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Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Edda Klipp
Systems biology2 – Reaction kinetics
Sommmersemester 2010
Humboldt-Universität zu BerlinInstitut für BiologieTheoretische Biophysik
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
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
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Classification of enzymatic reactions
irreversible - reversible
S PS P
1 2 3 4S
0
0.2
0.4
0.6
0.8
1
V
01
23
S0
1
2
3
P
- 0.50
0.5V
01
23
S
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Classification of enzymatic reactions
Number of substrates (and products)
S P
S1+S2 P
S1+S2 +S3 P1 2 3 4
S
0
0.5
1
1.5
2
2.5
V
1
S2 large (0.5)
S2 small (0)
uni
bi
ter
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Classification of enzymatic reactions
Type of kinetics
v = k SLinearMass action
HyperbolicMichaelis-Menten
SigmoidalHill kinetics,Monod, Koshland
.
mKS
SVv
max
nb
nb
SK
SKY
1v
1 2 3 4S
0
0.2
0.4
0.6
0.8
1
V
“Hyperbolic” and “Sigmoidal” show saturation, “Linear” involves unlimited reaction rates.
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
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)
Kinetics of Enzymatic Reactions
v(r,t) = v(S(r,t),t)
v(t) = v(S(t))
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
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
2CkBAkvvv
kk ,
eqeq
eq
BA
C
k
kq
2
Reaction rate
Rate constants
Equilibrium constant
The Mass Action Law
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Brown (1902): Mechanism for Invertase reaction (with sucrose), Which holds for one-substrate-systemes with backward reaction of effectors:
E+S ES E+P1+P2k1 k2
k-1
E – catalystS – substrateP – productki – kinetic constantcomplex
formation reversible
Michaelis, Menten (1913): rate equation under the assumptionThat second reaction will not influence the first equilibrium(Hypothesis of quasi-equilibrium)
21 kk
Briggs, Haldane (1925): more general derivation of Rate law under the assumption of a steady statefor the enzyme-substrate-complex (where )
0d
d
t
ES
ES 0
Michaelis-Menten Kinetics
complexdegradationirreversible
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
E+S ES E+P1+P2k1 k2
k-1Non-linear ordinary differential equation system
ESkSEkdt
dS11
ESkkSEkdt
dES211
ESkkSEkdt
dE211
VESkdt
dP 2
- The rate of product formation is equal to the reaction rate
(1)
(2)
(3)
(4)
- The sum of equations (2) and (3) is a conservation relation for the enzyme
ESEEdt
dE
dt
dESt ,0
- The whole set of equations cannot be solved analytically. Using quasi-steady state assumption
0d
d
t
ES
Michaelis-Menten Kinetics: derivation of rate law
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
E+S ES E+P1+P2k1 k2
k-1
1
21
2
kkk
S
SEkv t
Reaction rate
tEkV 2max
1
21
k
kkKm
mKS
SVv
max
Maximalvelocity
Michaelis constant
Michaelis-Menten-Rate expression S
vVmax
Vmax2
1
mK
Michaelis-Menten-Kinetics: The rate equation
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Integrated Form of MM rate law
t
S
t
PV
d
d
d
d
mKS
SV
t
S
max
d
d
SS
SKtV m ddmax
S
S
S
SmS
Smt
tS
S
SKS
S
SKtV
0 000d
dddmax
00
max ln SSS
SKtV m
Reaction rate = Product increase or substrate decrease per unit time
Integration from t0, S0 to t, S results in
and for t0 0
This is a function or . One can record a progress curve and estimate the kinetic constants
using non-linear regression.
tSS Stt
S(t)
S0
t
**
*
**
**
Henri-Michaelis-Menten-equation
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
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.
t
SS0
V=dS/dt
2. InterpretationPlot measurement results in (S,V)-Diagram; Compare with Michaelis-Menten rate law; Estimate parameters by non-lineare regression, for example least-squares methode
V
S
*
*
*
** *
*
*
*
*
*
*
*
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Linearizations of the MM rate law
Lineweaver-Burk-Plot
maxmax
m 111
VSV
K
V
1/S
Anstieg= Km/Vmax
1/Vmax
1/V
-1/Km
Eadie-Plot
S
VKVV mmax
V
V/S
Vmax
Vmax /Km
Hanes-Plot
max
m
max V
K
V
S
V
S
S
S/V
1/VmaxAnstiegVmax
Km
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Additional aspects
RT
G
eq0
Relation to thermodynamics
Vmax is related to turnover number, kcat
Condition: completely saturated enzymes, maximal rate:
totcat E
Vk max [1/(mol*s)]
Dissoziation constante KS of theenzyme-substrate-complex: 1
1
k
kKS
[mol]
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Here: the enzyme as target of effectors
Regulation of Enzyme Activity
- 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....
