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Bioinformatics Advance Access originally published online on March 6, 2007
Bioinformatics 2007 23(9):1049-1052; doi:10.1093/bioinformatics/btm074
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© The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Metabolic network properties help assign weights to elementary modes to understand physiological flux distributions

Qingzhao Wang *,{dagger}, Yudi Yang {dagger}, Hongwu Ma and Xueming Zhao

Metabolic Engineering Laboratory, Department of Biochemical Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072, People's Republic of China

*To whom correspondence should be addressed.


   ABSTRACT
TOP
 ABSTRACT
1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 

Motivation: Elementary modes (EMs) analysis has been well established. The existing methodologies for assigning weights to EMs cannot be directly applied for large-scale metabolic networks, since the tremendous number of modes would make the computation a time-consuming or even an impossible mission. Therefore, developing more efficient methods to deal with large set of EMs is urgent.

Result: We develop a method to evaluate the performance of employing a subset of the elementary modes to reconstruct a real flux distribution by using the relative error between the real flux vector and the reconstructed one as an indicator. We have found a power function relationship between the decrease of relative error and the increase of the number of the selecting EMs, and a logarithmic relationship between the increases of the number of non-zero weighted EMs and that of the number of the selecting EMs. Our discoveries show that it is possible to reconstruct a given flux distribution by a selected subset of EMs from a large metabolic network and furthermore, they help us identify the ‘governing modes’ to represent the cellular metabolism for such a condition.

Contact: diana_kingson{at}yahoo.com.cn(or) Wangqingzhao{at}eyou.com

Supplementary information: Supplementary data are available at Bioinformatics online.


   1 INTRODUCTION
TOP
 ABSTRACT
 1 INTRODUCTION
2 METHODS
 3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
Metabolic network analysis is one of the research focuses of systems biology. In recent years, a growing number of genome-scale metabolic networks of different species have been reconstructed with the aid of genome sequencing and high-throughput technologies (Covert et al., 2004), offering us a great opportunity to study them in an unprecedented manner, and acquire new knowledge about life science (Edward et al., 2000). Two aspects of metabolic networks-network topology and stoichiometry are what current researchers are most interested in, and both studies had revealed significant information. The study of network topology by the means of graph theory indicates that the metabolic network, organizing in a modular, hierarchical manner (Ma et al., 2004), resembles a small-world network. (Jeong et al., 2000). The research of the stoichiometric matrix of metabolic network has generated a series of powerful methodologies such as metabolic flux analysis (MFA), metabolic control analysis (MCA), flux balance analysis (FBA), etc. Considering both topological and stoichiometric characteristics of metabolic networks, metabolic pathway analysis (Schilling et al., 1999) may provide a more insightful and comprehensive means to study cellular metabolism than the above methodologies.

Recently, two related approaches for metabolic pathway analysis, elementary modes (EMs) (Schuster et al., 1994) and extreme pathways (ExPas) (Schilling et al., 2000), have demonstrated their power in studying the properties of metabolic networks (Klamt and Stelling, 2003; Papin, 2004). The robustness of the metabolic network, the potential of the species to convert a desired product, and even the gene regulation can be predicted by EMs analysis (Stelling et al., 2002). Regulatory structures for metabolic network of human red blood cell had been studied by ExPas analysis (Barrett et al., 2006). In addition to these applications, metabolic pathway analysis has been utilized to find the clue for strain optimization (Carlson et al., 2002). Since every steady-state flux distribution can be expressed as a non-negative linear combination of EMs (ExPas), understanding how probably and to what extent every mode devote to the real flux distribution would shed light on the complex cellular metabolism. For such a purpose, using EMs (ExPas) to reconstruct actual physiological flux distribution is the first and indispensable step.

