Parenclitic networks: uncovering new functions in biological data

Abstract

We introduce a novel method to represent time independent, scalar data sets as complex networks. We apply our method to investigate gene expression in the response to osmotic stress of Arabidopsis thaliana. In the proposed network representation, the most important genes for the plant response turn out to be the nodes with highest centrality in appropriately reconstructed networks. We also performed a target experiment, in which the predicted genes were artificially induced one by one, and the growth of the corresponding phenotypes compared to that of the wild-type. The joint application of the network reconstruction method and of the in vivo experiments allowed identifying 15 previously unknown key genes, and provided models of their mutual relationships. This novel representation extends the use of graph theory to data sets hitherto considered outside of the realm of its application, vastly simplifying the characterization of their underlying structure.

Figure 1. Schematic representation of the parenclitic network reconstruction method.

(a) Graphical represetation of the initial data set, composed of 20 instances (systems) and three features. Each instance is represented by a green sphere, located according to the value of its features in a 3-dimensional space. The constraint surface (gray wired surface) represents the overall standard relationship of the class. A generic unlabeled system is represented by a red sphere. (b–d) Data are then projected on each of the three possible planes. The green dashed lines represent the models extracted in each plane, i.e. . The red points are the positions of the unlabeled system, and the red lines indicate the distance of the system from the models. (e) The resulting parenclitic representation is a network where nodes are associated to features, and links are weighted according to the calculated distances (coded, in this Figure, into different line widths).

Figure 2. Parenclitic network reconstruction of a synthetic data set.

The left graph depicts the lineal fit of features 1 and 5; black squares represent instances 1–9, while the red dot represents instance 10 (the one with abnormal expression levels). The central (right) network is the result of the parenclitic reconstruction process for instance number 5 (10), i.e. for a normal (abnormal) instance. Nodes and links are coloured respectively according to their α-centrality and weight, from green (low) to red (high). For the sake of clarity, only links with weight greater than 1.5, and nodes connected to the giant component of the network are represented.

Figure 3. Parenclitic network for the response of Arabidopsis thaliana to osmotic stress after 3 h.

(a) Representation of the giant component of the network; for the sake of clarity, links with weight lower than 3 are not depicted. (b) Magnification of the neighborhood of the most central node, AT1G12610. Notice that labels are positioned in the lower right corner of each node - thus AT4G13980 is the label of the small one one the left. In both cases, color represents the link weight (from green to red), and node size is associated with the corresponding value of α-centrality.

Figure 4. In vivo experimental verification of the predictions.

(a) Mean root length corresponding to the wild type (WT, black column) and to 7 other transgenic lines in which a specific gene has been artificially induced. Whiskers represent the standard deviation corresponding to each group. Asterisks denote groups for which the distribution of root lengths is different with respect to the wild type with a 0.01 significance level. (b) Photos of one plant of each of the 8 lines, at the end of the full development process. (c) and (d) Photos of two vertical plates where plants are grown. In both cases, the left (right) photos refer to wild phenotypes (to phenotypes developed by the transgenic line).

Figure 5. Outcome of the experimental results.

Bars account for the 20 most central genes at each time step. For the six time steps considered, bar colors are coded according to the following stipulations: genes previously considered not to be involved in the plant's response to osmotic stress, that were respectively experimentally proven to develop (green) or to fail to develop (red) a statistically significant difference in the phenotype with respect to the wild-type phenotype; (cyan) genes predicted by the parenclitic analysis that were previously associated with the stress response in the Literature; and (gray) previously unknown genes, which could not be tested experimentally, due to their unavailability in the TRANSPLANTA collection.