### 2009 Workshop for Young Researchers in Mathematical Biology (WYRMB): Abstracts and Lecture Materials

#### Plenary Speakers

Multiscale Modeling in Biology
Mark Alber, Departments of Mathematics and Physics, Interdisciplinary Center for the Study of Biocomplexity, University of Notre Dame

Multiscale modeling approach typical of systems biology tends to mix continuous, discrete, deterministic, and probabilistic submodels.

To prevent the loss of blood following a break in blood vessels, components in blood and the vessel wall interact rapidly to form a thrombus (clot) to limit hemorrhage. In this talk we will describe a multiscale model of thrombus formation consisting of components for modeling viscous, incompressible blood plasma; coagulation pathway; quiescent and activated platelets; blood cells; activating chemicals; fibrinogen; the vessel walls and their interactions. Model has been tested for robustness with respect to fluctuations of basic parameters [1]. Simulation results demonstrate the development of an inhomogeneous internal structure of the thrombus which was confirmed by the preliminary experimental data. Also, the dependence of the thrombus size on the blood flow rate in simulations is close to the one observed experimentally [2]. As heterogeneous structural domains within the clot affect thrombus stability, understanding the factors influencing thrombus structure is of significant biomedical importance.

Swarming, a collective motion of many thousands of cells, produces colonies that rapidly spread over surfaces. In the second half of the talk a detailed cell- and behavior-based computational model of M. xanthus swarming will be used to show that reversals of gliding direction are essential for swarming and that the reversal period predicted to maximize the swarming rate is the same as the period observed in experiments [3]. This suggests that the circuit regulating reversals evolved to its current sensitivity under selection for growth achieved by swarming. Also, an orientation correlation function will be used to show that microscopic social interactions help to form the ordered collective motion observed in swarms.

1. Xu, Z., Chen, N., , Kamocka, M.M., Rosen, E.D., and M.S. Alber [2008], Multiscale Model of Thrombus Development, Journal of the Royal Society Interface 5, 705-722.
2. Xu, Z., Chen, N., Shadden, S., Marsden, J.E., Kamocka, M.M., Rosen, E.D., and M.S. Alber [2009], Study of Blood Flow Impact on Growth of Thrombi Using a Multiscale Model, Soft Matter 5, 769-779.
3. Wu, Y., Jiang, Y., Kaiser, D., and M. Alber [2009], Periodic reversal of direction allows Myxobacteria to swarm, Proc. Natl. Acad. Sci. USA 106 4 1222-1227 (featured in the Nature News, January 20th, 2009, doi:10.1038/news.2009.43).
Socially-induced reproductive synchrony in a seabird colony
Shandelle Henson, Department of Mathematics, Andrews University

Spontaneous oscillator synchrony has been documented in a wide variety of electrical, mechanical, chemical, and biological systems, including the menstrual cycles of women and oestrous cycles of Norway rats. In temperate regions, many colonial birds breed seasonally in a time window set by photoperiod; some studies have suggested that heightened social stimulation in denser colonies can lead to a tightened annual reproductive pulse. It has been unknown, however, whether the analogue of menstrual synchrony occurs in birds, that is, whether avian ovulation cycles can synchronize on a daily timescale within the annual breeding pulse. In this talk I will present data on every-other-day egg-laying synchrony in a breeding colony of glaucous-winged gulls (/Larus glaucescens/) and show that the level of synchrony declined with decreasing colony density. I will also discuss a discrete-time mathematical model based on the hypothesis that preovulatory luteinizing hormone surges synchronize through social stimulation.

Math Biology Experience A View from the IT Industry
Kirk Jordan, Computational Science Center, IBM T.J. Watson Research

Computation is playing an ever increasing and vital role in the biological and healthcare sciences. In many instances, scientists are developing mathematical models and using high performance computing to carry out analysis and simulations that provide insight into biological systems. The complexity of these models often demands increasing compute power and sophisticated mathematics for the solution. In collaboration, math bio scientists will play a pivotal role in using thousands of processors to look at biological problems in new ways, leading to science breakthroughs. In this talk, I will briefly describe some of the challenges facing computational community and outline through a couple of examples where I believe math biology experience may have impact. Briefly, I will mention areas of opportunity that a math biology background might be a fit. In conclusion, I will point out some of the computational trends that I believe hold opportunity for coupling high performance computing and mathematics to tackle multi-scale biological science problems.

Tumor-Immune Interaction, Surgical Treatment, and Cancer Recurrence in a Mathematical Model of Melanoma
Yang Kuang, Department of Mathematics and Statistics, Arizona State University, Tempe

Malignant melanoma is a cancer of the skin arising in the melanocytes. We present a mathematical model of melanoma invasion into healthy tissue with an immune response. We use this model as a framework with which to investigate primary tumor invasion and treatment by surgical excision. We observe that the presence of immune cells can destroy tumors, hold them to minimal expansion, or, through the production of angiogenic factors, induce tumorigenic expansion. We also find that the tumor-immune system dynamic is critically important in determining the likelihood and extent of tumor regrowth following resection. We find that small metastatic lesions distal to the primary tumor mass can be held to a minimal size via the immune interaction with the larger primary tumor. Numerical experiments further suggest that metastatic disease is optimally suppressed by immune activation when the primary tumor is moderately, rather than minimally, metastatic. Furthermore, satellite lesions can become aggressively tumorigenic upon removal of the primary tumor and its associated immune tissue. This can lead to recurrence where total cancer mass increases more quickly than in primary tumor invasion, representing a clinically more dangerous disease state. These results are in line with clinical case studies involving resection of a primary melanoma followed by recurrence in local metastases.

Optimal Control in Epidemic Models of Rabies in Raccoons
Suzanne Lenhart, Mathematics, Department, University of Tennessee

After introducing some basic background of optimal control, we discuss frameworks to investigate temporal/spatial controls for vaccine distribution as it impacts the spread of rabies among raccoons. The control gives amount and location of the food packets containing vaccine. The goal is to minimize the number of infected raccoons and the cost of distributing the packets.