To date, several means had been published to use EMs (ExPas) to reconstruct flux distributions of real metabolic networks. One way is to seek the minimal norm of weight vector by solving a quadratic programming problem (Schwartz and Kanehisa, 2005). The biological sense of this approach is apparent in that this algorithm finds the modes that are identical to the real flux distribution pattern. An earlier research used the Moore-Penrose generalized inverse of E (E is an n x m matrix, where m denotes the number of elementary modes, and n denotes the number of reactions of the metabolic network) to assign weights to each mode of the metabolic network (Poolman et al., 2004). A common defect of the two methods is that their application is confined to analyzing small metabolic networks whose limited number of EMs (ExPas) would not pose a threat to a normal PC's computation ability. However, in most situations, larger-scale metabolic networks considering not only the central metabolism but also the synthesis of precursors, the excretion of byproducts and (or) even the balance for co-enzymes are adopted to acquire a more systematic and precise information of cellular metabolism. The resulting huge numbers (from ten thousands to millions) of EMs (ExPas) of these systems (Gagneur and Klamt, 2004) make directly using these computation procedures impossible, since handling such a large number of variables simultaneously is currently beyond the ability of both software and hardware. Besides, to our knowledge, the ‘optimal’ solution of the weights acquired by the previous methods still needs more tests and improvements for interpreting the cells’ complex metabolic behaviors.

Eigenpathway, a definition from the singular value decomposition (SVD) of extreme pathway matrix, also had been used to reconstruct flux distribution (Price et al., 2003). For different patterns of flux distributions of the same metabolic network, the only difference of the reconstruction results is the coefficients of the eigenpathway vectors and the numbers of eigenpathway used for the calculation. Compared with the reconstruction by E x Pas the effects of gene manipulation and (or) transcriptional regulation (Covert and Palsson, 2003) cannot be easily evaluated in terms of eigenpathways, since the change of extreme pathway matrix may eventually alter eigenpathways. This methodology is therefore, short of biological sense compared with its mathematical convenience.

Instead of seeking for an optimized weights vector, we use an original approach to resolve the problem of assigning weights for a large set of EMs to reconstruct real flux distributions. First, we calculate the set of EMs of a revised Escherichia coli metabolic network (Stelling et al., 2002). Then, we devise a quadratic program to explore the possibility and performance of using a subset of the EMs to reconstruct flux distributions. We discover that it is possible to use a part of EMs set to well represent physiological flux distributions, and the number of non-zero weighted EMs calculated to reconstruct flux distribution increases logarithmically with the increasing scale of EMs subset. Further analysis shows that there do exist a special subset of EMs that receive non-zero weights with a more frequent rate than any other ones do, and the number of EMs belonging to this special subset is much smaller than the total number of EMs. The EMs belonging to this special subset can help us understand physiological flux distributions.


   2 METHODS
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
2.1 Models and fluxes data
A revised (the acetate uptake reaction was replaced by the succinate excretion reaction) E.coli central metabolic network and a previously constructed central metabolic network of purple non-sulfur bacteria were used as our models (Stelling et al., 2002). Flux Analyzer 6.0 (Klamt et al., 2003) was used to calculate all the EMs of the two networks. The metabolic network of E.coli had 89 metabolites, 110 reactions and a total of 711 984 EMs. The metabolic network of purple non-sulfur bacteria had 76 metabolites, 87 reactions and a total of 149 835 EMs. The flux distribution from anaerobic growth of E.coli (Schmidt et al., 1999) was used for the study of Formula (k) (average relative error) and Formula (k) (non-zero weight), and for the subsequent analysis to seek the dominating modes. Different patterns of flux distributions within the constraints of the two metabolic networks were used to evaluate the universality of the changing styles of Formula (k) and Formula (k) in terms of k (k denotes the number of selected EMs).

2.2 Calculation for Formula (k) and Formula (k)
The performance of using subset of EMs to reconstruct flux distribution was first evaluated. A total of k EMs were randomly selected and used for reconstruction. Since all the EMs have equal chance to be selected, for each k, the reconstruction procedure was repeated n times to guarantee that all the EMs can be selected and concerned for flux reconstruction. REi (k) denotes the ith relative error between the reconstructed flux vector vi (k) (where k denotes the number of EMs being used, i means ith reconstruction result) and the target vector v from the ith reconstruction result. The ith reconstructed flux vector vi (k) was calculated by solving the following non-negative constraints quadratic program(specifically, the function is lsqnonneg, which is provided by MATLAB):


Formula 1

(1)
Where xi(k) was the calculated weights for k modes, and Pi(k) was the matrix whose columns were composed of k modes. The ith relative error REi (k) was given by:


Formula 2

(2)
Ni (k) denotes the number of non-zero weights for the ith result when k EMs were selected to reconstruct the flux distribution. According to different number of k, n ranged from 1000 to 10 000 and the average relative error Formula (k) and number of non-zero weights Formula (k) were calculated.

