Consistency analysis of metabolic correlation networks

Background Metabolic correlation networks are derived from the covariance of metabolites in replicates of metabolomics experiments. They constitute an interesting intermediate between topology (i.e. the system's architecture defined by the set of reactions between metabolites) and dynamics (i.e. the metabolic concentrations observed as fluctuations around steady-state values in the metabolic network). Results Here we analyze, how such a correlation network changes over time, and compare the relative positions of metabolites in the correlation networks with those in established metabolic networks derived from genome databases. We find that network similarity indeed decreases with an increasing time difference between these networks during a day/night course and, counter intuitively, that proximity of metabolites in the correlation network is no indicator of proximity of the metabolites in the metabolic network. Conclusion The organizing principles of correlation networks are distinct from those of metabolic reaction maps. Time courses of correlation networks may in the future prove an important data source for understanding these organizing principles.


Background
In molecular biology, the consideration of biochemical processes as elements in an abstract network has become more and more important in the last few years [1,2]. These network-oriented approaches bridge the gap between single units and collective behavior. Metabolism is, in a sense, the mediator between organisms and their environment. Resources, external conditions (represented by control parameters like temperature and concentrations of external agents) and energy all affect the organism via its metabolism. At the same time, metabolism is a key field of application of network biology. The classical approach considers metabolic networks as the pattern of connec-tions of metabolites via enzyme-driven reactions. In this way, reaction networks are straightforward abstractions of what is commonly known as metabolic pathway maps. Global structural properties [3][4][5][6], statistical parameters like the degree distribution [3,7] and the diameter [3,8], and local properties like the motif content [9] are well studied. In combination with elementary flux mode analysis [10][11][12][13][14], possible routes between different metabolites are quantified within a metabolic map, while flux balance analysis (FBA) [15,16] is suitable to predict the whole-cell behavior by adding constraints to the regulation of metabolic transformations. These theoretical approaches constitute important steps towards dynamics and have the potential to elucidate the fundamental link between topology and dynamics even further. Particularly the recent work by Almaas et al. [17] points in this direction.
Other types of metabolic networks have been established as well in recent time as the orthogonal networks, where enzymes are connected to each other when they share a common metabolite [18], and the correlation networks. In correlation networks a connection between two metabolites (nodes) represents an above-threshold correlation in metabolite concentrations.
Due to their quality of being derived from metabolic concentrations, they constitute an interesting intermediate between topology and dynamics. Here we study the compatibility of this intermediate with its two antipodes: the topological structure given by the network of metabolic reactions and the dynamic behavior given by the time evolution of the correlations between metabolites.
The different relations between metabolites in both types of networks are illustrated in Fig. 1, which gives a qualitative view in an idealized situation, where all correlations between metabolites are produced by the reactions in a linear four-element chain. In this schematic example, the correlations are assumed to be high for immediate neighbors in the chain and slightly lower at higher distances. One sees that for small and intermediate thresholds in the correlation matrix the reconstructed network tends to compact the linear chain, while higher threshold values may break the chain. The precise pattern, how correlations decay along the chain, depends on details of the enzyme kinetics [19]. Though this concept offers an intuitive explanation for a distribution of correlation coefficients it is reasonable to ask how these correlations are influenced in a more complex network structure taking other aspects into account like superior regulatory mechanisms.
