• Tuesday Seminars
• Postdoc Seminars
• Recruiting Talks
• Probability and Biology


• Models in Immunology

In this talk I will describe a new mathematical approach for characterizing the behavior of the Bay based on observational and modeling data. Of particular interest to us is the development of predictive tools that help in assessing damage due to storm surges as well as determining the connection between hydrodynamics of the bay and its biological health. An overview of the mathematical challenges one faces in determining stable/coherent structures in a velocity field that is finite in time and discrete in space will be presented.

In this talk I will describe briefly some of the funding opportunities in applied mathemtics at ONR, especially in the context of how mathematics may interact with biology. I would also be happy to talk about some of the better practices in preparing a proposal to submit to ONR.

We can often generate a reduced model for a dissipative system by computing an invariant (or, for infinite-dimensional systems, inertial) manifold that rapidly attracts the flow in phase space. These manifolds have become known in the literature as slow invariant manifolds. In this talk, I review some methods for generating slow invariant manifolds, as well as some of the connections between them. I emphasize methods whose solutions converge on the exact slow invariant manifold. The discussion will be framed with examples from chemistry, biochemistry, and population biology.

Chlamydia, caused by the bacterium Chlamydia trachomatis, is the commonest sexually-transmitted bacterial disease in European countries and the United States. The annual burden of the disease and its consequences in the USA is estimated at $\$ $ 2 billion. If undetected and treated, the disease could inflict irreversible complications, such as chronic pelvic pain, infertility in females and potentially fatal ectopic pregnancy.

The seminar will address the problem of the transmission dynamics of chlamydia trachomatis in a population using a mathematical model. The population level impact of various treatment strategies, based on the use of suitable antibiotics, will be discussed.

Kinetic capillary electrophoresis (KCE) is defined as capillary electrophoresis of species that interact during electrophoresis. KCE can serve as a conceptual platform for development of homogeneous kinetic affinity methods for affinity measurements and affinity purification. Affinity measurements include two groups of applications: (i) measurements of kinetic and thermodynamic parameters (k_on, k_off, K_d, deltaH, and deltaS) of affinity interactions and (ii) quantitative measurements of concentrations using affinity probes (e.g. antibodies and aptamers). Affinity purification also includes two large groups of applications: (i) purification of known molecules and (ii) search of unknown molecules. A number of different KCE methods can be designed by varying initial and boundary conditions - the way interacting species enter and exit the capillary. The basic theory of KCE concept has been recently developed and a large number of practical applications have been proven. In this lecture, I will present the theoretical bases of KCE and explain how it can be used for: (i) measurements of rate and equilibrium constants of protein-DNA interaction, (ii) selection of smart aptamers, and (iii) using smart aptamers for protein detection in an ultra-wide range of concentrations.

In the microarray experiments, the measured mRNA levels are averaged over cell population thus imperfectly reflecting single cell expression profiles. We developed a deconvolution method, using Maximum Entropy principle, that allows for reliable estimations of the single cell mRNA levels from time course microarray data in cases the population synchrony can be modelled. Applying this method to data from a synchronized baker's yeast culture we were able to time the peaks of expression of transcriptionally regulated cell cycle genes to an accuracy of 2 min (~1% of the cell cycle time), an order of magnitude better resolution than in the original data. Our results reveal distinct subphases of the cell cycle undetectable by morphological observation, as well as the precise timeline of macromolecular complex assembly during key cell cycle events.

Kinetic capillary electrophoresis (KCE) is defined as capillary electrophoresis of species that interact during electrophoresis. KCE can serve as a conceptual platform for development of homogeneous kinetic affinity methods for affinity measurements and affinity purification. Affinity measurements include two groups of applications: (i) measurements of kinetic and thermodynamic parameters (k_on, k_off, K_d, \Delta H, and \Delta S) of affinity interactions and (ii) quantitative measurements of concentrations using affinity probes (e.g. antibodies and aptamers). Affinity purification also includes two large groups of applications: (i) purification of known molecules and (ii) search of unknown molecules. A number of different KCE methods can be designed by varying initial and boundary conditions - the way interacting species enter and exit the capillary. The basic theory of KCE concept has been recently developed and a large number of practical applications have been proven [1-10]. I this lecture, I will present the theoretical bases of KCE and explain how it can be used for: (i) measurements of rate and equilibrium constants of protein-DNA interaction, (ii) selection of smart aptamers, and (iii) using smart aptamers for protein detection in an ultra-wide range of concentrations.

