Identification of genome-scale metabolic network models using experimentally measured flux profiles.

Abstract

Genome-scale metabolic network models can be reconstructed for well-characterized organisms using genomic annotation and literature information. However, there are many instances in which model predictions of metabolic fluxes are not entirely consistent with experimental data, indicating that the reactions in the model do not match the active reactions in the in vivo system. We introduce a method for determining the active reactions in a genome-scale metabolic network based on a limited number of experimentally measured fluxes. This method, called optimal metabolic network identification (OMNI), allows efficient identification of the set of reactions that results in the best agreement between in silico predicted and experimentally measured flux distributions. We applied the method to intracellular flux data for evolved Escherichia coli mutant strains with lower than predicted growth rates in order to identify reactions that act as flux bottlenecks in these strains. The expression of the genes corresponding to these bottleneck reactions was often found to be downregulated in the evolved strains relative to the wild-type strain. We also demonstrate the ability of the OMNI method to diagnose problems in E. coli strains engineered for metabolite overproduction that have not reached their predicted production potential. The OMNI method applied to flux data for evolved strains can be used to provide insights into mechanisms that limit the ability of microbial strains to evolve towards their predicted optimal growth phenotypes. When applied to industrial production strains, the OMNI method can also be used to suggest metabolic engineering strategies to improve byproduct secretion. In addition to these applications, the method should prove to be useful in general for reconstructing metabolic networks of ill-characterized microbial organisms based on limited amounts of experimental data.

Full text

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Identification of Genome-Scale Metabolic
Network Models Using Experimentally
Measured Flux Profiles
Markus J. Herrga ˚rd, Stephen S. Fong¤, Bernhard Ø Ø. Palsson*
Department of Bioengineering, University of California San Diego, La Jolla, California, United States of America
Genome-scale metabolic network models can be reconstructed for well-characterized organisms using genomic
annotation and literature information. However, there are many instances in which model predictions of metabolic
fluxes are not entirely consistent with experimental data, indicating that the reactions in the model do not match the
active reactions in the in vivo system. We introduce a method for determining the active reactions in a genome-scale
metabolic network based on a limited number of experimentally measured fluxes. This method, called optimal
metabolic network identification (OMNI), allows efficient identification of the set of reactions that results in the best
agreement between in silico predicted and experimentally measured flux distributions. We applied the method to
intracellular flux data for evolved Escherichia coli mutant strains with lower than predicted growth rates in order to
identify reactions that act as flux bottlenecks in these strains. The expression of the genes corresponding to these
bottleneck reactions was often found to be downregulated in the evolved strains relative to the wild-type strain. We
also demonstrate the ability of the OMNI method to diagnose problems in E. coli strains engineered for metabolite
overproduction that have not reached their predicted production potential. The OMNI method applied to flux data for
evolved strains can be used to provide insights into mechanisms that limit the ability of microbial strains to evolve
towards their predicted optimal growth phenotypes. When applied to industrial production strains, the OMNI method
can also be used to suggest metabolic engineering strategies to improve byproduct secretion. In addition to these
applications, the method should prove to be useful in general for reconstructing metabolic networks of ill-
characterized microbial organisms based on limited amounts of experimental data.
Citation: Herrga ˚rd MJ, Fong SS, Palsson BØ (2006) Identification of genome-scale metabolic network models using experimentally measured flux profiles. PLoS Comput Biol
2(7): e72. DOI: 10.1371/journal.pcbi.0020072
Introduction
Constraint-based models [1] have been successfully used to
describe the steady-state functionality of genome-scale
metabolic networks in a variety of microbial organism as
well as specific mammalian cell types and organelles [2–4].
These models represent the metabolic network through a
series of physico-chemical constraints including stoichiomet-
ric network connectivity that delineate the space of allowed
metabolic fluxes in an organism. In addition to defining the
space of allowed flux distributions, constraint-based models
can be used to obtain a particular flux distribution by finding
the optimal distribution given a particular objective function
(e.g., growth or ATP production) using flux balance analysis
(FBA) [5,6]. These predicted flux distributions can then be
compared to experimental measurements in order to further
refine the model. For example, in [3] a genome-scale
metabolic model for yeast was used to predict growth
phenotypes of gene deletion strains, and mispredictions were
used to guide the iterative refinement of the model. However,
the approach used in [3] required careful manual evaluation
of the mispredictions, and in many cases clear reasons for
incorrect predictions could not be identified.
Compared to other types of modeling approaches, such as
kinetic models [7], constraint-based models have the advant-
age of having very few parameters that need to be determined
from experimental data. Due to the small numbers of
parameters required, genome-scale models of metabolic
networks have been built using primarily information in the
databases and literature without the direct use of exper-
imental data. There are, however, situations where they may
be gaps in our understanding of the metabolic network
structure in a particular organism. For less well-characterized
organisms there may be only limited direct experimental
evidence to determine which reactions should be present in
the metabolic network. Even for well-studied organisms such
as Escherichia coli or yeast there may be uncertainties about
specific cofactors or reaction mechanisms used in a particular
Editor: Chris Sander, Memorial Sloan Kettering Cancer Center, United States of
America
Received October 31, 2005; Accepted May 10, 2006; Published July 7, 2006
A previous version of this article appeared as an Early Online Release on May 10,
2006 (DOI: 10.1371/journal.pcbi.0020072.eor).
DOI: 10.1371/journal.pcbi.0020072
Copyright: ? 2006 Herrga ˚rd et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: FBA, flux balance analysis; MILP, mixed-integer linear program;
MOMA, minimization of metabolic adjustment; OMNI, optimal metabolic network
identification; PDH, pyruvate dehydrogenase; pgi, phosphoglucose isomerase; ppc,
phosphoenolpyruvate carboxylase; tpi, triose phosphate isomerase
* To whom correspondence should be addressed. E-mail: palsson@ucsd.edu
¤ Current address: Department of Chemical and Life Science Engineering, Virginia
Commonwealth University, Richmond, Virginia, United States of America
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reaction in a specific organism. Finally, there may be
regulatory effects that prohibit the use of all the possible
metabolic reactions simultaneously under any particular
growth condition.
