Research ArticleCOMPUTATIONAL BIOLOGY

Fault Diagnosis Engineering of Digital Circuits Can Identify Vulnerable Molecules in Complex Cellular Pathways

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Science Signaling  21 Oct 2008:
Vol. 1, Issue 42, pp. ra10
DOI: 10.1126/scisignal.2000008

Abstract

The application of complex system engineering approaches to cell signaling networks should lead to novel understandings and, subsequently, new treatments for complex disorders. In the area of circuit fault diagnosis engineering, there are various methods to identify the defective or vulnerable components of complex digital electronic circuits. In biological systems, however, knowledge is limited regarding the vulnerability of interconnected signaling pathways to the dysfunction of each specific molecule. By developing proper biologically driven digital vulnerability assessment methods, the vulnerability of complex signaling networks to the possible dysfunction of each molecule can be determined. To show the utility of this approach, we analyzed three well-characterized signaling networks—a cellular network that regulates the activity of caspase3, a network that regulates the activity of p53, and a central nervous system network that regulates the activity of the transcription factor CREB (adenosine 3′,5′-monophosphate response element–binding protein). We found important differences among the vulnerability values of different molecules. Most of the identified highly vulnerable molecules are functionally related and known key regulators of these networks. Experimental data confirmed the ability of digital vulnerability assessment to correctly predict key regulators in the CREB network. Because this approach may provide insight into key molecules that contribute to human diseases, it may aid in the identification of critical targets for drug development.

Introduction

Over the past few decades, a large amount of information has been collected regarding the function of individual signaling molecules and many detailed individual molecular mechanisms that regulate cellular function have been identified. Systems biologists have started to integrate these individual interactions and components to analyze the properties and functions that emerge from these complex biological systems (1).

Each cell in the human body includes many biomolecules that are interacting with each other through a network of many cellular signaling pathways. Dysfunction of some molecules involved in these pathways may interfere with the efficacy and efficiency of the signal transduction within the network, which can eventually result in a transition from the normal function (physiological condition) to a dysfunctional system (diseased or pathological condition). For some hereditary human diseases, such as Huntington’s disease, neurofibromatosis, and polycystic kidney disease, variations in a single gene cause the pathology. However, for some human diseases, including cancer as well as some neurodegenerative and psychiatric brain disorders, a single gene does not cause the disorder. Instead, the disease may result from the dysfunction of several molecules in different pathways. In such complex trait disorders, it is not clear which molecules have causative effects and how much each molecule may contribute to the development of the pathology.

The core idea of this article is to conceptualize a disorder at the molecular level as a faulty system in which one or more molecules in the complex intracellular signaling network are dysfunctional. Although genome- and proteome-wide expression analyses of biological systems provide a valuable picture of the “expression levels” of the molecules, it is the “functionality” of each molecule that determines the overall performance of the molecular system. We develop useful molecular fault models, similar to some fault models used in digital circuits (2), to quantify the functionality of different molecules in a network.

Results

There are many similarities between digital electronic circuits and genetic or signaling networks. In a manufacturing facility, a digital circuit is manufactured based on a particular design and is supposed to provide a specific function. However, during the fabrication process, physical defects—faulty transistors, open and short wires, and such—may occur, causing the manufactured circuit not to function according to the design specifications. Testing of digital circuits and systems allows defective manufactured parts to be separated from the nondefective ones to guarantee the shipment of fault-free products. The test is an assessment of the manufactured circuit according to a set of criteria. During the lifetime operation of electronic systems, the correct functionality is a key aspect and is typically referred to as reliability.

To determine the most vulnerable molecules in a network of molecules, we take advantage of a class of electronic circuit reliability analysis techniques known as vulnerability assessment methods. Such methods provide numerical values for the vulnerability of the operation of the entire molecular system to the dysfunction of each individual molecule. A high vulnerability for a molecule means that with high probability, the signaling network does not operate correctly if that particular molecule is dysfunctional. Identification of such important molecules, those with high vulnerability values, in signaling networks implicated in disease is a major step toward understanding the molecular basis of complex human diseases. From the drug development perspective, vulnerability assessment provides a set of candidate molecules to target.

