MBI Videos

2011 Workshop for Young Researchers in Mathematical Biology

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    Brittany Bannish
    The degradation of blood clots is a tightly regulated process. If the mesh of fibrin fibers securing the clot degrades too slowly, thrombi can form, leading to heart attack or stroke. If the fibrin degrades too quickly, excessive bleeding may occur. We study fibrinolysis (the degradation of fibrin by the main fibrinolytic enzyme, plasmin) using a multi-scale mathematical model intended to answer the following question: Why do coarse clots composed of thick fibers lyse more quickly than fine clots composed of thin fibers, despite the fact that individual thin fibers lyse more quickly than individual thick fibers? We use stochastic methods to model lytic processes on scales ranging from individual fiber cross section to whole clot. We find that while fiber number does have an effect on lysis rate, it is not simply "fewer fibers equals faster lysis", as many biologists suggest. In fact, the number of tissuetype plasminogen activator molecules (tPA, an enzyme that converts plasminogen to plasmin) relative to the clot surface area exposed to the tPA strongly influences lysis speeds. We also predict how many plasmin molecules can be produced by a single tPA molecule, how long it takes a given number of plasmin molecules to degrade a single fibrin fiber, and how patterns and speeds of lysis (both on an individual fiber and clot scale) vary under a range of conditions. This last point is of particular interest for development of treatments for occlusive blood clots. Often, a bolus of tPA is injected near the thrombus, in an attempt to initiate therapeutic lysis. Our model predicts other potential targets for future research on effective therapeutic strategies for degrading blood clots.
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    Robert McDougal, Robert McDougal
    Working memory is a process for the short-term storage and manipulation of information necessary for complex cognitive tasks. During the performance of working memory tasks, the prefrontal cortex (PFC) exhibits sustained persistent activity and is believed to play a key role in the process. Experiments have demonstrated that working memory performance is modulated by dopamine, which is known to be altered in certain pathological conditions, including schizophrenia.

    A number of models have been proposed for the maintenance of persistent activity in the PFC, often based on either intrinsic cellular bistability or recurrent excitatory connections formed via synaptic adaptation. Consistent with the observation that inhibitory connections dominate the PFC, we present a new approach: a network driven by excitatory-inhibitory interactions where the response to inhibition is modulated by intracellular calcium. Individual neurons fire irregularly, but our model network exhibits emergent properties, such as a clear gamma rhythm. The network is robust to noise and distracters. Only general assumptions about connection probabilities are assumed; the model can represent novel, unlearned stimuli.

    Dopamine modulates ion channel activity and synaptic conductances. We study the effects of this modulation on cellular and network behavior, and find the experimentally-observed inverted-U shaped relation between dopamine expression and working memory performance.
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    Elizabeth Allman
    Phylogenetics is the branch of biology concerned with inferring evolutionary relationships between currently extant species. For instance, are humans more closely related to chimpanzees or to gorillas on an evolutionary tree? A typical phylogenetic analysis from molecular data might consist of sampling gene sequences from a number of species, aligning them, and performing a statistical analysis to choose a tree that best displays the evolutionary relationships of taxa.

    While phylogenetic analyses are usually undertaken with standard statistical approaches such as Maximum Likelihood or MCMC in a Bayesian framework, these require formulating a probabilistic model of the DNA substitution process on a tree. Because many of these models are naturally given by polynomial parameterizations, by considering the algebraic varieties these maps define, the viewpoint of algebraic geometry can be used to gain theoretical understanding of the limits and advantages of such models.

