Scillation, adaptaion and bistability. The compressed forms of regulatory Naringin motifs in this paper are defined as regulatory motifs with minimal nodes.Regulatory Motif IdentificationFigure 2 presents a schematic diagram depicting the individual steps for regulatory motif identification. The details of the each step are described below. Step 1. The input network is compressed based on the regulatory relationship between neighboring edges. We used the node-based reduction part of a kernel identification algorithm, which replaces the neighborhood subnetwork of each node with a smaller one without disrupting the network dynamics [16]. (See File S2). Step 2. Using our subgraph search algorithm, we search the compressed network for the subgraphs matched with the compressed forms of regulatory motifs. This will be discussed in detail later. Step 3. The original paths from the compressed edge of matched subgraphs are recovered from the input network. The compressed edges are expanded by using a depth-first search with two constraints. The first constraint is that the path includes only eliminated nodes, except the start and end nodes. The second constraint is that total regulatory effect of the path is the same as that of the compressed edge. Step 4. The matched subgraphs and original path information are integrated into the input network and then, the individualKnown Regulatory MotifsWe collected regulatory motifs for representative bio-signaling such as oscillation, adaptation, and bistability from individual literatures. These are 12 oscillatory motifs [12], 11 adaptation motifs [13] and 12 bistable switch motifs [3], which were identified from an individual study based on mathematical modeling and simulation 1662274 (See File S1). These regulatory motifs are all 2- and 3node network topologies with signed directed edges, and they are parametrically robust in exhibiting dynamic behaviors. Since some of these regulatory motifs are in isomorphic relationship or can be compressed, we converted them into compressed forms of nonisomorphic networks as shown in Figure 1.RMOD: Regulatory Motif get BI-78D3 Detection ToolFigure 3. The construction of a path-tree for the adaptation motif as an example. doi:10.1371/journal.pone.0068407.gregulatory motifs are extracted by selecting one original path from each edge of the matched subgraphs.Subgraph Search AlgorithmThe searching (matching) process between a query regulatory motif and a given input network consists in the determination of mapping which associates nodes of the query regulatory motif to nodes of the input network. The solution to the searching problem could be obtained by computing all the possible partial mapping and selecting the ones satisfying the wanted mapping type. In order to reduce the number of paths to be explored during the search, our algorithm uses a novel data structure called a path-tree as feasibility rules for partial mapping and employs the ESU algorithm as a search strategy [19]. The path-tree is a novel data structure to evaluate the feasibility of adding newly explored node into the partial mapping. It is composed of all isomorphic graphs of query regulatory motifs and organized into a tree structure to directly evaluate the newly created edges. Figure 3 illustrates the construction of a path-tree for the adaptation motif as an example. It is constructed by loading canonical labels, which are the rearranged elements of adjacency matrices of isomorphic graphs. The isomorphic graphs of regulatory.Scillation, adaptaion and bistability. The compressed forms of regulatory motifs in this paper are defined as regulatory motifs with minimal nodes.Regulatory Motif IdentificationFigure 2 presents a schematic diagram depicting the individual steps for regulatory motif identification. The details of the each step are described below. Step 1. The input network is compressed based on the regulatory relationship between neighboring edges. We used the node-based reduction part of a kernel identification algorithm, which replaces the neighborhood subnetwork of each node with a smaller one without disrupting the network dynamics [16]. (See File S2). Step 2. Using our subgraph search algorithm, we search the compressed network for the subgraphs matched with the compressed forms of regulatory motifs. This will be discussed in detail later. Step 3. The original paths from the compressed edge of matched subgraphs are recovered from the input network. The compressed edges are expanded by using a depth-first search with two constraints. The first constraint is that the path includes only eliminated nodes, except the start and end nodes. The second constraint is that total regulatory effect of the path is the same as that of the compressed edge. Step 4. The matched subgraphs and original path information are integrated into the input network and then, the individualKnown Regulatory MotifsWe collected regulatory motifs for representative bio-signaling such as oscillation, adaptation, and bistability from individual literatures. These are 12 oscillatory motifs [12], 11 adaptation motifs [13] and 12 bistable switch motifs [3], which were identified from an individual study based on mathematical modeling and simulation 1662274 (See File S1). These regulatory motifs are all 2- and 3node network topologies with signed directed edges, and they are parametrically robust in exhibiting dynamic behaviors. Since some of these regulatory motifs are in isomorphic relationship or can be compressed, we converted them into compressed forms of nonisomorphic networks as shown in Figure 1.RMOD: Regulatory Motif Detection ToolFigure 3. The construction of a path-tree for the adaptation motif as an example. doi:10.1371/journal.pone.0068407.gregulatory motifs are extracted by selecting one original path from each edge of the matched subgraphs.Subgraph Search AlgorithmThe searching (matching) process between a query regulatory motif and a given input network consists in the determination of mapping which associates nodes of the query regulatory motif to nodes of the input network. The solution to the searching problem could be obtained by computing all the possible partial mapping and selecting the ones satisfying the wanted mapping type. In order to reduce the number of paths to be explored during the search, our algorithm uses a novel data structure called a path-tree as feasibility rules for partial mapping and employs the ESU algorithm as a search strategy [19]. The path-tree is a novel data structure to evaluate the feasibility of adding newly explored node into the partial mapping. It is composed of all isomorphic graphs of query regulatory motifs and organized into a tree structure to directly evaluate the newly created edges. Figure 3 illustrates the construction of a path-tree for the adaptation motif as an example. It is constructed by loading canonical labels, which are the rearranged elements of adjacency matrices of isomorphic graphs. The isomorphic graphs of regulatory.
