comparison and analysis of objective functions in flux balance analysis
TRANSCRIPT
Comparison and Analysis of Objective Functions in Flux Balance Analysis
Carlos Eduardo Garc�ıa S�anchezGrupo de Investigaci�on en Fluidos y Energ�ıa, Corporaci�on Centro de Desarrollo Tecnol�ogico del Gas, Parque Tecnol�ogico UISGuatiguar�a Km 2 v�ıa Refugio, Piedecuesta, Santander, Colombia
Rodrigo Gonzalo Torres S�aezGrupo de Investigaci�on en Bioqu�ımica y Microbiolog�ıa, School of Chemistry, Faculty of Sciences, Universidad Industrial de Santander(UIS), Carrera 27 Calle 9 Ciudad Universitaria, Bucaramanga, Santander, Colombia
DOI 10.1002/btpr.1949Published online in Wiley Online Library (wileyonlinelibrary.com)
Flux balance analysis (FBA) is currently one of the most important and used techniquesfor estimation of metabolic reaction rates (fluxes). This mathematical approach utilizes anoptimization criterion in order to select a distribution of fluxes from the feasible spacedelimited by the metabolic reactions and some restrictions imposed over them, assuming thatcellular metabolism is in steady state. Therefore, the obtained flux distribution depends onthe specific objective function used. Multiple studies have been aimed to compare distinctobjective functions at given conditions, in order to determine which of those functions pro-duces values of fluxes closer to real data when used as objective in the FBA; in other words,what is the best objective function for modeling cell metabolism at a determined environ-mental condition. However, these comparative studies have been designed in very dissimilarways, and in general, several factors that can change the ideal objective function in a cellu-lar condition have not been adequately considered. Additionally, most of them have usedonly one dataset for representing one condition of cell growth, and different measuring tech-niques have been used. For these reasons, a rigorous study on the effect of factors such asthe quantity of used data, the number and type of fluxes utilized as input data, and theselected classification of growth conditions, are required in order to obtain useful conclu-sions for these comparative studies, allowing limiting clearly the application range on anyof those results. VC 2014 American Institute of Chemical Engineers Biotechnol. Prog.,000:000–000, 2014Keywords: flux balance analysis, objective function, quality of flux distribution prediction
Introduction
Cellular metabolism consists of a complex network of bio-chemical reactions that transform raw materials in energyand compounds necessary for cell function.1 There are avariety of approaches for modeling cell metabolism, but allof them share a structure named stoichiometric matrix.2 Thismatrix represents the structure of the metabolic network,indicating the stoichiometry of all enzymatic reactions. Inthe stoichiometric matrix, every row denotes a metabolitethat is produced and/or consumed in the network reactions,and every column represents one of these reactions. Eachmetabolite involved in every reaction is indicated through anumber in the corresponding position; the magnitude of thenumber shows the stoichiometric proportion in which thatsubstance reacts, and the sign determines whether the metab-olite is produced (positive sign) or consumed (negative sign)in the reaction.3 An example of the construction of a stoichi-ometric network is shown in the Figure 1A.
When the stoichiometric matrix multiplies a vector ofreaction rates, it is obtained a vector containing the rate of
change in metabolites with respect to time. If it is assumedthat a cell is maintained at a steady state condition, that vec-tor of derivatives becomes a null vector, and the basic equa-tion of the stoichiometric modeling of cell metabolism isobtained (see Figure 1B). There are several arguments forjustifying this assumption, among these: time-scale of metab-olism is higher than other biological processes such as cellregulation or cell division,4 and a cellular system will endreaching a steady state under any given environmental condi-tion.5 Additionally, most of industrial processes are operatedunder quasi-steady state conditions, and a steady state can beconsidered as a temporal average of the values in the cellpopulation.6 In any case, stoichiometric modeling of cellmetabolism has a wide range of applications, including bothsystemic and structural analyses of metabolism, determina-tion of particular distribution of fluxes and construction ofpredictive models.1,3,7
Among stoichiometric modeling techniques predictingmetabolic fluxes under distinct conditions, flux balance anal-ysis (FBA) is the most popular method.8 In spite of assump-tion of steady state, the basic equation of the stoichiometricmodel is an underdetermined system of linear equations, inwhich vectors of feasible fluxes define a convex space ofsolutions.9 For this reason, a predictive technique must
Correspondence concerning this article should be addressed to C. E.Garc�ıa S�anchez at [email protected].
