Genetic intratumoral heterogeneity is a natural consequence of imperfect DNA replication. Any two randomly selected cells, whether normal or cancerous, are therefore genetically different. I will discuss the extent of genetic heterogeneity within untreated cancers with particular regard to its clinical relevance. While genomic heterogeneity within primary tumors is associated with relapse, heterogeneity among treatment‑naïve metastases has not been comprehensively assessed. We analyzed sequencing data for 76 untreated metastases from 20 patients and inferred cancer phylogenies for breast, colorectal, endometrial, gastric, lung, melanoma, pancreatic, and prostate cancers. We found that within individual patients a large majority of driver gene mutations are common to all metastases. Further analysis revealed that the driver gene mutations that were not shared by all metastases are unlikely to have functional consequences. A mathematical model of tumor evolution and metastasis formation provides an explanation for the observed driver gene homogeneity. Last, we found that individual metastatic lesions responded concordantly to targeted therapies in 91% of 44 patients. These data indicate that the cells within the primary tumors that gave rise to metastases are genetically homogeneous with respect to functional driver gene mutations and suggest that future efforts to develop combination therapies have the capacity to be curative.

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.

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.

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.

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.

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.

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.

It has long been accepted that type 1 diabetes results from a lack of insulin, as the insulin-secreting pancreatic beta cells are destroyed by an autoimmune process. In contrast, the cause of type 2 diabetes (T2D) is less clear. Most people with pre-diabetes or in the early stages of T2D have abnormally high plasma insulin concentrations, and insulin rises before glucose does. We show that these difficulties are resolved by a mathematical model in which the onset of T2D is represented by the crossing of a threshold. The threshold is atypical in some respects and requires consideration of the slow manifolds to avoid incorrect conclusions.

A conference in honor of Harold Layton's 60th Birthday (Sunday Session)

A conference in honor of Harold Layton's 60th Birthday (Afternoon Session)

A conference in honor of Harold Layton's 60th Birthday (Morning Session)

I will discuss some old and new results related to the analysis of stochastic SIR-type epidemics on a configuration model (CM) random graph having a fixed degree distribution p_k. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based process. I will also discuss the applications of the LLN approximation in building a "network-free" SIR Markov hybrid model which can be used for epidemic parameters inference. The hybrid model idea appears particularly relevant in the context of the recent Ebola and the Zika epidemics.

Fluid-structure interaction problems with composite structures arise in many applications. One example is the interaction between blood flow and arterial walls. Arterial walls are composed of several layers, each with different mechanical characteristics and thickness. No mathematical results exist so far that analyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structure is composed of several layers. In this talk we will summarize the main difficulties in studying this class of problems, and present a computational scheme based on which a proof of the existence of a weak solution was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of the thin fluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoretical results on modeling the human cardiovascular system will be discussed. This is a joint work with Boris Muha (University of Zagreb, Croatia), Martina Bukac (U of Notre Dame, US) and Roland Glowinski (UH). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL, Switzerland), and with A. Quaini (UH). Collaboration with medical doctors Dr. S. Little (Methodist Hospital Houston) and Dr. Z. Krajcer (Texas Heart Institute) is also acknowledged.

We report the results of our integrated experimental and theoretical investigations of biological systems dominated by interfacial effects. Particular attention is given to elucidating natural strategies for water-repellency, walking on water, underwater breathing, and drinking.

The evolution of resistance remains an elusive problem in the treatment of both cancer and infectious disease, and represents one of the most important medical problems of our time. While the illnesses are different on several non-trivial levels including timescale and complexity, the underlying biological phenomenon is the same: Darwinian evolution. To comprehensively approach these problems, I have focussed my attention on building a broad suite of investigations centered around the causes and consequences of the evolutionary process in these contexts. I will discuss my and my collaborator's efforts to; model the evolutionary process on the genomic scale in both an analytic (Markov process) and stochastic (individual based model and inference) format; to quantify in vitro competition and interaction between cancer cell lines through an evolutionary game theoretic lens using time-lapse microscopy and computer vision; and to understand the evolutionary contingencies inherent in collateral sensitivity in E. coli and ALK mutated non-small cell lung cancer.

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.

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.

Cells are highly organized and complex structures, with the capacity to segregate specific factors to spatially disjoint regions in a process called polarization. Polarization, which specifies a spatial axis in the cell, is a highly conserved biological process and is required for proper embryonic development, wound healing, and many other normal and pathological biological functions. Despite the importance of polarization, we do not fully understand how this protein segregation is initiated and maintained. In this talk, I will show how we can use numerical and analytical approaches to investigate how symmetry breaking begins the process of polarization, and how the geometry of the cell may play a role in the establishment and maintenance of spatial patterns associated with polarization.

