Lecture #14.a

Properties of the Null Space of S deciphered through the use of basis

vectors

Basis vectors can describe every point in a space

The null space describes flux states, that are candidate physiological states

Outline• Introduction

– The extreme pathway matrix P (SP=0)– 4 key properties– Example systems (simple, RBC, core E. coli,

genome-scale studies)

• Pathway “length” (size)• Reaction participation

– Co-sets; a ‘module’• Input-output feasibility

– Cross-talk• The effects of regulation

– Regulatory rules, expression profiling data– Elimination of ExPas

THE PATHWAY MATRIX (P) AND ITS FEATURES

Use of Basis Vector

• Basis vectors span a space • They can be used to determine all of its

properties; such as1. Pathway length2. Reaction participation/co-sets3. Input-output analyses4. Incorporation of regulation

111

111

222

001

100

110

010

101

222

P

EP1 EP2 EP3

v1

v2

v3

v4

v5

v6

b1

b2

b3

Pathway Matrix

000011100

100000110

010101000

000011100

000110010

000000221

001000001

S

ABCDEbypcof

v1 v2 v3 v4 v5 v6 b1 b2 b3

Stoichiometric Matrix

111

111

111

001

100

110

010

101

111

P~

EP1 EP2 EP3

v1

v2

v3

v4

v5

v6

b1

b2

b3

BinaryPathway Matrix

A simple example system

2 A 2 B C E

Dbyp

cof cof

byp

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

v2 v6

v3

v4

v5

b3

b1 b2v1

b3

EP3

EP1

EP2

The Red Blood CellA Model System for in silico Biology

• Relatively small metabolic network– 39 metabolites– 32 internal reactions

• Well studied, well understood system

• A full kinetic model has been developed in Mathematica® (BE 213)