VC 2014 American Institute of Chemical Engineers 1
distinguish what of those possible solutions represents thecellular metabolism under a determined condition in a betterway. In this sense, FBA converts this selection in an optimi-zation problem, so a vector of metabolic fluxes that maxi-mizes or minimizes a given objective function is chosen.7
Mathematically, FBA is represented according to thefollowing:
max Z5f ð m!Þ
subject to
S � m!5 0!
m!lower � m!� m!upper
m!j5 m!exp
8>>><>>>:
(1)
where Z is the objective function—normally linear—of themetabolic fluxes. The constraints are composed by the basicequation of stoichiometric modeling, inequality constraintsobtained from irreversibility from metabolic reactions andinformation about maximum and minimum rates, and equal-ity constraints that are used for introducing input data in themodel, that is, for indicating some metabolic fluxes repre-senting both environmental and growth conditions.
The quality of estimation of FBA, that is, the similaritybetween the vector of fluxes obtained from a FBA and themetabolic reaction rates that are really being produced in thecell, depends on the quality of metabolic model used, theconstraints imposed over the fluxes values, the fluxes used as
input data, and the objective function used in themodel.2,5,7,10–14
This review deals with comparative studies of objectivefunctions in FBA; that is, those aimed to determine the bestobjective function for predictive use of FBA. The utilizationof some different objective functions in FBA without a care-ful comparison is handled in a more general way.
Comparative Studies of Objective Functions
The most controversial characteristic of FBA is the utiliza-tion of an objective function for modeling the cellular meta-bolic behavior, which has been questioned in several ways.2
Independently of the validity of the assumption that cellshave objectives that can be mathematically represented, theselection of a solution among all flux distributions in the fea-sible space through FBA has become an important tool formetabolic engineering and systems biology.2,7,10,12,15–18 Themost used objective function in FBA has been maximizationof biomass production,19 but there has been some studies inwhich different objective functions have been used. Exam-ples of utilization of other objective functions include evalu-ation of production capability of different metabolites by acell organism, using as objective functions maximization ofproduction of some primary or secondary metabolites,20,21
and the study of metabolic modeling of mitochondria, usingas objective function maximization of ATP production.22
Nevertheless, in those studies distinct objective functionswere not compared in that the work selected an objectivefunction different to maximization of biomass; hence, theproblem was not aimed at finding the most suitable objectivefunction.
During the last decade of the past century, it was noticed
that there is a necessity of using appropriate objective func-
tions for a successful application of FBA. For this reason,
different studies with various objective functions were per-
formed,23–25 although all of them lacked a comparison of the
performance of objective functions from a quantitative point
of view. In one study, four different objective functions were
considered (minimization of ATP production, minimization
of nutrient molar uptake, minimization of mass nutrient
uptake, and minimization of NADH) for a hybridoma cell
line. The comparison of the functions was carried out
observing the influence of the objective on the distribution
of fluxes, and the authors concluded that minimization of
ATP production and minimization of mass nutrient uptake
seem to explain the behavior of the studied system,23
although in the second part of the study they fundamentally
used maximization of biomass production as objective func-
tion.24 A later study was focused on the introduction of a
biomass equation dependent on the growth rate, and a set of
equations of energetic requirements, in a metabolic model of
Escherichia coli; however, as part of the adjustment of the
biomass equation, two FBAs with minimization and maximi-
zation of biomass as objective functions were used for find-
ing lower and upper bounds of the growth rate, and were
solved with different supposed compositions of biomass until
obtaining a good adjustment to a set of experimental data.
Likewise, minimization/maximization of other fluxes—
mainly of metabolite secretion—was used in order to iden-
tify the range of appropriate values for every one of those
fluxes.25 This idea of computation of ranges for values of
fluxes has been subsequently used for analyzing multiple
Figure 1. Stoichiometric matrix and basic equation of stoichio-metric modeling of metabolism. (A) Example of theconstruction of the stoichiometric matrix; rows rep-resent metabolites and columns reactions in the met-abolic network. (B) The fundamental equation of thestoichiometric modeling of metabolism, and its appli-cation to the example previously shown. In steadystate condition, multiplying the stoichiometric matrixby the vector of fluxes must equal the zero vector, sothis equation constraints the feasible space for themetabolic fluxes.
