Enzymes

1
Enzymes

• Proteins, often complexed with cofactors
• Anorganic cofactors metall ions
• Organic cofactors (coenzymes) vitamin-derived
  complex groups

• remain unchanged after reaction as catalyst
• have a catalytical centre
• are in general highly specific
• are often pH- and temperature dependent

Turnover number 1000 /sec (100 /sec ... 10
million /sec) Acceleration (compared to
non-catalyzed reaction) by 106 to 1012 -
fold Thermodynamics Enzymes reduce the
necessary activation energy for the reaction
2
Classification of enzymatic reactions
irreversible - reversible
S
P
S
P
3
Classification of enzymatic reactions
Number of substrates (and products)
uni
S
P
S2 large (0.5)
bi
S1S2
P
S2 small (0)
ter
S1S2 S3
P
1
4
Classification of enzymatic reactions
Type of kinetics
.
v k S
Linear Mass action
Hyperbolic Michaelis-Menten
Sigmoidal Hill kinetics, Monod, Koshland
v
Hyperbolic and Sigmoidal show saturation,
Linear involves unlimited reaction rates.
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Kinetics of Enzymatic Reactions
Deterministic kinetic modeling of biochemical
reactions
Basic quantities
Concentration S number of molecules per unit of
volume Reaction rate v concentration change
per unit time
Postulat The reaction rate v at point r in
space at time t can be expressed as a unique
function of the concentrations of all substances
at point r at time t Simplifying
assumptions - spatial homogeneity
(well-stirred) - autonomous systemes (not
directly dependent on time)
v(r,t) v(S(r,t),t)
v(t) v(S(t))
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The Mass Action Law
The reaction rate is proportional to the
probability of collision of reactants, This is
in turn proportional to the concentration of
reactants to the power of their
molecularity. (Guldberg and Waage, 19. century)
AB
2 C
Reaction rate
Rate constants
Equilibrium constant
7
Michaelis-Menten Kinetics
Brown (1902) Mechanism for Invertase reaction
(with succrose), Which holds for
one-substrate-systemes with backward reaction of
effectors
E catalyst S substrate P product ki
kinetic constant
complex formation reversible
complex degradation irreversible
Michaelis, Menten (1913) rate equation under the
assumption That second reaction will not
influence the first equilibrium (Hypothesis of
quasi-equilibrium)
Briggs, Haldane (1925) more general derivation
of Rate law under the assumption of a steady
state for the enzyme-substrate-complex (where
)
8
Michaelis-Menten Kinetics derivation of rate law
Non-linear ordinary differential equation system
(1)

• The rate of product formation is equal to the
• reaction rate

(2)

• The sum of equations (2) and (3) is
• a conservation relation for the enzyme

(3)
(4)

• The whole set of equations cannot be solved
• analytically.
• Using quasi-steady state assumption

9
Michaelis-Menten-Kinetics The rate equation
Reaction rate
v
Vmax
Maximal velocity
Vmax
Michaelis constant
Michaelis-Menten- Rate expression
S
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Integrated Form of MM rate law
Reaction rate Product increase or substrate
decrease per unit time
Integration from t0, S0 to t, S results in
Henri-Michaelis- Menten-equation
and for
This is a function or
. One can record a progress curve and
estimate the kinetic constants using non-linear
regression.
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Estimation of Parameters Vmax and Km
2. Interpretation Plot measurement results in
(S,V)-Diagram Compare with Michaelis-Menten
rate law Estimate parameters by non-lineare
regression, for example least-squares methode
1. Measurement of initial rates Measure initial
rates for different initial concentrations ,
i.e. measure initial change of S.
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Linearizations of the MM rate law
Lineweaver-Burk-Plot
Eadie-Plot
Hanes-Plot
13
Additional aspects
Relation to thermodynamics
Vmax is related to turnover number,
kcat Condition completely saturated enzymes,
maximal rate
1/(mols)
Dissoziation constante KS of the enzyme-substrate-
complex
mol
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Regulation of Enzyme Activity
Important mechanism for the regulation of
cellular processes upon the adaptation to
internal and external changes.

• Regulation of enzyme amount (Gene expression /
  proteine degradation)
• Action of effectors (inhibitors, activators)
• Composition of mediums (pH, ions)
• Regulation of protein activity by kinases /
  phosphatases / methylases....

