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Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 1
Network Analysis
Graph theorety: Node + Edges Routes, (Substances + Reactions) Measure for connectivity
Stoichiometry:+ Molecule numbers Conservation relations,
Flux distributions, Elementary modes
Kinetics:+ Kinetics + Concentrations Control analysis
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 2
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 ?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 3
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 4
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.
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 5
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?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 6
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 7
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 8
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
?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 9
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 10
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
?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 11
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
? ?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 12
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.
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 13
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
? ?
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 14
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 15
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 16
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 17
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 18
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.
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 19
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:
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 20
The Summation Theorems
11
r
k
Jv
j
kC
The Flux control coefficients of a metabolic pathwayAdd up to 1.The enzymes share the control over flux.
01
r
k
Sv
i
kC
The concentration control coefficients for a substanceAdd up to zero.Some enzymes increase a metabolite concentration Others decrease it.
11JC 01SC
1
1
1
1Matrix notation:
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 21
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 22
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.
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 23
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:
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 24
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 25
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...
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 26
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 27
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 Ratio of two successive flux control coeff. Is positiv,are all flux control coeffizients in an unbranched pathway positiv.
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 28
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 29
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:
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 30
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 31
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 32
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 33
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
,,,,
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 34
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 35
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 36
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) !!
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 37
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 38
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 39
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 40
Glycolysis – Concentration Control Coefficients
1 2 3 4 5 6 7 8 9 101112131415161718192021222324
AMPNADNADHADPATPCNxACAxGlycxGlycEtOHxEtOHACAPyrPEPBGPDHAPGAPFBPF6PG6PGlcGlcx
- 3- 2- 1 0 1 2 3
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 41
Glycolysis – Flux 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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 42
Hierarchical Control
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 43
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 44
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 45
due to steady state:
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 46
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 47
Systems Equations: an Example
ODEs
d[S1]/dt = v1 v2
d[S2]/dt = v3 v4
d[S3]/dt = v5
d[S4]/dt = v3 + v4
S1
S2
S3
S4
S =
v1
v2
v3
v4
v5
v = N =
S1
S2
S3
S4
1 1 0 0 0
0 0 1 1 0
0 0 0 0 1
0 0 1 1 0
Stoichiometric Matrix
v1 v2 v3 v4 v5
1 1 0 0 0
0 0 1 1 0
0 0 0 0 1
0 0 1 1 0
X
v1
v2
v3
v4
v5
=
v1 v2 +0 +0 +0
0 +0 +v3 v4 +0
0 +0 +0 +0 +v5
0 +0 v3 +v4 +0
N v d[S]/dt X =
S1
S2
S4
S3
v1 v2
v3
v4
v5
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 48
Stoichiometric matrix N - Information
0pS,vN 0dt
dSorSteady state:
Linear equation system,Non-trivial solutions only for
NK 0
01100100000110000011
N
00011
1k
01100
2k
21 kkK
Feasible steady state fluxesElementary modesBalanced fluxes
2211 kkv
exam
ple
S1
S2
S4
S3
v1 v2
v3
v4
v5
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 49
Stoichiometric matrix N - Information
0GN
0d
d v
SGNG
t.S constG
Conservation relations:
1010G
.constSS 42
01100100000110000011
Nexam
ple
S1
S2
S4
S3
v1 v2
v3
v4
v5
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 50
Systemic Properties: Response and Control
010000
010000
001000
001000
000001
000001
JCS1
S2
S4
S3
v1 v2
v3
v4
v5
p1 p2
p3
p5
p4
001111
110000
001111
000011
SC
31
31
32
1
S
S1[0] = 0
S2[0] = 0
S3[0] = 0
S4[0] = 1
p1 = 1
p2 = 1
p3 = 1
p4 = 0.5
p5 = 0.5
p6 = 0.5
v6
p6
366
255
244
1433
122
11
Spv
Spv
Spv
SSpv
Spv
pv
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 51
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
p
v
p
S
s
vN
p
S tttt
t
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 52
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
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 53
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,….)
Max-Planck-Institut für molekulare Genetik
Metabolic control analysis 54
Skalare, Vektoren, Matrizen
1110
0111N
1
0
1
JS1 = 1
Rechengesetze:
Addition: a11 a12
a21 a22
b11 b12
b21 b22
a11+b11 a12 +b12 a21 +b21 a22 +b22 ( ( () ) )+ =
1 Spalte,n Zeilen
m Spalten,n Zeilen (m x n)
(m x n) (m x n) (m x n)
Multiplikation: a11 a12
a21 a22
b11 b12
b21 b22
a11b11 + a12 b21 a11 b12 + a12 b22 a21b11 + a22 b21 a21 b12 + a22 b22
( () ) ). =
(m x n) (n x p) (m x p)
(
a11 a12
a21 a22( )k . =
ka11 ka12
ka21 ka22( )
Inverse Matrix: A B = C A B B-1 = C B-1 A = C B-1
B B-1 = I =1 0 00 1 00 0 1
( ) B quadratisch, nicht singulär