Metabolic Control Theory
description
Transcript of Metabolic Control Theory
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Metabolic Control TheoryChange of activityof an enzyme, e.g. PFK
? Change of concentrationof metabolites, e.g. pyruvat ?
? Change of steady-statefluxes eg. within TCA cycle ?
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Example: Flux Control
110
1
1
01
1
1
0
0
1mSmP
mS
rm
mP
fm
K
S
K
P
SK
VP
K
V
v
Jvv 21
P0 S P1v1 v2
1
1
1
2
10
2211
21
21
10
PP
KKKK
VV
VV
mPmSmSmP
rm
rm
fm
fm
3
1,1
2
12
2
2
JSS
S
S
S
13
3,
4
5
1121
2
12
2
2
2
2
1
1
JS
KI
K
P
K
S
PK
VS
K
V
v
imPmS
mP
rm
mS
fm
14
3,
5
4
1111
0
1
1
01
1
1
0
0
JS
KI
K
S
K
P
SK
VP
K
V
v
imSmP
mS
rm
mP
fm
21
2
12
2
2
2
2
1
1
1mPmS
mP
rm
mS
fm
K
P
K
S
PK
VS
K
V
v
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Metabolic Control TheoryQuantification of metabolic controlQuantification of the impact of small parameter changes on the variablesof a metabolic system.
Problem: Relations between steady-state variables and parameters are usually non-linear and can not be expressed analytically. There exists no theorie, which permits quantitative prediction of the effect of large changes of enzyme activity on fluxes. Restriction to small (infinitesimal) changes. (Linearisation of the system in vicinity of steady state). Controlling parameters: kinetic constants, enzyme concentrations,...Controlled variables: fluxes, substrate concentrations Wanted: mathematical function quantifying control.
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Metabolic Control Theory
Relevant questions:
- Many mechanisms and regulatory properties of isolated enzyme reactions are known what is their quantitative meaning for metabolism in vivo?
-Which step of a metabolic systems controls a given flux? (Is there a rate-limiting step?)
-Which effectors or modifiers have the most influence on the reaction rate? Example: biotechnological production of a substance, Increase of turnover rate Question: which enzyme to activate in order to yield the most effect? Example: disease of metabolism, overproduction of a substanceQuestion: which reaction to modify to reduce overproduction in a predictable way?
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Coefficients used in Control Theory
Metabolic systems are networks; their behavior depends on the structure of the network and the properties of the individual components.
There are two types of coefficients : local and global ones
Elasticity coefficients Control coefficientsResponse coefficients
quantify the sensitivity of a ratefor the change of a concentrationOr a parameter value
directly, immediate(no steady state)
Quantitative measure for change of steady-state variables
Assume reaching new steady statesDepends on network structure
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Locally: Elasticity Coefficients
i
k
i
k
k
i
Si
k
k
iki S
v
S
v
v
S
S
v
v
S
iln
ln
0
S1 S2
v1 v2 v3
? ? ?
Question: How sensitive is a rateof an enzyme reaction with respect to small changes of a metabolite concentration?
Consider enzyme as isolated,Wanted: immediate effect
Elasiticity coefficient of reaction rate k with respect tometabolite concentration Si
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Parameter Elasticity
m
kkm p
v
ln
ln
-elasticities comprise derivatives with respect to a metabolite concentration (a variable!).
-elasticities comprise derivatives with respect to parameter values (kinetic constants, enzyme concentrations,...)
S1 S2
v1 v2(Km, Vmax) v3
?
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Globally: Control Coefficients
1. The system of metabolic reactions is in steady state.
J = v(S(p),p) S = S(p)
2. A small perturbation of a reaction is performed(Addition of enzyme, addition of metabolite,....)
3. The system approaches a new (nearby) steady state.
J J+J S S+S
What is the change of steady state-variables (fluxes, concentrations) due to the perturbation of a single reaction?
kkk vvv
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Definition of Control Coefficients
k
j
k
j
j
k
vk
j
j
kJk v
J
v
J
J
v
v
J
J
vC
k
j
ln
ln
0
Flux control coefficient
kv
jJ
jk Jv
- Change of rate of the k-th reaction under isolated fixed conditions
- Resulting change of steady state flux through the j-th reaction
- Normalization factor
S1 S2
v1 v2 v3
??
321 vvvJ
?
