Teaching Discrete and Algebraic Mathematical Biology to Undergraduates (July 29 - August 2, 2013)

Organizers: Raina Robeva (Sweet Briar College), Matt Macauley (Clemson University), Terrell Hodge (Western Michigan University)

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.

The highly interdisciplinary nature of the modern day work force is one of the greatest challenges facing high school and college education. Rarely is knowledge of just a single discipline adequate for today’s job market: a reality that is forcing students to explore means to better equip themselves with an understanding of the full gambit of disciplines needed to make an effective and valuable contribution. Attempting to bridge the current gap in the curriculum, between seemingly disparate subjects, The Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) is carrying out a portfolio of projects to improve student engagement in STEM fields and versatility for their transition from education to career. Through three series of curricular modules, DIMACS offers students insights into mathematics, computer sciences, and the sciences in general, and their interrelationships. In particular, one of these projects is the Challenge of Interdisciplinary Mathematics and Biology (IMB), which pioneered the linking of biology and mathematics in high schools. IMB includes twenty individual modules, which can be used in either biology or mathematics classes, as well as a senior-year (online book) on biomathematics, already in use across the country. A second project focuses on sustainability drop-in modules for college and high school classes. In both cases, many of the modules develop and utilize discrete and algebraic tools to solve specific problems at the interface of mathematics and biology. This talk will focus on how one develops curricular modules and tests their use in classrooms. Midge will help you draft an outline for modules to be developed this week, and will be available to assist participants as the week goes on.

One interesting example of a discrete mathematical model used in biology is a food web. The first biology courses in high school and in college present the fundamental nature of a food web, one that is understandable by students at all levels. But food webs as part of a larger system are often not addressed. This talk presents materials that can be used in undergraduate classes in biology and mathematics which provide students with the opportunity to explore mathematical models of predator-prey relationships, determine trophic levels, dominant species, stability of the ecosystem, competition graphs, interval graphs, and even confront problems that would appear to have logical answers that are as yet unsolved. Fundamental goals are for participants to be able to:

• Recognize various relationships between organisms, and look for patterns in food webs.
• Use graphs and directed graphs to model complex trophic relationships.
• Determine trophic levels and status within a food web, and the significance of these levels in calculating the relative importance of each species (vertices) and each relationship (arcs) in a food web.
• Use the food web and corresponding competition graph to determine the dimension of the community’s habitat.

Fitness landscapes are central in the theory of adaptation, and relate to most aspects of evolutionary biology including recombination, speciation, and evolutionary predictability [13]. Fitness landscapes and epistasis may be important for predicting, preventing and managing antimicrobial drug resistance problems. Mathematical topics include planar graphs, polygons, polytopes, and triangulations of graphs and polytopes.

Biochemical networks - such as gene regulatory, signal transduction or metabolic networks - describe our knowledge of the biomolecular processes in living organisms. They are made of networks of chemical reactions transforming or transporting biochemical compounds (genes, proteins, metabolites, ...). Such biochemical networks tend to be very large and dense. Thus, a crucial point is their concise and unambiguous representation, and the development of computational methods to model and analyze them in an efficient manner.

The talk describes a Petri net-based framework, which unifies the qualitative, stochastic, continuous, and hybrid paradigms. Each perspective adds its contribution to the understanding of the system. The methodology is supported by a sophisticated toolkit, comprising (1) Snoopy - tool for modelling and animation/simulation of hierarchical qualitative, stochastic, continuous, and hybrid Petri nets, recently extended by coloured counterparts of these net classes; (2) Charlie - analysis tool of standard Petri net properties; and (3) Marcie - symbolic reachability analysis of qualitative Petri nets, analytical and simulative model checking of stochastic Petri nets.

The talk will conclude with a brief outline of our approach to deploy coloured Petri nets to statically encode finite discrete space, possibly hierarchically structured.

