Department of Physics, Pennsylvania State University, University Park, PA 16802, USA

Department of Biology, Pennsylvania State University, University Park, PA 16802, USA

Abstract

Background

Understanding how signals propagate through signaling pathways and networks is a central goal in systems biology. Quantitative dynamic models help to achieve this understanding, but are difficult to construct and validate because of the scarcity of known mechanistic details and kinetic parameters. Structural and qualitative analysis is emerging as a feasible and useful alternative for interpreting signal transduction.

Results

In this work, we present an integrative computational method for evaluating the essentiality of components in signaling networks. This approach expands an existing signaling network to a richer representation that incorporates the positive or negative nature of interactions and the synergistic behaviors among multiple components. Our method simulates both knockout and constitutive activation of components as node disruptions, and takes into account the possible cascading effects of a node's disruption. We introduce the concept of elementary signaling mode (ESM), as the minimal set of nodes that can perform signal transduction independently. Our method ranks the importance of signaling components by the effects of their perturbation on the ESMs of the network. Validation on several signaling networks describing the immune response of mammals to bacteria, guard cell abscisic acid signaling in plants, and T cell receptor signaling shows that this method can effectively uncover the essentiality of components mediating a signal transduction process and results in strong agreement with the results of Boolean (logical) dynamic models and experimental observations.

Conclusions

This integrative method is an efficient procedure for exploratory analysis of large signaling and regulatory networks where dynamic modeling or experimental tests are impractical. Its results serve as testable predictions, provide insights into signal transduction and regulatory mechanisms and can guide targeted computational or experimental follow-up studies. The source codes for the algorithms developed in this study can be found at

Background

The normal functioning of biological organisms relies on the coordinated action of a multitude of components. The interactions between genes, proteins, metabolites and small molecules form networks that govern gene regulation, determine metabolic rates, and transduce signals

Given the topology (i.e. the nodes and edges) of a network, it is natural to wonder just how important (central) each node is to the network's connectivity and functionality. Not surprisingly the issue of node centrality and its correlation with node influence has attracted the attention of many researchers. A large number of graph measures have been developed for evaluating node centrality in complex networks, from degree centrality

Typically the functional significance of a gene or gene product is determined by cell viability after its knockout mutation, siRNA interference or inhibition by specific chemical inhibitors. Several recently introduced measures of node importance are based on the effects of the removal of that node on the network's efficiency _{p}, of which one, the edge from A to C_{p}, is negative (usually denoted as ---|). Moreover, combinatorial regulation is ubiquitous in biological networks; this means that multiple interactions that regulate a component may act in a synergistic (conditionally dependent) fashion _{p }requires the presence of B and C and the absence of A, the interactions B → C_{p}, C → C_{p }and A---|C_{p }will be conditionally dependent. This combinatorial nature of regulatory interactions is mostly neglected in graph-based methods developed so far. Even measures specifically designed for signal transduction networks, such as

Illustrative examples for essentiality of components in signaling networks

**Illustrative examples for essentiality of components in signaling networks**. I is the input node and O is the output node. All other nodes are intermediate nodes. → denotes activating regulations and ---| represents inhibitory regulations. We use three existing measures of node centrality: betweenness centrality, based on the node's participation in shortest paths between node pairs

In this work, we develop a novel method that addresses the shortcomings listed above. Our method expands a signaling network to a new representation that incorporates the sign of the interactions as well as the combinatorial nature of multiple converging interactions. We then simulate both knockout and constitutive activation of components as node disruptions, and determine the possible cascading effects of a node disruption by identifying indispensable regulatory effects. We introduce the new concept of elementary signaling mode (ESM), as being the minimal set of nodes that can perform signal transduction independently. The importance of each signaling component is then determined by comparing the number of ESMs following the cascading disruptions caused by the removal of the component with the number of ESMs in the intact network. We apply this method to several signaling networks including a network describing the immune response of a mammalian host to bacteria

