Existence and uniqueness of the total quasi-steady-state approximation for coupled systems of enzyme kinetics
by
Shev MacNamara
The Institute for Molecular Biosciences, The University Of Queensland, Australia
Coauthors: Alberto M. Bersani, Kevin Burrage, Roger B. Sidje

The total quasi-steady-state approximation (tQSSA) is obtained merely by introducing a simple change of variable into the conventional QSSA and has the benefit of being valid over a wider parameter range. The original work on the tQSSA (Segel et al., Bulletin of Mathematical Biology, 1996) demonstrated this for the quintessential example of Michaelis-Menten enzyme kinetics and recently interest has focused on being able to generalize the approach to more complicated networks of coupled enzymatic reactions (Ciliberto et al., PLoS Computational Biology, 2007). In special cases explicit formulae for the approximation may be derived but for more complicated systems these are not available. We provide a theorem guaranteeing the existence and uniqueness of the approximation for networks of coupled enzymatic reactions, as well as an accompanying numerical method. These results are applied to the Goldbeter-Koshland switch and the mitogen-activated-protein kinase cascade. One novel aspect of this work is the application to the chemical master equation to understand the dynamics of discrete and stochastic biochemical kinetics. Previously this has been felt infeasible because of difficulties involved with the computation of the exponential of a matrix of very high dimension but by using Krylov methods and the extra structure present in the reactions arising in enzymatic networks we show that significant progress can be made.

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Date received: January 30, 2008