Michaelis-Menten equation, derivation steady state assumption

The Michaelis-Menten equation can also be derived by applying the steady state assumption to the following scheme ... 

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and breakdown of ES (Eqns 6-7 and 6-8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse reaction, P—>S (described by k 2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to... 

Now substituting for E in the steady state assumption derivation of the Michaelis-Menten constant (equation 5.AS.5) gives ... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

Answer The basic assumptions used to derive the Michaelis-Menten equation still hold. The reaction is at steady state, and the overall rate is determined by... 

In deriving the rate law for a particular radical mechanism, the steady-state assumption in respect of all species with unpaired electrons, i.e. d[Ra ]/dt = 0, can usually be made, since the absolute concentrations are so low. The assumption parallels that made in steady-state enzyme kinetics (d[ES] / dt = 0), but whereas, in the absence of cooperativity, the steady-state approximation in enzyme kinetics always leads to the Michaelis-Menten equation (whatever the significance of or in terms of individual rate constants), the steady-state assumption in radical kinetics often leads to complex expressions. Dimerisations and dissociations in key steps can lead to fractional... 

Because the busy metabolic traffic of cells produces a steady state more often than an equilibrium, much use has been made of an equation devised by Briggs and Haldane (1925) who showed that it was unnecessary to assume an equilibrium between [E] and [S]. They derived Equation (v), formally similar to the Michaelis—Menten equation but free from this assumption and suitable for steady-state conditions. 

It is perhaps wise to begin by questioning the conceptual simplicity of the uptake process as described by equation (35) and the assumptions given in Section 6.1.2. As discussed above, the Michaelis constant, Km, is determined by steady-state methods and represents a complex function of many rate constants [114,186,281]. For example, in the presence of a diffusion boundary layer, the apparent Michaelis-Menten constant will be too large, due to the depletion of metal near the reactive surface [9,282,283], In this case, a modified flux equation, taking into account a diffusion boundary layer and a first-order carrier-mediated uptake can be taken into account by the Best equation [9] (see Chapter 4 for a discussion of the limitations) or by other similar derivations [282] ... 

Enzyme kinetics. The kinetics of transformation processes catalysed by a single enzyme are often described using the Micha-elis-Menten equation (1). The derivation of this equation is, however, based on two assumptions. The pseudo steady state hypothesis ( ) with respect to the intermediary enzyme-siibstrate complex is valid and the reverse reaction from product to substrate can be... 

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