April 9, 2024, 1:00-2:00pm

Speaker: Dr. Christopher Miles (University of California, Irvine)

Title: Decoding spatial stochastic RNA dynamics from static imaging data with point process inference 

Abstract: Advances in microscopy can now provide snapshot images of individual RNA molecules within a nucleus. Decoding the underlying spatiotemporal dynamics is important for understanding gene expression, but challenging due to the static, heterogeneous, and stochastic nature of the data. I will write down a stochastic reaction-diffusion model and show that observations of this process follow a spatial point (Cox) process constrained by a reaction-diffusion PDE. Inference on this data resembles a classical inverse problem but differs in the observations of individual particles rather than concentrations. We perform inference using variational Bayesian Monte Carlo with promising results. However, many open computational and modeling challenges remain in the development of scalable and extendable techniques for this inverse problem. This work is in collaboration with the Fangyuan Ding lab of Biomedical Engineering at UCI.

April 2, 2024, 1:00-2:00pm

Speaker: Haniyeh Fattahpour (Georgia State University) 

Title: Optimizing Cell Proliferation in Tissue Engineering: Interplay of Nutrient Dynamics, Scaffold Elasticity, and Cell Hunger Rate in Scaffold Channels.

Abstract: Tissue-engineering scaffolds contain channels lined by cells that allow nutrient-rich culture medium to pass through to encourage cell proliferation. Several factors have significant impacts on the tissue growth, including the nutrient flow rate and concentration in the feed, scaffold elasticity as well as cell properties. Recent studies have investigated these effects separately; however, in this work, we examine all of them simultaneously. Our objectives in this work are as follows: (i) developing a mathematical model describing the nutrient flow dynamics and concentration, scaffold elasticity and cell proliferation; (ii) solving the model and then simulating the cell proliferation process; (iii) optimizing the initial configuration of the scaffold channels to maximize the cell growth. The results of our study demonstrate that the rate of nutrient consumption by the cells (cell hunger rate) and the scaffold elastic compliance have an impact on tissue growth, with higher cell hunger rates leads to longer incubation periods, while scaffold elastic compliance slightly affects overall growth. Furthermore, decreasing the scaffold elastic compliance while maintaining a constant nutrient consumption rate results in an optimal funnel-shaped channel geometry, where the upper part of the channel is larger than the downstream, promoting enhanced tissue integration and functionality.

March 26, 2024, 1:00-2:00pm

Speaker: Dr. Radu Cimpeanu (Warwick Mathematics Institute) 

Title: From bouncing to making a splash: computational modelling of impact across scales

Abstract: The canonical framework of drop impact provides excellent opportunities to co-develop experimental, analytical and computational techniques in a rich multi-scale context. The talk will represent a journey across parameter space, as we address beautiful phenomena such as bouncing, coalescence and splashing, with a particular focus on scientific computing aspects and associated numerical methods.

To begin with, we consider millimetric drops impacting a deep bath of the same fluid. Experimental measurements of the droplet trajectory are compared directly to the predictions of a quasi-potential model, as well as fully resolved unsteady Navier-Stokes direct numerical simulations (DNS). Both theoretical techniques resolve the time-dependent bath interface shape, droplet trajectory, and droplet deformation. In the quasi-potential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to a set of coupled damped linear oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition, which we demonstrate to be an effective and efficient technique. The hybrid methodology has allowed us to unify and resolve interesting outstanding questions on the rebound dynamics of the multi-fluid system (Alventosa, Cimpeanu and Harris, JFM 957, 2023).

We then shift gears towards the much more violent regime of high-speed impact resulting in splashing, where a combination of matched asymptotic expansions grounded in Wagner theory and DNS allow us to produce theoretical predictions for the location and velocity of the ejected liquid jet, as well as its thickness (Cimpeanu and Moore, JFM 856, 2019). While the early-time analytical methodology neglects effects such as surface tension or viscosity (focusing on inertia instead), generalisations of the technique (Moore et al., JFM 882, 2020) and 3D extensions (ongoing work) will also be presented and validated against an associated computational framework. Should time allow it, recent results on coalescence and splashing for three-fluid setups (Fudge et al., PRE 104, 2021 and JCIS 641, 2023) will also be discussed.

