## KINETIC ANALYSIS MICHAELIS MENTEN EQUATION

### KINETIC ANALYSIS MICHAELIS MENTEN EQUATION

8.1.         Michaelis Menten Equation

The relation between substrate concentration and enzyme catalyzed reactions was given by Michaelis and Menten.

where, V0           =  Initial reaction velocity

km        =  Michaelis menten constant

Vmax    =  Maximum reaction velocity

S              =  Substrate concentration

It is an equation which describes how reaction velocity varies with substrate concentration. The rate or velocity of a reaction catalyzed by enzyme is measured with the varied substrate concentration. With increase in substrate concentration reaction rate increases linearly and attains the maximum velocity (Vmax), but do not increase further by increasing the substrate concentration.

Km is the Michaelis Menten constant i.e the substrate concentration at which the rate of reaction is half of the maximum rate of reaction. Lower the value of Km higher the enzyme affinity for the substrate. Km, the Michaelis Menten constant is a dynamic constant expressing the relationship between the actual steady-state concentrations rather than the equilibrium concentrations. Km depends on the particular substrate used, pH, temperature and ionic strength. Observed values of Km for different substrates and different enzymes vary widely.

Inspection of Michaelis-Menten equation shows that Km is equivalent to the substrate concentration that yields half maximal reaction rate. If v = Vmax /2 then,

Hence, Km = (S). Km is that substrate concentration at which the rate of reaction is half of maximum rate of reaction.

8.2.         Relation between Km and KES dissociation constant

Very often, Km is assumed to be equal to the dissociation constant for the enzyme substrate complex. This is true only if the rate constant for the formation of products, Kf2 is significantly smaller than Kb1. Consider a situation where Kb1 is much greater than Kf2.

The dissociation of the enzyme substrate complex to enzyme and substrate is much faster than the formation of enzyme and the product. Under these conditions,

The dissociation constant of the enzyme substrate complex (Kd) is

This means, Km is equal to the dissociation constant of the ES complex. Under these conditions (Kb1 >> Kf2), Km is a measurement of the strength of the ES complex. A high Km indicates weak binding between enzyme and substrate and a low Km means strong binding. Most often, Kf2 > > Kb1 in which case Km is not directly equivalent to a dissociation constant for ES. In any case Km is the concentration of substrate at which half the active sites are filled.

The numerical value of Km is of interest for several reasons,

1. The Km establishes an approximate value for the intracellular level of the substrate.
2. Km is a constant for a given enzyme, its numerical value provides a means of comparing enzymes from different organisms or from different tissues of the same organism or from the same tissue at different stages of development.
3. A ligand-induced change in the effective value of Km is one way of regulating the activity of an enzyme. By measuring the effects of different compounds on Km, it is possible to identify physiologically important inhibitors and activators.

The maximum velocity Vmax is not by itself a very useful comparative parameter because of its dependence on enzyme concentration.

A more useful parameter is turnover number or Kcat' which is equivalent to the rate constant kf2' for the breakdown of the ES complex to product when the enzyme is fully saturated with substrate. Since V max = kf2[Et], the term Vmax reveals the turnover number of an enzyme if the concentration of enzyme [Et] is known.

Kcat is a first order rate constant and therefore will have units of reciprocal time. The turnover number is a measure of the maximum potential catalytic activity of an enzyme. The reciprocal of the turnover (l/Kcat) is the time taken for a single round of catalysis to occur when the enzyme is saturated with substrate. Turnover numbers vary widely, the highest value observed is for carbonic anhydrase.

Another important parameter is the ratio of the turnover number to the Michaelis constant. The value Kcat/Km is a second order rate constant for the reaction of enzyme and substrate to form products.

Kcat is the maximum number of substrate molecules converted to product per active site per unit line. Kcat/Km is called as specific constant. At low substrate constredtion (S) the Michaelis-Menton equation can be with as

Kcat/Km  is an apparent second-order rate constant. It measures how the enzyme performs when substrate concertration is  low. The upper limit for Kcat/Km is the diffusion limit - the rate at which enzyme and substrate  diffuse together (~109/M for small substrates (glycerol) and 108/M  for larger substrates (nucleotides).

Kcat/Km  contains information about catalytic efficiency of an enzyme. How fast an enzyme performs its function and how much of the substrate is required to reach Vmax.

8.3.         Lineweaver-Burk equation (Double reciprocal plot) :

In 1934, Lineweaver and Burk made a simple mathematical alteration in the Michalies –Menten's equation by plotting a double inverse of substrate concentration and reaction rate.

Line weaver Burk plot is obtained by taking the reciprocal of both sides of the michalies–Menteris equation (MME).

In the line weaver Burk (LB) plot the Michaelis Menten equation is converted into straight line curve.

It is used to estimate the Vmax  from the position of intercept on the X-axis.

