Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly.

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.

The notochord, the defining feature of chordates, is a pressurized tube which actuates elongation of the chordate embryo. The zebrafish notochord consists of large vacuolated cells surrounded by a thin sheath. We characterized the patterns of the cells�?? packing, and their relationship to the known regular patterns from the study of foams, and irregular patterns in a gel bead system. Disruption of the wild type packing pattern leads to developmental defects. We characterize the bifurcations between the relevant regular patterns in terms of nondimensional geometrical and mechanical ratios, and suggest an important developmental role for the eccentric "staircase" pattern.

Cancers are caused by mutations that may be inherited, induced by environmental factors, or result from DNA replication errors (R). We studied the relationship between the number of normal stem cell divisions and the risk of 17 cancer types in 69 countries throughout the world. The data revealed a strong correlation (median = 0.80) between cancer incidence and normal stem cell divisions in all countries, regardless of their environment. The major role of R mutations in cancer etiology was supported by an independent approach, based solely on cancer genome sequencing and epidemiological data, which suggested that R mutations are responsible for two-thirds of the mutations in human cancers. All of these results are consistent with epidemiological estimates of the fraction of cancers that can be prevented by changes in the environment. Moreover, they accentuate the importance of early detection and intervention to reduce deaths from the many cancers arising from unavoidable R mutations.

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.

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.

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.

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.

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.

Many mathematical models in biology can be described by a system of hyperbolic conservation laws with nonlinear and nonlocal coefficients. In order to determine these coefficients one needs to solve auxiliary systems of equations, for example elliptic or parabolic PDEs, which are coupled to the hyperbolic equations. In this talk we give several examples: The growth of tumors, the shrinking of dermal wounds, T cell differentiation, the growth of drug resistant bacteria in hospitals, and the transport of molecules along microtubules in axon. In these examples, the auxiliary systems range from elliptic-parabolic free boundary problems to nonlocal ODEs. Motivated by biological questions, we shall describe mathematical results regarding properties of the solutions of the conservation laws. For example, we shall determine the stability of spherical tumors and the growth of �?fingers;�? we shall discuss conditions for shrinking of the wound; suggest how to reduce the growth of drug resistant bacteria, and derive biologically motivated asymptotic behavior of solutions.

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.

Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due  to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around.

New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection.

Abstract: The construction of flagellar motors in motile bacteria such as Salmonella is a carefully regulated genetic process. Among the structures that are built are the hook and the filament. The length of the hook is tightly controlled while the length of filaments is less so. However, if a filament is broken off it will regrow, while a broken hook will not regrow. The question that will be addressed in this talk is how Salmonella detects and regulates the length of these structures. This is related to the more general question of how physical properties (such as size or length) can be detected by chemical signals and what those mechanisms are. In this talk, I will present mathematical models for the regulation of hook and filament length. The model for hook length regulation is based on the hypothesis that the hook length is determined by the rate of secretion of the length regulatory molecule FliK and a cleavage reaction with the gatekeeper molecule FlhB. A stochastic model for this interaction is built and analyzed, showing excellent agreement with hook length data. The model for filament length regulation is based on the hypothesis that the growth of filaments is diffusion limited and is measured by negative feedback involving the regulatory protein FlgM. Thus, the model includes diffusion on a one-dimensional domain with a moving boundary, coupled with a negative feedback chemical network. The model shows excellent qualitative agreement with data, although there are some interesting unresolved issues related to the quantitative results.

The macroscopic diffusion constant for small ions in biological environments is in part dependent on the volume excluded by diffusional barriers and by long-range interactions between those barriers and the ion. Increasing excluded volume reduces diffusive transport of the solute, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. My colleagues and I have developed a computational methodology for studying these effects via a homogenized finite element method for the Smoluchowski equation. I will introduce the relevant background, both biological and mathematical, and present our recent results. This work is joint with Peter Kekenes-Huskey (U. Kentucky) and J. Andrew McCammon (UC San Diego).

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.

In a 2009 paper Yu, Etheridge and Cuthbertson proposed a model that was intended to address two questions. The first was a question related to Muller's ratchet, "What ratio of mutations must be beneficial for the meanfitness of a population to increase in time?" The second question was related to the Hill-Robertson effect, "If many beneficial mutations are introduced into a population, how much will competition slow the rate of adaptation?" They introduced a model of an asexually reproducing population of fixed size N and mutation rate mu and conjectured that the rate of adaptation is O(logN/(log logN)^2) for large N so long as there is some positive ratio of beneficial mutations. I will present an outline of my proof of this conjecture.

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.

Building models for biological, chemical, and physical systems has traditionally relied on domain specific intuition about which interaction and features most strongly influence a system. Statistical methods based in information criteria provide a framework to balance likelihood and model complexity. Recently developed for and applied to dynamical systems, sparse optimization strategies can select a subset of terms from a library that best describe data, automatically interfering model structure. I will discuss my group's application and development of data driven methods for model selection to 1) find simple statistical models to use wastewater surveillance to track the COVID pandemic and 2) recover chaotic systems models from data with hidden variables. I'll briefly discuss current preliminary work and roadblocks in developing new methods for model selection of biological metabolic and regulatory networks.

