Long-term growth rate of nonlinear autocatalytic networks

A flux network is a fundamental structure of many biochemical, ecological, and economical systems. Yet, the long-term behavior of general flux systems remains to be characterized, especially when nonlinear functions are involved. Here, using ergodic theory, we prove the convergence of the growth rate for a large class of nonlinear flux system. This class of flux systems exhibits not only steady-state growth but also allows complex dynamics such as limit cycles to coexist with robust growth.

Our mathematical framework provides explicit formulae to analyze the following types of questions: (Q1) Given a flux network, how much will growth rate vary if the underlying flux function is modified? (e.g., by reducing enzymatic activity or by changing the Hill coefficient) (Q2) if the system state is transiently altered (e.g., by introducing/removing a specific amount of molecules, or through noise fluctuations), what would be the long-term impact on the system size? We envision that this framework will stimulate future understanding on fundamental principles of general autocatalytic networks, including cellular systems and ecosystems.