Results

Integrative evaluation of the essentiality of signaling components

Signaling networks can be represented as directed graphs in which nodes denote signaling components, edges represent regulatory interactions, and the orientation of the edges reflects the direction of signal transduction

Expansion of signaling networks

We utilize two operation rules to expand a signaling network to a new representation that incorporates the regulatory logic (e.g. inhibition, synergistic regulations). First, to take into account inhibitory interactions, we introduce a complementary node for each component that negatively regulates other nodes or is being negatively regulated by other nodes (see Figure

Operation rules for the expansion of signaling networks

**Operation rules for the expansion of signaling networks **(a) A inhibits B, so two complementary nodes

Second, we introduce composite nodes to incorporate conditionally dependent relationships. We represent the combinatorial relationship of multiple regulatory interactions converging on a node

where _{ij }

Introducing complementary nodes and composite nodes increases the number of nodes and edges in the network, but the benefit is that ambiguity is eliminated. All the directed interactions in the expanded network represent activation. Multiple edges ending at a composite node are conditionally dependent on each other, and multiple edges ending at an original node or complementary node are independent. The expanded signaling network does not have ambiguous dependencies, and can serve as a substrate for augmented structural methods. In addition, expansion of a signaling network by decomposing complex Boolean rules into independent elements helps to untangle the network into individual modules.

Cascading effects of a signaling component's removal

As the expanded signaling network clearly represents the relationships among nodes and signaling interactions, we can evaluate the essentiality of a signaling component by examining the range to which its perturbation propagates. We determine the cascading effect of the removal of a node by an algorithm that iteratively finds and deletes the nodes that have just lost their indispensable regulators (see Methods). There are three cases for a regulator

Illustration of the cascading effects of a component's deletion and of the elementary signaling modes in signaling networks.

**Illustration of the cascading effects of a component's deletion and of the elementary signaling modes in signaling networks.** (a) The cascading effects of a component's deletion. (b) Elementary signaling modes. In (a) and (b), the ovals denote original nodes or complementary nodes, and the small circles mean composite nodes. In (a), the dashed edges and the dashed node contours indicate the nodes and edges that will be disrupted in the cascading failure following the removal of node A. In (b), the dashed shapes outline two elementary signaling modes (ESMs). Note that different ESMs can have common nodes.

Elementary signaling modes

The connectivity of a signaling network between the input node(s) (e.g. ligands) and the output node(s) (e.g. cellular responses) is most reflective of its signal transduction capacity. The concept of shortest paths is widely used to characterize the efficiency of information flow or communicability in networks

Elementary flux modes, minimal sets of enzymes that can make a metabolic system operate at a steady state, play an important role in metabolic network analysis

**Algorithms developed and used in this study**. This file contains the algorithms developed and used in this study, including the depth-first-search algorithm for enumerating all simple paths from the input node(s) to the output node(s) of a signaling network, the iterative integer linear programming algorithm for enumerating elementary signaling modes, the depth-first-search algorithm for estimating the number of elementary signaling modes, as well as the dynamic programming algorithm for finding shortest elementary signaling modes.

Click here for file

Our method ranks the importance of signaling components by the effects of their perturbation on the ESMs of the network. We characterize each node

Validation on three signaling networks

Several regulatory networks with documented synergistic and inhibitory interactions have been published ^{ESM}) and the simple path (SP) centrality measure (denoted by ^{SP}). We compare these measures to a traditional centrality measure, node betweenness centrality (denoted by BC)

**Essentiality of the components from dynamic models of the three signaling networks**. This file describes the essentiality of the signaling components obtained by dynamic simulation of Boolean models for the host immune response network and the guard cell ABA signaling network, and logical steady state analysis of the T cell receptor signaling network (Tables S1-S4).

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The host immune response network

Thakar

Using this network and its Boolean rules, we construct the expanded host immune response network shown in Figure

Results on the immune response network.

