This will be an introductory lecture on the auditory system for mathematicians. I'll start with sound waves, discuss the outer, middle, and inner ears, the VIIIth nerve, and up into the brain stem. Then I'll outline work on sound localization, stochasticity, hyperacuity, and the ocular-vestibular reflex. The emphasis will be on mathematical issues and the need for new conceptual constructs.

The talk is based on three chapters from a popular science book that I've just delivered to the publisher. The main aim of the book - and the talk - is to illustrate the broad range of mathematical ideas and methods now being used to provide insight into biological questions. The samples in the talk involve knot theory, multidimensional geometry, group theory (symmetry), and networks. As befits a popular science book, the mathematical technicalities will be omitted.

1. Virus from the Fourth Dimension (work of Reidun Twarock and others): Use of multidimensional geometry to understand the structure of virus protein coats.
2. Networking Opportunities (work of Atsushi Tero and others): Biological networks, including how to use slime mould to design the Tokyo railroad system.
3. Knots in DNA (work of Nicholas Cozzarelli and others): How the twisted topology of the DNA double helix can be used to work out how topoisomerases act, in particular Tn3 resolvase.

Identifying mechanisms for the onset of cardiac arrhythmias is an important component of ongoing research in electrophysiology. Mathematically, abnormal rhythms such as ventricular tachycardia and fibrillation can be identified with spiral waves and spatiotemporal chaos, respectively. Understanding the precursors of such arrhythmias is possible even if we restrict ourselves to an idealized one-dimensional fiber of cardiac cells. In this presentation, I will use asymptotic methods to reduce a standard Hodgkin-Huxley type PDE model of a cardiac fiber to a system of ODEs which is amenable to mathematical analysis. My calculations exploit a particular feature of cardiac tissue known as electrical restitution: the speed and duration of cardiac action potentials depends upon how [locally] well-rested the tissue is. The kinematic model that I will introduce is far less computationally expensive than standard PDE models, making it feasible to run repeated numerical experiments. I will discuss one such experiment: the use of far-field pacing and feedback control to terminate chaos.

Dynamics, or behavior of solution of nonlinear reaction-diffusion system is deeply related to that of its kinetic problem. In this lecture we will use the Lotka-Volterra competition model as an example to illustrate some connection between the two. The asymptotic behavior of the principal eigenvalue of cooperative system, as diffusion rates approaches zero, which plays a major role in our analysis, will also be discussed.

A complete classification for the global dynamics of a Lotka-Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.

The cancer stem cell hypothesis postulates that only a subpopulation of cancer cells in a tumor is capable of initiating, sustaining, and re-initiating tumors, while the bulk of the population comprises non-stem cancer cells that lack tumor initiation potential. If cancer stem cells are the engine of tumor progression, then it follows that non-stem cancer cells may naturally compete with and even hinder tumor-initiating cell dynamics. While cancer stem cells have received a lot of attention, the contribution to total tumor growth kinetics of their non-stem counterparts have been less thoroughly investigated, if not ignored outright. If cancer stem cells are the sub-population in a tumor that drives progression, then how do the non-stem cancer cells contribute to tumor fate? Clearly, elucidating such a strategy will require a means to identify and track the complicated interactions among the competing subpopulations. In this seminar I discuss experimentally validated hybrid mathematical-computational agent-based models that have proven to be powerful tools to quickly simulate complex, emerging system dynamics that depend on the behavior of the two cell populations and their interactions. I show that the interactions of these two phenotypically distinct populations can provoke various non-linear growth kinetics in the emerging tumor including prolonged phases of tumor dormancy and self-metastatic expansion. This model will then be used to specifically simulate cell kinetics in glioblastoma growth and radiation response, parameterized and validated with experimental data of the U87-MG human glioblastoma cell line. Simulations are performed to estimate glioma stem cell (GSC) symmetric and asymmetric division rates and explore potential mechanisms for increased GSC fractions after irradiation. Simulations reveal that, in addition to their higher radioresistance, a shift from asymmetric to symmetric division or a fast cycle of GSCs following fractionated radiation treatment is required to yield results that match experimental observations. We hypothesize a constitutive activation of stem cell division kinetics signaling pathways during fractionated treatment, which contributes to the frequently observed accelerated repopulation after therapeutic irradiation.

