## Seminars

The Center runs a roughly biweekly seminar series in which we invite researchers in mathematical biology to give a lecture, from around the country and beyond. Many of these seminar speakers are also long-term visitors to Penn, who will interact with a broad range of researchers across campus.

### Seminar speakers: Spring 2021

**January 26: Arnold Mathijssen (University of Pennsylvania)**

“Fluid mechanics of the respiratory system and active coating materials”

Our airways are continuously exposed to potentially harmful particles like dust and viruses. The first line of defence against these pathogens is a network of millions of cilia, whip-like organelles that pump flows by beating over a thousand times per minute. In this talk, I will first discuss the connection between local ciliary architecture and the topology of the flows they generate. We image the mouse airway from the sub-cellular (nm) to the organ scales (mm), characterising quantitatively its ciliary arrangement and the resulting flows. Interestingly, we find that disorder in the ciliary alignment can actually be beneficial for this pathogen clearance [1]. Second, I would also like to discuss how systems can be driven out of equilibrium by such active carpets. Combining techniques from statistical and fluid mechanics, I will demonstrate how we can derive the diffusivity of particles near an active carpet, and how we can generalise Fick’s laws to describe their non-equilibrium transport [2]. These results may be used for designing self-cleaning materials, much like our airways.

[1] Ramirez San-Juan, Mathijssen et al., “Multi-scale spatial heterogeneity enhances particle clearance in airway ciliary arrays”, Nature Physics 16, 958–964 (2020)

[2] Guzman-Lastra, Löwen & Mathijssen, “Active carpets drive non-equilibrium diffusion and enhanced molecular fluxes”, in press, Nature Communications (2021)

Note: We also have an informal discussion session after the seminar. Please stay in the Zoom seminar room to chat together with Professor Arnold Mathijssen!

**February 2: Hye-Won Kang (University of Maryland Baltimore County)**

“Stochastic Modeling of Reaction-Diffusion Processes in Biology”

Inherent fluctuations may play an important role in biochemical and biophysical systems when the system involves some species with low copy numbers. This talk will present the recent work on the stochastic modeling of reaction-diffusion processes in glucose metabolism. The first part of the talk introduces a compartment-based model for a simple glycolytic pathway using a continuous-time Markov jump process, which describes system features at different scales of interest. Then, we will see how the multiscale approximate method reduces the model complexity. We will briefly discuss how the compartment size in the spatial domain can affect the spatial patterns of the system.

In the second part of the talk, I will show another example for glucose metabolism where we see different-sized enzyme complexes. We hypothesized that the size of multienzyme complexes is related to their functional roles. We will see two models: one using a system of differential equations and the other using the Langevin dynamics.

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**February 9: Peter Hinow (University of Wisconsin Milwaukee)**

“Automated Feature Extraction from Large Cardiac Electrophysiological Data Sets”

A multi-electrode array-based application for the long-term recording of action potentials from electrogenic cells makes possible exciting cardiac electrophysiology studies in health and disease. With hundreds of simultaneous electrode recordings being acquired over a period of days, the main challenge becomes achieving reliable signal identification and quantification. We set out to develop an algorithm capable of automatically extracting regions of high-quality action potentials from terabyte size experimental results and to map the trains of action potentials into a low-dimensional feature space for analysis. Our automatic segmentation algorithm finds regions of acceptable action potentials in large data sets of electrophysiological readings. We use spectral methods and support vector machines to classify our readings and to extract relevant features. We show that action potentials from the same cell site can be recorded over days without detrimental effects to the cell membrane. The variability between measurements 24 h apart is comparable to the natural variability of the features at a single time point. Our work contributes towards a non-invasive approach for cardiomyocyte functional maturation, as well as developmental, pathological, and pharmacological studies.

This is joint work with Viviana Zlochiver, Stacie Kroboth (Advocate Aurora Research Institute), and John Jurkiewicz (graduate student at UWM).

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**February 16: Bhargav Karamched (Florida State University)**

“Mechanisms Underlying Spatiotemporal Patterning in Microbial Collectives: A Model’s Perspective”

We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example, we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise because of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap’s long side. This modeling approach and analysis can be extended to directionally growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior. As an example, we discuss how our model reveals how changes to a cell-shape describing parameter may manifest at the population level of the microbial collective. Specifically, we discuss mechanisms revealed by our model on how we may be able to control spatiotemporal patterning by modifying cell shape of a given strain in a multi-strain microbial consortium.