• Straight line is given by Y =  MX   +  C, where C is Y intercept of the regression of Y on X and M is slope.
• If M value increases then slope increases.
• If we make the value of Vmax equals to constant then Km value is high and if Km value is constant, Vmax value decreases.

In this the hyperbolic curve becomes a straight line and absolute value of the X-intercept of the line is the affinity (1/Km) of the enzyme for the substrate. The Y-intercept is 1/Vmax. The slope of the line is Km/Vmax. Again, originally this was done subjectively. It is the much finer method for finding the best-fit parameters for the untransformed Michaelis-Menten relationship.

Application

The Lineweaver–Burk plot use to  determine the Km and Vmax,. The Y-intercept of such a graph is equivalent to the inverse of Vmax; the X-intercept of the graph represents −1/Km. It gives a quick, visual impression of the different forms of enzyme inhibition.

Eadie–Hofstee diagram is a graphical representation of enzyme kinetics in which reaction rate is plotted as a function of the ratio between rate and substrate concentration:

In this Michaelis Menten equation is represented as

Invert and multiply with Vmax :

Rapid identification of Km  and  Vmax

8.5.         Hanes Woolf Plot

Graphical representation of the ratio of the initial substrate concentration [S] to the reaction velocity V is plotted against substrate concentration [S].

Michaelis-Menten equation is derived as

Invert and Multiply by [S] :

Rearrange :

This equation will give straight line of slope  a Y-intercept of  and an X-intercept of –km

Application :-

Used for determination of Km, Vmax and Vmax/Km parameters

8.6.         Enzyme Inhibition

Enzymes are proteins acting as a catalyst for the reactions. But there activity is inhibited or blocked by the molecules which are chemical substances (organic / inorganic) in nature. These molecules or compounds are called inhibitors and the process by which they inactivate the enzyme or block its activity is called enzyme inhibition.

They inhibit the enzyme catalytic activity reversibly or irreversibly by modifying the amino acid side chains required for enzymatic activity.

In drug discovering, drug analogs are design for detoxification of many antitoxins as they have inhibitory action.

8.6.1.     Rules followed by enzyme inhibition reactions

1. Enzyme binds with substrate in 1 : 1 ratio at active site in a lock-key arrangement or induced fit.
2. Inhibitor compounds compete with substrate for allosteric catalytic site on first come first basis to make enzyme inhibitor substrate complex or enzyme inhibitor complexes.
3. Enzyme and substrate or inhibitors react with each other in a kinetic manner which is expressed as kinetic constants of a catalytic reaction.
4. Physiological conditions like pH, temperature, concentration of substrate or reactants determines the rate of enzymatic reactions.
5. Intermolecular forms between enzyme subunits, substrate or inhibitor active group interactions, physical properties of binding nature : electrophilic, hydrophilic, nucleophilic and metalloprotien nature; hydrogen bonding affect the overall enzyme reaction rates and mode of inhibition.

8.6.2.     Types of Inreversible enzyme inhibition

8.6.2.1. Competitive inhibition (Reversible) : In this, competition present between inhibitor and the substrate for the active site.

Catalytic site of enzyme is occupied by inhibitor and its activity is inhibited. But the inhibition is reversible. In this case, both enzyme substrate and enzyme inhibitor complex are formed.

Effect on affinity :- When inhibitor concentration increases its affinity increases but the affinity of substrate decreases and Km value increases. But when substrate concentration increases then its affinity increase and affinity of inhibitor decreases and Km value decreases, i.e.,

The inhibitor binds to active sites of an enzymes that means the inhibitor competes with substrate for binding at active site, hence affinity of substrate towards enzyme decreases hence Km value increases. The new km is given by $\fn_jvn \alpha$Km, where

where

I = concentration of inhibitor

Kdi = dissociation constant of inhibitor.

When inhibitor concentration increases, the km value increases as affinity of substrate decreases. If dissociation constant of [EI] is more, enzyme inhibitor complex more, a is less thus km is less. The value of a is equal to 1 or greater then 1. New value of Km is aKm

Effect on Vmax

Vmax is calculated at infinite substrate concentration.

Vmax    = Kcat [ES]

At infinite substrate concentration all enzymes are in the form of enzyme substrate complex i.e. Vmax is not affected.

Lineweaver Burk plot of competitive inhibition

8.6.2.2. Examples of Competitive Inhibitors

(a)          Allopurinol :

Drug used for treatment of gout. Uric acid is formed in the body by oxidation of hypoxanthine by the enzyme xanthine oxidase. Allopurinol is structurally similar to hypoxanthine and inhibits the enzyme xanthine oxidase and reduced uric acid formation.

(b)          Methotrexate :

It is a competitive inhibitor of dihydrofolate reductase (DHFR). This Drug is used for cancer therapy. It is structurally similar to folic acid Thus, inhibits folate reductase competitively. It prevents formation of tetra hydrofolate. Hence, DNA synthesis is suffered.