2 Biotechnol. Prog., 2014, Vol. 00, No. 00
Tab
le1.
Su
mm
ary
of
Com
paris
on
Stu
die
sA
bou
tP
erf
orm
an
ce
of
Vari
ou
sO
bje
ctiv
eF
un
ctio
ns
inF
BA
Ref
eren
ceM
icro
org
anis
m,
Met
aboli
cM
odel
,an
dC
ondit
ions*
Input
Dat
aO
utp
ut
Dat
aC
om
par
edO
bje
ctiv
eF
unct
ions
Over
vie
wof
Concl
usi
ons
Burg
ard
and
Mar
anas
27
E.
coli
centr
alm
etab
oli
smm
odel
(62
reac
tions/
48
met
aboli
tes)
,ch
emost
atcu
lture
.A
erobic
gro
wth
(1),
anae
robic
gro
wth
(1).
Oxygen
and
glu
cose
upta
ke
rate
s.28
fluxes
det
erm
ined
by
isoto
pic
label
ing.
All
the
funct
ions
gen
erat
edby
sum
sof
the
model
’sfl
uxes
.In
both
cate
gori
es,
bio
mas
spro
duct
ion
isth
em
ost
import
ant
met
aboli
cfl
ux.
Knorr
etal
.28
E.
coli
gen
om
e-sc
ale
met
aboli
cm
odel
iJR
904
[931
reac
tions/
625
met
aboli
tes]
,bat
chcu
lture
,su
ccin
ate
assu
bst
rate
.M
ult
iple
tem
per
ature
san
dsu
ccin
ate
conce
ntr
atio
ns
(16).
Succ
inat
eupta
ke
rate
,an
dgro
wth
rate
.O
xygen
upta
ke
and
ace-
tate
pro
duct
ion.
Max
(bio
mas
spro
duct
ion),
min
(red
ox
pote
nti
alpro
duct
ion
rate
),m
in(A
TP
pro
duct
ion
rate
),m
ax(A
TP
pro
duc-
tion
rate
),m
in(s
ubst
rate
upta
ke
rate
)
Min
imiz
atio
nof
redox
pote
nti
alpro
duc-
tion
rate
isth
em
ost
pro
bab
leobje
c-ti
ve
funct
ion.
Sch
uet
zet
al.9
E.
coli
centr
alm
etab
oli
smm
odel
[98
reac
tions/
60
met
aboli
tes]
.B
atch
cult
ure
case
s:l
50.2
h2
1(1
);l
50.3
h2
1
(1);
l5
0.6
h2
1(1
).C
hem
ost
atcu
l-tu
reca
ses:
carb
on-l
imit
ed,l
50.1
h2
1
(1);
carb
on-l
imit
ed,l
50.4
h2
1(1
);nit
rogen
-lim
ited
,l
50.4
h2
1(1
).
Oxygen
and
glu
cose
upta
ke
rate
s,ca
rbon
dio
xid
eex
cret
ion.
10
fluxes
det
erm
ined
by
isoto
pic
label
ing.
Max
(bio
mas
syie
ld),
max
(AT
Pyie
ld),
min
(over
all
intr
acel
lula
rfl
ux),
max
(AT
Pyie
ldper
flux
unit
),m
ax(b
io-
mas
syie
ldper
flux
unit
),m
in(g
lu-
cose
upta
ke)
,m
in(r
eact
ion
step
s),
max
(AT
Pyie
ldper
reac
tion
step
),m
in(r
edox
pote
nti
al),
min
(AT
Ppro
-duci
ng
fluxes
),m
ax(A
TP
pro
duci
ng
fluxes
).
Inbat
chca
ses,
max
imiz
atio
nof
AT
Pyie
ldper
flux
unit
isth
ebes
tfu
nct
ion.
Inco
nti
nuous
case
sw
ith
nutr
ient
lim
i-ta
tion,
max
imiz
atio
nof
AT
Pyie
ld,
and
max
imiz
atio
nof
bio
mas
syie
ldar
eth
ebes
tfu
nct
ions.
Gia
nch
andan
iet
al.3
0S.
cere
visi
aece
ntr
alm
etab
oli
smm
odel
[62
reac
tions/
60
met
aboli
tes]
,bat
chcu
lture
,ca
rbon-l
imit
ed,l
50.3
7h
21
(1).
