Mathematical Modelling of Metabolic Networks

with Gene Regulation

Juan A. Asenjo

Chilean Academy of Sciences

IANAS, InterAmerican Network of Academies of Science

IAP, the Global Network of Science Academies

Centre for Biotechnology and Bioengineering (CeBiB)

University of Chile www.cibyb.uchile.cl; www.icdb.uchile.cl

Landmarks of Biological Evolution on Earth

Trips of Darwin: 1831-1836

Jemmy Button

Bacteria and Plasmids

Chromosome

Plasmid

Bacteria

Plasmids

Poration

“Recombinant” Bacteria with human DNA in Plasmids (e.g. insulin)

Systems Biology

Holistic Description of Cellular Functions

Connection

of "Modules"

Modular Aggregation

of Components

Single Component Analysis

Functional Analysis

Metabolic Networks

Regulatory Networks

Signalling Networks

Biological Information/Knowledge

Deductive

Inductive

Top-Down Bottom-Up

Centre for Biotechnology and Bioengineering

Main Research Activities

• Five interdisciplinary components:

• Metabolomics and Metabolic Engineering

• Protein Engineering

• Mathematical Modelling

• Bioinformatics

• Molecular Genetics, Extremophiles and Ecophysiology

Mathematical Modelling and Metabolic Engineering

Metabolomics

Metabolic Flux

Analysis

GLUCGLUC

GLUC6PGLUC6P

FRUC6PFRUC6P

3PG3PG

GAPGAP

PIR PIR

PEPPEPACETACETEtOHEtOH

ACAC

RIBU5PRIBU5P

XIL5PXIL5PRIB5PRIB5P

GAPGAPSED7PSED7P

FRUC6PFRUC6P

aaaa

aaaa

aaaa

aaaa

aaaaaaaaE4PE4P

CARBCARB

ATP ADPATP ADP

RNARNA

OO22EE OO22

COCO22 COCO22EE

u2

u3

u5

LIPLIP

AcCoAAcCoAmitmit

AcCoAAcCoAcitcit

FUMFUM AKGAKG

SUCCoASUCCoASUCSUC

MALMAL ISOCITISOCIT

OACOAC

SODSOD

SODSOD

SODSOD

SODSOD

SODSOD

PROTPROTPROTPROT

PROTPROT

PROTPROT

PROTPROT

u6

u7

u9

u13

u11

u10

u10

u76

u77

u 70-aa

OAC

u69

u71-aaOAC

u17

u16

u15

u14

u73-AcCoA

u30

u70-aaAKG

u71-aaAKG

u70-aaPIR

uPEP

uPIR

u74

u31

u3P G

u28

u27

u26

uE4P

u19 u20

u21

u22

u23

u18 u1

u25

u71-aaPIR

u70-aa3PG

u71-aaPE P

u70-aaPE P

u71-aa3PG

u71-aaE 4P

u70-aaE 4P

u70-aaRIB 5P

u71-aaRIB 5P

u72-nuOAC

u72-nuRIB5P

u72-nu3P G

NHNH44EE NHNH44

u78

LIPLIP

u 73-G

AP

PROTPROTaaaa

RNARNA SODSOD

nunu

uOAC

nunu

uRI B5P

aaaa

uAc CoAci t

u71-aaAcCoA

u70-aaAcCoA

uAK G

RNARNA

nunu

GLICGLIC

AcCoAAcCoAcitcit

u24

u75

u4

u8

Gonzalez, R., Andrews, B.A. Molitor, J.

and Asenjo, J.A. (2003) Biotechnol.

Bioeng., 82, 152-169.

dX/dt = S v - bdX/dt = S v - b

in SS: S v = b in SS: S v = b or or S r = 0 S r = 0 àà SScc r rcc + S + Smm r rmm = 0 = 0

Metabolic Flux AnalysisMetabolic Flux AnalysisMetabolic Flux BalanceMetabolic Flux Balance