#### Short Talks

A diffusion-activation model of CaMKII translocation waves in dendrites
Berton Earnshaw, Department of Mathematics, University of Utah

Ca2+-calmodulin-dependent protein kinase II (CaMKII) is a key regulator of glutamatergic synapses and plays a necessary role in the expression of many forms of synaptic plasticity. A recent study observed that stimulating dendrites locally with a single glutamate/glycine puff initially induces a local translocation of CaMKII into spines that subsequently spreads in a wave-like manner towards the distal dendritic arbor. Here we show that a minimal mathematical model of the diffusion, activation and translocation into spines of dendritic CaMKII can reproduce this wave of translocation. Using mathematical analysis and numerical simulations, we determine how the wave speed depends on the rates of diffusion, activation and translocation. In addition to these testable predictions, our model also predicts that translocation will fail to spread when the translocation rate into spines is greater than the activation rate. Our analysis provides a quantitative framework for understanding the spread of CaMKII translocation and its possible role in heterosynaptic plasticity.

The Dynamics of Exit from Mitosis
Baris Hancioglu, Department of Biological Sciences, Virginia Tech

Cell cycle events in eukaryotes are regulated by periodic activation and inactivation of a family of cyclin–dependent kinases (Cdks). Entry into mitosis is initiated by accumulation Cdk in complexes with B-type cyclins Clb1 and Clb2, and exit from mitosis requires inactivation of these Cdk-cyclin complexes and dephosphorylation of Cdk targets. The Cdks are inactivated by Cdc20- and Cdh1-dependent proteolysis of Clb1 and Clb2 and by binding with inhibitors Sic1, and dephosphorylation is carried out by Cdc14, an essential phosphatase promoting mitotic exit. We have developed a deterministic ODE model for the control of Cdc14 release from the nucleolus as budding yeast cells exit from mitosis. Our model provides a rigorous account of the factors affecting the dual exit pathways, called FEAR (Cdc14 early anaphase release) and MEN (mitotic exit network). The model captures the dynamics of mitotic exit in wild-type and mutant yeast cells, including many details of the ! physiology, biochemistry and genetics of the process. We propose a novel mechanism for multiphosphorylation of Net1 (an inhibitor of Cdc14) by several kinases: Cdk, Cdc5 (Polo) and Dbf2/Mob1 (through activation Cdc15). Understanding how Polo-like kinases fit into the exit pathways is important because Polo-like kinases are being actively pursued as therapeutic targets in the treatment of human cancer.

Modeling Cartilage Regeneration in the Extracellular Environment of a Cell-Seeded Hydrogel
Janine M. Haugh, Department of Mathematics, North Carolina State University

Articular cartilage is a connective tissue lining the surfaces of bones in diarthrodial joints such as hips, knees, and shoulders. As a natural biomaterial, cartilage is important for load support, energy distribution, and lubrication of joints, but can become damaged due to injury or osteoarthritis. Cartilage has a limited capacity for repair and growth that is regulated by cells, called chondrocytes, that are sparsely distributed throughout the tissue’s extracellular matrix (ECM). In recent years, the potential use of nutrient-rich hydrogels seeded with cartilage cells as biomaterials for tissue regeneration and repair has seen wide interest. In this study, we develop mathematical models for cartilage regeneration in the local environment of a single cell seeded in a hydrogel scaffold. A spherical geometry is employed with three concentric regions: the cell, its extracellular matrix, and the hydrogel. Radially symmetric reaction-diffusion equations are used to describe the coupling of nutrient and synthesized (unlinked) matrix concentrations in the region. Several models are considered to describe the process by which unlinked matrix proteins react with the hydrogel to form linked extracellular matrix. Numerical solutions are based on finite difference methods and capture motion of the evolving gel-tissue interface through a level set approach. The resulting models are used to conduct a parametric analysis that quantifies tissue regeneration times in terms of underlying biophysical parameters in the models.

Emergent Group Dynamics Produce a Robust Primary T Cell Response
Peter Kim, Department of Mathematics, University of Utah

The currently accepted paradigm for the primary T cell response is that effector T cells commit to autonomous developmental programs. This paradigm is based on experiments that study the dynamics of T cell responses over a wide variety of stimulation levels and initial conditions. This work studies the hypothesis that the dynamics of a primary response may also be governed by adaptive regulatory cells in addition to intrinsic developmental programs.

We formulate two mathematical models based on developmental programs. In one model, effector cells may undergo a fixed number of divisions before dying. In the second model, effector cells live for a fixed time during which they may divide. The study of these models suggests that developmental programs are highly sensitive to precursor frequencies.

Consequently we derive a third model based on the principle that adaptive regulatory T cells develop in the course of an immune response and suppress effector cells. Our simulations show that this feedback mechanism responds robustly over a range of at least four orders of magnitude of precursor frequencies.

We conclude that the proliferation program may be enhanced by an emergent group dynamic generated by interactions between effector and regulatory T cells. Our hypothesis is that additional feedback control mechanisms may be operating alongside the T cell proliferation program. This poster presents a connection between the analysis of experimental immunology and hypothesis formulation using the theory of dynamical systems.

Calculating tree distances: A divide-and-conquer approach
Simone Linz, Department of Computer Science, University of California, Davis

Calculating a distance between two phylogenetic trees is a central task in various evolutionary studies that aim at comparing trees. For example, the rooted subtree prune and regraft (rSPR) distance is often used to calculate a lower bound on the number of reticulation events, such as horizontal gene transfer or hybridization, that is needed to simultaneously explain two phylogenetic trees on the same set of present-day species. However, calculating this distance is an NP-hard problem and practical algorithms for computing it exactly are rare. We present a divide-and-conquer approach to calculate the rSPR distance which breaks the problem instance into a number of smaller and more tractable subproblems. By applying an efficient bottom-up strategy, we highlight an algorithm that makes use of this new theoretical result and calculates the rSPR distance exactly. This is joint work with Charles Semple (University of Canterbury, New Zealand).

