Metabolite coupling in genome-scale metabolic networks
- Scott A Becker^{1},
- Nathan D Price^{2} and
- Bernhard Ø Palsson^{1}Email author
DOI: 10.1186/1471-2105-7-111
© Becker et al; licensee BioMed Central Ltd. 2006
Received: 08 November 2005
Accepted: 06 March 2006
Published: 06 March 2006
Abstract
Background
Biochemically detailed stoichiometric matrices have now been reconstructed for various bacteria, yeast, and for the human cardiac mitochondrion based on genomic and proteomic data. These networks have been manually curated based on legacy data and elementally and charge balanced. Comparative analysis of these well curated networks is now possible. Pairs of metabolites often appear together in several network reactions, linking them topologically. This co-occurrence of pairs of metabolites in metabolic reactions is termed herein "metabolite coupling." These metabolite pairs can be directly computed from the stoichiometric matrix, S. Metabolite coupling is derived from the matrix ŜŜ^{T}, whose off-diagonal elements indicate the number of reactions in which any two metabolites participate together, where Ŝ is the binary form of S.
Results
Metabolite coupling in the studied networks was found to be dominated by a relatively small group of highly interacting pairs of metabolites. As would be expected, metabolites with high individual metabolite connectivity also tended to be those with the highest metabolite coupling, as the most connected metabolites couple more often. For metabolite pairs that are not highly coupled, we show that the number of reactions a pair of metabolites shares across a metabolic network closely approximates a line on a log-log scale. We also show that the preferential coupling of two metabolites with each other is spread across the spectrum of metabolites and is not unique to the most connected metabolites. We provide a measure for determining which metabolite pairs couple more often than would be expected based on their individual connectivity in the network and show that these metabolites often derive their principal biological functions from existing in pairs. Thus, analysis of metabolite coupling provides information beyond that which is found from studying the individual connectivity of individual metabolites.
Conclusion
The coupling of metabolites is an important topological property of metabolic networks. By computing coupling quantitatively for the first time in genome-scale metabolic networks, we provide insight into the basic structure of these networks.
Background
Cellular metabolism is an extensively studied process that is essential to the survival of any free-living organism. The metabolic process can be characterized as a set of biochemical transformations, each of which involves the consumption of one or more metabolites and the production of one or more metabolites. Subject to the law of mass conservation, the net sum of elements and electrical charge is conserved in each reaction and thus in the network as a whole. Each transformation by definition must involve more than one metabolite. Herein, we define "metabolite coupling" as the appearance of a pair of metabolites in the same biochemical transformation.
Any examination of carefully balanced biochemical transformations immediately displays the ubiquity of the proton (H^{+}) and water in metabolic reactions. The ubiquity of the proton in metabolic networks is interesting in light of the fact that at standard pH a typical bacterial cell contains approximately 60 free protons. Assuming pH = 7 [1] and the volume of the cell is equal to 1 μm^{3}, the number of free protons in solution is equal to (10^{-7} mol H^{+}/L)(6.022 × 10^{23} molecules/mol)(cell volume L); unit conversions yield the final answer. Thus, through the medium of water, the protons must be rapidly shuttled around the cell as metabolic transfers are taking place. Although the inclusion of protons is not complete in most metabolic pathway databases, protons have been included where appropriate in the networks analyzed in the present work. While the inclusion of protons in genome scale metabolic reconstructions is not frequent, it does allow for the computation of important effects of H^{+} balancing on network functions. For example, the change in pH of the growth media for cultures of Escherichia coli was predicted by a balanced model [2]. Careful elemental and charge balancing of the human mitochondrial metabolic network also allowed for the computational determination of ATP yield [3].