To calculate the amount of the vulnerability of a network of interconnected pathways to the dysfunction of each molecule, one needs a model for the network. There are different types of models, such as Boolean, Bayesian, differential equations, algebraic, and graph theory. We chose the Boolean framework, where each molecule is either active (on) or inactive (off) (3, 4); thus, by analogy with digital electronic circuits, the state of a molecule is either 1 or 0, respectively. The Boolean framework and binary logic has been extensively used to explore different characteristics of signaling and genetic networks (3). Through application of digital circuit fault and reliability analysis methods (2) and Boolean models of pathways, we show that the molecular system vulnerability to the dysfunction of each molecule in the system can be computed.

To illustrate this approach, we generated circuit diagrams for a “toy” molecular network (Fig. 1A), which has seven molecules. In the toy network, A and B are the input nodes (molecules), and G is the output node (molecule). Activation and inhibition are shown by lines ending in an arrowhead or blunt line, respectively. The activity of each molecule can be shown by a binary value assigned to the name of that molecule. For example, A = 0 indicates that A is inactive (off), whereas A = 1 means that A is active (on). To assess the vulnerability of a network, it is necessary to derive a binary logic equation for each molecule of the network. Based on the known physiological mechanisms by which different regulators control the activity of signaling molecules, we propose two rules to derive the logic equations of a molecular network. Rule 1 states that if a molecule has no inhibitory input, then the activation of at least one of its activatory inputs in enough to activate the molecule. Rule 2 states that if a molecule has at least one inhibitory input, then that molecule will be inactive if at least one of its inhibitory inputs is activated. Moreover, the molecule can be active only if all the inhibitory inputs are inactive.

Fig. 1

A toy molecular network. (A) The input molecules of the seven-node toy model are A and B, the intermediate molecules are C, D, E, F, and, finally, the output molecule is G. The activatory and inhibitory signals are shown by lines ending in → and ⊣, respectively. (B) Logic equations of the molecular network, which are used to generate the equivalent digital electronic circuit. (C) The digital electronic circuit derived for the molecular network based on the logic equations. The function of each logic gate (circuit component) is specified in the square box. Names of some molecules in the circuit diagram may appear multiple times to allow easy recognition of the inputs and outputs of the logic gates. A small black dot is used to show where a wire is branched out from another wire.

Based on the definitions of Boolean OR, AND, and NOT (inversion), Rule 1 implies the “OR of activatory inputs,” whereas Rule 2 is the “AND of inverted inhibitory inputs.” When a molecule has both activatory and inhibitory inputs, the two rules are combined, which means the “AND of inverted inhibitory inputs” with the “OR of activatory inputs”. Using the above two rules, we derived logic equations for the toy network (Fig. 1B), and based on the five logic equations, we generated a digital electronic circuit for the toy network (Fig. 1C). In digital circuits, there are four types of “gates”: AND, OR, BUFFER, and NOT. The following symbols are used to represent these gates: A semi-ellipse represents the AND operation, a bullet-like shape depicts the OR operation, a triangle with a bubble on the right vertex represents the NOT operation (inhibition), and a triangle with no bubble is called a BUFFER (activation). The input and output of a BUFFER have the same logic values; if the input is on or off, then the output is on or off, respectively.

To test the validity and feasibility of this approach in cellular signaling networks, we apply this logic to a well-characterized network (Fig. 2A) for which the interactions between the molecules are extensively characterized and experimentally verified (5, 6). Janes et al. (5) studied the individual and joint effects of the three input nodes, the ligands epidermal growth factor (EGF), insulin, and tumor necrosis factor (TNF), on the activity of the two output nodes caspase3 and forkhead-related transcription factor (FKHR). Their study also showed how the activity of several intracellular intermediate molecules is modulated to regulate the activity of caspase3 and FKHR. The caspase3–FKHR network investigated by Janes et al. has 22 nodes, with 27 interactions among them. Using the two rules that we previously defined, we derived a single logic equation for each node in the network (table S1) that includes all the regulatory inputs to each node.

Fig. 2

The caspase3–FKHR network. (A) The caspase3–FKHR network, based on (5, 6), has a total of 22 nodes. The input molecules are EGF, insulin, and TNF, and the output molecules are caspase3 and FKHR. The node ComplexI includes TNFR and TRADD-RIP-TRAF2 (5), whereas the node ComplexII stands for TRADD-RIP-TRAF2 and FADD (5). (B) The digital electronic caspase3–FKHR circuit, derived based on the logic equations (table S1) for the caspase3–FKHR molecular network.

