# Mathematical Biology

## Multiscale multicellular modeling of tissue function and disease using CompuCell3D

Multiscale multicellular models combine representations of subcellular biological networks, cell behaviors, tissue level effects and whole body effects to describe tissue outcomes during development, homeostasis and disease. I will briefly introduce these simulation methodologies, the CompuCell3D simulation environment and their applications, then focus on a multiscale simulation of an acute primary infection of an epithelial tissue infected by a virus like SARS-CoV-2, a simplified cellular immune response and viral and immune-induced tissue damage. The model exhibits four basic parameter regimes: where the immune response fails to contain or significantly slow the spread of viral infection, where it significantly slows but fails to stop the spread of infection, where it eliminates all infected epithelial cells, but reinfection occurs from residual extracellular virus and where it clears the both infected cells and extracellular virus, leaving a substantial fraction of epithelial cells uninfected. Even this simplified model can illustrate the effects of a number of drug therapy concepts. Inhibition of viral internalization and faster immune-cell recruitment promote containment of infection. Fast viral internalization and slower immune response lead to uncontrolled spread of infection. Existing antivirals, despite blocking viral replication, show reduced clinical benefit when given later during the course of infection. Simulation of a drug which reduces the replication rate of viral RNA, shows that a low dosage that provides only a relatively limited reduction of viral RNA replication greatly decreases the total tissue damage and extracellular virus when given near the beginning of infection. However, even a high dosage that greatly reduces the rate of RNA replication rapidly loses efficacy when given later after infection. Many combinations of dosage and treatment time lead to distinct stochastic outcomes, with some replicas showing clearance or control of the virus (treatment success), while others show rapid infection of all epithelial cells (treatment failure). This switch between a regime of frequent treatment success and frequent failure occurs is quite abrupt as the time of treatment increases. The model is open-source and modular, allowing rapid development and extension of its components by groups working in parallel.

## Binocular Rivalry; Modeling by Network Structure

Binocular rivalry explores the question of how the brain copes with contradictory information. A subject is shown two different pictures – one to each eye. What images does the subject perceive? Results from rivalry experiments usually lead to alternation of percepts and are often surprising. Hugh Wilson proposed modeling rivalry in the brain by using structured networks of differential equations. We use Wilson networks as modeling devices and equivariant Hopf bifurcation as a tool to both post-dict and predict experimentally observed percepts. This work is joint with Casey Diekman, Zhong-Lin Lu, Tyler McMillen, Ian Stewart, Yunjiao Wang, and Yukai Zhao.

## Gaps of saddle connection directions for some branched covers of tori

TBA

## Real-time modelling of the COVID-19 epidemic: perspectives from British Columbia

The COVID-19 global pandemic has led to unprecedented public interest in mathematical modelling as a tool to understand the dynamics of disease spread and predict the impact of public health interventions. In this pair of talks, we will describe how mathematical models are being used, with particular reference to the British Columbia epidemic.

In the first talk, Prof. Caroline Colijn (Dept. of Mathematics, Simon Fraser University) will outline the key features of the British Columbia data and focus on how modelling has allowed us to estimate the effectiveness of the provincial response. In the second talk, Prof. Daniel Coombs (Dept. of Mathematics and Inst. of Applied Mathematics, University of British Columbia) will describe forward-looking modelling approaches that can provide some guidance as the province moves towards partial de-escalation of measures. Each talk will be 30 mins in length and followed by a question and discussion period.

For more details on the group's work and to contact the team, please visit https://bccovid-19group.ca/

## What is epithelial-mesenchymal plasticity and why is it important for metastasis?

Until very recently most cancer biologists operated with the assumption that the most common route to metastasis involved cells of the primary tumor transforming to a motile single-cell phenotype via complete EMT (the epithelial-mesenchymal transition). This change allowed them to migrate individually to distant organs, eventually leading to clonal growths in other locations. But, a new more nuanced picture has been emerging, based on advanced measurements and on computational systems biology approaches. It has now been realized that cells can readily adopt states with hybrid properties, use these properties to move collectively and cooperatively, and reach distant niches as highly metastatic clusters. This talk will focus on the accumulating evidence for this revised perspective, the role of biological physics theory in instigating this whole line of investigation, and on open questions currently under investigation.

## Modeling strict age-targeted mitigation strategies for COVID-19

In this talk, we will use compartmental models to examine the power of age-targeted mitigation strategies for COVID-19. We will present evidence that, in the context of strategies which end with herd immunity, age-heterogeneous strategies are better for reducing direct mortalities across a wide parameter regime. And using a model which integrates empirical data on age-contact patterns in the United States and recent estimates of COVID-19 mortality and hospitalization rates, we will present evidence that age-targeted approaches have the potential to greatly reduce mortalities and ICU utilization for COVID-19, among strategies which ultimately end the epidemic by reaching herd immunity. This is joint work with Maria Chikina.

