max-planck-institut für molekulare genetik metabolic control analysis 1 network analysis graph...

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



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 ?



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



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.



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?



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



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



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

?



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



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

?



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

? ?



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.



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

? ?



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



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



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



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



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.



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:



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:



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



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.



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:



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



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




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 )


v k S k Si i i i i 1



Linear Pathway - Properties


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.




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





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:



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



Flux increase


v3

15 4321 ,,,EE

1 2 3 4

0.1

0.2

0.3

0.4

0.5

74411.J


v3

51 4321 EE ,,,

05631.J

1 2 3 4

0.1

0.2

0.3

0.4

0.5


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


v3

2J24321 EEEE



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


v3

14321 EEEE3331.J

8490.JP1 P2S1 S2v1 v2 v4S3

v3

7170.JP1 P2S1 S2v1 v2 v4S3

v3



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

,,,,



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



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



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) !!



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



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



Glycolysis – Concentration Control Coefficients

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

AMPNADNADHADPATPCNxACAxGlycxGlycEtOHxEtOHACAPyrPEPBGPDHAPGAPFBPF6PG6PGlcGlcx

- 3- 2- 1 0 1 2 3



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



Hierarchical Control



due to steady state:



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



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



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



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



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



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



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,….)



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

max-planck-institut für molekulare genetik metabolic control analysis 1 network analysis graph...

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