In all the cases described above one would like to identify
the correct active reactions to be included in the model from
a larger set of possible enzymatic reactions based on
comparison between model predictions and experimental
data. The experimental data types that are of particular
interest for the present application are growth rates,
substrate uptake, and byproduct secretion rates, as well as
intracellular flux distributions measured under defined
conditions [8–10]. Previously it has been found that in some
cases, such as E. coli grown on acetate [11] or yeast grown on
glucose [12], the FBA predictions are consistent with
experimentally measured growth rate and byproduct secre-
tion data. In other cases such as growth of E. coli on glycerol,
the FBA predictions were found to be comparable with
experimental data only after the strains had been adapted to
the growth environment by evolving them experimentally for
hundreds of generations [13]. However, it is also possible that
the FBA predictions remain inconsistent with experimental
data even after adaptation to a particular growth environ-
ment as is the case for some metabolic gene knock-out strains
of E. coli [14].
In cases where major discrepancies between model
predictions and experimental data exist even after adapta-
tion to a particular growth environment, methods need to be
developed to systematically find minimal modifications to the
metabolic network structure that improve model predictions.
The changes identified in this fashion can then be validated
by direct experimental techniques targeting specific novel
mechanisms suggested by the computational analysis. Effi-
cient methods for identifying parameters in small-scale
kinetic models of biochemical networks have been developed
[15–17]. Methods have also been developed for constraint-
based models to identify objective functions that are
consistent with experimentally measured metabolic flux data
[18,19]. However, approaches for identifying reaction com-
plements of genome-scale metabolic models have not been
presented before.
In this paper the development and application of computa-
tional methods for optimal metabolic network identification
(OMNI) based on in vivo measured growth rate, exchange flux
(substrate uptake and byproduct secretion rate), and intra-
cellular flux data is described. The model identification
method uses a bilevel mixed-integer optimization strategy
introduced in [20] to identify the optimal network structure
given one or more sets of experimentally determined
metabolic flux data. It is assumed that most of the reactions
in the network are active, and only a small fraction of the
reactions in the model can be either excluded from or
included in the list of active reactions. The task is then to find
which reactions need to be included in the model or removed
from the model to make the model predictions agree with the
experimental data as closely as possible (see Figure 1 for a
Figure 1. Bilevel Approach to OMNI
(A) Schematic illustration of the optimal metabolic network identification
approach. Changes in the model reaction set lead to changes in the FBA-
predicted optimal flux distribution (yellow) that can be compared to the
experimental fluxes (red).
(B) Bilevel optimization scheme for optimal metabolic network
identification.
DOI: 10.1371/journal.pcbi.0020072.g001
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Synopsis
One of the major uses of in silico models in biology is to identify
discrepancies between model predictions and experimental data
and use these discrepancies to drive discovery of novel biological
mechanisms. However, models only allow for identification of the
discrepancies; they do not necessarily provide any assistance in
discovering what are the missing or incorrect functionalities in the
model that cause these discrepancies. Herrga ˚rd et al. describe a new
in silico method, optimal metabolic network identification, or OMNI,
that performs this discovery process in an efficient and systematic
manner for genome-scale metabolic networks. Given a preliminary
metabolic network model and experimentally determined metabolic
flux data, OMNI finds the changes that need to be made to the
model so that its predictions match the experimental data as well as
possible. Herrga ˚rd et al. apply the method to identify metabolic
bottlenecks in experimentally evolved Escherichia coli strains and to
diagnose problems in strains designed through metabolic engineer-
ing strategies to overproduce specific desirable byproducts. The
OMNI method can also be adapted to number of other settings,
including identification of novel biochemical pathways in ill-
characterized organisms based on limited amounts of experimental
data.
Identification of Metabolic Networks

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schematic illustration of this process). In the present paper
we focus exclusively on applying OMNI to experimentally
evolved strains, but the method is not limited to this specific
application. Some of the potential applications of the OMNI
method are listed in Table 1.
The OMNI method is illustrated by using it to identify
potential flux bottlenecks based on experimental data for five
different E. coli knock-out strains evolved for 45–50 d on
glucose [14,21]. For each of these strains the original genome-
scale metabolic model [2] over predicts the growth rate
compared with the experimental data. The OMNI approach
allowsidentifying particular bottleneck reactions inthe model
whose removal improves the agreement between the model
predictions and experimental data. In order to provide
further support for the flux bottleneck role of the reactions
identified by OMNI, we also analyze gene expression data for
the evolved strains to find if the genes corresponding to the
bottleneck reactions are downregulated in these strains. In the
second study the OMNI approach is used as a diagnostic tool
toidentifyfluxbottlenecks inanE.colistraindesignedthrough
metabolic engineering to overproduce lactate [20]. We used
the OMNI method to identify potential in vivo flux bottle-
necks that could explain the lower-than-predicted perform-
ance of the strain. This application illustrates the utility of the
OMNI method in assisting strain development and identifying
potential novel metabolic engineering strategies.
Results
Evolved Metabolic Gene Knockout Strains
FBA applied to genome-scale metabolic network models
has been shown to correctly predict the physiological end
points of laboratory evolution of wild-type [13] and metabolic
gene knockout strains of E. coli [21]. However, for a small
fraction of the knockout strains, the model significantly
overpredicts growth rates compared to data obtained for the
endpoint strains after 45–50 d of experimental evolution
adaptation to glucose-minimal medium [21]. In these cases
one would like to characterize the behavior of the strains and
to identify reasons for discrepancies between model pre-
dictions and experimental data. Previously, metabolic flux
and gene expression profiling have been applied to charac-
terize the physiological and expression states of evolved
strains with behavior inconsistent with model predictions
[14]. In particular, two independently evolved endpoint
strains for each of the original deletion strains harboring
triose phosphate isomerase (tpi), phosphoenolpyruvate car-
boxylase (ppc), and phosphoglucose isomerase (pgi) gene
deletions were characterized. The growth rates of these
strains were overpredicted by the E. coli iJR904 genome-scale
metabolic model [2] by an average of 22% compared to the
endpoint strains obtained by experimental evolution. The
physiological characteristics of the parental deletion strains
and the evolved strains are described in Table 2.