Copyright Dennis Kunkel

GLU

vHK

vPGI

vPFK

vALD

vGAPDH

vPGK

vPGM

vLD

vDPGase

vDPGM

vTPI

G6P

F6P

FDP

GA3PDHAP

13DPG

23DPG

3PG

2PG

vEN

PEP

PYR

vPK

LAC

vG6PDH vPGL vPDGH

vXPI

vTK1

6PGL 6PGC RL5P

R5PX5P

vRPI

vTA

S7PGA3P

F6PE4P

vTK2

NAD

Pi

CO2

H+

NH3

NADPNADPH

NADH

vHGPRT

vPRM

vAdPRT

vADA

vAMPase

INO

IMP AMP

ADO

vIMPase

vAMPDAADP

ATP

vApK

vPNPasevAK

HXR1P

PRPP

vPRPPsyn

ADE

ADPATP

H2O

glycolysis pentose pathway bases nucleotides

GLU

vHK

vPGI

vPFK

vALD

vGAPDH

vPGK

vPGM

vLD

vDPGase

vDPGM

vTPI

G6P

F6P

FDP

GA3PDHAP

13DPG

23DPG

3PG

2PG

vEN

PEP

PYR

vPK

LAC

vG6PDH vPGL vPDGH

vXPI

vTK1

6PGL 6PGC RL5P

R5PX5P

vRPI

vTA

S7PGA3P

F6PE4P

vTK2

NAD

Pi

CO2

H+

NH3

NADPNADPH

NADH

vHGPRT

vPRM

vAdPRT

vADA

vAMPase

INO

IMP AMP

ADO

vIMPase

vAMPDAADP

ATP

vApK

vPNPasevAK

HXR1P

PRPP

vPRPPsyn

ADE

ADPATP

H2O

glycolysis pentose pathway bases nucleotides

Rapoport-Leubering shunt

GLU

vHK

vPGI

vPFK

vALD

vGAPDH

vPGK

vPGM

vLD

vDPGase

vDPGM

vTPI

G6P

F6P

FDP

GA3PDHAP

13DPG

23DPG

3PG

2PG

vEN

PEP

PYR

vPK

LAC

vG6PDH vPGL vPDGH

vXPI

vTK1

6PGL 6PGC RL5P

R5PX5P

vRPI

vTA

S7PGA3P

F6PE4P

vTK2

NAD

Pi

CO2

H+

NH3

NADPNADPH

NADH

vHGPRT

vPRM

vAdPRT

vADA

vAMPase

INO

IMP AMP

ADO

vIMPase

vAMPDAADP

ATP

vApK

vPNPasevAK

HXR1P

PRPP

vPRPPsyn

ADE

ADPATP

H2O

glycolysis pentose pathway bases nucleotides

GLU

vHK

vPGI

vPFK

vALD

vGAPDH

vPGK

vPGM

vLD

vDPGase

vDPGM

vTPI

G6P

F6P

FDP

GA3PDHAP

13DPG

23DPG

3PG

2PG

vEN

PEP

PYR

vPK

LAC

vG6PDH vPGL vPDGH

vXPI

vTK1

6PGL 6PGC RL5P

R5PX5P

vRPI

vTA

S7PGA3P

F6PE4P

vTK2

NAD

Pi

CO2

H+

NH3

NADPNADPH

NADH

vHGPRT

vPRM

vAdPRT

vADA

vAMPase

INO

IMP AMP

ADO

vIMPase

vAMPDAADP

ATP

vApK

vPNPasevAK

HXR1P

PRPP

vPRPPsyn

ADE

ADPATP

H2O

glycolysis pentose pathway bases nucleotides

Rapoport-Leubering shunt

Red Blood Cell Metabolic Network• 32 internal reactions• 19 exchange fluxes• 39 metabolites

Extreme Pathway Structure• 36 ‘Through’ Pathways (Type I)• 3 Futile Cycle Pathways (Type II)• 17 Reversible Reaction Pathways (Type III)

Currency exchanges

Biophys.J, 83(2): pp. 808-818 (2002).

P for the RBC

Biophys.J, 83(2): pp. 808-818 (2002).

The core E. coli model:

Number of rxns = 95Number of cmpds= 72

P for the core E. coli with glucose as the input

Anaerobic – Glucose input2006 extreme pathways•Number that produce acetate: 174•Number that produce co2: 506•Number that produce lactate: 249•Number that produce succinate: 1625

Aerobic – Glucose input16688 extreme pathways•Number that produce acetate: 1745•Number that produce co2: 11981•Number that produce lactate: 1420•Number that produce succinate: 7162

STAT1

rIFNγ

rIFNγ

JAK2

JAK2IFNγ

ADP

rIFNγ

JAK2

JAK2IFNγ STAT1

rIFNγ

JAK2

JAK2IFNγ STAT1

ADP

ADP

STAT1P

rIFNγ

JAK2

JAK2IFNγ

STAT2

JAK1

JAK1IFNα/β

ADP

rIFNα/β

JAK1

JAK1IFNα/β STAT2

rIFNα/β

JAK1

JAK1IFNα/β STAT2

ADP

ADP

STAT2P

rIFNα/β

JAK1

JAK1IFNα/β

rIFNα/β

STAT1P

STAT2P

P

P

P

P

Input

Input

OutputMetabolic Network

Energy GenerationTranscriptional

Regulatory Network

P PrIFNγP P

rIFNγP P

rIFNγP P

P PrIFNα/β

P PrIFNα/β

P PrIFNα/β

P PrIFNα/β

The JAK-STAT system in B cells

PATHWAY LENGTHProperty #1

Adjacency matrixPathway Length and Reaction Participation

EP1 EP2 EP3

EP1

EP2

EP3

7

56

546~~

PPT

Pathway Length Matrix

2 A 2 B C E

Dbyp

cof cof

byp

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

v2 v6

v3

v4

v5

b3

b1 b2v1

b3

EP1

EP2EP3

Distribution of Pathway Lengths for RBC

The figure shows the pathway lengths for the 39 Type I & II pathways

Example Extreme Pathways

ExPa with max ATP yield

Classical Glycolysis

ExPa that requires ATP as input

Example Extreme Pathways

3 optimal pathsfor NADPH yield of 6

Equivalent overall states

Pathway lengths from glucose input for the core E. coli

Mean pathway length = 35.6Median pathway length = 37

Anaerobic

Mean pathway length = 39.8Median pathway length = 40

Aerobic

Pathway Length from glucoseAnaerobic Aerobic

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

45

50Pathway Lengths

CO2 yield

path

way

leng

th

A set of ExPas with the same yieldThese particular pathways are optimality properties

