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:

k

_{1 }k_{2}E + S ES E + P

k

_{-1}the initial velocity V

_{0 }can be determined by the breakdown of ES complex into product which is give by: V

_{0 }= k_{2}[ES] Eq No. 1Since 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 = k

_{1}([E_{t}] – [ES])[S]Rate of ES breakdown = k

_{-1 }[ES] + k_{2 }[ES]At the steady state rate of [ES] formation will be equal to ES breakdown so above equation can be written as:

k

_{1}([E_{t}] – [ES])[S] = k_{-1 }[ES] + k_{2 }[ES]To solve the above equation multiply the left side and simplify the right side:

k

_{1}[E_{t}][S] - k_{1}[ES][S] = (k-_{1 + }k_{2})[ES]add the term k

_{1}[ES][S] to both sides of the equation:k

_{1}[E_{t}][S] = (k_{1}[S] + k_{-1}+ k_{2})[ES]and then solve the equation for [ES]:

[ES] = [E

_{t}][S] [S] + (k

_{2 }+ k_{-1})/k_{1}The term (k

_{2 }+ k_{-1})/k_{1 }is defined as Michaelis constant K_{m}Substitute k

_{m }into above equation:[ES] = [E

_{t}][S] [S] + K

_{m}Now V

_{0 }can be expressed in terms of [ES] by substituting above equation into equation no.1: [S] + K

_{m}Since maximum velocity occurs when enzyme is saturated with [ES] = [E

_{t}] so V_{max}can be defined as k_{2 }[E_{t}] so substitute the value of V_{0 }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 V

_{0 }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 V_{0 }= 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/V

_{0 }= 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/V

_{0}= K_{m }+ [S]V

_{max}[S] V_{max}This equation is known as double reciprocal plot or Lineweaver Burk equation.