Conservation of high-flux backbone in alternate optimal and near-optimal flux distributions of metabolic networks.

Abstract

Constraint-based flux balance analysis (FBA) has proven successful in predicting the flux distribution of metabolic networks in diverse environmental conditions. FBA finds one of the alternate optimal solutions that maximizes the biomass production rate. Almaas et al. have shown that the flux distribution follows a power law, and it is possible to associate with most metabolites two reactions which maximally produce and consume a given metabolite, respectively. This observation led to the concept of high-flux backbone (HFB) in metabolic networks. In previous work, the HFB has been computed using a particular optima obtained using FBA. In this paper, we investigate the conservation of HFB of a particular solution for a given medium across different alternate optima and near-optima in metabolic networks of E. coli and S. cerevisiae. Using flux variability analysis (FVA), we propose a method to determine reactions that are guaranteed to be in HFB regardless of alternate solutions. We find that the HFB of a particular optima is largely conserved across alternate optima in E. coli, while it is only moderately conserved in S. cerevisiae. However, the HFB of a particular near-optima shows a large variation across alternate near-optima in both organisms. We show that the conserved set of reactions in HFB across alternate near-optima has a large overlap with essential reactions and reactions which are both uniquely consuming (UC) and uniquely producing (UP). Our findings suggest that the structure of the metabolic network admits a high degree of redundancy and plasticity in near-optimal flow patterns enhancing system robustness for a given environmental condition.

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RESEARCH ARTICLE
Conservation of high-flux backbone in alternate optimal
and near-optimal flux distributions of metabolic networks
Areejit Samal
Received: 7 April 2009/Accepted: 12 May 2009/Published online: 30 May 2009
? The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract
has proven successful in predicting the flux distribution of
metabolic networks in diverse environmental conditions.
FBA finds one of the alternate optimal solutions that
maximizes the biomass production rate. Almaas et al. have
shown that the flux distribution follows a power law, and it
is possible to associate with most metabolites two reactions
which maximally produce and consume a given metabolite,
respectively. This observation led to the concept of high-
flux backbone (HFB) in metabolic networks. In previous
work, the HFB has been computed using a particular
optima obtained using FBA. In this paper, we investigate
the conservation of HFB of a particular solution for a given
medium across different alternate optima and near-optima
in metabolic networks of E. coli and S. cerevisiae. Using
flux variability analysis (FVA), we propose a method to
determine reactions that are guaranteed to be in HFB
regardless of alternate solutions. We find that the HFB of a
particular optima is largely conserved across alternate
optima in E. coli, while it is only moderately conserved in
S. cerevisiae. However, the HFB of a particular near-
optima shows a large variation across alternate near-optima
in both organisms. We show that the conserved set of
reactions in HFB across alternate near-optima has a large
overlap with essential reactions and reactions which are
both uniquely consuming (UC) and uniquely producing
(UP). Our findings suggest that the structure of the meta-
bolic network admits a high degree of redundancy and
plasticity in near-optimal flow patterns enhancing system
robustness for a given environmental condition.
Constraint-based flux balance analysis (FBA)
Keywords
analysis (FBA) ? Alternate optima and near-optima ?
Flux variability analysis (FVA) ? Flux plasticity
Complex network ? Flux balance
Introduction
The study of the structure and dynamics of complex bio-
logical networks is important for understanding the system-
level behaviour of living organisms (Hartwell et al. 1999;
Barabasi and Oltvai 2004). The metabolic network is one
such physicochemical system inside the cell which can be
viewed as a complex network consisting of metabolites
involved in enzyme catalyzed reactions. Recent advances
in the development of high-throughput data collection
techniques coupled with systematic analysis of fully
sequenced genomes has led to the reconstruction of
organism-specific metabolic networks (Feist et al. 2009).
Although the list of metabolic reactions along with the
involved stoichiometries is largely known for many
organisms, we currently lack the knowledge of most kinetic
rate constants and enzyme concentrations in these net-
works. In the absence of detailed knowledge of various
kinetic parameters, constraint-based modelling techniques
like Flux balance analysis (FBA), which utilizes primarily
structural information of the network in the form of stoi-
chiometry of various reactions, have proven useful in
predicting the quantitative behaviour of the cellular
metabolism under different environmental conditions
(Varma and Palsson 1994; Edwards et al. 2001; Kauffman
et al. 2003; Price et al. 2004).