Model formulation and optimal intervention strategies for cholera
Rachael Miller Neilan, Department of Mathematics, University Tennessee

Cholera is a diarrheal illness caused by infection of the intestine with the bacterium Vibrio cholerae. While cholera has been a recognized disease for about two hundred years, we still do not have a strategy for effective control of outbreaks. Previous studies have indicated three components essential for modeling the spread of cholera. These components are (1) a hyperinfectious, short-lived bacterial state; (2) a separate class for mild or inapparent human infections; and (3) a loss of one’s immunity to the disease over time. To better understand appropriate intervention strategies, we formulate a mathematical model that includes these components as well as control functions representing the effects of oral rehydration therapy, antibiotic treatment, vaccination, and sanitation. A new result quantifies contributions to the basic reproductive value from both mild and severe infections as well as both hyperinfectious and less infectious Vibrio cholerae. We use optimal control theory, parameter sensitivity analysis, and numerical simulations to illustrate the intervention strategies that minimize death due to disease and the cost of treatment during a cholera outbreak. Our results demonstrate that the ratio of mild to severe infections within a population along with the rates of recovery, death due to disease, and waning immunity are important factors in determining the optimal balance of multiple intervention methods. Specifically, these parameters affect the optimal duration of vaccination and sanitation efforts as well as the optimal quantity of antibiotic and rehydration treatment. We view our mathematical results as framework for policy makers designing cost-effective strategies for the control of a cholera outbreak.

The intermittency of synchronized activity in Basal Ganglia
Choongseok Park, Department of Mathematical Sciences, IUPUI

The basal ganglia (BG) are a group of interconnected subcortical nuclei which are involved in neural control of movement, as well as certain aspects of cognition and emotion. Dysfunction of the basal ganglia is associated with movement disorders such as Parkinson's disease (PD), a common and progressive age-related neurodegenerative disorder of movement. Recent studies indicate that patterns of oscillatory synchronous activity in BG are strongly relevant to BG physiology and BG disorders. In particular, neuronal activity in the beta-band significantly contributes to akinetic symptoms.

We use extracellular spiking activity and LFPs which are recorded from PD patients during surgery for implantation of DBS electrodes. We also use conductance-based models of subthalamic and pallidal cells in BG.

The results of the analysis indicate that the dynamics of beta-band oscillations in BG are marked by intermittency of synchronized episodes. Oscillations tend to be desynchronized for relatively short time while the desynchronizing events are quite frequent. Modeling work shows that the domain of the existence of intermittent dynamics is in between incoherent regime and synchronized regime, in the area which is characterized by the presence of different types of dynamics. These results suggest that in a disease (Parkinsonism), BG circuits are relatively close to the presumably healthy uncorrelated state. This closeness of the irregular healthy state to the pathological regular synchronized state may be justified by the efficiency of producing synchronized oscillations for movement generation.

Spatial Temporal regulation of epidermal development
Yuyu Luke Peng, Center for Mathematical and Computational Biology, UCI , Department of Mathematics

The epidermis (skin) is a stratified epithelium and regenerates itself throughout postnatal life. Self-renewing stem cells are located in the basal layer and they produce transiently amplifying (TA) progenitor cells that subsequently exit the cell cycle and terminally differentiate as they migrate toward the skin surface. Maintenance of stable cell population in the epidermis is important for robust regeneration of the stratified epithelium. The population size is usually regulated by cell secreted extra-cellular signaling molecules as well as intra-cellular molecules. In this poster, models incorporating both levels of regulation are developed to examine the balance between growth and differentiation for the stem cell population. In particular, the dynamics of known differentiation regulators Ovol1, Ovol2, TGFb and c-Myc, their threshold dependent differentiation, and feedback regulation on maintaining stable cell population are investigated.

How co-colonization affects competition of hospital-acquired and community-acquired MRSA
Joanna Pressley, Department of Mathematics, Vanderbilt University

Historically, Methicillin-resistant Staphylococcus aureus (MRSA) infections occurred in immunocompromised hospital patients. More recently, a new strain of MRSA has been detected in the general community (CAMRSA), differing genetically from the original strain (HAMRSA) and causing infections in healthy people. Initial data suggests that CAMRSA is becoming more common and is also possibly replacing HAMRSA in hospitals. Under the assumption that patients can only be colonized with one strain of MRSA at a time, previous modelers have found that competitive exclusion occurs between HAMRSA and CAMRSA strains in the hospital; the strain with the larger basic reproductive ratio will become endemic while the other is extinguished. But new studies suggest that patients can be co-colonized with multiple strains of MRSA. Here, we present a dynamical model composed of ordinary differential equations that allows patients to be colonized with both HAMRSA and CAMRSA (the co-colonization model). Converse to previous results derived from the assumption that co-colonization is impossible, the co-colonization model rarely exhibits competitive exclusion. More commonly, both strains become endemic in the hospital. Beyond competitive exclusion, we analyze the qualitative effects of increased risk factors, as well as two interventions (decolonization efficacy and hand-washing compliance) on the percentage of patients colonized with MRSA.

Microscopic and macroscopic models for chemotaxis
Zhian Wang, Institute for Mathematics ad Its Applications, University of Minnesota

The microscopic (kinetic) model describes the individual cell behavior and the macroscopic (keller-segel) model describes the cell population movement. The understanding of the connection of these two models with different spatial and temporal scales is an important issue in the multi-scale modeling of chemotaxis. We will discuss the parabolic limits, steady states as well as traveling waves of these two models to explore their correlations.

#### Posters

Improving Speed and Accuracy in Stochastic Simulation via Higher Order Leaping
Alexander V. Alekseyenko, Department of Statistics, Stanford University

Stochastic simulation methods are important in modeling chemical reactions and biological and physical stochastic processes describable as continuous-time discrete-state Markov chains with a infinite number of reactant species and reactions. An emerging area of application for such methods is in systems biology models, where stochastic component dominates the kinetics due to small molecule numbers or large separation of reaction scales. The current algorithm of choice, tau-leaping, achieves fast and accurate stochastic simulation by taking large time steps that leap over individual reactions. During a leap interval (t, t + tau) in tau-leaping, each reaction channel operates as a Poisson process with a constant intensity. We modify tau-leaping to allow linear and quadratic changes in reaction intensities, yielding a process with inhomogeneous kinetics. Because our version of τ-leaping accurately anticipates how intensities change over time, we propose calling it the step anticipation tau-leaping (SAL) algorithm. Empirical examples corroborate the theoretical advantages of SAL. In a typical simulation SAL is more accurate and/or takes fewer steps than ordinary tau-leaping. The degree of improvement varies with the situation. Near stochastic equilibrium, reaction intensities are roughly constant, and SAL and ordinary tau-leaping perform about equally well, while when the changes in reaction intensities are more likely SAL confidently dominates the ordinary tau-leaping.