REACTION PARTICIPATIONProperty #2

111

111

222

001

100

110

010

101

222

P

EP1 EP2 EP3

v1

v2

v3

v4

v5

v6

b1

b2

b3

Pathway Matrix

000011100

100000110

010101000

000011100

000110010

000000221

001000001

S

ABCDEbypcof

v1 v2 v3 v4 v5 v6 b1 b2 b3

Stoichiometric Matrix

111

111

111

001

100

110

010

101

111

P~

EP1 EP2 EP3

v1

v2

v3

v4

v5

v6

b1

b2

b3

BinaryPathway Matrix

Adjacency matrixPathway Length and Reaction Participation

Papin et al., Genome Research, 2002

EP1 EP2 EP3

EP1

EP2

EP3

7

56

546~~

PPT

Pathway Length Matrix

v1 v2 v3 v4 v5 v6 b1 b2 b3

v1

v2

v3

v4

v5

v6

b1

b2

b3

3

33

333

1111

11101

222012

1110011

22211102

333112123

~~ T

PP

Reaction participation matrix

2 A 2 B C E

Dbyp

cof cof

byp

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

v2 v6

v3

v4

v5

b3

b1 b2v1

b3

EP1

EP2EP3

Reaction Participation for the RBC

H,ATP, ADP and Pi – the primary currency

51 net reactions68 elementary reactions

Participations for the core E. coli: consumption of glucose (growth and no growth)

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

rxn number

frac

tion

of t

otal

ExP

as

Reaction Participation

Anaerobic – Glucose input Aerobic – Glucose input

ENO, FBP, GAPD, GLCpts, PFK, PGI, PGK, PGM, TPI, Glc exchange

GLCpts, Glc exchange

Never used under these growth conditions

95 net reactions133 elementary reactions

Reaction ParticipationAmino Acid Synthesis

Groups reactions into sets that are:

Always necessary (I)represent essential coreset of reactions

Sometimes necessary (II)Represent variability, redundancy in the metabolic network

Never utilized (III)

These groups each have important implications for metabolic engineering and understanding of biological systems

I

II

III

Papin et al., Genome Research, 2002

Reaction Participation

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 50 100 150 200 250 300

Reaction Number

Pe

rce

nta

ge

Reaction participation:JAK-STAT network

• ATP/ADP primary currency

• small number of reactions diversity in network function

• STAT1 and STAT3

• reactions with specific functions drug targeting

atp 100%adp 100%stat1 54%stat3 37%sd7 33%

S1_S3Pd 33%sd6 16%stat2 16%

S1_S2Pd 16%j1 14%

Papin, Palsson, Biophys. J., 2004.

If EPO is removed from culture, erythrocyte progenitors (CFC-Es) rapidly undergo apoptosis.

(Alberts, et al., Mol. Biol. Cell, 2004)

© 2004 Continuing Bioengineering Education, Inc.

Correlated Reaction Sets

Trends Biochem. Sci., 2003

A B C D

E

R1 R2 R7R3

R4 R6

R5

System Boundary

ABCDE

R1 R2 R3 R4 R5 R6 R7

S =

P =

R1

R2

R3

R4

R5

R6

R7

EP1 EP2

A B C D

E

R1 R2 R7R3

R4 R6

R5

System Boundary

EP1

EP2

Identical rows

Co-Sets for RBC

•9 Co-Sets in RBC

Anaerobic co-sets

1

99

2

3

4

4

5

6 7

8

Rhamnose

Rhamnose

Rhamnulose

Rhamnose 1-phosphate

Dihydroxyacetone phosphate

rhaA

rhaB

rhaD

rhaT

atp

adp

rhaDrhaArhaB

rhaSrhaR

rhaT

intracellular

extracellular

transcription & translation

operon

operon

operon

Lactaldehyde

h

h

Arginine

Arginine

N2-succinyl-L-arginine

N2-succinyl-L-ornithine

astA

astB

astC

arcD

h2o, h

co2, nh4

astEastBastDastAastC

intracellular

extracellular

transcription & translation

operonOrnithine

Ornithine

N2-succinyl-L-glutamate 5-semialdehyde

a-ketoglutarate

Glutamate

N2-succinyl-L-glutamate

h2o, nad

nadh, h

Glutamate

astE

Succinate

Succinyl-CoA

CoA, h

h2o

astD

operonspeBspeA

operonarcDydgB

Agmatine Putrescine

Ureah2oco2h

Urea

speA or adiA speB

glpF

operonglpKglpF

Known regulatory structure

Unknown regulatory structure

Correlated reaction sets

• E. coli metabolic network

• Example 1– 3 operons– 1 correlated

reaction set– 1 regulon

• Example 2– 4 operons– 1 correlated

reaction set– no known

regulatory rules– Genes are co-

expressed

Papin et al., Trends Biochem. Sci., 2004ref

Jamshidi, et al Molec. System Biol. (2006)

Co-Sets: A way to correlate SNPs?Co-Sets: A way to correlate SNPs?

