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.
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PA.. Jourdain. L. Lachkar.
from a comple:!:with colchicine and a stathminlike
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(R. COKhdc.P. ptrezMelero. C; Pelaez, R.: Medarde.M.Bioorg.
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........ t'J., iJc IJft ON, Dl.~[)c;,
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7D*C  A Free Database of Commercially AvailableCom'
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MultiObjective Optimization of Biological Networ
Prediction of IntracellularFluxes
JoseOscar H. Sendtn, Antonio A. Alonso, and Julio R. Bange
ProcessEngineering Group
lnstitutode Investigaciones Marinas (CSIC)
aEduardoCabello 6, 36208, Vigo, Spain
osendin@iim.csic.es, antonio@iim.csic.es,julio@iim.csic.es
Abs~act.In this contribution, we face the problem of predicting intracellular fluxes using a
multicriteria optimization approach. i.e. the simultaneous optimization of two or more cellular
~ncti?ns. Based on Flux Balance Analysis, we calculate the Pareto set of optimal flux distribu
uons ill E. coli for three objectives: maximization of biomass and ATP, and minimization of
mtracellular fluxes. These solutions are able to predict flux.distributions for differenl environ
mental conditions without requiring specific constraints, and improve previouS published
:esults.We thus illustrate lhe usefulness of multiobjective optimization for abetter understand
mgofcomplex biological networks.
Keywords: Multiobjective optimization, Pareto front, Flux Balance Analysis.
1 Introduction
Intracellular
su~ption
logtcal objective. Network capabilities
by ~sing, for example, Metabolic
wh~ch. can be found
stolC~ometricmodel of the network,
equattons at steadystate
(o~Jectives) must be defined, as well as other possible additional constraints. to find a
uruque solution.Successfulapplications
metabolic capabilities(Edwardset al. 200 1) and the genomescale
fluxes in biochemical
that cellular systems operate in an optimal way with respect to a certain bio
and flux distributions
flux Balance Analysis (FBA), the fundamentals
in e.g. (Vannaand Palsson
but since the linear system of mass balance
is generally underdetermined,
networks can be calculated
in silico under the as
have thus been predicted
of
1994). FBA only requires the
appropriate cellular functions
of FBA include the prediction of E. coli
reconstrUction of
the met~bolic network in S. cerevisiae
In t~IS context, a particularly
cen~y III detail (Schuetz et al. 2007; Nielsen 2007) concerns the principles behind the
?ptImal biochemicalnetwork operation,i.e.: "which are the criteria being optimized
l~ these systems?"By far, the most common objective considered is the maximiza
tiOn of. growth (or biomass yield),although other criteria, such as maximization
ATP yIeld (van Gulik and Heijnen1995) or minimization
flux (Bon'
!
. ~os et a . 1996), have been proposed for different systems an
Smc~ neither we nor nature have a single goal, a more desirable and realistic ap
p~oa.chISto consider the simultaneouSoptimization
flictmg. As a consequence,the solution will not be unique but instead this strategy
(Forster et al. 2003).
interesting question which have been addressed re
of
of the overall intraceHular
d d· . con ltlOns.
of twO or more criteria, often con
J.M. Corchado elaI (Ed
spnngerlink
).
......... s.. IWPACBB 2008,ASC 49. pp. 197205,2009 .
@SpringerVerlag BerlinHeidelberg 2009.com
Page 2
198
J.O.H. Sendrn, A.A. Alonso, and lR. Banga
. in the 0 timal tradeoffs
optimization is better ableto cope
bi \Y(Handl et al. 2007), bUIfewap
10 og
with other scientific and engmeenn
between the differ·
will result in a set o~ sol~tlo.ns rep;e~:~~}riteriar
ent objectives.MultlobJectIve
I·
f
odels from systems
with the comp exny
0 m
lications are found in literature 10 compallson
(0
.. i
..."
helds
(Sendin
et aL 2006).
we face the solution
from FBA. By simu1taneou.sl~ o.p~l.m~z~~~h:~
the aim of this study is to test the hcapabl~lrtol~mental conditions
. ddently from
lular fluxes
additional,casedependentand P?tentta
solution.After presenti~g t~e basl~ c~n~~ptsoli
centralcarbonmetabolIsm
10 Esc erJ~. '~ c
I·
. ciples can be generally
rna try pnn
.
f
hiobjective
m~ .'
optimization
veral common cellular functIons.
approach for predicting.inuac.el.
and without lmposfi"il
diuoning the lila
constraints can
and methods in MO, we will consider
as a case study to assess whetheropu
(MO) prob~ems
In this work
derived
0
t e eOVI
III epeo '.11
artificial
I.
the
y
app re .