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**February 23: Nancy Rodriguez (University of Colorado Boulder)**

“A story on relocation strategies, the Allee effect, and the Ideal Free Distribution”

It is well known that relocation strategies in ecology and in economics can make the difference between extinction and persistence. In this talk I present a unifying model for the dynamics of ecological populations and street vendors, an important part of many informal economies. I discuss the effects of chemotactic movement of populations subject to the Allee Effect by discussing the existence of equilibrium solutions subject to various boundary conditions and the evolution problem when the chemotaxis effect is small. On an interesting note, I present numerical simulations, which show that in fact chemotaxis can help overcome the Allee effect as well as some partial analytical results in this direction on a bounded domain. We can make this precise in unbounded domains.

I will conclude by making a connection to the Ideal Free Distribution and other movement strategies under competition.

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**March 2: Naoki Masuda (SUNY Buffalo)**

“Temporal network epidemiology: Impacts of concurrency and inter-event times”

Contact networks on which epidemic spreading occurs vary over time. Epidemic processes on such temporal networks are complicated by complexity of both network structure and temporal dimensions. We discuss two mathematical modeling topics on “temporal network epidemiology”. First, we analyze how concurrency, i.e., the number of partnerships that an individual (i.e., node of the network) simultaneously owns affects the epidemic threshold. We particularly use a temporal network model with which we can vary the degree of concurrency while preserving the structure of the aggregate, static network. Second, we analyze the epidemic threshold and dynamics when each node switches between a high-activity state and a low-activity state in a Markovian manner. This assumption facilitates theoretical analyses and also allows us to produce distributions of inter-event times resembling heavy-tailed distributions, which are prevalent in empirical data. We argue that it is not the tail of the distribution but the small values of inter-event time that impact epidemic dynamics.

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**March 9: Richard Bonneau (Flatiron Institute)**

“Contracting ML and probabilistic methods for navigating time and space in genomics”

I will describe new methods for spatial transcriptomics and spacial genomics and contrast these methods with previous single cell and longitudinal genomics analysis approaches. I will focus first on methods for determining differential expression for spatial transcriptomic methods. I will then contract these early probabilistic methods with new methods built on variational auto encoders and generative ML approaches. Lastly I will describe bottlenecks, such as integrating imaging and genomic data in these studies. Prospects for building computational pipelines to integrate time, space and genomic coordinate will not be discussed, but will become apparent after deep reflection following the talk (but only if you pay attention).

Zoom link: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**March 16: Adrian Lam (Ohio State University)**

“Some PDEs in Evolution of Dispersal”

In the first part of the talk, we discuss the multi-species competition in a spatial domain, particularly the result of A. Hastings and some recent progress on the conjecture by Dockery et al. concerning the evolution of slow dispersal, i.e. when other things are equal, the slowest diffuser can competitively exclude other competitors. We then discuss the effect of adding passive drift and how it changes the evolutionary dynamics so that fast/intermediate diffuser is selected. In the second part, we will discuss the effect of mutation, and the associated moving Dirac solutions in a nonlocal PDE model proposed by Perthame and Souganidis. This latter equation describes a population structured by space and a phenotypic trait, and can be understood as competition of infinitely many species with different rate of dispersal.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**March 23: Lisa Fauci (Tulane University)**

“Spinning helices, heaving panels, and waving tails: the role of flexibility in propulsion”

The observed gait of a swimmer arises from the interplay of internal force generation, the passive elastic properties of its body, and environmental features such as fluid viscosity, boundaries, and obstacles. One could question whether optimal swimming of a fish occurs when it is actuated at a frequency near a natural frequency determined by its material bending rigidity.