With the logic equations derived for the caspase3–FKHR network, we generated a caspase3–FKHR digital circuit (Fig. 2B) with AND, OR, BUFFER, and NOT gates. Using the logic equations, we can determine the binary logic values of the outputs, FKHR and caspase3, for any combination of inputs. The resulting “truth table” (Table 1) shows in the first three columns all possible input scenarios and in the last three columns, the calculated outputs, according to the circuit logic equations (table S1). Consistent with the experimental findings (5), digital circuit analysis showed that as long as EGF or insulin is active, caspase3 is inactive, indicating that apoptosis will not occur. However, when both EGF and insulin are inactive and TNF is active, then caspase3 is activated, which can lead to apoptosis. The consistency of the experimental data (5) with the results of the digital circuit analysis validates the biological application of this binary logic engineering approach and suggests that this level of abstraction, the Boolean on/off model, has a coarse predictive power that can be verified experimentally.

Table 1 Truth table of the FKHR-caspase3 digital circuit showing the binary logic values of the output FKHR–caspase3 network, when different inputs are applied to the circuit. The first three columns include all possible input scenarios, whereas the last three columns are computed according to the circuit logic equations (table S1).
View this table:

To perform the vulnerability assessment for any circuit, the fault model must be specified (2). We chose the class of “stuck-at-0” and “stuck-at-1” fault models, because they seem to be more biologically relevant than other types of fault models. To place this model into a biological perspective, consider the approximation that each molecule is supposed to change its active or inactive state according to its input signals (this is an assumption and biology does not precisely follow this logic). However, if the molecule is “stuck” at a particular state because of mutations or other structural or functional abnormalities, it cannot respond properly to the input signals. The state of such a dysfunctional molecule will not change, although it may receive stimulatory or inhibitory commands from its surrounding molecules. Thus, a node with the stuck-at-0 fault means that the logic value of that node (molecule) is always 0 (inactive), irrespective of its inputs. Similarly, a node with the stuck-at-1 fault implies that the node (molecule) is always 1 (active), and its state does not change if the inputs change.

In the caspase3–FKHR example, if AKT is stuck at 1, then according to the logic equation for caspase3 [caspase3 = AKT′ × (caspase8 + JNK1 + MK2); table S1], caspase3 is 0. This is because 1′ = 0 and 0 × (caspase8 + JNK1 + MK2) = 0. Thus, stuck-at-1 AKT removes the only chance for caspase3 activation, the presence of TNF in the absence of insulin and EGF, and the subsequent apoptosis under normal conditions (Table 1, compare last two columns). This dependence of caspase3 activation and by inference, apoptosis, on the state of AKT is supported by the experimental findings showing that the hyperactivity of AKT (stuck at 1 in the digital circuit fault analysis) is associated with malignant transformation (7).

The next step in digital circuit fault analysis is calculating the vulnerability of a node. By definition, the vulnerability value of a node is the probability that the system fails (incorrect system output) if that particular node is faulty (dysfunctional). To determine the vulnerability of the network to the dysfunction of each individual molecule, we considered the input signals to be statistically independent, such that each input molecule takes 0 and 1 with the same probability of 0.5. This assumption was made because of the lack of precise information regarding the actual biological probabilities of ligand bindings in the literature. By applying the error propagation probability (EPP) method (8, 9) to the caspase3–FKHR circuit (Fig. 2B), we computed the vulnerability values of all the molecules in the caspase3–FKHR network (Table 2).

Table 2 Vulnerablity values for each node in the FKHR-caspase3 circuit. Vulnerability (with FKHR) represents the calculated vulnerability values for the complete circuit. Vulnerability (without FKHR) represents the calculated vulnerability values for the three-input-to-one-output circuit that results if output node FKHR and its preceding BUFFER are removed from the circuit. The vulnerabilities are calculated with the use of the EPP algorithm (8, 9) and are sorted from high to low. In both circuits, AKT has the highest vulnerability value.
View this table:

