Mathematical Biology Seminar - Fall 2020

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. Single-particle tracking experiments have recently found that C. elegans zygotes rely on space-dependent 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 space-dependent 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 continuous-time 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).            

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, reaction-diffusion 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, reaction-diffusion 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.

Nicholas Russell, University of Delaware

Phytoplankton Aggregations: A Run-and-Tumble 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.

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 cell-size control, we show that an adder mechanism can lead to blow-up of cell size.  Finally, we develop stochastic counterparts to the classical deterministic aging dynamics theories. We show how the classic age-dependent 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.

Robert Rosenbaum, University of Notre Dame

Spatiotemporal Dynamics and Reliable Computations in Recurrent Spiking Neural Networks

Randomly connected networks of spiking neuron models provide a parsimonious model of neural variability, but are notoriously unreliable for performing computations. We show that this difficulty can be overcome by incorporating the well-documented dependence of synaptic connection probability on the distance between neurons. Using a Fokker-Planck formalism, we show that spatially extended spiking networks exhibit symmetry-breaking, Turing-Hopf bifurcations to generate intricate, high-dimensional spatiotemporal patterns. The resulting dynamics can be trained to perform dynamical computations using a reservoir computing approach.

Henry Shum, University of Waterloo

Modeling Differences in Motility of Flagellated Bacteria near Walls

It is well known that bacteria can be found in almost any environment on Earth and are very diverse. In particular, there are many differences between species of flagellated bacteria in morphology and patterns of motility; some have a single flagellum while others have more than 20, some have almost spherical cell bodies while others have long, cylindrical or helical bodies. To probe the physical implications of, and understand reasons for, these variations, we develop models for the mechanical problem of bacterial swimming. We focus on the hydrodynamics of swimming in Newtonian fluid with rigid flagella. Earlier studies showed that morphology was important in determining the behaviour of bacteria near interfaces, such as the walls of a channel. We will describe the behaviour expected for a "canonical" model bacterium with a single flagellum, which resembles typical experimental observations, and contrast this with models of bacteria with two flagella. The bacterium Magnetococcus marinus has an unusual morphology that necessitates a two-flagellum model whereas other bacteria with multiple flagella can be qualitatively described by a one-flagellum model.