We will share some insights that we have gained using computational models of a few systems that range from the Stokes regime to others where inertial forces are important.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**April 6: Jordan Rozum (Pennsylvania State University)**

“Boolean Networks: discrete dynamical models of high-dimensional nonlinear systems”

Boolean networks are a common modeling tool for studying the phenotypic changes that cells undergo in response stimuli. Examples include cell differentiation during embryogenesis, metastasis of cancer cells, and apoptosis. In these models, genes and proteins are represented by nodes in a network, and each node is assigned a Boolean activity variable that evolves in discrete time-steps such that the attractors of the resulting dynamics correspond to phenotypes of interest. In this talk, I will focus on the techniques we use to analyze these discrete dynamical systems. These techniques rely on the construction and iterative reduction of an auxiliary network that encodes dynamical information as graph structure (an example is diagrammed in the accompanying figure). Our goals include finding all attractors, identifying key feedback loops that govern attractor selection, and driving the system to a desired attractor from an arbitrary initial state. I will briefly cover several recent applications of these methods to empirical and statistical models. I will also discuss ways in which these discrete models — and the techniques we use to analyze them — are related to their ODE counterparts.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**April 20: Katrina Podsypanina (Institut des Hautes Études Scientifiques [IHES])**

“Cancer cells and their epithelial neighbors”

I am a cancer researcher working on a framework for a new mathematical model of metastasis. The reason for making a new model is that, in mice, metastases can grow from cells that are still normal at the time of arriving to the future metastasis site. If the same happens in humans, it would be pretty important for cancer therapy. First, it would explain why anti-cancer therapies fail to prevent metastases in some patients: if the cells are yet non-malignant at the time of therapy, they would be spared by the treatment. Second, it may be possible to identify improved treatments based on the ability to kill these non-malignant cells. I hypothesize that dissemination of non-malignant epithelial cells occurs in parallel with tumor cells, and subsequent transformation at the ectopic sites is a source of some metastases. This scenario ties together two contradictory observations: that metastases are associated with large and rapidly growing primary tumors, while metastatic tumors themselves often take a long time to appear. During my recent sabbatical at the Institut des Hautes Etudes Scientifiques, together with Misha Gromov and his colleagues I started to interrogate publicly available data on mutation rates and/or profiles from cancer patients and healthy individuals to determine whether the earliest common ancestor predicted using phylogenetic methods in some primary-metastasis pairs has features of a non-malignant cell.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**April 27: Cengiz Pehlevan (Harvard University)**

“Inductive Bias of Neural Networks”

Predicting a previously unseen example from training examples is unsolvable without additional assumptions about the nature of the task at hand. A learner’s performance depends crucially on how its internal assumptions, or inductive biases, align with the task. I will present a theory that describes the inductive biases of neural networks using kernel methods and statistical mechanics. This theory elucidates an inductive bias to explain data with “simple functions”, which are identified by solving a related kernel eigenfunction problem on the data distribution. This notion of simplicity allows us to characterize whether a network is compatible with a learning task, facilitating good generalization performance from a small number of training examples. I will present applications of this theory to artificial and biological neural systems, and real datasets.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

### Seminar speakers: Fall 2020

**December 8: Andreas Buttenschoen (University of British Columbia)****November 24: Qing Nie (University of California, Irvine): “Multiscale inference and modeling of cell fate via single-cell data”**

Cells make fate decisions in response to dynamic environmental and pathological stimuli as well as cell-to-cell communications. Recent technological breakthroughs have enabled to gather data in previously unthinkable quantities at single cell level, starting to suggest that cell fate decision is much more complex, dynamic, and stochastic than previously recognized. Multiscale interactions, sometimes through cell-cell communications, play a critical role in cell decision-making. Dissecting cellular dynamics emerging from molecular and genomic scale in single-cell demands novel computational tools and multiscale models. In this talk, through multiple biological examples we will present our recent effort to use single-cell RNA-seq data and spatial imaging data to uncover new insights in development, regeneration, and cancers. We will also present several new computational tools and mathematical modeling methods that are required to study the complex and dynamic cell fate process through the lens of single cells.

Zoom: https://upenn.zoom.us/j/96846928909?pwd=Q3JPTTc5dURmQk5xL01OMjZUc2FXUT09

**November 17: Mark Lewis (University of Alberta): “Population Dynamics in Changing Environments”**

Classical population dynamics problems assume constant unchanging environments. However, realistic environments fluctuate in both space and time. My lecture will focus on the analysis of population dynamics in environments that shift spatially, due either to advective flow (eg., river population dynamics) or to changing environmental conditions (eg., climate change). The emphasis will be on the analysis of nonlinear advection-diffusion-reaction equations and related models in the case where there is strong advection and environments are heterogeneous. I will use methods of spreading speed analysis, net reproductive rate and inside dynamics to understand qualitative outcomes. Applications will be made to river populations and to the genetic structure of populations subject to climate change.