March 19, 2024, 1:00-2:00pm

Speaker: Dr. Kristian Kiradjiev (University of Nottingham) 

Title: Modelling hormone distributions regulating root development

Abstract: Hormones are involved in many developmental processes in plants, including growth; however, how they are distributed within plant tissues is not yet fully understood. Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control development. In this talk, we present a series of mathematical models for transport of a specific growth hormone, gibberellin (GA), within plant roots. We begin with a multi-cellular mathematical model for GA transport, in which we consider both passive and active transport via the NPF membrane proteins. In addition, we consider the effect of the subcellular vacuole compartment on the transport dynamics. The model predictions reveal how the clade of NPF transporters control the spatio-temporal GA distribution, and how transport into the vacuole affects GA diffusion, enabling GA to be stored for later use.

We then use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to understand how sub-cellular compartments, growth and division affect the tissue-scale distribution. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cellular compartment. We allow the cells to grow at a rate that can depend both on space and time. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction–advection–diffusion equation and revealing how the effective diffusivity, effective velocity, and effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments can act as storage vessels, that significantly alter the tissue-scale transport. Furthermore, we show how cell growth and spatial variance across cell lengths induce an effective velocity at the tissue scale. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. Thus, the modelling provides unique insights into the GA distribution that regulates tissue patterning and plant root growth.

Using the upscaled model, we describe a more detailed model for the metabolic pathway of GA, incorporating the conversion between different GA forms. Finally, we discuss a hybrid three-dimensional model for GA transport combining the results from the discrete and continuum approaches to gain insight into hormone transport at the root scale. Whilst focussing here on GA, our model is general enough to be applied to transport of other hormones and substances, and in different types of cells. Thus, our research demonstrates how mathematical modelling enables us to test biological hypotheses and gain unique insights into experimental observations.

March 5, 2024, 1:00-2:00pm

Speaker: Dr. Amin Rahman (University of Washington) 

Title: Bouncing droplets as a damped-driven system

Abstract: Damped-driven systems are ubiquitous in science, however the damping and driving mechanisms are often quite convoluted.  This talk presents a fluidic droplet on a vertically vibrating fluid bath as a damped-driven system.  Fortunately, the damping and driving in the present system are relatively segregated. By separating the two mechanisms, we show that the droplet exhibits similar bifurcations present in other more complex damped-driven systems.  In this investigation we study a fluidic droplet in an annular cavity with the fluid bath forced above the Faraday wave threshold.  We model the droplet as a kinematic point particle in air and as inelastic collisions during impact with the bath.  In both experiments and the model the droplet is observed to chaotically change velocity with a Gaussian distribution leading to diffusion-like behavior.  In addition, the forcing above the Faraday wave threshold on the fluid bath simplifies the wave dynamics to that of a standing wave, which allows us to explicitly segregate the damping and driving mechanisms.  The energy gain comes from the kinematics of the droplet between impacts, and the energy loss comes from the hydrodynamic damping at impact.  We show that this energy gain-loss formulation exhibits dynamical behavior canonical to a wide range of damped-driven systems.  The bifurcations and route to chaos present in the droplet system reveals analogies with other well-studied systems from optics, detonation, and electronics. This also hints at analogies with many more systems that have yet to be analyzed in detail, and paves a framework with which other systems can be analyzed.

February 20, 2024, 1:00-2:00pm

Speaker: Dr. Jimmie Adriazola (Southern Methodist University)

Title: Optimal Control of Dispersive Waves

Abstract: This talk will address applications of optimal control theory to dispersive wave equations. We will discuss three technologically relevant experiments, each having its own unique challenge and physical setting including ultra-cold quantum fluids trapped by an external field, paraxial light propagation through a gradient index of refraction, and light propagation in periodic photonic crystals. In each of these settings, the physics can be modeled by dispersive wave equations, and the technological objective is to design the external trapping fields or propagation media such that a high fidelity or degree of coherence of the wave phenomena is achieved. Finding an optimal control can be numerically challenging since this amounts to solving a somewhat high-dimensional nonconvex optimization problem. To efficiently address the nonconvex nature of these problems, the program used is a global, nonconvex search based on a genetic algorithm which is then accelerated by a fast local method based on a projected gradient descent. This methodology is specifically tailored toward maintaining feasibility, via Tikhonov regularization, of implementing the computationally constructed control policies in technologically relevant settings.