1. Toward An Integrative Theory Of Within-host Disease Dynamics Using Principles of Biological Stoichiometry to Understand the Pathobiology of Cancer and Infectious Disease: A Collaboration of Empirical and Theoretical Biology, Theoretical Physics and Mathematics Yang Kuang 1 (Project Director), Jim Elser 2, Timothy Newman 3, John Nagy 4,2, Val Smith 5 and Marilyn Smith 6 Abstract: This multi-campus, interdisciplinary team is studying processes within a single biological host that can be described by models inspired by ecological stoichiometry, the study of the balance of energy and multiple chemical resources (usually elements) in ecological interactions. This work weaves together threads of theoretical and experimental research. Our primary aim is to construct predictive and verifiable theoretical models that can begin to explicitly deal with the effects of stoichiometric interactions in within-host disease dynamics. Fig. 1. Squamous cell cancer of the lung, H&E stain. Necrotic areas are marked ‘N’. Our early work seeks to understand the ecological collapse leading to necrosis. From Nagy (2005). Synopsis: Necrosis (Fig. 1), a common feature of cancer, represents a profound ecological collapse within a tumor, the cause of which remains unclear. Elser et al. (2003), applying the concept of biological stoichiometry, suggest that cancer cells should generally require more phosphorus than normal cells do, a prediction that we have subsequently confirmed (unpublished data), and shown with modeling to be have potentially significant influences on tumor growth dynamics (Fig. 2). In particular, an unfavorable C:P ratio may be a cause of necrosis—cells lacking P relative to C may “burn off” excess C through futile metabolic cycles, which produce lactic acid. A model of tumor metabolism supports the viability of this hypothesis for realistic parameter values (Fig. 3). This local acidosis may be the immediate cause of necrosis. Alternatively, necrosis may be caused by competition among tumor cells for a single nutrient. A variation of this hypothesis was proposed by Nagy (2004), in which necrosis arose when natural selection favored cells that traded off the ability to produced tumor angiogenesis factors for growth potential, producing a “hypertumor,” or a tumor growing parasitically on an established tumor (Figs. 4 and 5). Biological stoichiometry also suggests that pathogen growth potential should respond to changes in the ratios of carbon to various nutrients, including P, N and Fe. Indeed, larval mosquitoes infected with Beauvaria bassiana suffer a much higher mortality if given low-quality food (Fig. 6). Fig. 3. Local acidosis in a mathematical model of malignant neoplasia. Supply of glucose and nutrient (P) are adequate (left panels), although lactic acid builds up in the cells and interstitium (lower left, top right), which cell metabolism reaches an equilibrium with no cell growth (lower right). Fig. 2. Dynamics of a tumor growing in a healthy organ. Tumor growth is limited by phosphorus well below its “carrying capacity” for the given blood supply. In general, this model showed that tumors are very sensitive to changes in phosphorus supply. Fig. 5. Hypertumor dynamics. Resident cancer cells with growth and death rates shown in blue (upper left panel) are invaded by a fast-growing, cell line unable to secrete tumor angiogenesis factors. The fast growing cell line invades the tumor, causing it to become hypoxic (lower left), resulting in cessation of tumor growth (upper right) and eventual tumor destruction (lower right). Val Smith Synthesis: Synthesis: This research program takes a step toward new ways to understand pathology, aiming to develop robust and experimentally calibrated mathematical theories of disease-host interactions that can be applied to a wide variety of diseases. We firmly believe that such theories have a central role to play in present and future research. Fig. 6. Survivorship of larval mosquitoes given food of various quality. Mosquitoes infected with the fungus Beauvaria bassiana (right panel) survived equally well compared to uninfected larvae (left panel) when given high-quality food (blue triangles). However, infected larvae suffer much higher mortality than uninfected when food quality is low (green circles). References Elser, J.J., J.D. Nagy and Y. Kuang 2003. Biological stoichiometry: an ecological perspective on tumor dynamics. Bioscience 53: 1112-1120. Kuang, Y., J.D. Nagy and J.J. Elser 2004. Biological stoichiometry of tumor dynamics: mathematical models and analysis. Discrete and Continuous Dynamical Systems B 4: 221-240. Nagy, J.D. 2005. The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity. Mathematical Biosciences and Engineering 2: 281-418. Nagy, J.D. 2004. Competition and natural selection in a mathematical model of cancer. Bulletin of Mathematical Biology. 66: 663-687. Smith, V.H., T.P. Jones and M.S. Smith. 2005. Host nutrition and infectious disease: an ecological view. Frontiers in Ecology and the Environment. 3: 268-274. Acknowledgements: Acknowledgements: We gratefully acknowledge the support of the National Science Foundation and National Institutes of Health through grant number DMS/NIGMS 0342388. Fig. 4. Phase portrait from a mathematical model of cancer with two competing parenchyma (cancer) cell types, immature vascular cells (w) and mature blood vessel density (v). Variable u 1 represents the proportion of the parenchyma of cell type 1. (a) Dynamics on the boundary. (b) Dynamics in the interior. Yang Kuang Marilyn Smith 1 Department of Mathematics and Statistics, Arizona State University; 2 School of Life Sciences, Arizona State University; 3 Department of Physics and Astronomy, Arizona State University; 4 Department of Life Sciences, Scottsdale Community College; 5 Department of Ecology and Evolutionary Biology, University of Kansas; 6 Department of Microbiology, Molecular Genetics and Immunology, University of Kansas Medical Center Jim Elser Tim Newman John Nagy