Volume 8 Supplement 1
Selected articles from the Twelfth Asia Pacific Bioinformatics Conference (APBC 2014): Systems Biology
Comparison on extreme pathways reveals nature of different biological processes
 Yanping Xi^{1},
 Yue Zhao^{1},
 Li Wang^{2} and
 Fei Wang^{1}Email author
DOI: 10.1186/175205098S1S10
© Xi et al.; licensee BioMed Central Ltd. 2014
Published: 24 January 2014
Abstract
Background
Constraintbased reconstruction and analysis (COBRA) is used for modeling genomescale metabolic networks (MNs). In a COBRA model, extreme pathways (ExPas) are the edges of its conical solution space, which is formed by all viable steadystate flux distributions. ExPa analysis has been successfully applied to MNs to reveal their phenotypic capabilities and properties. Recently, the COBRA framework has been extended to transcriptional regulatory networks (TRNs) and transcriptional and translational networks (TTNs), so efforts are needed to determine whether ExPa analysis is also effective on these two types of networks.
Results
In this paper, the ExPas resulting from the COBRA models of E.coli's MN, TRN and TTN were horizontally compared from 5 aspects: (1) Total number and the ratio of their amount to reaction amount; (2) Length distribution; (3) Reaction participation; (4) Correlated reaction sets (CoSets); (5) interconnectivity degree. Significant discrepancies in above properties were observed during the comparison, which reveals the biological natures of different biological processes. Besides, by demonstrating the application of ExPa analysis on E.coli, we provide a practical guidance of an improved approach to compute ExPas on COBRA models of TRNs.
Conclusions
ExPas of E.coli's MN, TRN and TTN have different properties, which are strongly connected with various biological natures of biochemical networks, such as topological structure, specificity and redundancy. Our study shows that ExPas are biologically meaningful on the newborn models and suggests the effectiveness of ExPa analysis on them.
Background
Many largescale biological networks, including metabolic networks (MNs) [1], signaling networks [2], transcriptional regulatory networks [3] and transcriptional and translational networks [4] have been reconstructed along with the development of highthroughput technology in the past decades. These networks are then transformed into mathematical models for further analysis. ConstraintBased Reconstruction and Analysis (COBRA) is one of the most commonly used frameworks introduced to model and analyze steadystate biochemical networks [5]. In the past two decades, it has been successfully applied on MNs to study various phenotypes [6–9]. Recently, the same principles were also extended to other types of biochemical networks mentioned above [2–4, 10].
All the possible phenotypes, i.e. the flux distributions of feasible steady states, of a constraintbased biochemical model form a highdimensional cone. Networkbased pathways such as Extreme Pathways (ExPas) [11] are defined to study this cone. ExPas are vectors of fluxes that lie on the edges of the cone [12]. They constitute the minimal and unique vector set which generates the space of all feasible steady states through nonnegative linear combination. Since ExPas characterize the limits on the capabilities of a cell's metabolic system [13], ExPa analysis will reveal systemic properties of metabolism [14]. ExPa analysis as an approach to characterize the fundamental and timeinvariant topological properties of a given network [15] has been successfully applied to MNs, such as those of human red blood cells [16], Escherichia coli [17–19], Sacchoromryces cerevisiae [20, 21], Helicobacter pylori [22, 23], Haemophilus influenzae [11, 24] and Methylobacterium extroguens [25]. Besides, network models respectively describing a prototypic signaling system [10] and the JAKSTAT signaling system in the human BCell [2] have also been studied through ExPa analysis.
Recently, there emerged two COBRA models of biochemical systems with different types: E.coli transcriptional regulatory network (TRN) [3] and E.coli transcriptional and translational network (TTN) [4]. What should be clarified is whether ExPa analysis is still useful for new types of networks and whether ExPas of TRN or TTN show some properties different from those of MN. These questions are biologically significant because the answers determines whether we can rely on the existing analysis approaches to obtain novel and biologically meaningful findings in a brand new field. In this paper, we try to provide an anwser by comparing properties of ExPas among the E.coli TRN, MN and TTN. In the comparison, differences between biological processes were observed from multiple perspectives, including network structure, reaction participation, specificity and redundancy. The results indicate that ExPa analysis can be extended to biochemical systems of TRN and TTN, which helps researchers to further understand the corresponding biological systems. Besides an improved method was introduced to simplify the calculation and interpretation of ExPas on TRN models [3], which could also be useful.
Results
Firstly, we calculated extreme pathways of the three biological networks mentioned above. Since the number of ExPas grows exponentially with a networks' complexity [15], the enumeration of ExPas on the highly complex ones such as E.coli MN and TTN is computationally intractable. Fortunately ExPa calculation will be much more manageable if a MN or a TTN is divided into smaller subnetworks. Therefore, we chose the subnetworks with relatively complete and independent functions as the representatives of their belonging biologic systems. For the E.coli MN, two subnetworks were chosen: (1) Amino acid, Carbohydrate and Lipid metabolism (sACL) and (2) Membrane and Murein metabolism (sMM). For E.coli TTN, the two subnetworks were: (1) transcription (sTC) and (2) translation (sTL).
Then ExPa analysis was performed on each network/subnetwork and the properties from different aspects were obtained, including the total number of ExPas, the numberbased ratio of ExPa to reaction, ExPa length distribution, reaction participation distribution, correlated reaction sets (CoSets), and the interconnectivity of ExPas. Finally, a horizontal comparison on the properties was made among the five networks/subnetworks.
Moreover, some incompleteness and incorrectness in the E.coli TRN model which were stumbled through ExPa analysis are also reported in this section. This findings illustrate that ExPa analysis is capable of directing model refinement.
E.coli TRN model
The E.coli TRN model was published by Gianchandani et al. in 2009 [3]. It contains 147 environmental stimuli, 125 transcriptional factors and 503 downstream target genes which are represented in a matrix ${\mathbf{R}}^{*}$ [3]. The TRN model was improved to enhance the efficiency of ExPa calculation (Details are provided in Materials and Methods). The final TRN model contains 1009 components, 1106 internal regulatory reactions, and 1009 exchange reactions each corresponding to a component. All the extracellular metabolites were considered as inputs and all protein products were considered as outputs. There were 1599 ExPas, of which 9 were biologically infeasible because they employed conflicting input fluxes, and thus they were excluded from the ExPa set used in analysis.
Unused reactions in the E.coli TRN (Type I  Regulatory rules missing).
Reactions  Reuglatory rules  Reaction type 

