Often in biochemistry, if a single substrate \(\text S\) needs to be become a product \(\text P\) via a chemical reaction of the form \(\text S\to \text P\). Assuming this is a first-order elementary chemical reaction, it would merely have rate law \(\dot{[\text P]}=-\dot{[\text S]}=k[\text S]\) for some rate constant \(k>0\), and thus the usual exponential time evolution. However, nature has made use of enzymes \(\text E\), which are simply biological catalysts. As with any catalysts, the thermodynamics \(\Delta H, \Delta S, \Delta G\) are invariant because they are state functions, but the kinetics (and thus activation energy barrier \(\Delta E_a\)) are significantly reduced by providing an alternative reaction pathway to the direct conversion \(\text S\to\text P\)). Specifically, the enzyme \(\text E\) first binds onto the substrate \(\text S\) via an elementary chemical reaction of the form \(\text E+\text S\to\text{ES}\), forming an enzyme-substrate complex \(\text{ES}\). The enzyme \(\text E\) then desorbs from the enzyme-substrate complex \(\text{ES}\) to yield the desired product \(\text P\) and reforming the enzyme \(\text{E}\) again via a second elementary reaction of the form \(\text{ES}\to\text{E}+\text{P}\), thus being involved in but not consumed by the overall chemical reaction \(\text{S}\to\text P\), another defining property of catalysts. By applying the steady state approximation to the only reaction intermediate there is, namely the enzyme-substrate complex \(\text{ES}\), one has that \(\dot{[\text{ES}]}=0\). This eventually yields the Michaelis-Menten equation for the velocity \(\dot{[\text P]}\) enzyme \(\text E\)-catalyzed reactions on a single substrate \(\text S\) to form a product \(\text P\):
\[\dot{[P]}=\frac{k_{\text{cat}}[\text E]_0[\text S]}{[S]+K_M}\]
where \([\text E]_0\) is the initial enzyme concentration at time \(t=0\), and \(K_M\) is the Michaelis constant. There is also a somewhat misleading graph of \(\dot{[P]}\) as a function of \([\text S]\) often shown, where in the limit \([\text S]\to\infty\) of an infinite substrate concentration (i.e. near the start of the reaction), the velocity of product formation is at a global maximum \(\dot{[P]}^*=k_{\text{cat}}[\text E]_0\).