Important mechanism for the regulation of cellularprocesses upon the adaptation to internal and external changes.
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
1. Competitive inhibition: substrate and inhibitor compete for the binding place at the enzyme
E+S ES E+Pk1 k2
k-1+I
EI
k3k-3
EIESEEt
I3
3 Kk
k
EI
IE
SKI
K
SVv
Im
max
1
V
S
Vmax
1/2Vmax
Km K'm
mit Hemmung
Equilibrium for inhibitor binding
Conservation relation for the enzyme
Rate equation
Enzyme Inhibition
1/S
1/V
1/Vmax
1/Km1/K'm
i=1
i=2i=3
i=4
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Examples Competitive Inhibition
2. Acetylcholin esterasehas 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.
1. Succinic acid dehydrogenasehas as substrate succinic acid and is inhibited by Malonic acid.
Bernsteinsäuredehydrogenase
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
2. Uncompetitive inhibition: Inhibitor binds only to the enzyme-substrate-complex
3. Non-competitive inhibition: Inhibitor binds to free and bound enzyme
E+S ES E+Pk1 k2
k-1+I
EI+Sk3
k-3ESI
I+
KI KI
IS
max
1KI
SK
SVv
Im
max
1KI
SK
SVv
E+S ES E+Pk1 k2k-1
ESI
I+
KI
Enzyme Inhibition
1/V
1/Vmax
-1/Km
-1/V´max
1/S
I=0I=3x I=2x I=1x
Anstieg=Km/Vmax
-1/K´m
Vmax
V
S
Vmax/2
Km
1/V
1/Vmax
-1/Km
-1/V´max
1/S
I=0
I=3xI=2x
I=1x
Anstieg=Km/Vmax
Anstieg=Km/V´maxV´max
V´max/2
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
4. Irreversible inhibition : inhibitor binds the enzyme irreversibly, partial or complete loss of catalytic effectivity
EIIE 002max' IEkV
HICOCHSE
COCHISHE
22
22
Example: Reaction of Iod acetate with –SH groupsin cystein side chains of the reaction centre
Enzyme Inhibition, 3
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)
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Allosteric Inhibition: Inhibition by a molecule that does not bind to the reaction centre. conformation change of the enzyme, Change of reaction coordinate
Enzyme Inhibition , 5
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.
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Substrate Excess Inhibition
E+S ES
ESS
E+PKS k2
KI
+S
mS KKES
SE
IK
ESS
SES
ESSESEEt
Im
max2
1K
SSK
SVESkV
vopt
vopt
KI=Km
KI=100 Km
v
Vmax
S
0.5Vmax
Km
Imopt KKS
Im
maxopt
21 KK
VV
CO2-
CO2-
CH2
CH2 CH2 CH2CO2- CO2-
CO2-CO2- CH2CH2
Example: Succinic acid dehydrogenase
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
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
ActivationIncrease of the rate by
- Change of substrate binding
- Acceleration of product formation
E+S ES+A
EA+S
+A
EAS
E+P
EA+P
KS
KS
KAKA
k
k
,
,
,
EASkESkV
Enzyme Activation
SESkESkV
SESSEESEEt
ES
SEK S
A
ASS K
KKK
SE
SEK A
SES
SESK A
AA
SS
Att
K
S
K
K
S
K
K
SEkEk
V
1
Vmax
v
S
,
Example Substrate activation:
Substrats S acts as activator A. Reaction rate = Product formation rate
Enzyme conservation
Quasi-equilibrium condition
Reaction rate
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Activation and Inhibition for Mass Action Kinetics
A
PS ASkv
AK
ASkv 1+
I
PSI
KSkv I
IKI
Skv
1
1-
compulsory additional
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
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.