NOT_BirA    internal 
b0774_1  NOT(BirA)  internal 
b0775_1  NOT(BirA)  internal 
b0776_1  NOT(BirA)  internal 
b0778_1  NOT(BirA)  internal 
Ex_b0774    exchange 
Ex_b0775    exchange 
Ex_b0776    exchange 
Ex_b0778    exchange 
Unused reactions in the E.coli TRN (Type II  Contradictory regulatory rules)
Reactions  Regulatory rules  Reaction type 

b1814  ( gly(e)>0) AND (leuL(e)>0) leuL(e)>0 (NOT o2(e)>0) AND (leuL(e)>0) (gly(e)>0) AND (NOT leuL(e)>0) AND (leuL(e)>0) (leuL(e)>0) AND (NOT leuL(e)>0) (NOT o2(e)>0) AND (NOT leuL(e)>0) AND (leuL(e)>0)  Internal 
b3942_1  (Growth>0) AND (h202(e)>0) AND (NOT Growth>0)  Internal 
b4111_1  (NOT(Crp)) AND (Growth>0) AND (NOT Growth>0)  internal 
Ex_b3942    Exchange 
Ex_b4111    Exchange 
Reactions in Table 1 all relate to NOT_BirA (absence of protein BirA). However, no regulatory rule corresponds to the presence or absence of BirA, and therefore, the initial steps are unknown. As a result, the internal reactions using NOT_BirA (b0774_1, b0775_1, b0776_1 and b0778_1) and the corresponding exchange reactions (Ex_b0774, Ex_b0775, Ex_b0776 and Ex_b0778) will never be initiated. Furthermore, proteins BirA and the gene products of b0774, b0775, b0776 and b0778 do not participate in any other reaction except those in Table 1, so their invalidation will not affect other reactions in the network. In a word, these 9 reactions do not participate in any ExPa because their relevant reactions (either producing their substrates or consuming their products) are unavailable in the network. The unused reactions in Table 1 show the incompleteness of the E.coli TRN model and necessitate further refinement.
For the reactions in Table 2, the regulatory rule of b1814 can be divided by simple logical transformation into 6 rules, of which 3 contradict with each other (the shaded parts in Table 2). Since there are still 3 operational regulatory rules relating to the transcription of b1814, its corresponding exchange reaction can be initiated. Similarly, the regulatory rules of b3942 and b4111 are both contradictory and cannot be used in any ExPa. These reactions may imply some incorrect information in the model. Therefore, new biological knowledge is needed to improve E.coli TRN.
E.coli MN and TTN model
The MN model of E.coli K12 MG1655, iAF1260, was published by Feist et al, in 2007 [26]. It includes the activities of 1260 open reading frames (ORFs). It consists of 1688 metabolites and 2382 reactions. The E.coli TTN model was published by Thiele et al. in 2009 [4]. It consists of 11991 components and 13694 reactions which give rise to 423 functional gene products [4]. Given the critical inherent problem of combinatorial explosion during ExPa calculation, E.coli MN and TTN were divided into small subnetworks depending on the reactions' functions [11]. Subnetworks as representatives of important biological processes were chosen.
Sub networks of the E.coli MN
# Sub network  Name  Containing subsystems  No. of reactions 

1  A mino acid, C arbohydrat and L ipid metabolism (sACL)  Alanine and Aspartate Metabolism, Cysteine Metabolism, Folate Metabolism, Glutamate Metabolism, Glycine and Serine Metabolism, Histidine Metabolism, Methionine Metabolism, Threonine and Lysine Metabolism, Tyrosine Tryptophan and Phenylalanine Metabolism, Valine Leucine and Isoleucine Metabolism, Arginine and Proline Metabolism, Citric Acid Cycle, Pentose Phosphate Pathway, Pyruvate Metabolism, Glyoxylate Metabolism, Methylglyoxal Metabolism, Glycolysis and Gluconeogenesis, Oxidative Phosphorylation, Glycerophospholipid Metabolism, Anaplerotic Reactions  741 
2  Nitrogen and Nucleotide metabolism  Cofactor and Prosthetic Group Biosynthesis, Nitrogen Metabolism, Nucleotide Salvage Pathway, Purine and Pyrimidine Biosynthesis, tRNA Charging  445 
3  M embrane and M urein metabolism (sMM)  Cell Envelope Biosynthesis, Lipopolysaccharide Biosynthesis Recycling, Membrane Lipid Metabolism, Murein Biosynthesis, Murein Recycling  358 
4  Transport  Transport Inner Membrane,Transport Outer Membrane,Transport Outer Membrane Porin  849 
5  Others  Alternate Carbon Metabolism; Inorganic Ion Transport and Metabolism; Unassigned  408 
6  Exchange  Exchange reactions  304 
The E.coli TTN model comprises of 27 biological processes and the details are provided in [4]. Each process was treated as a discrete subnetwork. The largest two subnetworks, Transcription and Translation, were chosen for further ExPa analysis.
ExPa counting
Network characteristics and ExPa calculation results
Network name  No. of reaction  No. of ExPa  P/R ratio 

