Inference of a Boolean Network From Causal Logic Implications

Biological systems contain a large number of molecules that have diverse interactions. A fruitful path to understanding these systems is to represent them with interaction networks, and then describe flow processes in the network with a dynamic model. Boolean modeling, the simplest discrete dynamic modeling framework for biological networks, has proven its value in recapitulating experimental results and making predictions. A first step and major roadblock to the widespread use of Boolean networks in biology is the laborious network inference and construction process. Here we present a streamlined network inference method that combines the discovery of a parsimonious network structure and the identification of Boolean functions that determine the dynamics of the system. This inference method is based on a causal logic analysis method that associates a logic type (sufficient or necessary) to node-pair relationships (whether promoting or inhibitory). We use the causal logic framework to assimilate indirect information obtained from perturbation experiments and infer relationships that have not yet been documented experimentally. We apply this inference method to a well-studied process of hormone signaling in plants, the signaling underlying abscisic acid (ABA)—induced stomatal closure. Applying the causal logic inference method significantly reduces the manual work typically required for network and Boolean model construction. The inferred model agrees with the manually curated model. We also test this method by re-inferring a network representing epithelial to mesenchymal transition based on a subset of the information that was initially used to construct the model. We find that the inference method performs well for various likely scenarios of inference input information. We conclude that our method is an effective approach toward inference of biological networks and can become an efficient step in the iterative process between experiments and computations.


Supplementary File
Inference of a Boolean Network from Causal Logic Implications

Supplementary Table S1. List of interactions and their causal logic implications pertaining
to ABA induced stomatal closure.
The published data in (1) reports 206 regulatory relationships with literature references for each of the relationships. We review the literature references to find the causal logic implication of each of the regulations. These causal logic implications fall into six types: sufficient (s), necessary (n), sufficient inhibitory (si), necessary inhibitory (ni), sufficient and necessary (sn), and sufficient and necessary inhibitory (sni). This table lists the regulatory relationship, the nature of the effect: promoting or inhibiting, the type of interaction: direct or not direct, the corresponding references and the causal logic implication. The logic implication marked with an asterisk (*) shows a lower confidence in the logic implication compared to the ones without an asterisk. The last column is relevant to indirect causal relationships only and indicates whether there is at least circumstantial evidence that this relationship is independent from (not mediated by) other regulators of the target node (node B).

Supplementary Table S2. List of inferred edges in ABA induced closure
We use the causal logic analysis result on co-pointing subgraph to infer an edge. This table lists all the regulations that can be inferred using co-pointing subgraphs. Some of these regulations were already reflected in the network from direct regulations, and hence we do not add a new edge in those cases. In the remaining cases, we infer a new edge and add it to the network. The first column lists the regulator of the inferred regulation, the second column gives the target node, the third column gives the logic. The logic implication is either of sufficient (s), necessary (n), sufficient inhibitory (si), necessary inhibitory (ni), sufficient and necessary (sn), sufficient and necessary inhibitory (sni). The fourth column gives the references for the regulation/interaction that led to the inference. The fifth column gives the regulation observed from the references that can be used in the format of co-pointing subgraphs. The sixth column lists whether a new edge was added or not. A new edge is not added when a logically equivalent path or subgraph already exists. In the cases where a new edge is not added, i.e., the sixth column entry is "no", the seventh column lists the equivalent path. The last column indicates whether there is support for the inferred relationship being independent from other regulators of the target node.  Fig 1F of (88). The asterisk indicates that the confidence in this entry is less than in the others. The target node PA is underlined.
Most likely rule: PA* = PLD and PLD and PC and DAG and DAGK Footnotes: 1. Phosphatidylcholine is a substrate and assumed to always be present.
2. DAGK is an enzyme, and there are no knockout experiments or regulators known for it. We assume it is always ON.
Supplementary Table S4. Truth table for AnionEM   Table produced from information in (16,17,25). The target node AnionEM is underlined. Please note that the parsimonious extension assumes that the recessive double knockout mutant of abi2-hab1; and abi2-pp2ca1 will show higher stomatal responsiveness to ABA.