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Edda Klipp, Humboldt-Universität zu Berlin
Definition of Control Coefficients
kv
ik Sv
k
i
kk
ki
i
k
vk
i
i
kSk v
S
pv
pS
S
v
v
S
S
vC
k
i
ln
ln
0
Concentration control coefficient
- Change of rate of the k-th reaction under isolated fixed conditions
- Normalization factoriS - Resulting change of steady state concentration of Si
S1 S2
v1 v2 v3
? ?
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Edda Klipp, Humboldt-Universität zu Berlin
Choice of Perturbation Parameter
kk
kj
j
kJk pv
pJ
J
vC j
klp
v
p
v
k
l
k
k 0 , 0
The change of vk is based on a change of some parameters pk, which influences only this k-th reaction. (Enzymkonzentration, Inhibitoren, Aktivatoren, ....)
Extended expression of flux controlcoefficient:
Important: Perturbation of pk influences directly only vk and no further reaction
The flux control coefficients are then independent of the choice of the perturbed parameters pk.
The can be interpreted as measure for the degree to which reaction kcontrols a given flux in steady state.
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Edda Klipp, Humboldt-Universität zu Berlin
Response Coefficients, global
0ln
ln
tmm
tkk
m
kJp pp
JJ
p
JR k
m
0ln
ln
tmm
tii
m
iSp pp
SS
p
SR i
m
Consider: complete system in steady state. This state is determined by the values of the parameters; Parameter changes influence the steady state.The sensitivity of steady-state variables with respect to parameter Perturbations is expressed by response coefficients.
S1 S2
v1 v2(Km, Vmax, I) v3
?321 vvvJ
? ?
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Response Coefficients, Additivity 0ln
ln
tmm
tkk
m
kJp pp
JJ
p
JR k
m
j
mkj
km
vp
j
Jv
Jp CR Additivity: If several reactions are sensitive
for this parameter
if parameter m influences Only one reaction j :
S1 S2
v1 v2(p) v3(p)
321 vvvJ
mj
mkj
m
kmJ
v pv
pJRC k
j lnln
lnln
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Edda Klipp, Humboldt-Universität zu Berlin
Example 1
v
J
dJ/dI
dv/dI
ReferenzpunktJ, v
I
P1 S1 S2 P2v1 v2 v3
I
v2 wird sofort kleinerS2 sinkt, damit v3 kleiner,S1 staut sich, dann v1 sinkt.
IvvI lnln 2
2 IJR J
I lnln
Iv
IJRC v
I
JIJ
v lnln
lnln
222
Experimentelly measurable quantities:
Flux control coefficient
Add inhibitor to a biochemical reaction
Humboldt-Universität
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Edda Klipp, Humboldt-Universität zu Berlin
Non-normalized Coefficients
jkj k
k k
J p
v p
iki k
k k
S p
v p
Dv
Siji
j
iji
j
v
p
Non-normalized flux and Concentration control coefficients
Non-normalized elasticities
P1 S
P3
P2
v1
v2
v3
Examples
Control of second reaction?Non-normalized flux control coeff.
J2 0Be J dGluc dt1 J dLac dt2
Glycolysis:Glucose 2 Lactat
J J2 12Normalized coefficients
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Matrix Notation
ikjkSik
SJjk
J CC , , , CC
..
:::
..
..
21
22
2
2
1
11
2
1
1
r
r
rr
r
r
Jv
Jv
Jv
Jv
Jv
Jv
Jv
Jv
Jv
J
CCC
CCC
CCC
C ,...,1
,...,1
,...,1
ni
rk
rj
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Edda Klipp, Humboldt-Universität zu Berlin
Theorems of Control Theory
The problem:The fluxes J usually cannot be expressed as mathematical functions Of the reaction rates. How can one calculate the global controlcoefficients from the local (measurable) changes ??
The solution:Use of theoremes. Here, the theoremes are only given and described with examples.The mathematical derivation is given for your information on thefollowing pages.
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Edda Klipp, Humboldt-Universität zu Berlin
The Summation TheoremsThought experiment: What happens, if we induce by experimental manipulation the same fractional change in the local rate of all steps of the system?
3
3
2
2
1
1
v
v
v
v
v
v P0 S1 S2 P1v1 v3v2
Result: Flux J must also increase by factor . Since all rates increase in the same ratio, remain the concentration of the variable metabolites S1 and S2 unchanged.
