In this study, we propose a method for quantifying the importance of components in signaling and regulatory networks. This method incorporates synergistic and inhibitory regulation that is quite common in signaling networks but has received little attention so far in structural analysis. Our method can be easily adapted for evaluating the importance of genes in gene regulatory networks by considering the connectivity of the whole network instead of the connectivity from input to output. In addition, our graph measures can be readily adapted to evaluate the importance of edges (interactions). This allows the study of mutations of binding sites that do not knock components out but change their interactions [45].

While ESMs are the most concise and complete description of the signal transduction modes in a network, the combinatorial aspects of ESMs also make them difficult to count in large networks. Our results indicate that the simple path (SP) measure has a similar performance as the ESM measure as an indicator of node centrality. The reason is that both ESM and SP measures incorporate the cascading effects of a node's removal arising from the synergistic relations between multiple interactions. Either measure can serve the purpose of identifying a few most important components in a signaling network. The integer linear programming algorithm proposed in this study can be used by those researchers interested in individual signaling modes.

In addition to the application described in this study, ESMs can also be used to probe the relationship between the structure and dynamics of a signaling network. For example, if the dynamics of a signaling network is oscillatory, the state of at least one node needs to switch from 0 to 1 and vice versa, and thus it is possible that some ESMs contain both an original node and its complementary node. Thus one may predict the potential dynamics of the signaling network from the composition of its ESMs. The minimal intervention set, defined as a minimal set of important nodes whose simultaneous manipulation satisfies a user-defined goal (e.g. permanent deactivation of the output) [10, 46], identifies minimum failure modes for signaling networks and regulatory networks. One can conjecture that any node or minimal node combination whose deletion disrupts all the ESMs may be a minimal intervention set. For the example in Figure 3(b) a sustained signal (i.e. stable ON state of the input node) leads to a sustained response according to logical steady-state analysis [10]. There are three minimum intervention sets of size 1: {A}, {B}, and {E}, whose knockout (maintained OFF state) blocks the signal transduction and eliminates the response. Single-node deletion of A, B, or E disrupts all ESMs in this example, supporting the conjecture. Unlike the minimal failure modes defined by minimal intervention sets, the ESM measure gives quantitative importance values for all signaling components, regardless of whether they are important or not. The detailed relation between the dynamics of a signaling network and its ESMs is an interesting topic worth exploring in future research.

Our method requires less prior information such as initial conditions and timing, has less computational cost and performs as well as methods involving dynamic simulations such as Soni *et al.* 2008 [47]. Soni *et al.* constructed an ensemble of Boolean network simulations to estimate the frequency of active pathways and to rank interactions by their control effective flux (CEF). Since the same guard cell ABA signaling network was used as test example in their study, we can compare their results with ours. There are 7 intermediate signaling components involved in the five interactions with the highest CEF values. 5 of them have very high importance values (>0.98) according to our ESM measure, 4 of which are essential components according to dynamic simulation. In contrast, the remaining 2 signaling components have low importance values (<0.5) according to our ESM measure, and the dynamic simulation also shows that their knockout does not affect ABA signal transduction.

Another related work by Abdi *et al.* applies digital circuit fault diagnosis methods to generic Boolean representations of signaling networks to find vulnerable signaling components [48]. The method determines the probability that an error occurring at a signaling component propagates to the output(s) by calculating the signal probability (the probability of the state 1) of all nodes on the paths from the error site to the output(s). Vulnerable components are those nodes that have high error propagation probabilities to the output(s). A comparison on two signaling networks they used (the caspase3-FKHR network and the p53 network) indicates that signaling components identified as vulnerable (error propagation probability > 0.5) by Abdi *et al.* tend to have high essentiality (e.g. the sole vulnerable component AKT in the caspase3-FKHR network has essentiality 1.0, and all vulnerable components in the p53 network have essentialities larger than 0.9 according to our ESM measure). In our study, we propose ESMs as the basic unit of signal transduction. In addition to the systematic evaluation of essentiality of signaling components done here, the concept of ESM opens new avenues of research relating the structure and function of signaling networks, as discussed above.

The network expansion method proposed here has a potential limitation in handling overall activating input-output paths that have inhibitory edges separated by more than one activating edge. Such paths of the original network may be broken in the expanded network, because we introduce complementary nodes only for the nodes with direct inhibitory roles. If the nodes situated between the first (third,...) and second (fourth,...) inhibitory edge in the overall activating path already have complementary nodes in the expanded network due to their involvement in other paths, the path will be retained in the expanded network. If some of these intermediate nodes do not have complementary nodes, but these nodes are involved in other input-output paths, their importance may be somewhat underestimated. If the intermediate nodes are not involved in other paths, their essentiality may be seriously underestimated. A potential solution to this problem is to add a step in the network expansion procedure: after introducing complementary nodes for all nodes with direct inhibitory effects, we enumerate all activating input-output paths with inhibitory edges separated by more than one activating edge and introduce the complementary nodes necessary for the maintenance of these paths in the extended network. The edges of these complementary nodes are determined from the negation of the Boolean rules in which the original nodes participate in. The tradeoff of completeness is the increase in size and redundancy of the expanded network. The signalling networks evaluated in this study have no, one and two instances, respectively, of a pair of inhibitory edges separated by more than one activating edge, and applying the solution described above has negligibly minor effects on the results. Given the density of feedforward and feedback loops in signalling networks, and the propensity for direct "inhibit the inhibitor" structures [49, 50], we expect that our choice to focus on direct inhibitory effects is the more practical to make.

The aim of graph theoretical analysis of signaling networks is to provide primary clues for a better understanding of the signal transduction process [51]. For example, graph analysis of a large mammalian neuronal cell signaling network [14] revealed a separation of positive and negative feedback loops based on their graph distance from signals, suggesting an architecture that promotes dynamic stability and allows signals to persist. The shortest positive or negative paths among pairs of nodes can be used to determine a dependency matrix [10, 11, 32, 35] which reflects the long-range regulatory relationships among signaling components. The method proposed here augments graph theory and allows it to address important functional aspects of signaling components, leading to testable predictions of comparable accuracy as dynamic models.