References:

1. Wong, E.; Okhonin, V.; Berezovski, M.V.; Nozaki, T.; Waldmann, H.; Alexandrov, K.; Krylov, S.N. J. Am. Chem. Soc. 2008, 130, 11862-11863.
2. A.P. Drabovich, M. Berezovski, V. Okhonin, S.N. Krylov. J. Am. Chem. Soc. 2007, 129, 7260-7261.
3. M. Berezovski, M.U. Musheev, A.P. Drabovich, J. Jitkova, S.N. Krylov. Nat. Protoc. 2006, 1, 1359-1369.
4. M. Berezovski, M. Musheev, A. Drabovich, S.N. Krylov. J. Am. Chem. Soc. 2006, 128, 1410-1411.
5. A. Petrov, V. Okhonin, M. Berezovski, S.N. Krylov. J. Am. Chem. Soc. 2005, 127, 17104-17110.
6. A. Drabovich, M. Berezovski, S.N. Krylov. J. Am. Chem. Soc. 2005, 127, 11224-11225.
7. M. Berezovski, A. Drabovich, S.N. Krylova, M. Musheev, V. Okhonin, A. Petrov, S.N. Krylov. J. Am. Chem. Soc. 2005, 127, 3165-3171.
8. V. Okhonin, M. Berezovski, S.N. Krylov. J. Am. Chem. Soc. 2004, 126, 7166-7167.
9. M. Berezovski, S.N. Krylov. J. Am. Chem. Soc. 2003, 125, 13451-13454.
10. M. Berezovski, S.N. Krylov. J. Am. Chem. Soc. 2002, 124, 13674-13675.

The traditional way of modeling biochemical networks is to treat the system as a set of coupled ordinary differential equations, using experimentally-determined rate constants and initial concentrations. When sufficient experimental data is available, this approach is successful, though a major challenge is to extract general, global behaviors of the system.

We are exploring Game Theory as an alternative approach to modeling biochemical networks. Game Theory has been applied successfully to modeling ecological networks (eg work of J Maynard Smith), and in that setting yields insights into phenomena like predator/prey oscillations and criteria for evolutionary stability. The key Game Theory concepts whose counterparts may be sought in biochemical networks are Nash Equilibrium and Braess' paradox, which highlights how the inhibition of one step in a system producing quantity X may paradoxically increase the overall quantity of X produced by the system.

Host immune systems impose selection on pathogen populations which respond by evolving different antigenic signatures. Like many evolutionary processes, pathogen evolution reflects an interaction between different levels of selection: pathogens can win in between-strain competition by taking over individual hosts (within-host level), or by infection of more hosts (population level). Vaccination, which intensifies and modifies selection by protecting hosts against one or more pathogen strains, can drive the emergence of new dominant pathogen strains -- a phenomenon called {\it vaccine-induced pathogen strain replacement}. In this talk reports of increased incidence of subdominant variants after vaccination are reviewed and the current model for pathogen strain replacement, which assumes that pathogen strain replacement occurs only through the differential effectiveness of the vaccines, is extended. Our theoretical studies suggest that a broader range of mechanisms is possible including pathogen strain replacement even when vaccines are {\it perfect} -- that is, they protect all vaccinated individuals completely against all pathogen strains. Pathogen strain replacement with perfect vaccination occurs when strains interact through super-infection or co-infection but does not seem to occur in the simplest models when the strains interact through cross-immunity. Super-infection, co-infection, and sross-immunity are some examples of coexistence mechanisms -- mechanisms that lead to coexistence of pathogen variants. The question which coexistence mechanisms lead to strain replacement under perfect vaccination and which do not is also addressed.

In response to environmental signals, mammalian cells face constant cell fate choices, including quiescence, proliferation, differentiation, or cell death. In the process of cell proliferation, there is a critical time point when a mammalian cell makes an all-or-none growth decision -- known as the restriction point (R point). After passing the R-point, cell growth becomes autonomous and independent of continuous growth signals. Although the R-point has been linked to various activities involved in the regulation of the G1-S transition of the cell cycle, the underlying decision-making mechanism remains unclear. Using single-cell quantifications coupled with mathematical modeling, we have shown that the Rb-E2F signaling network functions as a bistable switch to convert graded growth inputs into all-or-none E2F responses. Once turned ON by sufficient growth stimulation, E2F can memorize and maintain this ON state even in the absence of continuous growth signals. We have further shown that, at each critical concentration and duration of growth stimulation, bistable E2F activation correlates directly with the ability of a cell to traverse the R-point and begin DNA replication. Lastly, we have shown that the Rb-E2F bistable switch also functions as a "digital counter". This counter mechanism enables precise quantitation of transient growth stimulation, and allocates corresponding numbers of cells to enter proliferation. The counting process can be described by a simple mathematical function, which is determined by the inherent stochasticity and bistability of the Rb-E2F pathway, and can be modified by altering the network dynamics with Cdk inhibitors. We are now extending this study to develop an integrated, quantitative understanding of the "design principles" of gene regulatory networks underlying diverse cell-fate decisions, particularly as they serve to connect the decisions of proliferation and cell death.