Such correlation networks have been reconstructed both from experimental data [20,21] and from numerical simulations [19]. Steuer et al. [19] used a stochastic system of linear equations based on an underlying metabolic network of biochemical reactions to discover correlations between metabolite fluctuations around a steady-state. This system is related to the metabolic control analysis (MCA) [10,[22][23][24][25][26], which also served as a groundwork for linear and non-linear perturbation studies of Camacho et al. [27] and Vance et al. [28]. Camacho et al. [27] point out that different sources of variability can, in principle, lead to the observed correlations. All these studies show that the relation between metabolic correlation networks and the actual network given by the metabolic reactions is far from trivial. We therefore look at the similarity of these two graphs with simple topological tools asking, if two nodes are close to each other in the one network, when they are close in the other.
From the experimental point of view, metabolomic technologies provide widely-used tools to identify com-Basic scheme of relations between the two networks Figure 1 Basic scheme of relations between the two networks. Schematic example of the relation between metabolic reaction and metabolic correlation networks. We assume a hypothetical scenario, where a simple chain of biochemical reactions produces strong correlations between the metabolites, which however decrease with the distance of two metabolites in the chain. The bottom part of the figure gives the (hypothetical) reconstructed correlation networks for different regimes of the threshold parameter κ. pounds in biological samples and to describe the current state of a system [29,30]. For details on the quantitative and qualitative analysis of the single metabolites see Weckwerth and Morgenthal [31] and Kell [32]. More than 1000 different compounds have been isolated and identified in a single tissue in this way.
The studies on different samples revealed strong correlations between certain pairs of metabolites, while most other combinations displayed little or no correlation. Such a correlation profile may serve as a basis for the construction of a metabolic correlation network. Because anti-correlations might have a physiological cause as well, they are treated equally to positive correlations. Beyond the identification of compounds via the concept of correlation networks, metabolomics is also capable to describe physiological processes in consequence of development and changing environmental conditions, since tissue samples can be analyzed at any point of time.
The interrelation between the architecture of a metabolic pathway map and the dynamic processes taking place upon it has sparsely been studied. Results concern the distribution of node degrees and of metabolic fluxes which have both been found to be scale-free [2,3,33] and the metabolic fluxes within the core of the metabolic network of E. coli, H. pylori, and S. cerevisiae [17], but most other studies deal with artificial networks. In the case of metabolic networks one observes discrepancies between theoretical graphs and the real-world networks' topologies.
One key publication on gene expression data in yeast [34] indicates that metabolic reaction networks display both scale-free and exponential degree distributions depending on whether one takes all potential paths or particular realizations under certain conditions into account. Furthermore, the biochemical modules derived from experimental data deviate from proposed modules in theoretical reaction networks. Here we will not focus on topological properties of metabolic correlation networks, but look at them as a mediator between topology and dynamics. Since a correlation network represents a dynamic aspect of the metabolic network, it is suitable for comparison with its topological counterpart. In the first step of our analysis, we investigated this tem- First, the small number of replicates, which forms the basis of the covariance matrix, provides a lower significance limit of the threshold κ. We used a p-value of 0.05 for this, leading to κ > 0.6. Second, above a certain level in κ the network is fragmented and some pair-wise distances are no longer defined. For our data fragmentation sets in for κ ≈ 0.75. In Fig. 2 and the consecutive figures we highlight the range of κ obeying both criteria. Analyzing statis-tically significant deviations from zero of θ in Fig. 2 for this range of κ, we find a p-value p < 0.09. The scatter diagram (inset in Fig. 2) exemplarily displays the relation between the time difference ∆t and network similarity σ. For κ = 0.6 the corresponding correlation coefficient (i.e. the consistency parameter θ) is θ = -0.48. If one repeats this consistency analysis with correlation networks taken at identical connectivities (as opposed to identical thresholds κ, as in Fig. 2), we also observe a systematic deviation of θ from zero in the interesting range of κ. In order to rule out the influence of any intrinsic network properties on σ we also checked that the consistency parameter θ is close to zero for the relevant range of κ when we maximally randomize the correlation networks.

