# Michaelis-Menten enzyme kinetics

## by PhilipJ on 27 September 2007

Enzymes are the workhorses of the cell. They are proteins which help facilitate reactions, such as RNA polymerase adding nucleotides to form new strands of RNA. The polymerase is the enzyme (*E*), the nucleotides are the substrate (*S*), which react to form a product (*P*), the newly formed RNA strand.

It can often be the case that there are very low copy numbers of enzymes in the cell, meaning only one to a handful of a specific kind of enzyme. It is also possible that there is significant substrate present. You might then wonder, how *fast* can the enzyme of interest, given basically unlimited substrate, carry out its reaction? Let’s simplify the situation and assume only one enzyme in a cell filled with substrate*. The below treatment was first outlined by Haldane, and then by Leonor Michaelis and Maud Menten.

The enzymatic reaction described above looks like this:

where the concentration of the substrate, *c _{S}*, comes into play, and it is assumed that the second step,

*ES*reacting to make product

*P*, is irreversible. Thinking of the reaction in terms of probabilities (which we will denote using lowercase

*p*), if the enzyme is unbound there is a probability per unit time

*c*of binding a substrate, if the enzyme is bound with the substrate there is a probability per unit time of

_{S}k_{1}*k*of reacting to form product, or probability per unit time

_{2}*k*of returning back to its unbound state. Explicitely,

_{-1}In the steady-state (*dp _{E}/dt = 0*), you can solve for the probability of being in the enzyme-substrate complex:

The rate at which product is created is then *k _{2}p_{ES}* as per our original reaction equation, and multiplying by the enzyme concentration (in our case just a single enzyme) gives you the

*velocity*of the reaction.

We can simplify the formula further by introducing these two substitutions:

where *K _{M}* is called the Michaelis constant, and is a measure of concentration, while

*v*is a measure of the rate of change of concentration. Substition gives the famous Michaelis-Menten rule:

_{max}This rule helps explain the observed kinetics of some enzyme reactions, such as the mechanochemical cycle RNA polymerase (free link to the paper at the bottom of the post).

So, why the primer on enzyme kinetics?

Because Maud Menten and I are schoolmates, so to speak!

** This treatment is based on P. Nelson’s Biological Physics, Ch. 10.*

Maud Menten and the Michaelis-Menten Equation

http://sandwalk.blogspot.com/2007/01/maud-menten-and-michaelis-menten.html

I had already seen that photo over at Prof Larry Moran’s Sandwalk blog, but I enjoyed the intro to the formula.

There is so much more to this formula than meets the eye. I recall learning how this formula was achieved and also all the other formulas related to enzyme kinetics.

My favorite method of determining the Km and Vmax is the Eisenthal Cornish-Bowden method

good n can do more on that