Decades of research in cancer have yielded scant results other than highlighting the need for new approaches that could go beyond the tried and tested molecular-based ones. Recent clinical studies show that tumour heterogeneity and selection, the ingredients of Darwinian evolution, can explain cancer progression towards malignancy as well as recurrence after treatment. In this talk I will describe mathematical and computational models that explore cancer evolutionary dynamics and that can explain how the interactions between the tumour with its environment (the tumour ecosystem) can yield a better understanding of cancer biology and lead to better and more efficacious treatments for cancer patients.

Traditional infectious disease epidemiology is built on the foundation of high quality and high accuracy data on disease and behavior. While these data are usually characterized by smallsize, they benefit from designed sampling schemes that make it possible to make population-level inferences. On the other hand, digital infectious disease epidemiology uses existing digital traces, re-purposing them to identify patterns in health-related processes. In this talk, I will discuss our work using data from small epidemiological studies as well as administrative “big data” to understand influenza transmission dynamics and inform disease surveillance and control.

There are two generally accepted models for the cell biological positive feedback loops that allow yeast cells to break symmetry and establish an axis of polarity. Both have been subjects of published mathematical analyses. Here I will argue that the models used to support a vesicle trafficking model incorporated a simplifying assumption that seemed innocuous but in fact was critical to their success. The assumption is not physically plausible, and its removal means that the model fails. I will show how changing other assumptions can make the model work, but there is no experimental support for those changes. And without them, the vesicle trafficking model perturbs polarity, rather than establishing polarity

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on evolutionarily stable strategies (ESS) for dispersal.

This talk will trace many years of work mathematical modeling hematological diseases. The ‘understand’ part talks about the use of mathematical to figure out what causes cyclical neutropenia, and the ‘treat’ part refers to work on treating cyclical neutropenia using recombinant cytokines. The ‘avoid’ part deals with current ongoing work trying to obviate the deleterious effects of chemotherapy on blood cell production—one of the major negative side effects of chemotherapy.

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.

We present two models of perpetrators' decision-making in extracting resources from a protected area. It is assumed that the authorities conduct surveillance to counter the extraction activities, and that perpetrators choose their post-extraction paths to balance the time/hardship of travel against the expected losses from a possible detection. In our first model, the authorities are assumed to use ground patrols and the protected resources are confiscated as soon as the extractor is observed with them. The perpetrators' path-planning is modeled using the optimal control of randomly-terminated process. In our second model, the authorities use aerial patrols, with the apprehension of perpetrators and confiscation of resources delayed until their exit from the protected area. In this case the path-planning is based on multi-objective dynamic programming. Our efficient numerical methods are illustrated on several examples with complicated geometry and terrain of protected areas, non-uniform distribution of protected resources, and spatially non-uniform detection rates due to aerial or ground patrols.

Graphs are becoming a favorite mathematical object for representation of data. Yet, statistical pattern recognition has focused almost entirely on vector valued data in Euclidean space. Graphs, however, live in graph space, which is non-Euclidean. Thus, most inference techniques are not even defined for graph valued data. Previous work in the classification of graph-valued data typically follows one of two recipes. (1) Vectorize the adjacency matrices of the graphs, and apply standard machine learning techniques. (2) Compute some number of graph invariants (e.g., clustering coefficient, or degree distribution) for each graph, and then apply standard machine learning techniques. We follow a different recipe based in the probabilistic theory of pattern recognition. First, we define a joint graph-class model. Given this model, we derive classifiers which we prove are consistent; that is, they converge to the Bayes optimal classifier. Specifically, we build two consistent classifiers for graph valued data, a parametric and a non-parametric version. In a sense, these classifiers span the spectrum of complexity, the former is consistent for graphs sampled from relatively simple random graph distributions, the latter is consistent for graphs sampled from (nearly) any random graph distribution. Although both classifiers assume that all our graphs have labeled vertices, we generalize these results to also incorporate unlabeled graphs, as well as weighted and multigraphs. We apply these graph classifiers to human brain data. Specifically, using diffusion MRI, we can obtain large brain-graphs (10,000 vertices) for each subject, where vertices correspond to voxels. We then coarsen the graphs spatially to obtain smaller (70 vertex) graphs per subject. Using <50 subjects, we are able to achieve nearly 85% classification accuracy, with results interpretable to neurobiologists with regard to the brain regions of interest.

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. One motivation for this work is to provide analytical solutions to help validate the accuracy of numerical methods for elastic membranes in flow. A second motivation is to clarify the steady response of capsules in some canonical flows. Finally, we investigate the formation of finite-time cusp singularities, of which there are only a few examples in interfacial Stokes flow, and where none involve elastic interfaces. This is joint work with Michael Booty and Michael Higley.

