Algebraic and Topological Indices of Molecular Pathway Networks in Human Cancers

Protein-protein interaction networks associated with diseases have gained prominence as an area of research. We investigate algebraic and topological indices for protein-protein interaction networks of 11 human cancers derived from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database. We find a strong correlation between relative automorphism group sizes and topological network complexities on the one hand and five year survival probabilities on the other hand. Moreover, we identify several protein families (e.g. PIK, ITG, AKT families) that are repeated motifs in many of the cancer pathways. Interestingly, these sources of symmetry are often central rather than peripheral. Our results can aide in identification of promising targets for anti-cancer drugs. Beyond that, we provide a unifying framework to study protein-protein interaction networks of families of related diseases (e.g. neurodegenerative diseases, viral diseases, substance abuse disorders).Comment: 15 pages, 4 figure

Predicting the Drug Release Kinetics of Matrix Tablets

In this paper we develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of partial differential equations. The predictions of both models show good qualitative agreement with experimental release curves. The models will provide tools for designing better controlled release devices.Comment: 17 pages, 7 figures; Elaborated at the first Workshop on the Application of Mathematics to Problems in Biomedicine, December 17-19, 2007 at the University of Otago in Dunedin, New Zealan

A mathematical model quantifies proliferation and motility effects of TGF--β\beta on cancer cells

Transforming growth factor (TGF) $\beta$ is known to have properties of both a tumor suppressor and a tumor promoter. While it inhibits cell proliferation, it also increases cell motility and decreases cell--cell adhesion. Coupling mathematical modeling and experiments, we investigate the growth and motility of oncogene--expressing human mammary epithelial cells under exposure to TGF--$\beta$. We use a version of the well--known Fisher--Kolmogorov equation, and prescribe a procedure for its parametrization. We quantify the simultaneous effects of TGF--$\beta$ to increase the tendency of individual cells and cell clusters to move randomly and to decrease overall population growth. We demonstrate that in experiments with TGF--$\beta$ treated cells \textit{in vitro}, TGF--$\beta$ increases cell motility by a factor of 2 and decreases cell proliferation by a factor of 1/2 in comparison with untreated cells.Comment: 15 pages, 4 figures; to appear in Computational and Mathematical Methods in Medicin

Modeling the bidirectional glutamine/ ammonium conversion between cancer cells and cancer-associated fibroblasts

Like in an ecosystem, cancer and other cells residing in the tumor microenvironment engage in various modes of interactions to buffer the negative effects of environmental changes. One such change is the consumption of common nutrients (such as glutamine/Gln) and the consequent accumulation of toxic metabolic byproducts (such as ammonium/NH4). Ammonium is a waste product of cellular metabolism whose accumulation causes cell stress. In tumors, it is known that it can be recycled into nutrients by cancer associated fibroblasts (CAFs). Here we present monoculture and coculture growth of cancer cells and CAFs on different substrates: glutamine and ammonium. We propose a mathematical model to aid our understanding. We find that cancer cells are able to survive on ammonium and recycle it to glutamine for limited periods of time. CAFs are able to even grow on ammonium. In coculture, the presence of CAFs results in an improved survival of cancer cells compared to their monoculture when exposed to ammonium. Interestingly, the ratio between the two cell populations is maintained under various concentrations of NH4, suggesting the ability of the mixed cell system to survive temporary metabolic stress and sustain the size and cell composition as a stable entity