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J. Biomedical Science and Engineering, 2008, 1, 121-126
Published Online August 2008 in SciRes. http://www.srpublishing.org/journal/jbise JBiSE
Constrain-based analysis of gene deletion on the
metabolic flux redistribution of Saccharomyces
Zi-Xiang Xu & Xiao Sun
State Key Laboratory of Bioelectronics, Southeast University, Nanjing 210096, China. Correspondence should be addressed to Z. X. Xu.( firstname.lastname@example.org,
Based on the gene-protein-reaction (GPR)
model of S. cerevisiae_iND750 and the method
of constraint-based analysis, we first calculated
the metabolic flux distribution of S. cere-
visiae_iND750. Then we calculated the deletion
impact of 438 calculable genes, one by one, on
the metabolic flux redistribution of S. cere-
visiae_iND750. Next w e analyzed the correlation
between v (describing deletion impact of one
gene) and d (connection degree of one gene)
and the correlation bet ween v and Vgene (flux sum
controlled by one gene), and found that both of
them were not of linear relation. Furthermore,
we sought out 38 important genes that most
greatly affected the metabolic flux distribution,
and determined their fu nctional subsystems. We
also found that many of these key genes were
related to many but not several subsystems.
Because the in silico model of S. cere-
visiae_iND750 has been tested by many ex-
periments, thus is credible, we can conclude
that the result we obtained has biological sig-
Keywords: Metabonomics, Metabolic engi-
neering, Metabolic networks, Gene deletion,
Genome-scale simulation, Flux balance analysis,
Gene-protein- reaction (GPR) model, Con-
Since various ‘omics’ datasets are becoming available,
biology has transited from a data-poor to a data-rich en-
vironment. This has underscored the need for systems
analysis in biology and systems biology has become a
rapidly growi n g fi el d as well  .
A change in mathematical modeling philosophy, i.e.,
building and validating in silico models is also necessi-
tated. Modern biological models need to meet new sets
of criteria: organism-specific, data-driven, easily scalable,
and so on. Many modeling approaches, such as kinetic,
stochastic and cybernetic approaches, are currently being
used to model cellular processes. Owing to the computa-
tional complexity and the large number of parameters
needed, it is currently difficult to use these methods to
model genome-scale networks. To date, genome-scale
models of metabolism have only been analyzed with the
constraint-based modeling philosophy [2, 3]. Ge-
nome-scale network models of diverse cellular processes
such as signal transduction, transcriptional regulation and
metabolism have been generated. Gene-protein-reaction
(GPR) associated models can keep track of associations
between genes, proteins, and reactions , and there
have been several genome-scale GPR models, such as E.
col i [4,5] , S. aureus , H. pylori , M. barkeri, S.
cerevisiae  and B. subtilis . A reconstruction is
herein defined as the list of biochemical reactions occur-
ring in a particular cellular system and the associations
between these reactions and relevant proteins, transcripts
and genes . A reconstruction can include the assump-
tions necessary for computational simulation, such as
maximum reaction rates and nutrient uptake rates.
Computer simulations of complex biological systems
began essentially as soon as the computational capability
became available. As reconstructed networks have been
made publicly available, researchers around the world
have undertaken new computational studies using these
networks. Many researches apply a core set of basic in
silico methods and often also describe novel methods to
investigate different models. An extensive set of methods
for analyzing these genome-scale models have been de-
veloped and have been applied to study a growing num-
ber of biological problems. As we have mentioned above,
genome-scale models of metabolism have only been
analyzed with the constrain t-based philosophy [2,3].
The in silico models can be applied to generate novel,
testable and often quantitative predictions of cellular
behavior. The impact of a gene deletion experiment on
cellular behavior can be simulated in a manner similar to
linear optimization of growth. The results can be used to
guide the design of informative confirmation experi-
ments and will be helpful to metabolic engineering.
Some gene deletion studies on genome-scale in silico
organism models have been made [4-10], most of them
are from the standpoints of distinguishing lethal and
SciRes Copyright © 2008
122 Z.X.Xu et al. / J. Biomedical Science and Engineering 1 (2008) 121-126
SciRes Copyright © 2008 JBiSE
non-lethal genes or growth rate [4-10, 15-19]. The im-
pact of gene deletion on flux redistribution, the charac-
ters and functions of key genes are still lacked research.