The combined effect of all changes in the local rates on the systems variables J, S1 and S2 can be describedAs the sum of all individual effects caused by the change of each local rate. For flux J holds:
JJJ
JJJ
CCC
v
vC
v
vC
v
vC
J
J
321
3
33
2
22
1
11
1321 JJJ CCCThus holds
Analog for S1 and S2
111
111
321
3
33
2
22
1
11
1
1
0 SSS
SSS
CCC
v
vC
v
vC
v
vC
S
S
0111
321 SSS CCC
0222321 SSS CCC
It follows:
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The Summation Theorems
11
r
k
Jv
j
kC
The flux control coefficients of a metabolic pathway add up to 1.The enzymes share the control over flux.
01
r
k
Sv
i
kC
The concentration control coefficients for a substance add up to zero.Some enzymes increase a metabolite concentration, others decrease it.
11JC 01SC
1
1
1
1Matrix notation:
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Connectivity Theorems – General Relations
Connectivity between flux control coefficients and elasticities
r
i
vS
Jv
i
j
m
iC
1
0 C 0J
r
ijk
vS
Sv
i
j
k
iC
1
IC S
kj
kjjk if ,0
if ,1
Connectivity between concentration control coefficients and elasticities
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Example: Calculate flux control coefficients
P0 S P1v1 v2
12121
JJJv
Jv CCCCSummation theorem:
Connectivity theorem: 022
11 S
JS
J CC
Result: 12
2
1SS
SJC
12
1
2SS
SJC
0ln
ln 11
S
vS 0
ln
ln 22
S
vSSince in general: and
01 JC 02 JC
follows (i.a.!):
Both reaction exert positive control over the flux.
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Example: Calculate concentration control coefficients
P0 S P1v1 v2
02121
SSSv
Sv CCCC
122
11 S
SS
S CC
1211
SS
SC
122
1
SS
SC
0ln
ln 11
S
vS 0
ln
ln 22
S
vSIt holds: and 01 SC 02 SC
Producing reactions have positive control,consuming reactions have negative control.
Summation theorem:
Connectivity theorem:
Result:
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Example: Linear pathway
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...
0 2 4 6 8 10
1
0.5
0
0.5
1
1.5
ConcentrationControlCoefficients
Producing reactions have positive control,consuming reactions have negative control.
Reaction
S5 S9
S1
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Linear Metabolic Pathway
iiii SSvv ,1
iiiii SkSkv 1
iii kkq
r
j
r
jmm
j
r
jj
qk
PqP
J
1
121
1
Each rate is a function of the concentrations of substrates and productes
Assuming mass action kinetics
With the equilibrium constants
One can derive an equation for theSteady state flux
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...
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Linear Metabolic Pathway – Flux Control
r flux control coefficients1 summation theoremr-1 connectivity theorems
Ck
q
kq
iJ i
mm i
r
jm
m j
r
j
r
1
1
1
General Expression for flux control coefficients(if )
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...
v k S k Si i i i i 1
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Linear Pathway - Properties
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...
Ck
q
kq
iJ i
mm i
r
jm
m j
r
j
r
1
1
1
Ratio of two successive flux control coeff.:0
1
1
1
11
i
i
ir
imm
i
r
imm
iJi
Ji q
k
k
qk
qk
C
C
Flux control coefficients: Summation theorem 11
r
k
JkC
Since sum of all flux control coeff is 1, and the ratio of two successive flux control coefficients is positiv,all flux control coefficients in an unbranched pathway are positiv.
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Linear Pathway - Properties
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...C
kq
kq
iJ i
mm i
r
jm
m j
r
j
r
1
1
1
1 1
1
1
1
1
11
1
ii
ikk
qi
i
ir
imm
i
r
imm
iJi
Ji q
k
k
qk
qk
C
CRatio of two successive flux control coeff.:
Flux control coefficients tend to be larger at the beginning than at the end.
Case 1: Be the kinetic constants of allinvolved enzymes equal and the
equilibrium constants larger than 1
kkkk ii ,
1 qkkq iii
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Linear Pathway - Properties
P1 P2S1 Si-1 Sr-1v1 vi+1 vrSi ...vi...