**November 10: Jasmine Foo (University of Minnesota): “Understanding the role of phenotypic switching in cancer drug resistance”**

Recent findings suggest that cancer cells can acquire transient resistant phenotypes via epigenetic modifications and other non-genetic mechanisms. Although these resistant phenotypes are eventually relinquished by individual cells, they can temporarily ’save’ the tumor from extinction and enable the emergence of more permanent resistance mechanisms. These observations have generated interest in the potential of epigenetic therapies for long-term tumor control or eradication. In this talk, I will discuss some mathematical models for exploring how phenotypic switching at the single-cell level affects resistance evolution in cancer. As an example, we will explore the role of MGMT promoter methylation in driving resistance to temozolomide in glioblastoma.

**November 3: Adriana Dawes (Ohio State University): “Antagonistic motor protein dynamics in contractile ring structures”**

Ring-shaped contractile structures play important roles in biological processes including wound healing and cell division. Many of these contractile structures rely on motor proteins called myosins for constriction. We investigate force generation by the Type II myosins NMY-1 and NMY-2 in ring channels, contractile structures in developing oocytes of the nematode worm C. elegans, as our model system. By exploiting the ring channel’s circular geometry, we derive a second order ODE to describe the evolution of the radius of the ring channel. By comparing our model predictions to experimental depletion of NMY-1 and NMY-2, we show that these myosins act antagonistically to each other, with NMY-1 exerting force orthogonally and NMY-2 exerting force tangentially to the ring channel opening. Stochastic simulations are currently being used to determine how NMY-1 and NMY-2 may be producing these antagonistic forces, with new tools from topological data analysis identifying persistent ring-like structures in the simulation data.

**October 27: Alexandria Volkening (Northwestern University): “Modeling and measuring pattern formation in zebuctures”**

Wild-type zebrafish (Danio rerio) are characterized by black and yellow stripes, which form on their body and fins due to the self-organization of thousands of pigment cells. Mutant zebrafish and sibling species in the Danio genus, on the other hand, feature altered, variable patterns, including spots and labyrinth curves. The longterm goal of my work is to better link genotype, cell behavior, and phenotype by helping to identify the specific alterations to cell interactions that lead to these different fish patterns. Using a phenomenological approach, we develop agent-based models to describe the behavior of individual cells and simulate pattern formation on growing domains. In this talk, I will overview our models and highlight how topological techniques can be used to quantitatively compare our simulations with in vivo images. I will also discuss future directions related to taking a more mechanistic approach to modeling cell behavior in zebrafish.

**October 20: Stefano Recanatesi (University of Washington): “Constraints on the dimensionality of neural representations: a mesoscale approach linking learning to complex behavior”**

In the domain of computational/theoretical neuroscience a recently revived question is about the complexity of neural data. This question can be tackled by studying the dimensionality of such data: is neural activity high or low dimensional? How does the geometrical structure of neural activity depend on behavior, learning or the underlying connectivity? In my talk I will show how it is possible to link these three aspects (animal behavior, learning and underlying network connectivity) to the geometrical properties of neural data, with an emphasis on dimensionality phenomena. My results depart from neural recordings and aim at building understanding of neural dynamics by means of theoretical and computational tools. Such tools are mainly borrowed from the domain of neural networks dynamics, using a blend of large scale dynamical systems and statistical physics approaches.

**October 13: Naomi Leonard (Princeton University): “Opinion Dynamics with Tunable Sensitivity: Consensus, Dissensus, and Cascades”**

I will present a general model of continuous-time opinion dynamics for an arbitrary number of agents that sense or communicate over a network and form real-valued opinions about an arbitrary number of options. Drawing from biology, physics, and social psychology, an attention parameter is introduced to modulate social influence and a saturation function to bound inter-agent and intra-agent opinion exchanges. This yields simply parameterized dynamics that exhibit the range of opinion formation behaviors predicted by model-independent bifurcation theory but not exhibited by linear models or existing nonlinear models. Behaviors include rapid and reliable formation of multistable consensus and dissensus states, even in homogeneous networks, as well as ultra-sensitivity to inputs, robustness to uncertainty, flexible transitions between consensus and dissensus, and opinion cascades. Augmenting the opinion dynamics with feedback dynamics for the attention parameter results in tunable thresholds that govern sensitivity and robustness. The model provides new means for systematic study of dynamics on natural and engineered networks, from information spread and political polarization to collective decision making and dynamic task allocation.

