model predicts how the clock filters out such noise, including fast light fluctuations, to keep time while remaining responsive to environmental shifts, revealing also that the wild-type operates at a noise optimum. Next, I will focus on how the clock interacts with the other major cellular cycle, the cell division cycle. Our single-cell data shows that the clock couples to the division rate and expression of cell cycle-dependent factors using both frequency modulation and amplitude modulation strategies, with implications for cell growth and cell size control. Our findings illustrate how simple systems can exhibit complex dynamics, advancing our understanding of the interdependency between gene circuits and cellular physiology.  
 

The question can be formulated as a statistical hypothesis asserting that the distribution of the shapes of closed curves representing outlines of cell nuclei in a spatial domain is independent of the distribution of their locations. The key challenge in developing a procedure to test the hypothesis from a sample of spatially indexed curves (e.g. from an image) lies in how symmetries in the data are accounted for: shape of a curve is a property that is invariant to similarity transformations and reparameterization, and the shape space is thus an infinite-dimensional quotient space. Starting with a convenient geometry for the shape space developed over the last few years, I will discuss dependence measures and their estimates for spatial point processes with shape-valued marks, and demonstrate their use in testing for spatial independence of marks in a breast cancer application.