With the exception of the proton and water, the most commonly coupled metabolites are those generally referred to as cofactors (i.e. ATP/ADP, NAD^{+}/NADH). Cofactors fulfil a range of roles in cells, from transferring energy and redox potential to carrying key metabolic intermediates. In some cases, different common cofactors can serve the same purpose – ATP and GTP can both deliver energy; NADH and NADPH can both supply reducing power – but one is often preferred for a given purpose in an organism. For example, NADH is more commonly used during the oxidative reactions that comprise catabolism whereas NADPH is preferred for the reductive reactions that are involved in anabolism [4]. The exact use of cofactors in metabolic reactions is typically not fully defined in genome annotations, and thus metabolic pathways derived from a sequence-based annotation may not correctly specify the cofactor used in a particular reaction in a network [5].
Topological analyses of the cellular metabolism of microorganisms have generated significant interest recently [6]. Metabolic networks have been shown to exhibit scale-free behavior, where the number of reactions in which a given metabolite participates follows a power-law distribution, meaning that the probability of a metabolite having k connections to other metabolites is approximately equal to k^{-γ}, where γ is constant for a given network [7, 8]. The reaction fluxes in the central metabolism of E. coli have been shown to behave in a similar manner; the probability that a reaction has a given flux v is proportional to (v+v_{0})^{-α}, with v_{0} and α constant [9]. The overall hierarchical and modular organization of metabolic networks has been demonstrated [10]. Protein domain networks also have been suggested to have a scale-free nature [11]. The methodology and principal results of the topological analysis of biological networks has been recently reviewed [6]. More recently, it has been asserted that a power law is a natural consequence of a high variability process, much as the central limit theorem dictates a Gaussian distribution for lower variability processes [12].
Metabolic networks of differing levels of detail and accuracy can be encoded using a variety of formalisms, including (hyper)graphs [7], petri nets [13], and stoichiometric matrices [14, 15]. The level of detail of a metabolic network is primarily determined by the method of reconstruction, independent of the specific encoding used. For example, networks generated in a semi-automated fashion principally from metabolic pathway databases enable high-throughput reconstructions for many organisms but sacrifice accuracy with regard to fine details such as cofactor preference (NADH vs. NADPH) and proton balancing. In contrast, the metabolic networks considered herein were manually reconstructed from diverse sources, including metabolic databases, legacy biochemical data, and physiological data [16]. They are curated to the point where they can be used for computation, and the subsequent comparison of these computations to experimental data provides further evidence as to the quality of the reconstructions [17, 18]. These networks were initially published using the stoichiometric matrix formalism. Furthermore, all of the metabolic networks analyzed herein were scrutinized for cofactor preference on a reaction-by-reaction basis as well as elementally and charge balanced before the networks were initially published, meaning that each reaction was examined to determine if the addition of protons was necessary to enforce the conservation of elements and charge at cellular pH. A study of metabolite coupling requires both this balancing and the most accurate assignment of cofactors to reactions possible for each organism.
In this paper, we present a topological analysis of metabolite coupling in genome-scale metabolic networks. We find that most pairs of metabolites never participate in a reaction together, smaller numbers of pairs occur together in a few reactions, and a select few pairs, such as cofactors, occur in a significant fraction of the reactions in the network, in many cases more often than would be expected based on random connectivity. This work is distinct from the previous analyses because we consider the consequences of pairs of metabolites occurring together on a genome scale, rather than individual metabolites. The motivation for studying metabolite pairs is that many biomolecules are of more interest as pairs than as single molecules. For example, ATP is most important because of its ability to couple with ADP, P_{i} and H^{+} in reactions that transfer the phosphate moiety as a form of energy currency for a cell. Studies of metabolite coupling can be used to highlight such chemically-based network properties as they appear throughout the entire network for multiple organisms. Because we focus on coupling, having essential detail in cofactor usage is essential. This necessitates that we use only the hand-curated metabolic networks that are available for a select few microorganisms and the human cardiac mitochondrion.