This is joint work with Alessio Franci (UNAM, Mexico) and Anastasia Bizyaeva (Princeton).

The talk is based on version 2 of the paper “A General Model of Opinion Dynamics with Tunable Sensitivity”, which will be available on Tuesday October 13, 2020 here: https://arxiv.org/abs/2009.04332v2

**October 6: Sarah Olson (Worcester Polytechnic Institute): “Dynamics of movement in complex environments”**

In this talk, we will highlight two different types of movement in viscosity dominated environments: sperm navigation and centrosome clustering in dividing cells. Sperm often interact with chemicals and other proteins in the fluid, changing force generation and emergent swimming trajectories. Recently developed computational methods and asymptotic analysis allow for insight into swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. We will also show how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties. For the case of centrosome movement, we explore how cancer cells can cluster additional centrosomes and proceed through either a bipolar or multipolar division. The models focus on understanding centrosome movement during cell division, which is the result of complex interactions between stochastic microtubule dynamics and motor proteins in the viscous cytoplasm of the cell.

**September 22: Leah Edelstein-Keshet (University of British Columbia): “From Cell polarity to intracellular networks in single and collective cell motility”**

Cell migration plays a central role in embryonic development, wound healing and immune surveillance. In 2008, Yoichiro Mori, Alexandra Jilkine and I published a model for the initial step of cell migration, the front-back chemical polarization that sets a cell’s directionality. (More detailed mathematical properties of this model were described by the same group in 2011.) Since then, progress has been made in investigating how that simple “wave-pinning” mechanism is shaped and tuned by feedback from other proteins, such as actin, from the cell’s environment (extracellular matrix), from interplay with larger signaling networks, and from cell-cell interactions. In this talk I will describe some of this progress, with emphasis on links to experiments on melanoma cell motility. If time permits, I will also briefly describe more recent work on collective cell migration that we are currently undertaking.

**Seminar speakers: Spring 2020**

**April 29: William Bialek (Princeton University)**– TALK CANCELLED**April 24: Anita Layton (University of Waterloo)**– TALK CANCELLED**April 10: Eric Brown (University of Washington)**– TALK CANCELLED**April 6: Qing Nie (University of California, Irvine)**– TALK CANCELLED**March 20: Robert Eisenberg (Rush Medical College)**– TALK CANCELLED

**February 28: Gillian Queisser (Temple University): “Ultrastructural 3D simulations of electrical and calcium dynamics in neurons and networks”**

Neurons make use of their complex cellular and intracellular architecture to process and guide electrical and biochemical signals. To study this structure-function interplay, computational methods are detremental, since many parameters are not directly accessible in an experimental setting. This also means that the detailed three-dimensional morphology of cells and organelles needs to be included in modeling and simulation, which results in complex-domain problems, described by systems of coupled, nonlinear, partial differential equations. We have developed numerical discretization methods and fast solvers to address this general type of biological problem set, with a focus on optimal weak scalability on High Performance Computing infrastructures. We present some of the important biological problems revolving around cellular calcium signaling, coupled to electrical models, and the use of our NeuroBox Toolbox and the multiphysics platform uG4 to solve such ultrastructural 3D neuron models. Selected results show how neurons are capable of using their (intra)cellular architecture to fine-tune their response to exterior/network input.

**February 14: Erick Matsen (Fred Hutchinson Cancer Research Center): “”Making Bayesian phylogenetics like training a neural network”**

Bayesian posterior distributions on phylogenetic trees remain difficult to sample despite decades of effort. The complex discrete and continuous model structure of trees means that recent inference methods developed for Euclidean space are not easily applicable to the phylogenetic case. Thus, we are left with random-walk Markov Chain Monte Carlo (MCMC) with uninformed tree modification proposals; these traverse tree space slowly because phylogenetic posteriors are concentrated on a small fraction of the very many possible trees.

In this talk, I will describe our wild adventure developing efficient alternatives to random-walk MCMC, which has concluded successfully with the development of a variational Bayes formulation of Bayesian phylogenetics. This formulation leverages a “factorization” of phylogenetic posterior distributions that we show is rich enough to capture the shape of posteriors inferred from real data. Our proof-of-concept implementation of variational inference using this method gives very promising results, and I will describe our ongoing efforts to develop an efficient implementation that integrates with modern modeling frameworks.

