The phenomenon of persistence is the focus of much modern biological and biomedical research. Persistence is found at all scales, from the molecular and cellular, to tissues, organisms, populations, and ecosystems. At the cellular level, persistence in gene and protein networks plays a key role in the establishment of homeostasis, which is the ability of the cell to regulate key variables, so that internal conditions remain relatively constant. At the intracellular level, the nodes of these interaction networks could be signaling molecules, genes, and gene products; at the ecosystem level, the nodes could be the various species and energy resources. Many diseases involve a disturbance of homeostasis in specific types of cells, which corresponds to a loss of persistence in the associated interaction networks. A more complete characterization of persistent systems may improve the understanding of these diseases. This project will create software that implements mathematical methods for understanding persistence, and will allow biologists and biomedical scientists to analyze persistence properties of diverse biological networks of interest.
Persistence and permanence refer to the capacity of a system to maintain all of its variables within some fixed limits in a robust way, and this capacity is one of the most important features of biological interaction networks. In order to understand the role played by specific biological interactions (e.g., a signaling pathway in a cell), one often faces great difficulties in trying to interpret the effect of positive and negative feedbacks, nonlinear interactions, and other complex signaling between the nodes of the network. These difficulties are due to the inherent complexity of the dynamics of nonlinear systems. There are significant mathematical challenges to be overcome, whose study will lead to rich new areas of biological insight as well as mathematical theory. This project will analyze persistence as a fundamental theoretical concept that traverses levels of biological complexity. Mathematical and computational tools will be developed to understand persistence in general biological interaction networks. As a concrete step in that direction, this project will focus on biochemical networks, with the aim to provide mathematical tools for drawing precise connections between reaction network structure and its persistence properties. These tools will be applicable to complex networks, and will be able to distinguish between very similar networks having rather different capacities for persistence.
