1. Systems biology 2 – Reaction kinetics Edda Klipp Sommmersemester 2010
Humboldt-Universität zu Berlin
Institut für Biologie
Theoretische Biophysik

2. Enzymes remain unchanged after reaction as catalyst
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 million /sec)
Acceleration (compared to non-catalyzed reaction) by 106 to fold
Thermodynamics:
Enzymes reduce the necessary
activation energy for the reaction

3. Classification of enzymatic reactions
irreversible reversible S P S P

4. Classification of enzymatic reactions
Number of substrates (and products)
uni S P
S2 large (0.5) bi
S1+S2 P
S2 small (0)
ter
S1+S2 +S3 P 1

5. Classification of enzymatic reactions
Type of kinetics .
Linear
Mass action
v = k S
Hyperbolic
Michaelis-Menten
Sigmoidal
Hill kinetics,
Monod, Koshland v
“Hyperbolic” and “Sigmoidal” show saturation,
“Linear” involves unlimited reaction rates.

6. 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))

7. The Mass Action Law A+B 2 C 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)
A+B
2 C
Reaction rate
Rate constants
Equilibrium constant

8. Michaelis-Menten Kinetics
Brown (1902): Mechanism for Invertase reaction (with sucrose),
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 )

9. 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

10. Michaelis-Menten-Kinetics: The rate equation
Reaction rate v
Maximal
velocity
Vmax
Vmax
Michaelis constant
Michaelis-Menten-
Rate expression S

11. 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
Wenn die Änderung der Reaktionsgeschwindigkeit nur von der Änderung der Substrat-sättigung des Enzyms und nicht von einer Hemmung durch das
Produkt oder das Erreichen des Gleichgewichtes abhängt, ist
This is a function or
One can record a progress curve
and estimate the kinetic constants
using non-linear regression.

12. Estimation of Parameters Vmax and Km
1. Measurement of initial rates
Measure initial rates for different initial
concentrations , i.e.
measure initial change of S.
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

13. Linearizations of the MM rate law
Lineweaver-Burk-Plot
Eadie-Plot
Hanes-Plot

14. Additional aspects Relation to thermodynamics
Vmax is related to turnover number, kcat
Condition: completely saturated enzymes, maximal rate:
[1/(mol*s)]
Dissoziation constante KS of the
enzyme-substrate-complex:
[mol]

15. 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

16. 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

17. Examples Competitive Inhibition
Bernsteinsäuredehydrogenase
1. Succinic acid dehydrogenase
has as substrate succinic acid and is inhibited by Malonic acid.
2. Acetylcholin esterase
has as substrate acetylcholin and is inhibited by Neostigmin. Note that obviously only the charged N(CH3)3+-group is active.
3. Sulfonamide(antibiotica)
block as competitive inhibitors the production of DNA,
Since they are used by the enzyme instead of the vitamine precursor p-Aminobenzoesäure.

18. Enzyme Inhibition 2. Uncompetitive inhibition:
Inhibitor binds only to the
enzyme-substrate-complex
3. Non-competitive inhibition:
Inhibitor binds to free and bound
enzyme

19. 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
1. Di-isopropyl-fluorophosphate (DFP)
and other alkylphosphates bind covalently to acetylcholinesterase. This enzyme is responsible for
Transmission of nerve stimuli. The organsims die of paralysis (Lähmung) of organ function.
(used in military gases and insektizids)

20. Enzyme Inhibition , 5 Allosteric Inhibition: Product 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.

21. 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

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 Kinetics+ S P
compulsory
additional I - S P

24. 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/heterotrope cooperativity :
Binding of a ligand molecule affects binding of further molecules
Of the same/ other ligands.