Recently a mathematical theory has been developed for spatial games with weak selection, i.e., the payoff differences between strategies are small. The key to the analysis is that when space an time are suitably rescaled the limit is partial differential equation (PDE). This approach can be used to analyze all 2 x 2 games, but there are a number of 3 x 3 games for which the behavior of the limiting PDE is not known. In this talk we will describe simulation results for two cases that are not considered by rigorous results: rock-paper scissors and bistable games. We will begin by describing results for a two strategy game that arises from studying pancreatic cancer and shows that theoretical predictions work even when selection is not very weak. This is joint work with Mridu Nanda, a student at North Carolina School for Science and Math.

I will briefly describe mathematical models of networks of neurons and chemical reactions within neurons that generate daily (circadian) timekeeping. The numerical and analytical challenges of these models as well as the benefits in terms of biological predications will be highlighted. I will then explain how models can be used to find schedules that decrease the time needed to adjust to a new timezone by a factor of 2 or more. These optimal schedules have been implemented into a smartphone app, ENTRAIN, which collects data from users and in return helps them avoid jet-lag. We will use the data from this app to determine how the world sleeps. This presents a new paradigm in mathematical biology research where large-scale computing bridges the gap between basic mechanisms and human behavior and yields hypotheses that can be rapidly tested using mobile technology.

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.

Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Of special importance are changes that occur early during malignant transformation because they may result in oncogene addiction and represent promising targets for therapeutic intervention. Here we describe a mathematical approach, called Retracing the Evolutionary Steps in Cancer (RESIC), to deduce the temporal sequence of genetic events during tumorigenesis from crosssectional genomic data of tumors at their fully transformed stage. When applied to a dataset of 70 advanced colorectal cancers, our algorithm accurately predicts the sequence of APC, KRAS, and TP53 mutations previously defined by analyzing tumors at different stages of colon cancer formation. We further validate the method with glioblastoma and leukemia sample data and then apply it to complex integrated genomics databases, finding that high-level EGFR amplification appears to be a late event in primary glioblastomas. RESIC represents the first evolutionary mathematical approach to identify the temporal sequence of mutations driving tumorigenesis and may be useful to guide the validation of candidate genes emerging from cancer genome surveys.

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.

A multiscale approach to characterize macromolecular dynamics and functions The understanding of emerging collective behaviors in biomolecular complexes represents a major challenge in modern biophysics. As a first step toward the study of such processes we have applied multi-resolution nonlinear dimensionality reduction and diffusion analysis to obtain reliable low-dimensional representations and models for the dynamics of apparently high-dimensional complex systems such as proteins in a biological environment. The results clearly show that the proposed methods can efficiently find low dimensional representations of a complex process such as protein folding.

Ductal carcinoma in situ (DCIS) is an early stage non-invasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. We will discuss a mathematical model that incorporates the cross-talk between stromal and tumor cells, and which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-$\beta$ signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation, and describe a hybrid model for the interaction of cells with the tumor microenvironment. The analysis sheds light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggests how epigenetic changes in transformed cells affect tumor progression.

Geodesic complexity of the d-dimensional boundary S of a convex polytope of dimension d+1 is intimately related to the combinatorics of nonoverlapping unfolding of S into a Euclidean space R^d following Miller and Pak (2008). This combinatorics is based on facet sequences, which are lists of adjacent facets traversed by geodesics in S. Our main result bounds the geodesic complexity of S from above by the number of distinct maximal facet sequences traversed by shortest paths in S. For d=2, results from the literature on nonoverlapping unfolding imply that this bound is polynomial in the number of facets. In arbitrary dimension d, a reinterpretation of conjectures by Miller and Pak (2008) leads to the conjecture that the geodesic complexity of S is polynomial in the number of facets. The theory and results developed here hold more generally for convex polyhedral complexes. This is joint work with Ezra Miller.

Two-sample tests are an important class of problems in statistics, with abundant applications ranging from astrophysics to zoology. However, much of the previous art assumes the data samples live in finite dimensional Euclidean space. Here, we consider a foray into two-sample testing when the objects live in a non-Euclidean space, with special emphasis on graph valued observations. Via embedding each graph into Euclidean space, and then learning a manifold along which the reside, we demonstrate the existence of a test such that for a given confidence level alpha, we obtain power > alpha. Simulations and real data applications demonstrate the pragmatic utility of our approach even for very large graphs.

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.

