The thesis is devoted to the study and application of constraint-based metabolic models. The objective was to find simple ways to handle the difficulties that arise in practice due to uncertainty (knowledge is incomplete, there is a lack of measurable variables, and those available are imprecise). With this purpose, tools have been developed to model, analyse, estimate and predict the metabolic behaviour of cells. Details: http://science.ensilicio.com

1. Interval and Possibilistic Methods
for Constraint-Based
Metabolic Models
Author
Francisco Llaneras Estrada
Supervisor
Jesús Picó i Marco
1

2. This thesis is focused on
Use models of living cells
considering uncertainty
1. Models useful
! Science and industry
2. Cells are complex
3. Uncertainty is present
Pichia pastoris cells
(M. Tortajada)
2

3. Outline
Context Constraint-based models
2 papers of living cells
(>30 citations)
Possibilistic ?
framework MFA
Methods
5
? ?
Estimation 0
4 papers
1
(>30 citations)
Applied to...
MFA 60 CO2
Monitoring
0
0h 120h 192h
FBA
?
? ? ?
Prediction ?
?
s v1
Model
1
Cell
wa
ll
1
v4
Validation
5
6 4
v5 p
1
v8 v6 v4
2
3
v2
v3 s 2
3

4. Outline
Context Constraint-based models
2 papers of living cells
(>35 citations)
Possibilistic ?
framework MFA
Methods
5
? ?
Estimation 0
4 papers
1
Applied to...
(>35 citations)
MFA 60 CO2
Monitoring
0
0h 120h 192h
FBA
?
? ? ?
Prediction ?
?
s v1
Model
1
Cell
wa
ll
1
v4
Validation
5
6 4
v5 p
1
v8 v6 v4
2
3
v2
v3 s 2
4

5. Constraint-based models
of living cells
«Use available knowledge as constraints
v3
to distinguish what is possible v1
from what is not.» v2
N v=0 + v 0 + v vM
Stoichiometry Irreversibility Capacity
[2] Llaneras F, Picó J (2008). Stoichiometric modelling of the cell metabolism.
Journal of Bioscience and Bioengineering. (38 citations)
[5] Llaneras F, Picó J (2010). Which metabolic pathways generate and
characterise the flux space? J. Biomedicine and biotechnology.
5

6. (1) Cell metabolism as a metabolic network
v1 Metabolites ~ Nodes ! s, p, M
s1
Ce
ll w
all
M1
Reactions ~ Arcs
v4
M5 reaction1 : s1 !v! M1
1
"
6
v5 p1 reaction 4 : M1 !v! M4 + M5
4
"
M4
v8 v6 v7 …
M2
M3
Fluxes ~ one per arc ! ! v
v2 (reaction rates)
v3 s2
Key variables
6

7. v1
s1
Ce
ll w
all
(2) steady-state assumption
M1
v4
M5
6
v5 p1
! ! for some intracellular reactions v8
M4
v6 v7
M2
M3
v2
v3 s2
Do not account for fast
intracellular dynamics
% v1 (
!
dM1
dt
= v1 (.) ! v4 (.) ! v8 (.) ! µ (.) " M1 v1 ! v4 ! v8 = 0 ' *
# v2 *
dM 2
= v2 (.) ! v6 (.) ! µ(.) " M 2 v2 ! v6 = 0 " N·' =0
dt ' ! *
# ' *
dM 3
= ...
... $ v8 )
dt &
Stoichiometric
equation
7

8. Knowledge as constraints,
v1
s1
Ce
ll w
all
M1
v4
a space of possible flux states
M5
6
v5 p1
M4
v8 v6 v7
M2
M3
v2
v3 s2
v3
v1
v2
N v=0 + v 0 v vM
Stoichiometry Irreversibility Capacity
Constraint-based model # N·v = 0
«To distinguish what is possible v !! : $
n
from what is not.» % D·v " 0
8

9. Ap
pli
c
Example: Pichia pastoris model at
io
n
A yeast for the expression of recombinant proteins
MET
CO2
GLCcyt O2 32
1 O2 H2O2
21
RU5Pcyt G6Pcyt
33
CO2 HCHO
23 22 2
Metabolic network (!small) R5Pcyt
24
XU5Pcyt F6Pcyt
3
XU5Pcyt
• 45 metabolites S7P
25
GA3P 26 FBPcyt
4 34
• 36 balanced metabolites F6P E4P GAPcyt
5
DHAPcyt
35
DHAcyt
• 44 reactions and fluxes
• 8 degrees of freedom 31 AKGmit AKGcyt
27
NADH
36 O2E iO2
6
NAD
37 GLCE GLCcyt
38 iCO2 CO2 E
28
39 ETHcyt ETHE PG3cyt GOLcyt O2 H2O
40 GOLcyt GOLE
41 CIT(E) ICITmit 7
42 PYR(E) PYRcyt
43 METE METcyt
PEPcyt PYRmit
HCOAmit 16, 17
14 ICITmit
8 30 AcCoAmit
CO2 CO2 15 aKGmit
10
11 9 29
ETHcyt ACDcyt PYRcyt OACcyt OACmit
18
CO2 20
Based on network by 12 HCOAcyt
13 MALmit SUCmit
(Dragosits, 2009) ACEcyt AcCoAcyt 19
9