Supplementary Text S1. Inferred Boolean rules of the ABA network
Here we list all the Boolean rules inferred using causal logic inference for the ABA network. The rules that are different compared to previously published data (1) (1) is below the explanation in black, bold font followed by text that is a quote from Text S2 of (1) relevant for the reasoning for the published rule and to the discrepancy. The citation numbers in the quote refer to the references in this file.
Group 1. The published and inferred functions contain the same regulators but there are differences in Boolean operators. The inferred function is equally or more consistent with the experimental information than the published function.

Actin Reorganization* = PtdInsP3 and PtdInsP4 and not AtRAC1 and ARP2/3 Complex and SCAB1
During the inference process the ARP complex is marked as necessary for Actin Reorganization; PtdInsP3

Actin Reorganization*= (PtdInsP4 or PtdInsP3) and not AtRAC1 and ARP Complex and SCAB1
"Expression of a dominant-positive mutant of AtRAC1 inhibits ABA-induced actin reorganization whereas expression of a dominant-negative mutant of AtRAC1 promotes actin reorganization in the absence of ABA (49). Inhibition of AtRAC1 is necessary for actin reorganization in response to ABA. ARP2 knockouts did not exhibit actin reorganization during ABA signaling. The arp2 phenotype was rescued upon application of an actin depolymerizing agent which illustrates that ARP2 is a positive regulator of the actin reorganization process (27). In this particular case, the behavior of one subunit is assumed to describe the behavior of the protein complex (27). In Arabidopsis, SCAB1 encodes a plant specific actin binding protein.
The scab1 loss-of-function mutant shows a delayed ABA response that is associated with delayed actin reorganization (100). Both PtdInsP4 and PtdInsP3 are implicated as positive regulators of actin reorganization in response to ABA (90)."

ABI1* = not PA and not RCARs and ROP11 and not ROS and pHc
According to causal logic inference, pHc is necessary for ABI1, and each of RCARs, ROS, PA is a sufficient inhibitor of ABI1. The existing information on the effect of ROP11 on ABI1 (93) does not lead to a strong logic implication. To achieve compatibility, the Boolean "and" operator is extended to ROP11.

Each of RCARs and ROS is a sufficient inhibitor of ABI2. As in the function of ABI1, the Boolean "and"
operator is extended to ROP11.

ABI2*= (not RCARS or ROP11) and not ROS
"ROP11 physically interacts with ABI2 and promotes its phosphatase activity (58). RCARs inhibit phosphatase activity of ABI2 by physical binding (6). ABI2 has been shown to be negatively regulated by ROS (98); we assume that the absence or low level of ROS is a necessary condition for ABI2 activity. In addition to this requirement, we assume that in order for ABI2 to be active, its positive regulator ROP11 must be active or its other negative regulators, RCARs, must be off."

KOUT*= (not NO or not ROS or pH c ) and Depolarization
"Membrane depolarization drives K + efflux from the guard cell. Outwardly rectifying K + channels are activated by cytosolic pH increase (82) and inhibited by ROS (95) and nitric oxide (70). K + efflux through outwardly rectifying K + channels requires membrane depolarization; thus we use an "and" function between "Depolarization" and other indicated positive or negative regulators of KOUT. In the absence of documented synergy, we use an "or" function between NO, ROS and pH c ."

PA* = PC and PLD and PLD and DAG and DAGK
The regulators of PA have logic implications with a lower confidence; hence we construct an incomplete truth

PA*= PC and (PLDδ or PLDα) or DAG and DAGK
"PC is the substrate needed by PLDα or PLDδ for PA production. DAG, a product of PLC, can be converted into PA by DAGK-mediated phosphorylation (52,113)."

