I read in my textbook that the assumptions of equilibrium and of steady state used by Michaelis and Menten were simplifying assumptions intended to make the following equation one that can be integrated: d[ES]/dt = k1[E][S] - k_-1[ES] - k_2[ES] (1) They go on to assume that k_-1 is much bigger than k_2 so that equilibrium is achieved and can now write: Ks= k_-1/k_1 = ([E][S])/[ES]. According to the textbook, the first equation can now be integrated. Can someone explain why it could not be integrated prior, why it can be integrated now and why it was necessary to integrate in the first place? The textbook is Fundamentals of Biochemistry third edition by Voet and Pratt Chapter 12 p.368
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1$\begingroup$ You can integrate the equations, but only numerically, the nature of the equations is such that an exact algebraic solution is not easy or feasible. It turns out that mathematically many types of differential equations are very, very hard to integrate. $\endgroup$– porphyrinCommented Nov 1, 2023 at 16:29
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$\begingroup$ I should add that to integrate you also need $d[E]/dt$ and $d[S]/dt$, by substituting for $k_s$ the term in $[E][S]$ can be changed to one in $[ES]$ alone so $d[ES]/dt$ can now be integrated by standard methods. Whether this is valid in a chemical sense is another matter. $\endgroup$– porphyrinCommented Nov 2, 2023 at 11:32
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I read in my textbook that the assumptions of equilibrium and of steady state used by Michaelis and Menten were simplifying assumptions intended to make the following equation one that can be integrated: d[ES]/dt = k1[E][S] - k_-1[ES] - k_2[ES]
No, you are assuming a steady state (i.e. d[ES]/dt = 0) so you can calculate the concentration of the enzyme substrate complex. The alternative would solve the system of differential equations ("to integrate them"), but that is more complicated and arguably gives you less insight.
They go on to assume that k_-1 is much bigger than k_2 so that equilibrium is achieved and can now write: Ks= k_-1/k_1 = ([E][S])/[ES].
That is not the classical derivation. I would expect a definition of $K_M$ and an expression of the initial rate dependent on $k_\mathrm{cat}, K_M, E_\mathrm{tot}$ and [S].
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$\begingroup$ (+1) Nice answer! For what it is worth, there is no particular difficulty in integrating the coupled ordinary differential equations with initial conditions and there is no need to simplify by assuming steady state or equilibrium or quasi-equilibrium, etc. There is even an analytic solution in terms of the Lambert W function. Despite all that, the standard simplifications do exactly what you say: they provide insight. $\endgroup$– Ed VCommented Dec 1, 2023 at 20:22
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1$\begingroup$ @EdV There is a historical subtlety I was not aware of. While the approximation is named after Michaelis and Menten, they did have an equilibrium approximation, and the steady state came a bit later by Haldane (according to Voet, Voet and Pratt textbook). $\endgroup$– Karsten ♦Commented Dec 1, 2023 at 23:23
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