A model of cardiac action potential that incorporates stochastic caveolar dynamics
Ian Besse, Applied Mathematical and Computational Sciences, University of Iowa

Recent investigations into the structure and function of cardiac caveolae, small invaginations of a cardiomyocyte's plasma membrane, reveal that treatment of a myocyte with a beta-agonist opens caveolar necks and presents the sarcolemma with additional sodium ion channels. As such, caveolae constitute a substantial and previously unrecognized source of sodium current that can significantly influence cardiac action potential morphology and the conduction velocity of the excitatory wave front. In this work, we formulate a model of cardiac action potential that incorporates stochastic caveolar dynamics and simulate the effects of caveolar sodium current on the action potential. In particular, we use this model to suggest that in pathological cases, caveolae may play a role the development of the late persistent sodium current known to cause the cardiac arrhythmia called long Q-T syndrome.

Discrete time optimal control of species augmentation: Augment then Grow
Erin N. Bodine, Department of Mathematics, University of Tennessee

Species augmentation is a method of reducing species loss via augmenting declining/threatened populations with individuals from captive-bred or stable, wild populations. We developed a difference equations model and optimal control formulation for discrete time augmentation of a general declining population. We numerically optimize our objective functional and show numerical results for scenarios of different illustrative parameter sets.

Phylogenetic Invariants
Julia Chifman, Department of Mathematics, University of Kentucky

Phylogenetics is the area of research concerned with finding the genetic relationship between species. The relationship can be represented by a phylogenetic tree, which is a simple, connected, acyclic graph equipped with some statistical information. This furnishes a certain polynomial map and we are interested in polynomials, called phylogenetic invariants, which vanish for every choice of model parameters. The set of phylogenetic invariants forms a certain algebraic object and we want to compute this object explicitly. One of the reasons that we want an explicit description of these polynomials is because it is claimed by Casanellas and Fernandez-Sanchez that using the entire set of phylogenetic invariants is an efficient phylogenetic reconstruction method. More importantly, phylogenetic invariants were used by Allman and Rhodes to study the problem of identifiability of tree topology for a number of phylogenetic models. In other words, given a distribution of observations that a certain model predicts, is it possible to uniquely determine all the parameters of the model? It is an important question since, if a tree is not uniquely determined by an expected joint distribution, then we cannot use that model for inference. This presentation/poster will explore in some detail different phylogenetic models, their invariants and the progress that has been made for a certain group of models. (The content is drawn from the joint work with Sonja Petrović, UIC , petrovic@math.uic.edu)

Mechanisms important in the formation of co-evolutionary patterns in mutualists
Carrie Diaz Eaton, Department of Mathematics, University of Tennessee

Ecological Network Theory is a rapidly growing field in Mathematical Biology. Most work is phenomenological and focuses on fixed nodes with varying connections strengths. I am interested instead in which mechanisms of network formation and evolution may be important mechanistically. Through simulations of trait evolution and co-evolution, I examine the role of intra-class competition and inter-class cooperation in determining network patterns between phylogenies of mutualists. Simple null models are used to illustrate the importance of co-evolution. Correlative co-evolution trajectories are derived from genetically explicit modeling work between classes of mutualistic species. This work is an important in linking Ecological Network Theory with Evolutionary Theory.

An analytically derived amplitude equation to describe alternans in cardiac tissue
Lisa Driskell, Department of Mathematics, Purdue University

An arrhythmia is any disturbance from the normal periodicity of the heart beat, and alternans is one type of arrhythmia which can be a precursor to potentially fatal conditions of the heart. Dynamical systems have the ability to describe cardiac action potentials and are an invaluable tool in the pursuit to understand cardiac electrical activity. In this research, I use existing models capturing the behavior of electrical wave propagation in the heart along with techniques of singular perturbation theory to derive and analyze an equation describing the onset and development of alternans in cardiac tissue.

Computational Analysis of an Integrated Model of the Dopamine Synapse
Betty Dulam-Banawa, Physics Department, Ludwig Maximillians University

Recent studies have reinforced the important role of synaptic processes for understanding brain function and pathologies. Despite its importance for understanding diseases such as Schizophrenia and Parkinson’s disease, detailed models of the dopamine synapse have been rare until recently. The poster will present on the one hand, work on an integrated model of the dopamine synapse, which combines work by Qi et al [1] on the dopamine metabolism in the pre-synapse and a post-synapse by Fernandez et al [2] which focused on DARPP-32 as a robust integrator of dopamine and glutamate signaling. The integrated model is formulated using the Biochemical Systems Theory (BST), in particular in Generalized Mass Action (GMA) form. The second aspect that will be presented is the work on developing a general reduction method for GMA models using singular perturbation theory as well as its application to the integrated model. Sensitivity and stability analyses of the reduced model will also be performed. Future work will involve model in the context of Schizophrenia, as an important characteristic of this disease is an oversupply of dopamine in the neurons in the mesolimbic area and the striatum. (This work will involve close collaboration with researchers in the LMU Department of Psychiatry and the Isar-Amper clinic in Munich, Germany)

1. Qi, Z., Miller G.W. and Voit, E.O. A Mathematical Model of Presynaptic Dopamine Homeostasis: Implications for Schizophrenia. Pharmacopsychiatry, 41, 2008.
2. Fernandez, E., Schiappa, R., Girault, J. and Le Novère, N. DARPP-32 is a Robust Integrator of Dopamine and Glutamate Signals. Plos Computational Biology, 2, 2006

Memory T cell dynamics under toxicant exposure
Horacio Gomez, UAMS, Pediatrics

Trichroloethylene (TCE) is a common environmental toxicant that induces several immunological changes. TCE is known to increase the expression the memory marker CD44 in T and also it stimulates the production of the proinflammatory cytokine IFN-g. We explore the changes on the dynamics of T cells exposed to TCE with a system of ordinary differential equations. Based on the experiments in mice, we obtain parameter estimates in the context of toxicant exposure. We also analyze the changes on the memory T cell dynamics based on the IFN-g levels.

