Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of (stochastic) genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carcinogenesis. In particular, two stochastic (mechanistic) models, the Armitage-Doll and the two-stage clonal expansion (TSCE) model, have been widely used in the past for cancer risk assessment and for the analysis of cancer population and experimental data. In this talk, I will introduce some of the biological and mathematical concepts behind the theory of multistage carcinogenesis, and discuss in detail the use of these models in cancer epidemiology and cancer prevention and control. Recent applications of multistage and state-transition Markov models to assess the potential impact of lung cancer screening in the US will be reviewed.

Experiments on macaque monkeys show that spike-triggered stimulation performed by a Bidirectional Brain-Computer-Interfaces (BBCI) can artificially strengthen synaptic connections between distant neural sites in Motor Cortex (MC) and even between MC and spinal cord, with changes that last several days. Here, a neural implant records from some neurons in MC and electrically stimulates others after set delays. The working hypothesis is that this stimulation procedure, which interacts with the very fast spiking activity of cortical circuits (on the order of milliseconds), induces changes mediated by synaptic plasticity mechanisms on much longer timescales (hours and days). The field of online, closed-loop BBCI's is rapidly evolving, with applications ranging from a science-oriented tool to clinical treatments of motor injuries. However, with the enhanced capability of novel devices that can record and stimulate an ever-growing number of neural sites comes growing complexity. It is therefore crucial to develop a theoretical understanding of the effects of closed-loop artificial stimulation in the highly recurrent neural circuits found in cortex, and how such protocols affect functional cotex-to-muscle mappings across a range of timescales. In parallel with ongoing experiments, we are developing a mathematical model of recurrent MC networks with probabilistic spiking mechanisms and spike-time-dependent plastic synapses (STDP) capable of capturing both neural and synaptic activity statistics relevant to BBCI protocols. This model successfully reproduces key experimental results and we use analytical derivations to predict optimal operational regimes for BBCIs. We make experimental predictions concerning the efficacy of spike-triggered stimulation in different regimes of cortical activity such as awake behaving states or sleep. Importantly, this work provides a first step toward a theoretical framework aimed at the design and development of next-generations applications of BBCI's.

Advances in high-throughput genomics have facilitated the development of pooled population sequencing techniques which involve the en masse sequencing of tens to hundreds of individual genomes in a single sequencing reaction. Pooled population sequencing methods have numerous applications in quantitative, population and evolutionary genetics. I will discuss some of the statistical and computational challenges associated with the analysis of pooled sequence data in the context of quantitative trait locus (QTL) mapping and detecting selection during experimental evolution.

Stochastic evolutionary models of biological sequences are widely used for phylogenetic inference and ancestral reconstruction. However, at long divergence times sequences enter the "twilight zone" of homology detection and reconstruction becomes very difficult. We describe a stochastic evolutionary model for protein 3D structure using elements of shape theory. This model significantly resolves this uncertainty and stabilizes evolutionary inferences. We also provide theoretical bounds on inferring evolutionary divergence times via connections to the probabilistic "cutoff phenomenon", in which a Markov chain remains far equilibrium for an extended period followed by a rapid transition into equilibrium. We show that this cutoff explains several previously reported problems with common default priors for Bayesian phylogenetic analysis, and suggest a new class of priors to address these problems.

Connecting dynamic models with data to yield predictive results often requires a variety of parameter estimation, identifiability, and uncertainty quantification techniques. These approaches can help to determine what is possible to estimate from a given model and data set, and help guide new data collection. Here, we examine how parameter estimation and disease forecasting are affected when examining disease transmission via multiple types or pathways of transmission. Using examples taken from the West Africa Ebola epidemic, HPV, and cholera, we illustrate some of the potential difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability. We also illustrate how even in the presence of large uncertainties in the data and model parameters, it may still be possible to successfully forecast disease dynamics.

