Using Elementary Flux Modes to Estimate the Distance to Regime Shifts in Kinetic Systems*

This paper studies regime shifts in dynamical systems with kinetic realizations. A regime shift occurs when small external disturbances shift the system's state from a nominal to an alternative qualitative behavior. Kinetic systems are dynamical systems found in biological networks that are defined with respect to a directed graph modeling mass/energy transport. This paper estimates the distance-to-regime-shift (DTRS) using robust stability concepts. A novel feature of the work is its parameterization of the network using elementary flux modes (EFM). EFMs represent fundamental pathways governing the fate of species in the network. The advantage of this parameterization is that the linearized system can be written as an affine parameter dependent (APD) system. The robust stability of APD systems can be checked through a linear matrix inequality (LMI) feasibility problem. The DTRS analysis, therefore, is computationally tractable since there exist efficient solvers for such LMI problems. This is demonstrated for an oscillator system that has been used to study the robustness of biochemical systems.