Bioinformatics Group, Department of Biology, Darmstadt University of Technology, 64287 Darmstadt, Germany

Max-Planck-Institute of Molecular Plant Physiology, Science Park Golm, 14424 Potsdam, Germany

School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany

Abstract

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

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

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.

Basic scheme of relations between the two networks

**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

Such correlation networks have been reconstructed both from experimental data

From the experimental point of view, metabolomic technologies provide widely-used tools to identify compounds in biological samples and to describe the current state of a system

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

Results and discussion

Time consistency of metabolic correlation topologies

When analyzing the metabolite correlations in different plant samples at different sampling points in time one has to consider three potential contributions to the data: Short-term fluctuations in the metabolites' concentrations represented by all plant samples at a given time point may either reflect intrinsic noise or may originate from plant-to-plant variability. Lastly, systematic changes of the steady-state correlation networks over time along a diurnal cycle may contribute to the data. Here we want to find out, whether the (steady-state) correlation networks obtained from different time points are systematically similar. Two main effects can be expected: (1) network similarity should on average be higher for networks at neighboring time points compared to more distant time points, (2) day-night and night-day transitions should be associated with substantially lower network similarity (compared to neighboring time points at constant illumination).

In the first step of our analysis, we investigated this temporal property by studying the similarity

Temporal consistency within the metabolic correlation networks

**Temporal consistency within the metabolic correlation networks**. The temporal systematics of different correlation networks has been determined via the consistency parameter

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

Are correlation networks and metabolic reaction networks related?

On the basis of the data set of Ma and Zeng

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

Selected common compounds from the metabolic reaction and the metabolic correlation networks

Amino acids

Carbohydrates

1

15

D-Ribose

2

L-Asparagine

16

D-Fructose

3

L-Aspartate

17

D-Mannose

4

L-Glutamate

18

D-Glucose

5

L-Glutamine

19

D-Fructose-6-phosphate

6

Glycine

20

D-Glucose-6-phosphate

7

L-Isoleucine

21

Sucrose

8

L-Methionine

22

Maltose

9

L-Serine

23

Raffinose

10

L-Threonine

24

D-Glucose-1-phosphate

11

L-Tyrosine

12

L-Valine

13

L-Phenylalanine

14

L-Cysteine

List of compounds belonging to the class of amino acids and carbohydrates which have been found in both metabolic networks of

Again, we generated correlation networks of varied connectivity for each sampling point in time by adjustment of _{C }

Relation of metabolic reaction and metabolic correlation networks (pair distances)

**Relation of metabolic reaction and metabolic correlation networks (pair distances)**. Comparison of metabolic reaction networks and metabolic correlation networks via the pairwise distances between nodes. The similarity between both types of networks (correlation network at

Relation of metabolic reaction and metabolic correlation networks (centrality)

**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

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 _{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.

Selected substances in the metabolic reaction network of A. thaliana

**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

Amino acids and carbohydrates in a metaboliccorrelation network

**Amino acids and carbohydrates in a metaboliccorrelation network**. The correlation network of

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

Methods

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.

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 10 different plant samples. Using 6 samples in time instead of 10 plant samples largely limits the statistically reliable range of

Correlation networks were reconstructed from the measured metabolite concentrations by pair-wise combination of the metabolite concentrations. In the correlation network, there is an edge between two compounds, if the absolute value of the correlation coefficient resulting from two concentration vectors exceeds a certain threshold

Example of a shortest path in a metabolic reaction network and a metabolic correlation network

**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,

By construction, the threshold (0 ≤

Data sets of metabolic networks

Various databases based on genomic analysis provide information on metabolic reactions. Among these the KEGG database _{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

The reconstruction of the metabolic network of

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 _{ij }is defined as the shortest path between the two nodes _{max }of pair distances of _{max }= _{1}_{2}) between the two networks _{1 }and _{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

In order to see whether our similarity indicator _{C }_{C }of a node is defined as the average path length of all shortest paths connecting this node to all other nodes.

Competing interests

The author declares that there are no competing interests.

Example of the similarity analysis in an artificial network

**Example of the similarity analysis in an artificial network**. The network similarity