Michaelis Menton Equation

ENZYME KINETICS

The Michaelis Menton Equation: a generalized theory of enzyme action was formulated by Leonor Michaelis and Maud Menton in 1913. The derivation starts with a basic step involving formation and breakdown of the ES complex.

The overall reaction is:

                    k₁                                        k₂

E + S           ES             E + P    

                        k_-1

the initial velocity V₀ can be determined by the breakdown of ES complex into product which is give by:

                                                V₀  =  k₂ [ES]           Eq No. 1

Since it is not easy to measure [ES] experimentally we should figure out an alternative expression of [ES]. So , we will consider [E_t ], that denotes for total enzyme concentration.

Free or unbound enzyme can be represented by:

                                                [E_t] – [ES]

The rate of formation and breakdown of ES can be determined by:

Rate of ES formation =  k₁ ([E_t] – [ES])[S]

Rate of ES breakdown = k_-1 [ES] + k₂ [ES]

At the steady state rate of [ES] formation will be equal to ES breakdown so above equation can be written as:

k₁ ([E_t] – [ES])[S]  = k_-1 [ES] + k₂ [ES]

To solve the above equation multiply the left side and simplify the right side:

k₁[E_t][S]  - k₁[ES][S] = (k-_{1 ­+} k₂)[ES]

add the term k₁[ES][S] to both sides of the equation:

k₁[E_t][S] = (k₁[S] + k_-1 + k₂)[ES]

and then solve the equation for [ES]:

[ES]  =                   [E_t][S]

               

                     [S] + (k₂ + k_-1)/k₁

The term (k₂ + k_-1)/k₁  is defined as Michaelis constant K_m

Substitute k_m  into above equation:

            

[ES]  =                   [E_t][S]

                           

                             [S] + K_m

Now V₀ can be expressed in terms of [ES] by substituting above equation into equation no.1:

V_{0   =}                                   k₂[E_t][S]

                          

                                            [S] + K_m

Since maximum velocity occurs when enzyme is saturated with [ES] = [E_t] so V_max can be defined as k₂ [E_t] so substitute the value of V₀ into above equation:

V_{0   =}                                     V_max [S]

                           

                                            [S] + K_m

The above equation is called as Michaelis Menton equation.

A numerical relationship exists in the Michaelis menton equation when V₀  is exactly one half of V_{max .} then:

      V_max   / 2  =                              V_max [S]

                                                

                                                       [S] + K_m

Divide above equation by V_max we get:

½  =    [S] / K_m + [S]

Solving for K_m we get K_{m  +} [S] = 2 [S],

Or  k_m  = [S] , when V₀  = ½ V_max

Transformation of Michaelis Menton equation: Double Reciprocal plot

V_{0   =}                                  V_max [S]

                          

                                         [S] + K_m

Take the reciprocal of above equation:

1/V₀   =                   K_m  + [S] / V_max [S]

Separate the components of numerator on the R.H.S of the equation and after simplification we get:

1/V₀  =     K_m           +      [S]

V_max[S]            V_max

This equation is known as double reciprocal plot or Lineweaver Burk equation.