In this paper, there are four parts. In part two, as a base
for later research, we firstly calculate flux distribution of
S. cerevisiae_iND750. Then we will calculate the dele-
tion impact of 438 calculable genes, one by one, on
themetabolic flux redistribution of S. cerevisiae_ iND750.
Next we will analyze the correlation between v (describ-
ing deletion impact of one gene) and d (connection de-
gree of one gene) and the correlation between v and
v (flux sum controlled by one gene). Furthermore,
we will seek out those important genes that most greatly
affected the metabolic flux distribution, determine their
functional subsystems. Because the in silico model of S.
cerevisiae_iND750 has been tested by many experiments
 , we can conclude that the result we got h as biological
significances; In part three, we introduce the GPR model,
some properties of the in silico model of S. cere-
visiae_iND750 (SBML format) and the method of con-
straint-based analysis; Part four is a simple discussion.
2.1. Metabolic flux distribution and of S. cere-
As a base for the later compared research, we here cal-
culated the flux distributio n of S. cerevisiae_ iND750 .
What we use is Sc_iND750_GlcMM.xml, the SBML file
that is presented with the reconstruction of S. cere-
visiae_iND750 . The computational method we used
is flux balance an alysis (FBA) , one of the most fun-
damental genome-scale phenotypic calculations, which
can simulate cellular growth. FBA is based on linear
optimization of an objective function, which typically is
bio-mass formation. Given an uptake rate for key nutri-
ents and the biomass composition of the cell (usually in
mmol component gDW-1 and defined in the biomass ob-
jective function), the maximum possible growth rate of
the cells can be predicted in silico. We use the COBRA
toolbox  to carry out this computation of FBA. The
flux distribution of S. cerevisiae_iND750 is illustrated in
0200 400 600 8001000 1200
the i th reaction in rxns
Figure 1. Flux distribution of S. cerevisiae _iND750. X-axis indi-
cating every re action in rxns (the order is as the same as in rxns,
total 1266) and y-axis indicating the value of its corresponding
flux (unit is mmol gDW-1h-1). Rxns is the reaction set in the model.
2.2 Impact of gene deletion on the metabolic flux
redistribution and key genes
2.2.1 Impact of gene deletion on the metabolic flux re-
There are 750 genes in the model of S. cere-
visiae_iND750, but we can not calculate the impact of
every gene deletion. If a single gene is associated with
multiple reactions, the deletion of that gene will result in
the removal of all associated reactions. On the other hand,
a reaction that can be catalyzed by multiple non-interact-
ing gene products will not be removed in a single gene
deletion. Among 750 genes of S. cerevisiae_ iND750,
there are 438 genes which have no “OR” relationship
with other genes in every reaction of S. cere-
visiae_iND750, and by the aid of the COBRA toolbox
, we can calculate the impact of their deletion. We
define the impact of one gene deletion on the whole
metabolic flux redistribution as v
ii vvv 2
v and i
are respectively the flux value of
i-th reaction of the model of S. cerevisiae_iND750 be-
fore and after a single gene deleting, and R is the whole
Figure 2 shows the deletion impact of these 438 genes.
From the figure, we can’t see what are important genes
which greatly affect, if deleted, the metabolic redistribu-
tion of S. cerevisiae_ iND750. This remains as a problem,
and we will settle it in section 4). In the following, we
will analyze the relationship between th e impact of every
gene deletion (v) and the connection degree of every
gene (d) and the connection degree of every gene
2.2.2. Correlation between v and d (connection degree of
We compute out the related reaction number d of every
gene in those 438 genes of the S. cerevisiae_iND750
model, as illustrated in Figure 3. From the figure, we can
find that some but not many genes have high d value, but
we don’t know whether they affect metabolic flux dis-
050100 150200250 300350400 450
t he ith gene in 438 genes
v va lue
Figure 2. The deletion impact of calculable 438 genes of the S.
cerevisiae_iND750 model. X-axis indicating every gene in 438
genes (the order is as the same as in genes, total 438) and
y-axis indicating its impact v.
Z.X.Xu et al. / J. Biomedical Science and Engineering 1 (2008) 121-126 123
SciRes Copyright © 2008 JBiSE
050100150 200 250300 350 400450
t he ith gene in 438 genes
number of reactions
Figure 3. The related reaction number of every gene in 438
genes of the S. cerevisiae_iND750 model. X-axis indicating every
gene in 438 genes (the order is as the same as in genes, total
438) and y-axis indicating the number of its related reactions.