2.3 Identification for the governing modes
Again, the steady-state flux distribution of E.coli for anaerobic growth was used to hunt for the governing modes. The same quadratic program was adopted, and k and n were chosen to be 5000 and 2000, respectively. All ten millions of the calculated weights and the corresponding indices of the selected modes were recorded. The elements of the 2000 weights vectors helped divide all the EMs into two subsets Z and NZ. The corresponding modes that acquire a zero weight at least one time belong to Z, and these that always acquire non-zero weights belong to NZ. The numbers of elements for both subsets, NZ and NNZ were counted by utilizing the corresponding relationship between the indices and the weights. The number of non-zero elements of the weights vectors was calculated, and WZ and WNZ were the numbers of the non-zero weights belonging to Z and NZ, respectively. Therefore, when used for reconstructing the physiological flux distribution, the average frequencies to receive a non-zero weight for the modes from each subset were given by:


Formula 3

(3)


Formula 4

(4)


   3 RESULTS
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
3.1 Randomly selecting EMs to reconstruct real flux distribution
We randomly selected a series of fixed numbers of EMs from a total number of 711 984 modes of the E.coli central metabolic network, and used a quadratic program to assign weights to the selected EMs to reconstruct the actual physiological flux distribution (see Methods section for details). We found a negative power function relationship between the decrease of average relative error Formula (k) (where k denotes the number of selected modes) and the increase of the numbers of selected modes k (Fig. 1). In order to exclude the possibilities of our results coming from the particular steady state being used, we further test more flux distributions and once again the same rules have been observed.


Figure 1
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  		Fig. 1. The distrubution of Formula for k. The circle denotes the distribution of Formula for k, and the asterisk represents a log–log plot for the distribution. The solid line is a linear fitting, and the relative coefficient is –0.9993. A non-negative constraints quadratic program was used to fit the anaerobic growth flux distribution of E.coli. Formula was calculated from 2000 times repeated simulations.

 
The relationship between Formula (k) and k implies us that for randomly selecting EMs to reconstruct flux distributions, there exists a boundary number of EMs by which the accuracy would not increase effectively when that number is exceeded. Another worth noting finding is the logarithmic relationship between the increase of the numbers of average non-zero weights Formula (k) and the increase of k (Fig. 2). According to Figure 2, if the entire EMs of E.coli central metabolic network could be used to reconstruct the same flux distribution, the number of non-zero weights would be no more than 16. We performed the same calculation procedures for different patterns of flux distributions of E.coli, and for those of purple non-sulfur bacteria to evaluate the universality of these rules. Despite differences between parameters obtained from linear fitting, all the results showed the same tendencies as above (data not shown).


Figure 2
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  		Fig. 2. The distrubution of Formula for k. The circle denotes the distribution of Formula for k, and the asterisk represents a lin–log plot for the distribution. The solid line is a linear fitting, and the relative coefficient is 0.9962.

 
It is a rational idea that using a larger subset of the total EMs would receive a better result for reconstructing flux distributions, since there is a greater possibility that the ‘more proper’ modes could be found within the larger subset. However, the power function relationship between Formula (k) and k, and the logarithmic relationship between Formula (k) and k cannot be simply explained by probability. As mentioned before, metabolic networks have a series of properties. They have a few so-called ‘hub metabolites’ (Ma and Zeng, 2003), a power law distribution of connection degree among the metabolite nodes, a log normal distribution of the flux distributions (Sariyar et al., 2006), and so forth, we think that our discoveries are also determined by the topological and stoichiometric properties of metabolic networks. In order to prove that we have used some counter conditions such as uniform distributions and normal distributions which are not within the constraints of metabolic networks to test this hypothesis (Supplementary Material I). These distributions do not observe such relationships.