AA

EE

BB

CC

DD FF

nn11

nn33

nn22

nn55

nn44

S r=0=S r=0=

1-0100D

01-010C

001-1-1B

54321 nnnnn

5

4

3

2

1

n

n

n

n

n

100D

010C

1-1-1B

321 nnn

3

2

1

n

n

n

1-0D

01-C

00B

54 nn

5

4

n

n

+

SS StoichiometricStoichiometric Matrix Matrix

rr Rate (Flux) vectorRate (Flux) vector

cc CalculatedCalculated

mm MeasuredMeasured

0

3

6

9

12

15

0 9 18 27 36 45

Tiempo, h

Glu

cosa

0.0

0.5

1.0

1.5

2.0

2.5

Cél

ula

s, E

tanol y S

OD

[GLUC] g/L [X], g/L

[SOD] g/L [EtOH] g/L

Fermentation Profiles: strain P+

0.0

0.3

0.6

0.9

1.2

0 9 18 27 36 45

Tiempo, h

Pro

teín

a T

ota

l y

Car

bo

hid

rato

s T

ota

les

0.00

0.06

0.12

0.18

0.24

RN

A T

ota

l

[CARB] g/L

[PROT] g/L

[RNA] g/L

Profiles of Cell Components: strain P+

P+ GLUC

GLUCGLUC

GLUC6PGLUC6P

FRUC6PFRUC6P

3PG3PG

GAPGAP

PIR PIR

PEPPEP

ACETACETEtOHEtOH

ACAC

RIBU5PRIBU5P

XIL5PXIL5PRIB5PRIB5P

GAPGAPSED7PSED7P

FRUC6PFRUC6P

aaaa

aaaa

aaaa

aaaa

aaaa

aaaa

E4PE4P

CARBCARB

ATP ADPATP ADP

RNARNA

OO22EE OO22

COCO22 COCO22EE

3.844

4.169

6.256

LIPLIP

AcCoAAcCoAmitmit

AcCoAAcCoAcitcit

FUMFUM AKGAKG

SUCCoASUCCoASUCSUC

MALMAL ISOCITISOCIT

OACOAC

RNARNA

GLICGLIC

SODSOD

SODSOD

SODSOD

SODSOD

SODSOD

PROTPROTPROTPROT

PROTPROT

PROTPROT

PROTPROT

6.151

6.122

1.470

8.850

3.564

0.079

8.988

0.025

0.121

0.102

0.166

0.097

0.023

0.069

0.029

0.138

0.208

2.232

0.105

0.137

4.130 4.267

0.029

0.234 0.325

0.177

0.148

0.559 4.611

0.247

0.017

0.048

0.004

0.025

0.028

0.004 0.025

0.006 0.006

0.022

0.042

0.019

NHNH44EE NHNH44

0.724

LIPLIP

0.002

PROTPROTaaaa

RNARNASODSOD

nunu

nunu

0.174

nunu

0.057

aaaa0.063

0.014

0.046

1.470

1.470

1.470

1.345

1.3491.349

1.397

1.397

0.177

P+ GLUC

GLUCGLUC

GLUC6PGLUC6P

FRUC6PFRUC6P

3PG3PG

GAPGAP

PIR PIR

PEPPEP

ACETACETEtOHEtOH

ACAC

RIBU5PRIBU5P

XIL5PXIL5PRIB5PRIB5P

GAPGAPSED7PSED7P

FRUC6PFRUC6P

aaaa

aaaa

aaaa

aaaa

aaaa

aaaa

E4PE4P

CARBCARB

ATP ADPATP ADP

RNARNA

OO22EE OO22

COCO22 COCO22EE

3.844

4.169

6.256

LIPLIP

AcCoAAcCoAmitmit

AcCoAAcCoAcitcit

FUMFUM AKGAKG

SUCCoASUCCoASUCSUC

MALMAL ISOCITISOCIT

OACOAC

RNARNA

GLICGLIC

SODSOD

SODSOD

SODSOD

SODSOD

SODSOD

PROTPROTPROTPROT

PROTPROT

PROTPROT

PROTPROT

6.151

6.122

1.470

8.850

3.564

0.079

8.988

0.025

0.121

0.102

0.166

0.097

0.023

0.069

0.029

0.138

0.208

2.232

0.105

0.137

4.130 4.267

0.029

0.234 0.325

0.177

0.148

0.559 4.611

0.247

0.017

0.048

0.004

0.025

0.028

0.004 0.025

0.006 0.006

0.022

0.042

0.019

NHNH44EE NHNH44

0.724

LIPLIP

0.002

PROTPROTaaaa

RNARNASODSOD

nunu

nunu

0.174

nunu

0.057

aaaa0.063

0.014

0.046

1.470

1.470

1.470

1.345

1.3491.349

1.397

1.397

0.177

P+ GLUC

GLUCGLUC

GLUC6PGLUC6P

FRUC6PFRUC6P

3PG3PG

GAPGAP

PIR PIR

PEPPEP

ACETACETEtOHEtOH