TRN  2115  1590  0.75 
ACL  741  24778  33.44 
sMM  358  11600  32.40 
sTC  1276  154  0.12 
sTL  7386  1871  0.25 
ExPa length
Summary of the statistical analysis of ExPa lengths
Network  No. of  ExPa length  L/R  

reactions  Avg  Max  Min  Most  Ratio(%)  
TRN  2115  5  21  3  4  0.24 
sACL  741  47  70  3  51  6.40 
sMM  358  64  115  4  82  17.97 
sTC  1276  31  34  26  32  2.43 
sTL  7386  101  121  11  109  1.37 
The length distributions of ExPas corresponding to those biological processes are very diverse. The longest ExPas consists 51, 82 32 and 109 reactions in sACL, sMM, sTC and STL, respectively, which is much longer than that in TRN (21). Reactions in E.coli TRN represent transcriptional regulatory rules rather than real biochemical reactions as in MN and TTN, and thus the ExPa length in TRN depicts the number of regulatory rules used for expressing certain genes. A regulatory rule describes how environmental stimuli affect transcriptional factors, which in turn affect downstream target genes. Therefore, the ExPa in TRN is reasonably shorter as the biological network has a relatively flat hierarchical structure [3]. Given the number of reactions, the ratio of average ExPa length to reaction number (L/R) was calculated for each biological network or subnetwork (Table 5). The L/Rs of the two representatives in MN are higher than those in TRN and their counterparts in TTN. Since ExPas convert substrates into products, ExPa length relates to how many reaction steps are needed to carry out the corresponding function. ExPa length can be characterized as the size and complexity of the corresponding flux distribution map [13]. The results indicate that the flux distribution map in MN is much more complex than those in TRN and TTN.
Reaction participation
The top 10 most frequently participated internal reactions
Part 1  