Glu
cose
upta
ke
rate
.An
upper
bound
of
0.3
7h
21
was
impose
dover
the
bio
mas
sfl
ux.
Flu
xes
det
erm
ined
by
isoto
pic
label
ing.
All
the
funct
ions
gen
erat
edby
linea
rco
mbin
atio
ns
of
the
model
’sfl
uxes
.M
axim
izat
ion
of
bio
mas
spro
duct
ion
isth
ebes
tobje
ctiv
efu
nct
ion.
Ow
etal
.31
Pla
smid
-bea
ring
E.
coli
gen
om
e-sc
ale
met
aboli
cm
odel
[931
reac
tions/
761
met
aboli
tes]
.C
hem
ost
atcu
lture
,ae
robic
,glu
cose
-lim
ited
(1).
Glu
cose
upta
ke
rate
.O
xygen
and
glu
cose
excr
etio
n,
24
intr
acel
-lu
lar
fluxes
det
erm
ined
by
isoto
pic
label
ing.
Max
(bio
mas
spro
duct
ion),
max
(pla
s-m
idpro
duct
ion),
max
(mai
nte
nan
ceen
ergy
expen
dit
ure
).
Max
imiz
atio
nof
mai
nte
nan
ceen
ergy
expen
dit
ure
(AT
Pdis
sipat
ion)
show
edth
ebes
tco
nsi
sten
cyw
ith
exper
imen
tal
obse
rvat
ion
for
pla
smid
-bea
ring
E.
coli
.G
arc� ı
aS� an
chez
etal
.11
S.ce
revi
siae
gen
om
e-sc
ale
met
aboli
cm
odel
iMM
904
[1,4
12
reac
tions/
1,2
28
met
aboli
tes]
.C
hem
ost
atcu
lture
case
s:an
aero
bic
(6);
aero
bic
,know
noxygen
upta
ke
(25);
aero
bic
,know
noxygen
upta
ke,
l�
0.1
5h
21
(19);
aero
bic
,know
noxygen
upta
ke,
0.1
5h
21�
l<
0.2
8h
21
(3);
aero
bic
,know
noxy-
gen
upta
ke,
0.2
8h
21<
l(3
);ae
robic
,know
noxygen
upta
ke,
glu
cose
assu
b-
stra
te(2
0);
aero
bic
,know
noxygen
upta
ke,
subst
rate
no
glu
cose
(5);
aero
-bic
,unknow
noxygen
upta
ke
(6).
Bat
chcu
lture
case
:ae
robic
,unknow
noxygen
upta
ke
(8).
Subst
rate
upta
ke
rate
.A
lso
oxygen
upta
ke
rate
,w
hen
know
n.
Spec
ific
gro
wth
rate
,an
dex
cret
ion
fluxes
that
wer
eex
per
imen
tall
ydet
erm
ined
.
All
the
funct
ions
gen
erat
edby
linea
rco
mbin
atio
nof
the
pro
pose
dco
mpar
t-m
enta
lobje
ctiv
es(1
,620
obje
ctiv
efu
nct
ions)
.
Inm
ost
case
s,m
axim
izat
ion
of
bio
mas
spro
duct
ion
isth
ebes
tobje
ctiv
e,al
ong
wit
hso
me
funct
ions
that
incl
ude
it.
When
the
oxygen
upta
ke
isunknow
nan
dth
ecu
lture
isco
nti
nuous,
max
imi-
zati
on
of
bio
mas
splu
sm
inim
izat
ion
of
carb
on
dio
xid
epro
duct
ion
isth
ebes
tobje
ctiv
e.W
hen
the
mic
roorg
an-
ism
isgro
win
gin
exponen
tial
phas
ean
dth
eoxygen
upta
ke
isunknow
n,
the
bes
tfu
nct
ion
ism
axim
izat
ion
of
bio
mas
spro
duct
ion
plu
sm
inim
izat
ion
of
NA
DH
pro
duct
ion
incy
toso
lplu
sm
inim
izat
ion
of
NA
D(P
)Hin
mit
och
ondri
on.
*T
he
num
ber
of
exper
imen
tal
dat
ase
tsuse
dfo
rco
mpar
ison
of
the
per
form
ance
of
the
obje
ctiv
efu
nct
ions
inev
ery
cate
gory
isin
dic
ated
inpar
enth
esis
.