pH c * = OST1 and not ABI2 and not ABI1 and Ca 2+ c and Vacuolar acidification During inference, we marked ABI1 and ABI2 as sufficient inhibitors of pH c and OST1, Vacuolar
Acidification, and Ca 2+ c as necessary for pH c increase based on experiments done in the presence of ABA.
The published function also uses specific evidence from external calcium induced closure to group regulators whose effect can be overcome in this process. pH c *= (OST1 and not ABI2 and not ABI1 or Ca 2+ c ) and Vacuolar acidification "Guard cells of abi1-1 (dominant negative), abi2-1 (dominant negative), and ost1-2 loss of function mutants show impaired cytosolic alkalization in response to ABA (31). Exogenous calcium application is assumed to increase Ca 2+ c concentration and can induce cytosolic alkalization in ost1-2 (loss-of-function), abi1-1 (dominant negative), and abi2-1 (dominant negative) mutants (31), indicating that Ca 2+ c -triggered pH c increase in guard cells does not require functional OST1 or the inactivation of ABI1 or ABI2; hence the OR relationship between Ca 2 c and these three proteins. External application of a calcium chelator, EGTA, reduces ABA-induced cytosolic alkalization (pH c increase) (105), indicating that Ca 2+ c is a positive regulator in ABA-induced cytosolic alkalization. Vacuolar acidification is a necessary condition for maintenance of ABA induced cytosolic alkalization state in guard cells (8). Thus, we use an "and" function between Vacuolar acidification and other indicated positive and negative regulators of pH c increase."

SLAH3* = (CKP6 or CPK23 or CPK3/21) and not ABI1
The causal logic of regulatory relationships indicates that any of the CPKs are sufficient for SLAH3 activity and that ABI1 is a sufficient inhibitor of SLAH3. The evidence for the effect of ABI1 on SLAH3 is stronger, hence we use the dominant regulators method with ABI1 as a dominant regulator. The published function uses a relationship between ABA and CPK3/21 that was not in the data source used for inference.

SLAH3*= (CPK6 or CPK23) and CPK3/21 and not ABI1
"All listed CPKs activate SLAH3 by physical interaction (16,37). All indicated CPKs have an independent positive effect on SLAH3. ABI1 inhibits CPK21-mediated activation of SLAH3 in oocytes indicating that ABI1 is a negative regulator of SLAH3 (16). Since CPK6 and CPK23 are weakly dependent on ABA (130), we implement the dependence of SLAH3 activation on ABA (16,37) by assuming that only the simultaneous activity of CPK6 and CPK3/21, or CPK23 and CPK3/21, is sufficient for SLAH3 activation." Group 2. The manually constructed regulatory function has more regulators. The inferred function is equally consistent with the experimental information as the published function.

PP2CA* = not RCARs
No regulatory relationship from ROS to PP2CA is in the data source used for inference. This relationship was assumed in the published version.

PP2CA*= not RCARS and not ROS
"In an in vitro study, it has been shown that soluble ABA receptors (RCARs, alternatively known as PYR/PYLs) PYR1, PYL1, PYL2, PYL4, PYL5, PYL6, PYL8 inhibit the phosphatase activity of PP2CA in the presence of ABA (92). ROS-mediated inhibition has been reported for three PP2C-type protein phosphatases that play negative regulatory roles in guard cell ABA signaling: ABI1, ABI2 and HAB1 (97-99). We assume that ROS would similarly inhibit the phosphatase activity of PP2CA."

We marked both V-ATPase and V-Ppase as necessary for vacuolar acidification. The self-regulatory relationship of Vacuolar Acidification assumed in the published function is based on a context other than
ABA-induced closure. We note that our recent work provides an alternative to making this assumption (131).

Vacuolar Acidification*= V-Ppase or V-ATPase or Vacuolar Acidification
"In yeast, the vacuolar proton ATPase (V-ATPase) proton pump plays an important role in vacuolar acidification (102,103). The proton pumping vacuolar pyrophosphatase (V-Ppase) uses the energy of PP i hydrolysis to acidify the vacuole (132). An Arabidopsis V-Ppase loss-of-function mutant, vhp1, shows delayed vacuolar acidification and slower stomatal closure in response to ABA (8,133). An Arabidopsis double knockout mutant of V-ATPase, vha1 vha2, exhibits a vacuolar pH of 6.4 rather than 5.9 (134). The double knockout mutant of the V-ATPase also shows delayed stomatal closure in response to ABA (8). We used an "or" function between V-Ppase and V-ATPase as the V-Ppase and V-ATPase play independent roles in vacuolar acidification (8). We assume that the vacuolar acidification state is sustained for a longer period and implement this assumption as a positive self-regulation. This assumption is necessary in order to allow the possibility of closure in response to internal closure signals (e.g. supply of S1P or Ca 2+ (135))" Calcium induces K + release through K + -permeable channels in the tonoplast (34). Vacuolar acidification also induces K + efflux from the vacuole (34). Group 3. The inferred regulatory function has more regulators. The inferred function is equally consistent with the experimental information as the published function.

