Enzymology Guide to Thermodynamics, Kinetics, and Regulation

Contenu
 Rappels thermodynamiques et bioénergétiques
 Cinétique chimique
 Enzymes: Généralités et principes
 Cofacteurs
 Nomenclature, classification enzymatique
 Influence de la température et du pH
 Cinétique enzymatique méchaelienne
 Inhibition enzymatique
 Cinétique à deux substrats
 Enzymes allostériques
 Unités de mesure de l’activité enzymatique
 Régulation de l'activité enzymatique

1. Inroduction
 Enzymes are biological catalysts that play a central role
in the living world. The essential reactions for the
functioning of a living beings are too slow and without the
presence of these catalysts, life as we know it today would
not be possible.
 Each chemical reaction taking place in a living being is
catalyzed by one or more specific enzymes. Enzymes are
macromolecules, mostly proteins or RNAs (ribozymes),
which specifically recognize certain molecules and
accelerate the transformation reactions of these molecules
sufficiently for their speed to become compatible with the
functioning of the organism.
 A second essential role played by the enzymes is to ensure
the physical coupling between endergonic reactions and

2. Bioenergetics and Thermodynamics
Overview
 What is matter?
Matter: any substance that takes up space
Matter is composed of elements, elements are
composed of atoms & atoms are composed of
elementary particles: protons, neutrons & electrons.
 What is energy?
Energy is the ability to do work. We can distinguish
five Types of Energy: Mechanical, Thermical ,
Electrical, Chemical and Light or Radiant energy
Energy can be stored (Potential energy ) or released
(Kinetic energy)
Energy is measured in biology as heat content with

Overview
Matter is Potential Energy:
Stored energy
• Chemical energy
 Food
 Coal
 Gasoline
 Wood
Kinetic Energy: is energy
in motion
• Radiant energy
(photons)
• Electricity (electrons
and ions)
• Mechanical (motion)
• Thermal (heat)
How are Energy and Matter Related?

Overview
Organisms
can be
classified
according to
their source
of energy
(sunlight or
oxidizable
chemical
compounds)
and their
source of

Overview
Biological Energy transformations obey the Laws of
Thermodynamics
 First law of thermodynamics: also known as laws of
conservation of energy states that energy can
neither be created or destroyed, one form of energy
can be converted to another form but the total
amount of energy remains constant. Energy and
Matter can be interconverted.
 Second law of thermodynamics: All processes
proceed toward equilibrium. In all natural
processes, the entropy of the universe increases. Any
system tends spontaneously to become disorganized.

Overview
 Living cells and organisms are open system,
exchanging both material and energy with their
surroundings;
 living systems are never at equilibrium with their
surrounding, and the constant transactions between
system and surrounding explain how organisms can
create order within themselves while operating
within the second law of the thermodynamics.
 Life is not 100% efficient. Even though energy can
be lost from a system to the surroundings, it is never
completely gone because of the 1st law of
thermodynamics. In all energy transformations,

Overview
Gibbs free energy: expresses the total amount of
energy capable of doing work, during a reaction
with constant temperature and pressure. Both
entropy (ΔS) and enthalpy (ΔH) contribute to ΔG
ΔG = Δ H - T Δ S
 If ΔG < 0, the reaction is exergonic.
The products have less free energy that the
reactants.
A “downhill” reaction
 If ΔG > 0, the reaction is endergonic
The products have more energy that the reactants
An “uphill” reaction that requires energy to go
forward

Overview
Enthalpy: is the heat content of the reacting system.
H= E + PV
 If ΔH is negative reaction is exothermic.
 If ΔH is positive reaction is endothermic,
 if ΔH is zero the reaction is isothermic
Entropy: Is a qualitative expression for the randomness and
disorder in a system
 When the product of the reaction is less complex & more
disordered than the reactant it is said to proceed with a gain
in entropy.
 The units of entropy are joules/mole.kelvin

Overview
Standard free energy change
 ΔG° : When the reactant and the products are
present in a concentration of 1mol /l, at 25°C, it is
known as standard free energy change
 ΔG°’ : For a biochemical reaction, a standard state
is defined as having a pH of 7, T: 25°C, pressure at
1 atm and reactants and products at 1 M,. It is
called standard transformed constant.

Overview
Calculating the free energy change for reactions not at
standard state ?
Consider a reaction:
Then
dDcCbBaA 
   
   ba
dc
BA
DC
Q 

Overview
 Remember, at equilibrium ΔG = 0, Q = Keq
so
 Since in biological systems the predominant pH is 7,
it makes more sense to adopt a modified standard
state – i.e., 1 M for all constituents except protons,
for which the standard state is pH 7
eqKRTGG ln0 0

eqKRTG ln0

'ln'0
eqKRTG 
Keq’ ΔG’ The reaction
>1.0 negative Proceeds forward
1.0 zero Is at equilibrium
<1.0 positive Proceeds in reverse

3. Chemical kinetics
 Kinetics is the study of how fast chemical reactions
occur.
 There are 3 important factors which affect rates of
reactions:
 reactant concentration
 Temperature
 action of catalysts
Reaction Rates
 Speed of a reaction is measured by the change in

 Concentration of reactant (A) and product (B) as
function of time
Conc.(M)