Birds, insects, and fish all exploit the fact that flexible wings or fins generally perform better than their rigid counterparts. Given the task of designing an optimal wing, though, it is not clear how to best distribute the flexibility: Should the wing be uniformly flexible along its length, or could some advantage be gained by making certain sections more rigid than others? I will discuss this question by using a 2D small-amplitude model for the fluid-structure interaction combined with an efficient Chebyshev PDE solver. Numerical optimization shows that concentrating flexibility near the leading edge of the wing maximizes thrust production, an arrangement that resembles the torsional-joint flexibility mechanism found in insect wings. I will discuss the possibility of extending into three dimensions to address the question of optimal wing architecture more generally.

In this talk, I will discuss a new result in fluid flows through channels with permeable membranes with simple pumping dynamics. Fluid will be exchanged and metabolized in a simple reservoir and I will demonstrate the existence of optimal reservoir properties that may either maximize or minimized the amount of fluid being extracted across the channel walls. The biological relevance of this work may be seen by noting that all living organisms of a sufficient size rely on complex systems of tubular networks to efficiently collect, transport and distribute nutrients or waste. These networks exchange material with the interstitium via embedded channels leading to effective permeabilities across the wall separating the channel interior from the interstitium. In many invertebrates, for example, respiratory systems are made of complex tracheal systems that branch out through the entire body allowing for passive exchange of oxygen and carbon dioxide. In many of these systems, certain animals utilize various pumping mechanisms that alter the flow of the air or fluid being transported. Although the net effect of pumping of the averaged rates of fluid flow through the channel is typically well understood, it is still a largely open problem to understand how, and in what circumstances, pumping enables and enhances the exchange of material across channel walls. It has been demonstrated experimentally, for example, that when certain insects flap their wings, compression of the trachea allow for more efficient oxygen extraction, however it is unclear if this pumping is optimized for flight, oxygen uptake or neither, and understanding this problem quantitatively will shed insight on this biological process. Many of these interesting scenarios occur at low Reynolds number and this regime will be the focus of the presentation.

Systems biologists seek to understand the higher-level organizational properties that a proposed system exhibits from the interactions of its many lower level components. It is then typically beneficial to learn, in a statistical sense, the nature of these interactions from experimental data; this is sometimes called reverse engineering. In this talk I will discuss some methods found in the literature for learning the kinetic parameters from stochastic system trajectories measured at discrete time points. I will also discuss estimating the system topology, or network structure, with the stoichiometric algebraic statistical model and detail its relationship with kinetic parameter estimation. Finally, I will discuss some ongoing work aimed at improving both parameter estimation and system topology estimation.

Considering cancer as an evolutionary disease, we aim at understanding the means by which cancer cell populations develop resistance mechanisms to drug therapies, in order to circumvent them by using optimised therapeutic combinations. Rather than focusing on molecular mechanisms such as overexpression of intracellular drug processing enzymes or ABC transporters that are responsible for resistance at the individual cell level, we propose to introduce abstract phenotypes of resistance structuring cancer cell populations. The models we propose rely on continuous adaptive dynamics of cell populations, and are amenable to predict asymptotic evolution of these populations with respect to the phenotypic traits of interest. Drug-induced drug resistance, the question we are tackling from a theoretical and experimental point of view, may be due to biological mechanisms of different natures, mere local regulation, epigenetic modifications (reversible, nevertheless inheritable) or genetic mutations (irreversible), according to the extent to which the genome of the cells in the population is affected. In this respect, the models we develop are more likely to be biologically corresponding to epigenetic modifications, although eventual induction of emergent resistant cell clones due to mutations under drug pressure is not to be completely excluded. From the biologist's point of view, we study phenotypically heterogeneous, but genetically homogeneous, cancer cell populations under stress by drugs. According to the cell populations at stake and to the exerted drug pressure, is drug resistance in cancer a permanently acquired phenotypic trait or is it reversible? Can it be avoided or overcome by rationally (model-guided) designed combinations of drugs? These are some of the questions we will try to answer in a collaboration between a team of mathematicians and another one of biologists, both dealing with cancer and Darwinian - possibly also Lamarckian - evolution of cell populations.