Results and discussion
Basic network properties
We characterized six metabolic networks based on topological features, with representatives from each of the three primary domains of life. The metabolic networks considered represent three bacteria (E. coli [2], Helicobacter pylori [19], Staphylococcus aureus [20]), one member of the archea (Methanosarcina barkeri [21]), one unicellular eukaryote (S. cerevisiae [22]), and one human cellular organelle (cardiac mitochondrion [3]). These are, to the best of our knowledge, the only genome-scale metabolic networks that are manually curated as well as elementally and charge balanced available to date.
Basic metabolic network statistics.
E. coli | S. cerevisiae | H. pylori | S. aureus | Mitochondrion | M. barkeri | |
---|---|---|---|---|---|---|
Number of compounds | 621 | 642 | 404 | 554 | 145 | 544 |
Number of reactions | 802 | 861 | 417 | 588 | 130 | 564 |
Density of M | 2.2% | 2.2% | 3.1% | 2.4% | 7.8% | 2.3% |
Metabolite coupling
Metabolite connectivity
Furthermore, the slope of each line for single metabolite connectivities (figure 6) is different than the slope of the corresponding line for metabolite coupling (figure 5). When the ratio of the slope for metabolite coupling to the slope for single metabolite participation is computed, the results range from 0.95 (mitochondrion) to 1.54 (S.aureus) with a mean of 1.11 and standard deviation of 0.10. Furthermore, when the slopes are rank ordered from greatest to least, it is observed that the networks are ordered differently when considering single metabolites than when considering metabolite coupling. Thus, the widely differing slope ratios and differential ordering indicates that the distribution of metabolite coupling (off-diagonal elements in M) is not simply a direct result of the distribution of metabolite connectivity (diagonal elements in M).
Relation between metabolite coupling and metabolite connectivity
Although the slopes of best-fit lines vary considerably for single metabolites and metabolite pairs, there is an important relationship between metabolite coupling and individual metabolite connectivity because a particular metabolite cannot be highly coupled with other metabolites if it does not appear in many reactions. Thus, each off-diagonal element of M (m_{ij}) is subject to both of the following criteria:
m_{ij} ≤ m_{ii}
m_{ij} ≤ m_{jj}
Coupling of prominent metabolites. The number of metabolites that couple with each compound. In the E. coli metabolic network, the proton participates in at least one reaction with 479 other unique metabolites. The ordering of metabolites is based on the number of reactions they participate in individually, averaged across all networks. Thus, the number of unique metabolites that couple with a given metabolite do not strictly have to decrease down a column, although it usually does.
Number of unique metabolites with which each prominent metabolite couples | ||||||
---|---|---|---|---|---|---|
Metabolite | E. coli | S. cerevisiae | H. pylori | S. aureus | mitochondrion | M. barkeri |
H^{+} | 479 | 490 | 291 | 400 | 97 | 402 |
H_{2}O | 368 | 361 | 214 | 275 | 65 | 265 |
ATP | 220 | 262 | 135 | 214 | 59 | 215 |
ADP | 179 | 161 | 114 | 142 | 42 | 134 |
P_{i} | 184 | 161 | 123 | 132 | 34 | 165 |
PP_{i} | 140 | 175 | 101 | 143 | 28 | 166 |
CO_{2} | 116 | 125 | 90 | 122 | 33 | 104 |
NAD^{+} | 126 | 121 | 44 | 108 | 34 | 86 |
NADP^{+} | 92 | 144 | 71 | 97 | 14 | 86 |
NADH | 123 | 116 | 37 | 111 | 34 | 78 |
CoA | 72 | 109 | 46 | 95 | 49 | 59 |
NADPH | 90 | 143 | 68 | 105 | 14 | 83 |
AMP | 72 | 126 | 39 | 80 | 24 | 108 |
glu | 84 | 80 | 57 | 71 | 17 | 91 |
NH_{4} | 80 | 88 | 58 | 68 | 14 | 64 |
The mitochondrion is a special case that does not conform well to the previous generalizations. An examination of the mitochondrial metabolic network shows that, relative to its size, the mitochondrion contains a disproportionately high number of metabolites that are highly coupled in other networks. Whereas the 15 most connected metabolites represent only 2.4% of the 621 metabolites present in E. coli, they account for over 10% of the mitochondrion's 145 metabolites. We present the results for the mitochondrion in the interest of completeness, but caution that it may be difficult to extract meaning from a direct comparison to other, more comprehensive, metabolic networks. While normalization of the results to some measure of network size would bring the data in figure 7 into closer agreement, it would not change the simple fact that the mitochondrial network is fundamentally more limited in scope and over-represents highly connected metabolites relative to the genome-scale matrices. This high connectivity and coupling within the mitochondrion suggests that the individual metabolic reactions therein are more interrelated and may affect each other to a greater extent, on average, than those within a genome-scale bacterial metabolic network.
