For each single interaction template (upper part of Fig. 3.1) a corresponding graph repre-sentation can be specified (lower part of Fig. 3.1). A graph G = (V, E) consists of nodes vi ∈V modeling the cells and edges eij ∈E connecting a node pair (vi, vj). In all graphs used here the edges are directed, i.e. if no multiple edges are allowed (see below) for any node pair two edges can be defined at maximum (one per direction).
Starting with the simplest graph model, the 1-comp template can be straightforward converted into thedirect graph model. “Direct” refers to the fact that the two cells, each represented as individual nodes (circles in Fig. 3.1), are directly connected, i.e. without considering any other component explicitly.
The direct graph model for signaling interactions can be further divided into a direct multiple and a direct unique model, employing multiple or single combined links between the cell type nodes respectively. In the direct multiple model, all different interactions that might exist between the same pair of cell types are represented as separate edges, whereas in the direct unique model all different interactions of the same direction between a pair of cell types are collapsed into one edge. That means in the direct multiple graph the number of edges can be arbitrarily large, whereas in the unique model this number is restricted to n2 edges at maximum (self-loops are allowed to allow autocrine signals). Each edge in the direct unique model can be additionally equipped with the number of the contained interactions and further information.
To translate the other two single interaction templates (2-comp and 3-comp respectively) into a corresponding graph model, r-partite graphs are applied. A graph G is called r-partite if the set of nodesV can be divided intorpartitions such that all node pairs satisfy the condition (vi, vj) ∈/ E with vi, vj ∈ Vk and 1≤ k ≤ r. That means, a graph is called r-partite if the graph can be divided into r distinct partitions where the nodes inside a partition are not connected and edges exist only between nodes of different partitions (see Chapter 1.6 in Diestel, 2000, and Fig. 3.2 for examples of 3-partite graphs).
It follows that nodes belonging to distinct partitions of the graph can be seen as pos-sessing different types. In the bipartite and tripartite models used here (bi andtri are used as prefix instead of 2- and 3-partite), these different node types are expressed by different
Figure 3.2: Two undirected 3-partite graphs as examples to demonstrate the definition ofr-partite graphs.
Each of the two graphs can be divided into three partitions. The nodes inside a partition are not connected, only edges between nodes of different partitions occur (figure adapted from Diestel, 2000).
symbols, i.e cells are still represented by circles, messengers and receptors as rectangles and diamond shapes respectively (Fig. 3.1, lower part).
Thus, by extending the set of node types in that way, both remaining templates for single cell-cell interactions (2-comp and 3-comp) can be converted immediately into a corresponding graph representation. Beside the already mentioned advantages that an explicit modeling of messengers and receptors has (as e.g. for combining information from different data sources, see Sec. 3.1), another major benefit of these representations becomes visible when several interactions are combined into a network (Fig. 3.1, bottom section):
any group of cells connected by the same messenger or messenger-receptor interaction can be combined into a bipartite representation in which the messenger (or the complete messenger-receptor interaction) is represented by a separate node. For a group ofs source and t target cells that is completely connected by the same interaction, the number of edges decreases from s·t in the direct multiple graph representation to only s+t in the bipartite model.
The addition of explicit receptor nodes in the tripartite model might further decrease the number of edges, though not as much as this is case for the transition from the direct to the bipartite representation. This further reduction is due to the fact that some mes-sengers might share the same receptor. Then each messenger needs only to be linked to one receptor, which in turn contains all edges to the target cells, instead of linking each messenger to all target cells repeatedly.
Note that any single interaction can be converted into a bi- and tripartite representation.
In the extreme case of only different messengers or messenger-receptor interactions nothing could be combined and the edge number would even increase, since any single interaction is then converted into a chain of two or three nodes (in the bi- or tripartite case respectively).
But this can be neglected in our case as the results show (see Sec. 4.2.2 and Sec. 4.2.4).
The transformation between the three graph models is possible in both directions:
mes-3.2 Graph representations to combine single interactions 43
senger and receptor nodes in the tripartite model can be combined into a single messenger node of the bipartite representation, which then can be resolved into all single interactions of the direct multiple model.
In bipartite and tripartite representations, however, adding and removing single inter-actions can not be performed easily. One possibility is to add or delete single interinter-actions in the direct model and then to recalculate the bi- and tripartite representation. To per-form the transper-formation in the other direction (from tripartite to direct representation) all interactions in the tripartite representation need to be stored separately. Then individual operations on single interactions can also be performed and the transformation process into the tripartite and bipartite model with combined messenger and receptor nodes performed afterwards. So, for any direction, if equal messenger or receptor nodes have been collapsed in order to reduce the amount of edges, operations on single interactions need recalculation.
Chapter 4
Reconstruction of cell-cell networks from CSNDB
Contents
4.1 Content and organization of CSNDB . . . 46 4.1.1 Relevant classes . . . 48 4.1.2 Assembly of intercellular signals . . . 49 4.2 Reconstruction approaches and results . . . 50 4.2.1 Reconstruction I: Accession of binary ligand-receptor interactions 51 4.2.2 Reconstruction I: Resulting interactions and networks . . . 52 4.2.3 Reconstruction II: Accession of any molecular interaction . . . . 53 4.2.4 Reconstruction II: Resulting interactions and networks . . . 59 4.3 Correlation of graph topology and biological behavior . . . . 63 4.3.1 Definition of distances . . . 63 4.3.2 Results . . . 65 4.4 Implementation . . . 66 4.5 Discussion . . . 66
This chapter presents how the Cell Signaling Networks Database (CSNDB) is applied to reconstruct cell-cell signals and to combine them into a network. As shown in Section 2.2, there are few databases available that contain relevant information that can be utilized.
The CSNDB provides information on interactions of signaling molecules and their locations in the human body. This data can be assembled to reconstruct complete intercellular signals.
Therefore, in the following section the data scheme of the CSNDB is shown and how it is applied to the extraction of intercellular signals (Section 4.1). Subsequently, two
Class Signal Molecule Cell Signaling ExtraCell Signaling Gene Expression
Fields Endogenous/Exogenous From molecule From tissue
Other Name To molecule To tissue
Is Synonym Interaction Signal Molecule
Species Type
Cell Signaling Tissue
Synthesis Target
Table 4.1: Class definitions in CSNDB. Only classes and fields used for the cell-cell signaling reconstruction are shown.
reconstruction approaches are then performed: in the first approach only information is accessed that can be directly detected as relevant, complemented by a second approach which is designed to exploit as much information from the CSNDB as possible. Both approaches and the resulting networks are shown in Section 4.2.
In Section 4.3 the subnetwork of organ-organ interactions resulting from the second reconstruction approach is used as an example how such networks can be used for further analysis. Section 4.4 briefly describes the implementation of the CSNDB extractions and finally, a discussion in section 4.5 closes this chapter.