In order to discover potential causes for the growth rate
prediction discrepancies we applied the OMNI method to
each of the endpoint strains. We did not include the pgiE2
strain in the analysis, since the model did not overpredict the
growth rate of this strain, and hence it was not expected that
the OMNI method could improve the model predictions.
Because all the discrepancies between experimental data and
in silico predictions were overpredictions, we assumed that
these were caused by reactions in the model that for some
reason could not operate at full capacity in the evolved
strains. In order to apply the OMNI method to the evolved
deletion strain data, we deleted in silico the reactions
corresponding to the genes deleted in each parental strain
(pgiD, ppcD, and tpiD) and set the uptake/secretion rates to the
mean values listed in Table 2. The set of measured target
fluxes used in the objective function of OMNI consisted of the
experimentally measured growth rate and 23 intracellular
fluxes reported in [14]. We applied the OMNI method to each
of the five evolved strains with one to four reaction deletions
allowed. Increasing the number of allowed reaction deletions
beyond four did not result in significant improvements in
model predictions.
The results from this study are summarized in Table 3,
which indicates the optimal one-to-four-reaction bottleneck
sets for each strain that was identified by the OMNI method.
This table also shows the values of the OMNI objective
function measuring the overall error in all flux predictions
and the error in growth rate prediction for each of the
modified models identified by OMNI. All the alternative
optimal reaction bottleneck sets with the same OMNI
objective value as well as all other suboptimal bottleneck sets
identified in the calculations are listed in Table S1. For all five
Table 1. Potential Applications of the OMNI Method
ApplicationInput Reaction Set Output Reaction Set
Filling in gaps in metabolic networks for
ill-characterized organisms
Evaluating functions of poorly annotated
enzymes
Identifying the correct alternative reaction
mechanism
Identifying bottleneck reactions in evolved
strains
Identify reasons for low byproduct
secretion in engineered strains
Library of inferred or candidate
metabolic reactions
Low-confidence reactions included
in the model
Set of alternative reaction mechanisms
Reactions missing from the model
Reactions that should be removed
from the model
Mechanism most consistent with
physiological data
Bottleneck reactions that limit growth
to less than predicted optimum
Bottleneck reactions that limit desired
byproduct secretion
A subset of reactions included in the model
A subset of reactions included in the model
The input reaction set is the set of reactions that the OMNI method is allowed to use to select reactions to be eliminated from or added to the model depending on the application. The
output reaction set is the set of reactions identified by the OMNI method as optimal modifications to the model in order to improve its predictive accuracy. The last two applications are
explored in this work.
DOI: 10.1371/journal.pcbi.0020072.t001
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strains studied, OMNI was capable of identifying a modified
model that predicts both growth rates and intracellular fluxes
significantly better than the parental strain model. Increasing
the number of reaction deletions also improved the overall
agreement between model predictions and experimental data
for all strains (Figure 2A). Across all five strains the average
improvement in the OMNI objective function in the best
possible modified model identified by the method compared
with the parental strain model was 48%. Increasing the
number of modifications generally decreased the growth rate
overprediction compared with the parental strain model
(Figure 2B), but for the tpiE2 strain there was a trade-off
between accurate prediction of intracellular fluxes and
growth rate.
We also investigated the correspondence between the
bottleneck reactions identified by the OMNI method and
the experimentally determined gene expression changes in
the evolved strains compared with the wild-type E. coli strain
[14]. For this purpose we identified the genes associated with
each of the bottleneck reaction sets in the iJR904 model. The
statistical significance of the overlap between the sets with
downregulated genes in each strain and the bottleneck gene
sets was quantified by the significance-based (expression
score 1) and sign-based (expression score 2) scores as
described in Materials and Methods. The expression scores
reported in Table 3 indicate that the gene sets corresponding
to the bottleneck reactions were indeed enriched in genes
that were downregulated at the expression level. However,
the downregulation was not always at a statistically significant
level (pgiE1 and tpiE1 strains). The gene expression changes
corresponding to the optimal bottleneck gene sets in all the
five strains studied are summarized in Figure 3. The gene
expression changes in all strains were quite similar despite
their different physiological characteristics, and despite the
significant differences in the bottleneck reaction sets identi-
fied by OMNI.
Lactate-Overproducing ptaA-pfk Strains
The OptKnock in silico strain design methods suggests
gene deletion strategies that link overproduction of a desired
byproduct to biomass production through stoichiometric
coupling [20]. The idea is that the designed strains could then
be evolved in laboratory conditions towards higher growth
rates, and at the same time, they could achieve higher
byproduct secretion rates. This approach was applied to the
iJR904 model to design two- and four-gene deletion strains of
E. coli that overproduce lactic acid [20]. Previously, three of
these designs were implemented in vivo, and three inde-
pendent experimental evolutions in anaerobic glucose
minimal media conditions of up to 60 d were performed
starting with each parental strain [22]. Strains derived from
two of the three designs achieved the predicted optimal
lactate secretion and growth rates after 60 d. However, after
60 d of evolution, strains derived from one of the parental
deletion strains (pfkA-pta double deletion) had growth rates
that were 18%–32% lower than what was predicted by FBA.
This strain actually evolved to the predicted optimal
phenotype after 20 d, but further experimental evolution
resulted in a suboptimal phenotype.