Preferentially coupled metabolite pairs
Preferentially coupled metabolite pairs in E. coli. The preferentially coupled metabolite pairs in the E. coli metabolic network, computed as described in the text. All of these pairs occur more often in the real network then in any of 10,000 randomizations, for an effective p value of 0.
H+ | ATP |
---|---|
H+ | ADP |
H+ | Nicotinamide adenine dinucleotide |
H+ | Nicotinamide adenine dinucleotide – reduced |
H+ | CO2 |
H+ | Nicotinamide adenine dinucleotide phosphate |
H+ | Nicotinamide adenine dinucleotide phosphate – reduced |
H2O | Phosphate |
H2O | Ammonium |
ATP | Phosphate |
ATP | ADP |
ATP | Diphosphate |
ATP | AMP |
Phosphate | ADP |
Nicotinamide adenine dinucleotide | Nicotinamide adenine dinucleotide – reduced |
Diphosphate | AMP |
Diphosphate | 5-Phospho-alpha-D-ribose 1-diphosphate |
Diphosphate | Farnesyl diphosphate |
CO2 | Malonyl-[acyl-carrier protein] |
CO2 | Acetoacetyl-ACP |
Nicotinamide adenine dinucleotide phosphate | Nicotinamide adenine dinucleotide phosphate – reduced |
Nicotinamide adenine dinucleotide phosphate | Malonyl-[acyl-carrier protein] |
Nicotinamide adenine dinucleotide phosphate | Acetoacetyl-ACP |
Pyruvate | Phosphoenolpyruvate |
Nicotinamide adenine dinucleotide phosphate – reduced | Malonyl-[acyl-carrier protein] |
Nicotinamide adenine dinucleotide phosphate – reduced | Acetoacetyl-ACP |
L-Glutamate | 2-Oxoglutarate |
L-Glutamate | L-Glutamine |
Coenzyme A | Acetyl-CoA |
Coenzyme A | Succinyl-CoA |
acyl carrier protein | Malonyl-[acyl-carrier protein] |
acyl carrier protein | Acetoacetyl-ACP |
acyl carrier protein | Myristoyl-ACP (n-C14:0ACP) |
acyl carrier protein | Hexadecenoyl-ACP (n-C16:1ACP) |
acyl carrier protein | R-3-hydroxy-myristoyl-ACP |
acyl carrier protein | Tetradecenoyl-ACP (n-C14:1ACP) |
acyl carrier protein | Octadecenoyl-ACP (n-C18:1ACP) |
acyl carrier protein | Dodecanoyl-ACP (n-C12:0ACP) |
acyl carrier protein | Palmitoyl-ACP (n-C16:0ACP) |
O2 | Hydrogen peroxide |
D-Glucose | Maltohexaose |
D-Glucose | Maltopentaose |
D-Glucose | Maltose |
D-Glucose | Maltotetraose |
D-Glucose | Maltotriose |
Ubiquinol-8 | Ubiquinone-8 |
Menaquinone 8 | Menaquinol 8 |
Succinate | Fumarate |
L-Aspartate | L-Asparagine |
CMP | CDPdiacylglycerol (E coli) |
CMP | CMP-3-deoxy-D-manno-octulosonate |
2-Demethylmenaquinone 8 | 2-Demethylmenaquinol 8 |
GTP | GDP |
Malonyl-[acyl-carrier protein] | Acetoacetyl-ACP |
Oxaloacetate | L-Malate |
Reduced thioredoxin | Oxidized thioredoxin |
Glyoxylate | Glycolate |
Flavin adenine dinucleotide oxidized | Flavin adenine dinucleotide reduced |
S-Adenosyl-L-methionine | S-Adenosyl-L-homocysteine |
tetradecanoate (n-C14:0) | Hexadecanoate (n-C16:0) |
tetradecanoate (n-C14:0) | tetradecenoate (n-C14:1) |
tetradecanoate (n-C14:0) | hexadecenoate (n-C16:1) |
tetradecanoate (n-C14:0) | octadecenoate (n-C18:1) |