FBA is a computational technique which can be used to
obtain a particular flux distribution for a given environ-
mental condition that maximizes certain objective function
which is usually taken to be the biomass production rate
A. Samal (&)
Max Planck Institute for Mathematics in the Sciences,
Inselstr. 22, 04103 Leipzig, Germany
e-mail: samal@mis.mpg.de
123
Syst Synth Biol (2008) 2:83–93
DOI 10.1007/s11693-009-9025-8

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(see ‘‘Methods’’ section for details). However, for large
scale metabolic networks, it has been shown that there are a
large number of different flux distributions or alternate
solutions for a given environmental condition with the
exact same value for the objective function (Lee et al.
2000; Mahadevan and Schilling 2003; Reed and Palsson
2004). Thus, FBA finds one of many possible alternate
optimal solutions for a given environmental condition.
Mixed integer linear programming (MILP) has been used
to enumerate different alternate optimal flux distributions
for a given environmental condition (Lee et al. 2000; Reed
and Palsson 2004). However, for large scale metabolic
networks, the number of alternate optimal solutions has
been shown to be extremely large, and it is computationally
infeasible using MILP to enumerate all possible alternate
optima (Reed and Palsson 2004). Although, it is not fea-
sible to enumerate all alternate optima, flux variability
analysis (FVA) (Mahadevan and Schilling 2003) can be
used to determine the minimum and maximum flux value
of each reaction across the set of alternate optimal solutions
for a given environmental condition. Specifically, using
FVA, we can determine reactions that can have a nonzero
flux in some alternate optima and the reactions whose flux
vary across different alternate optima for a given envi-
ronmental condition.
In the FBA approach, we obtain a particular solution by
imposing the optimality criterion wherein a flux distribu-
tion satisfying the governing constraints along with maxi-
mal biomass production rate is predicted as the phenotype
for a given environmental condition. However, the optimal
flux distribution with maximal biomass production may not
always explain the observed behaviour for a given envi-
ronmental condition (Ibarra et al. 2002; Segre et al. 2002;
Fong et al. 2005; Schuetz and Kuepfer 2007; Schuster
et al. 2008). Further, wet lab experiments have shown that
the organism may initially grow suboptimally under certain
media, i.e., the observed growth rate is less than the
maximum value predicted by FBA, and then evolve to
grow at the predicted optimal rate after several generations
(Ibarra et al. 2002; Fong et al. 2005). It is plausible that
there are multiple objectives that the cell is trying to
optimize to different extent under different environments.
Thus, the metabolic network may operate under suboptimal
growth conditions under certain environmental conditions.
Hence, it is important to study the suboptimal flux distri-
butions, especially, near-optimal flux distributions in met-
abolic networks for different environmental conditions.
Using the optimal solution predicted by FBA, Almaas
et al. (2004) have shown that the distribution of reaction
fluxes for a given environmental condition follows a power
law in the E. coli metabolic network. The observed power
law distribution implies an inhomogeneity in the reaction
network usage with most reactions having small fluxes and
a few reactions having high fluxes. Almaas et al. (2004)
showed that the heterogeneity of overall flux distribution is
also valid to a large extent at the level of individual
metabolites. Further, for most metabolites, they were able
to identity two reactions that dominate the production and
consumption of a metabolite, respectively. Using this
property at the level of an individual metabolite, Almaas
et al. (2004) have proposed a simple algorithm to construct
a subnetwork referred to as the ‘‘high-flux backbone’’
(HFB) which contains the locally maximal flow paths for a
given environmental condition in the network. The HFB
for a given environmental condition was shown to form a
giant connected cluster with large overlap with classical
pathways described in biochemical literature. It has also
been suggested that the HFB can be used as a network
reduction algorithm.
