The impact of external forcing on an enzymatic reaction system with a single finite stable state is investigated. External forcing impacts on the system in two distinct ways: firstly, the reaction system undergoes a series of discontinuous changes in dynamical state. Secondly, a critical level of forcing exists, beyond which all finite states become unstable. It is shown that the results stem from the conditions for global stability of the system. Competition between the attractor for stable states and the unbounded states leads to a loss of integrity and the fractal fragmentation of the attraction basin for the finite state. The consequences of a fractal basin in this context are profound. Initial states which are infinitesimally close diverge to a finite and an unbounded state where only the finite state is consistent with biological functionality. Furthermore, above a critical forcing amplitude, the system does not converge to a finite state from any initial state, implying that there is no configuration of metabolite concentrations that is consistent with sustained evolution of the system. These results point to opportunities for constraining uncertainty in cell networks where nonlinear saturating kinetics form an important component.
Liu, J., Crawford, J., & Leontiou, K. (2005). Collapse of single stable states via a fractal attraction basin: analysis of a representative metabolic network. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2060), 2327-2338. https://doi.org/10.1098/rspa.2004.1436