Systems Biology Research Systems Biology Research GroupGrouphttp://systemsbiology.ucsd.edu

components

small-scale modules

large-scale modules

phenotype (physiology)

mRNA

protein products

translation

A B C

A

B

C

D

E K

L

J

M

F G

H

cof cof*

O

O

genotype

Correlated reaction sets

Hierarchy in Hierarchy in biological biological networksnetworks

Papin et al., Trends Biochem. Sci., 2004

INPUT-OUTPUT ANALYSIS: THE IOFA

Property #3

© 2004 Continuing Bioengineering Education, Inc.

Network crosstalk: need to understand interactions

Dumont, et al., Cell. Signal., 2001

• e.g., cAMP inhibits proliferation in fibroblasts, and stimulates proliferation in epithelial cells

• Black lines – “textbook” pathways

• Green & red lines – interactions described over previous 2 years

• Localization, differentiated state, etc. need to be considered

• Overlap & specificity

• “Pathways are concepts, Networks are the reality” Uwe Sauer, 2005

Extreme Pathway 1

Pathway Redundancy

• These extreme pathways have the same “external state.”

input: 2 A

output: 1 E and 1 byp• However, the internal flux distribution is very different in all three

pathways.Pathway Redundancy = 3

for this network.

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

byp

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

byp

2 A 2 B CC EE

DDbyp

cofcof cofcof

byp

byp

Extreme Pathway 2 Extreme Pathway 3

Pathway Redundancy

Reconstructed Metabolic Map of H. influenzae

Reconstructed Metabolic Map of H. pylori

INPUTS OUTPUTS

AlanineArginineOxygen

FructoseGlutamateAmmonia

Oxygen

AcetateSuccinateCarbon dioxideAmmonia

Amino acidAcetateSuccinateCarbon dioxide

Reconstructed Metabolic

Map of H. influenzae

Reconstructed Metabolic Map of H. pylori

INPUTS OUTPUTS

AlanineArginineOxygen

FructoseGlutamateAmmonia

Oxygen

LysineAcetateSuccinateCarbon dioxideAmmonia

Amino acidAcetateSuccinateCarbon dioxide

Price et al., Genome Research, 2002

In silico organism Genome Size ORFs in Genome Reactions in Model # of PW / UExVH. influenzae 1.83 Mb 1740 461 46H. pylori 1.67 Mb 1590 381 2

461390

internal states external state 2

internal states external state46

Similar components – very different network properties!

Classifying the I/O signature

Overlaps between I/O signatures of ExPas

Crosstalk

Input/output relationshipsL

ATP

S

ADP

S_p

T

WW_p

T_p

bindLR

LRpS

SpT

TpW

L2

R2

ATP

S2

ADP

S2_p

T2

W2W2_p

T2_p

bindL2R2

L2R2pS2

S2pT2

T2pW2S

T

S2

T2

L3

R3

ATP

S3

ADP

S3_p

T3

W3W3_p

T3_p

bindL3R3

L3R3pS3

S3pT3

T3pW3

S3

T3

R

L4

bindL4R

R2

L5

bindL5R2

ATP S

ADP S_p

L3R3pS L3R3

ATP S2

ADP S2_p

LRpS2 LR

ATP T

ADP T_p

L2R2pT L2R2

WW_p

S2_p S2pW S2

WW_p

T3_p T3T3pW

R

S2T3pWW2W2

W

S2_p

T3_p

S2

T3

W2_p

W_p

SWpW2W3W2

W3

S_p

W_p

S2

T3

W2_p

W_p

W2_W3_ppW2_p W3_p+

2W_W3_ppp2 W_p W3_p+

+ WW2W3pppW2_p W3_pW_p +formWW2W3

formW2W3

form2WW3

: Internal to system

: Input to system

: Output of system

WW2W3pppWW2W3pppW_pW_p W2_pW2_p W3_pW3_p 2W_W3_ppp2W_W3_ppp W2_W3_ppW2_W3_pp

output of system for regulatory control

A C B

ED

Papin, Palsson, J. Theor. Biol., 2004.

