A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks

Complex molecular biological systems can be described in principle by reaction networks that explicitly take into account the sophisticated network of chemical interactions regulating cell life. Unfortunately, the effective utilization of such descriptions is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. They can also change with environmental variations. Thus, it is desirable to have a theoretical framework to robustly guarantee the behavior of such networks, based only on graphical knowledge and applying regardless of the particular form of kinetics. This paper introduces a class of networks that are “structurally (mono) attractive” by which we mean that they are incapable of exhibiting multiple steady states, oscillation, or chaos by the virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function which we call a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. We study basic motifs, three-body binding, and transcriptional networks. We focus on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signaling. We also study the Ribosome Flow Model. Arguably, the present approach identifies the largest class yet of biologically-relevant “attractive” networks. Author summary A theoretical and computational framework is developed for the identification of biochemical networks that are “structurally attractive”. This means they only allow global point attractors and they cannot exhibit any behavior such as multi-stability, oscillations, or chaos for any choice of the kinetics. They are characterized by the existence of energy-like functions. A computational package is made available for usage by a wider community. Many relevant networks in molecular biology satisfy the assumptions, and some are analyzed for the first time.