Results and discussion
Surprisingly, we find a high network similarity for the night-day transition and a low similarity for the day-night transition. Understanding this would, however, require more data. Omitting day-night transitions from this analysis and determining the consistency parameter θ at 0.6 <κ < 0.75 consequently yields a more pronounced result with comparable p-values.

Are correlation networks and metabolic reaction networks related?
On the basis of the data set of Ma and Zeng [35] and the data set of Weckwerth et al. [20,21], 39 common compounds have been identified, which are present in both networks. This value primarily depends on the number of identified compounds in the metabolomic analysis. Unidentified compounds are systematic signals in the metabolomic data, which cannot be unambiguously associated with a node in the reaction network. The effect of unidentified metabolites is difficult to estimate. Whether they influence the length scale (an average distance of identified metabolites) or not mainly depends on the individual structure of the network. The known metabolites show approximately the same distribution of pair distances as the complete reaction network and can therefore be regarded as a representative subset of the whole graph in this property. We also checked the influence of unknown nodes on the network similarity σ for two Erdős-Rényi (ER) random graphs [36] (with a starting value of σ = 0.3) when using less and less nodes for this computation. Similarity σ displays only a marginal dependency on the Temporal consistency within the metabolic correlation networks The common metabolites may serve as a basis for the computation of the pair distances, the statistical parameter already used in the previous section to determine the relation between two networks. This compound list primarily consists of two biochemical substance classes, namely the carbohydrates and amino acids and their corresponding derivatives (for details on the available amino acids and carbohydrates: see Table 1). There are a few compounds, which are assigned to the citric acid cycle, the photosynthesis, and the phospholipid and glycolipid metabolism, respectively.
Again, we generated correlation networks of varied connectivity for each sampling point in time by adjustment of κ between 0 and 1. In the following, only the results for ω = 8 are discussed, since only marginal differences were observed for ω < 8. Results of the network similarity σ for all sampling points are depicted in Fig. 3. Limits for the significance analysis were applied as described before. In this range of κ two sampling points display minor positive correlations and one sampling point minor negative correlations (with average p-values of 0.04, 0.07, and 0.08). The other three sampling points display no systematic deviation from zero (with p-values above 0.45). Thus, taking all sampling points together the results suggest no correlation between the two network types. Due to the three sampling points failing our significance analysis, however, we analyze this correlation also by means of another network property, namely the centrality σ C [4,35] of all common nodes. This topological parameter was tested accordingly (Fig. 4) and, here, no significant deviations from zero similarity (with average p-values above 0.3) and therefore from the pair distance results were observed.

Analysis of module-module interaction
The different distribution of metabolites in both types of networks is best illustrated through the comparison of class-specific compounds. We identified all metabolites in the networks, which belong to the group of amino acids and carbohydrates (Table 1), respectively, and calculated all pair distances within these groups. We checked all sampling points, different values of ω, and adjusted κ in a way to obtain sparsely connected networks, which in their connectivity resemble their topological counterpart. The metabolic reaction network (Fig. 5a) exhibits a distinct clustering within the biochemical classes, especially for the carbohydrates (with an average pair distance of 3.1). Fig. 5b depicts the distribution of these selected compounds across the reaction network. The correlation networks ( Fig. 6) display no significant difference between the intra-class and inter-class specific pair distances, if one takes all compounds into account. We performed a UPGMA cluster analysis on the distance matrices of the correlation networks in order to find the inter-class and intra-class specific clusters of metabolites with special regard to their day and night time-specific emergence. To insure that more than 95 % of the selected metabolites lie within the giant component of the network we restricted this analysis to values of 0.75 <κ < 0.8. We recovered a prescribed number n m = 5 of clusters each with at least two metabolites by horizontally cutting the tree at a certain hight. Therefore, one has to analyze the dependence of the cluster predictions on threshold variation. That way, several omnipresent groups of metabolites could be identified (e.g. a cluster of the amino acids 7, 11, and 12). Others appeared either at day or at night. Within the night samples, there were two distinct groups one solely containing metabolites of the class of carbohydrates (19, 20, and 24) and one inter-class specific group (4, 16, and 23), while the day samples contained several intra-class specific clusters (e.g. 21 and 22) and one inter-class specific cluster (9 and 21). All metabolites in the inter-class specific clusters display a pair distance of at least 7 in the reaction network.

Conclusion
In this work we investigated systematically the relationship between metabolic correlation networks and genome-wide predicted reaction networks. In recent studies we investigated how correlation networks are causally connected to the underlying biochemical reaction network and its regulation [19,20,30,37]. However, all these studies revealed the discrepancy between predicted pathway connectivity and a simple extrapolation of these hypothetical networks to correlation networks. We detected overall changes in the correlation network topol-  ogy, which are rather based on specific enzyme activity alterations [21,30]. Therefore a systematic comparison of predicted pathway map structure and the experimental metabolite covariance is crucial. Based on the prediction that alterations in biochemical regulation will change the correlation network structure [20] we have investigated different correlation network structures during a diurnal time course in Arabidopsis plant leaf samples. Intriguingly, the structure of these networks did not match the predicted pathway connectivity in plant metabolism, but nevertheless varies systematically in time. Moreover, compared to theoretical pathways in Arabidopsis the correlation network structure revealed different and novel clusters of compounds such as the correlation of aromatic amino acids and housekeeping sugars which is not predicted by the topology of theoretical pathways (Fig. 6). This fact supports our hypothesis that the correlation network largely reflects biochemical regulation. Particularly this observation that the systematics of correlation networks are somewhat orthogonal to those contained in pathways networks, suggests that the systematic and differential analysis of metabolite correlation network structures and metabolite covariances will in the future lead to novel insights into biochemical regulation and regulatory hubs in the in vivo system.

