The Michaelis-Menten Equation in biochemistry

The Michaelis-Menten Equation

Victor Henri, a French chemist in 1903, discovered that the enzyme-catalyzed reactions were initiated by a bond between the enzyme and the substrate (in general, a binding interaction). His research was undertaken by German biochemist Leonor Michaelis and Canadian physicist Maud Menten, who studied the kinetics of an enzymatic reaction system, invertase, which catalyzes sucrose hydrolysis into glucose and fructose. The mathematical model of the reaction was suggested in 1913. This equation is known as the Michaelis-Menten Equation.

Let us consider an Enzyme E is binding with a substrate S. It assumes the formation of an enzyme-substrate complex ES. After that, product P is released.

• The ES complex is assumed to be in fast balance with free enzyme E
• It is presumed that the breakdown of ES into products is slower than
  (1) formation of ES and
  (2) breakdown of ES to re-form E and S

This may be represented schematically as,

S + E  ↔ ES →  P + E  

rate of formation of product, V_o  = k₂  [ES]

ES formation rate =  k₁  [E][S]

= k₁  ([E_total] – [ES]) [S]

ES breakdown rate = k_-1 [ES]  +  k₂ [ES]

At steady state assumption

When the enzyme-catalyzed biochemical reactions are in the steady-state that means the formation and breakdown rate of the enzyme-substrate intermediate is equal.

ES formation rate = ES breakdow rate 

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

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

⇒ k1 [E_total][S]  =  (k₁[S] +  k_-1  +  k₂ )[ES]We know, V_o  = k₂ [ES]⇒ V_max  = k2  [E_total];

[Since, V_o = V_max   when  [E_total] = [ES] (at saturation).]

Enzyme Kinetics: Michaelis-Menton Equation

1.Characteristics of K_m

• K_m is called the Michaelis constant. It is characteristic of an enzyme and its specific substrate. The value of K_m represents the enzyme’s affinity to that substrate.
• K_m does not change with the changing value of enzyme concentration.

• Small K_m:
  A numerically tiny (low) value of k_m represents a strong substrate enzyme affinity.
• Large K_m:
  A numerically big (elevated)  value of k_m indicates a low substrate enzyme affinity.

2. Relationship of velocity to enzyme concentration:

The speed of an enzyme-catalyzed reaction at any substrate is directly proportional to the concentration of the enzyme. For example, if the enzyme concentration is halved, the initial rate of the reaction (v_o), as well as that of V_max (the maximum rate of reaction), is reduced to half that of the original rate of the reaction.

A Linear Form of the Michaelis-Menten Equation

The Michaelis-Menton Equation discussed above gives a carved graphical representation. From this carved graphical representation it is a little bit hard to determine the value of K_m and V_max. So the Michaelis-Menton Equation is further calculated to get an equation of a straight line. The calculation is shown as below-

The Michaelis-Menton Equation

Now, we invert the equation-

In this state, we factor them

Finally, we simplify the above equation

This simplified equation is called double-reciprocal or Lineweaver-Burk equation that gives a straight line in the graph.

Double-Reciprocal or Lineweaver-Burk Plot

If we plot the above double-reciprocal or Lineweaver-Burk equation in the graph then we will get a straight line such as the picture below-

The above plot has a very important role in biochemical reaction studies. The value of K_m and V_max are determined using the above double-reciprocal or Lineweaver-Burk equation as well as the graphical representation of it.

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