Treatment Strategies: Accounting for Resistance
Elsa Hansen, Mathematics and Statistics, Queen's University

Mathematical models are playing an increasingly important role in understanding the dynamics of epidemics and determining the best strategies for public health interventions. There are numerous issues that should be considered when deciding how best to treat an infectious disease. Using optimal control theory, we address this question for a deterministic SIR type model with mass action contact rates and characterize the treatment strategy that minimizes the total outbreak size.

The framework of our problem has been designed to focus on two issues. Firstly we investigate how the emergence of a treatment resistant strain changes the optimal treatment strategy. We show that the presence of resistance can, but does not always, change the optimal treatment strategy. We also detail how the basic reproduction numbers of the regular and resistant strains play a crucial role in determining the form of the optimal treatment strategy. Secondly we discuss how limited treatment resources change the optimal treatment strategy.

Autoimmunity selects for incomplete mimicry in parasites
Amy Hurford, Department of Mathematics and Statistics, Queen's University

The severity of an infection depends jointly on the traits of the infecting parasite and the nature of the host's immunological response. Parasitic molecular mimics share a high degree of structural homology with naturally occurring host peptides. Infection with a molecular mimic has been shown to later result in various autoimmune diseases of the host; for example, Campylobacter jejuni infections have been linked to Guillain-Barre syndrome. This may provide molecular mimics with a survival advantage when host selection against autoimmune disease eliminates the immune system receptors that destroy these parasites. This motivates the question, why are all parasites not molecular mimics? We formulate an infection-autoimmunity model describing the epidemiological dynamics and use optimization methods to understand how molecular mimicry affects the fitness of parasite strains. We show that even when the frequency of immune receptors decreases with increasing self-reactivity there is selection for incomplete mimicry in the parasite. This result is driven by an assumption that the total reactivity of the parasite in the space of possible immune receptors is constrained. Selection for incomplete mimicry depends on the number of secondary infections generated by hosts that have autoimmune diseases, chronic infections and acute infections and the durations of these types of infections. Darwinian medicine highlights that the true efficacy of medical interventions goes beyond the immediate alleviation of disease symptoms in a single individual. B-cell deletion therapies are currently under development as a means to prevent Rheumatoid arthritis (an autoimmune disease). We discuss the implications of this type of therapy in the context of selective forces and the resulting prevalence of disease.

Dynamics of a plant-herbivore model
Yun Kang, Department of Applied Mathematics, Arizona State University

We formulate a simple host-parasite type model to study the interaction of certain plants and herbivores. Our two dimensional discrete time model utilizes leaf and herbivore biomass as state variables. The parameter space consists of the growth rate of the host population and a parameter describing the damage inﬂicted by herbivores. We present insightful bifurcation diagrams in that parameter space. Bistability and a crisis of a strange attractor suggest two control strategies: Reducing the population of the herbivore under some threshold or increasing the growth rate of the plant leaves.

Length-induced switch-like behavior revealed in an open signaling cascade
Yongfeng Li, IMA, University of Minnesota

Signal transduction pathways are the wires of cellular information transfer that transmit messages to regulate all biochemical events. These pathways consist of series of chemical reactions called as cascades that enable the proteins to switch between an active and an inactive state. This ON-or-OFF(or All-or-None) phenomenon is referred as bistability in biology. It is widely accepted that the feedback is required to induce bistability. In this work, a GK type model is proposed to show that the bistability-type behavior can be produced without feedback when the cascade is sufficiently long.

Noise Attenuation through Non-receptor in Morphogen Gradients
Wing-Cheong Lo, Department of Mathematics, University of California, Irvine

Morphogen gradients orchestrate the patterning of developing tissues. Due to the perturbation of morphogen synthesis, randomness of binding processes, how to make morphogen gradients robust and precise become a major question for many researchers. Presence of non-receptor and positive feedback control on receptor clearance are suggested to answer this question. In this work, the constraints and the objectives of these strategies are studied by the mathematical and computational analysis.

Systems biology approach to mastitis
Nicoline van Loenen, Lincoln University, New Zealand, AGLS