ACAC

RIBU5PRIBU5P

XIL5PXIL5PRIB5PRIB5P

GAPGAPSED7PSED7P

FRUC6PFRUC6P

aaaa

aaaa

aaaa

aaaa

aaaa

aaaa

E4PE4P

CARBCARB

ATP ADPATP ADP

RNARNA

OO22EE OO22

COCO22 COCO22EE

3.844

4.169

6.256

LIPLIP

AcCoAAcCoAmitmit

AcCoAAcCoAcitcit

FUMFUM AKGAKG

SUCCoASUCCoASUCSUC

MALMAL ISOCITISOCIT

OACOAC

RNARNA

GLICGLIC

SODSOD

SODSOD

SODSOD

SODSOD

SODSOD

PROTPROTPROTPROT

PROTPROT

PROTPROT

PROTPROT

6.151

6.122

1.470

8.850

3.564

0.079

8.988

0.025

0.121

0.102

0.166

0.097

0.023

0.069

0.029

0.138

0.208

2.232

0.105

0.137

4.130 4.267

0.029

0.234 0.325

0.177

0.148

0.559 4.611

0.247

0.017

0.048

0.004

0.025

0.028

0.004 0.025

0.006 0.006

0.022

0.042

0.019

NHNH44EE NHNH44

0.724

LIPLIP

0.002

PROTPROTaaaa

RNARNASODSOD

nunu

nunu

0.174

nunu

0.057

aaaa0.063

0.014

0.046

1.470

1.470

1.470

1.345

1.3491.349

1.397

1.397

0.177

Microarrays of Gene Expression

- GeneChip from Affimetrix

(6,871 genes of S. cerevisiae)

eglu

etoH

cos eglu

etoH

cos eglu

etoH

cosP-

EtOH/Gluc

Glucose Ethanol

Central metabolic

pathway

82 genes

P- , Glucose Ethanol, Eth/Gluc

• 98 % of genes in central pathways are overexpressed

• 99% in PPP and biosynthetic pathways

e.g. fum 1 - 179x

• different from MFA (Met. Flux Anal.)

– μ = 40% lower (and also TCA cycle)

• Not possible to correlate in a direct function quantitative mRNA expression levels with cell function shown by MFA

–

• aminoacid synthesis pathways-highly overexpressed

e.g. asn1 – 433 x

gln 1 – 235 x

tkl 1 – 280 x

P -

Stat/EtOH

Ethanol Stationary

P+ / P-

Glucose

Microarrays of Gene Expression GeneChip from Affimetrix

(6,871 genes of S. cerevisiae)

Mathematical Modeling (continuous) of

metabolic networks with gene regulation

in yeast

GlcX0

F6P

G6P

GlcX

GlycDHAP

FBP

GlycX GlycX0

PPP CARB

PEP

BPG

Cit

ACoA

Pyr

Mal

Fum

Suc

Isocit

OAC

AKG

EtOHACA EtOHX EtOHX0

PROT

vPROT1

vinGlc

vstoragevPPP

vPGI

vPFK

vALD

vTIM

vlpGlyc

vGAPDH

vPK

vlpPEP

vPDH

vPDC vADH voutEtOH

vKGD

vPROT3

vPROT4

vPROT2

vCIT1

vPYC

vGlcTrans

GAP

vdifGlyc voutGlyc

vdifEtOH

vMDH2

vFUM

vSDH

vACO

vIDH

vFBP

vPCK1

vCIT2

CicloGlyoxy

vIDP

vALD2

AcetatvACS

vADH2

Glc

vHKATP

ADP

ATP

ADP ATP

ADP

NAD

NADH

ADP

ATP

ADP

ATP

NADH

NAD

NADH NAD

NADH NAD

CO2

ADP ATP

ATP ADP

NAD

NADH

ADP ATP

CoA

NAD

NADH

CoA

ATP

ADP

CO2

CoA

NAD

NADH

CO2

CO2

NADH

NAD

ADP

ATP

CO2

CoA

ATP

ADP

ADP ATP

CO2

NADH NADvconsNADH

ATP ADPvconsum

CO2 CO2X0VCO2

ADP

ATPO2

NADH NAD

Figure 1: Reaction network of the model

39 Fluxes

50 Enzymes

64 Genes

MalCicloGlyoxy

F6P

G6P

GlycDHAP

FBP

GlycX

PEP

BPG

Cit

ACoA

Pyr

Fum

Suc

Isocit

OAC

AKG

EtOHACA EtOHX

vCIT1

GAP

Acetat

Snf1

MIg1

Cat8

GlcX0

GlcX

Glc

Fluxes

Glucose induction Positive regulation: glycolytic genes.

