Tuesday, 22 November 2011

Michaelis Menton Equation

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    
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 [Et ], that denotes for total enzyme concentration.
Free or unbound enzyme can be represented by:
                                                [Et] – [ES]
The rate of formation and breakdown of ES can be determined by:
Rate of ES formation =  k1 ([Et] – [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 ([Et] – [ES])[S]  = k-1 [ES] + k2 [ES]
To solve the above equation multiply the left side and simplify the right side:
k1[Et][S]  - k1[ES][S] = (k-1 ­+ k2)[ES]
add the term k1[ES][S] to both sides of the equation:
k1[Et][S] = (k1[S] + k-1 + k2)[ES]

and then solve the equation for [ES]:
[ES]  =                   [Et][S]
                     [S] + (k2 + k-1)/k1

The term (k2 + k-1)/k1  is defined as Michaelis constant Km
Substitute km  into above equation:

[ES]  =                   [Et][S]
                             [S] + Km
Now V0 can be expressed in terms of [ES] by substituting above equation into equation no.1:
V0   =                                   k2[Et][S]
                                            [S] + Km

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

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 Vmax . then:
      Vmax   / 2  =                              Vmax [S]
                                                       [S] + Km

Divide above equation by Vmax we get:
½  =    [S] / Km + [S]

Solving for Km we get Km  + [S] = 2 [S],
Or  km  = [S] , when V0  = ½ Vmax

Transformation of Michaelis Menton equation: Double Reciprocal plot

V0   =                                  Vmax [S]
                                         [S] + Km
Take the reciprocal of above equation:

1/V0   =                   Km  + [S] / Vmax [S]

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

1/V0  =     Km           +      [S]
Vmax[S]            Vmax

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