Enzyme Kinetics and Regulation of Enzyme Activity

Enzyme Kinetics:

Kinetics means the motion ie; enzyme kinetics means the study of the chemical reactions which is catalyzed by the enzyme by measuring the rate of reaction. The rate of reaction means how fast or slow the reaction occurs. The rate of reaction and the enzyme kinetics helps us to study the catalytic mechanism of this enzyme ,its role in metabolism ,drug inhibition and control of enzyme activity.First of all the substrate bind to the active site of enzyme forming the enzyme-substrate complex and then it is transformed into the products through a series of steps known as the enzymatic mechanism. This mechanism can be studied under two groups ie;

a) Single substrate mechanism: In this type of mechanism enzymes bind to only one substrate for eg; triosephosphate isomerase. Phosphofructokinase and hexokinase are the important enzymes that play the key role in cellular respiration (Glycolysis).

b) Multiple substrate mechanisms: In this of mechanisms enzymes bind multiple substrates such as dihydrofolate reductase.The protease cleaves one protein substrate into two polypeptide products.Some enzyme joins two substrates together such as DNA polymerase links a nucleotide to DNA.

Although there are series of reaction occurs in the mechanisms but the overall kinetics is determined by the rate determining steps .The slowest step is the rate-determining step in the reaction. Here,we study only about the single substrate mechanism and the rate of reaction is determined by the equation called Michaelis-Menten Equation. The Michaelis-Menten equation is only basis for most single-substrate enzyme kinetics.

Michaelis-Menten Equation:

In the enzyme catalyzed reaction,as the substrate concentration is increased the reaction velocity increases linearly and reaches its maximum level known as maximum velocity (V_max). Further increase in substrate concentration usually have no effect on enzyme activity on the rate of reaction which is called as saturation effect (Kinetics) and this effect is exhibited by all types of enzymes.

This observation was first of all observed and postulated by Victor Henery in 1903. This saturation effect was further studied by LenovarMichaelis and Maul Menten and confirmed the Henrys observation and purposed such binding of an enzyme.

The Michaelis -Menten equation is an algebraic expression of the hyperbolic shape of the curve in which the important terms are substrate concentration {S],initial velocity [V₀] ,and maximum velocity [V_max] with Michaelis-Menten constant K_m. The equation is fundamental to all the studies of the enzyme kinetics because it makes possible for the quantitative calculation of enzyme characteristics and analysis of enzyme inhibition.

When substrate reacts with the enzyme ,there is the formation of coenzyme- substrate complex which is then broken down into product and free enzyme is released.

K₁

E + S ↔ ES — Eqn(1)

K-₁

k₂

ES ↔ E + P —Eqn (2)

k-₂

Then

,E_t =Total enzyme concentration( sum of free and combined enzyme)

E_s = Concentration of enzyme substrate complex

[s] = Substrate concentration

E= ( E_t- E_s)= Concentration of free enzyme or uncombined enzyme.

The derivation begins by considering the rate of formation and breakdown of ES complex.

1) Rate of formation of ES

ES= K₁ [ (Et)- (Es)].[S] — Eqn (3)

Where K₁ is the rate constant for the first reaction. The rate of formation of ES from E+P by the reversible reaction is very small so it is neglected.

2) Rate of breakdown of ES

The rate of breakdown of ES is

ES = K_-1 [ES] + k₂[ES]

3) Steady state condition :

When it attain the steady state condition the rate of formation of ES is equal to its breakdown.

ie; k₁[E_t-E_s] . [S] = k_-1[ES] + k₂[ES]

or,k₁ [E_t] [S] - k₁[ES] [S] = k_-1[ES] + k₂[ES]

or,k₁ [E_t] [S] = [ES] [ k_{-1 +}k₂] + k₁ [ES] [S]

or, k₁ [E_t] [S] = [ES] [k_-1+ k₂+ k₁ [S} ]

or, [ES] = K₁ [E_t] [S] / k_-1+ k₂ + k₁[S]

Dividing numerator and denominator of RHS by k1 of above equation we get,

[ES] = K1 [E_t] [S] /k₁ /k_-1+ k2 + k1 [S] /k1

or, [ES] = [Et] [S] / {(k_-1+k₂₎/k₁} + [S} --------(5)

• 4)Defination of initial velocity v_oin terms of ES

According to Michaelis-Menten theory the initial velocity is determined by the rate of dissociation of ES in equation--(4) whose rate constant is K₂so we get ,

V₀=k₂[E_s]

Or,[ES]=v₀/k₂

Substituting the value of [ES] in equation----(5)

V_o/k₂=[E_t][S]/(k₁+k₂/k₁)+[S]

V₀=K₂[E_t][S]/(K_-1+K₂/k₁)+[S]-------(6)

Now,let us simplify the equation------(6) further by defining Michaelis-Menten constant (k_m) which is equal to k_-1+k₂/k₁and by defining V_maxand k₂[Et] and substituting the term in equation------(6)we get,

V₀=V_max[S]/[S]+K_m

This is called as Michaelis-Menten equation and it is the rate equation for one substrate enzyme catalytic reaction.

Algebraic transformation of Michaelis-Menten Equation:

The direct measurement of the numeric value of V_maxand K_mis impractical because the high concentration of substrate is required to achieve the saturation point so the linear form of Michaelis-Menten equation is developed to avoid this difficulty which is more useful in the practical determination of V_maxand K_m.

Vo=Vmax[S]/[S]+km[From Michaelis Menten equction ]

1/v₀=[s]+[km]/V_max[s]

Or1/V₀₌[s]+[Km]/Vmax[S]

Or,I/Vo=[S]/Vmax[S]+Km/Vmax[S]

Or,1/Vo=1/Vmax+Km/Vmax[S]

Or,1/Vo=Km/Vmax1/[S]+1/Vmax

Or,Y=mx+C

The above equation for straight line is y=mx+c where,

Y=1/Vo and x=1/[S]

A plot of 1/Vo and 1/[S] gives a straight line whose y-intercept is 1/Vmax and slop is Km/Vmax

This plot is called line-Weaver-Burk double reciprocal plot or linear Weaver Burk plot

Significance of Km value

• It reflects the affinity of the enzyme to the substrate ie; higher the Km value, lower the affinity to the substrate and vice versa .From the plot small Km value for enzyme (I) reflects high affinity of the enzyme for the substrate and large Km value of the enzyme reflects a lower affinity of the enzyme to the substrate

2)Knowledge of Km value provides the information about the nature of inhibition of different inhibitors ie;whether competitive noncompitative and uncompetitive.

• 3) It is numerically equal to substrate concentration at which reaction velocity is equal to half Vmax( 1/2Vmax)

Note :When substrate is much less than Km then Vo =[S]

when the substrate is much greater than Km then Vo=Vmax

When substrate concentration is equal to Km

Vo=Vmax[S]/Km+[S]

Vmax[S}/[S]+[S](Km=S)

Vmax[S]/2[S]

= Vmax/2