## Christine Heitsch : The Combinatorics of RNA Branching

- Mathematical Biology ( 301 Views )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.

## Grzegorz A. Rempala, PhD DSc : Contact Processes and Stochastic Models of Epidemics

- Mathematical Biology ( 201 Views )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.

## Seth Sullivant : Statistically-Consistent k-mer Methods for Phylogenetic Tree Reconstruction

- Mathematical Biology ( 165 Views )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.

## Darryl Shibata : Reconstructing Human Tumor Ancestries from their Genomes: Making Human Tissues Talk

- Mathematical Biology ( 162 Views )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.

## Adriana Dawes : Protein localization at the single cell level: Interplay between biochemistry, mechanics and geometry

- Mathematical Biology ( 152 Views )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.

## Jacob Scott : Understanding the evolution of resistance: a comprehensive and integrated mathematical and experimental research program.

- Mathematical Biology ( 149 Views )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.

## Aziz Yakubu : Mathematical Models of Malaria with Applications to Mali and USA

- Mathematical Biology ( 148 Views )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.

## Jim Nolen : Sticky limit theorems for statistics in singular spaces.

- Mathematical Biology ( 144 Views )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.

## Phil Holmes : The neural dynamics of decision making: multiple scales in a single brain

- Mathematical Biology ( 140 Views )I will describe a range of models, from the cellular to cortical scales, that illuminate how we perceive stimuli and make decisions. Large networks composed of individual spiking neurons can capture biophysical details of neuromodulation and synaptic transmission, but their complexity renders them opaque to analysis. Employing methods of mean field and dynamical systems theory, I will argue that these high-dimensional stochastic differential equations can be reduced to simple drift-diffusion processes used by cognitive psychologists to fit behavioral data. This allows us to relate them to optimal methods from statistical decision theory, and prompts new questions on why we fail to make good choices.

## Jake Taylor-King : Generalized Jump Processes and Osteocyte Network Formation

- Mathematical Biology ( 139 Views )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.

## L. Ridgway Scott : Digital biology: protein-ligand interactions

- Mathematical Biology ( 136 Views )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.

## Jill Galagher : Targeting the phenotype: Treatment strategies for heterogeneous cancer

- Mathematical Biology ( 132 Views )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 tumors 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.

## Anita Layton : Myogenic Response to Systolic Pressure in the Afferent Arteriole

- Mathematical Biology ( 131 Views )Elevations in systolic blood pressure are believed to be closely linked to the pathogenesis and progression of renal diseases. It has been hypothesized that the afferent arteriole (AA) protects the glomerulus from the damaging effects of hypertension by sensing increases in systolic blood pressure and responding with a compensatory vasoconstriction. To investigate this hypothesis, we developed a mathematical model of the myogenic response of an AA smooth muscle cell, based on an arteriole model by Gonzalez-Fernandez and Ermentrout (Math Biosci 1994). renal hemodynamic regulation. The model incorporates ionic transport, cell membrane potential, contraction of the AA smooth muscle cell, and the mechanics of a thick-walled cylinder. The model represents a myogenic response based on a pressure-induced shift in the voltage dependence of calcium channel openings: with increasing transmural pressure, model vessel diameter decreases; and with decreasing pressure, vessel diameter increases. Further, the model myogenic mechanism includes a rate-sensitive component that yields constriction and dilation kinetics similar to behaviors observed in vitro. A parameter set is identified based on physical dimensions of an AA in a rat kidney. Model results suggest that the interaction of Ca2+ and K+ fluxes mediated by voltage-gated and voltage-calcium-gated channels, respectively, gives rise to periodicity in the transport of the two ions. This results in a time-periodic cytoplasmic calcium concentration, myosin light chains phosphorylation, and crossbridges formation with the attending muscle stress. Further, the model predicts myogenic responses that agree with experimental observations, most notably those which demonstrate that the renal AA constricts in response to increases in both steady and systolic blood pressures. The myogenic model captures these essential functions of the renal AA, and it may prove useful as a fundamental component in a multi-scale model of the renal microvasculature suitable for investigations of the pathogenesis of hypertensive renal diseases.

## Aaron Fogelson : Two Examples of Chemical Modulation of the Properties and Dynamics of Physiological Gels: Fibrin Formation and Mucin Swelling

- Mathematical Biology ( 129 Views )Gels formed from mixtures of polymers and solvent are ubiquitous in physiological systems. The structure and properties of a gel can change dramatically in response to chemical modulation. Two examples of the influence of chemistry on gel properties will be discussed. The structure of fibrin gels that form during blood clotting is strongly influenced by the concentration of the enzyme thrombin that produces the fibrin monomers from which the gel is built. Presumably at higher thrombin concentrations, fibrin monomers are produced more rapidly from the precursor fibrinogen molecules. I will present an analysis of a possible mechanism of fibrin branching that can explain the sensitivity of fibrin structure to the rate of supply of monomers. Mucin gel is released from vesicles in goblet cells. During this exocytotic process, the polyelectrolyte mucin gel swells to many times its original volume at a very rapid rate. I will present a model in which this swelling is triggered by an exchange of divalent calcium ions in the vesicle and monovalent sodium ions in the extracellular space, and in which the ion concentrations and the rheological properties of the mucin gel determine its equilibrium size and the dynamics of its swelling.

## Rafael Meza : Applications of stochastic models of carcinogenesis in cancer prevention

- Mathematical Biology ( 126 Views )Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of (stochastic) genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carcinogenesis. In particular, two stochastic (mechanistic) models, the Armitage-Doll and the two-stage clonal expansion (TSCE) model, have been widely used in the past for cancer risk assessment and for the analysis of cancer population and experimental data. In this talk, I will introduce some of the biological and mathematical concepts behind the theory of multistage carcinogenesis, and discuss in detail the use of these models in cancer epidemiology and cancer prevention and control. Recent applications of multistage and state-transition Markov models to assess the potential impact of lung cancer screening in the US will be reviewed.

