FFCA: a feasibilitybased method for flux coupling analysis of metabolic networks
 Laszlo David†^{1, 2, 3},
 SayedAmir Marashi†^{2, 4}Email author,
 Abdelhalim Larhlimi^{5},
 Bettina Mieth^{2, 6} and
 Alexander Bockmayr^{1, 2}Email author
DOI: 10.1186/1471210512236
© David et al; licensee BioMed Central Ltd. 2011
Received: 10 February 2011
Accepted: 15 June 2011
Published: 15 June 2011
Abstract
Background
Flux coupling analysis (FCA) is a useful method for finding dependencies between fluxes of a metabolic network at steadystate. FCA classifies reactions into subsets (called coupled reaction sets) in which activity of one reaction implies activity of another reaction. Several approaches for FCA have been proposed in the literature.
Results
We introduce a new FCA algorithm, FFCA (Feasibilitybased Flux Coupling Analysis), which is based on checking the feasibility of a system of linear inequalities. We show on a set of benchmarks that for genomescale networks FFCA is faster than other existing FCA methods.
Conclusions
We present FFCA as a new method for flux coupling analysis and prove it to be faster than existing approaches. A corresponding software tool is freely available for noncommercial use at http://www.bioinformatics.org/ffca/.
Background
Constraintbased analysis of metabolic networks has become increasingly important for describing and predicting the behavior of living organisms [1, 2]. While a growing number of metabolic network reconstructions have become available during the last years, the computational analysis of genomescale networks with hundreds or thousands of reactions may still be very timeconsuming. Therefore, there is a need for more efficient algorithms and tools (see e.g. [3–5]).
Flux coupling analysis (FCA) [6] is a useful method for finding dependencies between fluxes of a metabolic network at steadystate. If a nonzero flux through a reaction in steadystate implies a nonzero flux through another reaction, then the first reaction is said to be coupled to the second reaction. Several studies have used FCA for exploring various biological questions such as network evolution [7–9], gene essentiality [7], analysis of experimentally measured fluxes [10, 11] or gene regulation [12, 13]. Having a time efficient implementation of FCA is important in such studies.
In the rest of this section, we first recall some basic definitions. Then we briefly review the previously proposed FCA methods.
Mathematical definitions
Basic preliminaries
For a metabolic network with m (internal) metabolites and n reactions, the stoichiometric matrix S is an m × n matrix, where element S_{ ij } is the stoichiometric coefficient of metabolite i in reaction j. We denote by R = {1, ..., n} the set of reactions and by M = {1, ..., m} the set of metabolites.
There are two types of reactions in a metabolic network. Irr is the set of irreversible reactions which are assumed to always have nonnegative flux values. Rev is the set of reversible reactions which are allowed to have positive, negative or zero flux values. If a reversible reaction has a positive (resp. negative) flux, we say that it is working in forward (resp. backwar d) direction. Splitting a reversible reaction i means making reaction i irreversible and adding one more irreversible reaction i + n to the network, which works in the backward direction. Without loss of generality, we assume that in the numbering of reactions, the first Rev reactions are the reversible ones.
If v is the vector of metabolic fluxes through all the reactions, the steadystate condition holds for v if S · v = 0. The steadystate flux cone is defined as C = {v ∈ ℝ^{ n } S · v = 0, v_{ i } ≥ 0, for all i ∈ Irr}. The lineality space of C is given by lin.space(C) = {v ∈ ℝ^{ n } S · v = 0, v_{ i } = 0, for all i ∈ Irr}.
If for some reaction r ∈ R, v_{ r } = 0 for all v ∈ C, then r is called blocked, otherwise r is unblocked. The set of blocked reactions will be denoted by Blk. The set of unblocked reactions can be further partitioned based on the reversibility type of reactions [14]. We define Irev as the set of all reactions that can work only in one direction, i.e., those reactions that take only nonnegative or only nonpositive flux values at steadystate. If i ∈ Irev and v_{ i } ≤ 0 for all v ∈ C, one can multiply by 1 the ith column of the stoichiometric matrix, such that in the modified network, we have v_{ i } ≥ 0 for all v ∈ C. Without loss of generality, we assume that all reactions in Irev are operating in the forward direction.
The set of reactions that can work in both directions at steadystate is further partitioned into two subsets: (a) Prev, the set of pseudoirreversible reactions, and (b) Frev, the set of fully reversible reactions. A reaction i is in Prev if it can work in both directions, and additionally, for all flux vectors v in the lineality space of the flux cone, we have v_{ i } = 0. Accordingly, we define Frev as the set of those reactions, which can work in both directions, and additionally, can have a nonzero flux value if fluxes through all irreversible reactions are set to zero.
Flux coupling relations between a pair of reactions [6]
Consider two unblocked reactions i, j ∈ R. If for all v ∈ C, v_{ i } ≠ 0 implies v_{ j } ≠ 0, then i is directionally coupled to j. This will be denoted by i → j, or equivalently, j ← i. If for all v ∈ C, v_{ i } ≠ 0 implies v_{ j } ≠ 0 and vice versa, then we say that i and j are partially coupled, or i ↔ j. If i and j are partially coupled, and additionally there exists a constant k such that for all v ∈ C, v_{ i } ≠ 0 implies v_{ j } /v_{ i } = k, then i and j are fully coupled, or i ⇔ j. If neither i → j nor j → i holds, then i and j are uncoupled, or .
Elementary flux patterns
The set supp(v) = {i ∈ R  v_{ i } ≠ 0} of reactions active in v is called the support of v. Suppose v ∈ C is a flux vector and A ⊆ R is a subnetwork. The flux pattern of v for A is defined as A ∩ supp(v), which is the set of those reactions in A which have nonzero values in v[15]. A flux pattern is called an elementary flux pattern (EFP) if it cannot be written as the union of other flux patterns. For studying EFPs, Kaleta et al. [15] assume that the network contains only irreversible reactions. To achieve this, every reversible reaction should be split into two irreversible reactions (forward and backward).
Approaches to Flux Coupling Analysis
In this subsection, we briefly recall existing approaches to flux coupling analysis. For additional information and the technical details on the implementation of these algorithms, we refer to Additional file 1.
Flux Coupling Finder Algorithm (FCF)