 The rate for this reaction depends on the time. The
velocity, v, or rate, of the reaction A → B is the amount
of B formed or the amount of A consumed per unit time,
t.
 At beginning of reaction, rate is fast, and the rate slows
down as the reaction proceeds.
 At any time, the rate is given by either
 change in [A] with change in time,
 change in [B] with change in time.
 Instantaneous rate is mostly used for calculating rection
rates
Average rate = v
=
change in conc. of
B
change in time
=
Δ[B]
Δ t
- Δ[A]
Δ t
=

 The instantaneous rate is more precise than the
average rate
 It is the rate at some specific instance in time; the
concept is derived from the calculus.
 It is the slope of a curve at a specific instance in
time. This is obtained by constructing a tangent to
the curve at the indicated point.
Slope = Δy/Δx

0.6 M
38 min
Rate = Δ[B] /Δt = 0.6 M/38 min = 0.016 M/min

 Calculating Reaction Rates must take into
consideration the Stoichiometry of this reaction. A
product may appear faster than a reactant
disappear
For examlpe:
N2 + 3 H2 → 2 NH3
 In general rates increase as concentrations increase

Temperature and Rate
 As temperature increases, the rate increases.
 Since the rate law has no temperature term in it, the rate constant
must depend on temperature.
Activation Energy
 Arrhenius: molecules must possess a minimum amount of energy
to react. Why?
 In order to form products, bonds must be broken in the reactants.
 Bond breakage requires energy
 Activation energy, Ea, is the minimum energy required to initiate a
chemical reaction.

 The change in energy for the reaction is the difference in
energy between reactants and products.
 The activation energy, Ea , is the difference in energy
between reactants and transition state.
 The rate depends on Ea.
 Notice that if a forward reaction is exothermic, then the
reverse reaction is endothermic.
The Arrhenius Equation
 Arrhenius discovered most reaction-rate data obeyed the
Arrhenius equation:
 Both A and Ea are specific to a given reaction.
 k is the rate constant. R is the gas constant (8.314 J/K-mol)
 T is the temperature in K. A is the frequency factor.
a/RTE
A
e
k -
=

 Endergonic
 Exerginic

4. Reaction order
 Consider the general elementary reaction:
aA + bB + … zZ → P
 The rate of this process is proportional to the frequency with
which the reacting molecules simultaneously come together, that
is, to the products of the concentrations of the reactants. This is
expressed by the following rate equation
where k is a proportionality constant known as a rate constant.
The order of a reaction is defined as (a + b + … z), the sum of the
exponents. For an elementary reaction, the order corresponds to
the molecularity of the reaction, the number of molecules that
must simultaneously collide in the elementary reaction. Thus the
elementary reaction A → P is an example of a first-order or
unimolecular reaction. Unimolecular and bimolecular reactions

4. Reaction order
Zero-Order Reactions
A → products
Rrxn = k [A]0
Rrxn = k’
[k] = mol L-1 s-1

4. Reaction order
Then
-

dt= kd[A] 
[A]0
[
A
]t
0
t
-[A]t + [A]0 =
kt
[A]t = [A]0 -
kt
Δt
-Δ[A]
dt
= k
-d[A]Move to
the
infinitesim
al
= k
And integrate from 0 to time t

4. Reaction order
First-Order Irreversible Reaction
 The simplest possible reaction is the irreversible
conversion of substance A to product P (e.g.,
radioactive decay)
 The arrow is drawn from A to P to signify that the
equilibrium lies far to the right, and the reverse
reaction is infinitesimally small.
= - k dt
[A]
d[A ]

[A]0
[A]t

0
t

4. Reaction order
We get
= -ktln
[A]t
[A]0
ln[A]t = -kt + ln[A]0

4. Reaction order
 t½ is the time taken for one-half of a reactant to be
consumed. Since the rate is intrinsic to the reaction
in a first order reaction, the slope of the line in the
plot never changes and the half-life is the same
regardless of the starting concentration.
= - ktln
[A]t
[A]0
= - kt½
ln
½[A]0
[A]0
- ln 2 = - kt½
t½ =
ln 2
k
0.693
k
=

4. Reaction order
First-Order Reversible Reaction
Few reactions in biochemistry are as
simple as the first-order reaction described
above. In most cases, reactions are
reversible and equilibrium does not lie far
to one side. where k1 and k–1 are the rate
constants for the first-order, forward and
reverse, reactions respectively.
Therefore, the corresponding rate equation
is
For this reaction, where the forward and
reverse reactions are both first order, the
equilibrium constant (Keq) is equal to the
ratio of the rate constants for the forward
and reverse reactions. For a reaction to

4. Reaction order
Second Order Reaction
 A bimolecularor second-order reaction, which involves
two reactants, can be written
A + B → P + Q
 The velocity of this reaction can be determined from the
rate of disappearance of either A or B, or the rate of
appearance of P or Q:
 Since A and B must collide in order to react, the rate of
their reaction will be proportional to the concentrations
of both A and B. Because it is proportional to the
product of two concentration terms, the reaction is
k1

4. Reaction order
 Integration of rate equation where ‘t’ is dependent on
two variables, A and B. To solve this equation, either A
or B must be assumed to be constant.
 Experimentally, this can be accomplished by using a
concentration of B that is far in excess of requirements
such that only a tiny fraction of B is consumed during
the reaction and therefore the concentration can be
assumed not to change. The reaction is then considered
pseudo-first order.
 Alternatively, when the concentration of both A and B at
time zero are the same, i.e., [A]0=[B]0 , Equation can be
simplified