Ligand Binding and Cooperativity
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Case of 1 binding site:Binding of S (Ligand) to E (Protein) ESSE
Binding constante SE
ESKb
Definition: Fractional Saturation subunits ofnumber total
ligands bound that subunits, ofnumber Y
Fractional saturation for 1 subunit 10
SK
SK
EES
ES
E
ESY
b
b
Y
S
Plot of Y versus S is hyperbolic
YE
ES
V
V
0max
Fractional Saturation
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Positive, homotrope cooperativitySimplest case: dimeric protein - two similar ligand binding sites- Binding of first ligand increases affinity to second ligand
22schnell
2
2langsam
2
SMSSM
SMSM
SMSM 222
2222
22
2
22
SMSMM
SM
M
SMY
tot
Assumption: Binding of S increases affinityM2S reacts with S as soon as it is formed
Fractional saturation:
Complete cooperativity (each subunit is either empty or completey saturated) 222 SM2SM
22
22
SM
SMKb
2
2
222
22
2
22
1 SK
SK
SMM
SM
M
SMY
b
b
tot
M = monomere Untereinheit, M2 = Dimer
Binding constante
Hill-Kinetik
Fractional saturation:
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
For complete homotrope cooperativity of a protein with n subunits holds:This is a form of the Hill equation n
b
nb
SK
SKY
1
nn 22 OHbOHb
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
•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
Y
S
Hill Kinetics
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model for enzymes with sigmoidal kinetics
Model assumptions (J.Mol.Biol.(1965),12,88) : i) Enzyme consists of several identical subunits (SU)ii) each SU can assume one of two conformations (active = R or inactive = T)iii) all SU of an enzyme have the same conformation Conformation change for all SU at the same time (concerted transition).
R – Conc. active conformationR0 – R- Conc. without bound substrate
R1 – R- Conc. with 1 bound substrate
T – Conc. of inactive conformationT0 – Conc. without bound substrates
R - active T - inactive
0
0
R
TL
L
Conformation equilibrium
Allosteric constant
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model
n = 4 subunits
Binding constante for substrate S to one SU: KR or KT
(Assumption: Binding only to active form, S +
KR
00 TR KK ,
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 !!
!
iin
n
i
n
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model
RKSR
R4
0
1
SRKR R 01 4
RKSR
R
4
6
1
2
202 6 SKRR R
RKSR
R
6
4
2
3
303 4 SKRR R
RKSR
R
4
1
3
4
404 SKRR R 0
00 SKRR R
It holds:
iRi SKR
i
nR 0
General:
n
i
iR
n
ii SK
i
nRRR
00
0 n
RSKRR 10
YTRn
iRn
ii
1
TSKRn
SKRi
ni
Yn
R
n
i
iR
10
10
00
10
1
1
TSKR
SKSKRY
nR
nRR
Sum of all active states:
with binomic Formula:
Fractional saturation
Replacement of R and Ri
T exists only as T0
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model
TSKR
SKSKRY
nR
nRR
1
1
0
10
It follows
10
1
00
1
10
10
100
1
1
!!1
!1
:1 !1-!
!1
!1-!
!1
!!
!
nRR
jR
n
jR
iR
n
iR
iR
n
i
iR
n
i
iR
n
i
SKSKnR
SKjjn
nSKnR
jiSKiin
nSKnR
SKiin
nnR
SKiin
niRSKR
i
ni
inin
i
n bai
nba
1
LSK
SK
SK
SKY
nR
nR
R
R
1
1
1
nR
R
R
SK
LSK
SKY
11
1
1
YEkV t tEkV max
nR
R
R
SK
LSK
SKVV
11
1
1max
Reaction rate
Michaelis-Menten-Term
"Regulatory Term"
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1 0 102 103
104
activation
inhibition
S
v
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.
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
A R
I T
AI KK , Bindungskonstanten n
A
nI
AK
IKLL
1
1
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Monod-Wyman-Changeux model
F6 ATP FDP ADPPFKP
Example: Phosphofructokinase: experimentaly well studied system
Activators: Inhibitors: DPG, ATPTypical value for
AMP, NH K4+,
L 104
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
E+S ES E+Pk1 k2
k-1 k-2
Derivation of rate equation for steady state
0dtESd VdtdP
Relation between equilibrium constant q and kinetic constants of elementary steps 21
21
kk
kkq
121
2
221
1 1
k
P
kk
k
kkk
SkPSq
EV t
mPmS
mP
rück
mS
vor
K
P
K
S
PK
VS
K
V
V
1
maxmax
mSrück
mPvor
KV
KVq
max
max
Reaction rate
Kinetics of Reversible Reactions
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
E+S ES E+Pk1 k2
k-1 k-2
Relation to phenomenological quantities
S very high, P=0 P very high, S=0 Half-maximal forward rate Half-maximal backward rate
vorttt VkE
Sk
kSkE
kkSk
kSkEV max2
1
21
211
21
rücktt
t VkEPk
kPkE
Pkkk
kPkEV max1
2
21
221
21
mSvor K
k
kkSVV
1
21max
2
1
mPKk
kkPVV
2
21rückmax
2
1
For S and P very small holds
This resembles Mass action kinetics
(Also called linear kinetics).
