An alternative approach to Michaelis-Menten kinetics that is based on the renormalization group.

We apply to Michaelis-Menten kinetics an alternative approach to the study of Singularly Perturbed Differential Equations, that is based on the Renormalization Group (SPDERG). To this aim, we first rebuild the perturbation expansion for Michaelis-Menten kinetics, beyond the standard Quasi-Steady-State Approximation (sQSSA), determining the 2nd order contributions to the inner solutions, that are presented here for the first time to our knowledge. Our main result is that the SPDERG 2nd order uniform approximations reproduce the numerical solutions of the original problem in a better way than the known results of the perturbation expansion, even in the critical matching region. Indeed, we obtain analytical results nearly indistinguishable from the numerical solutions of the original problem in a large part of the whole relevant time window, even in the case in which the kinetic constants produce an expansion parameter value as large as ɛ=0.5.

Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation.

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis-Menten enzyme kinetics, double phosphorylation, the Goldbeter-Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

The total quasi-steady-state approximation for fully competitive enzyme reactions.

The validity of the Michaelis-Menten-Briggs-Haldane approximation for single enzyme reactions has recently been improved by the formalism of the total quasi-steady-state approximation. This approach is here extended to fully competitive systems, and a criterion for its validity is provided. We show that it extends the Michaelis-Menten-Briggs-Haldane approximation for such systems for a wide range of parameters very convincingly, and investigate special cases. It is demonstrated that our method is at least roughly valid in the case of identical affinities. The results presented should be useful for numerical simulations of many in vivo reactions.