3.2 Identifying the ‘governing’ modes
Former algorithms for assigning weights to EMs to reconstruct flux vector try to seek a special solution. However, a single weight vector of EMs cannot completely reveal the characteristics of a given phenotype, since at most conditions there are infinite combinations of modes to reconstruct the flux distribution. Due to the complexity and redundancy of cellular metabolism, the biological sense behind these algorithms is therefore not the proof of the accuracy and efficiency of these algorithms. Instead of searching an optimal solution of weight vector, we identified a subset of EMs that would be more easier to receive non-zero weights from the quadratic program than the other modes. Previous study about the relationship between Formula (k) and k helped us select a proper k for subsequent calculations, since although increasing k would improve the accuracy; it would lengthen the computation time as well (Table 1).


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  		Table 1. Statistical result for the repeated simulations

 
The frequency for the modes from the subset NZ to reconstruct the given flux distribution is ~400 times higher than that of the modes from the subset Z. Another important finding is that NZ has an extremely small size compared with that of the set of E.coli central metabolic network EMs, which makes it more easier to handle than the original one. We used the modes from the subset NZ to represent current flux distribution to evaluate its effectiveness, and the result showed a perfect fitting (the relative error is under 10–4).


   4 DISCUSSION
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
For model organisms whose relationship between mRNA abundance and DNA binding has been well studied such as E.coli, Saccharomyces cerevisiae and so on (Herrgard et al., 2004), transcriptional regulatory rules can be used to discard infeasible EMs for a given phenotype (Covert et al., 2003). However, for most organisms, such constraints are not easy to acquire compared with the topology and stoichiometry of their metabolic networks. Although, for some cases, special growth conditions can be used to dispose those EMs with improper substrate uptake or by-products excretion reactions, the original huge set of EMs can seldom be reduced to a handy scale for further analysis. In this work, we used only the information about network topology and stoichiometry on purpose to show how our method can lead to the identification of the ‘governing modes’ from the enormous set of EMs.

Our methods share some similarity with {alpha}-spectrum, which defines a weight range for each extreme pathway to reconstruct a given flux distribution (Wiback et al., 2003). The method of {alpha}-spectrum considers the weight ranges of all the ExPas when reconstructing physiological flux distribution. This method has been used to understand changing metabolisms brought by environment and (or) regulation. However, based on our study, we think that the number of non-zero weights, which derive from flux reconstruction calculation, is extremely limited, and it is therefore not necessary to consider all the modes. Especially, for a large set of ExPas with an identical scale of our model, the computation for {alpha}-spectrum may become time consuming or even impossible. Our method, on the contrary, can be easily performed without such a constraint. Therefore, it is especially a useful tool for dealing with large-scale metabolic networks.


   5 CONCLUSION
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
ACKNOWLEDGEMENTS
 REFERENCES
 
In this study, we used an original method to solve the problem of assigning weights to a large set of EMs to reconstruct real flux distribution. The relationship between Formula (k) and k determined by metabolic network properties helped us devise an algorithm to identify a subset of EMs whose elements modes appear much more frequently in reconstructing flux distribution than the rest ones do. Two characteristics distinguish our method. First of all, it bypasses the demanding requirement for both software and hardware brought by other algorithms when dealing with large set of EMs (ExPas). Second, it helps seek a set of proper modes rather than a single solution of weights to reconstruct flux distribution, which we think is more flexible and systemic to study cellular metabolism than the previous ideas.


   ACKNOWLEDGEMENTS
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
REFERENCES
 
This work is financially supported by the National Natural Science Foundation of China (NSFC-20536040), the State Key Development Program for Basic Research of China (No. 2003CB716003 and 2007CB707802) and the Program of Introducing Talents of Discipline to Universities (No. B06006 [GenBank] ).

Conflict of Interest: none declared.


   FOOTNOTES
 
{dagger}The authors wish it to be known that, in their opinion, the first two authors should be regarded as joint First Authors. Back

Associate Editor: Alfonso Valencia

Received on October 31, 2006; revised on February 21, 2007; accepted on February 24, 2007

   REFERENCES
TOP
 ABSTRACT
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 

    Barrett CL, et al. Network-level analysis of metabolic regulation in the human red blood cell using random sampling and singular value decomposition. BMC Bioinformatics, ( (2006) ) 7, : 132.[CrossRef][Medline].

    Carlson R, et al. Metabolic pathway analysis of a recombinant yeast for rational strain development. Biotechnol. Bioeng., ( (2002) ) 79, : 121–134.[CrossRef][ISI][Medline].