In this paper, we have studied the activity and flux
variability of different reactions across alternate flux dis-
tributions for a given medium under optimal and subopti-
mal growth conditions in metabolic networks of E. coli and
S. cerevisiae. Almaas et al. (2004) have used a single
solution obtained using FBA from the set of different
alternate optima to compute the HFB of the metabolic
network for a given medium. In this paper, we have
investigated in detail the effect of alternate optima on the
set of reactions in HFB computed using a single solution
for a given medium in E. coli and S. cerevisiae. We have
further investigated the effect of alternate suboptimal
solutions corresponding to near-optimal growth condition
on the set of reactions in HFB computed using a single
suboptimal solution for a given medium in the two
organisms.
Methods
Flux balance analysis
Flux balance analysis is a computational modelling tech-
nique which can be used to obtain a prediction for the
fluxes of all reactions in the metabolic network and growth
rate of the organism (Varma and Palsson 1994; Kauffman
et al. 2003; Price et al. 2004). The main assumptions made
in the technique of FBA are: (a) the network reaches a
steady state for any given environmental condition wherein
the concentration of all internal metabolites and velocities
of all reactions are constant, and (b) the cell tries to adjust
its intracellular machinery or reaction fluxes so as to
maximize its growth rate or biomass production. The key
requirement for implementing FBA is the information
about the stoichiometry of all metabolic reactions known
to occur in an organism. The complete list of reactions
along with their stoichiometry is encapsulated in the
84A. Samal
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stoichiometric matrix S of dimensions m 9 n where m
denotes the number of metabolites and n denotes the
number of reactions. In any metabolic steady state, each
metabolite achieves a dynamic mass balance and the vector
v = (v1,v2,...,vn)Trepresenting the fluxes through each
reaction must satisfy the equation
S ? v ¼ 0;
so as to not violate mass conservation. Equation 1 gives us a
system of linear equations relating various fluxes in the
network. In addition to linear stoichiometric constraints,
there exist thermodynamic constraints that may render
certain reactions irreversible and flux capacity constraints
which impose limitations on maximum flux through certain
reactions. These additionalconstraintscanbeusedtofurther
limit the space of allowable fluxes given by Eq. 1. To obtain
a particular solution from the space of allowable fluxes,
linear programming (LP) is used to find a fluxdistribution in
the allowable space that gives the optimal value for an
objective function. The most commonly used objective
function is the maximization of biomass production rate.
The LP formulation of the FBA approach is as follows:
ð1Þ
Maximize Z ¼ cTv
such that S ? v ¼ 0
a?vi?b
ð2Þ
where vector c corresponds to the objective function, a and
b are the lower and upper limits of individual flux values.
Mixed integer linear programming
For most large scale metabolic networks, FBA finds one of
many possible alternate optimal flux distributions that have
the same objective value for a given environmental con-
dition (Lee et al. 2000; Mahadevan and Schilling 2003;
Reed and Palsson 2004). We can use MILP to enumerate
different alternate optima. We have used here the recursive
MILP algorithm proposed by Reed and Palsson (2004) to
enumerate different alternate optima. In this method, in
addition to constraints in the LP formulation of FBA given
by Eq. 2, the following new constraints are imposed to
obtain different solutions:
X
X
yiþ wi?1;
a ? wi?vj?b ? wi
In this method each reaction i in the network is
associated with two binary variables yi and wi. NZJ
represents the set of reactions with nonzero flux value in
i2NZJ?1
yi?1
i2NZJ
wi? NZk
??
??? 1;
8i
k ¼ 1...J ? 1
8i
ð3Þ
the solution at iteration J. Given the constraints in Eq. 3, if
the binary variable yi=1 then wi=0, which forces both upper
and lower bounds for reaction i to be zero. Thus, setting
yi=1 forces the reaction i to be removed from the basis of
the next iteration. In this recursive algorithm, at each
iteration J, at least one of the nonzero fluxes of the previous
solution (NZJ-1) is removed from the basis of the current
iteration. The MILP problem given by Eqs. 2 and 3 is then
solved again to generate a different alternate solution.