In this type of analysis, the vulnerabilities of the output nodes are always 1, because if the output nodes are dysfunctional, by definition the network is not functioning correctly. The caspase3–FKHR network shows the highest vulnerability to the dysfunction of AKT, which may be because AKT is the immediate upstream regulator of both the output nodes (Fig. 2A). To clarify this issue, we removed the output node FKHR from the network and recalculated the vulnerabilities of all the nodes (Table 2). Even in this case, AKT still has the highest vulnerability value of all non-output molecules, which implies that AKT plays a critical role in this network. Using the three-input one-output caspase3 network, we find that a biological interpretation for the AKT vulnerability value 0.87 means that if AKT is dysfunctional, then on average, for 87% of all possible ligand-binding incidents (input signals), the cell will not correctly regulate the activity of caspase3 (the output node). The computed vulnerabilities (Table 2) are insensitive to the presence or absence of some nodes that have minor effects on the vulnerabilities of the rest of the nodes in the network. For example, removal of p38 from the caspase3–FKHR network, which allows the MKK3 node to activate the MK2 node directly, did not change the vulnerability values of the remaining components. Removal of the nuclear factor κB (NF-κB) node or the p38 and the NF-κB nodes also did not change the vulnerability values.

Using the approach of Ma’ayan et al. (10), we constructed a p53 cellular network containing numerous intermolecular interactions (Fig. 3A). The p53 network was constructed from pairs of experimentally verified molecular interactions reported in the literature (see Supplementary Materials for citations). This network was analyzed to test the validity and capability of circuit fault diagnosis engineering to correctly predict the main regulators of p53, which is a tumor suppressor that is a transcriptional activator of several genes that ultimately control cell cycle arrest, cellular senescence, or apoptosis. For more than half of all human cancers, p53 has been found mutated or functionally inactivated (11). The resulting network, called the p53 network, has 49 molecules, with 94 intermolecular interactions. The input nodes are the two ligands insulin and platelet-derived growth factor (PDGF), and the output node is the transcription factor p53. The specific interactions among the molecules of this network are derived from the literature and publicly available databases (fig. S1). A single specific logic equation for every individual molecule in the p53 network was derived on the basis of the logic rules (table S2); these logic equations were used to generate the p53 digital circuit (fig. S2) and the vulnerability values of all the molecules in the network were computed (Table 3).

Fig. 3

The p53 network. (A) This network has a total of 49 molecules: The input molecules are insulin and PDGF and the output molecule is p53. To make the figure less crowded, small circles depict signals coming from one or more molecule that targets one or several other molecules. All of the 94 intermolecular interactions of this network are listed in fig. S1 with literature citations. (B) The vulnerability bar chart of the p53 circuit. The vulnerabilities of PIP2, AKT, caspase3, and PP2A are greater than 0.5, whereas those of caspase8, PI3K, Abelson leukemia tyrosine kinase (Abl), phosphatidylinositol-(3,4,5)-triphosphate (PIP3), the lipid phosphatase PTEN, and protein kinase C (PKC) are between 0.1 and 0.5. The vulnerabilities of the rest of the molecules are less than 0.1. (C) The histogram of the vulnerability values of the p53 circuit. The number at the top of each bar represents the total number of molecules whose vulnerabilities fall within the range specified by the location of that bar.

Table 3 Vulnerablity values (Vul) for each node in the p53 circuit, calculated with the use of the EPP algorithm (8, 9) and sorted from high to low.
View this table:

The p53 network shows the highest vulnerability (more than 0.5) to the dysfunction of phosphatidylinositol 4,5-bisphosphate (PIP2), AKT, caspase3, and protein phosphatase 2A (PP2A) (Fig. 3B). Previous studies have experimentally shown that these molecules are known to be key regulators of p53 (1218). Thus, the analysis predicts that if any one of these molecules is dysfunctional, out of 100 incidents of ligand binding, over 50 times p53 will not function properly. The distribution of the vulnerability values of the molecules is nonuniform (Fig. 3C). There were 4 highly vulnerable nodes (>0.5) and 6 moderately vulnerable nodes (values between 0.1 and 0.5), and most of the nodes (38 out of the 49) exhibited low vulnerability values (less than 0.1) (Fig. 3B).

