UCSB Applied Math/PDE/Data Science Seminar

Abstract: Dynamic organization of the cytoskeletal filaments and rodlike proteins in the cell membrane and other biological interfaces occurs in many cellular processes, including membrane transport, and morphogenesis. Previous modeling studies have considered the dynamics of a single rod on fluid planar membranes. We extend these studies to the more physiologically relevant case of a single filament moving in a spherical membrane. First, we assume that the membrane is surrounded by 3D Newtonian fluids on the interior and the exterior. We use slender-body theory to compute the translational drag of the filament’s drag along its axis and in perpendicular direction, and its rotational drag as function of membrane viscosity, surrounding 3D fluid viscosity, membrane radius and the filament’s length. We find that the boundedness of spherical geometry gives rise to flow confinement effects that increase in strength with increasing the ratio of the filament’s length to membrane radius (L/R). These confinement flows result only in a mild increase in filament’s drag along its axis and its rotational drag. In contrast, we find that the drag in the perpendicular direction increases sharply with the filament’s length, when L/R > 1. Next, we extend these calculations to supported membranes. We assume the filament is embedded in the outer leaflet of the bilayer, as a model for monotopic proteins. The inner leaflet is supported by a rigid spherical boundary, and the two leaflets are coupled by a frictional force. We discuss the scaling behavior of translational and rotational drag coefficients at different asymptotic limits and compare these results against those of a freely suspended membrane. Finally, we consider the self-organization of a dilute suspension of active rods. Using stability analysis and continuum simulations, we show that the coupling between the membrane’s tangential flows to 3D bulk flows introduces several novel features that are absent in active suspensions in free space 3D and 2D geometries. Specifically, we show that pusher rods undergo a finite wavelength nematic order transition at sufficiently high activities and densities. The wavelength of the ordered domains decreases with increasing the strength of coupling between the membrane and 3D flows. 

Host: Paul Atzberger

Abstract: We propose an actor-critic framework to solve the time-continuous stochastic optimal control problem. A least square temporal difference method is applied to compute the value function for the critic. The policy gradient method is implemented as policy improvement for the actor. Our key contribution lies in establishing the global convergence property of our proposed actor-critic flow, demonstrating a linear rate of convergence. Theoretical findings are further validated through numerical examples, showing the efficacy of our approach in practical applications. 

Host: Ruimeng Hu

Abstract: Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modelling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. Nevertheless, little is known about the rate of convergence of this asymptotic. In this talk, I will address the question of estimating the rate in the presence of external drifts. In a joint work with Tomasz Dębiec and Benoit Perthame, we computed the rate in a negative Sobolev norm for generic bounded potentials, while in a work in progress with Alpár Mészáros and Filippo Santambrogio, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials. I will present these two results, which hold both for the barotropic pressure law (hence the porous medium equation) and for a singular pressure law with density constraints. 

Host: Matt Jacobs 

Abstract: We consider the dynamics of a closed inextensible interface immersed in a 2D Stokes fluid, a model that has been used for 2D simulations of vesicle dynamics.  In this model, a 1D closed interface exerts a bending force and the interface is subject to an inextensibility constraint. As part of the problem, one must solve for the unknown tension that ensures membrane inextensibility. Given a force exerted on the interface, we first show that the problem of determining the tension is solvable if and only if the interface is not a circle.  Using this result, we prove local-in-time well-posedness for this problem. We will finally discuss open questions and future directions.  

Host: Paul Atzberger 

Abstract: Molecular motors are proteins in biological cells which perform various sorts of biophysical work.  The microscale physics of their operation motivates inherently stochastic models, both for their binding kinetics as well as for their spatial motion.  The molecular motor kinesin, on which we will focus, carries a cargo load on its tail while its head walks along microtubule filaments.  We revisit two paradigms of cooperative action by kinesin molecular motors through analysis of coupled stochastic models for the biophysical dynamics.  First, we extend consideration of gliding assays to a situation where microtubules are crosslinked while being crowdsurfed by immobilized kinesin.  Second, for two dissimilar types of kinesin transporting a common cargo, we provide approximate analytical characterizations for how the motors cooperate in carrying the cargo, with attention to incorporating slack in the tether connecting the motor with the cargo.  The methodology combine multiscale asymptotic analysis, renewal theory, and first passage time calculations. 

Host: Paul Atzberger 

Abstract: In this talk, we explore a novel Immersed Boundary (IB) reformulation for flows with rigid bodies, called the Immersed Boundary Double Layer (IBDL) method. The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. However, when applied to problems with rigid structures, the method becomes prohibitively inefficient without preconditioning since it results in the need to solve an equation that corresponds to a poorly conditioned first-kind integral equation at every step in time. In this talk, we explore the IBDL method, a new well-conditioned IB formulation for boundary value problems. We explore the formulation and improved efficiency of the method for both Stokes and Navier-Stokes equations on various PDE domains. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions. This development, applied to the Helmholtz equation, is especially useful in the solution of reaction-diffusion equations on arbitrary domains. 

Host: Paul Atzberger