Biotechnol. Prog., 2014, Vol. 00, No. 00 3
optima that can be obtained in FBA, in a technique known
as flux variability analysis.26
Table 1 shows a summary of the different comparativestudies of objective functions for FBA presenting the orga-nism considered, the metabolic model used, the growth con-ditions for the experimentally determined fluxes, the amountof data, the fluxes used as input data (as equality constraints,for representing the growth condition) and output data (tocompare quality of obtained estimations), the objective func-tions submitted to comparison, and an overview ofconclusions.
The first study dealing with quantitative comparison ofobjective functions was aimed at developing an algorithm ofsearching for an objective function that could describe motorforces underlying cell metabolism.27 This algorithm wasnamed ObjFind, and it examined whether the maximizationof a weighted combination of fluxes could explain a set ofobserved experimental data. The fluxes considered for theoptimization were those that represented uptake of resourcesform the metabolic network, by drainage of metabolites orreducing power, or energy dissipation. The algorithm wastested with two subsets of experimental data determined byisotopic labeling in E. coli,32 with each one representing adifferent environmental condition. The metabolic networkwas represented by a central metabolism model. This studyconcluded that the most important reactions for explainingcell behavior are those related to biomass production. How-ever, some of the reactions that represents drainage ofmetabolites for biomass production were less important thanothers; the relevancy of some of those reactions was compa-rable with the importance of reactions related with metabo-lite excretion.27
In 2007, it was published a new study focused on thecomparison of five mathematical representations of the cellobjective, in which a selection strategy based on Bayesiandiscrimination of objective functions was used.28 Again,E. coli was selected as model organism, but its metabolismwas represented using a genome-scale network.33 In thiswork, the following five objective functions were compared:(i) maximization of cell growth rate, (ii) minimization ofproduction rate of redox potential (being interpreted as maxi-mization of energetic efficiency), (iii) minimization of ATPproduction rate (representing an efficient use of energy), (iv)maximization of ATP production (beforehand, authors con-sidered it an inadequate objective), and (v) minimization ofnutrient uptake (an efficient use of resources entails a longduration of these ones). Sixteen experimental flux distribu-tions for cell growth on succinate were used. Both substrateuptake rates and growth specific rate were used as inputsexcept for possible objective functions formed by one ofthese fluxes. Output data were oxygen uptake rate and ace-tate production rate. This study concluded that the bestobjective function is the minimization of the production ofredox potential.28 It is necessary to remark that number ofinput data was different for the five evaluated functions, sothat with two of the functions (maximization of biomass pro-duction and minimization of substrate uptake), FBA had onemore degree of freedom.
In 2007 another study was published that simultaneouslyevaluated eleven objective functions, and eight possibleadditional constraints for the metabolic model.29 In orderto determine the performance of the different functionsand constraints, six distributions of fluxes measuredthrough isotopic labeling were utilized; those data were
obtained from E. coli cultures at different dilution rates,oxygen uptake rates and mode of operation (continuous orbatch). These experimental data were obtained from threedifferent sources.34–36 Every one of the six datasets wasused for representing a different growth condition. Themetabolic model used in this work was one including onlythe central metabolism of E. coli. The model had 10degrees of freedom, and therefore 10 “split ratios” weredefined in key sites of the model and used as output data.The input data used were substrate uptake rate, oxygen (ornitrate, in one case) uptake rate and carbon dioxide releaserate. This work concluded that for aerobic batch cultures,the best function is the maximization of ATP yield perflux unit, and in the chemostat under substrate constraints,objective functions that produced best predictions weremaximization of ATP production and maximization of bio-mass production. However, in batch culture under anaero-bic conditions, all explored objective functions gaveestimation with similar quality, which could be attributedto the lower number of degrees of freedom obtained inthose growth conditions.29
