UCSB Applied Math/PDE/Data Science Seminar

Abstract: Elastic instabilities are abundant in natural and engineered structures across a wide range of scales, from supercoiled DNA and folded tissues to the waves of leaves and the petals of flowers. While much progress has been made over the last two centuries in understanding and predicting the equilibrium shapes of stressed materials, the non-equilibrium dynamics of buckling and wrinkling phenomena continues to pose interesting theoretical and computational challenges. In this talk, I will discuss our recent joint theoretical and experimental efforts to understand the evolution of elastic patterns that emerge from elastic and fluid dynamical instabilities. In the first part, by considering the evolution of wrinkle patterns of confined elastic membranes on fluid surfaces, I will show how integral constraints can slow down pattern selection dynamics, causing departures from self-similar behaviors frequently observed in fluid mechanics. In the second part, I will demonstrate how non-trivial buckling patterns may emerge under rapid quenching. Specifically, I will demonstrate how tuning external control parameter enables the targeted selection of specific buckling modes. This phenomenon, which is reminiscent of the Kibble-Zurek mechanism in continuous non-equilibrium phase transitions, promises novel approaches to dynamical pattern design.

Host: Björn Birnir

Abstract: In this talk, I will present the Nernst-Planck-Boussinesq (NPB) system, a coupled model that combines the Nernst-Planck equations with the Boussinesq approximation to describe ionic electrodiffusion in an incompressible fluid under non-isothermal conditions. A key feature of the NPB system is the nonlinear structure of the electromigration term, which is influenced by the reciprocal of the temperature. This distinguishes the NPB system from other electrodiffusion models, such as the Nernst-Planck-Navier-Stokes system, and introduces unique mathematical challenges. I will discuss several analytical properties of the system, focusing on the global existence of weak solutions in three dimensions, as well as the long-time behavior of these solutions, including their exponential convergence to steady states. This is a joint work with Elie Abdo and Ruimeng Hu.

Host: Ruimeng Hu

Abstract: Efforts to develop mathematical models often face a trade-off between interpretability and quantitative accuracy, which sometimes disfavors interpretability. Here we adopt an operational definition of interpretability, specifically that a model is described by an arrow diagram wherein each arrow corresponds to a positive effect or negative effect of one component upon a process, and fewer arrows is more interpretable than more arrows. We then develop a method to add flexibility — and thus accuracy in fitting data — to mathematical models by relaxing functional form assumptions, while constrained by the same arrow diagram and thus the same interpretability. We apply this method to the T cell, where quantitative models are particularly needed, in part because of ongoing efforts to engineer T cells as therapeutics. Recent experiments exposed T cells to pulsatile inputs and measured their frequency response, and found several nonlinear frequency responses: high-pass, low-pass, band-pass, and band-stop. Using our modeling approach with enhanced flexibility, we show that a simple signaling model quantitatively captures the frequency response of CD69 surface expression, one of the correlates of T cells activation, with accuracy within the experimental inter-replicate standard error of the mean. Specific qualitative behaviors map to specific parts of the arrow diagram, with qualitative interpretations: Band-pass behavior can be explained by refractory de-sensitizing circuit (we refer to this as “first-aid icing a wound”); Band-stop behavior can be explained by removal-inhibition (we refer to this as “roommate interrupts my studying”). Taken together, our results demonstrate the ability to achieve both quantitative prediction and interpretability in understanding cellular dynamics. Simple models may at first appear incapable of explaining complex data, but might indeed be able to by adding this modest flexibility.

Host: Paul J. Atzberger

Abstract: We present a comprehensive framework for the development of gradient flows of parameterization independent surface energies naturally expressed in terms of intrinsic quantities (curvature and metric). To this mix we add cartesian distance which allows adhesion-repulsion energies that guide folding flows of cellular organelles, and surface diffusion of embedded agents (eg proteins). Via a penalty method on membrane density, we derive a mechanism to generate locally incompressible flows of  “fluidic” membranes. In space dimension two, we show that stability analysis of surface patterns can be converted to an analysis of the second variation of the surface energy subject to the nonlinear constraints imposed on the first and second fundamental forms. We outline derivation of quasi steady dynamics, making application to adhesion-repulsion energies that guide folding flows of cellular organelles.

Host: Carlos Garcia-Cervera