The four molecules that had the highest vulnerability values in the p53 network are key regulators of the cellular functions for which p53 is responsible. For AKT and caspase3, there is clear evidence that these two molecules play crucial roles in regulating a number of p53-regulated functions, such as cell survival and apoptosis (11, 19). Experimental evidence supports a key role for AKT in proper activation of p53 (1214). Although PIP2 has not been directly connected to p53, it is a regulator of AKT and thus, this connection to AKT may explain its high vulnerability value. PP2A and caspase3 also serve a major role in regulating p53 activity (1517). Thus, molecules diagnosed as faulty by this approach are experimentally known to contribute to failure of the system and cause pathology in humans. This can confirm the reliability of this fault diagnosis methodology.

We also constructed a complex neuronal network following the same approach of Ma’ayan et al. (10). The output node is the transcription factor CREB [adenosine 3′,5′-monophosphate (cAMP) response element–binding protein] and the input nodes are seven major ligands in the nervous system—glutamate, dopamine, γ-aminobutyric acid (GABA), serotonin, acetylcholine (ACh), adenosine, and enkephalin. The CREB network (Fig. 4A) is composed of 64 molecules and 152 intermolecular interactions (see fig. S3 for citations). The logic equations for the CREB network (table S3) were derived with Rules 1 and 2, and the corresponding digital electronic circuit was constructed (fig. S4). Calculation of the vulnerabilities of all the molecules (Table 4) again revealed a nonuniform distribution (Fig. 4C). Dysfunction in 41 molecules out of 64 would not contribute to the failure of the CREB circuit (vulnerability values less than 0.1).

Fig. 4

The CREB network. (A) This network has a total of 64 molecules: The input molecules are glutamate, dopamine, GABA, serotonin, ACh, adenosine, and enkephalin and the output molecule is CREB. All the 152 intermolecular interactions are listed in fig. S3, along with literature citations. (B) The vulnerability bar chart of the CREB circuit. The vulnerabilities of calmodulin, Ca2+, cAMP, Gαi, AC2, AC1, AC5, PKA, P/Q-type calcium channels, and PP2A are greater than or equal to 0.5, whereas those of Gβγ, calcium/calmodulin-dependent protein kinase II (CaMKII), protein phosphatase 2B (PP2B), calcium/calmodulin-dependent protein kinase IV (CaMKIV), calmodulin-dependent protein kinase kinase (CaMKK), cAMP response element modulator (CREM), N-type calcium channels (abbreviated N CaCh in the network), N-methyl-d-aspartate–type glutamate receptor (NMDAR), PI3K, PIP3, the kinase PDK1, and the kinase RSK are between 0.1 and 0.5. The vulnerabilities of the rest of the molecules are less than 0.1. (C) The histogram of the vulnerability values of the CREB circuit. The number at the top of each bar represents the total number of molecules whose vulnerabilities fall within the range specified by the location of that bar.

Table 4 Vulnerablity values (Vul) for each node in the CREB circuit, calculated with the use of the EPP algorithm (8, 9) and sorted from high to low.
View this table:

The molecules with a vulnerability of more than 0.5, which indicates that their dysfunction can result in the failure of CREB function, are calmodulin, calcium, cAMP, Gαi (abbreviated Galphai in the network diagram), adenylate cyclase (AC) 1, AC2, AC5, protein kinase A (PKA), P/Q-type calcium channel (abbreviated P/Q CaCh in the network diagram), and PP2A (Table 4 and Fig. 4B). These molecules can be grouped into elements of the cAMP-dependent signaling (cAMP, Gαi, PKA, and the AC isoforms) and elements of calcium signaling (calcium, calmodulin, and P/Q-type calcium channels). Furthermore, the distribution of the vulnerability values in the CREB network is highly nonuniform (Fig. 4C). The molecules with the highest vulnerability values in the CREB network are functionally related molecules (Fig. 4B), and some of them are already known as main physiological regulators of CREB function. Indeed the name CREB is based on the identification of the protein as a cAMP responsive element–binding protein. This engineering analysis has identified cAMP and the molecules directly related to cAMP function, such as AC1, AC2, AC5, and PKA, as the most critical molecules for the regulation of CREB. The crucial role of PKA in the regulation of CREB is well known (20). Important functions of the cognitive and executive human brain, such as learning and memory, are directly regulated by cAMP-dependent CREB functions (20). In pathological terms, direct evidence for deregulation of PKA signaling has been reported in human disorders manifested by memory dysfunction, such as Alzheimer’s disease (21) or schizophrenia (22, 23). Vulnerability assessment of the CREB circuit has also identified some elements of calcium signaling as playing a major role in the function of CREB. This observation is also physiologically and pathologically relevant and consistent with experimental data (24). Furthermore, several pathological conditions associated with memory dysfunction can arise from deregulation of calcium-dependent signaling (2527). Although it is not yet clear how CREB is involved in the pathogenesis of these disorders, the physiological role of CREB in neuronal mechanisms underlying the memory function of mammalian brain has been known for many years (20).