5. Enzymes Chronology
 Biological catalysis was first recognized and
described in the late 1700s, in studies on the
digestion of meat by secretions of the stomach, and
research continued in the 1800s with examinations
of the conversion of starch to sugar by saliva and
various plant extracts
 In 185O, Louis Pasteur concluded that fermentation
of sugar into alcohol by yeast is catalyzed by
“ferments.” vitalism, prevailed for decades.
 in 1897, Eduard Buchner discovered that yeast
extracts could ferment sugar to alcohol, proving
that fermentation was promoted by molecules that

6. Enzymes cofactors
Enzymes Components
 Enzymes, like other proteins, have molecular weights
ranging from about 12,000 to more than 1 million. Some
enzymes require no chemical groups for activity other than
their amino acid residues. Others require an additional
chemical component called a cofactor, either one or more
inorganic ions or a complex organic or metalloorganic
molecule called a coenzyme
 A coenzyme or metal ion that is very tightly or even
covalently bound to the enzyme protein is called a prosthetic
group
 A complete, catalytically active enzyme together with its
bound coenzyme and/or metal ions is called a holoenzyme
 The protein part of such an enzyme is called the apoenzyme

7. What Does an Enzyme Do
 An enzyme, as catalist, provides a specific environment
within which a given reaction can occur more rapidly. The
distinguishing feature of an enzyme-catalyzed reaction is
that it takes place within the confines of a pocket on the
enzyme called the active site.
 The molecule that is bound in the active site and acted
upon by the enzyme is called the substrate.
 A simple enzymatic reaction might be written
where E, S, and P represent the enzyme, substrate, and product; ES and EP
are transient complexes of the enzyme with the substrate and with the product
 The function of a catalist is to increase the rate of a
reaction. Catalysts do not affect reaction equilibria.

 Any reaction, such as S → P, can be described by a
reaction coordinate diagram (free energy of the system is plotted
against the progress of the reaction)
 The starting point for either the forward or the reverse
reaction is called the ground state

 Enzymes enhance reaction rates
by lowering activation energies.
 Covalent interactions between
enzymes and substrates lower
the activation energy (and
thereby accelerate the reaction)
by providing an alternative,
lower-energy reaction path.
 Much of the catalytic power of
enzymes is ultimately derived
from the free energy released in
forming many weak bonds and
interactions between an enzyme
and its substrate. This binding
energy contributes to specificity

 Transition state theory suggests that as molecules collide
and a reaction takes place, they are momentarily in a less
stable state than either the reactants or the products.
During this transition state, the potential energy of the
activated complex increases, effectively creating an energy
barrier between the reactants and products.
 Products can only be formed when colliding reactants have
sufficient energy to overcome this energy barrier. The
energy barrier is known as the activation energy (ΔG‡
) of a
reaction.
 The greater the activation energy for a given reaction is,
the lower the number of effective collisions.
 When several steps occur in a reaction, the overall rate is
determined by the step (or steps) with the highest activation

 Two molecular models currently used to explain
how an enzyme catalyzes a reaction are the
induced-fit theory and Lock & Key theory.
Induced-fit theory Lock &
Key theory

8. Enzymes classification
1. Oxidoreductases: oxidases, peroxidases,
dehydrogenases
2. Transferases

3. Hydrolases: esterases, peptidases, lipases,
glycosidases
4. Lyases

5. Isomerases
6. Ligases (synthetases)

9. Enzymes Specificity
1. Absolute specificity
one enzyme acts only on one substrate
(example: urease decomposes only urea;
arginase splits only arginine)
2. Relative specificity
one enzyme acts on different substrates which
have the same bond type (example: pepsin
splits different proteins)
3. Stereospecificity
some enzymes can catalyze the transformation
of only substrates which are in certain

10. Enzymes nomenclature
Common names
 are formed by adding the suffix –aseto the name of substrate
Example: tyrosinase catalyzes oxidation of tyrosine; cellulase
catalyzes
the hydrolysis of cellulose
 Common names don’t describe the chemistry of the reaction
Trivial names
 Example: pepsin, catalase, trypsin. Don’t give information about
the substrate, product or chemistry of the reaction
La nomenclature officielle a établi codification internationale. Le
numéro de code spécifie:
 Le type de réaction (classe)
 Le type de fonction du substrat métabolisé (sous-classe)
 Le type de l’accepteur
 Le numéro d’ordre (dans la sous-sous-classe).
Example : EC 2.7.4.4. →
2 : Transferase, 7 : phosphate transferred, 4: transferred to another , phosphate, 4:
detaile’s acceptor

11. Factors Affecting Enzyme Activity
 Temperature affects enzyme activity.
Higher temperatures mean molecules
are moving faster and colliding more
frequently. Up to a certain point,
increases in temperature increase the
rates of enzymatic reactions. Excess
heat can denature the enzyme, causing
 Because protein nature of
enzymes, they are often affected
by pH. The “optimum pH”
refers to the pH at which the
enzyme exhibits maximum
activity. This pH varies from

12. Enzymes kinetics: Michaelis–Menten
Equation
 Kinetic
measurements of
enzymatically
catalyzed reactions are
among the most
powerful techniques
for elucidating the
catalytic mechanisms
of enzymes
 In enzyme kinetics,
it is customary to
measure the initial
rate v0 of a reaction to
minimize reversible
reactions and the
inhibition of enzymes