1 mPmS KPKS PkSkPK
VS
K
VV
mP
rück
mS
vor
maxmax
Kinetics of Reversible Reactions
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Edda Klipp, Humboldt-Universität zu Berlin
Several activated complexes
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Methode of King and AltmanEmpirical 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:
n
iitotal EXE
1
2. Relative concentration of each enzyme speciesis equal to ratio of two sums of terms,where every term Tij is the product of n-1 rate constantsand the related concentrations.
n
i jji
jji
total
i
T
T
E
EX
1,
,
3. Every term Tij contains the rate constants (times substrate conc.),which are associated with the steps leading individually or sequentially to EX i .The sum of all possible combinations (j) are the numerator, the sum of all numerators for all EXi is the denominator.
EXi - freies Enzym
4. The reaction is:
n
i jji
jjnn
jjnn
tnnnn
T
TPkTk
EEXPkEXkdt
dPv
1
1
1
,
,,
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
King-Altman for 3-Step reaction mechanism
E ES
EP
Sk1
k-1 k-2k2k3
Pk-3
1. Conservation of total enzyme amount: : EPESEEtotal
2., 3. Listing of all possibilities of n-1 = 2 lines leading to each enzyme species:
For Ek-1
k3
k-1
k-2 k3 k2
322113
3
1
kkkkkkTj
jE
,
For ESSk1
k3
Sk1
k-2 Pk-3 k-2
322131
3
1
PkkkSkkSkTj
jES ,
For EPSk1
k2 Pk-3 k2
322131
3
1
PkkkSkPkkTj
jEP ,
k-1
Pk-3
21232321322113
3
1
3
1
3
1
kkkPkkkkSkkkkkkkTTTNennerj
jEPj
jESj
jE ,,,
21232321322113
32132133
kkkPkkkkSkkkkkkk
kkPkkkSkEEPkEPkv total4. Reaction rate:
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Edda Klipp, Humboldt-Universität zu Berlin
E EA EAB EPQ EQ EAk1 Bk2 k3 k4
k-1 k-2 Pk-3 Qk-4
Ordered bi-bi-Mechanismus (Example: Kreatinkinase)
Further typical Mechanisms
iBmBiA
mAmP
iAmQ
eqr
f
mAiAmBiP
eq
f
KB
PKK
BKKQ
KA
PKKV
VBKKAK
KP
AB
KPQ
ABV
v
1111
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
E EA EAB EPQ EQ EAk1 Bk2 k3 k4
k-1 k-2 Pk-3 Qk-4
Ordered bi-bi-Mechanism (Example: Kreatinkinase)
Ping-Pong-Mechanism (Example : Transaminase, Nukleosid-Diphosphokinase)
E EA FB EQ EAk1 k2 Bk3 k4
k-1 k-3Pk-2 Qk-4FP F
Random bi-uni-Mechanism (Example : an Aldolase-Type)
E
EA
EAB
EB
E
Ak1 Bk2
Bk3 Ak4
k-1 k-2
k-4k-3
k5
Pk-5
Further typical Mechanisms
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Unbranched Reaction Chain
EXn
EX1
1k~
1k~
EX2
EX2
EXn-1
nk~
nk~
2k~
2k~
iii kRk ~
n
i i
i
k
kq
1~
~~
n
i
ji
ir r
rn
j i
n
i i
total
k
k
kk
qEv
1
11
1 11
11
1
~
~
~~
~
Apparent rate constants
Apparent equilibrium constants
General rate law
Holds for all sequential reaction mechanisms
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Example
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Convenience Kinetics(actually a generalized random kinetics….)
-200 -100 100 200
-400
-200
200
400
-200 -100 100 200
-400
-200
200
400
-400 -200 200 400
-400
-200
200
400
Convenience Kinetics
Ord
ere
d K
ineti
cs
r=0.946 r=0.975 r=0.983
Ordered KineticsPin
g-p
ong K
ineti
cs
Convenie
nce
Kin
eti
cs
Ping-pong Kinetics
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Other types of kinetics: S-SystemsIntroduced by M. Savageau, 1976 („synergistic systems“)
mn
j
hji
mn
j
gji
iii
jiji XX
VVX
11
,,
For i = 1...n n independent variablesm dependent variables
Steady state:
mn
jjiii
mn
jjiii
mn
j
hji
mn
j
gji
mn
j
hji
mn
j
gji
i
LogXhLogLogXgLog
XLogXLog
XX
X
jiji
jiji
11
11
11
0
0
,,
,,
,,
Xi
Xj2Xj3
Xj4
Xj5Xj1 Vi+ Vi
-
g, h – positive or negative, usually no integers
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Other types of kinetics: Lin-Log Kinetics
Sef Heijnen and others
0000lnln1
c
c
S
S
E
E
v
v