PLoS Biol. 2006 October; 4(10): e312.
Figure 1 Illustration of the Inference Rules Used in Network Reconstruction (1) If A → B and C → process (A → B), where A → B is not a biochemical reaction such as an enzyme catalyzed reaction or protein-protein/small molecule interaction, we assume that C is acting on an intermediary node (IN) of the A–B pathway. (2) If A → B, A → C, and C → process (A → B), where A → B is not a direct interaction, the most parsimonious explanation is that C is a member of the A–B pathway, i.e. A → C → B. (3) If A —| B and C —| process (A —| B), where A —| B is not a direct interaction, we assume that C is inhibiting an intermediary node (IN) of the A–B pathway. Note that A→ IN —| B is the only logically consistent representation of the A–B pathway. PLoS Biol. 2006 October; 4(10): e312.
PLoS Biol. 2006 October; 4(10): e312.
Network Synthesis and Path Analysis Logical organization of large-scale data sets is an important challenge in systems biology; our model provides such organization for one guard cell signaling system. As summarized in Table S1, we have organized and formalized the large amount of information that has been gathered on ABA induction of stomatal closure from individual experiments. This information has been used to reconstruct the ABA signaling network (Figure 2). Figure 2 uses different types of edges (lines) to depict activation and inhibition, and also uses different edge colors to indicate whether the information was derived from our model species, Arabidopsis, or from another plant species. Different types of nodes (metabolic enzymes, signaling proteins, transporters, and small molecules) are also color coded. An advantage of our method of network construction over other methods such as those used in Science's Signal Transduction Knowledge Environment (STKE) connection maps [61] is the inclusion of intermediate nodes when direct physical interactions between two components have not been demonstrated. As is evident from Figure 2, network synthesis organizes complex information sets in a form such that the collective components and their relationships are readily accessible. From such analysis, new relationships are implied and new predictions can be made that would be difficult to derive from less formal analysis. For example, building the network allows one to “see” inferred edges that are not evident from the disparate literature reports. One example is the path from S1P to ABI1 through PLD. Separate literature reports indicate that PLDα null mutants show increased transpiration, that PLDα1 physically interacts with GPA1, that S1P promotion of stomatal closure is reduced in gpa1 mutants, that PLD catalyses the production of PA, and that recessive abi1 mutants are hypersensitive to ABA. Network inference allows one to represent all this information as the S1P → GPA1 → PLD → PA—| ABI1—| closure path, and make the prediction that ABA inhibition of ABI1 phosphatase activity will be impaired in sphingosine kinase mutants unable to produce S1P. Another prediction that can be derived from our network analysis is a remarkable redundancy of ABA signaling, as there are eight paths that emanate from ABA in Figure 2 and, based on current knowledge (though see below) these paths are initially independent. The prediction of redundancy is consistent with previous, less formal analyses [62]. The integrated guard cell signal transduction network (which includes the ABA signal transduction network) has been proposed as an example of a robust scale-free network [62]. To classify a network as scale-free, one needs to determine the degree (the number of edges, representing interactions/regulatory relationships) of each node, and to calculate the distribution of node degrees (denoted degree distribution) [45,46]. Scale-free networks, characterized by a degree distribution described by a power law, retain their connectivity in the face of random node disruptions, but break down when the highest-degree nodes (the so-called hubs) are lost [46]. While the guard cell network may ultimately prove to be scale-free, the network is not sufficiently large at present to verify the existence of a power-law degree distribution; thus, the analogy with scale-free networks cannot be rigorously satisfied.
PLoS Biol. 2006 October; 4(10): e312.
