Teaching Discrete and Algebraic Mathematical Biology to Undergraduates

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  Discrete Dynamic Modeling of Signaling Networks
  Reka Albert

  The use of differential equation based modeling frameworks for intra-and inter-cellular signaling networks is greatly hampered by the sparsity of known kinetic detail for the interactions and processes involved in these networks. As an alternative, discrete dynamic and algebraic methods are gaining acceptance as the basis of successful predictive models of signal transduction, and a tool for inferring regulatory mechanisms. For example, Boolean models have been fruitfully used to model signaling networks related to embryonic development, to plant responses to their environment, and to immunological disorders. The construction of a Boolean model starts with a synthesis of the nodes (components) and edges (interactions) of the signaling network, followed by a distillation of the edges incident on each node into a Boolean regulatory function. The analysis of the model consists of finding its attractors (e.g. steady states), and the basin of attraction of each attractor (the initial states that converge into that attractor). The model can be used to look at "what if" scenarios, to analyze the effects of perturbations (e.g. node disruptions), and thus to predict which nodes are critical for the normal behavior of the network.

  Part one of this presentation will review the basics of Boolean modeling, with special attention to models that allow different timescales in the system (i.e. asynchronous models). I will then present an asynchronous Boolean model of the signaling network that governs plants' response to drought conditions. This model synthesizes a large number of independent observations into a coherent system, reproduces known normal and perturbed responses, and predicts the effects of perturbations in network components. Two of these predictions were validated experimentally. Part two of this presentation will present an asynchronous Boolean model of the signaling network that is responsible for the activation induced cell death of T cells (a type of white blood cell). Perturbations of this network were identified as the root cause of the disease T-LGL leukemia, wherein T cells aberrantly survive and then attack normal cells. The model integrates interactions and information on certain components' abnormal state, explains all the observed abnormal states, and predicts manipulations that can abolish the T-LGL survival state. Several of these predictions were validated experimentally. I will finish by presenting two methods for extracting useful predictions from Boolean models of signal transduction networks without extensive simulations.

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  Dynamics of Disease Transmission Networks
  Winfried Just

  The most basic models of transmission of infectious diseases assume a partition of the host population into a small number of compartments such as S (susceptible), I (infectious), and R (removed) and conceptualize the number or proportion of individuals in each compartment as variables in an ODE system. But disease transmission is inherently a stochastic event that may occur during a contact between a susceptible and an infectious individual. In real population, this probability will be different for different pairs of individuals. While classical compartment-based ODE models have only a limited capacity for dealing with these heterogeneities, it is, at least in principle, possible to build models of disease transmission based on the underlying contact networks.

  This contribution will introduce models of disease transmission dynamics on a given contact network and the problem of comparing the predictions of such models with coarser, compartment-based ODE models. In other words, we want to know which network properties most significantly influence the course of an epidemic. These questions are currently gaining prominence in research on disease dynamics. Meaningful numerical explorations are feasible at a level suitable for undergraduate research and will constitute a large part of the module. The topic is also well suited for exploring how the choice of modeling assumptions influences the model's predictions. Moreover, the model will introduce some strategies for collecting data on contact networks and building models of such networks based on limited and somewhat unreliable data.