AI sis (MOFBA)
2 MultiObjectiveFluxBalance na y
.d Basic Concepts
2 1 Problem Formulation an
.
AssumlOga bIOIOgl~al~e;:'~bjective
model
thfl
as findtnge
ux
junctionssubject
d If a stOlchlometnC
blem can be staled
Analys;s ~~ooor more obJeCll\'e
y
k operatmgat steadystate,
Flux Balance
h optlInlZeS simultaneouS
w IC.
s balanceequations
an
IS available. t euh
dlstnbutlon
th
toe mas.
MaxiMin Z ~ [Z,( v) Z, (v)
zJv))'
(I)
(21
Subject to:
s· v :::0(3)
L
< v"
V
:0:; V 
s (linear
.). Sisthe(m~
abolites an r
I~pe~er bounds';
\1
onhnear,
d
th<
..as we
of intracellul~
with lower an
d depending on the pr
n be Impose
I dge about the system.
data and the know ~ 'ves differs from tra lather,
tion of multiple obJectl
that if the objectives~e l~t neous1y all of them·
oPtimi~atl~ns~~ution whichopti~zes
as n
Z is the vector
.
r) stoichlOmetn~
of n. objective
atflX, where m
m.. the vector
f reactions,
v IS
o.
1
Additional
vU• respectIvey.
the availableexpen
Simultaneous
objective
fui:c:~~nnumber an odd
. obleOla
of r fluxe~,
.
numberconstramts ca d.tionalsingle'
'mental
.
oP.unu.za
there
.. conflict with each The key
slmU adoes not e~.ist
will not be a uruhq~of paretooptimal
Pt here ISt a
conce.• 'n the solution
A potnt
V.
I
anotherfeasible po
. In other
:~~iy
be a~hie~;~~~
MO problem
other
better than an
so~udtlOnbeparetooptimal
e is satto
Z (v*) for all i::: ,.. "
j
ense that irnprovemen
d v. is optimal
10 the s
s.. one or more
if thdeZre(Vl < Z/v$)
n an
j
t in one a J
the solution 0
for
1
b'ecuve
spacfa
iot v such that Z;(v). ~
of the others. T~USh' n be saidto be
f WhlC ca war
:ro;~~:~~i~lIY
family
infini;:~t~_~~~:~
is known as
~et or pareto front.
IS a
'T'l.;
IIUS
.
MultiObjective Optimization of Biological Networks
199
U
Methods for MultiObjectiveOptimization
Traditionally. multiple
pcsite function combining
In optimizing a weighted
relative importance of the associated
ucns has also been proposed,
and Sauer 2001: Schuetz et al. 2007). However,
timalsolution, overlooking
In this work we have combined
complete Paretofront (or at least a good representation
objectives
different
sum of the objectives .. w~ere
objective.
as e.g. maximization
are optimized
criteria.
simulta.neously
The most Widely use~ approach
each
within
FBA, this type of Ut.Llttyfunc
of ATP yield per flux urnt (Dauner
this approachwill yield only one op
between the objectives.
two wellknown techniques
of it):
by defining a c~~
consists
:velght repr~s.entsthe
the tradeoff
for generating the
• eConstraint
problem. In this approach.
objective linear programming
straints are linear) or a nonlinear programming
the objectives while the others are incorporated
the value of the parameter
straints), different Paretooptimal
difficulty to choose appropriate
good picture of the Pareto front, so that no regions are over or under represented.
• Normal BoundaryIntersection
(NBI): This
was developed to overcomethe drawbacks
Pro~ch in which it is difficult to obtain
opumat set. Startingfrom the individual
conve.rtsthe original MO problem into a set of LPslNLPs
tematic change in the method parameters
Pareto front. Thus, the completetradeoff
by SOlvlOga lesser number of optimization
the Pareto surface can be missed in problems
It should be not d th
lobal onrt.
.eat goal optlmlzahon
proaches Ifthe as"d ...
SOCiae slOgleobJective
(EC): This is also a common
the original
and intuitive
MO problem
(if the objective
(NLP) problem
as inequality
on the objectives
solutionscan be obtained.
values for the parameters
methodfor solving
into a single
functionsand the con
by optimizing
constraints.
converted
Its main drawback
of the method
a MO
is transformed
(LP) problem
one of
By changing
to con
E (i.e. the bounds
is the
to obtaina
technique
of methods
a complete
optima
(Des and
like the weighted
representation
for each
in such a way that a sys
an even spread
the objectivescan be captured
problems.However,
with more than two objectives.