Because many of the identified highly vulnerable molecules are already experimentally known regulators of CREB, we provide experimental evidence for only two, Gαi and P/Q-type calcium channels. We altered the activity or abundance of endogenous P/Q-type calcium channels and Gαi in primary cortical neurons from rats and then assayed for the changes in endogenous CREB activity or abundance (see Materials and Methods). We used primary neuronal culture as a model system because it has a CREB signaling network most similar to the in vivo signaling networks of the mammalian brain.

Short-term (2.5 hours) treatment of primary neurons with the selective P/Q-type calcium channel blocker ω-agatoxin IVA increased the phosphorylation of CREB at Ser133 without changing the total abundance of CREB (Fig. 5A). In contrast, long-term (12 hours) treatment decreased the phosphorylation of CREB at Ser133 as well as decreased the abundance of CREB (Fig. 5A). Furthermore, we targeted the expression of P/Q-type calcium channels with three unique adenoviral short hairpin–mediated RNA (shRNA) constructs and measured their effect on the endogenous total protein and the phosphorylation levels of CREB at Ser133. Three unique adenoviral shRNA expression vectors (CACNA1A V1, V2, and V3), which target the expression of transcript variants 1 and 2 of the alpha 1A subunit of P/Q-type calcium channels, were used to knockdown the expression of the channels in neurons. The V2 vector caused a substantial decrease in the abundance of P/Q-type calcium channels, the V1 vector caused a smaller decrease in the abundance, and V3 did not affect the abundance of the channel in primary neurons. We observed a decrease in the abundance of CREB and in the proportion of Ser133 phosphorylated CREB in cells with the V2 vector. However, the V3 vector, which did not alter P/Q-type calcium channel abundance, did not affect the abundance or phosphorylation of CREB. These experiments show that the activity or abundance of P/Q-type calcium channels can affect both the abundance and phosphorylation state of CREB. This observation is consistent with the findings of Sutton et al. (28), which showed that exogenous expression of P/Q-type calcium channels can induce transcription of specific genes related to the synaptic function. Whether P/Q-type calcium channels are altering calcium signaling to affect CREB activity or whether they are functioning through a nonconducting mechanism, similar to that described for other types of calcium channels (L and N type) (29), remains to be determined. It is also an open question as to why there are opposing effects of short-term and long-term blockage of P/Q-type calcium channel activity on CREB phosphorylation. A possible biological explanation for the different response between the short- and long-term treatments is related to the compensatory mechanisms that re-regulate the initial cellular response. The existing feedbacks in the circuit may explain the role of compensatory mechanisms that occur in different time intervals in biological systems. More detailed studies are needed to precisely address the differences among short- and long-term responses. Blockage of P/Q-type calcium channel activity was performed without applying input signals. Therefore, one cannot compare the observed CREB activity with the predictions of the model, as the status of the inputs is not known in this situation. The experiment was done merely to confirm the regulatory effect of P/Q-type calcium channel on CREB. To compare the prediction of the model with this experiment, the same set of input signals need to be applied to both the experiment and the model. This was not the purpose of this study. The intention was to show that P/Q-type calcium channel has regulatory effect on CREB.