Hexadecanoate (n-C16:0) | tetradecenoate (n-C14:1) |
Hexadecanoate (n-C16:0) | hexadecenoate (n-C16:1) |
Hexadecanoate (n-C16:0) | octadecenoate (n-C18:1) |
5,6,7,8-Tetrahydrofolate | 5,10-Methylenetetrahydrofolate |
5,6,7,8-Tetrahydrofolate | 10-Formyltetrahydrofolate |
tetradecenoate (n-C14:1) | hexadecenoate (n-C16:1) |
tetradecenoate (n-C14:1) | octadecenoate (n-C18:1) |
hexadecenoate (n-C16:1) | octadecenoate (n-C18:1) |
hexadecenoate (n-C16:1) | acyl-glycerophosphoglycerol (E.coli) |
octadecenoate (n-C18:1) | acyl-glycerophosphocholine |
UDPglucose | UDPgalactose |
Maltohexaose | Maltopentaose |
Maltohexaose | Maltoheptaose |
Phosphatidylglycerol (Ecoli) | acyl phosphatidylglycerol (E.coli) |
Maltopentaose | Maltotetraose |
Pyridoxal 5'-phosphate | Pyridoxamine 5'-phosphate |
Pyridoxal 5'-phosphate | Pyridoxal |
NMN | Nicotinamide |
Reduced glutathione | Glutathionylspermidine |
Reduced glutathione | (R)-S-Lactoylglutathione |
Isopentenyl diphosphate | Farnesyl diphosphate |
Isopentenyl diphosphate | Dimethylallyl diphosphate |
Isopentenyl diphosphate | Geranyl diphosphate |
D-Gluconate | 2-Dehydro-D-gluconate |
Spermidine | Glutathionylspermidine |
3-Methyl-2-oxobutanoate | L-Valine |
CMP-3-deoxy-D-manno-octulosonate | KDO-lipid IV(A) |
Trimethylamine N-oxide | trimethylamine |
Dimethyl sulfoxide | Dimethyl sulfide |
alpha,alpha'-Trehalose 6-phosphate | Trehalose |
1-Pyrroline-5-carboxylate | L-Proline |
UDP-2,3-bis(3-hydroxytetradecanoyl)glucosamine | 2,3-Bis(3-hydroxytetradecanoyl)-beta-D-glucosaminyl 1-phosphate |
acyl-glycerophosphoethanolamine (E.coli) | sn-Glycero-3-phosphoethanolamine |
acyl-glycerophosphoglycerol (E.coli) | Glycerophosphoglycerol |
Carnitinyl-CoA | crotonobetainyl-CoA |
Carnitinyl-CoA | L-Carnitine |
N-Acetyl-D-mannosamine 6-phosphate | N-Acetyl-D-mannosamine |
2-keto-4-methylthiobutyrate | 2,3-diketo-5-methylthio-1-phosphopentane |
Biotin | d-biotin d-sulfoxide |
Nitrite | Nitrate |
N1-(5-Phospho-D-ribosyl)glycinamide | N2-Formyl-N1-(5-phospho-D-ribosyl)glycinamide |
Orotate | (S)-Dihydroorotate |
4-(1-D-Ribitylamino)-5-aminouracil | 6,7-Dimethyl-8-(1-D-ribityl)lumazine |
Glycine betaine | Betaine aldehyde |
L-Glutamyl-tRNA(Glu) | tRNA (Glu) |
Furthermore, a number of less connected metabolites preferentially couple, demonstrating that this effect is not limited to the highly-connected metabolites. Although many of the preferentially coupled pairs are clustered toward the left side of figure 8 and correspond to the most connected metabolites, the remainder are spread throughout much of the figure, and as a result, the network as a whole. Thus, participating in many reactions is not necessary to detect non-random preferential coupling. This suggests that metabolites with low connectivity overall in the network are still tightly connected to certain other metabolites, often through shared