We discuss the question: What properties of solutions to coupled cell network systems are invariant under changes of coordinates that preserve network structure? This question was motivated by trying to understand the biological phenomenon of homeostasis in a mathematically satisfactory way. In its simplest mathematical form homeostasis can be described as follows. Given a stable equilibrium $x(lambda)$ of a system that depends on an input parameter $lambda$: When is some coordinate (say $x_j(lambda)$) approximately constant? First, we translate approximately constant to derivative approximately 0. This allows us to search for regions of homeostasis in a model using bifurcation theory like formulas. Second, we claim that there is a sense in which homeostasis can be thought of as a network invariant. This is joint work with Ian Stewart.

Certain strains of the influenza virus form coherent single, globally circulating viral populations. Multiple models of viral evolution have considered the virus on this level, identifying immune and structural factors underlying the observed changes in viral genotype. In this talk, I outline an alternative approach for studying viral evolution, considering events at the level of within-host viral growth and transmission. In particular, I describe statistical approaches to infer the extent to which natural selection acts upon the viral population within the course of a single infection, on the basis of genome sequencing data from Sanger sequencing, and next-generation technologies. Examining data from evolutionary experiments conducted with a reassortant H5N1 influenza virus, I discuss the potential for using data from evolutionary experiments to predict the potential evolution of this virus in a human population.

Cracking the neural code is one of the central challenges of neuroscience. Typically, this has been understood as finding the relationship between single neurons and the stimuli they represent. More generally, neural activity must also reflect relationships between stimuli, such as proximity between locations in an environment. Convex codes, comprised of activity patterns for neurons with classical receptive fields, may be the brain's solution to this problem. These codes have been observed in many areas, including sensory cortices and the hippocampus. What makes a code convex? Using algebra, we can uncover intrinsic signatures of convexity and dimension in neural codes. I will report on some recent results by multiple authors, including participants in my 2014 AMS Math Research Community.

Biochemical reaction networks can be represented as graphs, with chemical complexes serving as vertices and reaction channels serving as directed edges. To each such network there is an associated stochastic process that models the behavior of the underlying system of interest. In this talk I will present results that relate the structure of the reaction network to the possible dynamics of the model. I will pay particular attention to how the results pertaining to the stochastic models compare and contrast with similar results pertaining to deterministic models.

In the biopharma industry of drug development, figuring out the best doses to use is considered a high priority. It can mean the difference between having an effective drug and having one that gives no benefit. It can reduce toxicities that otherwise could prevent the drug from being used. And proposing a dose that later turns out to be excessive not only looks bad, it can also mean substantial revenue loss.

The type of control theory used in academia for many decades is now being looked at by industry as a potential way to address the problem of dose selection. The problem becomes even harder when a drug will be used in combination with one or more other drugs. I will explain some of the mathematics and show examples of how control theory can be used to optimize dose regimens.

Biological oscillators such as the cell cycle, circadian clocks, and metabolic rhythms are ubiquitous across the domains of life. These biochemical oscillators co-exist in the same cells, often sharing and competing for resources. Are there mechanisms and regulatory principles that ensure harmony between these oscillators? Recent studies have shown that in addition to the transcriptional circadian clock, many organisms (including Arabidopsis) have a circadian redox rhythm driven by the organism's metabolic activities. It has been hypothesized that the redox rhythm is linked to the circadian clock, but the mechanism and the biological significance of this link have only begun to be investigated. In the first half of my talk, I will describe our work (in collaboration with the Dong lab at Duke) on the coupling of redox rhythms and the plant circadian clock. In the second half of my talk, I will discuss our work on the coupling of yeast metabolic cycle and the cell division cycle.

John Hopfield pointed out, in his seminal paper on kinetic proofreading, that if a biochemical system operates at thermodynamic equilibrium there is a barrier to how well it can achieve high-fidelity in transcription and translation. Hopfield showed that the only way to bypass this barrier is to dissipate energy and maintain the system away from equilibrium. Eukaryotic gene regulation uses dissipative mechanisms, such as nucleosome remodelling, DNA methylation and post-translational modification of histones, which are known to play a critical regulatory role but have been largely ignored in quantitative treatments. I will describe joint work with my colleague Angela DePace in which we use the recently-developed, graph-theoretic “linear framework” to show that the sharpness with which a gene is turned “on” or “off” in response to an upstream transcription factor is limited if the regulatory system operates at equilibrium, even with arbitrary degrees of higher-order cooperativity. In contrast, if the regulatory system is maintained away from equilibrium, substantially higher degrees of sharpness can be achieved. We suggest that achieving sharpness in gene regulation exhibits a Hopfield Barrier, and uncover, along the way, a new interpretation for the ubiquitously used, but poorly justified, Hill function.