We applied the OMNI method as a diagnostic tool to
identify reasons for the suboptimal growth of the evolved
pfkA-pta strains. The physiological data for the three
independently evolved strains that were used as an input to
the OMNI approach are listed in Table 4. Glucose uptake and
lactate secretion were used as constraints in the OMNI
approach, and the objective consisted of the growth rate and
acetate/ethanol secretion rates. No intracellular flux data was
used in this application of the OMNI method. OMNI
identified the same one- and two-reaction bottleneck sets in
each of the three strains. Further reaction deletions in the
model reduced the objective function somewhat, but resulted
in higher growth rate prediction errors, and hence were
excluded from further analysis (all solutions are listed in
Table S2). The reaction bottlenecks identified in all the
strains were either PDH (pyruvate dehydrogenase) alone or
PDH together with ATPS4r (ATP synthase). The OMNI
objective values and growth rate prediction errors were both
reduced significantly for the modified models compared with
the parental strain model (Table 5). In particular, the error in
growth rate prediction is reduced to less than 10% of the
Table 2. Experimentally Measured Physiological Parameters of Unevolved and Evolved Knockout Strains and Predicted Growth Rates
from E. coli iJR904 Model
StrainDays
Evolved
GURASR PSR In Vivo
GR
GR Standard
Error
In Silico
GR
GR Prediction
Error (%)
Wild-type
pgiP
pgiE1
pgiE2
ppcP
ppcE1
ppcE2
tpiP
tpiE1
tpiE2
0
0
8.8
2.3
5.8
5.6
3
8.1
7.8
2.7
7.8
7.3
4.5
0.1
2.6
0
1.1
2.2
2.2
0.2
1
0.9
0
0
0
0
0
0
0
0
0
0
0.63
0.17
0.34
0.53
0.22
0.55
0.56
0.18
0.51
0.49
0.03
0
0.06
0.03
0.01
0.05
0.01
0.02
0.02
0.02
0.70
0.18
0.45
0.50
0.22
0.68
0.66
0.21
0.66
0.61
10.4
5.7
32.6
?6.5
1.9
24.4
17.0
15.3
29.0
25.3
50
50
0
45
45
0
50
50
Growth rates (GR) are reported in units of 1/h and uptake/secretion rates in units of mmol/gDW/h. P refers to the parental deletion mutant and E1/E2 to the two independently evolved
strains starting with the parental mutant. The error reported in the last column is the percentage error of the in silico growth rate prediction compared with the in vivo measured mean
growth rate.
GUR, glucose uptake rate; ASR, acetate secretion rate; PSR, pyruvate secretion rate.
DOI: 10.1371/journal.pcbi.0020072.t002
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experimental growth rate for all three independently evolved
strains. The three evolved strains were also subjected to gene
expression profiling using oligonucleotide arrays, and sig-
nificantly downregulated genes in each of the evolved pfkA-pta
strains compared with the wild-type E. coli were identified
(Materials and Methods). The bottleneck-associated genes
were significantly enriched in downregulated genes in all
three strains (Table 5).
Discussion
We have introduced a new approach, OMNI, for identifying
changes in the reaction complement of a genome-scale
metabolic model that are needed to minimize the discrepancy
between model predictions of optimal flux distributions and
experimentally measured flux data. Based on earlier exper-
imental work it is known that the optimal predictions should
be quantitatively comparable to measured uptake, secretion,
and growth rates for microbial strains adapted to specific
growth conditions through experimental evolution. The
application of the OMNI approach explored in this paper is
using it as a tool to identify potential sources of model
mispredictions when exchange or intracellular flux data for
experimentally evolved strains is available.
When applied to physiological data from five experimen-
tally evolved E. coli knockout strains, the OMNI method
identified potential bottleneck reactions in the iJR904 model;
removing those reactions from the model reduced the
discrepancy between experimentally measured and predicted
fluxes (Figure 2). The deletion of at least two reactions was
Table 3. Reaction Bottlenecks for the pgiE1, ppcE1, ppcE2, tpiE1, and tpiE2 Strains Identified Using the OMNI Approach
Reaction Number of
Reactions
OMNI
Objective
Growth Rate
Error (%)
Expression
Score 1
Expression
Score 2
Reaction DescriptionSubsystem
pgiE10
2
12.1
11.5
32.6
27.5
—
0.2
—
99.0
—
MTHFC
—
Methenyltetrahydrofolate
cyclohydrolase
NADH dehydrogenase
Deoxyribose-phosphate aldolase
NADH dehydrogenase
Fumarate reductase
—
Transketolase
2-oxogluterate dehydrogenase
Transketolase
2-oxogluterate dehydrogenase
NADH dehydrogenase
Fumarate reductase
—
Transketolase
2-oxogluterate dehydrogenase
Transketolase
NADH dehydrogenase
NADH dehydrogenase
Succinyl-CoA synthetase
(ADP-forming)
Formate dehydrogenase
2-oxogluterate dehydrogenase
NADH dehydrogenase
NADH dehydrogenase
—
6-phosphogluconolactonase
6-phosphogluconate dehydratase
6-phosphogluconolactonase
—
6-phosphogluconolactonase
6-phosphogluconate dehydratase
6-phosphogluconolactonase
Cytochrome oxidase bo3
2-dehydro-3-deoxy-phosphogluconate
aldolase
6-phosphogluconolactonase
6-phosphogluconate dehydratase
6-phosphogluconolactonase
UTP-glucose-1-phosphate
uridylyltransferase
Phosphopentomutase 2 (deoxyribose)
—
Folate metabolism
NADH6
DRPA
NADH6
FRD3
—
TKT2
AKGDH
TKT2
AKGDH
NADH6
FRD3
—
TKT2
AKGDH
TKT2
NADH6
NADH8
SUCOAS
Oxidative phosphorylation
Alternate carbon metabolism
Oxidative phosphorylation
Citrate cycle (TCA)
—
Pentose phosphate cycle
Citrate cycle (TCA)
Pentose phosphate cycle
Citrate cycle (TCA)
Oxidative phosphorylation
Citrate cycle (TCA)
—
Pentose phosphate cycle
Citrate cycle (TCA)
Pentose phosphate cycle
Oxidative phosphorylation
Oxidative phosphorylation
310.4 20.70.41.3
ppcE10
1
2
60.1
51.7
46.2
24.4
23.0
22.5
—
14.0
99.0
—
13.7
99.0
3 34.7 13.63.04.7
ppcE20
1
2
49.1
40.3
35.2
17.0
15.8
15.2
—
99.0
1.9
—
13.4
99.0
328.8 7.29.813.5
Citrate cycle (TCA)
Oxidative phosphorylation
Citrate cycle (TCA)
Oxidative phosphorylation
Oxidative phosphorylation
—
Pentose phosphate cycle
Pentose phosphate cycle
Pentose phosphate cycle
—
Pentose phosphate cycle
Pentose phosphate cycle
Pentose phosphate cycle
Oxidative phosphorylation
Pentose phosphate cycle
426.76.9 4.72.8FDH2
AKGDH
NADH6
NADH8
—
PGL
EDD
PGL
—
PGL
EDD
PGL
CYTBO3
tpiE10
1
2
33.6
25.5
9.8
29.0
21.9
20.2
—
1.2
1.0
—
13.0
13.2
tpiE20
1
2
33.1
24.3
10.7
25.3
18.5
16.8
—
13.8
99.0
—
13.0
13.4
310.4 12.71.9 13.0
EDA
PGL
EDD
PGL
GALU
Pentose phosphate cycle
Pentose phosphate cycle
Pentose phosphate cycle
Alternate carbon metabolism
49.916.2 99.099.0
PPM2Alternate carbon metabolism
The value of the OMNI objective function (described in detail in the text) measuring the overall agreement between predicted and observed fluxes is shown together with the error in
growth rate prediction (as percentage of the in vivo growth rate). In addition, the table also shows two different scores measuring the overlap between a particular bottleneck gene set
identified by OMNI and the set of downregulated genes in any strain (see Materials and Methods for details).