Flux variability analysis
The number of alternate optima in large scale metabolic
networks is extremely large, and it can be computationally
infeasible to enumerate all alternate optima. FVA (Mahad-
evan and Schilling 2003) is an alternative technique which
can be used to determine the flux variability associated with
each reaction in the complete set of alternate optima for a
givenmedium.Initially,themaximumvalueoftheobjective
function,i.e.,Zobj,isdetermined.Thisisfollowedbysolving
two LP problems for each reaction i in the network. The LP
formulation of the FVA approach is as follows:
Case 1:
Maximize vi
such that S ? v ¼ 0
cTv ¼ Zobj
a?vi?b
Case 2:
for i ¼ 1...n
ð4Þ
Minimize vi
such that S ? v ¼ 0
cTv ¼ Zobj
a?vi?b
In the first case, the maximum flux through each reaction
that supports optimal growth rate Zobjis determined. In the
second case, the minimum flux through each reaction that
supports optimal growth rate Zobj is determined. The
difference of maximum and minimum value that supports
optimal growth rate for each reaction gives the flux
variability of that reaction. A slight modification of the LP
problem given by Eqs. 4 and 5 is able to generate the flux
variability under suboptimal conditions. The suboptimal
condition corresponding to near-optimality studied here
requires the following change in Eqs. 4 and 5: Replace the
condition cTv = Zobjwith cTv C 0.99 Zobj.
for i ¼ 1...n:
ð5Þ
Reactions involved in futile cycles
We used a recently published algorithm by Wright and
Wagner (2008) to compute the reactions in E. coli and
Conservation of high-flux backbone85
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S. cerevisiae metabolic network that are involved in
internal cycles (Beard et al. 2002; Price et al. 2002) of size
C3. The number of reactions involved in internal cycles of
size C3 in E. coli and S. cerevisiae was obtained to be 22
and 60, respectively. Using METATOOL version 5.0 (von
Kamp and Schuster 2006), we obtained a total of 9 and 28
internal cycles of size C3 in the metabolic network of
E. coli and S. cerevisiae, respectively. In order to avoid
these internal cycles, we followed the standard procedure
of constraining at least one reaction flux in each internal
cycle to zero. By examining the different internal cycles of
size C3, we decided to constrain the flux of following
reactions: ABUTt2, ACCOAL, ADK1, GLUT4, PROt4,
SERt4, THRt4 and VPAMT in the E. coli metabolic net-
work iJR904 (Reed et al. 2003) and the flux of following
reactions: ASPTAm, ACONTm, DCMPDA, FRDcm,
GHMT2r, GLUT5m, MALt2r, NDPK1, NDPK8, NDPK9,
PHCDm, SHSL1, SUCCtm, SUCCFUMtm, SUCD2_u6m
and UGLT in the S. cerevisiae metabolic network iND750
(Duarte et al. 2004) to zero to avoid futile cycles. The
abbreviations of the reactions in the two metabolic net-
works mentioned above are same as in the databases
iJR904 and iND750, respectively.
High-flux backbone
The method to determine HFB for a given medium using a
particular solution is as follows (Almaas et al. 2004):
(a)For each metabolite in the network, remove all
reactions except the reaction that produces the
metabolite with largest flux and the reaction that
consumes the metabolite with largest flux. The
metabolites that are either not produced (consumed)
are not taken into account.
If metabolite A is an educt and metabolite B is a
product of reaction R, and further R maximally
consumes A and maximally produces B, then reaction
R is part of the HFB.
(b)
Since, the set of reactions in HFB of a particular solution
may change if it was obtained from a different alternate
solution, we here propose a simple method using FVA to
determine the set of reactions that are guaranteed to be in
HFB of all alternate solutions for a given medium. Our
method is as follows:
(a)Using FVA, we obtain the minimum and maximum
flux of each reaction R in the network that can support
optimal growth for a given minimal medium.
We designate reaction R as producing (consuming)
metabolite A with largest flux across all alternate
optima, if the minimum flux for R obtained using
(b)
FVA is greater than the maximum flux for each
reaction other than R that produces (consumes) A.
For each metabolite in the network, remove all
reactions except the reaction that produces the
metabolite with largest flux and the reaction that
consumes the metabolite with largest flux across all
alternate optima.
If metabolite A is an educt and metabolite B is a
product of reaction R, and further R maximally
consumes A and maximally produces B across all
alternate optima, then reaction R is contained in HFB
of all alternate optima.
(c)
(d)
Our method described here for alternate optimal solu-
tions can also be used to determine the set of reactions that
are guaranteed to be in HFB of all alternate near-optimal
solutions.