10. Ap
pli
c
Example: Pichia pastoris model at
io
n
A yeast for the expression of recombinant proteins
Irreversible 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1
Reaction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 O2 GLC CO2 ET GOL Cit Pyr MET BIO
1 GLCcyt -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
2 G6Pcyt 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,036
3 F6Pcyt 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,036
4 FBPcyt 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 DHAPcyt 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
6 GAPcyt 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
" N·v = 0
7 PG3cyt 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,017
8 PEPcyt 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,025
9 PYRcyt 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0
10 GOLcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
MOC = #
11 NADPHcyt 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,199
12 iCO2 0 0 0 0 0 0 0 0 -1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0,018
13 RU5Pcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14 R5Pcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,011
% D·v ! 0
15 XU5Pcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0
$
16 S7Pcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 E4Pcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,015
18 OAAcyt 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,024
19 PYRmit 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,029
20 ACCOAmit 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,008
21 OAAmit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,001
22 ICITmit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0
23 NADH 0 0 0 0 0 1 0 0 0 0 -1 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 -1 -1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0,063
24 AKGmit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
25 NADPHmit 0 0 0 0 0 0 0Stoichiometry (N)
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 Reactions reversibility (D)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,056
26 AKGcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -0,027
27
28
SUCmit
MALmit
0
0
0
0
0
0
0
0
0
0
0
0
0
036"44 matrix
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
-1
1
0
-1
0
0
0
0
0
0
0
0
0
0
0
0 12 reversible ones
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
29 ACDcyt 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
30 ACEcyt 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
31 ACCOAcyt 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,003
32 iO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 -0
33 EtOH cyt 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0
34 MeOHcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0
35 HCHOcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0
36 DHAcyt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0
10

11. Constraint-based models
are being used in different ways
Scientific works using these keywords
(by Google Scholar)
600
Metabolic flux analysis
(a) Simulate genetic modifications
400
(b) Study the modelled organisms (EMs)
Constraint-based models
(c) Estimate the cells behaviour (MFA) balance analysis
Flux
200 (d) Predict the cells behaviour (FBA)
Elementary modes
0
1996 1998 2000 2002 2004 2006 2008
11

12. Context Constraint-based models
of living cells
Possibilistic MFA
?
Methods framework Estimation
5
? ?
0
1
Applied to...
MFA 60 CO2
Monitoring
0
0h 120h 192h
FBA ?
?
?
Prediction ?
?
?
s1
Model
v1
Cel
lw
al l
1
Validation
v4
5
6 4
v5 p1
v8 v6 v4
2
3
v2
v3 s2
12

13. Pr
ob
lem
Goal
Handle uncertainty in constraint-based models
with computational efficiency
Means!! Intervalar framework (FS-MFA)
! ! ! Possibilistic framework (Poss-MFA)
13

14. We are dealing with
Constraint-satisfaction problems
Model constraints e.g. measurements Ideally CSP are easy
" N·v = 0 ! vm = wm
#
MEC = "
In practice, uncertainty
MOC = #
% D·v ! 0
$ #
$ » lacking knowledge !(many solutions)
» imprecision ! ! (no solution)
Limits of traditional approaches
(a) Only point-wise solutions Possibilistic
(b) Strong assumptions (normality) framework
(c) Computationally intensive (flexible & efficient)
14

15. Poss. framework: grade constraint-satisfaction
Possibility theory (Dubois, 96)
Model constraints e.g. measurements uncertainty via slack variables...
... weighted in a cost index
" N·v = 0 ! v m = w + ! 1 " µ1
$
# vm = wm
&
MOC = # MEC = "
MEC %
% D·v ! 0
$ #
&
$
'
! 1 , µ1 # 0
J (! ) = " ·#1 + $ ·µ1
1 For each solution, ! = {v, " , µ }
Possibility !(!)
grades constraint satisfaction
2
! (" ) = e # J(" )
Possibility calculus cast as
efficient optimisations (LP)
! (" ) : # $ [0,1]
" inf J(# )
! = 0 » Contradiction
! = 1 » Agreement Poss. of event, ! (A) = e # $A
15

16. Context Constraint-based models
of living cells
Possibilistic ?
framework MFA 5
Methods
? ?
Estimation
0
1
Applied to...
MFA 60 CO2
Monitoring
0
0h 120h 192h
FBA ?
?
?
Prediction ?
?
?
s1
Model
v1
Cel
lw
al l
1
Validation
v4
5
6 4
v5 p1
v8 v6 v4
2
3
v2
v3 s2
16

17. Pr
ob
lem
Metabolic flux analysis (MFA)
{Model + Measures} to estimate fluxes
?
Traditional MFA approaches
5
? ? (a) Point-wise estimates
(b) Require many measures
0
1 (c) Strong assumptions
(d) Computationally intensive
Proposal! Poss-MFA
! ! ! Flexible and efficient
17

