Michaelis-Menten equation

The usual equation to describe the conversion of a substrate in an enzymatic reaction. Let be the concentration of some substrate which is converted by an enzyme into a product . The reaction rate is proportional to for small values of , but there is a maximum rate, which is not surpassed even for large . These observations can be expressed by the equation

In this case the first reaction parameter, , describes the maximal conversion speed, and the second reaction parameter, , is equal to the substrate concentration at which the reaction speed is exactly . For small values of , the reaction rate is .

The equation can be derived from the chemical equilibrium reactions between the substrate and the enzyme , which combine to a compound . This compound is rearranged in an equilibrium reaction into a compound , which dissociates into the enzyme and product . In a formula:

In this reaction, the step from to is usually far slower than the other reactions, and if the reaction from to is irreversible, or if the product is removed by some transport mechanism, then the two reaction constants and can be discarded, and the rearrangement coefficient determines a simplified reaction equation:

Putting , the following system of differential equations emerges from the reactions:

These equations imply that is a constant, representing the maximal amount of either or . Usually, is large with respect to fluctuations in and so the reaction enters a steady state, in which the concentrations of and remain almost constant over a large time interval. In that case the proportion equals , which yields

and

The values of and can be inferred from the differential equation. This gives:

and so

The values can be fitted from observations of , but it is impossible to also find the parameters , , , and from the curve of .

The equation was derived by L. Michaelis and M.L. Menten [a1].

References

[a1]	L. Michaelis, M.L. Menten, "Die Kinetik der Invertinwirkung" Biochem. Zeitschrift , 2 (1913) pp. 333–369