In 2008, an algorithm was developed for an automaticsearch of the objective function that give better adjustmentto a particular dataset. The algorithm was termed BiologicalObjective Solution Search (BOSS).30 Production of objectivefunctions de novo could find the best objective functionalthough this one was not included among the possible func-tions. However, this can lead to overfitting to experimentaldata and to obtaining objective functions that lack biologicalsignificance. BOSS creates a new reaction from a combina-tion of fluxes included in the metabolic network, in such away that maximizing this reaction produces a vector offluxes that are as closer as possible to a vector of fluxesexperimentally measured. The case study used a dataset offluxes determined by isotopic labeling for the case ofSaccharomyces cerevisiae growing aerobically in batch cul-ture, under carbon limitation, and with a specific growth rateof 0.37 h21.37 The method was only applied to this dataset;therefore, it was neither evaluated at different experimentalconditions nor various distributions of fluxes for the sameexperimental conditions were used. Glucose uptake was usedas input data, and an upper limit of 0.37 h21 for the specificgrowth rate was imposed (without this restriction, the objec-tive “maximization of biomass production” could overesti-mate the cell growth rate12), and the measured intracellularfluxes were used as output data. The modeling of centralmetabolism consisted of 62 reactions and 60 metabolites.Under these conditions, the BOSS method drove to an objec-tive function practically equal to the biomass productionequation.30
In 2009, a study was aimed to compare three differentobjective functions, specifically for a recombinant E. coli.31
The compared functions were maximization of biomass,maximization of the replication of plasmids and maximiza-tion of ATP dissipation. The metabolism was representedwith a custom alteration of a genome-scale metabolic model.In this work, only one experimental situation and one datasetwere considered. Comparison of functions was not carriedout using a global measurement of the error in the predic-tion, but observing differences between estimated and exper-imental values for every one of the fluxes with determinedexperimental data. Authors concluded that the best functionfor representing the behavior of recombinant E. coli was themaximization of the ATP dissipation.31
4 Biotechnol. Prog., 2014, Vol. 00, No. 00
In 2012, a study compared the performance of multipleobjective functions resulting from combinations of objectivesproposed for different spatial compartments wherein intracel-lular reactions in eukaryotic cells are carried out.11 For thisstudy, S. cerevisiae was used as a model organism, takingadvantage from the availability of experimental data in multi-ple growth conditions. Cell metabolism was represented atgenomic scale with the model iMM904, which includes 1,228metabolites and 1,412 reactions distributed in 8 cellular com-partments.38 The input data used were uptake fluxes of meas-ured metabolites, while excretion fluxes and specific growthrate were used as output data. Experimental data were classi-fied in multiple categories, depending on the oxygen availabil-ity, mode of cell culture, growth rate and type of substrate. Ingeneral, “maximization of biomass” was found among thebest objective functions of most categories, generally compet-ing with few combinations of cell objectives that includedthat objective. It was also found that the absence of a knownvalue for oxygen uptake rate causes very poor fluxes estima-tions when using FBA. In the category of cell growth in con-tinuous culture with unknown oxygen uptake, “maximizationof biomass plus minimization of carbon dioxide production”was the best function for prediction cell behavior, obtainingremarkable differences regarding estimations obtained with“maximization of biomass”. In batch cultures with unknownoxygen uptake, the best function was “maximization of bio-mass plus minimization of cytosolic NADH production plusminimization of mitochondrial NAD(P)H consumption”, beingdetermined that “maximization of biomass” widely overesti-mated the true value of cell growth rate.11
Comparisons with other types of objective functions
Some other techniques use optimization for selecting oneof the feasible distributions of fluxes, but with objectivefunctions that are not combinations of the metabolic fluxes.The most famous are Minimization of Metabolic Adjustment(MOMA)39 and Regulatory On/Off Minimization (ROOM),40
and both of them were validated comparing the quality ofthe predictions with those obtained with a FBA using maxi-mization of biomass production as objective function.
MOMA is a technique designed for modeling cells thathave suffered a genetic modification, e.g., gene knockout,and uses as an objective function the minimization of theEuclidean distance between the cell fluxes in its wild condi-tion and the fluxes of the cell that have been modified bygenetic engineering.39 To evaluate MOMA’s performance,authors used a genome-scale model of E. coli, and performedlethality prediction experiments, and a quantitative compari-son for a E. coli pyk (pyruvate kinase) knockout mutantagainst the flux distribution obtained with a FBA. This quan-titative comparison was made using experimental measure-ments based in isotopic labeling, which represented threedifferent growth conditions: two with limitation in glucoseconcentration (one with low and other with high concentra-tion) and one with limitation in nitrogen availability.34
Authors concluded that MOMA gave better lethality predic-tions than the FBA, and that for the studied knockout mutantin the three conditions considered, MOMA yielded far supe-rior predictions than those obtained with the FBA. It isimportant to notice that application of MOMA requires hav-ing a wild state flux distribution, which is determined usingFBA, and the objective function used in this FBA can alterthe MOMA prediction.
