Modeling Networks of Coupled Enzymatic Reactions Using the Total Quasi-Steady State Approximation

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Abstract
In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis–Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme–substrate complex (C) is much less than the free substrate concentration (S0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S0. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter–Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter–Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis–Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively. The physiological responses of a cell to its environment are controlled by gene–protein interaction networks of great complexity. To understand how information is processed in these networks requires accurate mathematical models of the dynamical behavior of large sets of coupled chemical reactions. To avoid producing large and hardly manageable models, such reaction networks are often simplified using phenomenological reaction rate laws, such as the Michaelis–Menten rate law for an enzyme-catalyzed reaction. We show that, in regulatory networks where proteins swap places as enzymes and substrates, such simplifications must be carried out with care, keeping track of enzyme–substrate complexes. The risk is to provide a simplified description of the molecular networks that at best is correct for the long-term behavior but fails to represent the short-term dynamics of the real network. To avoid such a possibility, we suggest using an alternative approach called the total quasi-steady state approximation. We apply this alternative formalism to a model of the network controlling the entry into mitosis in the eukaryotic cell cycle, composed of three coupled protein modification cycles. Whereas the classical Michaelis–Menten formalism fails to represent the dynamics of this network correctly, the one we propose captures the behavior with economy and accuracy.