Fig. 5

Experimental verification of the fault diagnosis findings. Each blot shown in this figure is a representative blot of three independent experiments. (A) Western blot analysis of protein extracts from primary neuronal cultures after 2.5 and 12 hours treatment with either vehicle or 1 μM concentration of the selective P/Q-type calcium channel blocker, ω-agatoxin IVA (ωAga). (B) Forty-eight hours after adenoviral transduction with three unique shRNA constructs targeting P/Q-type calcium channels. Lane 1 shows the protein size by a protein marker (Magic Marker from Invitrogen). Lanes 2 and 6 are loaded with protein extracts from two control plates (mock) and lanes 3, 4, and 5 are treated with V1, V2, or V3 adenoviral vectors, respectively. The top blot shows a decrease in P/Q-type calcium channel abundance with V2 vector (lane 4) and a smaller decrease with the V1 vector (lane 3). Middle blots show the total protein and Ser133 phosphorylation of CREB in the same set of samples. The same membrane of the top blot was stripped and reprobed with antibody against CREB. The bottom blot shows total actin protein of the same membrane as the loading control. (C) After 2 hours treatment with 0.1 or 0.2 μg/ml of the Gαi antagonist, pertussis toxin (PTX), or 5 or 10 μM of Gαi agonist, MAS-7. Blots were first probed with an antibody against phosphorylated CREB (Ser133) (top) then stripped and reprobed with an antibody against CREB (middle). Actin (bottom) served as the loading control. (D) The activity of CREB after treatment with different neurotransmitter ligands, including serotonin (STN), glutamate (GLT), dopamine (DPM), GABA, or adenosine (ADN). The top blot shows phosphorylation of CREB at Ser133 15 min after treatment with vehicle, 10 μM forskolin (FSK), or the indicated ligands. The middle blot shows total CREB protein on the same blot. The bottom blot shows actin as loading control. (E) The effect of treatment with serotonin on the CREB activity when PKA is activated with forskolin or inhibited with H-89. Cells were treated with vehicle, 10 μM forskolin, or H-89 for 30 min followed by 15 min treatment with10 μM serotonin. (F) Immunofluorescent analysis of primary cortical culture after treatment with vehicle, forskolin, serotonin, or serotonin and forskolin. Red represents the phosphorylated CREB (Ser133) as a measure of CREB activity. Green represents the Map-2 staining as a specific neuronal marker and blue represents DRAQ-5 nuclear staining of all the cells in the primary culture (both neuronal and glial cells). Images were captured with the same confocal parameters for the four different treatment conditions.

To verify the importance of Gαi in the regulation of CREB, we inhibited Gαi with pertussis toxin (30) (see Materials and Methods) and found an increase in the abundance of CREB and an increase in Ser133 phosphorylation. In contrast, activation of Gαi by MAS-7, an active mastoparan analog that stimulates Gαi (30), caused a decrease in the abundance of CREB and a decrease in Ser133 phosphorylation (Fig. 5C). The decrease in abundance of CREB, along with the decrease in Ser133 phosphorylation, is consistent with an overall decrease in the activity of CREB, as phosphorylation of CREB at Ser133 is required for activity (20). Thus, the active/inactive Boolean framework used for modeling and vulnerability analysis lead to discovery of regulation by both changes in abundance and changes in phosphorylation state. These experiments show the ability of the fault diagnosis engineering approach to correctly identify key regulators of signaling pathways.

With the primary neuronal cultures, we tested the ability of the constructed Boolean network to correctly predict the output activity after an increase in the concentration of each input molecule. More specifically, we experimentally verified the accuracy of the CREB truth table (Table 5), which was obtained with the use of the logic equations of the CREB network (table S3).

Table 5 CREB circuit truth table. This table shows the binary logic values of the output CREB when different inputs are applied to the circuit. The first five columns include the input scenarios, whereas the last three columns are computed by simulating the circuit logic equations in ModelSim, a simulation and debug software tool for digital circuits.
View this table:

According to the truth table, serotonin is the only input that activates CREB when PKA is functional. To verify this prediction experimentally, we treated primary neurons, which are known to express these receptors (3137), with the activating concentration of 10 μM of serotonin (32), glutamate (33), dopamine (34), GABA (35), or adenosine (36) for 15 min then measured the activity of CREB by monitoring Ser133 phosphorylation (20). Treatment with serotonin had the most robust effect on the activity of CREB, stimulating phosphorylation to a similar extent as did the PKA activator forskolin (Fig. 5D). However, exposure of the neurons to the other ligand failed to robustly activate CREB (Fig. 5D). This is consistent with the output predicted in the truth table by the logic equations: Serotonin should be the most effective activator of CREB. Previous studies have also reported that serotonin can induce Ser133 CREB phosphorylation (32, 37); thus, the engineering model not only correctly predicts this effect of serotonin, but also shows the specificity of serotonin’s effect, compared to the other ligands.