T lymphocytes (T cells) are the main orchestrators of our adaptive immune system. T cells can be activated by a minute number of molecular signatures of the pathogen, and this, in turn, can ultimately lead to the clearance of an infection. In spite of enormous advances in experimental technology and the availability of data at an unprecedented level of detail, the principles that govern the emergence of an immune response have proven elusive. This is principally because the pertinent processes involve highly co-operative dynamic events that are further modulated by stochastic fluctuations. A synergistic combination of theoretical, computational and experimental approaches can glean mechanistic insights into such complex phenomena. In this talk, I will describe how such synergies can work by focusing on two issues pertinent to T cell signaling and activation. The first concerns membrane proximal signaling events leading to Ras activation in T cells. In this context, I will describe how positive feedback loops result in thresholding and signal stability, and how this relates to thymic selection, a key process that shapes our self-tolerant T cell repertoire. I will also describe how stochastic fluctuations and competing positive and negative feedback regulation enable cells to make binary decisions that would not be possible in a mean-field world. Related phenomena concerning signal spreading revealed by computer simulations and field-theoretic formulations will be mentioned.

Dynamics of multicellular systems is controled by the dynamics of intracellular signalling pathways and cell-to-cell communication. The description of biological processes often leads to nonlinear dynamical models with multiple steady states, which differ from the usual reaction-diffusion systems. Also, processes containing switching between different pathways or states lead to new types of mathematical models, which consist of nonlinear partial differential equations of diffusion, transport and reactions, coupled with dynamical systems controlling the transitions. In this talk I consider a systems of reaction-diffusion equations coupled with ordinary differential equations and approach the question of possible mechanisms of the formation of spatially heterogeneous structures. I study two mechanisms of pattern formation, diffusion-driven instability and hysteresis-driven mechanism, and demonstrate their possibilities and constraints in explanation of different aspects of structure formation in cell systems. Depending on the type of nonlinearities I show the existence of spatially inhomogeneous stationary solution of periodic type, the maxima and minima of which may be of the spike or plateau type, and the existence of transition layer solutions. These concepts are basic for the explanation of the morphogenesis of the fresh water polyp, hydra as well as growth of early lung cancers.

This talk will compare a few seemingly unrelated canonical models with noise-driven regular behavior, such as coherent oscillations, synchronization, and even quiescence. Different perspectives on coherence resonance show that the order in these models can be attributed to transients "stabilized" by stochastic effects. These perspectives suggest analysis on reduced models to better understand and predict these phenomena.

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with spotted pattern, and numerical results suggest that such spotted pattern is stable and multi-spots appear. On the other hand, it is shown by numerical simulations that the shadow system, which is a limiting system for the original Gierer-Meinhardt system, has no stable multi-spots. In this talk we study a linearized eigenvalue problem with a nonlocal term and prove the instability of multi-spots in a mathematically rigorous manner under suitable assumptions. In addition, we shall obtain profiles of eigenfunctions corresponding to the unstable eigenvalues.

Cell motility is an essential phenomena for almost all living organisms. It is natural to think that behavioral or shape changes of a cell bear information about the underlying mechanisms that generates the change. Reading the cell motion, namely, understanding these underlying biophysical and mechanochemical processes is of paramount importance. The quantitative technique developed in this paper determines some physical features and material properties of the cell locally and uses this information for making further inferences about the molecular structures and dynamics within the cell. The generality of the principals used in formation of the model ensures its wide applicability to different phenomena at different scales.

In this talk, I shall discuss two topics in mathematical biology. The first has to do with stacked fronts (see Roquejoffre, Terman, and Vitaly, SIMA 1996) I will show that for a simple monotone reaction-diffusion system with boundary equilibria, stacked fronts may occur and there is an explicit formula for the spreading speeds. This is joint work with Masato Iida and Hirokazu Ninomiya from Japan. The second topic has to do with reduction method for multiple time scale stochastic reaction network. This is joint work with Chang H. Lee from WPI. (see J. Math. Chem.) The paper assumes that the fast subsystem has unique equilibrium probability and the issue here is what to do when this assumption is not satisfied.

