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Enzyme Kinetics

Kinetic concepts Kinetics is the study of the

rates of chemical reactions. Unlike

Thermodynamics which tells us if a reaction can

occur, kinetics provides information on the rate

and mechanism of the reaction.

Chemical kinetics

- It is important to recognize that, although a

reaction may proceed with the stoichiometry, - A ? P
- The reaction may proceed via a series of

elementary reactions such as - A ? I1 ? I2 ? P
- A description of the elementary reactions and

intermediates is essential for defining the

mechanism of a reaction. - This is mostly basic chemical kinetics with a

biological twist to account for the problems of

enzymes.

Fundamental kinetic concepts

rate constant

k

- A P

Rate of disappearance of reactant

Rate of appearance of product

The instantaneous rate, or velocity (v), of the

reaction at any time is reflected in the

disappearance of reactant and/or in the

appearance of product. The rate is directly

proportional to the concentration of the

reactant, A. This proportionality is reflected

in the rate constant, k.

The above reaction is an example of a first-order

overall reaction

Fundamental kinetic concepts

k

- A A P

Rate Law

k

A B P

-dA -dB dP

kAB v

dt dt dt

Rate Law

The above reactions are both second-order overall

reactions. (The overall order of a reaction

comes from adding together the exponents in the

rate law)

Note Rate orders can NOT be deduced from the

balanced chemical equation. They can only be

determined experimentally.

Units for rate constants

- The units for rate constants differ depending on

whether they are first or second order - The first order differential rate equation has

units Ms-1, therefore k must have units s-1. - For second order rate equations (Ms-1) k(M2).

In order for the units to balance, the units for

k must be (M-1s-1). - The order of a reaction can be determined by

following the rate as a function of time, and

fitting to either a first or second order

equation.

v (Ms-1) k (s-1) A (M)

v (Ms-1) k (s-1M-1) A (M) B (M)

First order rate equation

- We want to rearrange the equation describing the

instantaneous reaction velocity into a more

useful form, where the change in A is expressed

as a function of time - Rearranging gives
- Which may be integrated from Ao the initial

concentration to A at time t - Which results in

1

dA

kdt

A

A

1

?

dA

k dt

A

A0

For a first-order reaction, there exists a linear

relationship between lnA and t

First order rate curves

- The half-time or half-life, t1/2 is constant for

a first order reaction. From the rate equation

The half-life for a first-order reaction is

independent of the initial concentration of the

reactant.

Second order rate equations for single reactant

systems

- In a second order reaction for the type 2A ? P

the variation of A is quite different from the

first order reaction. Rearranging and

integrating - Gives
- So that
- For a second-order reaction, there exists a

linear relationship between 1/A and t. Thus,

it is trivial to distinguish between first and

second order reactions by the nature of the rate

vs. concentration dependence (see the plot on the

following page)

Comparison of rate curves

- The t1/2 for a second order reaction is
- (try deriving this expression
- yourself)
- t1/2 for a second-order reaction is dependent on

the initial concentration of reactant and, thus,

differs from the first order reaction.

Enzyme kinetics The rapid-equilibrium

approach(The Henri-Michaelis-Menten Equation)

- A general scheme for a simple enzyme-catalyzed

reaction which converts a single substrate into a

single product is - E S ES EP

E P - This kinetic scheme is simplified significantly

when the reaction proceeds at initial velocity.

i.e. at the onset of the reaction, S 100

while P 0. While P remains very low, the

back reaction is negligible and the above scheme

may be simplified to - E S ES E

P - The assay of an enzyme under initial velocity

conditions is, therefore, an important

consideration in the practical design of enzyme

assays. We will revisit this later.

k1

k3

k2

k-1

k-2

k-3

k1

kcat

k-1

Derivation of the Henri-Michaelis-Menten equation

- The rate of the reaction is measured by the

appearance of product (or the disappearance of

substrate). The overall rate of the reaction is

governed by the rate of conversion of the final

intermediate (in this case, ES) into free enzyme

(E) and free product (P). Thus, the rate of the

above reaction is given by - eq. 1
- The preceding equation is not particularly

useful. Since ES is an intermediate in the

reaction, its concentration at any given time is

unknown and it is not practical to directly

follow its conversion into E P. It would be

much more useful to express the rate equation in

terms of the total enzyme concentration (Et)

and the initial substrate concentration (S),

both of which are known. - The derivation of such an expression requires

certain assumptions. The simplest derivation uses

the Rapid-Equilibrium assumption.