Models have long been of interest in the biological sciences, but traditionally very different kinds of models have been used for different purposes: simple models of ecological ideas to generate general insights, complex simulation models to explore whether such insights might stand up in the face of reality, and simple statistical models (often referred to as ‘methods’ or ‘routines’ – think ANOVA, PCA) were used to extract conclusions from data. And although all three kinds of models have occasionally been used to make predictions (for example, about dynamics into the future, or the response of something to an intervention), none are really suitable for this purpose because they either lack constraints from data (the first two kinds of model) or are unlikely to capture the non-linear dynamics observed in reality (the third kind). But thanks to relatively recent advances in computing power, data availability and computational statistics, it is now becoming possible to pursue a ‘joined up’ approach to modelling all across the biological sciences. This approach merges ideas-rich models with data via Likelihood / Bayesian approaches, to both increase our understanding of nature, and to begin to enable us to predict it.

In the first hour of this virtual seminar, Dr. Drew Purves will give an extended overview of the above, introducing the notion of Likelihood and Bayesian approaches, and try to provide an honest overview of the pros and cons of these approaches. Dr. Purves will explain ‘Metropolis Hastings MCMC sampling’, why it’s exciting (it is – really!) and introduce a particular sampler called ‘Filzbach’ that his group has developed, giving examples of the environmental science that it has enabled in his group (e.g. modelling the global carbon cycle) and in their sister group (e.g. modelling MHC Class I in the immune system, synthetic biology). Since many of you will be interested in environmental problems, Dr. Purves will also provide a tour of ‘FetchClimate’ an unnamed prototype browser-based tool for doing the whole data-model-predictions pipeline (available here if you want to try before hand: http://fetchclimate.cloudapp.net/ ).

In the second half, Dr. Mark Vanderwel, formerly a postdoc at MSRC in Cambridge and now at the University of Florida, will take you through every step of some example analyses using ‘FilzbachR’, which allows you to run your whole analysis from R, and even write the model itself in R. This seminar assumes some basic knowledge in R. By the end of the session you will feel equipped enough to at least try this approach on your own problems.

Directed cell migration, where cells migrate up/down chemical (e. g. chemokines, growth factors) or physical (e. g. stress, temperature) gradients, play important roles in a number of physiological processes; they include immune responses, tissue formation and cancer metastasis. In this talk, I will present work in my lab in understanding biophysical and biochemical mechanisms that cells use to migrate when subject to single or dual chemical gradients using an integrated experimental and theoretical modeling approach. Two examples will be given. First, I will describe how bacteria can sense chemical concentration gradients at a logarithmic scale; similar to sensory systems in high organism, such as human hearing and vision. I will also talk about how bacteria make movement decisions when subject to competing chemical gradients. Second, I will discuss about the roles of receptor-ligand binding kinetics in immune and cancer cell migration, and their implications in cancer metastasis.

I will discuss mathematical models of parts of liver metabolism that relate to public health including folate metabolism, neural tube defects, cancer chemotherapy, DNA methylation, colon cancer and folate supplementation, arsenic detoxification, and the toxicity of acetaminophen. What makes a model good? How do you determine parameters? How do you conduct biological experiments with models? As a mathematician, how do you figure out what problems to work on and what people to talk to? These hard questions are the subtext of the lecture.

Mathematical and computational neuroscience have contributed to the brain sciences by the study of the dynamics of individual neurons and more recently the study of the dynamics of electrophysiological networks. Often these studies treat individual neurons as points or the nodes in networks and the biochemistry of the brain appears, if at all, as some intermediate variables by which the neurons communicate with each other. In fact, many neurons change brain function not by communicating in one-to-one fashion with other neurons, but instead by projecting changes in biochemistry over long distances. This biochemical network is of crucial importance for brain function and it influences and is influenced by the more traditional electrophysiological networks. Understanding how biochemical networks interact with electrophysiological networks to produce brain function both in health and disease poses new challenges for mathematical neuroscience.