Vaginal microbicides (VMB) are currently among the few promising biomedical interventions candidates for preventing heterosexual transmission of HIV. They are specifically developed to help women reduce their risk of acquiring HIV infection. However, there are concerns that the next generation of microbicides containing ARV (ARV-VMB) may lead to the development of antiretroviral resistance and could paradoxically become more beneficial to men at the population-level. We aim to identify when and under what conditions the wide-scale use of VMB has a greater population-level impact among men than among women and to quantify the likelihood of male or female advantage in the benefits from the intervention under a wide array of utilization. We developed a deterministic model of HIV transmission to study the impact of a wide-scale population usage of VMB in a heterosexual population. Gender ratios of prevented infections and prevalence reduction are evaluated in 63 different introduction schedules of continuous and interrupted VMB use by HIV-positive women. The influence of different factors (VMB efficacy, transmission probabilities, etc.) on population-level benefits (e.g. infections prevented) among women and men is also studied through an analysis of 1,000 Monte Carlo simulations per scenario, using parameters sampled from ranges representative for the developing countries. Our analysis shows that VMB is still very much a female prevention tool. Effective control measures that restrict VMB-usage by HIV-positive women reduce significantly the risk of resistance development and further increase the likelihood of female advantage in VMB-prevented infections.

Evolutionary relationships among organisms are typically represented by a phylogenetic tree, which is mathematically described as an acyclic connected graph in which each internal node is connected through exactly three edges. In the field of phylogenetics, primary interest is on estimation of the species phylogeny, the tree that represents the actual sequence of speciation events that have led to the present configuration of species. However, numerous evolutionary processes can give rise to variation in the true evolutionary histories of individual genes, which are represented by gene trees. In this talk, we examine several distributions related to gene trees that arise when the coalescent process is used to model the relationship between gene trees and species trees. These distributions give insight into the challenges involved in using multi-locus data to estimate species-level phylogenies.

This talk is focused on two questions of chemotaxis modeling. One is how to establish the communications between microscopic and macroscopic chemotaxis models. The other is how information in the microscopic model is passed to the macroscopic model. For the first question, I use a novel approach to derive the macroscopic limits and express the microscopic quantities in terms of macroscopic quantities with the preservation of energy law. For the second question, I investigate the traveling waves of both microscopic and macroscopic models from which we see how traveling waves in the microscopic model are retained, lost or created during the transition from the microscopic to macroscopic models. Biological implications will be discussed along the talk.

A general branching process is a stochastic model for population dynamics that allows individuals to reproduce according to point processes during their lifetimes. The main difference between such a stochastic approach and deterministic population dynamics is that the latter directly models changes on the population level, whereas the former starts from individual reproduction and infers results about population behavior. We will describe the basic properties of a general branching process and consider three applications to cell biology: cell populations with quiescence, desynchronization of the cell cycle, and shortening of telomeres. The latter two represent work in progress.

Biological gels (such as collagen gels) used in tissue engineering have a fibrous microstructure which affects the way forces are transmitted through the material, and in turn affects cell migration and other behaviours. In order to understand the effects of mechanical interactions between the cells and the matrix on tissue architecture, we need to understand the mechanics of the gels themselves. In this talk, I will present a simple continuum model of gel mechanics, based on treating them as transversely isotropic viscous materials. Two simple canonical problems are considered involving thin two-dimensional films: extensional flow, and squeezing flow of the fluid between two rigid plates. Neglecting inertia, gravity and surface tension, in each regime we can exploit the thin geometry to obtain a leading-order problem which is sufficiently tractable to allow the use of analytic methods. Special cases in which the solution may be determined explicitly are considered and the physical interpretation of the results is discussed.

Tissues grow, change shape, and differentiate, function normally or abnormally, get diseased or injured, repair themselves, and sometimes atrophy. This complex suite of behaviors is governed by a complex suite of controls. Nonetheless, we can identify some general principles at work in the dynamics of tissues. Our goal is to understand how a tissue's mechanics and biology regulate each other.

Our models use a biologically-based continuum framework to track the mechanics, biology, and mechanobiology of the component cells, fluids, signaling molecules, and extracellular matrix materials. The presentation will describe our modeling approach, reveal some of the general principles we have identified, and discuss some of the questions our findings have raised about specific morphogenetic systems.