Expression arrays from combinations of IFN, -, or – stimulation indicated novel regulation (Der, et al., PNAS, 1998). Mathematical framework is needed for studying “combinations.”

Network Crosstalk

• Non-overlapping determined network

• Redundant output signals

• Significant network economization

Crosstalk: the non-negative linear combination of extreme signaling pathways.

Evaluate phenotypic effects of combinations of functional states, like conflicting cAMP signals.

Papin, Palsson, Biophys. J., 2004.

IOFA for RBC

IOFA for the core E. coli

CO2 CO2 CO2GLC GLC GLC GLC GLC GLC H2O H2O NH4 NH4

LAC H 27 0 0 0 0 0 ETOH FOR PYR H 27 0 0 0 0 0 ETOH FOR PYR SUCC H 8 0 0 0 0 0 ETOH FOR SUCC H 81 0 115 46 0 9 ETOH FOR LAC SUCC H 8 0 0 0 0 0 ETOH CO2 H 81 0 0 0 0 0 ACALD ETOH FOR CO2 H 18 0 0 0 0 0 ETOH FOR GLU CO2 H 0 33 0 0 0 0 ETOH FOR CO2 H 0 0 27 0 0 0 ETOH FOR SUCC CO2 H 69 0 89 0 0 0 ACALD FOR H 27 0 0 0 0 0AC FOR SUCC H 0 0 0 11 0 0 ACALD ETOH FOR SUCC H 8 0 0 0 0 0 ETOH FOR GLU H 0 51 0 0 0 0 ETOH FOR GLU SUCC H 0 15 0 0 0 0 ACALD SUCC H H2O 11 0 0 3 0 0 ETOH SUCC H H2O 57 0 0 8 0 0 PYR SUCC H H2O 0 0 0 14 0 0 ETOH FOR SUCC H H2O 8 0 0 10 0 0 LAC SUCC H H2O 0 0 0 27 0 0AC LAC SUCC H H2O 0 0 0 8 0 0AC SUCC H H2O 0 0 0 14 0 0 ETOH FOR GLU SUCC H H2O 0 21 0 0 17 0 ETOH GLU SUCC H H2O 0 66 0 0 10 0 ETOH LAC SUCC H H2O 31 0 0 0 0 0AC ETOH FOR SUCC H 35 0 8 0 0 0AC ETOH FOR CO2 H 0 0 18 0 0 0AC ETOH FOR H 0 0 27 0 0 0 ACALD FOR SUCC H H2O 0 0 0 26 0 0 FOR GLU LAC SUCC H H2O 0 0 0 0 17 0 AKG ETOH FOR SUCC CO2 H 23 0 0 0 0 0 AKG ETOH FOR CO2 H 0 0 18 0 0 0 ACALD ETOH FOR SUCC CO2 H 5 0 0 0 0 0 ETOH FOR GLU SUCC CO2 H 0 9 0 0 0 0 ACALD FOR GLU SUCC H H2O 0 0 0 0 17 0 ACALD AKG FOR SUCC H H2O 0 0 0 9 0 0 AKG ETOH FOR H 0 0 27 0 0 0 AKG ETOH FOR SUCC H 35 0 8 35 0 0 FOR LAC SUCC H H2O 0 0 0 17 0 0 FOR GLU SUCC H H2O 0 0 0 0 2 0 FOR SUCC H H2O 0 0 0 14 0 0 AKG FOR SUCC H H2O 0 0 0 1 0 0 ACALD AKG SUCC H H2O 0 0 0 1 0 0 ACALD GLU SUCC H H2O 0 0 0 0 2 0AC ETOH SUCC H H2O 35 0 0 0 0 0 FOR GLU PYR SUCC H H2O 0 0 0 0 2 0 AKG PYR SUCC H H2O 0 0 0 1 0 0 FOR PYR SUCC H H2O 0 0 0 1 0 0 GLU PYR SUCC H H2O 0 0 0 0 2 0 AKG FOR PYR SUCC H H2O 0 0 0 1 0 0 ETOH PYR SUCC H H2O 127 0 0 0 0 0 ETOH GLU PYR SUCC H H2O 0 42 0 0 0 0 AKG ETOH PYR SUCC H H2O 23 0 0 0 0 0AC ETOH FOR SUCC H H2O 5 0 0 5 0 0AC FOR GLU SUCC H H2O 0 0 0 0 2 0AC FOR SUCC H H2O 0 0 0 2 0 0AC AKG FOR SUCC H H2O 0 0 0 1 0 0 AKG ETOH SUCC H H2O 35 0 0 10 0 0 ETOH SUCC CO2 H H2O 46 0 0 0 0 0AC AKG SUCC H H2O 0 0 0 1 0 0AC GLU SUCC H H2O 0 0 0 0 2 0 ETOH GLU SUCC CO2 H H2O 0 42 0 0 0 0 GLU LAC SUCC H H2O 0 0 0 0 17 0 ETOH GLU LAC SUCC H H2O 0 42 0 0 9 0 AKG FOR LAC SUCC H H2O 0 0 0 9 0 0 AKG ETOH FOR SUCC H H2O 51 0 0 14 0 0 AKG ETOH SUCC CO2 H H2O 23 0 0 0 0 0 AKG LAC SUCC H H2O 0 0 0 9 0 0 AKG ETOH LAC SUCC H H2O 23 0 0 5 0 0 ETOH FOR PYR SUCC H H2O 5 0 0 0 0 0 ETOH FOR SUCC CO2 H H2O 5 0 0 0 0 0