The most widely used method for FCA is the Flux Coupling Finder (FCF) algorithm [6]. This approach is based on solving linear programs (LPs), and therefore is an optimalitybased method. After finding blocked reactions and splitting reversible reactions, for every pair of unblocked reactions i and j, two LPs are solved. Depending on the optimal values obtained, the coupling relation between i and j is determined. There is a postprocessing step in FCF. Since the reversible reactions have been split, flux coupling relations for these reactions have to be obtained from the coupling relations of the corresponding irreversible forward and backward reactions.
The FCF algorithm has been successfully used for finding coupling relations in a number of metabolic networks [6–13, 16]. However, this approach is rather timeconsuming for genomescale metabolic networks with thousands of reactions, although it is still one of the fastest FCA methods.
FCA based on Minimal Metabolic Behaviors (MMBFCA)
Larhlimi and Bockmayr [14, 17] have proposed a different strategy for flux coupling analysis. In this approach, a minimal set of generating vectors of the flux cone is computed. Then, the coupling relation for any pair of reactions is inferred based on the coappearance of nonzero fluxes in the generating vectors.
FCA based on elementary flux patterns (EFPFCA)
Recently, Kaleta et al. [15] introduced the concept of elementary flux patterns (EFPs) for the analysis of minimal active reactions in a "subnetwork", which account for possible steadystate flux distributions in a (much) larger metabolic network. They also presented a method based on mixedinteger linear programming (MILP) to compute EFPs. Kaleta et al. suggested that EFPs can be used for characterizing flux coupling relations (see Supplemental Material in [15]). Consider a subnetwork including two unblocked reactions i and j. If each of these reactions can have a nonzero flux independently of the other (i.e. ), {i} and {j} are the only EFPs in this subnetwork. On the other hand, if we assume (without loss of generality) that i is directionally coupled to j, then the EFPs of this subnetwork are {i, j} and {j}. Finally, if the reactions are partially coupled, we will have only one EFP, which is {i, j}. With this method, it is not possible to distinguish between partial and full coupling, since flux patterns only contain the information about the activity or inactivity of the fluxes, but not the flux values.
Further strategies for improving flux coupling analysis
In FCF, and some other FCA approaches, every reversible reaction is split into a forward and a backward reaction. This splitting procedure results in an increase in the number of LPs and also in the size of each LP to be solved. Moreover, a complicated postprocessing step is required to infer the flux coupling relations of the original reversible reactions. For all these reasons, it might be better to perform flux coupling analysis without splitting the reversible reactions. An implementation of FCF without splitting is presented in Additional file 1.
 1.
i, j ∈ Irev: In this case, i and j can be directionally, partially or fully coupled.
 2.
i ∈ Irev and j ∈ Prev: The only possibility is j → i.
 3.
i, j ∈ Prev: In this case, we can only have i ⇔ j.
 4.
i, j ∈ Frev: In this case, we can only have i ⇔ j.
Therefore, it is enough to determine the reversibility types of i and j, and then check if the corresponding coupling relation holds. This will be referred to as "ReversibilityType prunings" (or simply, RTprunings). RTprunings were originally used in MMBFCA as presented in [14].
After RTprunings, a further improvement can be applied to LPbased FCA methods. Assume that in the stoichiomtric matrix S, the columns corresponding to the blocked reactions have been removed. According to Proposition 6.13 in [18], for two reactions i and j such that i, j ∈ Prev or i, j ∈ Frev, the following two statements are equivalent:
• i ⇔ j
• For all v such that Sv = 0, v_{ i } = 0 implies v_{ j } = 0.
The two reactions i and j will be fully coupled if and only if in the above LP . Note that the irreversibility constraints are not included in the above LP. Therefore, not only solving one LP is sufficient, but also the LP becomes smaller. This improvement will be called Prev/Frevbased improvement (or simply, PFimprovement).
An improved version of FCF, which takes into account all of the abovementioned improvements, has been suggested in [18]. Here we implemented this improved version of FCF and compared it with the other FCA approaches (see the Results and Discussion section).
Goals of the present work
FCA is a promising tool for metabolic network analysis. However, to perform FCA most authors seem to use their personal implementation of the FCF algorithm [9, 13]. Only very recently, an implementation of FCF has been included in the latest versions of the FASIMU software [19, 20]. On the other hand, several approaches for FCA have been proposed in the literature. It is not known which of these methods is the fastest in practice. In this paper, we present a novel "feasibilitybased" flux coupling analysis method (FFCA) and compare it to previously existing approaches. A corresponding software tool is freely available for noncommercial use.
Results and Discussion
FFCA: Feasibilitybased Flux Coupling Analysis
A linear programming problem maximize(c^{ T } x  Ax ≤ b) can be solved by enumerating a sequence of feasible solutions, i.e., solutions of the system of linear inequalities Ax ≤ b such that at each solution x the objective function value c^{ T } x improves. Accordingly, finding one feasible solution could be faster than computing an optimal solution (see [21] for the theoretical background).
 1.i, j ∈ Irev: In this case, we check the feasibility of two systems of linear inequalities:(P1)
 2.i ∈ Irev and j ∈ Prev: The only possible coupling relation is j → i (but not i → j). Hence, (P1) will always be feasible, because feasibility of (P1) means that i → j does not hold. However, we need to check the feasibility of (P2). Additionally, since j can take negative values, one more system should be checked for feasibility:(P3)
 3.i, j ∈ Prev or i, j ∈ Frev: In this case, the following system of linear inequalities should be checked for feasibility:(P4)
If (P4) is infeasible, then i ⇔ j (because feasibility of (P4) implies j → i, which in turn implies i ⇔ j based on Proposition 2 in [14]). If (P4) is feasible, then i and j are uncoupled.
Since v_{ i } is constant, the optimal value exists if and only if this problem is feasible. Similar LPs can be solved to check the feasibility of (P2) and (P3).
General comparison of different approaches to flux coupling analysis
Method name  Preprocessing  Type of linear program and solution  Further distinguishing of partial and full coupling required?  Postprocessing for reversible reactions? 