To model acute lung injury we previously developed a partial differential equation (PDE) model of gas exchange and inflammation within a cluster of approximately 25 alveoli along a capillary. We refer to this cluster as a respiratory unit (RU). We increased biological fidelity of this model by incorporating metabolism. We created a closed loop lung-scale model by linking multiple RUs under various anatomical and physiological conditions with metabolism effects.

Once metabolism was modeled, our original PDEs model for gas exchange and inflammation on a single RU was simplified mathematically to a more computational feasible model. The model was reduced such that arterial PO₂ and PCO₂ reflect the combined effects of metabolism and inflammation.

Combining multiple RUs we created a closed loop lung-scale model. Computational gains on the single RU allow the implementation of more accurate heterogeneity within the lung. Therefore, we vary both blood and tidal volumes on the RUs simultaneously.

In the closed loop lung-scale model we see that shunting (the closing of alveoli) is a major contributor to the reduction of PO₂ during inflammation. Including metabolism gives rise to more accurate drops in PO₂ than in the previous model. This model will be used with a more accurate inflammation model to simulate PO₂ changes during specific diseases, i.e. pneumonia.

In modern biomedical research, it is increasingly common to encounter high-dimensional and complex data, such as gene expression profiles over time, longitudinal trajectories in biomarkers and images. Increasing the complexity is the common interest in combining information across data of different sources. Bayesian hierarchical models provide a useful paradigm for addressing these problems, but parametric assumptions and the curse of dimensionality present difficulties. To address these challenges, this talk presents a general class of local partition mixture models, which facilitate sparse modeling of high-dimensional random effects distributions. These models provide a generalization of commonly-used latent class models, finite mixture models and Dirichlet process mixture models, with some clear advantages in terms of favoring a simultaneous reduction in dimensionality and improvement in fit. The methods are illustrated through applications to modeling of reproductive hormone curves, gene expression data and joint modeling of images and captions.

Results of Nowak and collaborators concerning the onset of cancer due to the inactivation of tumor suppresor genes give the distribution of the time until some individual in a population has experienced two prespecified mutations, and the time until this mutant phenotype becomes fixed in the population. Here we apply these results to obtain insights into regulatory sequence evolution in Drosophila and humans. In particular, we examine the waiting time for a pair of mutations, the first of which inactivates an existing transcription factor binding site and the second which creates a new one. Consistent with recent experimental observations for Drosophila, we find that a few million years is sufficient, but for humans with a much smaller effective population size, this type of change would take more than 100 million years. In addition, we use these results to expose flaws in some of Michael Behe's arguments concerning mathematical limits to Darwinian evolution.

Much of what is known about single cells has been inferred from averaged population measurements that often mask the individual cell dynamics and lead to a distorted view of cell behavior. The resulting distorted picture may be difficult to reconcile with single-cell gene regulatory network models, and may lead to incorrect constraints on biochemical parameters. A large contributor to the distortion is a significant amount of asynchronous variability in the population: at any given time the constituent cells can be found at different points in their respective cell cycles. It is then difficult to extract 'true' temporal data from mixed populations as contributions from cells in different stages average out. However, with an accurate model of population asynchrony and an established regularization method, it is possible to remove artifacts due solely to the asynchrony and/or uncover features masked by the population averaging. The resulting data is thus better suited for comparison with single-cell network models and parameter estimation.

I propose a simple two-parameter model of the distribution of Caulobacter crescentus cells in a population as a function of time following synchronization. This model accurately matches observed distributions of Caulobacter and can be generalized to any organism for which the synchronization state can be characterized particularly those that undergo asymmetric division. I then show how a generalization of deconvolution along with the Caulobacter distribution model can be used to extract the "single-cell"-like average of gene expression profiles from published microarray data. The resulting high-resolution profiles more accurately predict the cell-cycle position and size of gene expression peaks, and display new features not evident in the original population-level data set.

Mitochondria have long been known to sequester cytosolic Ca$^{2+}$ and even to shape intracellular patterns of endoplasmic reticulum -based Ca$^{2+}$ signaling. Accumulating evidence suggests that the mitochondrial network is an excitable medium which can demonstrate Ca$^{2+}$ induced Ca$^{2+}$ release via the mitochondrial permeability transition. The role of this excitability remains unclear, but mitochondrial Ca$^{2+}$ handling appears to be a crucial element in diverse diseases as diabetes, neurodegeneration and cardiac dysfunction. We extend the modular Magnus-Keizer computational model for respiration-driven Ca$^{2+}$ handling to include a transition pore and we demonstrate both excitability and Ca$^{2+}$ wave propagation that is accompanied by depolarizations similar to those reported in cell preparations. These waves depend on the energy state of the mitochondria, as well as other elements of mitochondrial physiology. Our results support the concept that mitochondria can transmit state dependent signals about their function in a spatially extended fashion.