Equation
 The rate increases rapidly and linearly with [S] at
low substrate concentrations, but it gradually
levels off toward a limiting value at high
concentrations of the substrate. In this region, all
the enzyme molecules are saturated, and the rate
becomes zero order in substrate concentration.
 Mathematical analysis shows that the relationship
between v0 and [S] can be represented by an
equation of a rectangular hyperbola

Equation
 In 1913, the German biochemist Leonor Michaelis and the
Canadian biochemist Maud L. Menten proposed a
mechanism to explain the dependence of the initial rate of
enzyme-catalyzed reactions on concentration. They
considered the following scheme, in which ES is the enzyme–
substrate complex
 According to this model, when the substrate concentration
becomes high enough to entirely convert the enzyme to the ES
form, the second step of the reaction becomes rate limiting
and the overall reaction rate becomes insensitive to further
increases in substrate concentration. The initial rate of
product formation, v0, is given by

Equation
 The overall rate of production of ES is the difference
between the rates of the elementary reactions leading to its
appearance and those resulting in its disappearance:
 Integration of this equation is made easy under two
assmptions
I. Assumption of equilibrium: assumed that k–1>>k2, this
means that ES association/dissociation is assumed to be
a rapid equilibrium, and KS is the enzyme-substrate
dissociation constant.
II. Assumption of steady state: Briggs and Haldane, in
1925, assumed the concentration of the enzyme–substrate
complex ES quickly reaches a constant value in such a

Equation
 That is, the change
in concentration of
ES with time, t, is 0.
The Figure below
illustrates the time
course for
formation of the
ES complex and
establishment of the
steady-state
condition,

Equation
 According to the steady-state assumption, the rate of ES
formation must therefore balance the rate of ES consumption:
 the total enzyme concentration, [E]T, is usually known:
 So we get
 Dividing both sides by [ES] and k1
 At this point, we can define the Michaelis constant, KM as:

Equation
 So or
 Solving for [ES] yields
 Initial vilocity becomes
 The maximal velocity of a reaction, Vmax, occurs at high
substrate concentrations when the enzyme is saturated, that
is, when it is entirely in the ES form:
 so

Equation

Equation
 In practice, however, we find that the plot of v0 versus [S] is
not very useful in determining the value of Vmax because
locating the asymptotic value Vmax at very high substrate
concentrations is often difficult.
 A more satisfactory approach, suggested by the American
chemists Lineweaver and Burk, is to employ the double-
reciprocal plot of 1/v0 verus 1/[S], as follow:

Equation
 Lineweaver-Burk plot has the disadvantage of compressing
the data points at high substrate concentrations into a
small region and emphasizing the points at lower substrate
concentrations, which are often the least accurate.
 Among other ways of plotting the kinetic data, we shall
mention the Eadie–Hofstee plot. It shows a plot of v0 versus
v0/[S].

Equation
Key Parameters of the Michaelis–Menten
Km(mol.l−1)
 The magnitude of KM varies widely with the identity of the
enzyme and the nature of the substrate It is also a function
of temperature and pH.
 as KS decreases, the enzyme’s affinity for substrate
increases. KM is therefore also a measure of the affinity of
the enzyme for its substrate providing k2/k1 is small
compared with KS , that is, k2 < k–1. In other words, it
approximates the dissociation constant of the ES complex:

Equation
 Any variation in KM (for the same enzyme and
substrate) is often an indication of the presence of
an inhibitor or activator.
 For the majority of enzymes, KM lies between 10-
1M and 10-7M.
Vmax (s–1)
 The Vmax is the maximum velocity that an enzyme
could achieve. The measurement is theoretical
because at given time, it would require all enzyme
molecules to be tightly bound to their substrates.
Vmax is approached at high substrate
concentration but never reached.

Equation
kcat(s–1)
 The kcat, also thought of
as the turnover number
of the enzyme, is a
measure of the maximum
catalytic production of
the product under
saturating substrate
conditions per unit time
per unit enzyme. The
larger the value of kcat,
the more rapidly catalytic
events occur.

Equation
Enzyme Efficiency, kcat/KM (s–1.M–1)
 An estimate of "how perfect" the enzyme is, and can be
taken as a measure of substrate specificity.
 It measures how the enzyme performs when [S] is low
 The upper limit for kcat/KM is the diffusion limit, the rate at
which E and S diffuse together (108 to 109 s–1.M–1)
 When [S] << KM, very little ES is formed. Consequently,
[E]≈[E]T
 kcat/KM is the apparent second-order rate constant of the
enzymatic reaction

13. Enzymes inhibition
 Enzyme inhibitors are molecular agents that
interfere with catalysis, slowing or halting
enzymatic reactions.
 The study of enzyme inhibitors also has provided
valuable information about enzyme mechanisms
and has helped define some metabolic pathways.
 Inhibition is defined as a reduction in enzyme
activity through the binding of an inhibitor to a
catalytic or regulatory site on the enzyme, or, to the
enzyme–substrate complex.
 There are two broad classes of enzyme inhibitors:
reversible and irreversible.