(c)           MAO inhibitors (Mono Amine Oxidase) :

They are, first class of antidepressants to be developed. MAO inhibitors increase the level of serotonin dopamine by inhibiting the MAO. eg. Catecholamines (epinephrin and norepinephrines).

(d)          Dicumarol :

This drug is similar to vitamin K. Drug warfarin act as an anticoagulant by competitively inhibiting vitamin K.

8.6.3.     Uncompetitive Inhibition

There is no competition between inhibitor and the substrate as sites of attachments of the substrate and inhibitor are different. Inhibitor has no structural similarity to substrate therefore cant bind to free enzyme. Inhibitors binds with enzyme substrate complex that expose inhibitor binding site. Binding of inhibitor can cause distortion of the active site or allosteric site that inactivates the catalysis.

Effect on affinity :

High affinity of inhibitor means low dissociation of enzyme substrate complex to enzymes substrate. In this, inhibitor binds to other then active site on enzyme substrate complex. That means inhibitor show affinity for ES complex rather then enzyme. Thus in the presence of inhibitor the affinity of enzyme toward substrate increases. This decrease the Km. Therefore

Effect on Vmax

Vmax is calculated at infinite substrate concentration. At infinite substrate concentration all enzymes are in the form of enzymes substrate complex. The inhibitor show affinity for enzymes substrate complex. Thus inhibitor binds to enzymes substrate complex and prevent the catalysis of enzyme substrate complex into enzyme and product. That's why Vmax decreases and new Vmax is given by

Inhibitor concentration increases a value increases and Vmax decreases

On putting the values of new Km and  new Vmax in lineweaver burk plot, The equation is as follows :-

Uncompetitive inhibitor causes different intercepts on both Y and X-axis but same slope.

8.6.4.     Mixed (Non-Competitive) Inhibition

This inhibitor is not similar to substrate structurally but can binds to free enzyme and the enzyme substate complex both.

When inhibitor binds to the enzyme away from the active sites. It induces the conformational changes and reduces its catalytic activity. Thus, enzyme inhibitor [EI] and enzyme substrate inhibitor [ESI] complexes become non productive. The substrate concentration does not reverse the reaction. Hence, inhibition leads to unaltered Km but reduced Vmax.

Lineweaver Burk plot is used to determine Km and Vmax in enzyme kinetics. The Y-intercept of such a graph is equivalent to the inverse of Vmax, X intercept of the graph represents competitive inhibitors hence the same Y-intercepts (as Vmax is unaffected by competitive inhibitors) but there are different slops.

Non competitive inhibitor produces plot with same X-intercept as Km is unaffected but different slopes with Y-intercepts.

8.7.         Kinetics of Multisubstrate Reaction

In the enzyme kinetics, simple reactions involve one substrate binding to an enzyme and undergoing catalytic reactions. This condition is not common. A majority of biochemical reactions catalyzed by two or more substrates taking part in the reactions. For example, an enzyme E, catalyzed the reaction involving two substrates A and B and yield the product P and Q.

This type of reaction is called as Bi-Substrate reaction . These reaction can proceed in two ways:

8.7.1.     Sequential

Both the substrates A and B, bind to the enzyme E, and then reactions proceeds to yield products P and Q

This type of reaction is called as sequential or simple-displacement reactions which are further divided into following groups.

Ordered substrate binding or ordered sequential mechanism – In this type, one substrate must bind before a second substrate.

This reaction indicates the sequential binding of substrates as well as sequential release of product. This type of mechanism is observed in the reactions catalyzed by lactate dehydrogenes involving NAD+ and lactate.

8.7.2.     Random substrate binding- In this type either A or B may bind to the enzyme first, followed by the other substrate and the release of the product.

This type of mechanism is observed in reactions catalyzed by transferases enzyme as hexokinase catalyzed phosphorylation of glucose by ATP.

8.7.3.     Theorell-Chance Sequential mechanism

It is a type of ordered sequential bisubstrate reaction in which the ternary complex does not accumulate.

8.7.4.     Ping pong mechanism

The other possibility in bi-substrate reaction is that one substrate, A, binds to the enzyme and on reacting with it a product, P, is released and enzyme turns into a modified form, E′. The second substrate, B, comes in and binds with modified enzyme to yield second product, Q and regenerate the enzyme, E.

The reactions following the above mechanism are called Ping-Pong or double-displacement reactions. This type of mechanism is observed in reactions catalyzed by aminotransferases.

These enzymes catalyze the transfer of an amino group from an amino acid to an α-keto acid.The products formed are a new amino acid corresponding to keto acid and a new keto acid corresponding to carbon skeleton of amino acid such as:-

Another example of ping pong reaction is phosphoglycerate mutase. The enzyme get phosphate from one substrate and after phosphorylation of enzyme, the phosphate is transferred to second substrate.