    The talk begins with an introduction to phylogenetics, and then addresses how algebraic techniques are being used to advance the theoretical end of this field. Surprising connections will be made between seemingly disparate areas of mathematics.
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    Wanda Strychalski
    Blebbing occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. Recent interest has focused on cells that use blebbing for migrating through three dimensional fibrous matrices. In particular, metastatic cancer cells have been shown to use blebs for motility. A dynamic computational model of the cell is presented that includes mechanics of and the interactions between the intracellular fluid, the actin cortex, and the cell membrane. The computational model is used to explore the relative roles in bleb formation time of cytoplasmic viscosity and drag between the cortex and the cytosol. A regime of values for the drag coefficient and cytoplasmic viscosity values that match bleb formation time scales is presented. The model results are then used to predict the Darcy permeability and the volume fraction of the cortex.
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    Marcos Nahmad
    The specification of cell identities during development is orchestrated by signaling molecules named morphogens that establish spatial patterns of gene expression within a field of cells. In the classical view, the interpretation of morphogen gradients depends on the equilibrium morphogen concentrations, but the dynamics of gradient formation are generally ignored. The problem of whether or not morphogen gradient dynamics contribute to developmental patterning has not been explored in detail, in part, because genetic experiments that selectively affect signaling dynamics while maintaining unchanged the steady-state morphogen profile are difficult to design and interpret. Here, I present a mathematical approach to identify genetic mutations in developmental patterning that may affect the transient, but leave invariant the steady-state signalling gradient. As a case study, I illustrate how these tools can be used to explore the dynamic properties of Hedgehog signalling in the developing wing of the fruit fly, Drosophila melanogaster. This analysis provides insights into how different properties of the Hedgehog gradient dynamics, such as the duration of exposure to the signal or the width of the gradient prior to reaching the equilibrium, can be genetically perturbed without affecting the local steady-state distribution of the gradient. I propose that this method can be generally applicable as a tool to design experiments to probe the role of transient morphogen gradients in developmental patterning and discuss potential applications of these ideas in other problems.
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    Jared Barber
    Necrotizing enterocolitis is a severe inflammatory disease in premature infants that is characterized by wounds in the intestinal wall. The ongoing dynamics of the disease depend upon a complex interplay between the immune system, intestinal bacteria, and intestinal epithelium. We have developed a three-dimensional computational model that examines this complex interplay and its dependence on the spatial structure of the intestine. The model reproduces expected physiological results and shows that the spatial structure of intestinal wounds may affect the outcome of necrotizing enterocolitis.
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    Leon Glass
    Even understanding the way we perceive very simple images presents a major challenge for both neurophysiologists and computer scientists. In this talk I will discuss two visual effects. In one random dots are superimposed on themselves following a linear transformation. In the second, a rotating disk with radial spokes is viewed under stroboscopic illumination, where the frequency and duration of the stroboscopic flash are varied. Though these phenomena are very different, in both correlation plays a major role in defining the structure of the image. In this talk, I will give demonstrations of these phenomena and discuss related experimental and theoretical work by ourselves and others. In particular, I focus on recent theory that uses the theory of forced nonlinear oscillations to predict the percept of rotating disks during stroboscopic illumination over a wide range of disk rotation speeds and strobe frequencies.

    Finally, I suggest that the anatomical structure of the human visual system plays a major role in enabling the amazingly rapid and accurate computation of spatial and time dependent correlation functions carried out by the visual system.
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    Julien Arino

    Infectious diseases have been spreading across vast distances for milenia as a result of the movement of both human and animal hosts. In the past, both types of hosts had limited movement ranges, and one observed travelling waves of infection slowly expanding across space. Nowadays, the movement of humans has considerably accelerated and expanded, so that one observes another kind of spread, which appears less coherent.

    In this talk, I will discuss the mechanisms that give rise to the spatialization of an infectious disease. I will then present metapopulation models, one of the methods that can be used to describe the spatio-temporal spread of infections between distant locations. I will review some mathematical properties of these models, and will illustrate with a stochastic application in the context of the spread of infections via the global air transportation network.

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    David Murrugarra
    This talk will introduce a new modeling framework for gene regulatory networks that incorporates state dependent delays and that is able to capture the cell-to-cell variability. This framework will be presented in the context of finite dynamical systems, where each gene can take on a finite number of states, and where time is also a discrete variable. The state dependent delays represent the time delays of activation and degradation. One of the new features of this framework is that it allows a finer analysis of discrete models and the possibility to simulate cell populations. Applications presented will use one of the best known stochastic regulatory networks, that is involved in controlling the outcome of lambda phage infection of bacteria.
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    Lauren Childs
    A novel bacterial defense system against invading viruses, known as Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR), has recently been described. Unlike other bacterial defense systems, CRISPRs, are virus-specific and heritable, producing a form of adaptive immune memory. Specific bacterial DNA regions, CRISPR loci, incorporate on average 25 copies of unique short (30 base pair) regions of viral DNA which allow the bacteria to detect, degrade and have immunity against viruses with matching sub-sequences. Ideally, the number of unique viral-copied regions a CRISPR loci contains would grow indefinitely to allow immunity to accumulate to a large number of viruses. However, the number of these viral-copied regions in the CRISPR loci of any bacteria is limited in length and number. We use a birth-death master equation model to explore the growth and decay of the length of the CRISPR loci and thus the number of viral-copied regions. Additionally, we use a simple probabilistic model to determine bounds on the length of viral-copied region within the CRISPR locus.
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    Orou Gaoue
    Over the past two decades, modeling the ecological impacts of harvesting wild plants, as source of food and medicine, has used stationary population growth rate as the metric to measure effects of harvest. In this talk, I show that using asymptotic rather than the transient growth rates may underestimate the effect of harvest and of other disturbances. The transient growth rate and its variation between population-level harvest intensities (high versus low) were smaller than their asymptotic equivalent. Patterns of elasticity of transient growth rates to perturbation of vital rates were different from those of the asymptotic elasticity. Asymptotic growth rates were more elastic to perturbation of late life stages; however, transient growth rates were more elastic to early life perturbations. These results suggest that the more than fifty published studies on the effects of harvest on wild plant population dynamics using only asymptotic growth rates may have been underestimating such effects in the short-term.
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    Alan Lindsay
    Determining whether a habitat with fragmented or concentrated resources is a benefit or hindrance to a species' well-being is a natural question to ask in Ecology. Such fragmentation may occur naturally or as a consequence of human activities related to development or conservation. In a certain mathematical formulation of this problem, one is led to study an indefinite weight eigenvalue problem, the principal eigenvalue of which is a function of the habitat's makeup and indicates the threshold for which the species either persists or becomes extinct. For a particular but general class of fragmentation profiles, this threshold can be calculated implicitly and optimized to reveal an definitive strategy for minimizing the persistence threshold and thereby allowing the species to persist for the largest range of physical parameters.

