Abstract
Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. Here, we describe a general theoretical framework that reveals underlying principles of long-term growth: scalability of flux functions and ergodicity of the rescaled systems. Our theory shows that nonlinear fluxes can generate not only balanced growth but also oscillatory or chaotic growth modalities, explaining nonequilibrium dynamics observed in cell cycles and ecosystems. Our mathematical framework is broadly useful in predicting long-term growth rates from natural and synthetic networks, analyzing the effects of system noise and perturbations, validating empirical and phenomenological laws on growth rate, and studying autocatalysis and network evolution.
Highlights
Growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems
We identify an important class of biological reaction networks, scalable reaction networks (SRNs), whose long-term growth properties can be studied using powerful ergodic theory tools
We mathematically demonstrate two basic principles of exponentially growing systems: scalability of the underlying flux functions and ergodicity of the rescaled system
Summary
Wei-Hsiang Lina,b,c,d,e,1 , Edo Kussellf,g, Lai-Sang Youngh,i,j , and Christine Jacobs-Wagnera,b,c,d,e,k,1. Natural systems (e.g., cells and ecosystems) generally consist of reaction networks (e.g., metabolic networks or food webs) with nonlinear flux functions (e.g., Michaelis–Menten kinetics and density-dependent selection) Despite their complex nonlinearities, these systems often exhibit simple exponential growth in the long term. By applying ergodic theory on rescaled systems, we show mathematically that scalability and ergodicity ensure that a large class of reaction networks have well-defined long-term growth rates (λ), which can be positive (exponential growth), negative (exponential decay), or zero (subexponential dynamics) This theoretical framework opens the door to the study and characterization of processes that exhibit different growth modalities, including static equilibrium, steady state, and balanced growth and oscillatory and nonperiodic growth (Fig. 1A). It enables one to construct scalable networks of arbitrarily high complexity, predict the growth rate and other dynamical features of the system, and identify autocatalytic networks associated with positive exponential growth
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