# Michaelis-Menten Kinetics

## What is Michaelis Menten Hypothesis?

Michaelis Menten hypothesis or Michaelis Menten kinetics is a model that is designed to generally explain the velocity and the gross mechanism of the reaction that is carried out by enzyme catalysts. Michaelis Menten hypothesis is one of the best known models in biochemistry to determine the catalyst kinetics of a reaction.

This Michaelis Menten kinetics was first stated in 1913, where it assumes the rapid formation of a complex that is reversible in nature formed between the enzyme and its substrate. A substrate is the substance on which the catalyst acts to form a desired product. It also assumes that the concentration of the product (p) is directly proportional to the rate of formation of the product.

### Michaelis Menten Equation

The kinetics or the velocity of such reaction is highest when the active sites of the enzyme molecules on which the catalyst activity taking place is filled with substrates. In other words, as the concentration of the substrates increases the kinetics of the reaction also becomes higher. This relationship has been established as the basis of all kinetic studies of enzymes. Thus, the Michaelis Menten hypothesis or the kinetics theory has been deduced to a mathematical formula relating the concentration of the substrate [S] to the rate of formation of product [P] or reaction rate [v]. The formula is stated below that is known as the Michaelis-Menten equation.

{V = d[P] / dt } = Vmax{ [S] / KM + [S] }

Where,  Vmax is the maximum rate of reaction achieved by the system occurring at the saturated  substrate concentration. KM  is equal to the concentration of the substrate when the value of rate of reaction is half of Vmax. A more clear understanding of the hypothesis can be drawn by plotting a graph showing the relationship between the reaction rate and the concentration of the substrate in an enzyme catalysed reaction.

### Mechanism of Enzyme-catalyzed Reactions

The mechanism of enzyme-catalyzed reactions are the series of reactions that takes place when an enzyme attracts the substrates to attach to its active site and then catalyses the reaction to form a desired product which then dissociates itself from the active site of the enzyme where the actual reaction took place. Thus the combination of substrate and the active enzyme is known as substrate complex.

If one substrate and one enzyme is involved in the reaction then it is known as binary complex, if two substrates and one enzyme is involved in the reaction then the complex is known as ternary complex. The bonding that is shared between the substrate and the complex is not chemical bonds and they are attached to each other by electrostatic pull or hydrophobic force. Hence the nature of bonding is physical and they are non-covalent in nature.

In many biochemical reactions, it has been observed that introduction of an enzyme as a catalyst for a reaction actually increases the reaction rate by a greater fraction which is about 106 times the normal rate of reaction in the absence of a catalyst. Also from the  mechanism of enzyme-catalyzed reactions it was studied that enzymes have the quality to separate two very similar substrates and greatly enhance the rate of reaction of one without having much effect  on the second substrate.

The mechanism of enzyme-catalyzed reactions can be explained by a simple model popularly known as lock and key model. The model can be clarified by the idea that the enzyme and the substrate that is involved in the chemical kinetics are three dimensional in nature. Both substrate and the enzyme compliments one another in a way that the structure of the enzyme can fit tightly with the substrate and the active catalytic site of the enzyme comes in the closest proximity to the substrate and its chemical bonds that are altered during the reaction. Just as the key is shaped to fit into the keyhole  of the lock, in the same way the active site of the catalyst is shaped to fit perfectly with the chemical structure of the substrates.

### Mathematical Expression of Michaelis Menten Kinetics Mechanism

In 1913, a mathematical model of Michaelis Menten kinetics mechanism was proposed by two scientists, a German biochemist Leonor Michaelis and a Canadian physicist Maud Menten. In this model an enzyme [E] binds physically with a substrate [S] to form a complex [ES] on the surface of the enzyme catalyst which under chemical reaction converts to a product [P] and the original form of enzyme [E] is retrieve

In this equation kf is forward rate of reaction, kr is the reverse rate of reaction and kcat is the catalytic rate of reaction. The complex formed from the substrate and the enzyme is reversible in nature but the product formed from the complex is irreversible in nature. Thus, in certain assumptions where the concentration of enzyme is much lesser than the concentration of the substrate, then the mathematical equation is such case would be-

{V = d[P] / dt } = Vmax{ [S] / KM + [S] } = kcat [E0] { [S] / KM + [S] }

Now, in low substrate concentration where [S]<< KM, the rate of reaction  v =  kcat [E0] { [S] / KM } will vary linearly with the concentration of the substrate [S]. This is called first-order kinetics, but when KM<< [S], the reaction becomes independent of the concentration of substrate and progresses asymptomatically to its maximum reaction rate Vmax = kcat / [E0], where [E0] is the concentration of the initial enzyme. This rate is achieved when the total concentration of substrate is bounded with the enzyme. Here,  kcat is the turnover number which denotes the maximum no. of substrate molecules that have been converted to product per enzyme molecule per second.