Graph Theory has enormous applicability to biology. Each of the following graph theoretic concepts will be instantiated with biological applications: (1) ordering problems with One-dimensional Interval Graphs, Maximal Cliques, Transitive Orientability, Adjacency matrices, Incidence matrices; (2) Planar Graphs with Two-dimensional Voronoi Tessellations, Delaunay Triangulations, Lewis, Desch, and Aboav-Weaire Laws, Topological charge; (3) Three-dimensional Polytopes, Polyhedral Graphs, Delaunay tetrahedra, Euler on vertices, edges, and faces, T-numbers, quasi-crystals, Kepler’s face-centered cubic packing, voxels; (4) x.y-dimensional Graphs, Fractals; (5) Graph Grammars – Lindermayer Systems; (6) Trees: binary branching trees, minimal spanning trees, Ulam trees, polytomies; (7) Networks: Random, Scale-free, Small World, diameter, connectance, complexity, clustering coefficients, hubs, fragility, robustness, Maximal Cliques and other sub-graphs; (8) Paths and Circuits: Eulerian and Hamiltonian, de Bruijn graphs; and (9) Bipartite Graphs: Matching, Pancaking flipping problem, Hungarian algorithm, Gale-Shapley algorithm & 2012 Nobel Prize. In many cases we have developed software for handling biological data using these graph theoretic techniques: Ka-me: A Voronoi Image Analyzer, javaBENZER, 3D FractaL Tree, Split Decomposition, and BioGrapher. Basically I will argue that many biological problems are best addressed with a discrete perspective and that topological reduction is a good way to investigate complex data sets.

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.

Our understanding of cancer has been aided by a network centric view. The fundamental relevance of systems biology to the understanding and treatment of cancer is the insight that genes and proteins do not act in isolation, but rather as nodes in complex interactive networks that include multiple feedback mechanisms and redundancies. The design of effective drugs to battle cancer will depend on the understanding of these networks and of the specific network alterations present in an individual tumor. And an understanding of characteristic changes in metabolic networks can lead to new prognostic and diagnostic methods. The complexity of these dynamic networks makes it difficult or impossible to study them without the aid of computer models based on mathematical analysis. This talk will discuss systems biology and mathematical models as an approach to cancer biology by way of a case study, focused on intracellular iron metabolism and its relationship to breast cancer.

Determining the structure of the RNA molecule for a given RNA sequence is an important problem in molecular and computational biology. The molecule structure depends on different types of intra-sequence base pairings. Unfortunately, many aspects of the 3D interactions between the nucleotides are still not well-understood. Since the prediction of the 3D structure is currently out of our reach, a lot of effort has been devoted to the prediction of the secondary structure as an important intermediate step towards the complete solution. Various combinatorial methods and models have been used to compare, classify, and otherwise analyze secondary structures and thereby contribute to our understanding of the base pairing of RNA sequences. This talk will survey some of the graph-theoretic and formal language approaches that are used to analyze secondary structures with emphasis on those accessible to undergraduate students.

RNA structures can be presented as diagrams, i.e. their backbone drawn horizontally and their bonds as arcs in the upper halfplane. The diagram presentation shows clearly the difference between secondary and pseudoknot RNA structures. Namely the former have no crossing arcs and the latter exhibit crossings. In this lecture we enrich diagrams to fatgraphs, endowing the vertices with an cyclic ordering of their incident halfedges. Such an ordering can be motivated by the chemistry of the RNA nucleotides and expresses an additional organization of the bonds. Fatgraphs are then shown to have an associated compact, oriented surface and we discuss the notion of topolgical genus. We show how to compute genus from the diagram. We then employ genus introducing topological RNA structures and discuss salient features. We analyze in detail the case of genus one, where we show that there emerge only four new "shapes". Finally we show how to rewire genus one diagrams into certain planar trees and discuss the implications for RNA folding.

Gene regulatory networks are ubiquitous in molecular systems biology and contribute to the control of major biological processes including metabolism and development. Given the abundance of gene data sets, the ability to reconstruct the regulatory network underlying the data has become one of the prime objectives in systems biology research. We will introduce various methods for reverse-engineering or inferring the structure of gene regulatory networks. These methods use techniques from computational algebra to build models of polynomial dynamical systems, which provide a rich backdrop within which to perform network analyses. We will build some algebraic models and simulate how one could use them to design biologically meaningful models in an experimental setting.

Due to the size and complexity of biological systems and models, it is sometimes necessary to consider a "core" model that keeps the key dynamical properties of the original model, but is simple enough for analysis. In this talk we will present a systematic way to reduce discrete models while keeping important dynamical features such as steady states. We will focus on Boolean models of biological systems. A similar approach for discrete-time continuous-space models may be discussed.

Recently there have been much work and collaborations between modern biology and higher mathematics. A number of important connections have been established between computational biology and the emerging field of "algebraic statistics", which applies tools from combinatorics, computational algebra, and polyhedral geometry to statistical computational problems and statistical modeling. Phylogenetics has provided an abundant source of applications for algebraic statistics, with research areas including phylogenetic invariants, the geometry of tree space, and analysis of Phylogenetic reconstruction. The purpose of this review is to provide the audience with an introduction to this subject, a non-comprehensive guide to further reading, and a collection of more detailed case studies that provide examples of how algebraic methods have been used in this the context of molecular phylogeny.