Construction of correlation networks
The analysis of the correlation data and the respective correlation networks was based on the experimental data sets of Weckwerth et al. [20,21]. In their studies, they recorded the concentrations of 257 different metabolites of Arabidopsis thaliana. In principle, there are two practicable approaches to construct a correlation network from the data given here. Considering long-term fluctuations one would use the plant-specific changes in time given by six concentration values. We preferred a statistically more reliable method which considers the short-term fluctuations provided by i.e. positive correlations and anti-correlations are treated likewise. Fig. 7 displays the connections between glucose and glucose-6-phosphate (G6P) in one of the correlation networks of Arabidopsis thaliana [20,21] (Fig. 7 right). In the reaction network both metabolites characterize the first reaction step in glycolysis, which is catalyzed by hexokinase under consumption of ATP (Fig. 7 left). The correlation coefficient of glucose and G6P is close to zero, which means that there is nearly no correlation between these two compounds under the studied conditions. Con-sidering very strong correlations only, multiple links are required to get from the one compound to the other. In the example given in the right-hand part of Fig. 7 the corresponding part of the correlation network is shown for the sampling point 0.5 h with ω = 8 and κ = 0.85; in this case, the correlation coefficient of glucose and G6P is 0.13. Two alternative shortest paths using 5 links exist here, one is depicted here with the corresponding correlation strengths of each metabolite pair. Some of the compounds have been observed in the experiment but not identified yet.

Relation of metabolic reaction and metabolic correlation networks (pair distances)
By construction, the threshold (0 ≤ κ ≤ 1) regulates the connectivity of a correlation network. For κ = 0 one obtains a complete network, i.e. every node is connected to every other node. Occasionally, in the measurements some metabolites have not been registered. In these cases gaps occur in the data matrix. We account for these missing measurements by introducing a second threshold ω into our analysis and restrict the range of metabolites in the network to those pairs with at least ω measurement results. In this way we qualitatively vary the reliability of the data entering our analysis. We changed this parameter Relation of metabolic reaction and metabolic correlation networks (centrality) Figure 4 Relation of metabolic reaction and metabolic correlation networks (centrality). Another method of quantifying network similarity uses an individual node parameter, the centrality for comparison of both types of metabolic networks. Computing the similarity of the two networks based upon this parameter, which is the average path length of all shortest paths connecting a node to all other nodes, shows no significant relation between metabolic reaction and correlation networks. The regime enclosed by the gray-shaded box describes the statistically reliable region of κ. Selected substances in the metabolic reaction network of A. thaliana. We investigated the common compounds of both types of metabolic networks belonging to the classes of amino acids and carbohydrates, respectively. a) Color-coded are the pair distances of these metabolites. As before, the pair distance is the number of connections in the shortest path between two compounds. Most of the carbohydrates are characterized by small internal pair distances, while there are more links needed to connect them to the group of amino acids, which display a relatively small average internal pair distance as well. b) This graph representation shows the giant component of the metabolic reaction network of A. thaliana. All metabolites, which have also been found in the correlation network, are color-coded according to their biochemical class (see Fig. 5a).
in an interval of 4 ≤ ω ≤ 8 in order to validate the robustness of the results with regard to the choice of data.