chemical structural properties. Examination of the list presented in table 3 supports this assertion. The proton couples preferentially with several metabolites that gain or lose a proton during the course of balanced biochemical transformations. The adenine and phosphate containing metabolites preferentially couple in a number of cases. Sugars preferentially couple with other sugars, and fatty acids preferentially couple with other fatty acids.
Preferentially uncoupled metabolite pairs
Preferentially uncoupled metabolite pairs in E. coli. The top preferentially uncoupled metabolite pairs in the E. coli metabolic network.
Metabolite | Metabolite | p-value |
---|---|---|
H+ | Pyruvate | 0.0001 |
ATP | Nicotinamide adenine dinucleotide phosphate | 0.0001 |
ATP | Nicotinamide adenine dinucleotide phosphate – reduced | 0.0001 |
ADP | Nicotinamide adenine dinucleotide | 0.0001 |
ADP | Diphosphate | 0.0001 |
ADP | Nicotinamide adenine dinucleotide – reduced | 0.0001 |
ADP | Nicotinamide adenine dinucleotide phosphate – reduced | 0.0001 |
ATP | Nicotinamide adenine dinucleotide – reduced | 0.0002 |
Phosphate | Nicotinamide adenine dinucleotide – reduced | 0.0002 |
H2O | 2-Oxoglutarate | 0.0003 |
Phosphate | Nicotinamide adenine dinucleotide | 0.0003 |
H+ | Phosphoenolpyruvate | 0.0004 |
ADP | Nicotinamide adenine dinucleotide phosphate | 0.0004 |
ADP | Pyruvate | 0.0005 |
H+ | D-Glucose | 0.0006 |
ATP | Pyruvate | 0.0006 |
H+ | 2-Oxoglutarate | 0.0007 |
ATP | Nicotinamide adenine dinucleotide | 0.0007 |
Phosphate | Pyruvate | 0.0009 |
Conclusion
This study presents the first calculations of metabolite coupling for curated, genome-scale metabolic networks. As expected, metabolite coupling is dominated by a few highly connected pairs (H^{+}/H_{2}O, ATP/ADP, etc.). However, it was also shown that the coupling of metabolites with lower individual connectivity demonstrated a striking amount of regularity, with the number of reactions shared by pairs of metabolites in a given metabolic network closely approximating a line on a log-log plot. We also probed the network to discover which metabolite pairings occurred much more or less than would be expected from their individual connectivities alone. We found that the highly connected metabolites contributed a disproportionate percentage of the enriched coupling interactions in that highly connected metabolites are more connected to each other than would be expected from their individual connectivities alone. These preferentially coupled pairs of metabolites highlight chemical relationships between molecules. In this study, it is interesting that all of the top 15 most coupled metabolites have at least one other metabolite with which they preferentially couple in E. coli, highlighting the importance of metabolite coupling to the emergence of highly-connected compounds in metabolic networks.