Under anaerobic conditions

(there are many more I/O combinations for aerobic)

Glucose is the primary input

27 ways to make lactate and a proton from glucose

Only a fraction of ExPas give a growth like I/O signature

INCORPORATING REGULATIONProperty #4

COBRA View: Regulation is a Constraint (a restraint)

ExternalGlucose

ExternalSignal

(-)

(+) CRP

Mlc

Regulatory Proteins

ptsHI, crr(+)

X

glpK

(-)

Transcriptional Regulation

(-)

(+)

Altered NetworkCapabilities

(+)

(-)

Restricted Solution Space

Solution Shifts Biomass

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 2 4 6 8 10

Time (hrs)

Conce

ntr

atio

n (

g/L

)

ExpReg/MetMet OnlyKremling

10

10

10

10

10

10

1010 0.5

10 0.0

10-0.5

10-1.0

10-1.5

10-2.0

10-2.5

0 2 4 6 8 10

New Growth Behavior

Time (hrs)

Con

cent

rati

on (

g/L

)

Growth Prediction Shifts

Metabolic Network

Reconstruction:Reconstruction:DatabasesLiterature

General Solution Space

Constraints:Constraints:Mass Balance

S.v = 0

Capacityi ≤vi≤ i

Particular Solution

Glucose

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

Acetate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

ExpReg/MetMet

Biomass

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Time (hrs)

Concentration (g/L)

Met Only

Exp

Reg/Met

A.

8:00 8:27

8:00 8:27

8:00 8:27

Biomass

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Time (hrs)

Concentration (g/L)

Met Only

Exp

Reg/Met

A.

8:00 8:27

8:00 8:27

8:00 8:27

0 2 4 6 8 10 12

Acetate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

ExpReg/MetMet

A.

8:00 8:27

8:00 8:27

8:00 8:27

Acetate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

ExpReg/MetMet

A.

8:00 8:27

8:00 8:27

8:00 8:27

ExprFBAFBA

Acetate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

ExpReg/MetMet

A.

8:00 8:27

8:00 8:27

8:00 8:27

Acetate

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12

Time (hrs)

Concentration (m

M)

ExpReg/MetMet

A.

8:00 8:27

8:00 8:27

8:00 8:27

E x p

r F B A

F B A

Biomass

Co

ncen

tratio

n (

mM)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Time course of growth(phenotype)

Dynamics:Dynamics:Quasi Steady-

State Assumption

Integration

Time (hrs)

Con

cent

rati

on (

g/L

)