MMBFCA  computing MMBs + reaction classification  n/a  No  No 
EFPFCA  splitting reversible reactions  MILP, optimal  Yes  Yes 
FCF  splitting reversible reactions  LP, optimal  No  Yes 
WFCF  n/a  LP, optimal  No  No 
WRFCF/WRPFCF  reaction classification  LP, optimal  No  No 
FFCA  reaction classification  LP, feasible  Yes  No 
Comparison of the four FCA approaches
To compare the different approaches, namely FCF, MMBFCA, EFPFCA and FFCA, we implemented all of them in Matlab (see Additional file 1). A benchmark set of six metabolic network models was used for the evaluation. The number of unblocked reactions in these models ranges from 18 to 765. Table 2 summarizes the running times, while Table 3 reports on the resulting coupling relations. One can see that in all cases FFCA is 2 to 3 times faster than FCF and orders of magnitude faster than EFPFCA. Table 2 also shows that FFCA is more appropriate for FCA in genomescale networks. MMBFCA is the fastest method for the three smallest networks. However, for the middlesized H. pylori network and especially for the large networks of S. cerevisiae and E. coli, FFCA proves to be faster than MMBFCA. The computation time required for MMBFCA rapidly grows when the number of reactions increases. This is due to the possibly exponential size of the set of generating vectors which has to be computed before finding the coupled reactions (see Section 4.4 in [18]). EFPFCA, which is based on solving mixedinteger linear programs, turns out to be much slower than other methods. Although the concept of elementary flux patterns is very useful in the analysis of subnetworks, its applicability in full FCA seems to be limited.
CPU running time (in seconds) required for flux coupling analysis of the benchmark networks when CLP [32] is used as the LP solver
no. of unblocked reactions  MMBFCA  EFPFCA  FCF  WFCF  WRFCF  WRPFCF  FFCA  