Glucose repression Negative regulation: gluconeogenic genes.

Figure 2: Interactions between metabolic network and

regulatory genetic network.

- Continuous lines represent reactions in the metabolic

network

- Broken lines represent transcription factors or

enzymes

- Nodes represent transcription factors in the genetic

network.

Different colours represent different genetic

regulation mechanisms.

- Blue: Glucose repression (Negative regulation.

Gluconeogenic genes)

- Red: Glucose induction (Positive regulation.

Glycolytic genes)

v k[E][S]

v ek'[S]

dCS v

dt

Method in detail

enz1 enz2

d[E]K [mRNA] K [E]

dt

mrna1 mrna2

d[mRNA]K K [mRNA]

dt

r1

r2 r3

1 k [G]

k k [G]

1 inGlcv X0 XGlc Glc

X0vinGlc X

conv

[Glc ]K [Glc ]

K

2

GlcTransv XGlc Glc

vGlc1 X

vGlc2 X

K [Glc ]

K [Glc ]

3 HKv Glc ATP G6P ADP

1 vHKe K [Glc][ATP]

4 PGIv G6P F6P

vPGI,f vPGI,rK [G6P] K [F6P]

5 PFKv F6P ATP FBP ADP

1 vPFKe K [F6P][ATP]

6 FBPv FBP ADP F6P ATP

2 vFBPe K [FBP][ADP]

7 ALDv FBP GAP DHAP

vALD,f vALD,rK [FBP] K [DHAP]

8 TIMv DHAP GAP

vTIM,f vTIM,rK [DHAP] K [GAP]

9 GAPDHv GAP NAD BPG NADH

vGAPDH,f vGAPDH,rK [GAP][NAD] K [BPG][NADH]

10 lpPEPv BPG ADP PEP ATP

vlpPEP,f vlpPEP,rK [BPG][ADP] K [PEP][ATP]

11 PKv PEP ADP Pyr ATP

vPKK [PEP][ADP]

12 PDCv 2Pyr ACA CO

1 vPDCe K [Pyr]

13 ADHv ACA 2NADH EtOH 2NAD

21 vADHe K [ACA][NADH]

GlcX0

F6P

G6P

GlcX

GlycDHAP

FBP

GlycX GlycX0

PPP CARB

PEP

BPG

Cit

ACoA

Pyr

Mal

Fum

Suc

Isocit

OAC

AKG

EtOHACA EtOHX EtOHX0

PROT

vPROT1

vinGlc

vstoragevPPP

vPGI

vPFK

vALD

vTIM

vlpGlyc

vGAPDH

vPK

vlpPEP

vPDH

vPDC vADH voutEtOH

vKGD

vPROT3

vPROT4

vPROT2

vCIT1

vPYC

vGlcTrans

GAP

vdifGlyc voutGlyc

vdifEtOH

vMDH2

vFUM

vSDH

vACO

vIDH

vFBP

vPCK1

vCIT2

CicloGlyoxy

vIDP

vALD2

AcetatvACS

vADH2

Glc

vHK

NADH NADvconsNADH

ATP ADPvconsum

CO2 CO2X0vCO2

4,8910

4,8910

4,8910

0,2430,211

0,7490,437

3,899-0,648

00,648

3,899-0,648

3,8990

2,655-0,930

1,2440,281

1,2440,281

1,2440,281

4,272-3,065

4,272-3,065

03,065

4,2720

03,065

03,065

1,9110

4,2720

6,555-1,578

6,457-1,638

6,4570,054

0,1860

1,9111,259

1,9111,259

1,9111,259

1,8012,998

1,8012,998

1,8012,998

1,8012,998

01,806

01,806

01,692

0,0970,060

0,0800,054

0,0760,047

0,1100,067

2,0967,295

6,3517,100

10,4606,386

Distribution of metabolic fluxes at different growth

phases:

1) exponential growth on glucose (top values) and

2) exponential growth on ethanol (bottom values in italics).

