Enzyme kinetics
Enzyme kinetics studies how rapidly enzyme-catalysed reactions occur and how reaction rates arise from molecular binding, unbinding and conversion processes.
A basic enzyme mechanism
Let an enzyme \(E\) bind a substrate \(S\), forming an enzyme–substrate complex \(ES\). The complex can either dissociate or form product \(P\):
The enzyme is released after product formation, so it acts as a catalyst rather than being permanently consumed.
Mass-action reaction rates
| Reaction | Meaning | Mass-action rate |
|---|---|---|
| \(E+S\to ES\) | binding | \(k_1[E][S]\) |
| \(ES\to E+S\) | dissociation | \(k_{-1}[ES]\) |
| \(ES\to E+P\) | product formation | \(k_2[ES]\) |
Square brackets denote concentration. The binding rate depends on both \([E]\) and \([S]\) because an enzyme and a substrate molecule must encounter one another.
Building the differential equations
The complex is created by binding and removed by dissociation or product formation:
\[\boxed{\frac{d[ES]}{dt}=k_1[E][S]-(k_{-1}+k_2)[ES].}\]Product is created by the catalytic step:
\[\boxed{\frac{d[P]}{dt}=k_2[ES].}\]The substrate balance is
\[\boxed{\frac{d[S]}{dt}=-k_1[E][S]+k_{-1}[ES],}\]and the free-enzyme balance is
\[\boxed{\frac{d[E]}{dt}=-k_1[E][S]+(k_{-1}+k_2)[ES].}\]Enzyme conservation
In this closed enzyme mechanism, each enzyme molecule is either free or in the complex. Therefore
\[\boxed{[E]_T=[E]+[ES]},\]where \([E]_T\) is the constant total enzyme concentration. Hence
\[[E]=[E]_T-[ES].\]This reduces the number of independent variables and makes clear that enzyme capacity is limited.
Reaction velocity
The instantaneous product-formation rate is
\[\boxed{v=\frac{d[P]}{dt}=k_2[ES].}\]Thus the reaction velocity depends directly on how much enzyme is currently present as enzyme–substrate complex.
The initial transient
Immediately after substrate is introduced, \([ES]\) may be small and then rise as binding occurs. The full mass-action equations describe this transient directly.
Under suitable experimental conditions, the complex can subsequently change much more slowly than the individual formation and removal fluxes. This observation motivates a useful approximation developed in the next lesson.
Units of the rate constants
The unimolecular constants \(k_{-1}\) and \(k_2\) have units of inverse time. Because \(k_1[E][S]\) must have units of concentration per unit time, \(k_1\) has units of inverse concentration per unit time.
What the mechanism already tells us
Even before reducing the model, the mechanism shows why enzyme concentration matters, why binding depends on both enzyme and substrate, why enzyme is conserved, and why product formation depends on the amount of bound complex.
It also provides the mechanistic starting point for more complicated models involving inhibition, cooperativity, multiple substrates, reversible product formation or several enzyme states.
From mechanism to a reduced rate law
The next lesson asks whether the four-variable mechanism can be reduced to a simple relationship between substrate concentration and initial reaction velocity. Under the classical quasi-steady-state assumptions this leads to the Michaelis–Menten law
\[v=\frac{V_{\max}[S]}{K_m+[S]}.\]The derivation, parameter interpretation, limiting behaviour and parameter estimation belong to that dedicated lesson.