18. ?
How Poss-MFA works?
5
? ?
0
1
A model " N·v = 0
MOC = # ! A Estimate fluxes
#
MOC $ D·v ! 0
%
The most possible v
Uncertainty # " MOC
$
"
Measurements $ v m = w + ! 1 " µ1
& min J s.t. #
% MEC
$
! , µ ,v
MEC = %
MEC & ! 1 , µ1 # 0
'
#
# B
Consistency analysis
J = ! ·"1 + # ·µ1 {MOC vs MEC}
Poss.
Framework ! (v, " , µ ) = e # J( " , µ )
$ Maximum possibility
! mp = exp("J min ) #[ 0,1]
T
18

19. Other richer estimates ... all efficient
(LP problems)
«Similar» to Monte Carlo
1
Distributions
By cuts of for !=0.1, 0.2...
res
Marginal possibility
su
#MOC ! MEC
Mea
Poss. vi, p = max vi s.t. $
M
% J < " log p
Conditional vi, p = min vi s.t. ...
m
possibility
0
vx
Value of v Intervals
Of conditional possibility "
%MOC " MEC
0.1, 0.5 and 0.8 possibility intervals vi,! = max vi s.t. &
M
' J # log $ mp < # log !
vi,! = min vi s.t. ...
m
19

20. Ap
pli
c at
Case study: C. glutamicum cultures io
n
Irreversible 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0
Reaction GLU 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 O2 NH3 BIO LYS TRE CO2 H2O NADPH
1 AC 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 AKP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0
(ACCOA) 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -332 0 0 0 0 0 0 0 0
3 AKG 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 364 0 0 0 0 0 0 0 0
4 ALA 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -54 0 0 0 0 0 0 0 0
" N·v = 0
5 ASP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 -80 0 0 0 0 0 0 0 0
6 ATP (ADP) 0 -1 0 -1 1 0 1 0 0 0 0 0 1 0 1 0 -1 0 0 0 0 0 0 0 0 4 2 0 -1 0 0 -1 -3000 0 0 0 0 0 0 0 0
7 BIO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 0 0 -1 0 0 0 0 0
8 CO2 0 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 143 0 0 0 0 0 -1 0 0
MOC = #
9 COA 0 0 0 0 0 0 0 0 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 332 0 0 0 0 0 0 0 0
10 E4P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 FADH (FAD) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
12 FRU6P 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -7 0 0 0 0 0 0 0 0
% D·v ! 0
13 G3P 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -150 0 0 0 0 0 0 0 0
14 GAP 0 0 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 -13 0 0 0 0 0 0 0 0
$
15 GLC6P 1 -2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -21 0 0 0 0 0 0 0 0
16 GLUM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -25 0 0 0 0 0 0 0 0
17 GLUT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -446 0 0 0 0 0 0 0 0
18 H2O 0 0 0 0 0 1 0 0 0 0 -1 0 0 -1 0 1 0 0 1 -1 0 0 0 0 0 2 2 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0
19 LAC 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
20 LYSI 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -33 0 0 0 -1 0 0 0 0
21 MDAP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0
22 NADPH (NADP) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 -1 2 0 0 0 0 0 0 0 0 -2 0 0 0 -100 0 0 0 0 0 0 0 1
24 NH3 Stoichiometry (N)
23 NADH (NAD) 0 0 0 0 1 0 0 -1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Reactions reversibility (D)
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
25 O2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
26 OAA
27 PEP
36"41 matrix
0 0 0 0 0 0 0 0 1 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0
-1 0 0 0 0 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 reversible ones 0
-52
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
28 PYR 1 0 0 0 0 0 1 -1 0 -1 0 0 0 0 0 0 0 -1 -2 0 0 0 0 0 0 0 0 0 -1 0 0 0 -30 0 0 0 0 0 0 0 0
29 RIB5P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 -126 0 0 0 0 0 0 0 0
30 RIBU5P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
31 SED7P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
32 SUC 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
33 SUCCOA 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0
34 TREHAL 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0
35 VAL 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -40 0 0 0 0 0 0 0 0
36 XIL5P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
(Vallino et al., 1996) 20

21. Ap
pli
(rich) Measurements are imprecise, c at
io
n
but Poss-MFA estimates are rich
1
!
0
-50 v1 100 v8
1
!
0
v9 v16
1
!
0
v17 v24
1
!
0
v25 v32
1
!
0
v33 40 80 0 40 0 0.08 -5 5 40 80 v40
6 measured fluxes Poss. distributions Box-plots cond. possibility
v1, v34, v36, v37, v38, v39 Poss. distributions ! = 1, 0.8, 0.5, 0.1
(measured fluxes)
21

22. Ap
pli
(reliable) Poss-MFA gives estimates, c at
io
n
even if measurements are scarce
70 3 measurements
6 measurements (with Poss-MFA)
vGLC (1), vCO2 (39) vBio (36)
60
6 required by trad. MFA
50
Flux value [mM/h]
Flux value [mM/h]
40
30
20
10
0
0
1
1 3
3 5
5 7
7 9
9 11
11 13
13 15
15 17
17 19
19 21
21 23
23 25
25 27
27 29
29 31
31 33
33 35
35 37
37 39
39
Flux [#]
22