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Enzyme Kinetics


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 ... – PowerPoint PPT presentation

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Title: Enzyme Kinetics

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

Fundamental kinetic concepts
rate constant
  • 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
  • A A P

Rate Law
-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

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

k dt
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
Second order rate equations for single reactant
  • In a second order reaction for the type 2A ? P
    the variation of A is quite different from the
    first order reaction. Rearranging and
  • 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
  • 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.

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,

Note KS is a dissociation constant

eq. 4
ES eq. 5
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


multiply the numerator and denominator by KS
Enzymologists define
(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

(Note that k2 is analogous to kcat)
k1ES k-1ES k2ES 0
ES eq. 6
k-1 k2
Derivation of the Briggs-Haldane equation
  • As before,

vi k2ES Et E ES

Substituting in eq. 6, we get
vi k2k1ES (k-1 k2) Et E
ESk1 (k-1 k2)

eq. 7
k-1 k2
Derivation of the Briggs-Haldane equation
The Michaelis constant, Km, has units of M and is
defined as
(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
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
(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

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
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