Mathematical Biology Seminar  Fall 2020
Seminars are held on Tuesdays at 2:30PM on Webex, unless noted otherwise.
Seminars are held on Tuesdays at 2:30PM on Webex, unless noted otherwise.
For questions about the seminar schedule, please contact James Maclaurin
Date  Speaker, Affiliation, and Title 

September 8 
Patrick Murphy, Rice University Modeling Rapid Diffusion State Switching During Cellular Polarization of a C. Elegans Zygote Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. Singleparticle tracking experiments have recently found that C. elegans zygotes rely on spacedependent switching diffusivities to form intracellular gradients during cell polarization. We analyze a model of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how the presence of switching diffusivities modifies the standard theory and show that spacedependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations. We further consider how this mechanism of switching diffusive states interacts with a locally varying periodic microstructure in the cell, and use homogenization theory to show that at the typical cellular scales involved, such a microstructure does not need to be resolved in fine detail in order to accurately capture the dynamics of the system. 
September 22

Gregory Handy, University of Chicago Digging through DiRT: Investigating how Trap Recharge Time Influences the Statistics of Particle Diffusion Many diverse biological systems are described by randomly moving particles that can be captured by traps in their environment. Examples include neurotransmitters diffusing in the synaptic cleft before binding to receptors, the delivery of nanoparticles to targeted receptors, and prey roaming an environment before being captured by predators. We will investigate this stochastic process, referred to as diffusion with recharging traps (DiRT), in multiple ways: by considering the full stochastic process, via a continuoustime Markov process with discrete states approximation, and a with deterministic PDE approximation. Each framework yields interesting insights into the DiRT process, such as identifying the conditions under which trap recharge time can drastically affect key statistics (e.g., the average number of captured particles). 
October 6 
Tom Wooley, University of Cardiff Patterns, Cellular Movement and Brain Tumours I present three pieces of work that illustrate the power of mathematics as a tool for understanding biology. Although the applications appear to be disparate the underlying mathematics is very similar. I begin by looking at theoretical and experimental pattern formation, with emphasis on whisker formation in mice. Here, reactiondiffusion equations are used to provide insights into how the wavelength of the whiskers are controlled. Next, I consider the phenomena of blebbing cells. Initially, I use a diffusion equation to understand the motion of muscle stem cells and illustrate how old cells fundamentally move differently to old cells. This is then extended to include solid mechanics, which allows us to link the structural properties of the cell to their motion. Finally, reactiondiffusion equations are used to understand the formation of brain tumours. Critically, the cells move at different speeds in white and grey matter, including this information can lead to very different migration patterns of the tumours. 
October 13 
Nicholas Russell, University of Delaware Phytoplankton Aggregations: A RunandTumble Model with Autochemotaxis A specific species of phytoplankton, Heterosigma akashiwo, has been the cause of harmful algal blooms (HABs) in waterways around the world causing millions of dollars in damage to farmed animals and destroying ecosystems. Developing a fundamental understanding of their movements and interactions through phototaxis and chemotaxis is vital to comprehending why these HABs start to form and how they can be prevented. We create a complex and biologically accurate mathematical and computational model reflecting the movement of an ecology of plankton, incorporating phototaxis, chemotaxis, and the fluid dynamics that may be affecting the flow. We present and analyze a succession of models together with a sequence of laboratory and computational experiments that inform the mathematical ideas underlying the model. Lastly, we discuss further experiments and research necessary for our continued insight into problems that we are encountering. 
October 27 
Tom Chou, University of California Los Angeles Dynamics of Structured Populations: From Aging Demographics to Cell Size Control I will review PDE models of population dynamics structured according to age, size, or added size. A few new models will be presented including deterministic descriptions of population control through delayed births and of cell division through sizer and adder mechanisms. We show how a softer, staggered birth policy can be effective in population control. In cellsize control, we show that an adder mechanism can lead to blowup of cell size. Finally, we develop stochastic counterparts to the classical deterministic aging dynamics theories. We show how the classic agedependent population models are connected to a hierarchy of equations for reduced probability distributions, the lowest order of which is a master equation for the total stochastic population. Differences in the stochastic description of birth through budding or splitting are explored. 
November 10 
Robert Rosenbaum, University of Notre Dame Balanced Neural Networks Abstract: TBA 
November 24 
Henry Shum, University of Waterloo Dynamics of Flagella Abstract: TBA 
Updated: October 13, 2020