This talk is about extending classical limit theorems of probability (law of large numbers, central limit theorem) to a non-Euclidean setting. I'll talk about new and interesting phenomena observed when sampling independent points from certain singular geometric spaces. The main result is a limit theorem -- the "sticky central limit theorem" -- which applies to the mean or barycenter of a family of independent samples as the number of samples grows. The theorem shows that the geometry of the underlying space may have an interesting effect on the asymptotic fluctuations of the sample means, in a way that does not occur with independent samples in Euclidean space. One motivation for thinking about statistics in singular geometric spaces comes from evolutionary biology; one can consider phylogenetic trees as points in a metric space of the sort discussed in this talk. Apart from this basic motivation, however, the talk will have little biological content and will be mainly about probability.

Indirect measurements are ubiquitous in the life sciences because it is often impossible or impractical to directly measure the process of interest. I will show how dynamic mathematical models of biological systems can aid interpretation of biological markers, focusing on the dynamics of acetaminophen (APAP, Tylenol) overdose and stable isotope signatures. APAP is one of the most common drugs on the planet. While safe in therapeutic doses, APAP is the leading cause of acute liver failure in the United States. I will present a mathematical model of APAP overdose that can be used to estimate time since overdose, overdose amount, and need for liver transplant from measurable markers of liver injury at the time of hospital admission. Similarly, stable isotopes are used by ecologists and forensic scientists as markers of diet and movement patterns. I will show how, with suitable mathematical models, stable isotope ratios of the hair of a murder victim were used to reconstruct the victim's movement history in the time preceding death.

The job of embryogenesis is to diversify cells into the hundreds of specialized cellular functions required by an animal or plant and to place those cells in the correct location in the embryo. To accomplish that complex job transcriptional regulation provides control circuits directing each dichotomy, each cell movement, and the patterning of cellular assemblies so that the animal that emerges from embryogenesis can feed and perform the necessary functions for survival of the species. The questions being addressed in this talk are to ask how gene regulatory networks are assembled, how do they change with time, and how do they accomplish the enormous regulatory task needed for building an embryo.

Recent studies of cholesterol-rich membrane microdomains, called caveolae, reveal that caveolae are reservoirs of recruitable sodium ion channels. Caveolar channels constitute a substantial and previously unrecognized source of sodium current in cardiac cells. In this talk, I will present a family of DE and PDE models to investigate caveolar sodium currents and their contributions to cardiac action potential morphology. We show that the b-agonist-induced opening of caveolae may have substantial impacts on peak overshoot, maximum upstroke velocity, and conduction velocity. Additionally, we show that prolonged action potentials and the formation of potentially arrhythmogenic afterdepolarizations, can arise if caveolae open intermittently throughout the action potential.

Persistent homology measures geometric structures using topological invariants. When the structures are magnetic resonance images of branching arteries, for example, persistent homology records the connectedness of an increasing subset of the vessels. Although the theory of persistent homology is relatively well developed, and many aspects of its behavior are understood in synthetic examples, only recently have applications to genuine experimental data begun. This talk explains what we have learned about the geometry of blood vessels in aging human brains, as well as lessons this exploration has taught us about applications of persistent homology in general. These lessons inform further potential applications of persistent homology in statistical problems from biological and medical imaging. The main results are joint with Paul Bendich, Steve Marron, Aaron Pieloch, and Sean Skwerer (Math junior faculty, Stat faculty, Math undergrad, and Operations Research grad student). The talk will be accessible to advanced mathematics and statistics undergraduates, medical and biological researchers, statistics and mathematics faculty, and everybody in between.

A major focus of synthetic biology is to engineer gene circuits to perform user-defined functions. These gene circuits can serve as well-defined models to probe basic biological questions of broad significance. In this talk, I will discuss our efforts along this line of research, whereby we have engineered gene circuits to program bacterial dynamics in time and space, guided by quantitative modeling and experiments. Insights learnt from these circuits have implications for developing new strategies to combat bacterial pathogens or to fabricate new materials.