Ck
q
kq
iJ i
mm i
r
jm
m j
r
j
r
1
1
1
Case 2: qi 1 Ck
k
k
k k kiJ i
jj
ri
r
1
1
1
1 1 1
1
1 2 . . .with
Using Relaxation time as measure for the velocity of an enzyme:
ii ik k
1with or holds and therefore qi 1 k ki i ii k21
CiJ i
r
1 2 . . .
All enzymes are involved in control.Slow enzymes exert more control. There is no „rate-limiting step“
holds:
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Flux increase – how?
P1 P2S1 S2v1 v2 v4S3
v3 iiiiii SkSkEv 1
iii kkq
Simple case:
1iE
2,1,2 iii qkk
121 PP
1J1 2 3 4
0.10.20.30.40.5
Flux control coefficients
Reaction
E1 E1 + 1% J J + C1 * 1% 1.0053
k
kJk E
J
J
EC
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Flux increase
P1 P2S1 S2v1 v2 v4S3
v3
15 4321 ,,,EE
1 2 3 4
0.1
0.2
0.3
0.4
0.5
74411.J
P1 P2S1 S2v1 v2 v4S3
v3
51 4321 EE ,,,
05631.J
1 2 3 4
0.1
0.2
0.3
0.4
0.5
P1 P2S1 S2v1 v2 v4S3
v3
total
iJi E
EC
28712.J
1051562120921243 4321 .,.,.,. EEEE1 2 3 4
0.1
0.2
0.3
0.4
0.5
1 2 3 4
0.1
0.2
0.3
0.4
0.5
P1 P2S1 S2v1 v2 v4S3
v3
2J24321 EEEE
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Irreversibility and Feedback
1 2 3 4
0.10.20.30.40.50.6
1 2 3 4
0.10.20.30.40.50.6
1 2 3 4
0.10.20.30.40.50.6
P1 P2S1 S2v1 v2 v4S3
v3
14321 EEEE3331.J
8490.JP1 P2S1 S2v1 v2 v4S3
v3
7170.JP1 P2S1 S2v1 v2 v4S3
v3
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• Bis hier am 7. Juni 2010
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Branching System
255
144
233
122
011
Skv
Skv
Skv
Skv
Pkv
10110
01011N
P0 S1 S2 P3
P4 P5
v1 v2 v3
v4 v5
53
3
42
4
53
3
42
4
42
2
42
2
53
5
42
4
53
5
42
4
42
4
42
4
1
001
1
001
00001
kk
k
kk
k
kk
k
kk
kkk
k
kk
kkk
k
kk
k
kk
k
kk
kkk
kkk
k
JvC
5342
0212
42
011 kkkk
PkkS
kk
PkS
,
5342
05215
42
0414
5342
03213
42
0212011 kkkk
PkkkJ
kk
PkkJ
kkkk
PkkkJ
kk
PkkJPkJ
,,,,
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Branching Systemwith ATP/ADP-Exchange
3323333
2232222
11111
SAkAPkEv
SAkAPkEv
SkPkEv
110
110
111
N
1
1
2
K
110
111rN
111NG 0
Rang(N) = 2 < r
Conservation relation ATP + ADP = const.
Reduced stoichiometric matrix
Basis vector for admissible steady state fluxes
P1 S
P3
P2
v1
v2
v3
ATPADP
ADPATP
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Mathematical Derivation of the Theorems
0p,pSvN
0p
vN
p
S
S
vN
r
r
p
v
p
vp
v
00
00
00
2
2
1
1
....p
v
S
vNM
SRp
vNM
p
vN
S
vN
p
S-
1
1
es.elasticiti
1
CCC
NS
vN
p
v
p
S-1
Start with equation
Implicite Differentiation w.r.t. parameter vector p
regular Jacobi matrix M
Rearrange to
Rearrange to
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Mathematical Derivation of Theorems, 2
Start with equation
Implicite differentiation w.r.t. parameter vector p
Rearrange to
p,pSvJ
JRp
vN
S
vN
S
vI
p
S
S
v
p
v
p
J
1
NS
vN
S
vI
pv
pJ
1
Non-normalized
Flux CC
1
NS
vN
S
vIN
S
vN
S
vI
1
Both non-normalized CC are independent of the choice of perturbed parameter. They depend only on stoichiometry (N) and kinetics (dv/dS) !!