It is well-known that genomes encode ancestry through replication errors - on average the greater the numbers of differences between two genomes, the greater the time since they shared a common ancestor ("molecular clock hypothesis"). This approach is commonly used to infer ancestries of species and populations, and these same tools can be applied to somatic cell evolution, in particular to better infer ancestries of normal and neoplastic tissues. For example, by sampling from opposite sides of the same human colorectal tumor, one can coalesce back to the earliest tumor cells. Such studies reveal that many human colorectal tumors are simple single "Big Bang" expansions, with evidence of neutral evolution during growth. It may be possible to understand in detail what is impossible to directly observe - the first few divisions of human tumors.

Frequencies of k-mers in sequences are sometimes used as a basis for inferring phylogenetic trees without first obtaining a multiple sequence alignment. We show that a standard approach of using the squared-Euclidean distance between k-mer vectors to approximate a tree metric can be statistically inconsistent. To remedy this, we derive model-based distance corrections for orthologous sequences without gaps, which lead to consistent tree inference. The identifiability of model parameters from k-mer frequencies is also studied. Finally, we report simulations showing the corrected distance out-performs many other k-mer methods, even when sequences are generated with an insertion and deletion process. These results have implications for multiple sequence alignment as well, since k-mer methods are usually the first step in constructing a guide tree for such algorithms. This is joint work with Elizabeth Allman and John Rhodes.

In this talk, we will introduce a deterministic malaria model for determining the drug administration protocol that leads to the smallest first malaria episodes during the wet season. To explore the effects of administering the malaria drug on different days during wet season while minimizing the potential harmful effects of drug overdose, we will define 40 drug administration protocols. Our results fit well with the clinical studies of Coulibaly et al. at a site in Mali. In addition, we will provide protocols that lead to small number of first malaria episodes during the wet season than the protocol of Coulibaly et al. In the second part of the talk, we will use our malaria model to "capture" the 2013 Centers of Disease Control and Prevention (CDC) reported data on the 2011 number of imported malaria cases in the US. Furthermore; we will use our "fitted" malaria models for the top 20 countries of malaria acquisition by US residents to study the impact of protecting US residents from malaria infection when they travel to malaria endemic areas, the impact of protecting residents of malaria endemic regions from mosquito bites and the impact of killing mosquitoes in those endemic areas on the 2013 CDC malaria surveillance data.

My talk will have two parts. PART I, From Birds to Bacteria: Generalised Velocity Jump Processes. There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. In this talk, I will give a mathematical description of this process, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. We derive an expression for the mean squared displacement, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution. We also consider the Levy regime where the variance of the running distribution tends to infinity. This leads to a fractional diffusion equation for superdiffusive Levy walks and can be solved analytically. Our theory opens up new perspectives both for the systematic derivation of such equations, and for experimental data analysis of intermittent motion. I will also briefly discuss recent developments (by other researchers) within the field of velocity jump processes. PART II: Modelling Osteocyte Network Formation: Healthy and Cancerous Environments. Advanced prostate, breast, and lung cancer can metastasize to bone. In pathological bone, the highly regulated bone remodeling signaling pathway is disrupted. Within bone dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone formation. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, preliminary studies measuring the number of osteocytes per unit area using pathology slides show that the number density of osteocytes change from healthy to metastatic prostate and breast cancer xenografted mice. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and number density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We consider variations of our model to test specific physiological hypotheses related to osteoblast differentiation; for example we can predict how changing measurable biological parameters, such as rates of bone secretion, rates of dendrite growth and rates of osteoblast differentiation can allow for qualitatively different network morphologies, and vice versa. We thenuse our model to hypothesize reasons for the limited efficacy of zoledronate therapy on metastatic breast cancer.

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.

The digital nature of biology is crucial to its functioning as an information system, as well in building hierarchical components in a repeatable way. We explain how protein systems can function as discrete components, despite the importance of non-specific forces due to the hydrophobic effect. That is, we address the question of why proteins bind to ligands predictably and not in a continuous distribution of places, the way grease forms into blobs. We will give a detailed description of how data mining in the PDB can reveal how proteins interact. We highlight the role of the hydrophobic effect, but we see that it works inversely to the usual concept of hydrophobic interaction. Our work suggests the need for a more accurate model of the dielectric effect in the vicinity of a protein surface, and we discuss some advances in this direction. Our research also provides an understanding of how molecular recognition and signaling can evolve. We give an example of the use of our ideas in drug design.

Targeted cancer drugs attack pathway specific phenotypes and can lead to very positive outcomes when a particular phenotype dominates the population of a specific tumor. However, these drugs often fail because not all cells express the targeted phenotype to the same degree. This leads to a heterogeneous response to treatment, and ultimate recurrence of the cancer as sensitive cells die off and resistant cells take over. We explore how treatment strategies informed by a tumor’s phenotypic mix, can help slow the emergence of resistance and stave off tumor recurrence. We use an off-lattice agent-based model that incorporates inheritance of two phenotypes – proliferation rate and migration speed – and is modulated by a space limiting selection force. We find how and when distinct distributions of phenotypes require different treatment strategies.

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.