Systems Biology Approach to Mastitis Bovine mastitis is the result of an inflammation in the mammary gland caused by a variety of bacteria in the udder and is one of the three major causes of disease in the Dairy industry world wide. Around parturition and during early lactation, E. coli mastitis is leading to several cow deaths each year. In the USA, mastitis has been estimated to cost the industry US$1.7 billion annually (Kerr et al., 2001). The host's immune reaction to mastitis is an important factor for the outcome of the disease. Isolates of bovine mammary epithelial cells were cultured and challenged with pathogenic E. coli bacteria for 1, 3, 6 and 24 hours. In order to obtain an insight in the host's immune reaction, microarray analyses were performed. As microarray analysis typically surveys the expression profile of more than 20,000 genes at once, it can greatly assist in the identification of key genes and pathways involved in the cow's immune defense system. Eight clusters with distinctly different expression profiles indicated the involvement of immune related biological networks of several genes. The complexity of the innate immune system requires a systems approach using mathematical modelling to elicit the dynamical interaction between the different genes in biological networks. This approach assumes that genes function through biological networks and investigates the dynamic interaction of the elements in a particular network. Here we propose a mathematical model of 78 ordinary differential equations to increase understanding of qualitative and quantitative aspects of the interactions of the complex innate immune system reactions. The insights obtained can be used for the identification of new strategies to combat mastitis; the identification of informative biological experiments and the development of pharmaceutical products, breeding targets and vaccines. The effect of random movement on the design and functioning of MPAs Urmila Malvadkar, Mathematics, Statistics, and Computer Science, St. Olaf College Marine protected areas (MPAs) are a conservation tool growing in popularity due to claims of a vast array of benefits, including protecting fish populations, protecting habitat, ensuring fishery yield against uncertainty, and increasing fishery yield in adjacent areas. Fisheries focuses on the last item: under what circumstances will establishing an MPA increase fishing yield? Much modeling work on this topic considers sedentary species with a single larval dispersal phase; this work extends the discussion to mobile species, describing the population using nonlinear PDEs. The size of a yield-maximizing MPA generally decreases with increasing growth rates and increases asymptotically with increasing fishing mortalities. For species with mobile adults, MPAs produce a larger yield than effort control under a few circumstances: fish that can leave the habitat and have a small birth rate, large death rate, and small habitat sizes. Generally, the indirect e! ffects of adult movement on yield through decreased birth rate are greater than the direct effects of adult movement on yield through increased catch. Except for species with very high (>4) ratios of fishing mortality to birth rates, optimal reserve sizes for many species are frequently in the proposed and established ranges of MPA sizes, indicating that these likely produce close to optimal yield. Transmission Dynamics and Underreporting of Kala-azar in the Indian State of Bihar Anuj Mubayi, University of Texas, Arlington "Kala-azar" (or Visceral Leishmaniasis) is a vector borne infectious disease affecting primarily poor communities in tropical and subtropical areas of the world. Bihar, a state in India, has one of the highest prevalence and mortality levels of Kala-azar but the magnitude of the problem is difficult to assess because most of the cases are handled by private health providers who are not required to report them. We study the impact of underreporting using district level reported incidence data from the state of Bihar. We derive expressions for, and compute estimates of Kala-azar's reproduction numbers as well as levels of underreporting for 21 districts of Bihar. The average reproduction number estimates for the state of Bihar range from 1.1 (2003) to 4.3 (2005) with some districts' estimates supporting values less than one in the two years. It is estimated that the proportion of underreported cases declined from about 88% in 2003 to about 73% in 2005. However, our estimates show that at least 5 districts had still over 90% levels of underreporting in both years. Estimated underreporting is adjusted to reported incidence data and high-risk districts are identified. Four out of eight (in 2003) and three out of nine (in 2005) districts are miss-identified as high-risk by reported data. Total of seven (in 2003) and five (in 2005) districts are not even there in the list of high-risk districts according to reported incidence suggesting significantly different targeting of resources. Population dynamics in hemi-parasitic plants: A model Jennifer A. Nelson, Department of Mathematics and Statistics, Cornell College Hemiparasitic plants, such as the Indian Paintbrush or Mistletoe, are plants that both produce their own sugars through photosynthesis and gain nutrients through parasitism of other nearby plants. The degree to which a hemiparasitic plant acts as a parasite can vary between species and individuals, indicating possible tradeoffs between the plant’s ability to parasitize and photosynthesize. Population dynamics in these plants will vary based on the degree to which an individual is parasitic, as well as the degree of competition it suffers from nearby plants. Hemiparasitic plants both compete with their hosts and prey upon them. The possible existence of an optimal trade-off between parasitism and competition will be explored, as well as the effect of this trade off on long-term dynamics. It is predicted when competition between the host and hemiparasite will reduce the density, and therefore impact, of the hemiparasitic plant than situations with lower competi! tion since hemiparasitic plants have been shown to be poorer competitors than their hosts. Fluid Dynamics of the Dinoflagellate Transverse Flagellum Hoa Nguyen, Department of Mathematics, Tulane University The action of the transverse flagellum encircling the dinoflagellate body is investigated from a hydrodynamical point of view. The flagellum is modeled as a closed circular helix and its self-propulsion is achieved by propagating waves along its length. Waves that propagate counter-clockwise along the flagellum give rise to its clockwise motion as well as forward thrust. The grid-free method of regularized Stokeslets is used to understand the fluid dynamics of the flagellum at low Reynolds number. The resulting motion is also analyzed by slender-body theory. On multiple regulatory mechanisms in the tryptophan operon system in Escherichia coli: in silico study of perturbation dynamics Lan Khuyen Nguyen, Center for Advanced Computational Solutions, Agriculture & Life Sciences, Lincoln University, New Zealand Living organisms often exist in uncertain environments where changes are the norm. Cellular systems therefore require resilient regulatory mechanisms for timely and stable adaptation. Among various regulation motifs, multiple feedback control emerges as a common theme. The tryptophan operon system in Escherichia coli regulates the production of intracellular tryptophan using an apparatus of three feedback mechanisms: repression, attenuation and enzyme inhibition; each provides essentially the same function but operates in distinctly different ways. In this study, we carry out a large-scale perturbation/response analysis of the tryptophan (trp) operon system, in order to explore the roles of the individual feedback mechanisms governing this regulatory network [1]. To do this, we first develop an S-systems approximation model of an existing model (proposed by Santillan and Mackey [2]) for the system. To characterise transient responses of dynamical systems, we propose two new measurable quantities: maximum disturbance (MD) and recovery time (RT). Using the developed S-system model and the transient dynamics metrics, we systematically design different large-scale sets of trp “in silico” mutant systems with alternative structures and carry out extensive simulations on these sets. We expose individual mutants within each set to perturbations of different kinds and quantify the system’s respective transient response. Analysis on the resulting quantitative data allows individual as well as combined effects of the core control mechanisms governing the trp system to be unravelled. Our simulation results showed that combined regulation using all three feedback mechanisms significantly increases system stability, broadening the range of kinetic parameters for stable behaviour. Enzyme inhibition was shown to directly control the disturbance level in system variables after perturbations. Attenuation, on the other hand, was found to speed up system recovery whereas repression lengthens recovery time. The method developed in this paper and the defined transient dynamics measurements can be applied to other cellular systems. References 1. Nguyen, L.K. and D. Kulasiri, On multiple regulatory mechanisms in the tryptophan operon system in Escherichia coli: in silico study of perturbation dynamics In Silico Biol, 2008. 