25. 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

26. 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:

27. 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 h=2.8
Hintergrund: experimentelle Befunde zur Bindung des Sauerstoffs ans Hämoglobin fanden
Bohr und Mitarb., dass die Auftragung der fraktionellen Sättigung des Hämoglobin mit Sauerstoff
gegen den Sauerstoffpartialdruck eine sigmoide Kurve ergab.Hill (1909) erklärte das auf der
Grundlage von Wechselwirkungen zwischen den Bindungsstellen, die positive Kooperativität
bewirken. Damals wußte man schon, dass jedes Häm ein Sauerstoffmolekül bindet, allerdings nicht
aus wieviel UE ein oligomeres Protein besteht. Hill leitete seine Gleichung ab für
Er nahm vollständige Kooperativität an und fand experimentellen Hillkoeff. von h=2.8. Heute
weiß man, dass es vier Bindungsstellen an jedem Hämoglobinmolekül gibt, so dass keine voll-
ständige Kooperativität vorliegt. Praktischer Nutzen der sigmoiden Binungscharakteristik: In der
Lunge ist der Sauerstoffpartialdruck hoch, dort kann Hb den Sauerstoff gut binden; im Körper ist
der Sauerstoffpartialdruck geringer und Hb kann O2 leicht abgeben
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

28. 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).
R - active
T - inactive
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

29. Monod-Wyman-Changeux model
n = 4 subunits KR
Binding constante for substrate S to one SU: KR or KT
(Assumption: Binding only to active form,
S +
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

30. Monod-Wyman-Changeux model
It holds:
General:
Sum of all
active states:
with binomic
Formula:
Fractional saturation
Replacement of R and Ri
T exists only as T0

31. Monod-Wyman-Changeux model
It follows
Reaction rate
Michaelis-Menten-
Term
"Regulatory Term"

32. Monod-Wyman-Changeux model
For S∞ :
Monod-Kinetics approaches Michaelis-Menten-Kinetics
small S: regulatory term important depending on L
L = 0: MM-Kinetics
L >> 0: sigmoidal curve, shifted to right.
102
103 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

34. Kinetics of Reversible Reactions
Derivation of rate equation
for steady state
Relation between equilibrium constant q
and kinetic constants of elementary steps
Reaction rate

35. Kinetics of Reversible Reactions
Relation to phenomenological quantities
S very high, P=0
P very high, S=0
Half-maximal forward rate
Half-maximal backward rate
For S and P very small holds
This resembles Mass action kinetics
(Also called linear kinetics).

36. Several activated complexes

37. 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:

38. King-Altman for 3-Step reaction mechanism
Sk1
1. Conservation of total enzyme amount: : E ES
k-1
k-2
Pk-3 k2 k3 EP
2., 3. Listing of all possibilities of n-1 = 2 lines leading to each
enzyme species:
k-1
k-1
For E k3
k-2 k3 k2
Sk1
Sk1
For ES k3
k-2
Pk-3
k-2
k-1
Sk1
For EP
Pk-3 k2
Pk-3 k2
4. Reaction rate:

39. Further typical Mechanisms
Ordered bi-bi-Mechanismus (Example: Kreatinkinase)

40. Further typical Mechanisms
Ordered bi-bi-Mechanism (Example: Kreatinkinase)
Ping-Pong-Mechanism (Example : Transaminase, Nukleosid-Diphosphokinase)
Random bi-uni-Mechanism (Example : an Aldolase-Type)

41. Unbranched Reaction Chain
EXn
Apparent rate constants
EXn-1
EX1
Apparent equilibrium constants
EX2
EX2
General rate law
Holds for all sequential reaction mechanisms

42. Example

43. Convenience Kinetics (actually a generalized random kinetics….)
Ordered Kinetics
Ping-pong Kinetics
Convenience Kinetics
Convenience Kinetics
Ordered Kinetics
Ping-pong Kinetics
r=0.946
r=0.975
r=0.983

44. Other types of kinetics: S-Systems
Introduced by M. Savageau, 1976 („synergistic systems“)
Xj4
Xj3
Xj2
Xj1 Xi
Xj5
Vi+
Vi-
For i = 1...n
n independent variables
m dependent variables
g, h – positive or negative, usually no integers
Steady state:

45. Other types of kinetics: Lin-Log Kinetics
Sef Heijnen and others