**Results on the immune response network. **(a) The expanded host immune response network. (b) The shortest elementary signaling mode in this network. (c) Importance values of nodes in this network obtained by single-node deletions. (d) Prediction accuracy. Bt: bacteria, EC: Epithelial cells, PIC: Pro-inflammatory cytokines, Th1RC: T helper cell type 1 related cytokines, Th2RC: T helper cell type 2 related cytokines, DC: Dendritic cells, T0: T0 cells, Th1C: T helper cell type 1, Th2C: T helper cell type 2, MP: Macrophages, BC: Antibody-producing B cells, Cab: Complement-fixing antibodies, Oab: Other antibodies, AgAb: Antigen-antibody complex, RP: Recruited PMNs, Cp: Complement, AP: Activated phagocytes, PH: Phagocytosis. In (a) and (b), composite nodes are represented by small gray solid circles, original nodes and complementary nodes are represented by rectangles. The labels of complementary nodes are denoted by the labels for the corresponding original nodes prefixed by the symbol '~'. The color coding of the nodes in (b) indicates the level of their participation in the 15 ESMs of the network. In (c) and (d), triangles represent the importance values or prediction accuracy obtained by the ESM measure, circles represent the simple path measure, plus signs denote the

The importance values of signaling components based on the ESM measure and the SP measure are shown in Figure

We rerun the Boolean dynamic model of Thakar

The guard cell ABA signaling network

Plants take up carbon dioxide for photosynthesis and lose water by transpiration through pores called stomata, which are flanked by pairs of specialized guard cells. The size of stomata is regulated by the guard cells' turgor ^{2+}, phosphatidic acid, as well as ion channels.

Using this network and the Boolean rules, we construct the expanded ABA signaling network shown in Figure _{c}, H^{+}ATPase, Ca^{2+}
_{c}, or KOUT leads to a strong reduction of the signal transduction connectivity. In addition, single-node knockouts of SphK and S1P will increase the length of the shortest ESM by more than 60%, suggesting that these signaling components are critical for the efficiency of ABA signal transduction. The important components predicted by our method are validated by numerous experimental observations (Additional file

The expanded guard cell ABA signaling network

**The expanded guard cell ABA signaling network**. Composite nodes are represented by small gray solid circles, original nodes are represented by large empty ovals, and complementary nodes are represented by rectangles. The labels of complementary nodes are denoted by the labels for the corresponding original nodes prefixed by the symbol '~'.

**Essentiality of the guard cell ABA signaling components from our method**. This file contains the importance values of the guard cell ABA signaling components obtained by single-node deletions (Figure S1) and two-node deletions (Figure S2), and literature support of the uncovered essential components.

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A detailed comparison of prediction accuracy by the four methods is given in Figure

Comparison of different methods applied to the guard cell ABA signaling network in terms of prediction accuracy

**Comparison of different methods applied to the guard cell ABA signaling network in terms of prediction accuracy**. Triangles represent the prediction accuracy of the ESM measure, circles represent the simple path measure, plus signs denote the

The T cell receptor signaling network

T cells (lymphocytes) play a central role in the immune response. T cells detect antigens by a special receptor on their surface called T cell receptor (TCR), which is triggered by Major Histocompatibility Complex (MHC) molecules and induces a series of intracellular signaling cascades. CD28 provides an essential co-stimulatory signal during T-cell activation, which increases T cell proliferation and prevents the induction of anergy and cell death. Saez-Rodriguez

We use CD28 antigen and the ligand of the T cell receptor (denoted by TCRlig) as the two inputs of the T cell signaling network and use NFκB and AP1 as the two outputs. The other outputs studied by Saez-Rodriguez

**The expanded T cell receptor signaling network**. This file contains the expanded T cell receptor signaling network (Figure S3) and the importance values of the T cell receptor signaling components found by our method with AP as the input node (Figure S4).