SLAC1* = (CKP6 or CPK23 or CPK3/21) and MPK and OST1 and GHR1 and not ABI1 and not PP2CA and not ABI2 and pHc and MRP5
MRP5 is necessary for Ca 2+ c activation of SLAC1 (123). Because there isn't a path or subgraph that can mediate this effect, MRP5 appears as necessary regulator in the inferred rule. MRP5 is not present as regulator in the published rule; instead, it is assumed to be a regulator of CaIM.

CaIM* = Actin Reorganization or (NtSyp121 and GHR1 and MRP5) or not ABH1 or not ERA1 or OST1
The inferred rule has the regulator OST1 which is not present in the published rule. The inference process finds that OST1 is sufficient for CaIM. The network has no sufficient path or subgraph from OST1 to CaIM (the path OST1→RBOH→ROS→GHR1→CaIM is not sufficient), hence we add this as an edge.

Closure* = Microtubule Depolymerization and H 2 O efflux and cADPR and NtSyp121 and S1P and CIS and not H+ ATPase
The list of regulatory relationships in the inference process has multiple regulators that are found to be necessary for closure. Some of the regulators are reduced if a necessary path or subgraph to closure already exists but several regulators are not reduced and hence appear in the Boolean rule for stomatal closure. The published function is based on the biological knowledge that two independent processes are responsible for the shape and volume changes of the guard cells needed for stomatal closure.

Closure*= Microtubule Depolymerization and H 2 O efflux
"Microtubule depolymerization and H 2 O efflux are both needed for stomatal closure (66)."

Supplementary Table S6. Derived logic observations for the inference of the EMT network
Each row lists a logic observation denoted by a regulator node, target node, the corresponding logic implication (s, n, si, ni, sn, or sni) and a Boolean marker (Y/N) for whether the edge is expected to be direct or indirect. Highlighted in blue are all the indirect edges. These are reduced by the code in the inference process. Each of them has a mediator between the regulator and the target, and for each, regulator → mediator as well as mediator → target inference information exists in this list. Highlighted in green are all the cases in which we infer the mediator node. The mediators for the three cases are: CDC42, CD44, and betaTrCP, respectively. Highlighted in pink is the case where an incomplete path from the regulator (TCF/LEF) to the target (SHH) is known to be transduced via a potential mediator node (GLI) helping us infer an edge between the regulator and mediator thus completing the indirect observation of regulator (TCF/LEF) effect on target (SHH). Highlighted in sea green is an example of the co-pointing theorem potentially being applied to this network. Assuming the presence of betacatenin_memb, RAS is sufficient inhibitory for E-cadherin but TWIST1 is necessary inhibitory for E-cadherin. Sufficient inhibitory and necessary inhibitory are incompatible logic implications hence extending the co-pointing theorem, RAS must be sufficient for TWIST1. The rest of the list helps us infer that RAS is sufficient for TWIST1, and hence the co-pointing theorem application here just acts as supporting evidence.

Regulator
Target

Supplementary Text S5. Inferred Boolean functions for the modified EMT network
This text presents the inferred Boolean functions using our inference method for the reduced input information as provided in Table S10. The resulting Boolean functions are identical to the expected set of functions (see Text S9) except in the case of 18 nodes that are highlighted below in bold. The majority of cases of discrepancy (16) consist of missing a regulator; in 6 of these cases this omission creates a source node. There is one function (that of SNAI2) from which 2 regulators are missing. Indeed, in all of these cases the regulator-target relationship was missing from the input information. Finally, there is one function (that of RAS) in which one regulator is replaced (it contains "or AKT" instead of "or not