DNA methylation has been shown to play an important role in the silencing of tumor suppressor genes in various tumor types. As it is desirable to have a system-wide understanding of the methylation changes that occur in tumors, we have developed the DMH protocol that can simultaneously assay the methylation status of all known CpG islands. As is common with all microarray technologies, a large percentage of the signal obtained from the array can be attributed to various measurable and unmeasurable confounding factors unrelated to the biological question at hand. In order to correct the bias due to the noise, we have implemented a quantile regression model for assessing significance of signal. As a proof of concept, we have applied this model to the methylation signature analysis of breast cancer cell lines which were obtained from the LBNL-ICBP (Lawrance Berkeley National Lab, Integrative Cancer Biology Program). We have been able to correctly identify some known methylated and unmethylated genomic regions. In this paper, we will describe how to use quantile regression to model DNA methylation microarrays and discuss how to summarize the regression results to identify all regions that are commonly methylated among all breast cancer cell lines. Finally, we validate our results using known methylated genes and housekeeping genes.

Sleep is not a simple process but instead is the result of a dynamic interplay between several different brain regions. As such, mathematical and computational modeling are useful in understanding how interactions between different neuron groups give rise to observed sleep and wake states. Recent experiments have shown that the brain contains mutually inhibitory connections between sleep-active and wake-active regions. I will present a biologically-based mathematical model of these brain regions consistent with this mutual inhibition concept. The model is able to account for several features of the human sleep/wake cycle including the timing of sleep and wakefulness under normal and sleep-deprived conditions, rapid eye movement (REM) rhythms, and the circadian dependence of several sleep characteristics. Additionally, if the input from the neuropeptide orexin is removed, the system exhibits more frequent switching between sleep and wakefulness, consistent with the sleep disorder narcolepsy. Our model demonstrates that the mutual inhibition concept, with the addition of a sleep homeostat and circadian rhythm, is sufficient to account for several features of the sleep/wake cycle. Phase-plane analysis of the model gives insight into the mechanisms of sleep transitions, as well as what might be going wrong in certain sleep disorders.

In the first part of the talk a simple HIV model that is relevant to the T-tropic phase of the disease (symptomatic HIV infection or pre-AIDS) will be presented. Treatment with imperfect drugs leads to an endemic equilibrium state with the following properties: Treatment causes the stable endemic equilibrium point to collide with the disease free state, but the two states do not bind. In fact, the endemic equilibrium may exist in the "disease free zone" as an unstable endemic equilibrium state which is capable of bouncing back to a stable endemic equilibrium point when the conditions are right.

In the second part of the talk a more comprehensive HIV model will be presented, which addresses how TB infection can accelerate the progression of the disease. This model is relevant to the asymptomatic and symptomatic HIV infection. Bacterial, macrophage, T cell and cytokine dynamics will be explained and future directions will be outlined.

Glioblastoma is the most common primary tumor of the brain, and has a dismal prognosis, with a mean survival of around 1 year from diagnosis. Invasion of surrounding brain tissue is one of the main hallmarks of gliomas, and is a major reason for treatment failure, because tumor cells remaining after surgical resection cause tumor recurrence. Although tumors show preferred invasion routes in the brain, at present it is not possible to predict patterns of invasion and recurrence for a given tumor. Variations are seen in the numbers of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3-D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell-cell adhesion. We develops a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on critical parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally.

Living cells are enveloped by membranes, made up of lipid molecules arranged in a bilayer configuration. These lipid bilayers are composed of different types of lipid molecules and, rather than being a homogeneous mixture, the lipids exhibit partial "phase separation", with the formation of cholesterol- and sphingomyelin- enriched microdomains (also known as "lipid rafts") within the membrane. It is thought that certain proteins or other reactants associate preferentially with these phase-separated microdomains, and thus that rafts can act to prevent interactions with other reactants in the rest of the membrane, or conversely, bring desired reactants into close proximity to promote certain reactions. Lipid rafts are therefore thought to play many very important roles in cell biology, but the basic principles that govern their formation and function remain poorly understood.

