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.