.
(GO) solversWIll be needed
NLPs are nonconvex.
Dennis
1998)
sum ap
of the Pareto
objective,NBIalso
generates
between
of points on the
someregionsof
for both ap
3 CaseStudy
Here Weconsider the ce traI
iedin (5 h
e ue\Zet al 2007) '0
""wolk ob'"
~ecllves The
and 10spli,.
rattosR(I=I
Tak."
109 as reference the
threerelevant
ee u arfunctlOns
bo ......
.
ncar n metabolIsm
.
h
examme t e pre Ictlve capaCIty of 11 hnear and nonlinear
t . hi'.
S OlC
omemc model conSIsts of 98 reacttons
10)a'·'1bh·
"'"
plvoamncpomts were defined (Figure
b.
~ Ove mentionedwork,we address
are optimizedsimultaneously:
to Eschenchw colt, whICh has been stud
d".
.
..and 60 metabolites.
.
I)
the problem
r
.
II I
in which
Find v to
maxZl(V)~V
maxz2(v)=VATP
.
BIOmass
(4)
Page 3
200
J.O.H. Sendrn, A.A. Alonso, and J.R. Banga
CCME. ~tH1
~I

,~.

co,l<l
I'11l<l
I.ACrl
,<,cEdFOflxl
ETHxt
nt the spill
. Red arroWS represe
(for furtherexpl.C tral carbon metabolism pathways tfr~~~~;/~~i~h~O~~twork
~n
ribe the systerruc degree 0
WhlC.h descSchuetz et al. 2007).
and abbrevtanonssee
FIg. 1.
ratios
diti ns considered
. (R) for the con ruo_
.. tal flux split ratios
Table 1. Expenmen Anaerobic
Aerobic
ContinuouS
Climited
ContinuOUS
N_limited
4.0mM/g·h
"5
0,82 ±0.02
Batch
ContinuouS
Climited
5.0 mM!g·h
Ex C2
8.3 mM/g·h
Ex C1
Ex C4
O.96±O.14
0.00 to.OS
0.64 ± 0,05
O.19±O.11
0.00 t 0.05
0.70 ± 0,06
O.84±O.14
0.85 + 0.09
0.00 ±0.05
a.llia.03
0.00 ± 0.05
0.00+0.05
0.00 ±O.05
O,72±O.lO
a.90ta.IS
0.50 ±O.06
0.00 ±0.05
D.OI ±O.O!
0.04 ±O.OJ
0.00 ±0.05
0.69 ±O.12
0.23 ±O.lO
0.00 t 0.05
O.84±O.14
0.91 ±O.ll
0.64tO.13
0.46±0.13
0.35 to.08
0.00 t 0.05
0.00 ±0.05
0.70 ± 0.02
0.13tO.06
0.00 ± 0.05
0.78 to.02
0.81 tOm
0.24 ± 0.02
0.OOtO.05
0.00 ±0.05
0.58 ±0.03
0.ootO.05
RI
R2
R3
R4
R5
R6
R7
R8
R9
BalCh
NO)1e5 .
\.77 mt>lIg·h
O.OO±O,05
0.00±0.05
0.96±0.02
0.96±0.02
0.02 ± 0.01
o.OO±o.05
0.00 ±0,05
0.65 ±0,01
0.30 ±O.02
RIO
MultiObjective Optimization of Biological Networks
201
.I
uer and lower bounds
be noted
~:cellular
on fluxes (Eqs.
that objective
) is nonlinear,
tlbJCCllO the mass balance .equatton.s and td. I PPhould
2·). No additional constraints
tlonS Zl and Z:z are linear, and the overa
func
butare Impos~l'.flux (2
3
In
convex.
Peeteoptimal
abovewill be compared Wit expenme
di .
environmental con mons
[hemostats). The overall agreement
Wlcebetween the computed
".
h
ornbination
of the methods
E I· (Table
. co /
. . g batch
respmn
if d usistandardized
re
uSI~g a _
described
I) under
andaerobic
solutionsobtained
. h
(
oxy.
WIt
ntal flux data
.
gen or nitrate
IS quanti
split ratios and the experimental
a c
f
five
. rom
<
cultures
Euclidean dis
,
ones.