## Multiple fission cycles in Chlamydomonas

In this talk I will present a "dynamical paradigm" for modeling networks of interacting genes and proteins that regulate every aspect of cell physiology. The paradigm is based on dynamical systems theory of nonlinear ODEs, especially one- and two-parameter bifurcation diagrams. I will show how we have used this paradigm to unravel the mechanisms controlling "multiple fission" cycles in the photosynthetic green alga Chlamydomonas. While most eukaryotic cells maintain a characteristic size by executing binary division after each mass doubling, Chlamydomonas cells can grow more than eight-fold during daytime before undergoing rapid cycles of DNA replication, mitosis and cell division at night, which produce up to 16 daughter cells. We propose that this unusual strategy of growth and division (which is clearly advantageous for a photosynthetic organism) can be governed by a size-dependent bistable switch that turns on and off a limit cycle oscillator that drives cells through rapid cycles of DNA synthesis and mitosis. We show that this simple ‘sizer-oscillator’ arrangement reproduces the experimentally observed features of multiple-fission cycles and the response of Chlamydomonas cells to different light-dark regimes. Our model makes unexpected predictions about the size-dependence of the time of onset of cell-cycle oscillations after cells are transferred from light to dark conditions, and these predictions are confirmed by single-cell experiments.

Collaborators: Stefan Heldt and Bela Novak (Oxford Univ) on the modeling; Fred Cross (Rockefeller Univ) on the experiments.

## In Progress COVID-19 modelling

A variety of strategies and approaches have been proposed, and implemented by governments, for COVID mitigation. In this presentation, I introduce some of these, briefly discuss some of the resulting difficulties - in particular in the context of the northern Netherlands, where I have been working most recently. We then take a preliminary look at the possibility of `targeted quarantine' . Many questions, both mathematical, clinical, logistical and ethical remain to be answered, and as such, this presentation will be closer to a discussion session than the usual Mathbio Works in progress seminars. All feedback appreciated and welcome

## Variation in the descent of genome: modeling and inference

In meiosis, DNA is copied from parents to offspring, so that individuals who share common ancestors may have identical DNA copies from those ancestors through repeated meiosis. This identical-by-descent (IBD) DNA underlies the similarities between relatives, at both the family level and at the population level. However, the process of meiosis is quite variable, and DNA is inherited generation-to-generation in large segments. The patterns of IBD genome among relatives are complex, and in remote relatives segments of IBD DNA are rare but not short. Modern genetic data on millions of markers across the genome allows estimation of shared DNA, but accurate estimation requires modelling the processes that give rise to these complex IBD patterns. IBD must be estimated jointly among individuals and across the genome. Pedigree information, if available, provides prior probabilities of IBD patterns. Where inferred IBD is discordant with pedigree information, there is potential to detect selection or other processes distorting the outcomes of the meiotic process.

**Speaker Biography:** Elizabeth Thompson received her B.A. and Ph.D. in mathematics from Cambridge University, UK. After postdoctoral work in genetics at Stanford University, she joined the mathematics faculty of the University of Cambridge in 1976. She was a Professor of Statistics at the University of Washington from 1985 until her (semi-) retirement in 2018. Her research is in the development of methods for model-based likelihood inference from genetic data on both humans and other species, including inference of relationships among individuals and among populations. Dr. Thompson has received an Sc.D degree from the University of Cambridge, the Jerome Sacks award for cross-disciplinary statistical research, the Weldon Prize for contributions to Biometric Science, and a Guggenheim fellowship. She is an honorary fellow of Newnham College, Cambridge, and an elected member of the International Statistical Institute, the American Academy of Arts and Sciences, and the US National Academy of Sciences.

## If Space Turned out to be Time: Resonances and Patterns in the Visual Cortex

When subjects are exposed to full field flicker in certain frequencies, they perceive a variety of complex geometric patterns that are often called flicker hallucinations. On the other had, when looking at high contrast geometric patterns like op art, shimmering and flickering is observed. In some people, flicker or such op art can induce seizures. In this talk, I describe a simple network model of excitatory and inhibitory neurons that comprise the visual area of the brain. I show that these phenomena are reproduced and then give an explanation based on symmetry breaking bifurcations and Floquet theory. Symmetric bifurcation theory also shows why one expects a different class of patterns at high frequencies from those at low frequencies.

On the other hand, the visual system is also very sensitive to specific spatial frequencies and this sensitivity can be pathological in the case of so-called pattern-senstive epilepsy. It has been shown that certain types of "op art"can cause visual discomfort. We show that the network that we used in flicker is also sensitive to spatially periodic inputs and suggest that a Hopf bifurcation instability is responsible for the discomfort and seizures.

#### Speaker Biography

Bard Ermentrout received his PhD in Theoretical Biology at the University of Chicago and was a postdoctoral fellow at the NIH from 1979-1982. He has been at the University of Pittsburgh since then. He is the author of over 200 papers and two books as well as the simulation package, XPPAUT. He is a Sloan Fellow and a SIAM Fellow and received the Mathematical Neuroscience Prize in 2015.