Flux values are expressed in

mmol gr.cell hr

Determination of kinetic parameters and metabolite concentrations

g gV K,C f

e eV K,C f

Total of 72 equations and 120 unknowns in this system

2

2i,f i,f i,b i,b2 2 2 2w w k w l

i k lMFA,i

K r K rR 1 apriori C K 5 conservation

f

2

2 2GlcX0,g2 2 2 2w GlcX0,g GlcX0,g GlcX,g EtOHX0,e EtOHX0,e w GlcX0,e GlcX,e

conv

Cˆ ˆapriori C C C 0,995 C C C C

k

2 2 22

w ATP,g ADP,g ATP,e ADP,e NAD,g NADH,g NAD,e NADH,e CoA,g ACoA,g CoA,e ACoA,econservation C C C C C C C C C C C C

Weight coefficients

• ε (“preserved”: ATP + ADP; NAD + NADH; ) = 150

• φ (measured glucose + ethanol) = 10,000

• γ (intracellular concentrations) = 0.01

• δ (kinetic constants) = 0.0001

GlcX0

F6P

G6P

GlcX

GlycDHAP

FBP

GlycX GlycX0

PPP CARB

PEP

BPG

Cit

ACoA

Pyr

Mal

Fum

Suc

Isocit

OAC

AKG

EtOHACA EtOHX EtOHX0

PROT

vPROT1

vinGlc

vstoragevPPP

vPGI

vPFK

vALD

vTIM

vlpGlyc

vGAPDH

vPK

vlpPEP

vPDH

vPDC vADH voutEtOH

vKGD

vPROT3

vPROT4

vPROT2

vCIT1

vPYC

vGlcTrans

GAP

vdifGlyc voutGlyc

vdifEtOH

vMDH2

vFUM

vSDH

vACO

vIDH

Glc

vHK

NADH NADvconsNADH

ATP ADPvconsum

CO2 CO2X0vCO2

Figure 4: Qualitative behavior of the model during the

exponential growth on glucose phase. Arrows

correspond to active metabolic fluxes.

F6P

G6P

GlycDHAP

FBP

GlycX GlycX0

PPP CARB

PEP

BPG

Cit

ACoA

Pyr

Mal

Fum

Suc

Isocit

OAC

AKG

EtOHACA EtOHX EtOHX0

PROT

vPROT1

vstoragevPPP

vPGI

vALD

vTIM

vlpGlyc

vGAPDH

vPK

vlpPEP

vADH2 voutEtOH

vKGD

vPROT3

vPROT4

vPROT2

vCIT1

GAP

vdifGlyc voutGlyc

vdifEtOH

vMDH2

vFUM

vSDH

vACO

vIDH

vFBP

vPCK1

vCIT2

CicloGlyoxy

vIDP

vALD2

AcetatvACS

NADH NADvconsNADH

ATP ADPvconsum

CO2 CO2X0vCO2

Figure 5: Qualitative behavior of the

model during the exponential growth on

ethanol phase, without glucose.

Arrows correspond to active metabolic

fluxes.

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35

Glu

cose

[gr/

l]

Time [hr]

Glucose Regulation

Glucose

Enzyme HK

Enzyme FBP

Re

lative C

on

cen

tration

of

Enzym

e

Figure 6a: Enzymatic expression regulated by glucose during the batch

fermentation. HK: Hexokinase (induced); FBP: Fructose bisphosphatase (repressed).

Figure 6b: Model simulation. Biomass, glucose and ethanol profiles during a whole

batch fermentation. Profiles given by the model (continuous lines) and its

comparison to the experimental. Experimental results are from González et al., 2003.

Mathematical Modeling of metabolic

networks with gene regulation in

Escherichia coli

Figure 1: Reaction network of the model

43 Fluxes

47 Enzymes

52 Genes

Figure 2: Interactions between metabolic network and

regulatory genetic network.

- Continuous lines represent reactions in the metabolic

network

- Broken lines represent transcription factors or

enzymes

Different colours represent different genetic

regulation mechanisms.

- Blue: Glucose repression (Negative regulation. Lac

operon, Acetate consumption)

- Red: Glucose or Galactose induction (Positive

regulation)

Distribution of metabolic fluxes at different growth

phases:

1) exponential growth on glucose (top values),

2) exponential growth on lactose (middle values) and

3) exponential growth on galactose (bottom values).

Flux values are expressed in

mmol gr.cell hr

Figure: Model simulation. Biomass, glucose, lactose, galactose and acetate profiles

during a whole batch fermentation. Profiles given by the model (continuous lines)

and its comparison to the experimental (Bettenbrock et al, 2006).

Metabolic Network with Gene Regulation

• The model was able to simulate a fermentation of E. coli during the exponential growth phase on glucose and the exponential growth phases on lactose and on galactose using only one set of kinetic parameters.

• The “Reverse Engineering” methodology allowed the obtention of realistic fluxes and concentrations using only 115 equations in a system with 177 unknowns.

• All fluxes in the model follow the behaviour shown by MFA obtained from experimental results.

• Furthermore, intracellular metabolite concentrations obtained by the model are in the range of those obtained experimentally by previous authors.