Derivation of the Henri-Michaelis-Menten equation

- Assume Rapid Equilibrium (k-1 gtgt kcat)
- E S ES KS

eq. 2 - The total enzyme concentration (Et) is the sum

of all enzyme species - Et E ES eq. 3
- Divide eq. 1 by eq. 3
- vi kcatES
- Et E ES and, from rearranging

eq. 2,

k1

ES

k-1

ES

k1

k-1

Note KS is a dissociation constant

eq. 4

ES

ES eq. 5

KS

Derivation of the Henri-Michaelis-Menten equation

- Now, substitute eq. 5 into eq. 4
- vi kcat ES KS
- Et E ES KS
- vi kcat S KS
- Et 1 S KS
- (kcat Et)S
- S KS

KS

?

multiply the numerator and denominator by KS

KS

Enzymologists define

vi

(kcat Et) Vmax

where KS is a true dissociation constant

The Briggs-Haldane steady state approach

- The Henri-Michaelis-Menten equation was

originally developed by Victor Henri (1903) and

later confirmed by Leonor Michaelis and Maud

Menten (1913). - The assumption of rapid equilibrium in the

derivation of the Henri-Michaelis-Menten equation

requires that the rate of dissociation of the ES

complex (k-1) far exceed the rate of conversion

of the ES complex into E P (kcat).

Unfortunately, this assumption is invalid for

many (if not most) enzymes. - In 1925, Briggs Haldane developed an initial

velocity rate equation that did not require the

assumption of rapid equilibrium. Rather, the

Briggs-Haldane approach was to assume a

steady-state for the ES complex.

The Briggs-Haldane steady state approach

- Steady state assumption
- When S gtgt E, the level of ES stays constant

after an initial burst phase. - i.e. dES/dt 0

Note The accompanying figure is somewhat

deceptive. In fact, steady state is reached very

quickly. The initial phase of an enzyme-catalyzed

reaction, prior to the onset of steady state, can

only be followed using specialized equipment in

combination with rapid sample mixing techniques.

The kinetics are much more complex but they can

yield important information about the individual

kinetic steps in an enzyme-catalyzed reaction.

This type of kinetics is referred to as

pre-steady state

Derivation of the Briggs-Haldane equation

- As before, this derivation deals only with

initial velocity kinetics. We can treat the

reverse reaction as neglible and simplify the

scheme to - E S ES E P
- The overall rate of production of ES is the sum

of the elementary reaction rates leading to its

appearance minus the sum of those leading to its

disappearance.

k1

k2

k-1

(Note that k2 is analogous to kcat)

dES

k1ES k-1ES k2ES 0

dt

k1ES

Rearranging

ES eq. 6

k-1 k2

Derivation of the Briggs-Haldane equation

- As before,

and

vi k2ES Et E ES

Substituting in eq. 6, we get

vi k2k1ES (k-1 k2) Et E

ESk1 (k-1 k2)

k2EtS

vi

eq. 7

k-1 k2

S

k1

Derivation of the Briggs-Haldane equation

The Michaelis constant, Km, has units of M and is

defined as

and

(k2 Et) Vmax

Substituting these into eq. 7 gives the final

form of the Briggs-Haldane equation

Some special cases

When S gtgt Km, vi Vmax (i.e. velocity is

independent of S The enzyme is said to be at

saturation) When S ltlt Km, vi (Vmax/Km)S

(i.e. the velocity is linear with respect to S)

The meaning of Km

- Substitution of ?Vmax/2 into the Briggs-Haldane

equation shows that

The value of Km does NOT necessarily give a

measure of the affinity of the enzyme for the

substrate. When k2 is small relative to k-1, Km

approximates to the dissociation constant of the

ES complex, KS. Since KS is a true dissociation

constant, only KS gives a true measure of the

enzyme-substrate binding affinity.