The top 10 reactions with highest participation rate in the TRN  
Order  Reaction  Participation Rate (%)  
1  CRP_noGLC_1  9.18  
2  Crp_1  8.99  
3  NOT_PdhR_1  7.55  
4  Fis_1  6.16  
5  Lrp_1  4.09  
6  Fnr_1  3.90  
7  NOT_PurR_1  3.27  
8  NOT_ArcA_1  3.02  
9  NOT_Cra_1  2.83  
10  NOT_Lrp_1  2.45  
The top 10 reactions with highest participation rate in the sACL  
Order  Abbr.  Reaction Name  Participation Rate (%) 
1  ASPTA  aspartate transaminase  91.42 
2  ASAD  aspartate semialdehyde dehydrogenase  90.40 
3  ASPK  aspartate kinase  90.40 
4  HSDy  homoserine dehydrogenase NADPH  90.31 
5  GHMT2r  glycine hydroxymethyltransferase reversible  75.82 
6  GLYAT  glycine C acetyltransferase  68.22 
7  HSK  homoserine kinase  67.27 
8  THRS  threonine synthase  67.27 
9  THRD  L threonine dehydrogenase  59.23 
10  FUM  Fumarase  57.71 
The top 10 reactions with highest participation rate in the sMM  
Order  Abbr.  Reaction Name  Participation Rate (%) 
1  ACCOAC  acetyl CoA carboxylase  75.08 
2  MCOATA  Malonyl CoA ACP transacylase  75.08 
3  3HAD100  3 hydroxyacyl acyl carrier protein dehydratase n C100  74.48 
4  3HAD40  3 hydroxyacyl acyl carrier protein dehydratase n C40  74.48 
5  3HAD60  3 hydroxyacyl acyl carrier protein dehydratase n C60  74.48 
6  3HAD80  3 hydroxyacyl acyl carrier protein dehydratase n C80  74.48 
7  3OAR100  3 oxoacyl acyl carrier protein reductase n C100  74.48 
8  3OAR40  3 oxoacyl acyl carrier protein reductase n C40  74.48 
9  3OAR60  3 oxoacyl acyl carrier protein reductase n C60  74.48 
10  3OAR80  3 oxoacyl acyl carrier protein reductase n C80  74.48 
Part 2  
The top 10 reactions with highest participation rate in the sTC  
Order  Abbr.  Reaction Name  Participation Rate (%) 
1  tscr_elo_TU8389_ini  Formation complex for elongation of TU8389'  0.65 
2  tscr_elo_TU8390_ini  Formation complex for elongation of TU8390'  0.65 
3  tscr_elo_TU8397_ini_rho_dep  Formation complex for elongation of TU8397 (RHO DEPENDENT TERMINATION)'  0.65 
4  tscr_elo_TU8407_ini_rho_dep  Formation complex for elongation of TU8407 (RHO DEPENDENT TERMINATION)'  0.65 
5  tscr_elo_TU01181_ini_stab  Formation complex for elongation of TU01181 (stable RNA)'  0.65 
6  tscr_elo_TU01182_ini_stab  Formation complex for elongation of TU01182 (stable RNA)'  0.65 
7  tscr_elo_TU01183_ini_stab  Formation complex for elongation of TU01183 (stable RNA)'  0.65 
8  tscr_elo_TU01186_ini_stab  Formation complex for elongation of TU01186 (stable RNA)'  0.65 
9  tscr_elo_TU01187_ini_stab  Formation complex for elongation of TU01187 (stable RNA)'  0.65 
10  tscr_elo_TU01189_ini_stab  Formation complex for elongation of TU01189 (stable RNA)'  0.65 
The top 10 reactions with highest participation rate in the sTL  
Order  Abbr.  Reaction Name  Participation Rate (%) 
1  IF2_RECHARG  recharge of IF2 with GTP'  100 
2  Rib_30_ini_FORM  formation of 30S translation initiation complex (30S subunit, IF1, IF2GTP, IF3)'  100 
3  Rib_70_DISS  70S ribosome dissociation'  100 
4  EFG_RECHARG  recharge of EFG with GTP'  99.95 
5  tl_elo_b3461_16_rib1  Translation elongation 1 b3461 16 ribosome(s)'  0.75 
6  tl_elo_b3461_16_rib2  Translation elongation 2 b3461 16 ribosome(s)'  0.75 
7  tl_elo_b3461_1_rib1  Translation elongation 1 b3461 1 ribosome(s)'  0.75 
8  tl_elo_b3461_1_rib2  Translation elongation 2 b3461 1 ribosome(s)'  0.75 
9  tl_elo_b3461_8_rib1  Translation elongation 1 b3461 8 ribosome(s)'  0.75 
10  tl_elo_b3461_8_rib2  Translation elongation 2 b3461 8 ribosome(s)'  0.75 
In TRN, the two most active reactions CRP_noGLC_1 and Crp_1 relate to the regulation rules of the transcription factor (TCF) Creactive protein (CRP). Other high rank reactions Fis_1, Lrp_1, Fnr_1, and NOT_ArcA_1 relate to the regulation rules of the TCFs Fis, Lrp, Fnr and ArcA, respectively. In E.coli, the above TCFs belong to the seven global regulators that control most of the regulated genes [28]. The reaction NOT_Cra_1 is relevant to the regulation rules of the TCF Cra, a pleiotropic regulatory protein that controls carbon and energy fluxes in enteric bacteria [29, 30]. The reaction NOT_PdhR_1 concerns the regulation rules of PdhR, a TCF that controls the respiratory electron transport system in E.coli. Its regulation target, the pyruvate dehydrogenase (PDH) multienzyme complex, plays a key role in the metabolic interconnection between glycolysis and the citric acid cycle [31].
In sACL, the most active reaction is ASPTA. It transfers oxoglutarate and aspartate to corresponding ketoacid, which are indispensable in glyoxylate cycle, an anabolic metabolic pathway occurring in E. coli [32]. The second one is ASAD which is the second step in the biosynthesis of amino acids in prokaryotes, fungi, and some higher plants. ASAD forms an early branch point in the metabolic pathway producing lysine, methionine, leucine and isoleucine from aspartate as well as diaminopimelate which plays an essential role in bacterial cell wall formation [33]. Deletion of gene asd (encoding ASAD) is lethal to the organism as demonstrated by experiments with Legionella pneumophila, Salmonella typhimurium, and Streptococcus mutans, which indicates that ASAD may also be an essential reaction in the metabolism of E.coli [34]. Another active reaction is ASPK, which is the commitment step in the pathway to the synthesis of lysine, methionine, threonine and isoleucine.
In sMM, the reaction ACCOAC is most active. It is a ratedetermining step in the fatty acid synthetic pathway and may play a pivotal role in regulating fatty acid oxidation [35]. The second most active reaction MCOATA transfers Malonyl CoA to acylcarrier proteins (ACPs). The product Malonyl ACP provides malonyl groups for biosynthesis of fatty acid and polyketide. On the other hand, Malonyl CoA, the substrate of MCOATA, is a highlyregulated molecule in fatty acid synthesis as it inhibits the ratelimiting step in betaoxidation of fatty acids [36]. Flux change in MCOATA affects the consistency of Malonyl CoA and guarantees the biosynthesis of fatty acid.
In sTC, all the top reactions relate to the formation of the transcription elongation complex, an extremely complicated and highly regulated molecular machine that can sense signals coming from numerous regulatory protein factors, as well as those encoded in the DNA sequence. They are the basis of transcription elongation, because transcription can run smoothly and continuously only depending on their precise work.
In sTL, the reactions IF2_RECHARG, Rib_30_ini_FORM and Rib_70_DISS are used by all ExPas. IF2_RECHARG recharges the initiation factor 2 (IF2) with GTP and Rib_30_ini_FORM produces 30S translation initiation complex which consists of 30S subunit, IF1, IF2GTP and IF3. In bacteria, the correct mRNA starting site and the reading frame are selected when, with the help of IF1, IF2 and IF3, the initiation codon is decoded in the peptidyl site of the 30S ribosomal subunit by the anticodon fMettRNAfMet. Furthermore, Rib_30_ini_FORM is also proved to be the intermediate step in the formation of 70S initiation complex (70SIC) which regulates translation initiation, the ratelimiting step in protein synthesis [37]. The other reaction Rib_70_DISS dissociates 70S ribosomes to 30S ribosomal subunit/IF1/IF3 complex (rib_30_IF1_IF3) and 50S ribosomal subunit (rib_50_inact). This is an essential step before a ribosome can participate in a new round of translation since the initiation complex for protein synthesis involves a 30S subunit. The dissociation of 70S ribosomes contributes to the efficiency and sustainability of protein synthesis [38].
Reportedly, RPRs help to find important reactions in MN [13]. Our results further indicate that RPR can also be extended to TRN and TTN to evaluate the relative importance of a given reaction.
Correlated reaction set
A correlated reaction set (CoSet) comprises reactions that always participate in the same ExPa set in a given network [13]; namely if one reaction functions, the others in the same CoSet function simultaneously.
A CoSet can be transformed to a graph by treating each reaction as a node and adding an edge between two reactions that involve a common substance. In a certain CoSet, some member reactions are topologically connected while others are not. The correlationship of the second type of reactions often indicates a transcriptional coregulation by the corresponding genes [11] while that of the first type has relatively trivial biological meaning. Therefore, a CoSet is defined as a trivial set if all its member reactions are connected in topology. A trivial CoSet provides less novel information, and thus it is unworthy of deep study. In this paper, the adjacent ratio is used to represent the percentage of trivial CoSets.
Summary of CoSets
Network  No. of  No. of  Size of CoSet  Adjacent  