13. Enzymes inhibition
 Irreversible inhibition nearly always involves the
covalent binding of a toxic substance that permanently
disables the enzyme
 Reversible inhibition involves the noncovalent binding
of an inhibitor to the enzyme which results in a
temporary reduction in enzyme activity.
 Inhibitors differ in the mechanism by which they
decrease enzyme activity. There are three basic
mechanisms of inhibition: competitive,
noncompetitive, and uncompetitive inhibition

13.1 Competitive Inhibition
 A competitive inhibitor is usually a close analogue of the
substrate. It binds at the catalytic site but does not undergo
catalysis.
 the presence of an inhibitor decreases the ability of the
enzyme to bind with its substrate

13.2 Mixed noncompetitive Inhibition
 A noncompetitive inhibitor does not bind to the catalytic
site but binds to a second site on the enzyme and acts by
reducing the turnover rate of the reaction
 Inhibitor binding does not prevent substrate binding but
alters the catalytic activity of the enzyme (thereby
decreasing the apparent Vmax)

13.2 Mixed noncompetitive Inhibition

13.3 Uncompetitive Inhibition
 An uncompetitive inhibitor does not bind to the enzyme but
only the enzyme-substrate complex.
 As a result, Vmax and KM appear to be reduced by the same
amount.

13. 3 Uncompetitive Inhibition
 Inspection of this equation indicates that at high values of
[S], vo asymptotically approaches Vmax/α’, so that, in
contrast to competitive inhibition, the effects of
uncompetitive inhibition on Vmax are not reversed by
increasing the substrate concentration.
 However, at low substrate concentrations, that is, when
[S]<<KM, the effect of an uncompetitive inhibitor becomes
negligible, again the opposite behavior of a competitive
inhibitor

14. Bisubstrate reactions kinetics
 More than half of all known biochemical reactions involve
two substrates.
 Most of these bisubstrate reactions are either oxidation–
reduction reactions or transferase reactions.
 Cleland has devised a standardized way of referring to
bisubstrate enzymatic reactions. The substrates, products
and stable enzyme forms are denoted as follows:
 Substrates are lettered A, B, C and D, in the order that they are
added to the enzyme
 Products are lettered P, Q, R and S, in the order that they leave the
enzyme
 Stable enzyme forms are designated E, F, and G in the order that they
occur, with E being the free enzyme
 The numbers of reactants and products in a given reaction are
specified, in order, by the terms Uni (one), Bi (two), Ter( three), and
Quad (four). A reaction requiring one substrate and yielding three

There are two types of bisubstrate reactions
1. Single displacement reactions, or Sequential
Reactions: The enzyme must bind all
substrates before a reaction occurs and
products are released. Sequential reactions
can be classified into those with a compulsory
order of substrate addition to the enzyme,
which are said to have an Ordered mechanism,
and those with no preference for the order of
substrate addition, which are described as
having a Random mechanism.
2. Double displacement reactions: Some products

Random mechanism for group transfer (G)

14.1 Rapid Equilibrium Random Bi Bi
For the derivation of the simplest form of the rate equation
within this mechanism, one must assume that both substrates
are at independent and rapid equilibrium with the enzyme. A
corollary of this is that the EAB →EPQ conversion is rate-
limiting. The equation rate is:

 Under rapid equilibrium approximation random bibi scheme
is:
 KsA for the dissociation of the complex EA: KsA =[E][A]/[EA],
 KsB for the dissociation of the complex EB: KsB =[E] [B]/[EB],
 KAm for the dissociation of A from the complex EAB: KAm = [EB]
[A]/[EAB],
 KBm for the dissociation of B from the complex EAB: KBm
=[EA][B]/[EAB].
Since the system is at equilibrium, the four constants are related by the
thermodynamic relationship:

 All parameters are determined experimentally
 The equation of each saturation curve as a function of the
concentration of substrate A for a fixed concentration of B
is:
 The equation of each saturation curve as a function of the
concentration of substrate B for a fixed concentration of A is
:
 For each concentration of A and B, the maximum apparent
speed is

 with SATURANT concentration in A or B
 For each concentration of B, the apparent Michaelis constant
for substrate A is :
 With saturant concentration of B:

 For each concentration of A, the apparant Michaelis
constante for substrate B is :
 With saturant concentration of A:

 Two cases must be considered concerning the binding of A and B
 Either the binding of A and B to the enzyme is dependent, i.e. the
binding of A modifies the affinity of the enzyme for B and vice versa; or
it is independent; the binding of one substrate occurs in the same way
in the presence or absence of the second.
 Primary plots can distinguish the different binding was
 (a) positive dependence between the substrate-binding sites
 (b) negative dependence between the substrate-binding sites
 (c) independent substrate-binding sites.

 One can notice that primary plots can not determine all
kinetics parameters. We need secondary plots.
 En traçant 1/v contre 1/[A] à [B] fixe donne une ligne
avec pente et avec l'intercepté (ordonnées à l’origine) sur
l'axe 1/v :

14.2 Ordered BiBi mechanism
The substrates must bind the enzyme in an ordered way. This
also is called sequential mechanism. In the Cleland notation
this can be written:
The conversion of EAB to EPQ is as rapid as the other steps in
catalysis, steady state assumptions must be used in the derivation
of the velocity equation. The rate equation for the Ordered BiBi
mechanism

 Primary plots for an ordered Bi Bi mechanism at steady state are given
below
 X-intercept is -1/KA
M . Y intercept is (1/Vmax)(1+KB
M/[B]). In all these
cases the apparent KM is extracted from the graphs. The actual KA
M is
obtained in the presence of a saturating concentration of B.