The complexity of the human airway coupled with advances in computational technology have led to the growing interest in the use of computational fluid dynamics (CFD) techniques to simulate airway physiology in order to collect objective data due to inter-individual anatomy and pre- & post-surgical changes. Outcomes from airway surgery are sometimes difficult to predict a priori, and it is not known whether topical medications are reaching target sites within the human air passage. This talk will give an overview of how CFD is used to explore these issues, as well as demonstrate the potential ability of this methodology in pre-surgical planning.

The ability of animals to self-organize into remarkable patterns of movement is seen throughout nature from herds of large mammals, to flocks of birds, schools of fish, and swarms of insects. Remarkably, patterns of collective movement can be seen even in the simplest forms of life such as bacteria. M. xanthus are common soil bacteria that are among the most “social" bacteria in nature. In this talk clustering mechanism of swarming M. xanthus will be described using combination of experimental movies and stochastic model simulations. Continuous limits of discrete stochastic dynamical systems simulating cell aggregation will be described in the form of reaction-diffusion and nonlinear diffusion equations. Surface motility such as swarming is thought to precede biofilm formation during infection. Population of bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. This collective mechanism of cell- cell coordination was recently shown to moderate swarming direction of individual bacteria to avoid a toxic environment. In the last part of the talk a three-dimensional multiscale modeling approach will be described for studying fluid–viscoelastic cell interaction during blood clot formation.

Insulin resistance, a major factor in type 2 diabetes development, is a systemic defect characterized by reduced intracellular insulin signaling. Although there are many proposed causes of insulin resistance, the precise mechanisms that influence its long-term progression remain unclear. In this talk, we develop mathematical models to study the hypothesized role of oxidative stress and mitochondrial dysfunction in skeletal muscle insulin resistance. Simulation results suggest that a perfect storm of environmental and genetic factors leading to oxidative stress can confer protection on the individual cell via insulin resistance.

The jellyfish has been the subject of numerous mathematical and physical studies ranging from the discovery of reentry phenomenon in electrophysiology to the development of axisymmetric methods for solving fluid-structure interaction problems. In this presentation, we develop and test mathematical models describing the pulsing dynamics and the resulting fluid flow generated by the benthic upside down jellyfish, Cassiopea spp., and the pelagic moon jellyfish, Aurelia spp. The kinematics of contraction and distributions of pulse frequencies were obtained from videos and used as inputs into numerical simulations. Particle image velocimetry was used to obtain spatially and temporally resolved flow fields experimentally. The immersed boundary method was then used to solve the fluid-structure interaction problem and explore how changes in morphology and pulsing dynamics alter the resulting fluid flow. For Cassiopea, significant mixing occurs around and directly above the oral arms and secondary mouths. We found good agreement between the numerical simulations and experiments, suggesting that the presence of porous oral arms induce net horizontal flow towards the bell and mixing. For Aurelia, maximum swim speeds are generated when the elastic bell is resonating at its natural frequency. Alternating vortex rings can also enhance swimming speed and efficiency.

Cellular motility arises from the interaction of numerous components: actin filaments, ligands, adhesion complexes to name a few. Each individual component exhibits stochastic behavior, yet overall cooperative behavior is observed, and leads to motility. Linking overall cellular motion to the stochastic behavior of its components is a remarkable mathematical challenge. This talk introduces a method to cyclically extract surrogate motility models from detailed stochastic simulation of the components. The starting point is a sampling of the detailed cytoskeleton dynamics over a short time interval. This detailed information is repeatedly projected onto a lower dimensional statistical manifold to obtain a coarse-grained model of the cytoskeleton, from which a surrogate cell model is obtained. The projection operation corresponds to transport along the geodesics of embedded statistical manifolds. The surrogate model is advanced over a larger time interval and then used to recreate the detailed microscopic cytoskeleton state needed to start the next cycle. The procedure is applied to study the behavior of Listeria monocytogenes bacterium and inert models of the motility behavior of this bacterium