The quadratic nature of the objective function of MOMAtends to favor a big number of small flux changes over afew large variations. In response to some genetic alterationsin S. cerevisiae, it has been observed that only few metabolicfluxes changes, but the ones that do change have large alter-ations.41 These considerations led to the proposition of atechnique known as ROOM.40 This method is also aimed tomodel cells that have undergone a gene deletion, but itsobjective function consists of minimization of the number offluxes that suffer a significant change (compared with thewild-type distribution) after the genetic modification. It isimportant to take into account that ROOM predictionsdepend on the supposed wild-type flux distribution, and alsoon the quantities used to define the threshold of “significantchange” in the flux values. Authors used nine different datasets from diverse E. coli knockout strains, and concludedthat ROOM gave better correlation with the experimentallydetermined fluxes than FBA and MOMA. Also, using experi-mental data from altered E. coli, they declared that MOMAcan model better the metabolism of a cell in a short termafter the deletion, but the subsequent adaptation is better rep-resented by the FBA and ROOM techniques. Similarly,authors established that ROOM and FBA are superior toMOMA in predicting essentiality of genes, using a genome-scale metabolic model of S. cerevisiae. However, MOMAshowed better results than FBA and ROOM when predictinglethality of genes in the same E. coli genome-scale metabo-lism model used by MOMA’s authors.
Discussion
The comparative studies of different objective functionshave not considered all the factors that influence the qualityof the obtained estimation. Additionally, a design protocolhas not been created for this kind of work, and this causesremarkable differences among different studies. Some of thedifferences among the research protocols are necessary inorder to analyze distinct cells and diverse situations, such asmultiplicity of studied organisms and variety of growth con-ditions. On the other hand, other divergences affect the com-parative investigations in a fundamental manner, such asdifferences in size and type of metabolic models, in amountand nature of fluxes used as input and output data, in theconsideration of the probable multiplicity of solution of theFBA, and in the sources of experimental data.
Regarding the differences in the modeling of metabolism,the quality of the metabolic model has a high impact on theestimates obtained with the FBA. Metabolic models withvery different characteristics have been used in the compara-tive studies of objective functions, from very simplified mod-els of central metabolism up to the much more complexgenome-scale models, in which inevitably there is less confi-dence in many reactions. Currently it is unknown if it is bet-ter (aiming at an adequate prediction of the metabolic fluxesdistribution) to use a genome-scale metabolic model, withthe uncertainty associated to many of the included reactions,or a central metabolism model (or some other type ofmedium-sized model), which has smaller scope but morereliability in the incorporated reactions, because there are notstudies comparing the quality of FBA estimations with dif-ferent kinds of metabolic models. Another source of discrep-ancies is the inclusion of equality restrictions over certainreactions, like setting the value of the maintenance flux orthe P/O ratio. A lack of coherence between those equality
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restrictions and the real metabolic behavior degrades thequality of the estimations. In any case, it can be expectedthat advances in the creation of new metabolic models andthe generation of consensus models will allow for overcom-ing these issues.
Fluxes used as input (i.e., reactions rates whose values areintroduced as equality constraints) and output (i.e., reactionrates that are utilized to evaluate the quality of the estima-tion) data are properties that needs more attention, because ithas been demonstrated that depending of the fluxes selectedas input and output data in the study, the best objective func-tion can change.13 Different sets of input and output datacan be needed for modeling different configurations of cellcultures (for example, in a chemostat is set the growth rate,so that l must be an input data in order to model this modeof culture) or for pursuing different objectives (e.g., for eval-uating the estimation of intracellular metabolic fluxes isrequired to use intracellular reaction rates as output data).However, as long as every study uses different combinationsof input and output data, the level of generalization andapplication of the results of these works will be very limited.In this matter, the use of additional fluxes—other thanuptake fluxes—as input data decreases in a large extent theusefulness of the conclusions pursuing the computationalmodeling of the cell as a biological machine. It is alsonoticeable that some studies have showed strong evidencesabout the existence of dependence between the best objectivefunction and the mode of operation of the culture (e.g., batchor chemostat),11,13,29 although it is likely that the real factorsbehind this dependence are the amount of available substrateand the specific growth rate.