05 10 15 20 25
Figure 4. The scatters diagram (d, v). X-axis indicating d (con-
nection degree of every gene) and y-axis indicating the corre-
sponding gene impact v.
Figure 4 is the scatters diagram (d, v), total 438 data
pairs. From the diagram, we can easily find that the rela-
tionship between d and v is not of linear relation. So
high-d genes and low-d genes are equally important to
the metabolism of S. cerevisiae_iND750.
2.2.3. Correlation between v and gene
v (flux sum con-
trolled by every gene)
We define the flux sum controlled by every gene as
jgene vv (2)
v is the flux valu e of j-th reaction of the model
of S. cerevisiae_iND750 before a single gene deleting,
and where Rgene is the reaction set controlled by the given
gene. We can easily compute out the flux sum gene
every gene in those 438 genes of the S. cere-
visiae_iND750 model, as illustrated in Figure 5. From
the figure, we can find that some but not many genes
v, but will they affect metabolic flux dis-
Figure 6 is the scatters diagram (gene
v, v), total 438
data pairs. From the diagram, we can easily find that the
relationship between gene
v and v is also not of linear
relation. So high-gene
v genes and low-gene
v genes are
equally important to the metabolism of S. cerevisiae_
050 100 150 200250 300 350 400 450
the ith gen e in 43 8 g en es
flux sum controlled
Figure 5. The controlled reaction number of every gene in 438
genes of the S. cerevisiae_iND750 model. X-axis indicating every
gene in 438 genes (the order is as the same as in genes , total
438) and y-axis indicating the number of its controlled reactions.
00.5 11.5 22.5 3
Figure 6. The scatters diagram (vgene, v). X- axis indicating vgene
(the flux sum controlled by every gene) and y-axis indicating the
Table 1. Gene number (GN) and v scope
v (×107) 0 0-0.1 0.1-1.5 1.0-1.51.5-1.8
GN 4 100 80 71 46
v (×107) 1.8-2.02.0-2.5 2.5-3.0 >3.0
GN 13 43
2.2.4. Key genes that affect metabolism most greatly
In this section, we seek out what are important or key
genes which greatly affect the metabolic redistribution of
S. cerevisiae_iND750, and furthermore in next section,
we will give their belonged functional subsystems. Table
1 provides the corresponding relationship between gene
number (GN) and v scope, and as an example, GN=100
& v= (0-0.1)×107 means that there are 100 genes while
the v scope which these genes control is (0~0.1)×107.
We define those genes with v>3.0×107as key genes, and
there are 38 genes.
2.2.5. Functional subsystems to which these key genes
If a gene catalyze a reaction and while the reaction be-
long to a certain subsystem, we will say that the gene
belong to the subsystem. Functional subsystems about
important genes in the metabolic system of micro organ-
ism are seldom reported. We list the functional subsys-
tems to which every key gene belong, and several genes
124 Z.X.Xu et al. / J. Biomedical Science and Engineering 1 (2008) 121-126
SciRes Copyright © 2008 JBiSE
Table 2. The functional subsystems and their related genes of S. cerevisiae_iND750
Peroxisomal Fatty Acid Bio-
synthesis Pentose Phosphate
'YHL016C' 'YBR035C' 'YBR041W' 'YBR041W' 'YCR036W'
system Citric Acid
and CoA Bio-
Pyruvate Metabo -
genes 'YEL058W' 'YER014W'
'YOR278W' 'YGR267C' 'YIL155C' 'YKL106W' 'YKL106W'
system NAD Biosyn-
Metabolism Cysteine Me-
genes 'YLR209C' 'YOR040W' 'YOR100C' 'YPL214C' 'YPR167C'
appear in more than one subsystem, shown in Tabl e 2.
We will find that many of 38 key genes are related to 26
but not several subsystems.
3. MATERIALS AND METHODS
3.1. Gene-protein-reaction (GPR) associated
The association between genes and reactions is not a
one-to-one relationship. Many genes may encode suunits
genes that encode so-called promiscuous enzymes that
can catalyze several different reactions. So it is necessary
to keep track of associations between genes, proteins,
and reactions and to distinguish “&” and “OR” associa-
tions in GPR models. Examples of different types of
GPR associations are illustrated in th e figures of Ref. [4,
3.2. GPR model structure of S. cerevisiae_
The in silico model that we used is S. cerevisiae_iND750
, a metabolic reconstruction consistin g of th e ch emical
reactions that transport and interconvert metabolites
within yeast. This network reconstruction was based on a
previous reconstruction, termed S. cerevisiae_iFF708
. The general features of S. cerevisiae_iND750 are
shown in the Table 1 of Ref. .