Click here for file

The importance values of signaling components obtained by single-node deletions are summarized in Figure

Comparison of different methods applied to the T cell receptor signaling network with NFκB as the output

**Comparison of different methods applied to the T cell receptor signaling network with NFκB as the output**. (a) Importance values obtained by single-node deletion of original nodes. (b) Importance values obtained by single-node deletion of complementary nodes. (c) Prediction accuracy. Rectangles indicate the importance values or prediction accuracy obtained by the ESM measure, circles represent the simple path measure, diamonds denote the

We calculate the steady states of this T cell receptor signaling network

Discussion

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

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)

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

Another related work by Abdi

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

The aim of graph theoretical analysis of signaling networks is to provide primary clues for a better understanding of the signal transduction process

Conclusions

Quantitative dynamic modelling of signaling networks helps to understand complex signal transduction processes, but it depends heavily on known mechanistic details and kinetic parameters. At the same time, structural analysis is emerging as a feasible and useful alternative for interpreting signal transduction. Aiming to overcome the limitations of existing structure-based approaches, we present an integrative computational method for evaluating the essentiality of components in signaling networks. The main steps of our method are expanding an existing signaling network to a richer representation that incorporates the positive or negative nature of interactions and the synergistic behaviors among multiple components and ranking the importance of signaling components by the cascading effects of their perturbation on the elementary signaling modes of the network. Validation on several signaling networks shows that this method can effectively uncover the essentiality of components mediating a signal transduction process. We conclude that while some properties of a dynamic model may depend on initial conditions and update time scales, other properties are encoded in the combinatorial regulations represented by Boolean rules and do not depend on the details of the dynamic simulation. Our method distils the most important ingredients of a dynamic model and integrates them into the network topology without the necessity of dynamic simulation. This method can be effectively used for exploratory analysis of large signaling networks where dynamic modeling or experimental tests are impractical and its results can guide targeted computational or experimental design.

Methods

Synthesizing evidence for inhibition and synergy

If the inhibitory regulations and combinatorial regulations in a signaling network are known, as was the case in

Determining cascading effects of component disruption

Given an expanded signaling network

• Step 1. Remove an original node or a complementary node

• Step 2. For each direct target of

• Step 3. Take a node

We need to check at most each node and each edge of the expanded signaling network to determine the cascading effects of the removal of a node. Therefore, the worst time complexity of the iterative algorithm is

Shortest elementary signaling modes

Multiple edges ending at a composite node in a signaling network are conditionally dependent and the activation of this node requires the activation of all its regulators. Thus, a composite node's activation follows the regulator that is activated last. In contrast, the activation of an original node or a complementary node can be done by any of several independent regulators and thus follows the regulator activated first. If we use the distance of signaling components from the input as a proxy for the sequence of events in signal transduction, the distance from the input node to a node

where

Essentiality of signaling components

Elementary signaling modes (ESMs) can be used to define an importance measure for the essentiality of signaling components in two different ways. First, we can rank the effect of a node's removal on the length of the shortest ESM. Second, we can determine the reduction in the number of ESMs following the removal of a node

where _{ESM}(_{ESM}(_{Δv}
_{Δv }

We also consider a more traditional graph measure for the essentiality of signaling components based on the number of all simple paths (SPs) from inputs to outputs:

where _{SP}(_{SP}(_{Δv}
_{Δv }

Implementation of the method

All the algorithms in this study were coded and implemented in Matlab 7.6 (The Mathworks, Inc.). The depth-first algorithms are the Matlab implementation of the pseudo code given in the Additional file

Authors' contributions

RSW and RA conceived and designed the study. RSW performed the study. RSW and RA analyzed the results. RSW and RA wrote, read, and approved the final manuscript.

Acknowledgements

This work was supported by NSF grants MCB-0618402 and CCF-0643529 (CAREER). We thank Dr. Juilee Thakar for sharing the code for the dynamic model of the immune response and Prof. István Albert for helpful comments on the manuscript.