Here the enzyme as target of effectors
15
Enzyme Inhibition

• Competitive inhibition substrate and inhibitor
  compete for
• the binding place at the
  enzyme

Equilibrium for inhibitor binding
Conservation relation for the enzyme
Rate equation
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Examples Competitive Inhibition
Bernsteinsäuredehydrogenase
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Enzyme Inhibition
2. Uncompetitive inhibition Inhibitor binds
only to the enzyme-substrate-complex
3. Non-competitive inhibition Inhibitor binds
to free and bound enzyme
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Enzyme Inhibition, 3
4. Irreversible inhibition inhibitor binds the
enzyme irreversibly, ? partial or complete
loss of catalytic effectivity
Example Reaction of Iod acetate with SH
groups in cystein side chains of the reaction
centre
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Enzyme Inhibition , 5

• Allosteric Inhibition
• Inhibition by a molecule that does not bind to
  the reaction centre.
• conformation change of the enzyme,
• Change of reaction coordinate

• Product Inhibition
• Inhibition by the product due to allosteric
  inhibition
• (prevents excess production)
• Reduction of the net reaction rate, due to an
  accumulation of product
• which is substrate of the backward reaction.

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Substrate Excess Inhibition
Binding a further substrate molecule to
ES-complex ? Enzyme-Substrate-Complex ESS, Which
does not transforms to reaction products.
Reversible inhibition, if one molecule
dissociates.
Equilibrium assumptions
Enzyme conservation
Reaction rate
Optimum
Example Succinic acid dehydrogenase
21
Inhibition by Reaction Inhibitor-Substrate
Rate reduction by binding of inhibitor I and
substrate S to complex SI, which cannot be
processed by the enzyme. Formation of SI reduces
the effective amount of substrate. If S0 and I0
are the initial concentrations Then holds
according to mass action law Effective
substrate concentration At high substrate
concentrations is the maximal reaction rate In
presence of the inhibitor equal to the maximal
rate without Inhibitor. The Lineweaver-Burk-Plot
is not linear.
22
Enzyme Activation
Activation Increase of the rate by - Change of
substrate binding - Acceleration of product
formation
Example Substrate activation Substrats S acts
as activator A.

Reaction rate Product formation rate
Enzyme conservation
Quasi-equilibrium condition
Reaction rate
23
Activation and Inhibition for Mass Action
A

P
S
compulsory
additional
I
-
P
S
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Ligand Binding and Cooperativity
Ligand compound that binds to enzyme /
protein Here Binding of ligands to monomeric
und oligomeric proteins.
several ligand binding sites at a
protein Possibility of interactions between
these sites during binding This phenomenon is
called cooperativity
Positive/negative cooperativity Binding of a
ligand molecule increases/reduces the affinity of
the protein for further ligands. Homotrope/heter
otrope cooperativity Binding of a ligand
molecule affects binding of further molecules Of
the same/ other ligands.
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Fractional Saturation
Case of 1 binding site Binding of S (Ligand) to
E (Protein)
Binding constante
Definition Fractional Saturation
Fractional saturation for 1 subunit
Plot of Y versus S is hyperbolic
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Hill-Kinetik
Positive, homotrope cooperativity Simplest case
dimeric protein - two similar ligand binding
sites - Binding of first ligand increases
affinity to second ligand
M monomere Untereinheit, M2 Dimer
Assumption Binding of S increases affinity M2S
reacts with S as soon as it is formed
Fractional saturation
Complete cooperativity (each subunit is either
empty or completey saturated)
Binding constante
Fractional saturation
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Hill Kinetics
For complete homotrope cooperativity of a
protein with n subunits holds This is a form of
the Hill equation
Y
S
Hemoglobin sigmoid bindung curve of oxygen
against oxygen partial pressure Hill (1909)
Interaction between binding sites - positive
cooperativity Known hem binds oxygen
molecules Unknown number of subunits per
protein Assumption complete cooperativity -
experimental Hill coefficient h2.8

• Four Binding sites per hemoglobin molecule
• No complete cooperativity
• High oxygen partial pressure in lungs good
  binding of oxygen to Hb
• Low oxygen partial pressure in body easy
  delivery of O2

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Monod-Wyman-Changeux model for enzymes with
sigmoidal kinetics

• Model assumptions (J.Mol.Biol.(1965),12,88)
• Enzyme consists of several identical subunits
  (SU)
• each SU can assume one of two conformations
  (active R or inactive T)
• all SU of an enzyme have the same conformation
• Conformation change for all SU at the same time
  (concerted transition).