?Km KS

The meaning of Vmax , kcat , and Specific Activity

Vmax (Ms-1) is the maximal velocity of an

enzyme-catalyzed reaction. The maximal velocity

is reached when the substrate is saturating. Vmax

is dependent on Et.

kcat (s-1) is the catalytic constant, also called

the turnover number. It is a pseudo-first order

rate constant and is independent of the total

enzyme concentration.

(kcat Et) Vmax

Specific Activity (U/mg total protein) is often

used to characterize enzyme activity when the

enzyme solution is impure. It is, most always, a

quick but less accurate means of kinetically

characterizing an enzyme. 1 International Unit

(1 U) is the amount of enzyme which catalyzes the

formation of 1 mmole of product per minute under

defined conditions. i.e. 1 U 1 mmole/min

The kinetic significance of kcat

kcat, a pseudo-first order rate constant,

includes the individual rate constants for all

steps leading from the ES complex to product

release. For example, in a more complex kinetic

scheme such as

kcat

ES

E P

It can be shown that kcat is comprised of all the

individual rate constants between ES and E P

(k2, k-2 and k3)

k2 k3

kcat

(k2 k-2 k3)

The kinetic significance of kcat/Km

When the substrate concentration is very low (S

ltlt Km)

vi (Vmax/Km)S

Recall that Vmax kcatEt

So, vi (kcat/Km)SEt

Also, when S is very low, very little of the

total enzyme species will be tied up in the ES

complex (or any other intermediate

complex). Et E ES But at low substrate

concentrations, Et E (because ES

0) thus, vi (kcat/Km)SE when S ltlt Km

kcat/Km (catalytic efficiency) is a second-order

rate constant that describes the conversion of

free E and free S into E P. The rate at low

S is directly proportional to the rate of

enzyme-substrate encounter.

Catalytic Perfection

k1 when k2 gtgt k-1

- How quickly can an enzyme convert substrate into

product following enzyme-substrate encounter?

This depends on the rates of the individual steps

in the reaction. The rate is maximal when k2gtgtk-1

which means that the reaction proceeds whenever a

collision occurs. - Many enzymes in metabolic pathways have evolved

to function at substrate concentrations less than

Km to optimally and efficiently turnover

metabolic intermediates. - Some enzymes are so incredibly efficient that

they instantaneously convert S into P following

enzyme-substrate encounter. The reaction rate for

these enzymes is limited only by the rate of

diffusion ( 108 109 M-1s-1). These enzymes are

said to have reached evolutionary perfection.

Examples of rate constants

- There is a wide variation in kinetic parameters

reflecting the interplay between KM and kcat.

Because of the central role of the

EnzymeSubstrate complex, there is also large

variability depending on the nature of the

substrate.

Analysis of Kinetic Data

- The rate equations are non-linear, so it is

convenient to reformulate the equation to give a

linear relationship. The most common

linearization is the Lineweaver-Burk or double

reciprocal plot, obtained from the reciprocal of

the Michaelis-Menten equation. - DISADVANTAGE the data usually involve large

ratios of KM so the data is crowded. At low S

values the errors are often large. - There are several other linear forms of the

Michaelis-Menten equation. However, most data is

now easily analyzed directly on computer by a

direct non-linear regression fit to the

Michaelis-Menten equation.

Michaelis-Menten Kinetics

Km is the substrate concentration at which the

rate of the reaction is half the maximum rate

(Vmax)