To shed light on these fundamental issues experiments have been conducted on simple in vitro systems, with microdomains forming in membranes composed of controlled lipid mixtures (e.g. 20% cholesterol and 80% phosphatidylcholine (PC)). This talk will briefly outline the background biology and the experimental measurements, and then describe how a mathematical model can be built to explain the experimental findings. The model relies on the Smoluchowski theory of coagulation and fragmentation, applied to an idealized system in which a large number of cholesterol molecules in 2D clusters of varying sizes (the rafts) are diffusing around in an otherwise inert 2D fluid (the PC bilayer). The key step in the modeling lies in studying the physics of the cluster- cluster interactions to deduce how the rate coefficients for the coagulation and fragmentation events depend on the cluster size, something on which the model results depend sensitively. We find remarkably good agreement with the experiments, and moreover, the model provides us with a large amount of information that the experiments cannot measure.

The bacterium Proteus mirabilis is known for its ability to swarm over hard surfaces and form spectacular concentric ring patterns. During pattern formation, the colony front is observed to move outward from the inoculation site either continuously or periodically, due to collective movement of elongated, hyper-flagellated swarmer cells at the leading edge. The formation of the rings was thought to arise from periodic colony expansion. However, recent experimental results show that swimmer cells stream inward toward the inoculation site, and form a number of complex patterns, including radial and spiral streams, rings and traveling trains. To understand the underlying mechanism of these complicated patterns, we developed a hybrid cell-based model which incorporates a simplified single cell signal transduction model with both the adaptation and excitation components. By assuming that swimmer cells respond to a chemoattractant that they produce, we are able to predict the formation of radial streams as a result of the modulation of the local attractant concentration by the cells. We further predict the spiral streams by incorporating a swimming bias of the cells near the surface of the medium. The hybrid cell-based model becomes computationally expensive because of the large number of cells due to cell division, therefore a higer level description is needed. We also present a moment-closure method for deriving macroscopic evolution equations from the hybrid cell-based model using perturbation analysis, and compare the solutions of the cell-based model and the derived continuum model.

Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow is investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying model of blood flow.

We obtain that for the certain form of the pressure data one can guarantee smooth solution, but for the physiologically relevant data one has shock formation. This is a joint work with Suncica Canic.

Temperate zone bats are subject to serious energetic constraints due to their high surface area to volume relations, the cost of temperature regulation, the high metabolic cost of flight, and the seasonality of their resources. We developed a model for a female bat that is primarily based on life history and energetics. It describes the growth of an individual female bat using a system of differential equations modeling the dynamics of two main compartments: storage (lipids) and structure (proteins and arbohydrates). The model is parameterized for the little brown bat, Myotis lucifugus, because of information available on energy budgets and changes in body mass throughout its life history. However, with appropriate modifications the conceptualization might be applied to other species of bats with similar life histories.

The individual model is integrated into a structured population model.Characteristics of the individuals determine the structure and, subsequently the dynamics of the population. This methodology uses and integrates the information on bat biology and physiology that has been collected primarily at the individual level. Survival and reproductive rates estimated from simulated populations under varying density dependence are comparable to those reported in the literature for natural populations of M. lucifugus. The population model provides insight into possible regulatory mechanisms of bat population sizes and dynamics of survival and extinction. A better understanding of population dynamics can assist in the development of management techniques and conservation strategies, and to investigate stress effects.

Life is thought to have arisen on Earth approximately 3.85 billion years ago, emerging from chemicals in the air, water, and rock. Though the process is likely to have obeyed all the known physico-chemical laws, the exact details of the origin of life pose one of the fundamental questions of science. How did the transformation from non-living chemicals to living cells occur? While the question is (deceivingly) simple, the answers are proving to be unquestionably complex. Scientific principles inevitably play a key role in any discussion of life's origins, dealing less with the question of why life appeared on Earth than with where, when, and how it emerged on the seemingly hostile surface of our primitive planet.

In this talk, I will extend the Turing diffusion-driven instability conditions for reaction- diffusion systems from fixed domains to continuously deforming or growing domains for some growth profiles satisfying the condition that the divergence of the domain velocity is a constant. By using the arbitrary Lagrangian-Eulerian formulation (ALE), the model equations on a continuously deforming domain are transformed to a fixed domain at each time, resulting in a conservative system. Linearising the conservative ALE formulation around a time-independent solution u_s we show that u_s is a solution of a system of nonlinear algebraic equations. We further derive and prove the conditions that generalise the classic Turing parameter space inequalities for the case of growing domains and we show that these conditions are a function of not only the model parameters in the reaction terms but also are a function of the divergence of the domain velocity. A fundamental structural difference between our diffusion-driven instability conditions and those obtained on fixed domains, is that our results do not enforce cross/pure kinetics thereby demonstrating that domain growth enlarges the potential range of kinetics which can lead to patterning.