A SBML format file to the model S. cere-
visiae_iND750 can be downloaded from the supplemen-
tary information of Ref. . SBML file properties are
given in the supplementary of Ref. . The dimensions
of rxns, mets, and genes are respectively 1266, 1061,
750. We can use Cytoscape  to draw the
GPR-network (Figure 7) of S. cerevisiae_ iND750,
which contains 3077 nodes and 6666 edges.
The minimal media of in silico model is an important
aspect. The computational minimal media of S. cere-
visiae_iND750 is also included in Ref. . In the method
of constraint-based analysis, the biomass objective func-
tion (BOF) should be defined. The BOF was generated
by defining all of the major and essential constituents
that make up the cellular biomass content of S. cerevisiae
. Gene-protein-reaction associations embodied in
rxnGeneMat matrix, which is a matrix with as many
rows as there are reactions in the model and as many
columns as there are genes in the model. The i-th row
and the j-th column contains a one if the j-th gene in
genes is associated with the i-th reaction in rxns and
3.3. Methodology of constraint-based analysis
3.3.1. Constraint-based analysis
In silico modeling and simulation of genome-scale bio-
logical systems are different from that practiced in the
physicochemical sciences. A network can fundamentally
have many different states or many different solutions.
Which states (or solutions) are picked is up to the cell
and based on the selection pressure experienced, and
such choices can change over time. Therefore, con-
straint-based approaches [2, 3] to the analysis of com-
plex biological systems have proven to be very useful.
This difference between the physicochemical sciences
and the physical sciences or engineering is illustrated in
Ref. . All theory-based considerations (i.e., engi-
neering and physics) lead one to attempt to seek an “ex-
act” solution, and typically computed based on the laws
of physics and chemistry. However, constraint- based
considerations (as in biology) are useful. Not only can a
network have many different behaviors that are picked
based on the evolutionary history of the organism, but
Z.X.Xu et al. / J. Biomedical Science and Engineering 1 (2008) 121-126 125
SciRes Copyright © 2008 JBiSE
Figure 7. GPR-Network of S. cerevisiae_ iND750, 3077
nodes and 6666 edges, Created by Cytoscape with Layout
also these networks can carry out the same function in
many different and equivalent ways.
3.3.2. Representation of reconstructed metabolic network
Before calculation and simulation, the reconstructed
metabolic network must be represented mathemati-
cally.The stoichiometric matrix, S, is the centerpiece of a
mathematical representation of genome-scale metabolic
networks. It represents each reaction as a column and
each metabolite as a row, where each numerical element
is the corresponding stoichiometric coefficient. A
graphical form of the first few reactions of glycolysis and
the corresponding stoichiometric matrix are shown in the
Figure 2 of Ref. .
An upper and lower bound for the allowable flux
through each reaction also requires defining. This repre-
sents the lowest and highest reaction rate possible for
each reaction. The set of upper and lower bounds is rep-
resented as two separate vectors, each containing as
many components as there are columns in S, and in the
same order. An example is shown in Figure 2 of Ref.
. In many cases, reversible reactions are defined to
have an arbitrary large upper bound and an arbitrarily
large negative lower bound. Irreversible reactions have a
lower bound that is nonnegative, usually zero.
In order to predict meaningful fluxes, setting upper
and lower bounds is especially important for exchange
reactions which serve to uptake compounds to the cell or
secrete compounds from the cell. The lower bound of the
exchange reaction column must be a finite negative
number using this orientation (e.g., glucose). The upper
bound of the exchange reaction column must be greater
than zero. At least one of the reactions in the model must
have a constrained lower /upper bound, and typically, the
substrate (e.g., glucose or oxygen) u ptake rates are set to
experimentally measured values. The upper and lower
bounds for exchange reactions are quantitative in silico
representations of the growth media environment.
3.3.3. Biomass objective function (BOF) and minimal
The constraint-based approach is based on the assump-
tion that cells strive to maximize their growth rate. This
assumption which provides an acceptable starting point
for many types of computations is satisfied by simulating
maximal production of the molecules required to make
new cells (biomass precursor molecules). In spite of their
limitations, th e predictive pow er of genome-scale models
of metabolic networks has been demonstrated in diverse
situations through careful experimentation .