Ordered BiBi mechanism when approximating a quasi-
equilibrium
If conversion of the EAB complex to EPQ is the rate-limiting
step in catalysis, then E, A, B, and EAB are all in
equilibrium, and the velocity of the reaction will be given by

The kinetic parameters in the equations describing
bisubstrate reactions significance are:
 Vmax is the maximal velocity of the enzyme obtained
when both A and B are present at saturating
concentrations
 KAM and KBM are the respective concentrations of A and
B necessary to achieve Vmax/2 in the presence of a
saturating concentration of the other
 KAS and KBS are the respective dissociation constants of
A and B from the enzyme, E.

Double displacement reaction mechanism
 Mechanisms in which one product is released
before the other substrates has been added is
known as Ping Pong reactions. it is represented
by:

 The main
difference between
single
displacement
reaction
mechanism and
double
displacement
reaction
mechanism is the
absence of the
ternary complexes
EAB and EPQ

 It is very useful to have another way of corroborating a
mechanism by using a method that does not rely on kinetic
data.
 It’s a hard task to distinguish between random and
compulsory ordered bibi mechanisms. It is necessary to
resort to the use of product inhibition pattern or using
isotope incorporation studies.
Product inhibition pattern
 By measuring the initial velocity of the reaction in the presence of
several concentrations of inhibitor, and varying separately the
concentrations of P and Q, one can identify the reaction
mechanism from the pattern of double reciprocal plots and
reference to these tables.

 An alternative means of distinguishing among reaction
mechanisms is to look at the rate of exchange between a
radio-labeled substrate and a product molecule under
equilibrium conditions, it became obvious that such an
exchange could take place only for a double-displacement
reaction
 For random or compulsory ordered reactions, the need to
proceed through the ternary complex before initial product
release would prevent the incorporation of radiolabel into
one product in the absence of the second substrate.

 when the rate of isotope exchange is measured under
equilibrium conditions for a general group transfer
reaction
 Under these conditions the forward and reverse reaction
rates are equivalent, and the equilibrium constant is given
by:
 If under these conditions radio-labeled substrate B is
introduced in an amount so small that it is insufficient to
significantly perturb the equilibrium, the rate of formation
of labeled BX can be measured. The measurement is
repeated at increasing concentrations of A and AX, to keep

Suppose that the reaction proceeds through a
compulsory ordered mechanism in which B is the first
substrate to bind to the enzyme and BX is the last
product to be released. If this is the case, the rate of
radiolabel incorporation into BX will initially increase
as the concentrations of A and AX are increased.
As the concentrations of A and AX increase further,
however, the formation of the ternary complexes E · AX
· B and E · A · BX will be favored, while dissociation of
the EB and EBX complexes will be disfavored. This will
have the effect of lowering the rate of isotope exchange
between B and BX. Hence, a plot of the rate of isotope
exchange as a function of [AX] will display substrate
inhibition at high [AX], as illustrated in Figure A
The effect of increasing [AX] and [A] on the rate of
radiolabel exchange between B and BX will be quite
different, however, in a compulsory ordered reaction
that requires initial binding of AX to the enzyme. In this
case, increasing concentrations of AX and A will
disfavor the free enzyme in favor of the EAX and EA
forms. The EAX form will react with B, leading to
formation of BX, while the EA form will not. Hence, the

 The enzymes sucrose phosphorylase and maltose phosphorylase provide
two clear-cut examples of how enzymatically catalyzed isotopic exchange
reactions are used to differentiate kinetic mechanisms.
 Sucrose phosphorylase catalyzes the reaction:
glucose-fructose + phosphate ↔ glucose-1-phosphate + fructose
 If the enzyme is incubated with sucrose and isotopically labeled fructose
in the absence of phosphate, it is observed that the label passes into the
sucrose:
glucose-fructose + fructose* ↔ glucose-fructose* + fructose
 or the reverse reaction, if the enzyme is incubated with glucose-1-
phosphate and 32P-labeled phosphate, this label exchanges into the
glucose-1-phosphate:
glucose-1-phosphate + phosphate* ↔ Glucose-1-phosphate* +
phosphate
 These observations indicate that a tight glucosyl–enzyme complex is
formed with the release of fructose, thereby establishing that the sucrose

 One class of enzymes has kinetics that do not obey the
Michaelis-Menten description. Instead of the usual
hyperbolic curve, the rate equations of these enzymes
produce a sigmoidal curve.
 This behavior is typically exhibited by enzymes that possess
multiple binding sites and whose activity is regulated by the
binding of inhibitors or activators.
 In the simplest case, the active sites on these different
subunits act independently, as if each represented a separate
catalytic unit. In other cases, however, the binding of
ligands at one active site of the enzyme can increase or
decrease the affinity of the active sites on other subunits for
ligand binding
 When the ligand binding affinity of one active site is
affected by ligand occupancy at another active site, the
active sites are said to be acting cooperatively. In positive
cooperativity ligand binding at one site increases the affinity
of the other sites, and in negative cooperativity the affinity of

 Hemoglobin is an example of a protein where allosteric
effects play an important role. Each heme prosthetic group,
one on each of the four subunits, can bind an oxygen
molecule. We can get an idea of what one subunit on its own
can do by looking at myoglobin, a related molecule that
moves oxygen within the cytoplasm. Myoglobin has just one
polypeptide chain and one heme.

