## The Michaelis-Menten limit

In a previous post I wrote about the Michaelis-Menten reduction of reactions catalysed by enzymes in which a single equation (effective equation) is the limit of a system of two equations (extended equations) as a parameter $\epsilon$ tends to zero. What I did not talk about is the sense in which solutions of the effective equation approximate those of the extended ones. I was sure that this must be well-known but I did not know a source for it. Now I discovered that what I had been seeking is to be found in a very nice form in a book which had been standing on a shelf in my office for many years. This is the book ‘Asymptotic Expansions for Ordinary Differential Equations’ by Wolfgang Wasow and the part of relevance to Michaelis-Menten reduction starts on p. 249. Michaelis-Menten is not mentioned there but the key mathematical result is exactly what is needed for that application. The theorem is due to Tikhonov but the original paper is in Russian and so not accessible to me. For convenience I repeat the equations from the previous post on this subject. $\dot u=f(u,v),\epsilon\dot v= g(u,v)$. This is the type of system treated in Tikhonov’s theorem, including the possibility that $u$ and $v$ are vector-valued.

The statement of the theorem is as follows. On any finite time interval $[0,T]$ the function $u$ in the extended system converges uniformly to the solution of the reduced system as $\epsilon\to 0$. Given a solution of the reduced system it is possible to compute a corresponding function $v$. On the time interval $(0,T]$ the function $v$ in the extended system converges to the function $v$ coming from the reduced system uniformly on compact subsets. Of course this conclusion  requires some hypotheses on the functions $f$ and $g$. The key thing is that for a fixed value of $u$ we have an asymptotically stable stationary solution of the equation for $v$ (with $\epsilon\ne 0$).

With this result in hand it is possible to compute higher order corrections in $\epsilon$. This was first done by Vasileva and is also explained in the book of Wasow. The result was extended to a statement global in $t$ by Hoppensteadt, Trans. Amer. Math. Soc. 123, 521. I expect that there are more modern treatments of these things in the literature but I find the sources quoted here very helpful for the beginner like myself. There remains the question of the relation to the usual Michaelis-Menten procedure. This is nicely discussed in a paper by Heineken et. al., Math. Biosci. 1, 95.