The Michaelis-Menten equation is a mathematical equation used to describe the relationship between the rate of an enzyme-catalyzed reaction and the concentration of substrate molecules. It is named after the biochemists Leonor Michaelis and Maud Menten, who first proposed the equation in 1913.

The Michaelis-Menten equation is:

V = (Vmax [S]) / (Km + [S])

where:

- V is the rate of the reaction
- [S] is the concentration of substrate
- Vmax is the maximum rate of the reaction, which occurs when all enzyme molecules are bound to substrate
- Km is the Michaelis constant, which is the substrate concentration at which the reaction rate is half of Vmax.

The Michaelis-Menten equation is widely used in biochemistry to analyze enzyme kinetics and to determine enzyme parameters such as Vmax and Km. It assumes that the enzyme-substrate complex is in rapid equilibrium with the free enzyme and the free substrate, and that the rate-limiting step of the reaction is the conversion of the enzyme-substrate complex to product and enzyme.

To derive the Michaelis-Menten equation, we start with the basic assumptions of enzyme kinetics:

- Enzyme and substrate react reversibly to form an enzyme-substrate complex.
- The formation of the enzyme-substrate complex is in rapid equilibrium with the free enzyme and the free substrate.
- The rate-limiting step of the reaction is the conversion of the enzyme-substrate complex to product and enzyme.

Based on these assumptions, we can write the following reaction scheme:

E + S ↔ ES → E + P

where E is the enzyme, S is the substrate, ES is the enzyme-substrate complex, and P is the product.

The rate of the reaction, V, is defined as the rate of formation of product, or the rate of disappearance of substrate:

V = -d[S]/dt = d[P]/dt

Assuming that the reaction is at steady-state, we can write:

d[ES]/dt = 0

This means that the rate of formation of the enzyme-substrate complex is equal to the rate of its breakdown:

k1[E][S] = k-1[ES] + k2[ES]

where k1 is the rate constant for the formation of the enzyme-substrate complex, k-1 is the rate constant for its breakdown, and k2 is the rate constant for the conversion of the enzyme-substrate complex to product and enzyme.

We can now write the rate of the reaction, V, in terms of the concentration of enzyme-substrate complex, [ES]:

V = k2[ES]

To eliminate [ES], we use the assumption that the formation of the enzyme-substrate complex is in rapid equilibrium with the free enzyme and the free substrate. This means that:

k1[E][S] = k-1[ES]

or

[ES] = (k1/[k-1][S])[E][S]

Substituting this expression for [ES] into the equation for V, we get:

V = k2(k1/[k-1][S])[E][S]

Simplifying this equation and rearranging, we obtain the Michaelis-Menten equation:

V = (Vmax [S]) / (Km + [S])

where

Vmax = k2[E] is the maximum rate of the reaction, which occurs when all enzyme molecules are bound to substrate, and Km = (k-1 + k2) / k1 is the Michaelis constant, which is the substrate concentration at which the reaction rate is half of Vmax.