Data sets of metabolic networks
Various databases based on genomic analysis provide information on metabolic reactions. Among these the KEGG database [38] has been used by several research groups for the reconstruction of metabolic networks. This comparatively reliable group within the biochemical networks opens up a reaction space, which is used for metabolic transformations. Basically two approaches to interpret a reconstructed metabolic network exist so far, which differ in the consideration of ubiquitous compounds like H 2 O, ATP, and NADH, often referred to as current (or currency) metabolites. These hub forming compounds drastically reduce the average path length and, consequently, the pathways in such a metabolic network do not resemble the conventional order of reactions. Ma and Zeng [35] addressed this issue by eliminating all connections from a current metabolite, if the connections represent particular reactions, like the transfer of electrons or of certain functional groups. Additionally, they corrected mistakes and inconsistences in the original data set and checked the reversibility and direction information of each reaction.
The reconstruction of the metabolic network of A. thaliana, used in this paper, is based on the data from Ma and Zeng (http://genome.gbf.de/bioinformatics/ index.html [35]). This connection table displays 2070 dif-Amino acids and carbohydrates in a metaboliccorrelation network Figure 6 Amino acids and carbohydrates in a metaboliccorrelation network. The correlation network of A. thaliana displays other distance relations of amino acids and carbohydrates compared to the reaction network in Fig. 5a. This representative example from sampling point 4 h light shows the color-coded pair distances of all metabolites belonging to the biochemical classes of the amino acids and of the carbohydrates. The typical separation of compounds according to their biochemical classes as found in the metabolic reaction network (see Fig. 5a) is not seen in the correlation networks independently of the sampling point and the parameter ω(data not shown). A distinct intra-class correlation which has been observed in most of the sampling points exists for Isoleucine (7), Tyrosine (11), and Valine (12). Some sampling points, however, exhibit correlations which hint to inter-class specific relations (e.g. serine (9) and sucrose (21)).

Similarity analysis of networks
Various methods for the determination of the similarity of two networks exist so far. A prominent example is the graph alignment [39,40], which is a suitable method for related networks. The great variability of the considered networks here asks for other statistical parameters. Thus, Example of a shortest path in a metabolic reaction network and a metabolic correlation network Figure 7 Example of a shortest path in a metabolic reaction network and a metabolic correlation network. The transformation of glucose to glucose-6-phosphate in glycolysis is represented by one connection in the pathway network. In this example of a sparsely connected but non-fragmented correlation network (parameter constellation: sampling point 0.5 h light, ω = 8 and κ = 0.85) the shortest path between this pair of metabolites consists of 5 links (numbers next to the links denote the individual correlation strength of the involved metabolite pairs; here, the correlation strength between glucose and glucose-6phosphate is 0.13.). The deviation of path lengths shown in this example just demonstrates the possible structural differences within these networks. can be computed in both networks. The similarity σ = σ (G 1 , G 2 ) between the two networks G 1 and G 2 is the Pearson correlation coefficient of the respective pair distance vectors.
We tested this method both with real networks of different size but similar structure and with artificial graphs of the same size and connectivity which were altered systematically. Therefore, we compared the reaction network of A. thaliana with six reaction networks of other eukaryotes from the Ma and Zeng database. Despite their different size (332 <n < 625) the networks displayed an average network similarity σ of 0.81.
In order to see whether our similarity indicator σ is capable of capturing the monotonous decrease in similarity under increasing randomization of one of the networks we analyzed σ for an ER graph [36] with n = 250 nodes and m = 1245 edges and its rewired counterparts as a function of the randomization depth r which corresponds to the number of rewired edges (Fig. 8). In each step of this randomization procedure one end of a randomly chosen edge is rewired to new randomly chosen node. With increasing r the continuous destruction of the original network structure is reflected in an explicit reduction of the network similarity σ until all networks display zero similarity. In our studies on the relation of correlation networks and metabolic reaction networks, we additionally used the centrality σ C [4,35] to compare both types of networks. The centrality σ C of a node is defined as the average path length of all shortest paths connecting this node to all other nodes.
Example of the similarity analysis in an artificial network Figure 8 Example of the similarity analysis in an artificial network. The network similarity σ for an ER graph (n = 250 nodes, m = 1245 edges) and its rewired versions has been computed as a function of the randomization depth r (number of rewired edges). In each randomization step one endpoint of an edge is rewired to another randomly chosen node.