ILLUSNET  18  0.01  26.3  0.25  0.14  0.09  0.06  0.06 
RBC  38  0.05  152  1.39  0.80  0.68  0.64  0.62 
EC core  63  0.22  585  6.58  3.03  3.13  2.19  2.15 
H. pylori  217  69.8  >1 day  196  83.6  67.0  63.3  60.7 
S. cerevisiae  639  >1 day  >1 day  8.5 × 10^{3}  4.0 × 10^{3}  3.4 × 10^{3}  3.3 × 10^{3}  3.1 × 10^{3} 
E. coli (iJR904)  765  >1 day  >1 day  1.2 × 10^{4}  7.4 × 10^{3}  6.3 × 10^{3}  5.9 × 10^{3}  5.6 × 10^{3} 
Number of flux (un)coupling relations for each of the benchmark networks (Table 2)
no. of unblocked reactions  total no. of reactions  fully coupled  partially coupled  directionally coupled  uncoupled  

ILLUSNET  18  19  6  0  9  138 
RBC  38  42  21  0  15  825 
EC core  63  63  19  0  9  1925 
H. pylori  217  480  164  3  309  22960 
S. cerevisiae  639  1144  692  90  2214  200845 
E. coli (iJR904)  765  930  2567  68  6208  283387 

When an LP is solved in FFCA, finding the first feasible solution is sufficient, while the LPs should be solved to optimality in case of the FCF algorithm.

In the FCF method, in contrast to FFCA, every reversible reaction is split into two (forward and backward) irreversible reactions. This step slows down the procedure and increases the size of the LPs to be solved.

For computing the flux coupling relation between any pair of reactions, we always need two LPs in FFCA, while in FCF sometimes more LPs have to be solved. For example, for computing the coupling relation between an irreversible and a reversible reaction (after splitting), four LPs are solved (see Additional file 1).

The ReversibilityType prunings [14] to reduce the number of possible coupled reaction pairs are only considered in FFCA.