It is common to use a multitype branching process to model the accumulation of mutations that leads to cancer progression, metastasis, and resistance to treatment. In this talk I will describe results about the time until the first type k (cell with k mutations) and the growth of the type k population obtained in joint work with Stephen Moseley, and their use in evaluating possible screening strategies for ovarian cancer, work in progress with Duke undergraduate Kaveh Danesh. The point process representation of the limit, which is a one-sided stable law, together with results from 10-60 years ago leads to remarkable explicit formulas for Simpson's index and the size of the largest clone. These results are important in understanding tumor diversity which can present serious obstacles to treatment. The last topic is joint work with Jasmine Foo, Kevin Leder, John Mayberry, and Franziska Michor

The cellular cytoskeleton ensures the dynamic transport, localization, and anchoring of various proteins and vesicles. In the development of egg cells into embryos, messenger RNA (mRNA) molecules bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive the effective velocity and diffusivity of cargo at large time for a general class of problems. Including the geometry of the microtubule filaments allows for better prediction of particle localization and for investigation of previously unexplored mechanisms. Our numerical studies incorporating model microtubule structures suggest that anchoring of mRNA-molecular motor complexes may be necessary in localization, to promote healthy development of oocytes into embryos.

In butterflies and moths, wing scales provide the cellular basis for coloration. Many times these scales are filled with pigments providing colors such as black, brown, red, and yellow. In some cases, however, color is generated structurally, which is often the case for green and blue coloration. Optical physicists have done a remarkable job defining how wing nanostructures create structural color, but we know little about the developmental basis for creating the patterned nanostructures. I will describe our work to understand the developmental and cellular basis for scale formation and the generation of structural color, focusing on a group of butterflies known as the Achillides swallowtails. The scales of these butterflies use a combination of multilayer reflection and scale geometry to create a range of colors. Developmentally, the scale geometry appears to be controlled by cytoskeletal reorganization, and evolutionary changes in geometry appear to contribute to variation between species, between populations, and between seasonal variants.

Despite the impressive performance of rituximab (R) containing regimens like R-CHOP in CD20+ Non-Hodgkin’s Lymphoma (NHL), 30-60% of R-naïve NHL patients are estimated to be resistant, and approximately 60% of those patients will not respond to subsequent single agent R treatment. Given that antibody dependent cell mediated cytotoxicity (ADCC) and phagocytosis (ADCP) are thought to be the major mechanisms of action of Rituximab, increasing the activation levels of natural killer (NK) and macrophage (MP) cells may be one strategy for overcoming R resistance.

During (and after) the Fields Institute Industrial Problem Solving Workshop in August 2019, academic participants and industry mentors developed and calibrated to literature data a quantitative systems pharmacology (QSP) model of ADCC/ADCP to interrogate which mechanisms of R resistance could be overcome by increased NK or MP activation, and how much effector cell activation would be required to overcome a given degree and mechanism of R resistance.

This work was motivated by a real-world pharmaceutical drug development question, and the academic-industry interactions during and after the workshop resulted in sharknado plots as well as a published QSP model (presented at American Association of Cancer Research Annual Meeting, 2021) that was able to address some of the key questions around overcoming R resistance. The published model was then incorporated into an in-house QSP model supporting the development of a Takeda investigational drug which is being developed to restore R sensitivity in an R-resistant patient population.

Visual processing begins in the outer plexiform layer of the retina, where
bipolar, horizontal, and photoreceptor cells interact. In vertebrates, the
onset of dim backgrounds can enhance small spot flicker responses of
retinal horizontal cells. This flicker response is called background-
induced flicker enhancement. The underlying mechanism for the feedback
is unclear but competing hypotheses have been proposed. One is the GABA
hypothesis, which states that the inhibitory neurotransmitter GABA,
released from horizontal cells, mediates the feedback by blocking calcium
channels. Another is the ephaptic hypothesis, which contends that calcium
entry is regulated by changes in the electrical potential within the
intersynaptic space between cones and horizontal cells. In this study, a
continuum spine model of cone-horizontal cell synaptic circuitry is
formulated. The model captures two spatial scales - the scale of an
individual synapse and the scale of the receptive field involving hundreds
to thousands of synapses. We show that the ephaptic mechanism produces
reasonable qualitative agreement with the temporal dynamics exhibited by
flicker enhancement experiments. We find that although GABA produces
enhancement, this mechanism alone is insufficient to reproduce the
experimental results. We view this multiscale continuum approach as a
first step in formulating a multi-layer mathematical model of retinal
circuitry, which would include the other ‘brain nuclei’ within the retina:
the inner plexiform layer where bipolar, amacrine, interplexiform, and
ganglion cells interact.