Derivation of the Michaelis-Menten Equation

The Michaelis-Menten equation is a fundamental tool used in biochemistry and enzymology to study enzyme kinetics. The equation was first proposed by biochemists Leonor Michaelis and Maud Leonora Menten in 1913, and has since become a vital concept in understanding enzyme-substrate interactions. It takes the form of an equation relating reaction velocity to substrate concentration for a system where a substrate 'S' binds reversibly to an enzyme 'E' to form an enzyme-substrate complex 'ES', which then reacts irreversibly to generate a product 'P' and to regenerate the free enzyme 'E'. This system can be represented schematically as follows-
S + E ⇌ ES → P + E

Conditions

The Michaelis constant K_M is the substrate concentration at which V_i is half the maximum velocity(V_max/2) attainable at a particular concentration of enzyme.
K_M thus has the dimentions of substrate concentration. The dependence of initial reaction velocity on [S] and K_M may be illustrated by evaluating the Michaelis Menten equation under three conditions-

Condition-1
Point-A in figure-
When [S] is much less than K_M, the term K_M + [S] is reduced to K_M.
Thus Michaelis Menten equation becomes-
V_i = V_max [S] / K_M
or, V_i = (V_max / K_M)[S]
Since, V_max and K_M are both constants, their ratio is a constant. In other words, when [S] is considerably below K_M,
V_i ∝ K[S]
Thus the initial velocity therefore, is directely proportional to [S].

Condition-2
Point-C in figure-
When [S] is much greater than K_M, then the term K_M + [S] is reduced to [S].
Thus Michaelis Menten equation becomes-
V_i = V_max [S] / [S]
or, V_i = V_max
Thus, when [S] greatly exceeds K_M, the reaction velocity is maximal (V_max) and unaffected by further increase in substrate condition.

Condition-3
Point-B in figure-
When [S] = K_M
Then Michaelis Menten equation becomes-
V_i = V_max [S] / 2[S]
or, V_i = V_max / 2
This equation states that when [S] is equal to K_M, the initial velocity is half maximal.
This equation also reveals that K_M is and may be determined experimentally fron- the substrate concentration at which the initial velocity is half maximal.

What is the K_M value ?

K_M value is the concentration of substrate at which half of the V_max value is attained.
K_M value value is equal to the substrate concentration at which half of the enzyme active sites are saturated with the substrate. It tells about the affinity of enzymes for their substrate.

What does Michaelis-Menten model assume ?

a. Enzyme, Substrate, and Enzyme-Substrate complex are in equilibrium
b. Enzyme-Substrate complex decomposes to product only
c. No intermediate is formed
d. There are no enzyme catalyzed reactions

Which of the following is false about the Michaelis-Menten equation ?

a. Velocity is inversely proportional to enzyme concentration.
b. Velocity is proportional to the turnover number.
c. Velocity is proportional to substrate concentration.
d. The maximum rate of reaction is reached as the substrate concentration increases indefinitely.

Which of the following will increase the reaction rate of an enzymatic reaction ?

I. Adding a competitive inhibitor
II. Increasing the substrate concentration
III. Decreasing the affinity of substrate to the enzyme
a. I only
b. II only
c. I and II
d. I and III

Which of the following is a correct statement with regards to an enzyme-catalyzed reaction that obeys Michaelis-Menten kinetics ?

a. The reaction is first-order with respect to substrate at low substrate concentrations, and zero-order with respect to substrate at higher substrate concentrations.
b. The reaction is always first-order with respect to substrate regardless of substrate concentration
c. The reaction is first-order with respect to substrate at high substrate concentrations, and zero-order with respect to substrate at lower substrate concentrations
d. The reaction is always zero-order with respect to substrate regardless of substrate concentration

Based on Michaelis-Menten enzyme kinetics, if a system has a substrate concentration that is significantly more then the K_M, which of the following is a sound inference ?

a. The system is not saturated, adding more enzyme would increase the Vmax.
b. The substrate is the limiting factor. The system is not saturated.
c. The enzyme concentration is the limiting factor of the reaction rate, adding more substrate would not increase the rate.
d. The system is fully saturated and the rate is below V_max.