The PFimprovements are only considered in FFCA.
One can think of implementing FCF without splitting reversible reactions and/or with the RTprunings and/or PFimprovement. In order to assess the importance of these issues, three improved versions of FCF were implemented (see the Additional file 1 and also Table 1): (i) FCF was reimplemented without splitting reactions (WFCF); (ii) FCF was reimplemented without splitting reactions and with the RTprunings (WRFCF); and (iii) FCF was reimplemented without splitting reactions and with the RTprunings and the PFimprovement (WRPFCF), as suggested in [18].
In Table 2 the computational running times of these methods are also shown. As expected, the three versions of the improved FCF algorithm are better than the classical FCF algorithm, while they are still slower that FFCA.
Computational Properties of FFCA
Flux coupling analysis of new generation genomescale metabolic networks
no. of unblocked reactions  total no. of reactions  fully coupled  partially coupled  directionally coupled  uncoupled  running time (sec)  

H. sapiens (Recon 1)  2018  3284  1555  4  3503  2030091  1.59 × 10^{5} 
E. coli (iAF1260)  1878  2075  2563  143  12791  1747006  1.66 × 10^{5} 
Human hepatocyte (HepatoNet1)  2309  2309  1463  201  1593  2661329  1.35 × 10^{5} 
Since FFCA includes the RTprunings and PFimprovement, it might be interesting to see how the number of coupling relations (and also the running time of FFCA) depends on the number of reversible reactions in a network. This problem is studied in Additional file 2. Briefly, the flux coupling relations have been computed for the original metabolic network of E. coli[23], for the modified metabolic network of E. coli[24] (which includes a smaller number of irreversible reactions), and for some random networks in between. As expected, the numbers of uncoupled pairs increase (and the running times generally decrease) with the increase in the number of reversible reactions.
Conclusions
We introduced FFCA as a new method for flux coupling analysis, and proved it to be faster than any other available approach. Our implementation of FFCA is fast enough to perform FCA for every pair of reactions in S. cerevisiae and E. coli genomescale networks in a few hours. A corresponding software tool is available for noncommercial use at http://www.bioinformatics.org/ffca/ and we recommend it for FCA of genomescale networks.
Methods
Metabolic network models
Nine metabolic networks were used in this study: (i) ILLUSNET network from [14]; (ii) RBC : metabolic network of red blood cell [25]; (iii) EC core: central metabolic network of E. coli[26]; (iv) H. pylori genomescale metabolic network [27]; (v) yeast (S. cerevisiae) genomescale metabolic network [28]; (vi) the 2003 genomescale metabolic network of E. coli (iJR904) [23]; (vii) the 2007 genomescale metabolic network of E. coli (iAF1260) [29]; (viii) the metabolic network of human (Recon 1) [30]; and (ix) the metabolic network of human hepatocyte (HepatoNet1) [31].
For FCA, all uptake reactions were assumed to be able to carry nonzero fluxes.
Implementation of different FCA methods
Unless indicated otherwise, all tools were implemented in Matlab v7.4. Details about the implementation of the different FCA approaches, together with the corresponding pseudocodes are presented in Additional file 1.
Comparison of different FCA methods
All computations were performed on a 64bit Debian Linux system with Intel Xeon 3.0 GHz processor. The running times include the CPU time for preprocessing, computation of flux coupling relations, and postprocessing (where necessary).
Notes
List of abbreviations
 FCA:

Flux Coupling Analysis
 FFCA:

Feasibilitybased Flux Coupling Analysis
 FCF:

Flux Coupling Finder
 MMBFCA:

Flux Coupling Analysis based on Minimal Metabolic Behaviors
 EFPFCA:

Flux Coupling Analysis based on Elementary Flux Patterns
 LP:

Linear Program
 MILP:

MixedInteger Linear Program
 RTprunings:

ReversibilityType prunings
 PFimprovement:

Prev/Frevbased improvement.
Declarations
Acknowledgements
We would like to thank K. Kelleher (MPI for Molecular Genetics) for her comments on the manuscript.
Authors’ Affiliations
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