This line of work was started by Cheng Zhang (now faculty at Peking University) when he was in my group; ongoing work is by Michael Karcher, Seong-Hwan Jun, Andy Magee, and Mathieu Fourment (University of Tech, Sydney).

**February 7: Chuan Xue (Ohio State University): “Mathematical models in biological pattern formation”**

I will discuss two mathematical problems in biological pattern formation. The first is on modeling concentric ring patterns formed in engineered bacterial colonies. I will first present a hybrid model that incorporates a detailed description of cell movement and cell signaling and explains the underlying mechanism of the ring pattern. I will then present a PDE model derived from the hybrid model that captures the biological phenomena equally well. The second concerns spatial pattern formation in reaction-diffusion systems. I will discuss a computational method that can be used to discover potentially all nonuniform steady states of the PDE system, describing the structure of the spatial patterns. The method uses techniques from numerical algebraic geometry. This talk is based on joint work with Min Tang from Shanghai Jiaotong University and Wenrui Hao from Penn State University.

**January 27: Daniel Gomez (University of British Columbia): “Localized Patterns in Bulk-Membrane Coupled Models”**

Turing instabilities in reaction diffusion systems describe potential mechanisms for pattern formation in qualitative models of microbiological processes. A recent direction of research has been to incorporate bulk-membrane coupling (BMC) into these models which introduces a process of attachment and detachment to and from the cell membrane. In these models chemical species can therefore undergo periods of bulk- and membrane-bound diffusion in addition to prescribed kinetics. Linear stability analysis and numerical simulations have revealed that differences between membrane and cytosol diffusivities can trigger Turing-like pattern-forming instabilities in BMC models. We further investigate the role of bulk-membrane coupling by analyzing its effect in a singularly perturbed model where the diffusivity of one membrane-bound species is asymptotically small. In this context, localized solutions are known to exist and can be approximated using asymptotic methods. Additionally, the linear stability and long time dynamics of these localized solutions leads to novel non-local eigenvalue problems and differential-algebraic systems. In this talk we will outline this asymptotic framework and highlight the role of bulk-membrane coupling in the stability properties of localized solutions.

**January 24: Daniel Cooney (Princeton University): “PDE Models of Multilevel Selection: The Evolution of Cooperation and the Shadow of Individual Selection”**

Here we consider a game theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on the Prisoners’ Dilemma: a game in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. Comparison principles and an invariant property of the tail of the population distribution are used to characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: if groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.

**January 23: Eduardo Garcia-Juarez (University of Pennsylvania): “Analysis of Moving Interfaces in Incompressible Flows”**

Interfaces that evolve with a fluid flow abound in nature and engineering. They are subject to intense research in many different fields, from meteorology to medical sciences or the petroleum industry. However, the mathematical analysis of these free boundary problems is still emerging, which impedes a deeper understanding necessary for accurate numerical methods. In this talk, we will give an overview of certain fluid-fluid and fluid-structure interfaces problem (inhomogeneous Navier-Stokes, Boussinesq, Muskat and Peskin models). The focus is placed on global-in-time results (versus finite-time singularities), with initial interfaces that are not just small perturbations. The analysis is thus purely nonlinear. Particular attention is given to initial data in critical spaces and the challenging non-local effects of viscosity contrasts.

**January 17: HyunJoong Kim (University of Utah): “Communicating by touch”**

During development, cells figure out their location and fate through morphogen concentration gradients. The most commonly accepted mechanism for morphogen gradient formation is diffusion from a local source combined with degradation. Recently, however, there has been growing experimental evidence for an alternative mechanism, based on the direct delivery of morphogens via thin and long cellular protrusions known as cytonemes. In this talk, we will address the effects of various cytoneme transport mechanisms. We then explore the stochasticity from the discrete nature of transport. To deal with the complex nature of the stochastic process, we introduce the strong Markov property and queuing theory.

