## Time-scale separation in enzyme kinetics

The year 2013 is the 100th anniversary of Leonor Michaelis and Maud Menten's paper that introduced their famous mathematical formula for the rate of an enzymatic reaction [1, 2]. There are many instructive lessons in this paper [3], although the focus of the present review is on one particular aspect of what they did, which has ramified through biochemistry, pharmacology, molecular biology and, now, systems biology. Michaelis and Menten considered the reaction scheme:

in which free enzyme, *E*, binds reversibly to a substrate, *S*, to form an intermediate enzyme–substrate complex, *ES*, which then irreversible breaks down to free the enzyme and yield the product, *P*. The labels on the reactions are the rate constants, assuming mass-action kinetics. Michaelis and Menten derived from this scheme their rate formula:

in which is the maximal rate of the reaction, , and is the Michaelis–Menten constant.

There is something quite odd about the relationship between the reaction scheme in Eqn (1) and the rate formula in Eqn (2). The former involves the free enzyme *E* and the enzyme–substrate complex *ES* but these components have disappeared from the latter. The only vestige left of the enzyme is its total amount in the expression for the maximal rate. The total amount does not change over the course of the reaction, and so is a conserved quantity, not a dynamical variable. All other enzyme-related components have been eliminated.

To pull off this sleight-of-hand, Michaelis and Menten used a time-scale separation. They assumed that, under their *in vitro* conditions, in which substrate was in considerable excess of enzyme, the enzyme–substrate complex would rapidly form and reach a quasi-steady state, in which *d*[*ES*]/*dt* = 0. We might say, informally, that the enzyme-related components are assumed to be fast variables, which rapidly reach steady state, whereas the substrate and product are slow variables, which adjust to this steady state. (Formally, in biochemical systems, it is the reactions which are fast or slow, relatively speaking, not the components, a point to which we will return below.) With a little algebra, which has struck terror into the hearts of generations of students, the enzyme-related components can be eliminated in favour of the total amount of enzyme , from which Eqn (2) falls out.

Two historical points should be made here. First, this is not quite what Michaelis and Menten did. They used a different time-scale separation (a rapid equilibrium assumption) and it was Briggs and Haldane who suggested the more appropriate steady-state assumption that is now standard [4]. Second, they were not quite the first to use time-scale separation, as we will discuss below, although they were certainly the first in terms of influence.

Enzymologists rapidly took up the method of time-scale separation to analyze more complicated reaction schemes than that in Eqn (1) and there are now enzyme-rate formulas that cover a wide-range of enzymological contexts and include the impact of inhibitors and other kinds of effectors [5]. An interesting feature of these formulas is that they are always rational functions in the slow variables. That is, the right hand side is a ratio in which the numerator and the denominator are both sums of products (polynomials) in the concentrations of the slow variables. This may not appear to be particularly remarkable for the original Michaelis–Menten formula in Eqn (2) but it is a striking and universal feature of more complex formulas. The necessary algebraic manipulations, which get very intricate very quickly, were eventually codified in the King–Altman method [6], to which we will return.

Eliminating variables such as *ES* is, of course, a very good thing because, at the time of Michaelis and Menten, nobody knew anything about them. They were theoretical entities, suggested by the experimental data. It is often forgotten that Michaelis and Menten never characterized the enzyme–substrate complex (for the enzyme invertase which they studied) and they never measured its rates of assembly and disassembly ( and in Eqn (1)). The first person to do so, for the enzyme peroxidase, was Britton Chance, no less than 30 years after Michaelis and Menten [7]. This did not stop enzymologists from enthusiastically using such theoretical entities in the intervening years. Biology is actually more theoretical than physics; biologists just like to pretend otherwise [8].

The Michaelis–Menten formula has been hugely important [1]. It is perhaps the one quantitative mathematical statement that any biologist working at the molecular level would be expected to know. Unfortunately, its very familiarity has bred, not respect, but, rather, ignorance. The elimination of the enzyme–substrate complex has meant that such complexes have been lost from view, so that enzyme sequestration is all too readily overlooked [9, 10]. The formula is also widely used in the wrong contexts. In particular, Michaelis and Menten assumed that product formation was irreversible because they measured initial reaction rates when product was negligible. Yet, the formula is habitually used in contexts, such as phosphorylation and dephosphorylation cycles, in which the amount of product could be substantial [11]. Michaelis and Menten would have been horrified. One of the goals of this review is to explain how we can start to do justice to what Michaelis and Menten taught us a century ago. Before setting out to do so, let us examine some of the other contexts in which time-scale separation has been used.