A graphical representation of the change of concentration of enzyme, substrate and the complex with time is given below for more clarity.

### Michaelis Menten Kinetics Mechanism Derivation:

The Complex Equilibrium Equation:

Substrate Transformation Equation:

ES → E + P

Where, K is the dissociation constant for the dissociation of the substrate complex back to the initial substrate and enzyme form, whereas, k is the rate constant of formation of product from the substrate complex in a non reversible reaction.

From the product forming step, the rate of reaction of the first-order decay of enzyme-substrate complex expected to be

d[P]/dt = k[ [ES]

Assuming the product concentration [P] = x and the initial concentration of substrate be S0 and enzyme be E0. thus, the material balance will be given by the equation,

E0 =n [E] + [ES] and S0 = [S] + [ES] + x

Since most of the enzyme-catalysed experiments are performed by taking the concentration of substrate much higher than the enzyme, therefore, S0 >> E0. therefore, in this case [ES] concentration of the enzyme-substrate complex is negligible for the equation and the concentration of the initial substrate becomes, S0 = [S]+ x

From the above conclusion drawn, the enzyme-substrate equilibrium imposes the relationship

K ={ [E] [S] / [ES] } ={ [E](S0 - x) } / [ES]

Using the above equation, eliminating [E] from the expression of E0 will give

E0 ={ k [ES] / (S0 -x) } + [ES] ={( K / S0 -x)} / [ES]

Thus the law of rate of product formation becomes

dx/dt = k [ES] ={ kE0 (S0 -x) } / { k + (S0 -x) }

Now, if k << S0 -x, then the dependence on the substrate cancels and the law will tend to order zero

dx/dt = k E0 = kobs

The above equation for the zero-order rate constant depends on E0.

If k >> S0 - x, then the rate of law becomes first-order kinetics and the equation becomes

dx/dt = { kE0 / K } (S0 -x) =  kobs (S0 -x)

With the first-order rate constant  kobs = kE0 / K.

As mentioned earlier, the kinetics of enzyme is studied by using the initial rate method, therefore, by this logic, the rate is approximated by dx/dt ≈ ߡx / ߡt, that measures the small amount of product formed in the small interval of time where, x << S0. Thus, the final rate as well as equilibrium constant can be calculated by incorporating various rates taken at different concentrations of S0 and E0

ߡx / ߡt ≈ k S0 E0 / K + S0

But since the concentration of enzyme-substrate complexes is always much smaller than the concentration of the substrate in real laboratory analysis, therefore the Michaelis Menten kinetics mechanism can be analyzed by applying steady-state approximation to [ES].

Different parameter values of various enzymes are provided in the table below

## Application of Michaelis Menten Kinetics

 Enzymes KM (M) Kcat (S-1) Kcat / KM (M-1 S-1) Chymotrypsin 1.5 * 10-2 0.14 9.3 Pepsin 3.0 * 10-4 0.50 1.7 * 103 T-RNA synthetase 9.0* 10-4 7.6 8.4 * 103 Carbonic anhydrase 2.6 * 10-2 4.0 * 105 1.5 * 107 Ribonuclease 7.9 * 10-3 7.9 * 102 1 * 105

Kcat / KM is the efficiency of the catalyst that demonstrates how efficiently an enzyme catalyst can convert substrate into the product. Thus diffusion enzyme catalysts, fumarase that works on the theoretical upper limit of 108-1010 M-1 S-1 actually diffuses the substrate into the active site of the enzyme catalyst. Other than biochemical reactions, it has also been applied in the field of alveolar clearance of dust, clearance of blood-alcohol, the richness of species pool, bacteriophage infection and photosynthesis-irradiance relationship.