Methods
Basics
We study the distribution of metabolite coupling in the chemically-detailed and highly curated metabolic networks of E. coli [2], S. cerevisiae [22], Helicobacter pylori [19], Staphylococcus aureus [20], M. barkeri [21], and the human cardiac mitochondrion [3]. We derived the stoichiometric matrix S from the publicly-available reaction lists for each organism. Each row of S represents a metabolite, each column represents a reaction, and each element is a stoichiometric coefficient. If a compound existed in more than one cellular compartment (for example, the lysosome), the row(s) representing a given compound in any compartment(s) other than the cytosol were added to the cytosolic row and then eliminated. This pre-processing of S assured that the coupling of two metabolites was considered identically regardless of the sub-cellular localization of the metabolites.
Calculating the metabolite coupling matrix
The binary form of S, termed Ŝ, was formed for each network by replacing every non-zero element in S with unity. The symmetric "metabolite coupling matrix" M was then computed for each network by Eq. 1.
M=ŜŜ^{T} (1)
The density of each M is defined as the number of non-zero elements divided by the total number of elements in the matrix. In order to visualize the entire range of data with meaningful size and greyscale differences differences, particularly between zero and small numbers, each element of M was transformed for figure 1 such that new value = log_{2}(old value + 1)/(number of unique metabolite pairs in network).
Figure 2 uses the binary form of M, created by replacing all non-zero elements with 1, and was constructed in Matlab (The Mathworks, Inc.) using the imagesc function. All numerical calculations were done with the same software.
Metabolite coupling and connectivity (Figures 3, 4, 5, 6)
For each network, the number of metabolite pairs (y-axis) that participate together in a given number of reactions (x-axis) was plotted against that number of reactions. The leftmost 10 points were used to calculate a best-fit line on a log-log scale for the less connected metabolites (those that participate in 1 to 10 reactions together). The same procedure was done for individual metabolites, except that points 2 to 11 were used to compute the best-fit line because relatively few metabolites participate in only one reaction. The best-fit lines were calculated in Excel (Microsoft).
The number of unique coupling interactions for each metabolite was computed by first sorting the rows and columns of each M separately, according to the ordering of the diagonal elements, such that the most connected metabolites occupy the first rows and columns. The number of non-zero elements in each row of the sorted M was plotted against the row number in a cumulative fashion and normalized by dividing by the total number of non-zero elements in the matrix.
Preferentially coupled/uncoupled metabolites
Monte Carlo sampling was used to find the distribution of possible off-diagonal elements of M, given the diagonal elements. This procedure elucidates the range of reactions in which two metabolites couple given the number of reactions in which they occur individually. For each off-diagonal element m_{ij}, we randomly computed a value based on the diagonal elements m_{ii} and m_{jj}. We constructed two vectors of m_{ii} and m_{jj} random unique integers between 1 and the total number of reactions in the network. These vectors are taken to represent the random reactions in which each metabolite individually participates. The overlap between them gives the randomized value for m_{ij}. For example, take a hypothetical network with a total of 10 reactions, m_{11} = 4, and m_{22} = 3. Then, we pick 2 sets of random, unique integers between 1 and 10, one of 4 integers and one of 3: [1 4 2 8]. [2 3 1]. The integers 1 and 2 appear in both of these vectors, so the random value for m_{12} is 2. When this process is repeated many times (we used 10,000 randomizations for each off-diagonal element of M for each network), the random value converges to (m_{ii})(m_{jj})/(number of reactions). Importantly, actually performing the randomizations produces a distribution of all possible values for each m_{ij}. By sorting the distribution of 10,000 random coupling values and noting how many random values are greater (less) than m_{ij}, we estimate the probability of all pairs of two metabolites coupling as often (rarely) as they do by random chance.
Declarations
Acknowledgements
We would like to thank Neema Jamshidi and Andrew Joyce for assistance with calculating random matrix elements.
Authors’ Affiliations
References
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