Cholera was one of the most feared diseases of the nineteenth century, and remains a serious public health concern today. Here we describe a mathematical model for waterborne disease dynamics that is motivated by cholera. The model partitions the population into susceptible (S), infectious (I), and recovered / removed (R) individuals, and includes pathogen concentration in a water reservoir (W). Susceptible individuals can become sick either through contact with an infectious individual, or by drinking contaminated water. Infectious individuals can further contaminate the water by shedding pathogen into the reservoir. Our analysis of the resulting "SIWR" model includes expressions for the basic reproductive number, the final epidemic size, and the growth rate of the epidemic. In particular, we consider the effect of varying the contribution of the different transmission routes on the reproductive number and growth rate. We consider the SIWR model in the context of the nineteenth century London cholera outbreaks, including the famous 1854 outbreak investigated by John Snow.

We construct our understanding of the world solely from neuronal activity generated in our brains. How do we do this? Many studies have investigated how the electrical activity of neurons (action potentials) is related to outside stimuli, and maps of these relationships -- often called receptive fields -- are routinely computed from data collected in neuroscience experiments. Yet how the brain can understand the meaning of this activity, without the dictionary provided by these maps, remains a mystery. I will present some recent results on this question in the context of hippocampal place cells-i.e., neurons in rodent hippocampus whose activity is strongly correlated to the animal's position in space. In particular, we find that topological and geometric features of the animal's physical environment can be derived purely from the activity of hippocampal place cells. Relating stimulus space topology and geometry to neural activity opens up new opportunities for investigating the connectivity of recurrent networks in the brain. I will conclude by discussing some current projects along these lines. This is joint work with Vladimir Itskov (Columbia, CTN).

The dynamics of biochemical reaction systems can be modeled either deterministically or stochastically. Typically, the equations governing the dynamics of these models are quite complex. Further, there is oftentimes little knowledge about the exact values of the different system parameters, and, worse still, these system parameter values may vary from cell to cell. However, the network structure of a given system induces the corresponding equations (up to parameter values) governing its dynamics. I will show in this talk how this fact may be exploited to infer qualitative properties of large classes of biochemical systems and, most importantly, to learn which properties are independent of the details of the system parameters. I will give results for both stochastically and deterministically modeled systems. I will also discuss some recent work on numerical methods for the simulation of sample paths for stochastically modeled systems. The use of such methods is quickly increasing throughout the biology and biochemistry communities and therefore warrants more careful study.

Rhythmic oscillations in brain voltage activity characterize both healthy and pathological brain states. In this presentation, we consider a particular brain pathology - epilepsy - and how the quantitative analysis of voltage rhythms may suggest novel surgical therapies. In addition, we describe brain voltage rhythms in health, and a novel mechanism of rhythm generation: period concatenation. We conclude with a discussion of interacting rhythms, and show how a simple (yet biophysical) model of these rhythms motivates new mathematics.

In our first meeting of the quarter, we will discuss chapters 1 and 2 of Wakeley's book (background material). We will discuss chapter 3 on January 21st.

The first three chapters of the book are available on the publisher's website:
http://www.roberts-publishers.com/wakeley/

According to Dr Das: "I will keep the level of the talk at an introductory level where I will introduce the key players of our immune system and describe how they communicate and work together, ending with the key questions that remain to be understood. In the second talk I will be more specific show how mathematical modeling can help in understanding those questions in the context of T cell and NK cell signaling."

In the lung, alternatively activated macrophages (AAM) form the first line of defense against microbial infection. Due to the highly regulated nature of AAM, the lung can be considered as an immunosuppressive organ for respiratory pathogens. However, as infection progresses in the lung, another population of macrophages, known as classically activated macrophages (CAM) enters; these cells are typically activated by IFN-gamma. CAM are far more effective than AAM in clearing the microbial load, producing pro-inflammatory cytokines and anti-microbial defense mechanisms necessary to mount an adequate immune response. In this talk, I will outline the main differences between these two macrophage phenotypes, specifically in response to Mycobacterium tuberculosis (MTb) infection in the lung and present a mathematical model that determines the first time when the population of CAM becomes more dominant than the population of AAM. We call this event the "switching time" and use the model to propose strategies for reducing it.