4 Resultsand Discussion
4.1 Optimization Settings
The threeobjective optimization
bon of the aconstraint
lowing steps:
problem
and NBI.
defined
Th
above
I ti
is solved
st ategy consists of the fol
usi~g a combina
techniquee so u Ionr
I. Maximize VIIi",""" using LP
2. Choose different values bm, for VSiom",H in the range (0, vS;o",,,,,]
J. For each value bm.; the following
ing NBI: maximizationof ATP and minimization
subject to the econstraint:
rTl"~
.
IS solved hiobjectiveoptimization
of the overall
problemus
flux intracellular
VB;o",,,.<_<;:: bm,
The resulting NLPs from application
clustering algorithm, GLOBALm (Sendfn et al. 2008).
method which can detect the potential existence
thesame value of the objective
comparisonwith the results reported in (Schuetz et al. 2007), we have made use of the
solvers included in the MATLAB® Optimization
linprogfor the LPs andftnincon
as local solver within GLOBALm
of NBI are solvedby means
This is a global
optima
flux profiles).
of a I~ul.tis~art
optmuzanon
(i.e. solutions
For the sake of
of multiplewith
function and different
Toolbox(The MathWorks,
for the NLPs.
lnc.):
4.2 ParetoOptimal
Sets
The resulting Pareto surfaces (interpolated)
are showed in Figures 2 and 3, respectively.
eral
. intracellular flux is also depicted
~pOn~lng 10 the experimental
rbrid ap.proach EConstraint_NBI
for both aerobic
The tradeoff
for each one of the biomass
Both Paretooptimal
are representedin Figure 4.
and anaerobic
between
conditions
ATP yield and the
fluxes
sets obtained
OV
1
corre
conditions. using the
From Inspection of these figures
~~p prod",rion and the Overall intracellUlar
Ih/ Yields(higher in theaerobic
~n~Ymeusage and with low growth
maxImIzedWhile.
pa ihi . malOtamlOg t e overall
Y n t s case ISa decrease in the ATP yield.
is clearly evident
flux for a given biomass
case) are achieved
rates.On the other
the existing conflict
flux. Maximum
of an increase
biomass
between
at Ihe expensein
sidecan be
'.
h'
intracellUlarflux at low levels. The cost to
Page 4
202J.O.H. Sendfn, AAAlonso, and J.R. Banga
MultiObjective Optimization of Biological Networks
203
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•
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7.5
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~5
Oeeran Rux ()(lo5)
Fig. 2. Pareto front in aerobic conditions
Fig. 4. Comparison of Paretooptimal sets
4.3 Analysisof Solutions
~plit ratios for each Pareto solution
Ingthose which yield the closest flux predictions
are comparedwith the experimental
(Table 2).
data, select
,
,
,
,
'~',',
"','l',
""T_
"
,
c. __ L_
: ,;'
'
'
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,
,
,
,
,,
•i
:
,
' ~;"
'"
.
,
,
,,,
,
,
,
,,,,,
 ",
Table 2. Selected Paretooptimal points
ExpCI
ExpC2
ExpC3
C
ExpC4
ExpCS
A
B
D
E
VB'""""",
8.3
0.74
0.42
0.00
0.80
0.81
0.31
000
0.00
5.0
11.0
7.0
1.75
RI
R2
R3
R4
R5
R6
R7
R8
R9
RIO
0.98
0.00
000
0.98
0.92
0.79
0.12
0.00
004
0.00
0.64
0.09
00
0.72
0.70
0.59
0.0
0.0
0.05
00
0.98
000
0.00
0.85
0.86
0.77
0.00
0.00
009
0.00
0.55
tOO
000
0.97
0.52
0.01
000
0.00
0.74
0.22
0.50
000
15
10
,
5
. anaerobic conditions
F·
3 Pareto front In
.g..
Page 5
204
J.O.H. Sendfn, A.A. Alonso, and JR. Banga
For the continuous
(2007) when maximizing
Somewhat similar flux distributions were obtained here, but we wantto stressthe fact
that no additional, casespecific,constraints were imposed. For example.solution 8
(for Climited continuous cultures) is similar to that resulting from maximization of
ATP subject to an overproductionof 35% of NADPH relative to the NADPH e
quirementfor biomass production, and the flux profile C maximizes biomassv.'hllt
satisfying a constraint on intracellular fluxes (limited to a 200% of theglucoseUplake
rate), and an upper bound on the oxygen uptake of 150% of the glucose uptake.Fa'
Nlimited continuouscultures, point D also improves the prediction obtainedwhen
only one single objective is considered (with or without additional constraints).
cultures,
biomass or ATP yield coupled with several constraints.
the best predictions were found in Schuetzd al
5 Conclusions
In this work we have addressed the question of whether intracellular fluxescanbe
predicted consideringoptimality principles. The assumption here is that.nu~esart
distributed to optimize not only one single cellular function but severalobJectlves SI'
multaneously (multiobjectiveoptimization).