Genome

Extreme Pathways

Extreme Pathways and Regulatory ConstraintsFlu

x C

Flux B

Flux A

P2

P1 P3

P4

Consider the entire solution space of a metabolic network,

bounded by extreme pathways P1-P4…

P1

P2

P3

P4

P1 is not permitted due to regulatory constraints

One or more of these pathways

may not be feasible, depending on the environment and

corresponding regulatory effects…

Flu

x C

Flux B

Flux A

P1

P2

P3

P4

P2

P3

P4

This leads to a reduced solution

space bounded by fewer extreme

pathways

Covert et al., Journal of Theoretical Biology, 2002

Sample NetworkOxygen

A

B

ATP

Tc1

R1

C

Biomass

R4

R2a

Rz

2 NADH

0.2 C

ATPATPuseRatp

2 ATP3 NADH

10 ATP

2 ATP

3 D

3 E4 NADH

R6

R7

ATPNADH

O2

Rres

Fext

F

G

R5a,b

R3

Hext

Carbon2

Tc2 Tf

ThH R8a,b

2 ATP3 NADH

R2b

Dext

Eext

Td

Te

1 ATP2 NADH

TO2

Carbon1

Characteristics21 metabolic reactions4 regulatory proteins7 regulated reactions

Boolean representation

Regulation modeledCatabolite repression

Amino acid biosynthesisOxygen-dependent

Carbon storage

Analysis80 Extreme pathwaysForced growth output 5 environmental inputs25 = 32 environments

Example: C1, C2, O2 (ExPA)

6137

4117

1 2 3 4212223245 6 7 825262728910111229303132

1314151633343536181920 383940424344 626364

4546474865666768495051526970717253545556737475765758596077787980

All possible extreme pathways

Pathway reduction- Remove all inconsistent pathways

46

78746270

585042

3430

Environment-specific regulation: R5b, Tc2

o Environmental-dependent constraints

Environment-specificity: C1, C2 and O2

9101112293031321314151633343536171819203738394041424344616263644546474865666768

78

1 2 3 4212223245 6 7 825262728

495051526970717253545556737475765758596077 7980

o Environmental inconsistencies

Environment-independentregulation

9101112293031321314151633343536171819203738394041424344616263644546474865666768

1 2 3 4212223245 6 7 825262728

495051526970717253545556737475765758596077787980

o Environment-independent constraints

- Constrained solution spaceo 4 extreme pathwayso Corresponds to Phenotypic Phase Plane

Carbon1 Uptake Rate (mmol/gDCW/hr)

Oxygen Uptake Rate

(mmol/gDCW/hr)

Gro

wth

Rat

e (1

/hr)

Carbon1 Uptake Rate (mmol/gDCW/hr)

Oxygen Uptake Rate

(mmol/gDCW/hr)

Gro

wth

Rat

e (1

/hr)

P30

P34

P46

P50

LO

Covert et al., Journal of Theoretical Biology, 2002

Complex medium: Regulation of pathways

Environment-specific regulation: R2a, R5b, R7,

R8a, Tc2

9101112293031321314151633343536171819203738394041424344616263644546474865666768

78

1 2 3 4212223245 6 7 825262728

495051526970717253545556737475765758596077 7980

Environment-independentregulation

9101112293031321314151633343536171819203738394041424344616263644546474865666768

1 2 3 4212223245 6 7 825262728

495051526970717253545556737475765758596077787980

Overall (C1 + C2 + F + H) Uptake Rate

(mmol/gDCW/hr)

Oxygen Uptake Rate

(mmol/gDCW/hr)

Gro

wth

Rat

e (1

/hr) P49,52

P50,51

P38,45,48

P37,46,47

P33,36

P34,35

P4

P2

P12

P10

LO

P5,6,30,31

P8,29,32

P9

• Number of extreme pathways is only reduced to 26• More flexibility in the system

Covert et al., Journal of Theoretical Biology, 2002

Regulation for the core E. coli

-10 0 10 20 30 40 50 60 700

2

4

6

8

10

12

14Pathway Lengths - Regulation

pathway length

num

ber

of p

athw

ays

With regulation, the reactions D_LACt2, FUMt2_2, ICL, MALS, MALt2_2, MDH, NADH16, and SUCCt2_2 are inactivated under anaerobic conditions

w/regulation:118 ExPas are feasible

w/o regulation: 2006 ExPas are feasible Mean pathway length = 28.9

Median pathway length = 31

Summary• Basis vectors span a space and can describe all of

its contents• Some of the properties of P are:

– Pathway lengths– Reaction participation

• Co-sets

– I/O redundancy characteristics• Cross talk

– Incorporation of regulation• Shrinking the space and excluding possible states

EXTRAS

Reaction participations in the simple example