reactions  CoSet  Max  Min  Avg  Most  Ratio (%)  
TRN  2115  481  8  2  2  2  98 
sACL  741  157  11  2  2  2  78 
sMM  358  49  25  2  4  3  47 
sTC  1276  161  18  2  6  6  93 
sTL  7386  1718  23  2  2  2  85 
Crosstalk analysis
Crosstalk analysis was first raised to illustrate the relationship between multiple inputs or outputs of a signaling pathway [39]. The whole ExPa set was compared pairwise to build the simplest form of crosstalk [2, 10]. A pair of ExPas may have identical, overlapped or disjoint inputs (or outputs). There are 9 categories of crosstalk with their biological meanings described in [10]. Here, crosstalk analysis is applied to other biological processes to detect the relationships between fundamental functional states. Various forms of crosstalk in the five networks/subnetworks above were characterized. As several exchange reactions participate in most ExPas of sACL, MN, sTC and sTL, almost all of the ExPa pairs have overlapped inputs or outputs. A close look at the highly participating exchange reactions reveals that most of them relate to small molecules such as H_{2}O, ATP and NADP commonly seen in various biochemical reactions. In order to further elucidate the difference in crosstalk between ExPa pairs, all the exchange reactions in the four subnetworks were sorted in a descending order depending on RPR and the top 20% ExPa pairs were neglected in the subsequent crosstalk analysis.
Discussion
ExPa analysis were applied to two new models, the E.coli TRN and TTN. A horizontal comparison was performed for the five networks/subnetworks: TRN, sACL, sMM, sTC, sTL from five aspects: (1) Total number of ExPas and the P/R ratios; (2) ExPa length distribution and L/R ratios; (3) Reaction participation rates; (4) Correlated reaction sets and adjacent ratios; (5) Interconnectivity of ExPas.
Reactions in TTN represent actual biochemical reactions like those in MN, and thus, ExPas in TTN characterize the steadystates of the corresponding biological systems. In contrast, columns in TRN represent the transcriptional regulatory rules and coefficents only reflect the qualitative information describing the presence or absence of the corresponding components rather than the quantitative information describing reaction stoichiometries as in TTN and MN. Therefore, an ExPa in TRN characterizes a specific transcriptional regulatory state, namely which transcriptional regulatory rules are activated and which genes are expressed in a specific environmental state.
ExPa analysis emphasizes the functional and systemic properties of biologcial process as ExPas are systemically independent functional units. The total number of ExPas and the P/R ratios characterize the flexibility of the networks/subnetworks. ExPa length corresponds to the reaction steps needed to form a steady state, therefore showing a close relation to network complexity. Crosstalk enables the analysis of pathway redundancy and network determinacy. Comparisons from these aspects indicate that MN is more flexible but less deterministic than TRN and TTN. Environmental cues affect transcriptional regulation, which controls the following transcription and translation processes. Then the resulting gene products (enzymes) enter the metabolic system to catalyze the corresponding reactions. It is necessary for a cell to respond accurately to the environment and produce the required enzymes. MN is more robust to environmental changes, which reflects the struggle of a cell to achieve an alternative steadystate to provide substance support for TRN and TTN and maintain life.
The distributions of reaction participation in the five networks/subnetworks are similar except that there are more reactions participating in more than 10% ExPas in sACL and sMM. Only a small percent of the reactions participate in a large number of ExPas, which indicates the phenotypic potentials of TRN, TTN and MN are affected greatly by a small number of important reactions. Evaluations on the representatives show that reactions with high participation rates often play an important role in certain biological processes. These reactions are the relatively weak part of the networks because a large number of ExPas will be destroyed when these reactions become invalid, which may cause the loss of various functions. These reactions may be used as drug targets and further direct the design of new drugs.
CoSets were identified via the calculation of reaction participation. Besides the expected topological connections, the topologically unconnected reactions in a CoSet may indicate the information of transcriptional coregulation in MN. However, most Cosets of TRN and TTN are trivial, and thus have few chances to be a clue giving novel information like in MN.
Last but not least, an improved approach was introduced to calculate the ExPas on TRN models. Compared to the existing method, the biggest advantage of ours is the high efficiency in calculating all the extreme pathways of a TRN, especially for the one which may work under huge amount of environmental conditions. For example, the E.coli TRN model which we studied in the paper has 776 components whose availability (i.e., presence or absence) constitute the environmental condition, including environmental stimuli, transcription factors or proteins. It is impossible to enumerate all the possible conditions due to "combination explosion" without mentioning the calculation of the ExPas under each condition. However, using the approach we proposed, it took only about 45 seconds to computing the whole ExPa set on a PC with four 3.2GHz Intel(R) XEON processors and 16GB RAM (in fact, only one processor and 15MB RAM are used for the calculation). We believe that this approach could be helpful for readers who are also interested in the ExPas of TRNs.
Conclusions
This study presents the first horizontal comparison among the E.coli TRN, MN and TTN through ExPa analysis. The results show that ExPa also has biological meanings in TRN and TTN. Different properties of ExPas reflect the biological nature of each biological process. Along with the the increase of reconstructed models on TRNs and TTNs as well as the development of new methods, ExPa analysis may reveal more biological properties and get larger space of application in the medical and biochemical fields.
Methods
COBRA framework and ExPa analysis
where ${\mathbf{\text{S}}}_{m\times n}$ is the stoichiometric matrix of a biochemical network with $m$ components and $n$ reactions and ${\mathbf{\text{v}}}_{n\times 1}$ is a vector of the fluxes through each reaction in the system [40].