    Covert MW, Palsson BO. Constraints-based models: regulation of gene expression reduces the steady-state solution space. J. Theor. Biol., ( (2003) ) 221, : 309–325.[CrossRef][ISI][Medline].

    Covert MW, et al. Integrating high-throughput and computational data elucidates bacterial networks. Nature, ( (2004) ) 429, : 92–96.[CrossRef][Medline].

    Edwards JS, Palsson BO. Metabolic flux balance analysis and the in silico analysis of Escherichia coli K-12 gene deletions. BMC Bioinformatics, ( (2000) ) 1, : 1.[CrossRef][Medline].

    Gagneur J, Klamt S. Computation of elementary modes: a unifying framework and the new binary approach. BMC Bioinformatics, ( (2004) ) 5, : 175.[CrossRef][Medline].

    Herrgard MJ, et al. Reconstruction of microbial transcriptional regulatory networks. Curr. Opin. Biotech., ( (2004) ) 15, : 70–77.[CrossRef][ISI][Medline].

    Jeong H, et al. The large-scale organization of metabolic networks. Nature, ( (2000) ) 407, : 651–654.[CrossRef][Medline].

    Klamt S, Stelling J. Two approaches for metabolic pathway analysis? Trends Biotechnol., ( (2003) ) 21, : 64–69.[CrossRef][ISI][Medline].

    Klamt S, et al. FluxAnalyzer: exploring structure, pathways, and flux distributions in metabolic networks on interactive flux maps. Bioinformatics, ( (2003) ) 19, : 261–269.[Abstract/Free Full Text].

    Ma H, Zeng A. Reconstruction of metabolic networks from genome data and analysis of their global structure various organisms. Bioinformatics, ( (2003) ) 19, : 270–277.[Abstract/Free Full Text].

    Ma H, et al. Decomposition of metabolic network into functional modules based on the global connectivity structure of reaction graph. Bioinformatics, ( (2004) ) 20, : 1870–1876.[Abstract/Free Full Text].

    Papin JA, et al. Comparison of network-based pathway analysis methods. Trends Biotechnol., ( (2004) ) 22, : 400–405.[CrossRef][ISI][Medline].

    Poolman MG, et al. A method for the determination of flux in elementary modes, and its application to Lactobacillus rhamnosus. Biotechnol. Bioeng., ( (2004) ) 88, : 601–612.[CrossRef][ISI][Medline].

    Price ND, et al. Analysis of metabolic capabilities using singular value decomposition of extreme pathway matrices. Biophys. J., ( (2003) ) 84, : 794–804.[ISI][Medline].

    Sariyar B, et al. Monte Carlo sampling and principal component analysis of flux distributions yield topological and modular information on metabolic networks. J. Theor. Biol., ( (2006) ) 242, : 389–400.[CrossRef][ISI][Medline].

    Schilling CH, et al. Metabolic pathway analysis: basic concepts and scientific applications in the post-genomic era. Biotechnol. Prog., ( (1999) ) 15, : 296–303.[CrossRef][Medline].

    Schilling CH, et al. Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J. Theor. Biol., ( (2000) ) 203, : 229–248.[CrossRef][ISI][Medline].

    Schmidt K, et al. Quantitative analysis of metabolic fluxes in Escherichia coli, using two-dimensional NMR spectroscopy and complete isotopomer models. J. Biotechnol., ( (1999) ) 71, : 175–190.[CrossRef][ISI][Medline].

    Schuster S, Hilgetag C. On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst., ( (1994) ) 2, : 165–182.[CrossRef].

    Schwartz JM, Kanehisa M. A quadratic programming approach for decomposing steady-state metabolic flux distributions onto elementary modes. Bioinformatics, ( (2005) ) 21, : 204–205.[CrossRef][ISI].

    Stelling J, et al. Metabolic network structure determines key aspects of functionality and regulation. Nature, ( (2002) ) 420, : 190–193.[CrossRef][Medline].

    Wiback SJ, et al. Reconstructing metabolic flux vectors from extreme pathways: defining the {alpha}-spectrum. J. Theor. Biol., ( (2003) ) 224, : 313–324.[CrossRef][ISI][Medline].


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This Article
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