In addition, the likely existence of multiple optima inFBA (i.e., multiple distributions of fluxes that produce thesame optimum value of the objective function42) has beenwell documented. Some methods for the analysis of the mul-tiplicity of solutions has been developed,26 but, in general(excluding the work of Schuetz et al.29) the comparativestudies of objective functions have not kept in mind the mul-tiplicity of solutions. The existence of multiple optima in thetechniques based on optimization for the analysis of theintracellular metabolism is still an important obstacle,because it is not unique the solution of the optimizationproblem, and the searching for techniques and approachesfor analyzing (or selecting one of) those multiple solutionsrequire better attention.
Regarding experimental data, there is dissimilarity in thenumber of fluxes measured in the various datasets used forthe different studies, the number of experimental datasetsused for evaluating the quality of the estimations in everycategory, and the techniques used for obtaining these data.Concerning measured fluxes, both metabolic model and num-ber of fluxes used as input data alter the degrees of freedomof the optimization problem, which influences the size of thefeasible region and the complexity of the solution process.On the other hand, the number of fluxes used for evaluatingthe quality of the predictions can affect the measurement ofobtained errors, because having few fluxes for the measure-ment of the error generally leads to an artificially lower error(the small error would be due to comparison with insufficientvalues and not because the estimation is really good).
A fundamental step in the searching for a good representa-tion of the cellular objective by means of an objective func-tion in FBA is the comparison with the experimental data.For this reason, the reliability and general applicability of
these studies require a thorough evaluation of the effect ofnoise and experimental error on the data used. A set of cellsgrowing in a determined environment is an ensemble oforganisms at different stages in the cell cycle, consumingnutrients and using them for different aims. Therefore, whenrepresented with the techniques of stoichiometric modeling,an average of the cell behavior (that is, of the fluxes) is rep-resented of all the modeled cells.6 Cell culture data can thushave experimental noise that is added to the inherent uncer-tainty of experimental techniques and random error, which itmakes necessary to use multiple sets of experimental data tocompare the performance of the different mathematical rep-resentations. This approach has been used for the studies car-ried out by Knorr et al.28 and Garc�ıa S�anchez et al.11
Another option for the analysis of the effect of noise on thedetermination of the best function is to analyze this problemin an explicit way, as done in the study of Gianchandaniet al.30
All these differences in the structure and analysis of stud-ies cause problems regarding generalization of the obtainedresults. Some of the conclusions of the studies are usedindiscriminately, in a wider way than allowed by the scopeand design of the original research. For this reason, it isrequired to gain more depth with respect to the importanceof each one of the above-mentioned factors, in order toknow exactly which culture conditions can be modeled by aFBA with a “best objective function” obtained in a particularcomparative study.
Conclusions
The determination of the distribution of fluxes of a cell atdetermined conditions has a lot of degrees of freedom,although that determination is done supposing steady state.Transformation of this issue into an optimization problemallows selection of a distribution of fluxes in the feasibleregion, and depending on the quality of the model and con-straints, representation of the growth conditions, and objec-tive function used for optimization, this selection can orcannot represent the real behavior of the cell.
Several studies have compared the performance of differ-ent objective functions with regards to accuracy of the esti-mation obtained respect to sets of fluxes measuredexperimentally. However, in general the studies have notconsidered factors that have great effect on the quality of theobtained estimations with the FBA, and even can change theobjective function that best models a determined growth con-dition. In the same way, most of the studies have used onlyone set of experimental data for representing a specificgrowth condition, which prevents the appreciation of theeffect of the errors and can lead to overfitting the data.These facts limit the reproducibility of the results and limitthe usefulness of the conclusions obtained in every study,because it is not clear if those conclusions can be general-ized to other growth conditions or other types of organismsor cells.
In order to be able to use FBA as a predictive technique,advances are required in both the modeling of metabolic net-works and the modeling of restrictions over the reaction ratesvia constraints. Additionally, it would be helpful to gainimproved knowledge about what particular objective func-tions describe every environmental condition. Finally, thedetermination of those objective functions requires a best
6 Biotechnol. Prog., 2014, Vol. 00, No. 00
analysis of the effect of factors such as experimental error,amount of measured data, experimental values used as equal-ity constraints, and multiplicity of optimal functions in FBA.
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