This project focuses on monotone input/output systems as a tool in mathematical biology. Monotone i/o systems originated in the analysis of mitogen-activated protein kinase cascades and other cell signaling networks, but later proved useful in the study of a broad variety of other biological models. Their theory was and continues to be developed in response to the analysis of concrete biological models. The work includes: (a) The formulation and solution of new, biologically-motivated mathematical problems concerning signaling and regulatory biomolecular networks, including the study of memory, switching behavior, oscillations, and other dynamical characteristics, built upon the foundations of previous work on monotone and almost-monotone i/o systems; (b) The development of a theoretical and practical framework for the analysis of biomolecular networks using a combination of qualitative and quantitative techniques, and in particular the study of network topology as a constraint on possible dynamic behaviors and the study of the interplay of structure and parameters, by breaking up systems into well-behaved building-blocks components such as monotone subsystems, and using only input/output data for these subsystems in order to characterize global behavior.
The biological sciences are in the midst of revolutionary developments. Literally each day brings new discoveries, and proposals for novel organizing principles, which hold the promise of altering the fundamental understanding of life and diseases. Concomitant with these advances, leading biologists have recognized that new basic mathematcial and systems-level knowledge is urgently required. This work deals with mathematical questions motivated by these developments.