Understanding and predicting cancer progression requires detailed interacting models of tumor and stromal cells, all calibrated to experimental data. Work to date has been limited by a lack of standardization of data representations of multicellular systems, though this is now being addressed through MultiCellDS (MultiCellular Data Standard) and digital cell lines, which are standardized representations of microenvironment-dependent cell phenotypes. Computational cancer modelers require biologically and mathematically consistent initialization routines to seed simulations with cells defined in digital cell lines. In this talk, we will briefly introduce a 3-D agent-based model designed for use in integrative computational biology. We introduce a �?snapshot generator�? that can take a digital cancer cell line and produce for the agent-based model an initial cell arrangement and a phenotypic state based upon analyses of the digital cell line data elements. We demonstrate 2-D monolayer and 3-D hanging drop simulations up to 500k MCF7 cells, a common breast cancer cell line. We additionally demonstrate the production of digital snapshots, standardized simulation output that will facilitate computational model comparison with a common core of analytical tools. With an early version of these tools, we assess the match between simulations and in vitro experiments. In the future, this work will be used to create and simulate combinations of tumor and stromal cells from appropriate digital cell lines in realistic tissue environments in order to understand, predict, and eventually control cancer progression in individual patients.

Prior to cell division, chromosomes are segregated by mitotic spindle. This molecular machine self-assembles remarkably fast and accurately in an elegant process of search and capture, during which dynamic polymers from two centers grow and shrink rapidly and repeatedly until a contact with chromosomes is established. At the same time, the chromosomes interact with each other. I will show computer simulation and experimental microscopy results illustrating how cell deploys a number of redundant mechanisms optimizing the assembly by keeping it accurate yet rapid. Very similar strategies are used to assemble Golgi apparatus. The talk will illustrate how computer modeling assists experiment in unraveling mysteries of cell dynamics.

A preliminary examination of a very rich data set consisting of a detailed survey of individuals in male-male sex networks in South India. The motivation for the study is to understand the spread of HIV in male-male sex networks in South India. The data contains survey information from participants, as well as their cell phone contacts and incomplete information on the contacts by participants. We provide predictive models of attributes of contacts given participant attributes, as well as predictive models of the attributes, such as sexual position. We study how model parameters vary as a function of connectedness of individuals and how modeling network interactions has an effect on the model.

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.

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.

Heterogeneity in cancer is an observed fact, both genotypically and phenotypically. Cell-cell variation is seen in almost all aspects of cancer from early development all the way through to invasion and subsequent metastasis. Our current understanding of this heterogeneity has mainly focused at the genetic scale with little information on how this variation translates to actual changes in cell phenotypic behavior. Given that many genotypes can lead to the same cellular phenotype, it is important that we quantify the range and scope of this heterogeneity at the phenotypic scale as ultimately this variability will dictate the aggressiveness of the tumor and its treatability. Central to our understanding of this heterogeneity is how the tumor cells interact with each other and with their microenvironment. Since it is these very interactions that drive selection and that ultimately define the ecology of the tissue in which the tumor is developing. Considering an organ as an ecological system, means that we should view normal tissue homeostasis as an equilibrium that cancer cells must disrupt if they are to be successful. Disruption of this equilibrium is often one of the first events in cancer development, as the normal control mechanisms of the tissue are damaged or ignored. We will discuss the interplay between homeostasis, heterogeneity, evolution and ecology in cancer progression and treatment failure with an emphasis on the metabolism of breast cancer.

We discuss two types of partial differential equation models of fishery harvesting problems. We consider steady state spatial models and diffusive spatial-temporal models. We characterize the distribution of harvest effort which maximizes the harvest yield, and in the steady state case, also minimizes the cost of the effort. We show numerical results to illustrate various cases. The results inform ongoing debate about the use of reserves (regions where fishing is not allowed), and are increasingly relevant as technology enables enforcement of spatially structured harvest constraints.

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

A major feature of biological sciences in the 21st Century is their transition from phenomenological and descriptive disciplines to quantitative and predictive ones. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while preserving the fundamental physics in complex biological systems. We discuss a multiscale multiphysics and multidomain paradigm for biomolecular systems. We describe macromolecular system, such as protein, DNA, ion channel, membrane, molecular motors etc., by a number of approaches, including macroscopic electrostatics and elasticity and/or microscopic molecular mechanics and quantum mechanics; while treating the aqueous environment as a dielectric continuum or electrolytic fluids. We use differential geometry theory of surfaces to couple various microscopic and macroscopic domains on an equal footing. Based on the variational principle, we derive the coupled Poisson-Boltzmann, Nernst-Planck, Kohn-Sham, Laplace-Beltrami, Newton, elasticity and/or Navier-Stokes equations for the structure, function, dynamics and transport of protein, protein-ligand binding and ion-channel systems.

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.