 Some enzymes have a sigmoidal kinetic curve,
indicating that initially, at low concentrations of
substrate, the enzyme is not very responsive to
binding the substrate. Only as the substrate
concentration is significantly increased does the
enzyme begin to show normal activity.
 The sigmoidal kinetic demonstrates that the
enzyme itself, in the absence of any ligands,
appears to be in an inactive state. Only as the
concentration of substrate increases significantly
does the enzyme show a proportional increase in
activity. This sigmoidal kinetic curve is a direct
demonstration that this enzyme can exist in two
different conformational states: an inactive (or

 Cooperative conformational changes depend on
variations in the structural stability of different
parts of a protein.
 When the modulator is a molecule other than the
normal ligand the interaction is heterotropic. The
interaction of 2,3-bisphosphoglycerate with
hemoglobin provides an example of heterotropic
allosteric modulation. Feedback inhibition is
another example of heterotropic allosteric
inhibition.
 In homotropic enzymes, the active site and
regulatory site are the same.

Cooperative Ligand Binding: Hill
equation
 Consider a protein E consisting of
n subunits that can each bind a
molecule S
 Assume that the ligand binds with
infinite cooperativity so that there
are no observable intermediates
ES1, ES2, etc. The dissociation
constant for this reaction is

 We get from both equations
 After algebraic
rearrangement becomes the
Hill equation
 Hilh equation describes the degree of saturation of a multi-subunit
protein as a function of ligand concentration. The quantity n, the Hill
constant, increases with the degree of cooperativity of a reaction and
thereby provides a convenient characterization of a ligand-binding
reaction. The upper limit of n is the number of binding sites, which is 4 for
hemoglobin
 If n = 1, ligand binding is not cooperative, a situation that can arise
even in a multisubunit protein if the subunits do not communicate
 An n > 1 indicates positive cooperativity in ligand binding
 An n < 1 indicates negative cooperativity, in which the binding of one
molecule of ligand impedes the binding of others. Well-documented cases

Grafic determination of Hill equation parameters
The extrapolation of the extreme slopes
theoretically allows the determination
of the association constants K1and Kn,
corresponding to the binding of the
substrate to the first and the nth site,
respectively

Mechanisms for Cooperative Binding

 Microscopic and macroscopic equilibrium
constants

 When all the microscopic K's are identical with an
independent binding on the n sites of a macromolecule
comes back to a single site; Adair and the equation
collapses :
 Aggregated sites should not be independent to
generate non-Michaelian behavior
 If the microscopic dissociation constants of the Adair
equation are not equal, the fractional saturation curve
will describe cooperative ligand binding. Decreasing
and increasing values of these constants lead to
positive and negative cooperativity, respectively.

14. Models of allosteric behavior
 When the ligand binding sites of an oligomeric
enzyme interact cooperatively, we need to modify
the existing kinetic equations to account for this
intersite interaction. Two theoretical models have
been put forth to explain allostery in enzymes and
other ligand binding proteins
 The concerted transition or symmetry model
(MWC), is based on the work of Monod, Wyman,
and Changeux (1965) and has been widely applied
to proteins such as hemoglobin, to explain ligand
binding cooperativity.
 The sequential interaction model (KNF), proposed

The MWC Model for concerted allosteric transitions
 based on the following postulates:
 In the absence of any ligand, a protein which displays
cooperative effects exists as an equilibrium of two
conformations:
 The equilibrium is defined by the allosteric constantL0, such
that
L = [T0]/[R0].
 The ligand can bind to a protomer in either conformation.
Only the conformational change alters the affinity of a
protomer for the ligand.
 The molecular symmetry of the protein is conserved during

 For a ligand S and an allosteric protein consisting of n
protomers, The microscopic dissociation constant for the R
state is kR and The microscopic dissociation constant for
ligand binding to the T state is kT
 We can put Ti ≡ TSi and Ri ≡ RSi
 The fractional saturation, Ys, for ligand binding is

 α = [s]/KR the reduced substrate concentration;
 c =KR/KT the ratio of the microscopic dissociation
constants. The ratio is named the non-exclusion coefficient
because it takes into account the fact that S can also bind
to the T state, which is the opposite of what occurs in a so-
called exclusive system.
 Symmetry model of allosterism for homotropic interactions
is given by:
 In the case of an exclusive system, c → 0.
Where Vmax = n kcat [ET]

 Symmetry model saturation function curves for tetramers
(a) Their variation with L when c= 0.
(b) Their variation with c when L = 1000.