T - inactive
R - active
Conformation equilibrium
R Conc. active conformation R0 - R- Conc.
without bound substrate R1 - R- Conc. with 1
bound substrate T Conc. of inactive
conformation T0 Conc. without bound substrates
L
Allosteric constant
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Monod-Wyman-Changeux model
n 4 subunits
KR
S
Binding constante for substrate S to one SU KR
or KT (Assumption Binding only to active form,
For each enzyme there are the following possible
bound states R0 - Concentration of R without
substrate binding, R1 - Conc. of R with 1 bound
molecule of S R2 - Conc. of R with 2 bound
molecules of S R3 - Conc. of R with 3 bound
molecules of S R4 - Conc. of R with 4 bound
molecules of S
1 possibility
4 possibilities
6 possibilities.
4 possibilities
1 possibility
General Possibilities of substrate binding for Ri
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Monod-Wyman-Changeux model
It holds
General
with binomic Formula
Sum of all active states
Fractional saturation
Replacement of R and Ri
T exists only as T0
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Monod-Wyman-Changeux model
It follows
Reaction rate
Michaelis-Menten- Term
"Regulatory Term"
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Monod-Wyman-Changeux model
0
102
103
For S?8 Monod-Kinetics approaches
Michaelis-Menten-Kinetics small S regulatory
term important depending on L L 0
MM-Kinetics L gtgt 0 sigmoidal curve, shifted
to right.
v
activation
104
inhibition
S
Explanation of the action of activators and
inhibitors - Activators bind to active
conformation - Inhibitors bind to inactive
conformation -Shift of equilibrium to R or T
Bindungskonstanten
33
Monod-Wyman-Changeux model
Example Phosphofructokinase experimentaly well
studied system
Activators Inhibitors DPG, ATP Typical
value for
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Kinetics of Reversible Reactions
Derivation of rate equation for steady state
Relation between equilibrium constant q and
kinetic constants of elementary steps
Reaction rate
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Kinetics of Reversible Reactions
Relation to phenomenological quantities
S very high, P0   P
very high, S0   Half-maximal forward
rate   Half-maximal backward rate
For S and P very small holds This resembles
Mass action kinetics (Also called linear
kinetics).
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Several activated complexes
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Methode of King and Altman
Empirical methode to derive steady-state rate
equations for reactions, Which are catalyzed by
an enzyme (no interaction between enzymes!)
1. Conservation of total enzyme amount
EXi - freies Enzym
2. Relative concentration of each enzyme
species is equal to ratio of two sums of
terms, where every term Tij is the product of n-1
rate constants and the related concentrations.
3. Every term Tij contains the rate constants
(times substrate conc.), which are associated
with the steps leading individually or
sequentially to EXi . The sum of all possible
combinations (j) are the numerator, the sum of
all numerators for all EXi is the denominator.
4. The reaction is
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King-Altman for 3-Step reaction mechanism
Sk1
1. Conservation of total enzyme amount
E
ES
k-1
k-2
Pk-3
k2
k3
2., 3. Listing of all possibilities of n-1 2
lines leading to each enzyme species
EP
k-1
k-1
For E
k-2
k3
k3
k2
Sk1
Sk1
For ES
k-2
Pk-3
k3
k-2
Sk1
k-1
For EP
Pk-3
k2
Pk-3
k2
4. Reaction rate
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Further typical Mechanisms
Ordered bi-bi-Mechanismus (Example
Kreatinkinase)
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Further typical Mechanisms
Ordered bi-bi-Mechanism (Example Kreatinkinase)
Ping-Pong-Mechanism (Example Transaminase,
Nukleosid-Diphosphokinase)
Random bi-uni-Mechanism (Example an
Aldolase-Type)
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Unbranched Reaction Chain
EXn
Apparent rate constants
EXn-1
EX1
Apparent equilibrium constants
EX2
EX2
General rate law Holds for all sequential
reaction mechanisms
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Example
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Convenience Kinetics
(actually a generalised random kinetics.)
Ping-pong Kinetics
Ordered Kinetics
Convenience Kinetics
Convenience Kinetics
Ordered Kinetics
Ping-pong Kinetics
r0.983
r0.946
r0.975
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Other types of kinetics S-Systems
Introduced by M. Savageau, 1976 (synergistic
systems)
Xj4
Xj3
Xj2
Xj5
Xj1
Xi
Vi
Vi-
For i 1...n n independent variables m
dependent variables
g, h positive or negative, usually no integers
Steady state
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Other types of kinetics Lin-Log Kinetics
Sef Heijnen and others