For a given network, the ExPa set has the following properties: (1) It is unique; (2) Each ExPa uses fewest reactions to form a function unit; (3) It is systemically independent which means an ExPa cannot be represented by a nonnegative linear combination of other ExPas [42, 43].
ExPa calculation on the MN and TTN
ExPas were calculated using an open source tool 'expa' [44]. The E.coli MN and TTN models were divided into small subnetworks using the method proposed in [11].
An improved approach to compute the ExPas of TRN models
A TRN is composed of a set of transcriptional regulatory rules which describe cells' transcriptional responses to environmental signals. A regulatory network matrix $\mathbf{R}$ was used by Gianchandani et al. to represent the components (environmental cues, metabolites, genes and proteins) and reactions (regulatory rules and exchange reactions of products) of a TRN [3]. It was further combined with an environmental matrix $\mathbf{\text{E}}$, which characterizes a particular environmental state, yielding a complete regulatory state matrix ${\mathbf{R}}^{*}=\left[RE\right]$. Each column of $\mathbf{\text{E}}$ delineates the availability of a unique environmental cue, transcription factor, target gene or protein [3, 45]. Different environmental states correspond to different $\mathbf{\text{E}}$s, thus forming different ${\mathbf{R}}^{*}$s.
where A, B, C and D are four metabolites enacting as signalling stimuli.
In order to calculate all the ExPas of the TRN, all the environmental states, namely all possible $\mathbf{\text{E}}$s, need be enumerated. Then ExPas participating in each possible environmental state are generated and the unique ones are grouped to form the complete ExPa set. Since the number of possible environmental states grows exponentially with the number of extracellular metabolites, it is inefficient to enumerate all possible environmental states for a TRN with numerous envionmental cues [45]. Therefore, an improved method is introduced here to simplify the ExPa calculation on the COBRA model of TRN.
The gist of the method is to improve Gianchandani's method by employing two columns instead of one to delineate the presence and absence of a unique envionment cue respectively, by which a new environment matrix ${E}_{new}$ is constructed. The matrix ${E}_{new}$ covers all possible environmental states. Without loss of generality, we assume that the top $n$ rows in $\mathbf{R}$ and ${E}_{new}$ represents the present state of n environmental inputs ${m}_{e}\left(e=1,\cdots \phantom{\rule{0.3em}{0ex}},n\right)$ onetoone and the following $n$ rows represents the absent state of them. The original regulatory state matrix is ${\mathbf{R}}^{*}=\left[\mathbf{R}E\right]=\left[{\mathbf{R}}_{1},{\mathbf{R}}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k}{\mathbf{R}}_{k+1},{\mathbf{R}}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+{n}_{}}\right]$ and the new matrix is ${\mathbf{\text{R}}}_{new}^{*}=\left[\mathbf{R}{E}_{new}\right]=\left[{\mathbf{R}}_{1},{\mathbf{R}}_{2}\cdots {\mathbf{R}}_{k}{\mathbf{R}}_{k+1},{\mathbf{R}}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+n}{\mathbf{R}}_{k+n+1},{\mathbf{R}}_{k+n+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+2n}\right]$($k$ is the number of columns in $\mathbf{R}$, and ${E}_{new}=[{r}_{k+1},{r}_{k+2},.\; .\; .,{r}_{k+n}{r}_{k+n+1},{r}_{k+n+2},.\; .\; .,{r}_{k+2n}]$). For an input ${m}_{\text{e}}$, column ${\mathbf{R}}_{k+e}$ represents its presence and column ${\mathbf{R}}_{k+n+e}$ represents its absence under the environmental condition, where ${\mathbf{R}}_{k+e}\left(e\right)$ and ${\mathbf{R}}_{k+n+e}\left(n+e\right)$ equal to 1 and the other elements are all zeros. For example, the ${\mathbf{\text{R}}}_{new}^{*}$ matrix of the above toy model is illustrated in Figure 5B. The shaded columns constitute ${E}_{new}$. Obviously, the space and time complexity for constructing ${E}_{new}$ is $\mathrm{O}\left(n\right)$, where $n$ is the number of components of a TTN model. The convex basis of the right null space of ${\mathbf{\text{R}}}_{new}^{*}$ comprises the ExPa set of the TRN which could then be enumerated by the tool 'expa' [44].
Figures 6B and 6C show two ExPas resulting from the matrixs in Figures 5A and 5B respectively. The two vectors represent the same steady state of the TRN in which gene G1 is inhibited because of lack of metabolite B. In Figure 6B, the exact meaning of "$\text{1}$" in element ${B}_{AV}$ cannot be decided directly from ExPa without referring to the shaded part of matrix in Figure 5A. However, in Figure 6C, "$\text{1}$" in column ${B}_{AB}$ clearly means the absence of metabolite $\text{B}$. Namely, the interpretation of an ExPa resulting from the improved method is independent from the environmental matrix, which makes an ExPa easier to understand.
Validation of the approach of ExPa calculation on TRNs
Given $n$ environmental cues, there are ${2}^{n}$ possible environmental states, each corresponding to a matrix ${E}_{i}$ and the corresponding ${\mathbf{\text{R}}}_{i}^{*}$ (${\mathbf{\text{R}}}_{i}^{*}=[\mathbf{\text{R}}{E}_{i}]$, $i=1,\cdots \phantom{\rule{0.3em}{0ex}},{2}^{n}$). The ExPa set obtained from ${\mathbf{\text{R}}}_{i}^{*}$ is denoted as ${\text{P}}_{i}$ and the feasible ExPa set calculated from ${\mathbf{\text{R}}}_{new}^{*}$ is denoted as ${\text{P}}_{new}$. Since the meaning of the environmental part of ${\text{P}}_{i}$ is dependent on the environmental states, ExPas of different environmental states should be normalized to eliminate the dependence before being grouped up. We normalized a ExPa ${\mathbf{\text{p}}}_{i}^{j}$ in the set ${\text{P}}_{i}$ by expanding its dimension of the input part from $n$ (${\mathbf{\text{p}}}_{i}^{j}$) to $2n$ (${\widehat{\mathbf{\text{p}}}}_{i}^{j}$). Details of the normalization are described in Algorithm 1.
Data: ${\mathbf{\text{p}}}_{i}^{j}$,${A}^{i}$,${\overline{A}}^{i}$.
// ${\mathbf{\text{p}}}_{i}^{j}$ represents the $j$th ExPa in the $i$th environment, where $i=1,\cdots \phantom{\rule{0.3em}{0ex}},{2}^{n}$;
// ${\mathbf{\text{p}}}_{i}^{j}=\left[{v}_{1},{v}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k}^{}{v}_{k+1},{v}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k+n}\right]$;
// ${A}^{i}$ is a set which consists of all the absent inputs;
// ${\overline{A}}^{i}$ is a set which consists of all the present inputs;
Result: ${\widehat{\mathbf{\text{p}}}}_{i}^{j}$. // ${\widehat{\mathbf{\text{p}}}}_{i}^{j}=\left[{v}_{1},{v}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k}^{}{v}_{k+1},{v}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k+n}^{},{v}_{k+n+1},{v}_{k+n+2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k+2n}\right]$;