The biomass objective function (the function vgrowth ,
see below) is a special reaction taking as substrates of all
biomass metabolites, ATP and water, and producing
ADP, protons, and phosphate (as a result of the
non-growth associated ATP maintenance require-
The minimal media is determined computationally
with the systematic testing of distinct inputs. Different
combinations of molecules were allowed to enter the
reaction network until the minimal group that allowed
biomass production, or non-zero Z (see below), was
found . It is only concerned that some amount of
biomass production is calculated but do not discriminate
between extremely slow, inefficient growth and rapid
3.3.4. Computation of phenotypic states
In genome-scale metabolic networks, the fluxes within a
cell usually cannot be uniquely calculated because a
range of feasible values exist when fluxes are subjected
to known constraints. Flux-balance analysis (FBA) was
used to find optimal growth phenotypes. Briefly, a
large-scale linear programming was used to find a com-
plete set of metabolic fluxes (v) that are consistent with
steady-state condition (eq. 3) and reaction rate bounds
(eq. 4), and at the same time maximize the biomass
objective function in the defined ratio. This corresponds
to the following linear programming problem :
Max Z = vgrowth
Subject to S • v = 0 (3)
αi < vi < βi (4)
where S is the stoichiometric matrix, and where αi and βi
define the bounds through each reaction vi. The flux
range was set arbitrarily high for all internal reactions so
that no internal reaction restricted the network, with the
exception of irreversible reactions, which have a mini-
mum flux of zero. The inputs to the system were re-
stricted to a minimal media.
The value of Z computed with the above procedure
can either be zero (predicting no growth) or greater than
zero (corresponding to cellular growth) depending on the
inputs and outputs that are allowed, according to the nu-
trients provided in the media.
3.3.5. Ge ne deletion study
The effect of a gene deletion experiment on cellular
126 Z.X.Xu et al. / J. Biomedical Science and Engineering 1 (2008) 121-126
SciRes Copyright © 2008 JBiSE
growth can be simulated in a manner similar to linear
optimization of growth [5, 11]. Gene–reaction associa-
tions model the logical relationship between genes and
their corresponding reactions. If a single gene is associ-
ated with multiple reactions, the deletion of that gene
will result in the removal of all associated reactions, i.e.
to simultaneously restrict the fluxes (upper and lower
flux bounds) of these reactions to zero prior to comput-
ing maximal biomass objective function. On the other
hand, a reaction that can be catalyzed by multiple
non-interacting gene products will not be removed in a
single gene deletion. The possible results from a simula-
tion of a single gene deletion are unchanged maximal
growth (non-lethal), reduced maximal growth or no
growth (lethal). Those genes were considered essential if
no biomass could be pro d uced without their usage.
Based on the gene-protein-reaction model of S. cere-
visiae_iND750 and the methodology of constraint- based
analysis and by the aid of the COBRA toolbox and
MATLAB software, we have calculated the deletion im-
pact of 438 calculable genes, one by one, on the meta-
bolic flux re di stributio n of S. cerevisiae_iND750.
We found that both of the v-d correlation and the
vcorrelation were not of linear relation. Although
some properties about the metabolic network of micro-
organisms have been reported in literatures [15-19], our
research will provide further evidences to the properties
about the metabolic network, because the measure we
defined is different.
Furthermore, we sought out 38 important genes that
most greatly affected the metabolic flux distribution,
determined their functional subsystems and found that
many of 38 key genes were related to many but not sev-
eral subsystems. From these results, we speculate that
many but not several subsystems are important subsys-
tems in the metabolism of S. cerevisiae and that this may
increase the robustness of the metabolic network.
As a next step, we will do similar research on other
organisms and compare them with the case of E. coli.
Although it is theoretically possible to attempt double
deletion of every possible gene pair experimentally, the
sheer number of possible two-gene deletions makes this
virtually impossible. However, computational predict-
tions of double gene deletion phenotypes can be made in
a matter of hours . This will also become our work in
Support for this work was provided by China Postdoctoral Science
Foundation (20070420960), Jiangsu Planned Projects for Postdoctoral
Research Funds (0701026B), Southeast University Foundation of Sci-
ence and Technology (XJ2008318). We thank systems biology research
group at UCSD (University of California, San Diego) for providing the
COBRA Toolbox and BIGG database, and thank Dr. Nicolo Giorgetti at
IEEE for providing the Glpkmex program which is used to solve linear
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