**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} = k_{2} [ES]

**ES** formation rate = k_{1} [E][S]

= k_{1} ([E_{total}] – [ES]) [S]

**E****S** breakdown rate = k_{-1 }[ES] + k_{2} [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] + k_{2} [ES]

⇒ k1 [E_{total}][S] – k_{1}[ES][S] = ( k_{-1} + k_{2} )[ES]

⇒ k1 [E_{total}][S] = (k_{1}[S] + k_{-1} + k_{2} )[ES]We know, V_{o} = k_{2 }[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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Allosteric Enzyme Regulation and Covalent Enzyme modification

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