# Michaelis-Menten kinetics

This year is the one hundred anniversary of the Michaelis-Menten equation, which was published in 1913 by German born biochemist Leonor Michaelis and Canadian physician Maud Menten. Menten was one of the first women to obtain a medical degree in Canada and travelled to Berlin to work with Michaelis because women were forbidden from doing research in Canada. After spending a few years in Europe she returned to the US to obtain a PhD from the University of Chicago and spent most of her career at the University of Pittsburgh. Michaelis also eventually moved to the US and had positions at Johns Hopkins University and the Rockefeller University.

The Michaelis-Menten equation is one of the first applications of mathematics to biochemistry and perhaps the most important. These days people, including myself, throw the term Michaelis-Menten around to generally mean any function of the form

$f(x)= \frac {Vx}{K+x}$

although its original derivation was to specify the rate of an enzymatic reaction.  In 1903, it had been discovered that enzymes, which catalyze reactions, work by binding to a substrate. Michaelis took up this line of research and Menten joined him. They focused on the enzyme invertase, which catalyzes the breaking down (i.e. hydrolysis) of the substrate sucrose (i.e. table sugar) into the simple sugars fructose and glucose. They modelled this reaction as

$E + S \overset{k_f}{\underset{k_r}{\rightleftharpoons}} ES \overset{k_c}{\rightarrow }E +P$

where the enzyme E binds to a substrate S to form a complex ES which releases the enzyme and forms a product P. The goal is to calculate the rate of the appearance of P.

Assuming mass action kinetics, the concentrations of the chemicals obey the following differential equations

$\frac{d[E]}{dt} = k_r [ES]-k_f[E][S]+k_c[ES]$   (1)

$\frac{d[S]}{dt} = k_r [ES]-k_f[E][S]$  (2)

$\frac{d[ES]}{dt} = -k_r [ES]+k_f[E][S]-k_c[ES]$  (3)

$\frac{d[P]}{dt} = k_c [ES]$     (4)

What Michaelis and Menten did was to assume that the enzyme-substrate reaction is in quasi-equilibrium so that the rate of appearance of P is then just simply given by equation (4).  To solve for the equilibrium concentration, note that if we add equations (1) and (3) we obtain

$\frac{d[E]}{dt} +\frac{d[ES]}{dt}=0$

Integrating this equation then leads to

$[E]+[ES]=E^T$  (5)

where $E^T$ is the total available concentration of the enzyme.  In order to reach equilibrium, we must assume that the forward reaction from the complex ES to E and P is small compared to the direct enzyme substrate reactions.  We can then set the derivatives in equations (1) – (3) to zero, which all yield the same result:

$[ES]= \frac{k_f}{k_r}[E][S]$  (6)

Substituting [ES] from equation (6) into equation (5) and solving for [E] gives

$[E]=\frac{E^T}{1+(k_f/k_r)[S]}$

which when substituted back into equation (6) gives the result we need

$[ES]=\frac{E^T[S]}{K+[S]}$

where $K=k_r/k_f$ is the equilibrium constant of the enzyme-substrate reaction.  This then gives the famous Michaelis-Menten result for the rate or velocity of the reaction as a function of the concentration of substrate

$\frac{d[P]}{dt} = \frac{V_{\max}[S]}{K+[S]}$

where the maximal velocity $V_{\max}=k_cE^T$ and K is the concentration of substrate that attains half maximal velocity.

A decade later in 1925, Briggs and Haldane came up with a quasi-steady-state approximation to calculate the reaction velocity. Instead of assuming quasi-equilibrium, which can only occur in a reversible process, they considered ES to be approximately constant. Setting the derivative to zero in equation (4) gives

$k_f[E][S]=k_r[ES]+k_c[ES]$

Performing the same steps but with equation (6) replaced by this equation then yields

$\frac{d[P]}{dt} = \frac{V_{\max}[S]}{K_m+[S]}$

where $K_m=(k_r+k_c)/k_f$ is called the Michaelis constant.