Complex interactions occur between tumor and host immune system during cancer development and treatment, and a weak systemic immune response can be prognostic of poor patient outcomes. We strive to not only better understand the dynamic behavior of circulating immune cell populations before and during cancer therapy, but also to monitor these dynamic changes to facilitate real-time prediction of patient outcomes and potentially therapy adaptation. I will provide examples of both theoretical (mathematical) and data-driven (epidemiological) approaches to incorporating established systemic immune markers into clinical decision-making. First, coupling models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy, in turn identifying the optimal treatment target for maximum reduction of global tumor burden. Second, improved understanding of how circulating immune markers vary both within and between individual patients can allow more accurate risk stratification at diagnosis, and personalized prediction of patient response to therapy. The importance of multi-disciplinary collaborations in making predictive and prognostic models clinically relevant will be discussed.

Israel experienced an outbreak of wild poliovirus type 1 (WPV1) in 2013-14, detected through environmental surveillance of the sewage system. No cases of acute flaccid paralysis were reported, and the epidemic subsided after a bivalent oral polio vaccination (bOPV) campaign. As we approach global eradication, polio will increasingly be detected only through environmental surveillance. However, we have lacked the theory to translate environmental surveillance into public health metrics; it is a priori unclear how much environmental surveillance can even say about population-level disease dynamics. We developed a framework to convert quantitative polymerase chain reaction (qPCR) cycle threshold data into scaled WPV1 and OPV1 concentrations for inference within a deterministic, compartmental infectious disease transmission model. We used differential algebra and profile likelihood techniques to perform identifiability analysis, that is, to assess how much information exists in the data for the model, and to quantify inference uncertainty. From the environmental surveillance data, we estimated the epidemic curve and transmission dynamics, determining that the outbreak likely happened much faster than previously thought. Our mathematical modeling approach brings public health relevance to environmental data that, if systematically collected, can guide eradication efforts.

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.

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.

Tyrosine kinase inhibitors (TKIs), such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). However, the majority of patients are not cured for undetermined reasons. It turns out that many patients who otherwise responded well to IM therapy still show variations in their BCR-ABL transcripts. In this talk we will overview mathematical models for leukemia, drug resistance, and stem cells. Our main focus will be on our recent results concerning mathematical models that integrate CML and an autologous immune response.  This is a joint work with G. Clapp, T. Lepoutre, and F. Nicolini.

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.

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.

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.

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.

The use of delay differential equations (DDEs) to study biological phenomena has a long history, when the rate of change of model variables depends their previous history. Today, DDEs occupy a central place in models of infectious disease dynamics, epidemiology, ecology and tumor growth. In this talk, I will present a delayed partial differential equation (PDE) model of tumor growth and treatment. The model accounts for cell cycle arrest and cell death induced by chemotherapy, and explicitly includes intracellular signaling pathways relevant to drug action. The model is simplified to give a 1D hyperbolic PDE, which is further reduced to a nonlinear, non-autonomous DDE by projecting along characteristics. Necessary and sufficient conditions for the global stability of the cancer-free equilibrium are derived and conditions under which the system evolves to periodic solutions are determined. This has clinical implications since it leads to a lower bound for the amount of therapy required to affect a cure. Finally, I will present a clinical application of the model, by applying it to the treatment of ovarian cancers. Two types of drugs are considered – platinum-based chemotherapeutic agents that are the current standard of care for most solid tumors, and small molecule cell death inducers that are currently under development. The model is calibrated versus in vitro experimental results, and is then used to predict optimal doses and administration time scheduling for the treatment of a tumor growing in vivo.

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.

In an experimental study of single enzyme reactions, it has been proposed that the rate constants of the enzymatic reactions fluctuate randomly, according to a given distribution. To quantify the uncertainty arising from random rate constants, it is necessary to investigate how one can simulate such a biochemical system. To do this, we will take the Gillespie's stochastic simulation algorithm for simulation the evolution of the state of a chemical system, and study a modification of the algorithm that incorporates the random rate constants, using in part the Metropolis-Hastings algorithm to enact the distribution on the random rate constants. This modified algorithm, when applied to the single enzyme reaction system, produces simulation outputs that are corroborated by the experimental results. This project is in its early stages, and it is hoped that it can subsequently be used as a tool for the estimation or calibration of parameters in the system using experimental data.

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.

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.

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.

RNA-Seq is a new technology for measuring the content of a transcriptome using high-throughput sequencing technology. I will provide a self-contained introduction to the technology, and proceed to discuss some interesting mathematical questions we have had to address in order to realize the potential of "comparative transcriptomics" for comparing and contrasting transcriptomes. We will start with the "freshman's dream", and proceed to examine issues related to maximum matching, the (phylogenetic) space of trees and Simpson's paradox. This is joint work with my current and former students Natth Bejraburnin, Nicolas Bray, Adam Roberts, Cole Trapnell and Meromit Singer.

ABSTRACT: The past decade has seen a number of engineering innovations that make construction of devices of micro- and even nanometric dimensions feasible. Hence, there is a growing interest in exploring new and efficient ways to generate propulsion at these small scales. Here we explore optimization of one particular type of low Reynolds number propulsion mechanism – flagella. Beyond the general challenges associated with optimization, there are a number of issues that are unique to swimming at low Reynolds numbers. At small scales, the fluid equations of motion are linear and time-reversible, hence reciprocal motion – i.e., strokes that are symmetric with respect to time reversal – cannot generate any net translation (a limitation commonly referred to as the Scallop Theorem). One possible way to break this symmetry is through carefully chosen morphologies and kinematics. One symmetry-breaking solution commonly employed by eukaryotic microorganisms is to select nonreciprocal stroke patterns by actively generating torques at fixed intervals along the organism. Hence, we will address the question: For a given morphology, what are the optimal kinematics? In this talk we present optimal stroke patterns using biologically inspired geometries such as single-tailed spermatozoa and the double-tail morphology of Chlamydomonas, a genus of green alga widely considered to be a model system in molecular biology.

Dinoflagellates swim due to the action of two eucaryotic flagella - a trailing, longitundinal flagellum that propagates planar waves, and a transverse flagellum that propagates helical waves. The transverse flagellum wraps around the cell in a plane perpendicular to the trailing flagellum, and is thought to provide both forward thrust along with rotational torque. Motivated by the intriguing function of this transverse flagellum, we study the fundamental fluid dynamics of a helically-undulating ring in a viscous fluid.  We contrast this biofluiddynamic study, where the kinematics of the waveform are taken as given, with a model of mammalian sperm hyperactivated motility.  Here, our goal is to examine how the complex interplay of fluid dynamics, biochemistry, and elastic properties of the flagellum give rise to the swimming patterns observed.  

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.