8, 0037 . 2. Santillan, M. and M.C. Mackey, Dynamic regulation of the tryptophan operon: a modeling study and comparison with experimental data. Proc Natl Acad Sci U S A, 2001. 98(4): p. 1364-9. Examination of Ductal Carcinoma Morphology using Computational Modeling and Imaging Techniques Kerri-Ann Norton, Computational Biology (BioMaPS), Rutgers University Ductal Carinoma in situ (DCIS) is a pre-invasive carcinoma of the breast characterized by increased proliferation of abnormal cells that have not breached the basement membrane. Currently, the architectural morphology is used in diagnosis of the disease, but a strong correlation between architectural pattern and outcome of the disease is lacking. We use modeling and imaging techniques to better understand the mechanisms that influence the architectural development of DCIS, with the goal of elucidating the paths of progression into the known morphologies. For this research, we have developed a 2D computational model of DCIS progression under the hypothesis that bio-mechanical interactions play a role in cancer development. The model simulates DCIS progression shaped by cellular interactions, reproduction, cell death, migration, and intra-ductal pressure. We find that the model accurately reproduces the four common architectures of DCIS: cribriform, micropapillary, solid and comedo, governed primarily by changes in reproduction, death and pressure. We also find three different time progressions occur in separate regions of the computational parameter space: this is used to generate predictions for conditions that lead to the different architectures. Additionally, we have produced preliminary 3D reconstructions from biopsy specimens. These indicate that microlumens found in certain morphologies of DCIS may form in two different ways: they may form from merging fingers (papillae) or they may form as localized pockets. Thes! e results indicate that tumors that look similar architecturally may in fact evolve along very different paths because of variations in their growth parameters. Hepatitis C virus drug resistance in patients treated with a new protease inhibitor Libin Rong, Theoretical Biology and Biophysics, Los Alamos National Laboratory Telaprevir, a novel HCV protease inhibitor, has demonstrated substantial antiviral activity in patients infected with HCV genotype 1. However, drug resistance variants appear very rapidly after treatment initiation. The exact mechanism underlying the rapid emergence of drug resistance during dosing is not fully understood. We develop a mathematical model to address this issue, and then extend the simple model to a general model with multiple viral strains. We examine the effects of backward mutation and liver cell proliferation on the pre-existence of the mutant virus and the competition between wild-type and drug resistant virus during therapy. Model analysis suggests that mutations during therapy do not play a significant role in the dynamics of various viral strains, although they are capable of generating low levels of HCV variants that would otherwise be completely suppressed because of fitness disadvantages. Liver cell proliferation may not affect the pretreatment frequency of mutant variants, but is able to influence the quasi-species dynamics during therapy. It is the relative fitness of each mutant strain compared with wild-type that determines which strain(s) will dominate the virus population. Our study provides a theoretical framework for exploring the prevalence of pre-existing mutant variants and the evolution of drug resistance during treatment with other HCV protease or polymerase inhibitors. A Theoretical Quantitative Model for Evolution of Cancer Chemotherapy Resistance Ariosto Silva, H Lee Moffitt Cancer Center and Research Institute, Integrated Mathematical Oncology Disseminated cancer remains a nearly uniformly fatal disease. While a number of effective chemotherapies are available, tumors inevitably evolve resistance to these drugs ultimately resulting in treatment failure and cancer progression. We propose that in order to understand the evolutionary dynamics that allow tumors to develop chemo resistance, a comprehensive theoretical quantitative model must be used to describe the interactions of cell resistance mechanisms and tumor microenvironment during chemotherapy. In order to achieve this goal we built a computer model of tumor microenvironment as an avascular spherical tumor mass embedded in a volume of well vascularized tissue and simulated different tumor progression and therapy strategies (chemotherapy protocol, use of pH buffers and glucose restriction) and selected those that led to tumor eradication or prolonged survival in case of resistant tumors. Tumor population was composed of cells presenting different values for the following phenotypes: glucose consumption, acid resistance, drug resistance and proliferation rate. We predict that the use of minimum amount of drug sufficient to keep tumor size stable will yield better results than Maximum Tolerated Dose in tumors presenting a resistant subpopulation. We also expect that restriction of glucose availability through 2-Deoxyglucose or other non-metabolizable glucose analog will reduce viability and ATP-dependent chemo resistance of cells in hypoxic regions of tumor and, together with systemic dministration of pH buffers, reduce glycolysis mediated acidification and acid mediated apoptosis of host cells in tumor-host interface. This work shows that understanding and manipulating tumor microenvironment’s selective forces-during tumorigenesis and treatment- can significantly improve the outcome of chemotherapy. Minimal Moduler Clustering of Protein Interaction Networks Nassim Sohaee, University of Texas Southwestern Medical Center, Clinical Science Biological systems are composed of interacting , separable, functional modules. Identifying these modules is essential to understand the organization of biological systems. In this paper we present a framework to identify minimal modules within biological networks. Evolution of Mutation Rates Christopher C. Strelioff, Department of Microbiology & Molecular Genetics, Michigan State University For any organism, an elevated mutation rate is a double-edged sword, resulting in an increased rate of mutations which can be neutral, beneficial or deleterious. Most mutations are neutral or deleterious, which explains the generally low mutation rates observed across organisms. However, notable and important exceptions have been documented where high mutation rates have been observed in connection with pathogens and cancer. In addition, the emergence of high mutation rates in E. coli has been observed in laboratory populations undergoing adaptation to a new environment. In the present research, we consider an extension of the discrete time quasispecies model to allow for two mutation rates: normal and mutator (high mutation rate). Using this model of mutation and selection in an infinite population we are able to investigate conditions which select for elevated mutation rates. We find two conditions which favor the rise of mutator strains: (1) initial adaptation to a constant and smooth fitness landscape and (2) alternating smooth or rugged fitness landscapes which provide a continuing pressure to rapidly adapt. In the first case, normal (low) mutation rates will dominate in the long-term. However, in the second case, changing environments allow for the long-term dominance of mutators. Prediction of genome-wide novel enhancers/promoters and analysis or histone signature Cenny Taslim, Department of Statistics and Department of Molecular Virology, Immunology & Medical Genetics, The Ohio State University Previous studies have shown many transcription factors