    This relates to work contained in the publication: A.E. Lindsay, M.J.Ward, (2010) An Asymptotic Analysis of the Persistence Threshold for the Diffusive Logistic Model in Spatial Environments with Localized Patches Discrete and Continuous Dynamical Systems Series B, Volume: 14, Number: 3, pp.1139-1179
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    Jim Cushing
    Discrete time matrix models for the dynamics of structured populations provide one way to study the dynamic consequences of different life history strategies. One fundamental strategy is semelparity. Mathematically, semelparity can be associated with a high co-dimensional bifurcation at R0 = 1 which results in a dynamic dichotomy between persistence equilibrium states (lying in the interior of the positive cone) and synchronous cycles and cycle chains (lying on the boundary of the cone). Biologically, the dynamic alternative is between equilibration with overlapping generations and periodic oscillations with non-overlapping generations. I will describe what has been proved about the bifurcation at R0 = 1 for lower dimensional models. It remains a difficult mathematical challenge to describe the nature of the bifurcation at R0 = 1 for higher dimensional models. Time permitting I will discuss the bifurcation at R0 = 1 for matrix models extended to an evolutionary setting (by evolutionary game theory).
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    Jing Li
    Government investment in public health management can elicit strong responses from individuals within communities. These responses can reduce and even reverse the expected benefits of the policies. Therefore, projections of individual responses to policy can be important ingredients into policy design. Yet our foresight of individual responses to public health investment remains limited. This paper formulates a population game to explore how individual investment through behavior and government investment through taxation impact the health commons. We model the problem of infectious disease management through reductions in transmission risk for a disease that does not elicit immunity in a population without demographic structure. We identify three common modes of government and individual investments and describe how each mode relates to policy responses and health outcomes. We also provide general bounds on the magnitude of practical investment by individuals. The methods we present can be extended to address specific policy problems where public responses are expected to impose key feedbacks.

    Work done in collaboration with Darla Lindburg, Rachel A. Smith, and Timothy C. Reluga.
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    Calistus Ngonghala
    A a new SIS model for malaria that incorporates mosquito demography is developed and studied. This model differs from standard SIS models in that the mosquito population involved in disease transmission (adult female mosquitoes questing for human blood) are identified and accounted for. The main focus of this model is disease control. In the presence of the disease, we identified a trivial steady state solution, a nontrivial disease-free steady state solution and an endemic steady state solution and showed that the endemic steady state solution can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. The model therefore captures natural oscillations known to exist in malaria prevalence without recourse to external seasonal forcing and/or delays. Besides the basic reproduction number, we also identified a second threshold parameter that is associated with mosquito demography. These two threshold parameters can be used for purposes of disease control. Analysis of our model also indicates that the basic reproduction number for malaria can be smaller than previously thought and that the model exhibits a backward bifurcation. Hence, simply reducing the basic reproduction number below unity may not be enough for disease eradication. The discovery of oscillatory dynamics and the re-interpretation of the basic reproduction number for malaria presents a novel and plausible framework for developing and implementing control strategies. Model results therefore indicate that accounting for mosquito demography is important in explaining observed patterns in malaria prevalence as well as in designing and evaluating control strategies, especially those interventions that are related to mosquito control.

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