In general terms, Paretooptimalflux distributio.ns i~Prove th~be.stpredlc~ons lllld
tained with traditional FBA using different combinations of obJecttve~~cttons
constraints. The advantage of the multiobjective approach is that no addltt~~~~~:~:
specific, constraints are needed, and it can ?e a powerful tool for a betteru
ing of the factors that influence the metabolic flux.
.' 00
Acknowledgments
, B' LSHG CT_2006037469

This work has been supported by EU project BaSys
10
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MultiObjective
PredictionofIntracellular Fluxes
Optimization of Biological Networks for
~.Qscar
H. Sendfn.Antonio
A. Alonso. and Julio R. Banga
Process Engineering Group
IMllIuto de Investigaciones
a Eduardo Cabello 6, 36208, Vigo, Spain
olendiniHim.cs Ic .es . antonio@iim.csic.es,
Marinas (CSIC)
julio@iim.csic.es
Abstract. In this contribution,
IlVJlticriteriaoptimization
functions. Based on Flux Balance Analysis,
tions in E. coli for three objectives:
Intracellular fluxes. These solutions
mental conditionswithout
results.We thus illustrate the usefulness
Ing of complex biological networks.
we face the problem
approach,i.e. the simultaneous
of predicting
optimization
the Pareto
of biomass
flux distributions
constraints,
of multiobjective
intracellular
of two or more cellular
set of optimal
and ATP, and minimization
for different
and improveprevious
optimizationfor a better
fluxes using
a
we calculateflux distribu
maximization
are able to predict
specific
of
environ
published
understand
requiring
Keywords:
Multiobjective optimization,Paretofront, Flux Balance Analysis.
I Introduction
IntracellUlar fluxes i bi
su ion th
mpllOnt at cellular
logica bi
... 0 ~ectlve
byus'
f
whic~n~,'ocbex~mple,.Metabolic
ne lound
stoichiometric mod I f h' .
equations at stead
(Objectives) must~s dt~IS:enerallyunderdeter~ned, appropriate
unique solun'onS 'lllfe,as well
. llCcess ul applic ti
metabolic capabilities(Edwards et a Ioos
the metabolic netWork in S.. a . 200 I) and the genomescale
In lhi. cereVISlae (Forster
s Context, a particUlarly
int r Ii
cen.tlyin detail (Schuetz
et al. 2007.eN~s 1ng queslion
°Plimal biOChemical
t
k
'h ne woroperatIon
I~t ese systems?"
By farth
liOn of growth(or biomas~ ~ I~OSI common
ATP yield (van Gulikand HY~~), j'glthough
~(B .
~n~
onanoset al1996)havb
Sinc~ neither
h.I
n
IQC erruca
s st
y ems operate
Netwk
or capa t Illes and flux distributions
Flux Balance Analysis
tn eg
(Varm
a an
eta0
.1 e network, but since the linear system of mass
networks
.
In an optrmal way with respect
biliti.
can be calculated
.
.
in silico under
to a certain
have thus been predicted
(FBA), the fundamemals
1994). FBA only
the as
bio
.
.
.
of
the
d P j
a sson
requires
balance
cellular functions
constraintsas other posslble additional
f FBA
0
to find a
of E. coli
'
mclude
,
the
prediction
reconstruction of
et al. 2(03)
...
which
have
the principles
criteria
considered
criteria,such
...
IllJruzatlOnof the Overall
for different
been
addressed
behind
being
is the
maximization
re
the
'.
le ~en 2007)
,I.e..
concerns
h
are
."h'
W Ie
objective
other
theoptimized
maximiza_
as of
~)~~
intracellUlar
w~nor n~ture eha::n
the simultaneous
:~?OSed systemsand conditions.
~~~ch ts to consider
e tng. As a consequence,
op~gl~g~al, a more
t~o
desirable
or more
andrealistic
often
ap
the solutionw~Zatl~~fcriteria,con
I M
s' : COrchado et at. (&Is ) lwP
Pnogerlink.com
no uOlquebutinstead thisstrategy
' .~CB~ 2008, ASC 49, pp. 197205,2009.
SpnngerVcrlag Berlin Heidelberg 2009
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