For $q=1$ to $k$ do ${\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(q\right)={\mathbf{\text{p}}}_{i}^{j}\left(q\right)$; End for
For $q=1$ to $n$ do
If ${\mathbf{\text{p}}}_{i}^{j}\left(k+q\right)=0$ do
${\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+q\right)=0;\phantom{\rule{0.3em}{0ex}}{\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+n+q\right)=0;$
Else if ${\mathbf{\text{p}}}_{i}^{j}\left(k+q\right)=1$ do
If $q\in {A}^{i}$ do
${\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+q\right)=0;\phantom{\rule{0.3em}{0ex}}{\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+n+q\right)=1;$
Else if $q\in {\overline{A}}^{i}$
${\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+q\right)=1;\phantom{\rule{0.3em}{0ex}}{\widehat{\mathbf{\text{p}}}}_{i}^{j}\left(k+n+q\right)=0;$
End if
End if
End for
Algorithm 1: Procedure of normalizing ${\mathbf{\text{p}}}_{i}^{j}$ to ${\widehat{\mathbf{\text{p}}}}_{i}^{j}$ by dimension expanding.
In a normalized ExPa $\widehat{\mathbf{\text{p}}}=\left[{v}_{1},{v}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{v}_{k+2n}\right]$, "$\text{1}$" on ${v}_{k+e}\left(e=1,\cdots \phantom{\rule{0.3em}{0ex}},n\right)$ indicates that ${m}_{e}$ is present on the ExPa while "$\text{1}$" on ${v}_{k+n+e}\left(e=1,\cdots \phantom{\rule{0.3em}{0ex}},n\right)$ indicates that ${m}_{e}$ is absent, and $0$ indicates that ${m}_{e}$ does not affect the transcriptional states characterized by this Expa. The normalized ExPa set of ${\text{P}}_{i}$ is denoted as ${\widehat{\text{P}}}_{i}$ and the union of ${\widehat{\text{P}}}_{i}$$\left(i=1\dots {2}^{n}\right)$ is denoted as $\widehat{\text{P}}$. As explained above, the ExPas in set ${\text{P}}_{new}$ are already in the normalized form, hence no normalization are needed.
Here we prove that ${\widehat{P}}_{new}$ equals to $\widehat{\text{P}}$:
Statement 1: each ExPa in $\widehat{\text{P}}$ can be obtained from ${\mathbf{\text{R}}}_{new}^{*}$.
Proof: given $n$ extracellular metabolites ${m}_{p}\left(p=1,\cdots \phantom{\rule{0.3em}{0ex}},n\right)$, each ${\mathbf{\text{R}}}_{i}^{*}$ can be transformed to ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$ as follows (Algorithm 2):
Data: ${\mathbf{\text{R}}}_{i}^{*}$,${A}^{i}$,${\overline{A}}^{i}$.
// ${\mathbf{\text{R}}}_{i}^{*}$ represents TRN in the $i$th environment, where $i=1,\cdots \phantom{\rule{0.3em}{0ex}},{2}^{n}$;
// ${\mathbf{\text{R}}}_{i}^{*}=\left[{\mathbf{R}}_{1},{\mathbf{R}}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k}{\mathbf{R}}_{k+1},{\mathbf{R}}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+n}\right]$;
// ${A}^{i}$ is a set which consists of all the absent inputs;
// ${\overline{A}}^{i}$ is a set which consists of all the present inputs;
Result: ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$.
// ${\widehat{\mathbf{\text{R}}}}_{i}^{*}=\left[{\mathbf{R}}_{1},{\mathbf{R}}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k}{\mathbf{R}}_{k+1},{\mathbf{R}}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+n},{\mathbf{R}}_{k+n+1},{\mathbf{R}}_{k+n+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+2n}\right]$;
For $q=1$ to $2n$ do ${\mathbf{R}}_{k+q}=0$; End for
For $q=1$ to $n$ do
If $q\in {\mathbf{\text{A}}}^{i}$ do
${\mathbf{R}}_{k+n+q}\left(n+q\right)=1$;
Else if $q\in {\overline{A}}^{i}$
${\mathbf{R}}_{k+q}\left(q\right)=1$;
End if
End for
Algorithm 2: Procedure of transforming ${\mathbf{\text{R}}}_{i}^{*}$ to ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$.
For ${\widehat{\mathbf{\text{R}}}}_{i}^{*}=\left[{\mathbf{R}}_{1},{\mathbf{R}}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k}{\mathbf{R}}_{k+1}\text{,}{\mathbf{R}}_{k+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+n},{\mathbf{R}}_{k+n+1},{\mathbf{R}}_{k+n+2},\cdots \phantom{\rule{0.3em}{0ex}},{\mathbf{R}}_{k+2n}\right]$ ($i=1,\cdots \phantom{\rule{0.3em}{0ex}},{2}^{n}$) resulted from Algorithm 2, if $\exists j\in \left\{k+1,\cdots \phantom{\rule{0.3em}{0ex}},k+2n\right\}$ such that ${\mathbf{R}}_{j}=0$, then a constraint ${v}_{j}=0$ is added. Then the resulting network is a subnetwork of that represented by ${\mathbf{\text{R}}}_{new}^{*}$. As proven in [46], $G$ and ${G}^{\prime}$ are two MNs whose reactions are all irreversible and whose ExPa sets are $\text{EP}$ and $\text{E}{\text{P}}^{\prime}$, respectively. If $\text{E}{\text{P}}^{\prime}$ is a subnetwork of $\text{EP}$, then $\text{E}{\text{P}}^{\prime}\subseteq \text{EP}$. Therefore ${\widehat{\text{P}}}_{i}\subseteq {\widehat{\text{P}}}_{new}$, because $\text{P}={\displaystyle \bigcup _{i=1}^{n}}{\widehat{\text{P}}}_{i}$, $\widehat{\text{P}}\subseteq {\widehat{\text{P}}}_{new}$.
Statement 2: each feasible ExPa in ${\widehat{\text{P}}}_{new}$ can be obtained by some ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$.
Proof: Since any environmental cue is impossible to be both present and absent in a specific environment, ${v}_{k+e}\times {v}_{k+n+e}=0$ ($e=1,\cdots \phantom{\rule{0.3em}{0ex}},n$) is true for each ExPa in ${P}_{new}$. For any ExPa $\mathbf{\text{p}}\in {\widehat{P}}_{new}$, let $T={\mathbf{\text{R}}}_{new}^{*}$. For any $e$, $T$ is modified as follows: (1) If ${v}_{k+e}=0$ and ${v}_{k+n+e}\ne 0$, ${t}_{k+e}=0$; (2) If ${v}_{k+n+e}=0$ and ${v}_{k+e}\ne 0$, ${t}_{k+n+e}=0$; (3) If ${v}_{k+e}=0$ and ${v}_{k+n+e}=0$, ${t}_{k+e}=0$, where ${t}_{i}$ is the $i$th column of $T$. As can be shown easily, $\mathbf{\text{p}}$ is an ExPa of the right null space of $T$. According to Algorithm 2, a legal ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$ contains one zero column and one nonzero column corresponding to the two input reactions of a certain input component respectively. Therefore, $T$ is a legal ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$, and each ExPa in ${\widehat{\text{P}}}_{new}$ can be obtained by some ${\widehat{\mathbf{\text{R}}}}_{i}^{*}$, or in other words, ${\widehat{\text{P}}}_{new}\subseteq \widehat{\text{P}}$.
From statements (1) and (2), we conclude that ${\widehat{\text{P}}}_{new}=\widehat{\text{P}}$, and thus all possible ExPas of a TRN can be obtained using our new representation.
Classification of ExPas
ExPas fall into three classes, in which class III stands for internal reaction cycles with no exchange flux [12]. Class III ExPas were proven to be thermodynamically infeasible [47] and thus were not considered in our analysis.
List of abbreviations used
 COBRA:

Constraintbased Reconstruction and Analysis
 MN:

Metabolic Network: ExPa: Extreme Pathway
 TRN:

Transcriptinal Regulatory Network
 TTN:

Transcriptional and Translational Network
 sACL:

The subnetwork of Amino acid, arbohydrate and Lipid metabolism
 sMM:

The subnetowrk of Membrane and Murein metabolism
 sTC:

The subnetwork of Transcription in the TTN
 sTL:

The subnetwork of Translation in the TTN
 ORF:

Open Reading Frame
 P/R:

the Numberbased Ratios of ExPa to Reaction
 L/R:

the Ratio of Average ExPa Length to Reaction Number
 RPR:

the Reaction Participation Rate
 TCF:

Transcription Factor
 CRP:

Creactive Protein
 PDH:

Pyruvate Dehydrogenase
 ACP:

Acylcarrier Protein
 IF:

Initiation Factor
 70SIC:

70S Initiation Complex
 rib_30_IF1_IF3:

30S Ribosomal Subunit/IF1/IF3 Complex
 rib_50_inact:

50S Ribosomal Subunit
 CoSet:

Correlated Reaction Set.
Declarations
Acknowledgements
This work is supported by Chinese National Natural Science Foundation (61073068) and the Graduated Students' Innovation Fund of Fudan University. The authors would also like to thank Ying Wang and Dongqiang Xie for helpful discussions on the work.
Declarations
Publication of this article was funded by the corresponding author.
This article has been published as part of BMC Systems Biology Volume 8 Supplement 1, 2014: Selected articles from the Twelfth Asia Pacific Bioinformatics Conference (APBC 2014): Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/8/S1.
Authors’ Affiliations
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