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required in order to improve agreement between model
predictions and experimental data significantly. This result
indicates that a full enumeration of all possible reaction
deletions would be highly inefficient compared with the
bilevel optimization approach used in the OMNI method.
Typically, the bottlenecks identified for the two independ-
ently evolved strains, starting with the same parental strain,
were exactly the same or at least in the same metabolic
subsystem. Even in cases where the optimal bottleneck sets
for the two independently evolved strains were different, such
as ppcE1 and ppcE2, there were slightly suboptimal solutions
for each strain that were more similar with each other than
the optimal solutions (Table S1).
The comparison between gene expression changes in
evolved strains and the bottlenecks identified by the OMNI
approach showed that in many cases the genes corresponding
to the bottleneck reactions were downregulated in the
evolved strains compared with the wild-type strain. Even
when only genes that were statistically significantly down-
regulated (false discovery rate of 5%) in the evolved strain
were considered, most of the gene sets corresponding to the
bottleneck reactions reported in Table 3 were significantly
enriched in downregulated genes. Only in the pgiE1 and
tpiE1 strains did the genes corresponding to most optimal
bottleneck sets fail to show a significant degree of down-
regulation. The actual expression profiling data shown in
Figure 3 indicates a source of potential discrepancies
between gene expression changes and bottleneck reactions
identified using OMNI. In many cases, genes corresponding
to a reaction identified as a bottleneck by OMNI act as
isozymes or are part of a complex catalyzing the same
reaction. In these cases only one of the isozymes or complex
members may be downregulated, and this downregulation
Figure 2. The Dependence of the Model Prediction Errors on the Number of Reactions Deleted from the Parental Strain Model
(A) The overall combined intracellular and growth rate prediction error as measured by the percentage of the OMNI objective value for the optimal
modified model compared with the objective value for the parental strain model.
(B) The growth rate prediction error (percentage of the experimental growth rate) for the optimal modified model identified by the OMNI method.
DOI: 10.1371/journal.pcbi.0020072.g002
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may be sufficient to restrict the flux through the correspond-
ing reaction significantly.
The combination of the OMNI approach applied to
experimentally measured flux data together with gene
expression profiling data allows us to identify potential
regulatory constraints that may limit the ability of strains to
evolve to achieve predicted optimal growth phenotypes. For
example, in the case of the tpiE1 and tpiE2 strains, the key
bottlenecks identified by OMNI were in the pentose
phosphate pathway (PGL/G6PDH2r and EDA/EDD), and the
expression data strongly supported the downregulation of
this pathway in the evolved strains. It is not clear why the
expression of the genes in the pentose phosphate pathway is
lower in the evolved strain than in the wild-type strain, since
allowing maximal possible flux through this pathway should
be advantageous for the cell based on the model predictions.
One possibility is that the cells are trying to reduce NADPH
Figure 3. Correspondence between Bottleneck Reactions Identified by the OMNI Approach and Gene Expression Changes in the Evolved Knockout
Strains Relative to the Wild-Type E. coli Strain
Expression changes, reported as log2ratios and statistically significant expression changes (see Materials and Methods for details) in each evolved strain,
are shown byþ/?signs depending on the direction of the change. The reaction names are listed in the first column and the corresponding genes are
listed in the second column. The genes corresponding to the best combination of reaction bottlenecks of up to three reactions identified by the OMNI
approach (Table 3) for each strain are shown by white boxes.
DOI: 10.1371/journal.pcbi.0020072.g003
Table 4. Physiological Parameters of the 60-d Evolved pfkA-pta
Strains under Anaerobic Conditions
Strain GUR LSRIn Vivo
GR
In Silico
GR
GR Prediction
Error (%)
pfkA-ptaE1
pfkA-ptaE2
pfkA-ptaE3
12.9
17.3
12
12.4
17.9
12.3
0.2
0.26
0.18
0.24
0.34
0.21
18.3
31.6
17.8
Growth rates (GR) are reported in units of 1/h and uptake/secretion rates in units of
mmol/gDW/h. The acetate and ethanol secretion rates were determined to be negligible
in all three endpoint strains based on the experimental data.
DOI: 10.1371/journal.pcbi.0020072.t004
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production by this pathway in order to compensate for the
partial loss of ability to consume this cofactor by reactions
linked to the deleted tpi gene reaction.