Results and discussion
In this section, we present results from our study of the two
metabolic networks: E. coli [version iJR904 (Reed et al.
2003)] and S. cerevisiae [version iND750 (Duarte et al.
2004)]. The two networks can be downloaded from the
BiGG database: http://bigg.ucsd.edu/. The E. coli meta-
bolic network iJR904 contains 761 metabolites and 931
reactions and the S. cerevisiae metabolic network iND750
contains 1,061 metabolites and 1,149 reactions. For each
organism, the two databases additionally provide a biomass
production reaction which gives the relative contribution of
key metabolites towards cellular biomass. We have
investigated here the two metabolic networks under dif-
ferent environmental conditions corresponding to 89 and
43 aerobic minimal media for E. coli and S. cerevisiae,
respectively. Using FBA, it has been predicted elsewhere
that this set of 89 and 43 aerobic minimal media for E. coli
and S. cerevisiae, respectively, considered here are the only
aerobic minimal media that can support nonzero growth in
the two networks (Samal et al. 2006).
Activity and flux variability of reactions under optimal
and near-optimal conditions
We can determine the maximum and minimum flux
through each reaction that supports optimal growth in a
minimal medium using FVA (see ‘‘Methods’’ section for
details). A reaction is considered to be possibly active in a
medium across alternate optima, if the maximum flux value
that supports optimal growth is nonzero. A reaction is
considered to have a variable flux in a medium across
alternate optima, if the difference between maximum and
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minimum flux value that supports optimal growth is
nonzero.
Using FVA, we determined the number of reactions that
can be possibly active and the reactions with variable flux
across alternate optima for each of the 89 different minimal
media in the E. coli metabolic network. The number of
reactions that can be possibly active in a medium varies
between 278 and 332 reactions across the 89 minimal
media. The number of reactions with variable flux in a
medium varies between 34 and 83 reactions across the 89
minimal media. These results for the E. coli metabolic
network are shown in Fig. 1a. We find that 11–25% of the
reactions that can be possibly active in a given medium to
have variable flux across alternate optima for that medium.
The results obtained here for the E. coli metabolic network
using FVA for 89 different aerobic minimal media are
consistent with the results obtained earlier by Reed and
Palsson (2004) using a larger set of alternate optima for
136 different aerobic and anaerobic minimal media.
We also determined the number of reactions that can be
possibly active and the reactions with variable flux across
alternate optima for each of the 43 different minimal media
in the S. cerevisiae metabolic network. These results for the
S. cerevisiae metabolic network are shown in Fig. 2a. We
find that 22–69% of the reactions that can be possibly
active in a given medium to have variable flux across
alternate optima for that medium in S. cerevisiae. For one
of the 43 minimal media, i.e., arginine aerobic minimal
medium, we find a much larger number of possibly active
reactions and reactions with variable flux across alternate
optima compared to other minimal media (cf. Fig. 2a). A
closer look at the data for arginine minimal medium
showed that many transport reactions involved in excretion
of ammonia, guanine, urea and xanthine have a variable
flux across alternate optima. This observation explains the
larger flexibility under arginine minimal medium within
the S. cerevisiae metabolic network to achieve optimal
growth rate compared to other media. Even if we exclude
the results for the arginine minimal medium, we find that
the flux variability for a typical minimal medium in
S. cerevisiae is greater than the flux variability for a typical
minimal medium in E. coli under optimal growth
Optimal
0
200
400
600
800
Near-optimal 99%
Activity and flux variability
0
200
400
600
800
Suboptimal 5%
Different media
1214161 81
0
200
400
600
800
(a)
(b)
(c)
Fig. 1 Activity and flux
variability of reactions under a
optimal growth, b near-optimal
growth (C99% of the optimal
rate) and c suboptimal growth
(C5% of the optimal rate) for 89
different minimal media in the
E. coli metabolic network. Each
bar corresponds to one of the 89
different minimal media. The
height of the bar gives the
number of reactions that can be
possibly active and the height
upto which the bar is filled with
grey colour gives the number of
reactions whose flux vary across
alternate solutions for that
medium
Conservation of high-flux backbone 87
123