regulate gene transcription by binding not only to proximal promoters but also to DNA elements distant from gene promoters (Carroll et al. 2005). It has been demonstrated that distal-proximal binding sites are involved in gene transcriptional activity through chromatin interactions and previous studies have suggested that different cells have different chromatin signatures (Heintzman et al. 2007). However, histone modifications, their locations and roles in the genome have remained unclear. We develop a quantitative approach to characterizing chromatin signature and to predict de novo promoters/enhancers. First, ChIP-seq profiles in approximately 10kb windows around TSS and distal binding sites for six histone modifications are examined. K-means is used to cluster the binding signatures of known promoters/enhancers. The characteristic signature within each cluster will then be modeled. Statistical inference on model-based classification will then be used to accurately predict the location and function of novel regulatory elements. Our results will give insights into the difference of distal/proximal chromatin features in different cells and the relationships between histone modifications and gene transcriptions activity in the human genome. References: 1. Carroll JS et al. (2005). Chromosomewide mapping of estrogen receptor binding reveals longrange regulation requiring the forkhead protein FoxA1. Cell; 122: 33-43. 2. Heintzman et al. (2007). Distinct and predictive chromatin signatures of transcriptional promoters and enhancers in the human genome. Nature Genetics; 39: 311-318. Mathematical Modeling of the Effectiveness of Facemasks on Reducing the Spread of Novel Influenza A (H1N1) Samantha Tracht, Capital University On June 11, 2009, the World Health Organization declared the outbreak of novel influenza A (H1N1) a pandemic. With limited supplies of vaccines and antivirals, countries and individuals are looking at other ways to reduce the spread of novel H1N1, particularly options that are cost effective and relatively easy to implement. Recent experiences with the SARS and 2009 H1N1 epidemics show that people would wear facemasks to protect themselves against infection; however, little research has been done to quantify the impact of the use of facemasks in reducing the spread of disease. We construct and analyze a mathematical model in which a portion of the population wears a facemask during the pandemic. We look at two scenarios, one in which N95 respirators are worn and one in which surgical masks are worn. To estimate the parameter values used for the effectiveness of facemasks, we used available data from studies done on N95 respirators and surgical masks. We conclude from our model that, if worn properly, facemasks are an effective intervention strategy in reducing the spread of novel H1N1. A two strain SIS model with diffusion Necibe Tuncer, Department of Mathematics, University of Florida In this work, we consider a two-strain SIS model with diffusion and coefficients depending on space. One way to understand the role of spatial effects in epidemiology is to consider models with diffusion. Spatial dependence of the coefficients is necessary to account for transmissibility, recovery and other epidemiological characteristics that vary with location. We introduce the basic reproduction number, and invasion numbers for the SIS model and study the properties of the disease-free equilibrium and endemic equilibria. We show that although in the corresponding space-independent SIS model the two strains will exclude each other, nich-partitioning mechanisms in the diffusion model may allow for coexistence of the strains. Our two strain model is based on the single strain SIS model considered in L. J. S. Allen, B. M. Bolker, Y. You, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21, (2008) 1-20. Optimal integral expressions for effective radii in the generalized Born model Zhenli Xu, Department of Mathematics and Statistics, University of North Carolina at Charlotte Fast and accurate calculation of solvation energies has become an essential task in molecular dynamic and Monte Carlo simulations of biomolecules in aqueous environments. The generalized Born theory is one of the most successful macroscopic approaches for electrostatic interactions, which approximately reproduces the result from Poisson or Poisson-Boltzmann equation models through a semi-analytical pairwise summation. The goal of this Poster is to derive better approximations for effective Born radii in the generalized Born model of molecular solvation by analyzing the Kirkwood-series solution of the Poisson equation for a spherical solute. The main focus is on solutes under non-conducting boundaries, where the solute dielectric constant and solvent dielectric constant lead to a non-vanishing ratio$\delta$. A series approximation is developed for the Born radius, in which a dominant leading term is corrected by terms of the order of$O(\delta)$. The series is further approximated by an expansion of volume integrals over the solute domain. Optimal combinations of integrals are then proposed, based on computational cost and accuracy criteria. The combinations are tested on solute molecules with non-spherical geometry including a prolate spheroidal model and a protein molecule. For finite values of$\delta$, the proposed formulas are seen to work better than the models developed previously. The relative accuracy of integral expressions is seen to vary among spherical and non-spherical solutes. Fus3-triggered Tec1 degradation modulates mating transcriptional output during the pheromone respons Su Zhao, Department of Mathematics, University of California, Irvine The yeast transcription factor Ste12 controls both mating and filamentation pathways. Upon pheromone induction, the mitogen-activated protein kinases, Fus3 and Kss1, activate Ste12 by relieving the repression of two functionally redundant Ste12 inhibitors, Dig1 and Dig2. Mating genes are controlled by the Ste12/Dig1/Dig2 complex through Ste12-binding sites, whereas filamentation genes are regulated by the Tec1/Ste12/Dig1 complex through Tec1-binding sites. The two Ste12 complexes are mutually exclusive. During pheromone response, Tec1 is degraded upon phosphorylation by Fus3, preventing cross-activation of the filamentation pathway. Here, we show that a stable Tec1 also impairs the induction of mating genes. A mathematical model is developed to capture the dynamic formation of the two Ste12 complexes and their interactions with pathway-specific promoters. By model simulations and experimentation, we show that excess Tec1 can impair the mating transcriptional output because of its ability to sequester Ste12, and because of a novel function of Dig2 for the transcription of mating genes. We suggest that Fus3- triggered Tec1 degradation is an important part of the transcriptional induction of mating genes during the pheromone response. Mathematical modeling of immune responses in tissues Wen Zhou, Department of Mathematics, Iowa State University A self-regulated comprehensive mathematical framework to model MHC class II-mediated immunity is developed. The MHC II mediated immune response plays an essential role in generating an immune response to different types of pathogens. The function of T cells, APCs, B cells, antibodies are captured to recapitulate the various immune responses against different pathogens based on recognizing them through the phenotypic "pathogen characteristics". The main system is composed of chemotaxis equations and reaction diffusion equations. Threshold effect and delay mechanism is model from both spatial and temporal perspective. Homogenization is used to estimate the cell motility in capillary, which is a reticulated structure. We demonstrate using numerical simulations that the model can successfully respond to broad classes of pathogens. A highly skewed$T_H1$response is generated against some virtual pathogens (e.g. those modeled after Mycobacterium tuberculosis, Leishmania major etc.) and granuloma formation is observed \cite{SuZhouJones}, other virtual pathogens lead to an unskewed or mixed response (e.g. such as leishmania amazonensis etc.) and some virtual pathogens lead to a$T_H1$to$T_H2$switch (modeled after M. avium paratuberculosis), and a$T_H2\$ responses is generated against sole extracellular pathogens (e.g. parasitic worms such as helminths etc.). Both acute and chronic infections are handled by our system with realistic responses. Using this framework, we can study the fundamental mechanism of MHC class II mediated responses to various pathogens.