As described above the pfkA-pta deletion strain studied here
was designed using the OptKnock computational strain
design to maximize the production of lactate while still
maintaining reasonable growth rate [20]. Here, we used the
OMNI approach, which is a modification of the same bilevel
optimization procedure that OptKnock uses to diagnose why
the designed strain did not perform as well as the model
predicted. In particular, the results obtained here suggest that
the primary bottleneck in the evolved pfkA-pta strains would
be the PDH reaction. This conclusion is also supported by the
expression profiling data that for all three independently
evolved strains shows significant downregulation of the
subunits of the PDH complex (Table 5). It is possible that
overexpressing components of the PDH complex in the
evolved pfkA-pta strain might remove the growth rate
limitation, but since this also would potentially result in
higher ethanol production, more complex strain engineering
might be required to improve the growth rate of the pfkA-pta
strain. The OMNI approach proposed in this work would
then not only be useful for model building, but could also be
used to aid in developing improved strains for metabolic
engineering applications.
The application of the OMNI method to evolved E. coli
strains demonstrated only one potential use of the method.
The use of the OMNI method is not limited to experimentally
evolved strains, as variants of FBA, such as minimization of
metabolic adjustment (MOMA) [23], have been shown to allow
accurate prediction of metabolic phenotypes for nonevolved
metabolic gene deletion strains. The inner FBA problem in
the bilevel optimization procedure that is used in the OMNI
approach can be replaced by a variant of the MOMA method
[20]. With this modification the OMNI method should also
prove to be useful in characterizing sources of mispredictions
for nonevolved strains. For ill-characterized organisms the
combined OMNI/MOMA method would allow refinement of
the network structures of specific subsystems based solely on
systems level data such as exchange fluxes. For this type of
application it may be necessary to use flux data measured in
multiple different media conditions that require the use of
different reaction sets. The inner problem of the OMNI
approach can include multiple independent FBA or MOMA
problems representing different media conditions or gene
deletion strains so that the OMNI approach can be adapted to
simultaneously use multiple flux datasets.
Using multiple datasets generated under different con-
ditions would also avoid potential issues with over fitting the
model to only one flux profile. In particular, some of the flux
datasets could be used to train the model using the OMNI
approach while the predictive ability of the modified models
identified by OMNI could be tested against a set of
independent flux datasets. In the applications to evolved
strains discussed in this paper it would be possible to apply
the OMNI method to multiple flux profiles from the
independently evolved endpoint strains for each parental
strain (e.g., ppcE1 and ppcE2). However, these strains are not
necessarily genetically identical, and hence it may not be
appropriate to try to identify common flux bottleneck sets
for both strains.
Conclusions
We have developed a method, OMNI, to identify potential
changes to genome-scale metabolic models based on compar-
ing model predictions of fluxes to experimental measure-
ments. The method uses mixed-integer linear programming
to efficiently search through the space of potential metabolic
model structures. We applied the method to intracellular flux
data for five experimentally evolved E. coli strains to identify
why the model predictions for growth rates of these strains
were an average of 22% higher than the actual experimental
Table 5. Reaction Bottlenecks Identified Using the OMNI
Approach for the Evolved Lactate-Overproducing Strains pfkA-
ptaE1–3
StrainOMNI
Objective
Growth Rate
Error (%)
Expression
Score 1
Expression
Score 2
Reactions
pfkA-ptaE1 184.0 18.3
10.8
?8.4
31.6
26.4
4.5
17.8
10.9
?8.3
—
99.0
3.2
—
99.0
13.2
—
PDH
PDH/ATPS4r
—
PDH
PDH/ATPS4r
—
PDH
PDH/ATPS4r
133.8
91.6
pfkA-ptaE2 214.8
169.6
103.1
99.0
3.1
—
99.0
2.3
99.0
14.2
—
99.0
1.9
pfkA-ptaE3 158.6
116.8
78.8
The table omits bottleneck sets that did not improve the growth rate prediction even if
they decreased the overall objective value. For each strain the same bottlenecks (PDH and
PDH/ATPS4r) were identified by OMNI. PDH is the pyruvate dehydrogenase reaction and
ATPS4r the ATP synthase reaction. See the legend of Table 2 for explanation of the
column headers.
DOI: 10.1371/journal.pcbi.0020072.t005
Table 6. Descriptions of Variable Names Used in Equations 1–4
Symbol DescriptionSymbol Description
v
S
c
vmax
w
vopt
vexp
Flux distribution vector
Stoichiometric matrix
Vector of objective coefficient
Vector of maximum flux rates
Vector of weights for measured fluxes
Optimal flux distribution
Experimentally measured flux distribution
y
F
D
K
M
E
Rn
Binary vector indicating whether a reaction is part of the model or not
Set of reactions that are fixed (i.e., that can not be removed from the model)
Set of reactions that can be deleted from the model
Number of reaction deletions allowed in the model
Set of reactions with measured fluxes
Set of reactions with constrained fluxes (e.g., exchange fluxes)
nth set of previously identified reaction bottlenecks
DOI: 10.1371/journal.pcbi.0020072.t006
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measurements. The OMNI method identified specific bottle-
neck reactions in the iJR904 metabolic model that, when
removed from the model, would reduce the discrepancies
between predicted and observed fluxes significantly. Genes
corresponding to these reactions were often found to be
downregulated in the evolved strains relative to the wild-type
strain. We also applied OMNI to strains created by metabolic
engineering approaches and experimentally evolved to
optimize their metabolite production potential
[20,22,24,25]. In this application, OMNI could be used to
identify sources of lower-than-predicted secretion of desired
byproducts and to suggest potential ways to improve the
strain designs. In summary, the OMNI method provides an
efficient and flexible way to study and refine genome-scale
metabolic network reconstructions using limited amounts of
experimental data.
