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 (S 0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S 0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S 0. 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.
Highlights
A major goal of molecular systems biology is to build, simulate, and analyze mathematical models of complex molecular regulatory systems comprising genes, proteins, and metabolites [1,2,3]
A and B are numbers referring to the concentrations of the particular chemical species. [A] and [B] are alternative ways to express the concentrations of chemical species.) This formulation leads to many differential equations with many separate terms on the right-hand sides
To understand how information is processed in these networks requires accurate mathematical models of the dynamical behavior of large sets of coupled chemical reactions
Summary
A major goal of molecular systems biology is to build, simulate, and analyze mathematical models of complex molecular regulatory systems comprising genes, proteins, and metabolites [1,2,3]. [A] and [B] are alternative ways to express the concentrations of chemical species.) This formulation leads to many differential equations (one for every chemical species in the network, including all intermediary complexes formed, transiently) with many separate terms on the right-hand sides (one for every reaction in which the species participates). Some of these reactions are very fast, some are very slow, and some are in between. The large differences of timescales in the network (typically many orders of magnitude) create huge difficulties for simulating the temporal evolution of the network and for understanding the basic principles of its operation
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