Dynamic Modeling Our model differs from previous models employed in the life sciences in the following fundamental aspects. First, we have reconstructed the signaling network from inferred indirect relationships and pathways as opposed to direct interactions; in graph theoretical terminology, we found the minimal network consistent with a set of reachability relationships. This network predicts the existence of numerous additional signal mediators (intermediary nodes), all of which could be targets of regulation. Second, the network obtained is significantly more complex than those usually modeled in a dynamic fashion. We bridge the incompleteness of regulatory knowledge and the absence of quantitative dose-response relationships for the vast majority of the interactions in the network by employing qualitative and stochastic dynamic modeling previously applied only in the context of gene regulatory networks [53]. Mathematical models of stomatal behavior in response to environmental change have been studied for decades [63,64]. However, no mathematical model has been formulated that integrates the multitude of recent experimental findings concerning the molecular signaling network of guard cells. Boolean modeling has been used to describe aspects of plant development such as specification of floral organs [65], and there are a handful of reports describing Boolean models of light and pathogen-, and light by carbon-regulated gene expression [66–68]. Use of a qualitative modeling framework for signaling networks is justified by the observation that signaling networks maintain their function even when faced with fluctuations in components and reaction rates [69]. Our model uses experimental evidence concerning the effects of gene knockouts and pharmacological interventions for inferring the downstream targets of the corresponding gene products and the sign of the regulatory effect on these targets. However, use of this information does not guarantee that the dynamic model will reproduce the dynamic outcome of the knockout or intervention. Indeed, all model ingredients (node states, transfer functions) refer to the node (component) level, and there is no explicit control over pathway-level effects. Moreover, the combinatorial transfer functions we employed are, to varying extents, conjectures, informed by the best available experimental information (see Text S1). Finally, in the absence of detailed knowledge of the timing of each process and of the baseline (resting) activity of each component, we deliberately sample timescales and initial conditions randomly. Thus, an agreement between experimental and theoretical results of node disruptions is not inherent, and would provide a validation of the model. The accuracy of our model is indeed supported by its congruency with experimental observation at multiple levels. At the pathway level, our model captures, for example, the inhibition of ABA-induced ROS production in both ost1 mutants and atrboh mutants [12,19,21] and the block of ABA-induced stomatal closure in a dominant-positive atRAC1 mutant [22]. In our model, as in experiments, ABA-induced NO production is abolished in either nos single or nia12 double mutants [13,18]. Moreover, the model reproduces the outcome that ABA can induce cytosolic K+ decrease by K+ efflux through the alternative potassium channel KAP, even when ABA-induced NO production leads to the inhibition of the outwardly-rectifying (KOUT) channel [70]. At the level of whole stomatal physiology, our model captures the findings that anion efflux [35,71] and actin cytoskeleton reorganization [22] are essential to ABA-induced stomatal closure. The importance of other components such as PA, PLD, S1P, GPA1, KOUT, pHc in stomatal closure control [8,20,31,43,58,72], and the ABA hypersensitivity conferred by elimination of signaling through ABI1 [28,29], are also reproduced. Our model is also consistent with the observation that transgenic plants with low PLC expression still display ABA sensitivity [73]. The fact that our model accords well with experimental results suggests that the inferences and assumptions made are correct overall, and enables us to use the model to make predictions about situations that have yet to be put to experimental test. For example, the model predicts that disruption of all Ca2+ ATPases will cause increased ABA sensitivity, a phenomenon difficult to address experimentally due to the large family of calcium ATPases expressed in Arabidopsis guard cells (unpublished data). Most of the multiple perturbation results presented in Figure 5 and Table 2 also represent predictions, as very few of them have been tested experimentally. Results from our model can now be used by experimentalists to prioritize which of the multitude of possible double and triple knockout combinations should be studied first in wet bench experiments. Most importantly, our model makes novel predictions concerning the relative importance of certain regulatory elements. We predict three essential components whose elimination completely blocks ABA-induced stomatal closure: membrane depolarization, anion efflux, and actin cytoskeleton reorganization. Seven components are predicted to dramatically affect the extent and stability of ABA-induced stomatal closure: pHc control, PLD, PA, SphK, S1P, G protein signaling (GPA1), and K+ efflux. Five additional components, namely increase of cytosolic Ca2+, Atrboh, ROS, the Ca2+ ATPase(s), and ABI1, are predicted to affect the speed of ABA-induced stomatal closure. Note that a change in stomatal response rate may have significant repercussions, as some stimuli to which guard cells respond fluctuate on the order of seconds [74,75]. Thus our model predicts two qualitatively different realizations of a partial response to ABA: fluctuations in individual responses (leading to a reduced steady-state sensitivity at the population level), and delayed response. These predictions provide targets on which further experimental analysis should focus. Six of the 13 key positive regulators, namely increase of cytosolic Ca2+, depolarization, elevation of pHc, ROS, anion efflux, and K+ efflux through outwardly rectifying K+ channels, can be considered as network hubs [45], as they are in the set of ten highest degree (most interactive) nodes. Other nodes whose disruption leads to reduced ABA sensitivity, namely SphK, S1P, GPA1, PLD, and PA, are part of the ABA → PA path. While they are not highly connected themselves, their disruption leads to upregulation of the inhibitor ABI1, thus decreasing the efficiency of ABA-induced stomatal closure. Similarly, the node representing actin reorganization has a low degree. Thus the intuitive prediction, suggested by studies in yeast gene knockouts [76,77], that there would be a consistent positive correlation between a node's degree and its dynamic importance, is not supported here, providing another example of how dynamic modeling can reveal insights difficult to achieve by less formal methods. This lack of correlation has also been found in the context of other complex networks [78]. Comparing Figure 3 and Figure 6C, one can notice a similar heterogeneity in the measured stomatal aperture size distributions and the theoretical distribution of the cumulative probability of closure in the case of multiple node disruptions. While apparently unconnected, there is a link between the two types of heterogeneity. Due to stochastic effects on gene and protein expression, it is possible that in a real environment not all components of the ABA signal transduction network are fully functional. Therefore, even genetically identical populations of guard cells may be heterogeneous at the regulatory and functional level, and may respond to ABA in slightly different ways. In this case, the heterogeneity in double and triple disruption simulations provides an explanation for the observed heterogeneity in the experimentally normal response: the latter is actually a mixture of responses from genetically highly similar but functionally nonidentical guard cells.
