Question

derive michaelis menten equation

please show steps

Answer 1

Michaelis Menten equation provides an idea on enzyme kinetics when the enzyme concentration is low, which relates to in vitro kinetics. When an enzyme acts on its substrate, the enzyme initially forms an Enzyme-substrate complex, which is then converted into product and free enzyme is released. In the initial stages of a reaction, the product concentration is very little, and the reverse reaction can be ignored. We also assume that concentration of substrate is much higher than the enzyme concentration. This is given in the equation shown below:

Where E = enzyme

S = substrate

ES = enzyme substrate complex

K₁, k₂, and k₃ are rate constants

The rate of the process (v) to the change in product formation is written as a function of time. It is written as follows:

In a reaction, the concentration of enzyme-substrate complex (ES) cannot be measured experimentally. It is alternatively represented as the enzyme bound (ES) to the enzyme unbound (E) to the substrate. The fraction of bound enzyme is expressed as follows:

Where E_t is the total enzyme concentration. Multiply both sides with E_t, we get the following equation:

If both the numerator and denominator of the right-hand side of equation 4 is divided by 1/[ES], we come to the following equation:

Equation (5) says that the rate of change of [ES] with time is zero:

Another steady state assumption is that the rate of formation of [ES] equals the rate of breakdown of [ES]. This can be mathematically represented as follows:

This shows the rate of formation of enzyme substrate complex and the two ways of break down of the complex. Equation (6) can be rearranged to get the ratio of:

The Michaelis constant (k_m) is expressed as given below:

Substitute K_m into equation 7, we get:

Substitute equation 9 in equation 5, we get the following:

Multiply the numerator and denominator of the right hand side of equation 10 by [S], we get:

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Now substitute the value of [ES] in equation 2, we get

When the substrate concentration is far higher than the enzyme concentration, all the catalytic sites are saturated and k₃[E_t] equals V_max; the maximum velocity of the reaction that can be obtained. Equation 12 can be rewritten to get the Michaelis-Menten equation:

The equation 13 represents the typical Michaelis-Menten equation when the substrate concentration is far higher than the enzyme.