## Carina Curto : Convex neural codes

- Mathematical Biology ( 124 Views )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.

## Ezra Miller : Applying persistent homology to brain artery and vein imaging

- Mathematical Biology ( 124 Views )Persistent homology measures geometric structures using topological invariants. When the structures are magnetic resonance images of branching arteries, for example, persistent homology records the connectedness of an increasing subset of the vessels. Although the theory of persistent homology is relatively well developed, and many aspects of its behavior are understood in synthetic examples, only recently have applications to genuine experimental data begun. This talk explains what we have learned about the geometry of blood vessels in aging human brains, as well as lessons this exploration has taught us about applications of persistent homology in general. These lessons inform further potential applications of persistent homology in statistical problems from biological and medical imaging. The main results are joint with Paul Bendich, Steve Marron, Aaron Pieloch, and Sean Skwerer (Math junior faculty, Stat faculty, Math undergrad, and Operations Research grad student). The talk will be accessible to advanced mathematics and statistics undergraduates, medical and biological researchers, statistics and mathematics faculty, and everybody in between.

## Avner Friedman : Conservation laws in mathematical biology

- Mathematical Biology ( 122 Views )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.

## Susan Holmes : Computational Tools for Evaluating Phylogenetic and Hierarchical Clustering Trees

- Mathematical Biology ( 121 Views )Inferential summaries of tree estimates are useful in the setting of evolutionary biology, where phylogenetic trees have been built from DNA data since the 1960's. In bioinformatics, psychometrics and data mining, hierarchical clustering techniques output the same mathematical objects, and practitioners have similar questions about the stability and `generalizability' of these summaries. I will present applications of the Billera, Holmes, Vogtman (2001) distance to inferential problems both in the frequentist (bootstrap) and Bayesian contexts. I will compare the tree of trees representation to the Euclidean approximations of treespace made available through Multidimensional Scaling of the matrix of distances between trees. We also provide applications of the distances between trees to hierarchical clustering trees constructed from microarrays and phylogenetic trees of metagenomic data of bacteria in the gut. This talk contains joint work with John Chakerian and Alfred Spormann.

## Tom Kepler : Microevolution in the Immune System: A Computational Systems Approach

- Mathematical Biology ( 121 Views )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.

## L. Mahadevan : Mechanochemistry and motility: individual and collective behavior

- Mathematical Biology ( 120 Views )I will describe three different types of problems inspired by the dynamics of the cytoskeleton: (i) the structural and kinetic aspects of dynamic instability in macromolecular assemblies, (ii) the collective flagella-like dynamics of ordered assemblies of active particles and cells, and (iii) the onset of oscillations and the persistence of strain in disordered aggregates of motors and filaments. In all cases, I will show how simple aspects of geometry, chemical kinetics and statistical and continuum mechanics allow us to explain biological observations in a minimal setting.

## Katarzyna Rejniak : Fluid dynamics in cancer cell biology

- Mathematical Biology ( 119 Views )Eukaryotic cell microenvironment (inner and outer) is composed in large parts from fluids that interact with solid and elastic bodies, whereas it is the cell cytoplasm, cytoskeleton and basal membrane; the interstitial fluid interpenetrating the stroma and tumor cells; or blood flow carrying the immune or circulating tumor cells. I will discuss the use of two fluid-structure interactions methods, the immersed boundary and the regularized Stokeslets, in applications dealing with the tumor development and treatment. First model operates on the cellular scale and will be used to model various cell processes, such as cell growth, division or death, during the cellular self-organization into a normal mammary acinus, a 3D in vitro structure recapitulating the morphology of breast cysts (acini). I will discuss model development, parameterization and tuning with the experimental data, as well as their subsequent use to investigate the link between morphogenesis of epithelial mutants and molecular alterations of tumor cells. Second model acts on the tissue level, and will be used to investigate the relation between tumor tissue structure and efficacy of anticancer drugs in the context of interstitial fluid flow. I will present simulation results showing non-linear relation between tumor tissue structure and effectiveness of drug penetration. I will also discuss how tumor tissue metabolic state(its oxygenation and acidity) becomes modified due to actions of chemotherapeutic drugs leading to the emergence of tumor zones with potentially drug-resistant cells and/or to tumor areas that are not exposed to drugs at all. Both of these phenomena can contribute to the moderateclinical success of many anticancer drugs.

## Joshua Goldwyn : Analysis of stochastic auditory nerve models with applications to cochlear implant psychophysics

- Mathematical Biology ( 119 Views )Cochlear implants are neural prostheses that restore a sense of hearing to individuals with severe to profound deafness. Two fundamental theoretical questions that we face are: How does the auditory nerve respond to electrical stimulation? And how is sound information represented in the spike trains of auditory nerve fibers? We will discuss model-based efforts to investigate these questions. I will focus on the development of reduced models that incorporate essential biological features of this complicated system, and remain useful tools for analyzing neural coding.

Using a point process model of the auditory nerve, I simulate amplitude modulation detection, a common test of temporal resolution. I find that the temporal information in the simulated spike trains does not limit modulation sensitivity in cochlear implant users, and discuss how the point process framework can be extended to include additional biophysical mechanisms. Next, I illustrate how spatial spread of excitation and neural degeneration can lead to of within- and across-patient variability in listening outcomes. This points toward an important goal of computational modeling: to develop patient-specific models that can be used to optimize stimulation strategies for individual cochlear implant users.

## Jim Keener : Mechanisms of length regulation of flagella in Salmonella

- Mathematical Biology ( 117 Views )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.