 In the second model, the sequential model, proposed in
1966 by Daniel Koshland, Nemethy and Filmer, is referred
to as the KNF model, ligand binding can induce a change
of conformation in an individual subunit. A
conformational change in one subunit makes a similar
change in an adjacent subunit, as well as the binding of a
second ligand molecule, more likely. There are more
potential intermediate states in this model than in the
concerted model.
 The induced-fit theory is based on three postulates:
 the substrate binding to an enzyme provokes a reversible and
discrete change in the conformation of the enzyme;
 in order for enzymatic activity to occur, a suitable and very
precise orientation of the enzyme catalytic groups in relation
to those of the scissile bond of the substrate is necessary;
 the substrate induces its own orientation in relation to the
enzyme via the change it provokes in the conformation of the

During the
dynamic process,
the substrate
“teaches” the
enzyme the
conformation it
must adopt. The
sequential model
does not at all take
into account the
symmetry
conservation and
assumes the
existence of hybrid

 For the case of an enzyme with nactive sites
displaying a high degree of cooperativity, this
model has the same form as the well-known Hill
equation.
 The Hill constant is related
 to the enzyme–substrate dissociation constants
(k’=∏kn) and provides an estimate of the affinity
of the enzyme for a particular substrate

 Example, in case we have a tetramer

14. Enzymatic catalysis
 Catalysis is a process that increases the rate at
which a reaction approaches equilibrium.
 What apparently make enzymes such powerful
catalysts are two related properties: their
specificity of substrate binding combined with
their optimal arrangement of catalytic groups.
 ‘‘The real puzzle is why the enzyme reaction with
the specific chemical groups (e.g., acids and bases)
is so much faster than the reaction with the same
groups in solution.’’
 nearly all explanations focus on the stability of
enzyme transition states and/or the dynamic

Catalysis mechanisms include:
 Stabilization of Reaction Transition States
 Electrostatic Stabilization of Transition States
 Intrinsic Binding Energy
 Reacting Group Approximation, Orientation and Orbital
Steering
 Acid/Base Catalysis
 Covalent Catalysis
 Catalytic Facilitation by Metal Ions
 Promotion of Catalysis via Enzyme Conformational
Flexibility
 Promotion of Catalysis via Force Sensing and Force-
Gated Mechanisms

Stabilization of Reaction by lowering activation energy
 As stated by Pauling (1947), the idea was that each enzyme
becomes structurally complementary to the transition state,
such that the geometry, polarity, and electrostatic charge of
the enzyme and the transition-state configuration of the
substrate are mutually stabilizing. Pauling (1947)
 Note that little advantage would be gained if an enzyme
were to stabilize both the ES and EX ‡.
 By mimicking the transition state, these analogues can
bind to an enzyme with extraordinary affinity

Electrostatic Stabilization of Transition States
 As the name implies, electrostatic catalysis is the consequence of
the strong local Coulombic interactions that stabilize ionic and
polarized transitions states. The presence of such charged
groups actually makes the active site’s local environment
significantly more polar than water
 The nucleophilic and electrophilic properties of functional
groups on the catalyst and reactant are also increased by
dehydration of the catalytic center.
 Another advantage of electrostatic effects is that they are
‘‘tunable,’’ meaning that the local environment can alter the pKa
values of acidic and basic groups. For example, when placed
into a hydrophobic environment, acids tend to exhibit higher
pKa values (i.e., formation of the –COO– is disfavored), whereas
bases tend to have lower pKa values (i.e., formation of cationic –
NH3
+ groups is disfavored)

 These charge distributions apparently serve to guide polar
substrates toward their binding sites so that the rates of these
enzymatic reactions are greater than their apparent diffusion-
controlled limits
Intrinsic Binding Energy
 Binding energy effects arise from the sum total of favorable
non-covalent interactions between an enzyme and its
substrate(s), including a substantial contribution from van der
Waals interactions associated with structural complementarity of
the enzyme and its substrate as well as desolvation.
 The favorable enthalpy of substrate binding is thought to
overcome the unfavorable entropy associated with bringing two
(or more) molecules together. Once formed, the E$S complex
allows the catalysis to be effectively an intramolecular process.
 the loss in entropy in going from a bimolecular to a
unimolecular reaction (i.e., E + S ↔ ES) results in the loss of
translational, rotational and vibrational degrees of freedom, thus
8

Reacting Group Approximation and Orientation
 Converting multi-substrate
reactions from bimolecular
rate processes to what
essentially becomes a
unimolecular rate process
 By arranging and orienting
reactant functional groups
with respect to each other
 Impact of proximity
demonstrated in
 experiments involving the
non-enzymatic hydrolysis
of p-bromo-phenylacetate

Acid-Base Catalysis
 In the active site of an
enzyme, a number of
amino acid side
chains can similarly
act as proton donors
and acceptors. These
groups can be
precisely positioned in
an enzyme active site
to allow proton
transfers, providing
rate enhancements of
the order of 102 to 105.
This type of catalysis
occurs on the vast

Covalent catalysis
 In covalent catalysis, a
transient covalent
bond is formed
between the enzyme
and the substrate.
Consider the
hydrolysis of a bond
between groups A and
B, In the presence of a
covalent catalyst (an
enzyme with a

Metal catalysis
 There are two classes of metal ion–requiring enzymes
that are distinguished by the strengths of their ion–
protein interactions
 Metalloenzymescontain tightly bound metal ions, most
commonly transition metal ions such as Fe2+, Fe3+, Cu2+,
Zn2+ , Mn2+
 Metal-activated enzymesloosely bind metal ions from
solution, usually the alkali and alkaline earth metal ions
Na+, K+, Mg2+, or Ca2+ .
 Metal ions participate in the catalytic process in three
major ways
 By binding to substrates so as to orient them properly for
reaction.

Metal catalysis: Carbonic anhydrase

Enzymology Guide to Thermodynamics, Kinetics, and Regulation

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Enzymology Guide to Thermodynamics, Kinetics, and Regulation