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Limitations of the Current Analysis Network topology. Our graph reconstruction is incomplete, as new signaling molecules will certainly be discovered. Novel nodes may give identity to the intermediary nodes that our model currently incorporates. Discovery of a new interaction among known nodes could simplify the graph by reducing (apparent) redundancy. For example, if it is found that GPA1 → OST1, the simplest interpretation of the ABA → ROS pathway becomes ABA → GPA1 → OST1 → ROS, and the graph loses one edge and an alternative pathway. As an effect, the graph's robustness will be attenuated. Among likely candidates for network reduction are the components currently situated immediately downstream of ABA because, in the absence of information about guard cell ABA receptors [94], we assumed that ABA independently regulates eight components. It is also possible that a newly found interaction will not change the existing edges, but only add a new edge. A newly added positive regulation edge will further increase the redundancy of signaling and correspondingly its robustness. Newly added inhibitory edges could possibly damage the network's robustness if they affect the main positive regulators of the network, especially anion channels and membrane depolarization. For example, experimental evidence indicates that abi1 abi2 double recessive mutants are more sensitive to ABA-induced stomatal closure than abi1 or abi2 single recessive mutants [29], suggesting that ABI1 and ABI2 act synergistically. Due to limited experimental evidence, we do not explicitly incorporate ABI2, but an independent inhibitory effect of ABI2 would diminish ABA signaling. While it is difficult to estimate the changes in our conclusions due to future knowledge gain, we can gauge the robustness of our results by randomly deleting entries in Table S1 or rewiring edges of Figure 2 (see Texts S2 and S3). We find that most of the predicted important nodes are documented in more than one entry, and more than one entry needs to be removed from the database before the topology of the network related to that node changes (Text S2). Random rewiring of up to four edge pairs shows that the dynamics of our current network is moderately resilient to minor topology changes (Text S3 and Figure S1). Dynamic model. In our dynamic model we do not place restrictions on the relative timing of individual interactions but sample all possible updates randomly. This approach reflects our lack of knowledge concerning the relative reaction speeds as well as possible environmental noise. The significance of our current results is the prediction that whatever the timing is, given the current topology of regulatory relationships in the network, the most essential regulators will not change. Our approach can be iteratively refined when experimental results on the strength and timing of individual interactions become available. For example, we can combine Boolean regulation with continuous synthesis and degradation of small molecules or signal transduction proteins [95,96] as kinetic (rate) data emerge. Our model considers the response of individual guard cell pairs to the local ABA signal; however, there is recent evidence of a synchronized oscillatory behavior of stomatal apertures over spatially extended patches in response to a decrease in humidity [97]. Our model can be extended to incorporate cell-to-cell signaling and spatial aspects by including extracellular regulators when information about them becomes available (see [51]). Node disruptions. A knockout may either deprive the system of an essential signaling element (the gene itself), or it may “set” the entire system into a different state (e.g., by affecting the baseline expression of other, seemingly unrelated signaling elements). Our analysis and current experimental data only address the former. Because of this caveat, in some ways rapid pharmacological inhibition may actually have a more specific effect on the cell than gene