Materials and Methods
FBA. The FBA approach applied to genome-scale constraint-based
metabolic models can be used to make predictions of flux
distributions based on linear optimization [5,6]. In mathematical
terms, a prediction for the metabolic flux distribution vector v is
obtained as the solution of
max cTv
subject toSv ¼ 0
0 ? vi? vmax
i
:
ð1Þ
All the definitions of the variables used in this section are
summarized in Table 6. The objective coefficients are usually derived
from the cellular biomass composition so that the maximum value of
the objective function corresponds to the predicted optimal growth
rate given the network stoichiometry and media composition
represented by upper bounds on exchange reactions. The linear
nature of these problems allows developing methods for model
structure identification that are significantly different from methods
that would have to be used in modeling frameworks based on systems
of ordinary differential equations [17,26].
OMNI. The problem of finding the best possible metabolic
reaction set to match model predictions with experimental flux data
can be formulated as a bilevel optimization problem, as shown in
Figure 2. The outer optimization problem searches through a set of
reactions to include in the model, and the inner optimization
problem produces a flux distribution as a solution to a FBA problem
(Equation 1) given a particular model structure. Mathematically this
problem can be expressed as (see Table 6 for definitions of the
variables used):
yopt¼ argmin
y
i2M
vopt¼ argmaxvcTv
subject to
>
vl¼ vexp
vopt
biomass
X
8
>
wivopt
i
? vexp
i
?? ??
subject to
Sv ¼ 0
0 ? vj? vmax
0 ? vk? vmax
j
j 2 F
k
ykk 2 D
>
:
>
<
ð2Þ
l
l 2 E
biomass? vmin
yk¼ f0;1g 8k 2 D
X
k2D
ð1 ? ykÞ ¼ K:
The nonzero elements of the vector y indicate the reactions that
are included in the model and the zero elements the reactions that
are excluded from the model. The bilevel optimization formulation is
similar to the one used in the OptKnock computational strain design
approach [20]. The major difference is the objective function for the
outer problem, where OptKnock only uses a single flux instead of a
distance between predicted and experimentally determined flux
profiles. In addition, the set of constraints E includes measured
exchange fluxes that are not used in the OptKnock approach.
In the applications described in this paper the measured exchange
fluxes (set E) are not used as part of the objective except in the case of
the pfkA-pta strains; instead, the exchange fluxes are set explicitly to
their measured values. This is done because at least one of the fluxes
has to be bounded for the inner optimization problem to have a
bounded solution. By fixing the exchange fluxes we ensure that the
flux distributions satisfy at least these constraints and reduce the
problem to finding a model that minimizes the discrepancy between
the remaining experimentally measured and predicted fluxes. In
general, it is possible to move fluxes between sets E (constraints) and
M (objective function). The fluxes in set E are assumed to be more
accurately measured than those in set M so that in most applications
intracellular fluxes would be included in set M and exchange fluxes in
set E.
The reactions in the model are assumed to be partitioned into a set
of fixed reactions that cannot be removed from the model (F), and a
set of reactions that can be deleted (D). For example, the set F could
be the set of reactions included in the model with high confidence
and D could be a set of potential reactions inferred (e.g., by sequence
homology only or all known biochemical reactions not included in set
F from a large database such as the Kyoto Encyclopedia of Genes and
Genomes [KEGG]) [25,27]. The stoichiometric matrix S contains the
stoichiometric coefficients for all reactions in both sets D and F, but
only parts of this matrix would be used as constraints depending on
the nonzero elements in y. In the inner optimization problem a
standard FBA problem is solved for a particular metabolic network
structure defined by the set of binary variables yk. The outer problem,
on the other hand, treats the elements of the vector y as variables in
its own optimization problem that aims to minimize the discrepancy
between observed (vexp) and predicted optimal flux (vopt) distributions.
The outer problem will systematically search over the space of
possible integer vectors y to find the one that minimized the
discrepancy, and hence discover the best possible metabolic model
given the experimental flux data.
Solving the bilevel optimization problem directly using combina-
torial approaches would be time consuming due to the large number
of possible combinations of reactions that could be added to the
model or removed from the model. Fortunately, the linear program-
ming nature of the inner problem allows formulating the overall
problem as a single mixed-integer linear program (MILP) following
the approach developed in [20]. This formulation is based on duality
theory and amounts to converting the inner linear program to a
larger set of equalities and inequalities that every optimal solution to
the linear program has to satisfy [28]. By applying the primal–dual
approach to the bilevel optimization problem (Figure 2), the original
bilevel problem can be written as a single large optimization problem
as shown in [20]. The conversion of the inner problem from an
optimization problem to a set constraint also bypasses the issue with
alternative optimal solutions to the inner FBA problem [29]. The
OMNI method will always evaluate the distance between predicted
and experimentally determined flux vectors (i.e., the OMNI objective
function) based on the alternative optimal flux vector that is closest
to the experimental one.
The objective function in Equation 2 can be expressed in an
equivalent form that only involves linear terms and constrains as:
X
whereDþ
D?
Dþ
i2M
wiðDþ
iþ D?
i? vopt
i? vexp
i; D?
iÞ
i
? vexp
? vopt
i
ii
i? 0:
ð3Þ
Using this form of the objective function converts the overall
problem (Equation 2) into a single large MILP. After identifying one
potential set of model changes we also identified other reaction
deletions that would result in the same prediction by repeatedly
running the OMNI algorithm and requiring it to find a solution that
is different from all of the previously found ones. This requirement
can be implemented by adding the following additional constraints to
problem (Equation 2):
X
Here N is the total number of previously obtained solutions. The
OMNI algorithm was run with these additional constraints until no
further solutions that would reduce the error between the predicted
and observed flux distribution could be found.
This MILP can be solved in most cases to optimality in a reasonable
time (a few hours) using standard MILP solvers run on a single
workstation. For genome-scale models the solver time can be reduced
significantly by first identifying the tightest possible upper and lower
k2Rn
yk.0 n ¼ 1...N:
ð4Þ
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bounds for each reaction using flux variability analysis [30]. For many
reactions these bounds turn out to be zero under in a particular
environment so that these blocked reactions and their corresponding
integer variables can be removed from the model without any loss of
information. In this work the CPLEX 9.0 MILP solver was used
through the GAMS interface (GAMS Development Corp., Washing-
ton, D. C., United States). The Matlab scripts used to set up the OMNI
optimization problems and to solve the problem using CPLEX
through GAMS are provided in Dataset S1.
