Seven competing ways to recover the Michaelis–Menten equation reveal the alternative approaches to steady state modeling

Abstract

The Michaelis-Menten enzymatic reaction is sufficient to perceive many subtleties of network modeling, including the concentration and time scales separations, the formal equivalence between bulk phase and single-molecule approaches, or the relationships between single-cycle transient probabilities and steady state rates. Seven methods proposed by different authors and yielding the same famous Michaelis-Menten equation, are selected here to illustrate the kinetic and probabilistic use of rate constants and to review basic techniques for handling them. Finally, the general rate of an ordered multistep reaction, of which the Michaelis-Menten reaction is a particular case, is deduced from a Markovian approach. Preliminary tools: the helpful scale separationsThe approximation of concentration and time scales separations is often reasonable in cellular biochemistry, but some discernment is necessary for its proper application, in particular to define pseudo-first order constants and decide which reactions can be considered as in quasiequilibrium compared to others. Concentration scale separationThe wide differences of molecular concentrations in the cell greatly facilitate network modeling. The concentration of the more concentrated reactant, generally called the ligand or the substrate in enzymology, can be associated to second-order constants to give so-called pseudofirst order constants. This approximation strongly simplifies elementary treatments, for example to define hyperbolic saturation functions through equating the concentrations of total and free ligand. To apply the concentration scale separation, it is important to decide which reactant should be fused to the second-order constant. Depending on the cases, the same molecule can behave as either the leading macromolecule or as the ligand. This is the case, for example, for a transcription factor, say the estrogen receptor (ER), activated by the estrogen hormone (E2) and then capable of binding to a given unique gene (G) from the X chromosome. Even if ERs are not very numerous in the cell, they are however much more abundant than the single gene. Hence, ER can be considered as a diffusible ligand whose concentration varies slowly compared to the dynamics of its interaction with the gene. Conversely, when studying the activation of ER by E2, the ligand is now E2, which should be integrated in pseudo-first order rates of ER state changes. If the binding of E2 to ER and the binding of ER to G are to be mixed in the same model, a new approximation intervenes: the time scale separation. Time scale separationIn the example introduced above, the interactions between E2 and ER can be considered as more dynamic than those occuring between ER and G. Time scale separation is particularly important to obtain smooth graded interactions between a ligand and a very unique binding site [1]. This approximation allows kinetic and equilibrium constants to coexist in the same equation, as long proposed [2]. Once built, the first-order network can be treated through different m...

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