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Theorems, Normalized Control Coefficients
JJC dgdg 1J
JSC dgdg 1S
SDJ dgdg 1
pv dgdg 1
jJ
J
J
00
00
00
meint dg 2
1
J
2
1
2
2
1
2
2
1
1
1
2
2
2
2
1
2
2
1
2
1
1
2
1
1
1
1
2
1
0
0
0
0
S
S
S
v
S
vS
v
S
v
S
v
v
S
S
v
v
SS
v
v
S
S
v
v
S
v
v
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Reaction System with Conservation Relations
Problem: Jacobi-Matrix is not regularM N v S
0NL
ILNN
vNL
I
S
S
b
a 0
dt
d
0000 p
vN
p
S
S
S
S
vN
p
S
S
vN a
a
b
b
a
a
LSvNM 00
010 NML
010 NMLS
vI
Rearrange rows of N and S, Such that dependent rows are at bottom.
Implicite Differentiation of independentsteady-state equationsw.r.t parameter vector p
The non-singular Jacobi matrix Of the reduced systems
:
Non-normalized concentrations cc
Non-normalized flux cc
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Real System – GlycolysisConcentration control coefficients
1 2 3 4 5 6 7 8 9 101112131415161718192021222324
AMPNADHNAD+ADPATPCNxACAxGlycxGlycEtOHxEtOHACAPyrPEPBGPDHAPGAPFBPF6PG6PGlcGlcx
- 3- 2- 1 0 1 2 3
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„Real System“ – GlycolysisFlux control coefficients
1 2 3 4 5 6 7 8 9 101112131415161718192021222324
24- AK23- consum22- storage
21- inCN20- lacto19- outACA18- difACA17- outGlyc16- difGlyc15- lpGlyc14- outEtOH13- difEtOH
12- ADH10- PDC10- PK
9- lpPEP8- GAPDH7- TIM6- ALD5- PFK4- PGI3- HK
2- GlcTrans1- inGlc
- 3- 2- 1 0 1 2 3
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Concentration control in the MAP K cascade
MAPKKK
1 2 3 4 5 6 7 8 9 10 11 12
MAPKKK
MAPKKK- P
MAPKKK- PP
MAPKK
MAPKK- P
MAPKK- PP
MAPK
MAPK- P
MAPK- PP
MAPKKK-P MAPKKK-PP
MAPKK MAPKK-P MAPKK-PP
MAPK MAPK-P MAPK-PP
12
34
56
78
910
1112
All parameter values: k, p=1
- 3- 2- 1 0 1 2 3
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Concentration control in Wnt signaling
2003
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
0 1 2 3 4 5
0
0.5
1
0 1 2 3 4 5
0.5
0
0.5
1
1.5
2
0 1 2 3 4 50
0.5
1
Non-Steady State Trajectories
What is the effect of parameter perturbations on time courses ? S[t]
S1
S2
S3 S4
Time
p2p4
p1,3
p4
p5
S1[0]
S3[0]
S2[0] S4[0]
p2
S2[0]
S4[0]p1,3
S1[0]
p5S3[0]
RS3
RS2
0pp
Sp p
ptStR
,
B.P. Ingalls, H.M. Sauro, JTB, 222 (2003) 23–36
S1[0] = 0
S2[0] = 0
S3[0] = 0
S4[0] = 1
p1 = 1
p2 = 1
p3 = 1
p4 = 0.5
p5 = 0.5
S1
S2
S4
S3
v1 v2
v3
v4
v5
p1 p2
p3
p5
p4
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Time-dependent Response:
0pp
Sp p
ptStR
,
B.P. Ingalls, H.M. Sauro, JTB, 222 (2003) 23–36
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Experimental Methods to Determine Control Coefficients
- Titration with purified enzyme
- Addition of specific inhibitors
- Overexpression of an enzyme using genetic techniques
- Downregulation of individual genes / Reduction of enzyme amount
Humboldt-Universität
Zu Berlin
Edda Klipp, Humboldt-Universität zu Berlin
Metabolic Control Analysis - History/People
1973 Kacser /Burns1974 Heinrich /Rapoport - Definition of coefficients
about 1980 Discovery by Experimentalists (Westerhoff)
1988 Reder – Matrix Formulation
BTK - Models and Experiments(Fell, Cornish-Bowden, Hofmeyr, Bakker, Schuster,….)