E. coli model. The E. coli iJR904 genome-scale metabolic model
containing 1,075 reactions catalyzed by 904 distinct enzymes [2] was
used as the base model for this study. For each particular gene
deletion strain studied, the reactions catalyzed by the deleted genes
where removed form the model to obtain the parental strain model.
For the purposes of the OMNI approach we considered all the
reactions involved in central metabolism, including oxidative
phosphorylation and alternative carbon source metabolism (227
reactions total), to be potential candidate bottleneck reactions (set D
in Equation 2). In the OMNI objective each of the discrepancies
between predicted and measured fluxes was weighted in proportion
to the inverse of the experimental standard deviation of the flux
measurement (wiin Equation 2). Furthermore, in order to bias the
search towards network structures that would result in correct
growth rate predictions, the growth rate error was weighted as 80%
of the overall objective, and the cumulative error over all the other
fluxes in the objective function was weighted as 20% of the overall
objective. We also performed the OMNI calculations with other
weight values (90% and 100% growth rate weight), and these are
reported in Protocol S1. The optimal bottleneck sets for the
calculations where the intracellular flux data was used (80% and
90% growth rate weight) were generally consistent with each other.
Gene expression profiling and data analysis. The gene expression
profiling for the three independently evolved pfkA-pta deletion strains
was done as described in [14] using Affymetrix (Santa Clara,
California, United States) E. coli Antisense Genome Arrays. Three
biological replicates were used for each of the three evolved pfkA-pta
deletion strains and six biological replicates for the wild-type strain
all grown aerobically on glucose-minimal media. Probe intensity
calculations and normalization was performed using the Robust
Multichip Average (RMA) method [31] and differentially expressed
genes between each evolved strain and wild-type were detected using
a standard t test applied to log-transformed data. The p value cut-off
for significance was determined for each strain separately using the
Benjamini-Hochberg approach [32], with a false discovery rate of 5%.
For each set of bottleneck reactions the corresponding genes were
identified based on the gene-reaction associations in iJR904 [2]. In
order to assess whether the genes in a particular bottleneck gene set
were significantly enriched in the set of downregulated genes we used
the hypergeometric distribution to calculate the probability p that
the observed overlap between the gene sets occurs by random chance.
The gene expression overlap scores reported in Tables 2 and 4 are
then defined as ?log10(p). For the expression score 1 (significance-
based score), the set of downregulated genes was determined by using
the p value threshold determined as described above and requiring
that the log2ratio between mean evolved deletion strain and wild-
type expression levels was negative (i.e., gene is downregulated). For
the expression score 2 (sign-based score), the downregulated genes
were determined based on the sign of the log2ratio between the
evolved knockout strain and the wild-type strain, ignoring the
significance of the expression change. If the p value used to calculate
either score was exactly equal to zero (corresponding to case where
all the bottleneck genes are downregulated), the score was assigned
the value 99 in Tables 2 and 4.
Supporting Information
Dataset S1. Compressed Archive of the Matlab and GAMS Scripts
Needed to Run OMNI Calculations
Found at DOI: 10.1371/journal.pcbi.0020072.sd001 (40 KB ZIP).
Protocol S1. Supplementary Results Document Describing Sensitivity
Analysis of the Results Presented in the Paper
Found at DOI: 10.1371/journal.pcbi.0020072.sd002 (33 KB DOC).
Table S1. Optimal and Suboptimal Bottleneck Reaction Sets
Identified for the Five Single-Deletion Strains
Found at DOI: 10.1371/journal.pcbi.0020072.st001 (30 KB XLS).
Table S2. Optimal and Suboptimal Bottleneck Reaction Sets
Identified for the pfkA-pta Strain
Found at DOI: 10.1371/journal.pcbi.0020072.st002 (20 KB XLS).
Table S3. Gene Expression Datasets for the Three Evolved Strains
Derived from the pfkA-pta Parent Strain
Found at DOI: 10.1371/journal.pcbi.0020072.st003 (1.6 MB XLS).
Accession Numbers
The EcoCyc (http://biocyc.org/ECOLI/NEW-IMAGE?type¼GENE&
object¼b1851) accession numbers for the genes discussed in this
paper are tpi (b3919), ppc (b3956), pgi (b4025), pfkA (b3916), pta
(b2297), lpdA (b0116), sucA (b0726), sucB (b0727), pgl (b0767), galU
(b1236), fdnG (b1474), fdnH (b1475), fdnI (b1476), edd (b1851), nuoN
(b2276), nuoM (b2277), nuoL (b2278), nuoK (b2279), nuoJ (b2280), nuoI
(b2281), nuoH (b2282), nuoG (b2283), nuoF (b2284), nuoE (b2285), nuoC
(b2286), nuoB (b2287), nuoA (b2288), fdoI (b3892), fdoH (b3893), fdoG
(b3894), fdhF (b4079), frdD (b4151), frdC (b4152), frdB (b4153), frdA
(b4154), deoC (b4381), and deoB (b4383).
Acknowledgments
We thank Trey Ideker, Shankar Subramaniam, Bing Ren, Kenneth
Kreutz-Delgado, Costas Maranas, Tony Burgard, and Jennie Reed for
valuable discussions.
Author contributions. MJH, SSF, and BØP conceived and designed
the experiments. MJH and SSF performed the experiments. MJH and
SSF analyzed the data. SSF contributed reagents/materials/analysis
tools. MJH and BØP wrote the paper.
Funding. Support for this work was provided by the National
Institutes of Health (RO1 GM071808) and National Science Founda-
tion (BES-0331342).
Competing interests. The authors have declared that no competing
interests exist.
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Identification of Metabolic Networks