Seminar speakers: Fall 2019

November 22: Alexander Mogilner (New York University): “Many-body problem of classical mechanics in cell biology”Many-body problem of celestial mechanics revolutionized applied mathematics and continues to provide inspiration. Math/physical communities are much less aware that there are numerous example of fascinating many-body problems of classical mechanics arising in live cells at drastically different scales: instead of years and millions of kilometers, in the cell we deal with minutes and microns. Another big difference is: rather than Newtonian mechanics in empty space, when acceleration is proportional to force, in the cells filled with viscous cytoplasm, we deal with Aristotelian mechanics, in which velocity is proportional to force. Yet another difference is a great diversity of complex inter-body forces in the cell, compared to pleasingly simple gravitational force of celestial mechanics. Because of this diversity, in cell biology we often need to solve the ill-posed inverse problem – reverse-engineering forces from the observed patterns and movements – contrasted with the well-posed direct problem of predicting patterns and movements from known forces.

I will discuss two many-body problems of cell biology – assembly of mitotic spindle from two centrosomes and tens of chromosomes, and nuclei positioning in multi-nucleated muscle cells. Three approaches – solutions of ODEs of ‘particle’ models, solutions of PDEs of continuous approximation, and energy minimization, complemented by computer screening – shed light on the molecular origins of the intracellular forces that ensure proper and robust cellular architecture.

November 15: Samuel Isaacson (Boston University): “Control of membrane-bound tethered signaling reactions”Many membrane-bound T cell receptors have long, unstructured cytoplasmic tails that contain tyrosine sites. These sites can serve as regulators of receptor activation when phosphorylated or dephosphorylated, while also serving as docking sites for cytosolic enzymes. Interactions between receptors then involve the in-membrane diffusion of the receptor proteins, and reactions between proteins tethered to the receptors’ tails (and hence diffusing within the three-dimensional cytosolic space near the membrane). We develop a particle-based stochastic reaction-diffusion model based on the Convergent Reaction-Diffusion Master Equation to study the combined diffusion of individual receptors within the cell membrane, and chemical reactions between proteins bound to receptor tails. The model suggests a switch-like behavior in the dependence of the fraction of activated receptors on both receptor diffusivity, and on the molecular reach at which two receptor tails can interact. A simplified, analytically solvable model is developed to approximate the more complicated multi-particle system, and used to illustrate how the switch-like behavior arises.

October 25: Sean Sun (John Hopkins University): “Water Dynamics in Cells and Tissues”The mammalian cell surface is highly permeable to water. The cell can also actively control the water flux across the cell surface by pumping solutes (mostly ions), and thereby controlling the cell water content and the cell volume. In this talk, we will explore how the cell also uses active water fluxes to move and change cell shape. The same players in the cell volume control system are involved in driving cell movement, especially in high viscosity environments. Mathematical modeling shows that the water-driven cell movement is energetically costly, but is necessary when the hydraulic environment is viscous. Finally, we will discuss how epithelial cell layers such as the kidney tubule pump water and generate mechanical force.

September 27: Gustavo Martínez-Mekler (Instituto de Ciencias Físicas, UNAM): “Fertilization Regulatory Networks“Fertilization is one of the fundamental processes of living systems. Here I will address marine external fertilization and comment on recent work on mammals. I will show experiments that substantiate that sea urchin sperms exhibit chemotaxis as they swim towards the ovum. They are guided by flagellum internal [Ca2+] concentration fluctuations triggered by the binding of chemicals from the oocyte surroundings. Based on experiment, I present a family of logical regulatory networks for the [Ca2+] fluctuation signaling-pathway that reproduce previously observed electrophysiological behaviors and provide predictions, which have been confirmed with new experiments. These studies give insight on the operation of drugs that control sperm navigation. In this systems biology approach, global properties of the [Ca2+] discrete regulatory network dynamics such as: stability, redundancy, degeneracy, chaoticity and criticality can be determined. Our models operate near a critical dynamical regime, where robustness and evolvablity coexist. This regime is preserved under a class of strong perturbations. Based on global dynamics considerations, we have implemented a network node-reduction method. The coincidence of this reduced network with our bottom-up step-by-step, continuous differential equation modeling is reassuring. For the case of mammals our research has centered on the understanding of capacitation and acrosomal reaction. The first is a process by means of which roughly one third of the spermatozoa acquire the “capacity” to fertilize; the second enables the spermatozoa to penetrate the egg´s surrounding zona pellucida. Overall, our studies might contribute to fertility issues such as the development of male contraception treatments, which is an area of intense research.