We also tried to verify the biological relevance of the stuck-at-1 and stuck-at-0 fault models (Table 5, last two columns). We tested the effect of serotonin on the activity of CREB when PKA, a highly vulnerable molecule in this network, is either activated (stuck at 1) or inhibited (stuck at 0), by forskolin or H-89, respectively. When PKA is stuck at 1 or 0, the model predicts that serotonin should no longer activate CREB. To verify this experimentally, we treated primary neurons with serotonin in the presence or in the absence of either forskolin or H-89 (Fig. 5E). Whereas individual treatments with serotonin or forskolin activate CREB, treatment with serotonin after activation of PKA by forskolin attenuated the effect of either agent alone on the activity of CREB, as compared to treatment with serotonin only. This result is counterintuitive; one might expect that addition of two activators of CREB—serotonin and forskolin—should have an enhanced stimulatory effect. Thus, when the highly vulnerable molecule PKA is stuck at 1, the network output is not correctly regulated by the input. Inhibition of PKA by H-89 also prevents serotonin from stimulating the activity of CREB (Fig. 5E). Immunofluorescent analysis of CREB phosphorylation in primary neuron cultures exposed to vehicle, forskolin, or serotonin individually, or to serotonin after treatment with forskolin, was consistent with the results obtained by Western blotting for CREB phosphorylation. The cells were triple labeled with antibodies against Ser133 CREB (as a measure of CREB activity), Map-2 (as a marker that specifically labels neurons), and DRAQ-5 (as a nuclear marker that stains the nucleus of neuronal and nonneuronal cells in primary culture). Treatment of the neuronal primary cultures with forskolin increased the activity of CREB in both neuronal and non-neuronal (glial) cells in primary culture. However, treatment with serotonin only increased the activity of CREB in neurons, not in the nonneuronal cells. This specific activation in the neuronal cell population by serotonin was expected, because neurons have receptors for serotonin, whereas the nonneuronal cells in the culture do not. Consistent with the Western blot data (Fig. 5E), the effect of serotonin on the activity of CREB in neurons is attenuated, when serotonin was added after treatment with forskolin. These two experiments suggest that dysfunction of the network output occurs when PKA is stuck in either an active or an inactive state. Thus, the data support the prediction of the model that serotonin fails to activate CREB when PKA is faulty, which is entirely consistent with PKA having a high vulnerability value in the network.

Discussion

Taken together, our experimental data indicate that fault diagnosis engineering can identify new critical regulators and correctly predict previously known regulators of the output molecules. Furthermore, we have provided experimental evidence that a reconstructed Boolean network can correctly predict the activity of output molecules based on the activity of input signals. Finally, our experimental data support the theoretical finding of the proposed fault diagnosis approach; specifically, the experiments confirm that for proper propagation of the input signals to regulate the activity of output molecules, the normal function and activity of highly vulnerable molecules are necessary. When these highly vulnerable molecules become faulty, stuck at 1 or 0, the interconnected pathways cannot correctly propagate the signals from the input to the output, and the molecular network does not function properly.

Boolean modeling has certain applications and it cannot be used to precisely model all characteristics of signaling networks. The usefulness of Boolean modeling depends on the goal of the study. In the present study, Boolean framework provides a model that captures those essential characteristics of a molecular network necessary to determine the vulnerability of the network function to the dysfunction of each molecule. The application of a Boolean approach has provided biologically relevant results that are consistent with the experimental findings of other research groups and the experiments reported here. The proposed approach can be extended in several directions to broaden its scope. For example, two rules were introduced to build digital circuit equivalents of large signaling molecular networks and these were found sufficient for the purposes of vulnerability assessment. However, the activation or inhibition of a particular molecule may differ from what is considered in the two rules. Examples may include molecules that need sequential phosphorylation to become activated or competitive inhibition mechanisms of some molecules. In such scenarios, certainly an appropriate mixture of AND/OR/NOT operations can be used to properly model the activity of a molecule in terms of its inputs.

We chose a Boolean modeling framework from the many different modeling options because it provided a picture of cell signaling that was sufficient for vulnerability assessment analysis that we performed and for the creation of truth tables for prediction of biological outputs. The results of the analysis were either consistent with experimental data reported here or consistent with previous studies. Other methods that may be applicable for identifying critical molecules in signaling pathways include graph theory and sensitivity analysis. Graph theoretical approaches make conclusions based on the topology and connections that exist among the nodes of a graph that represents a molecular network. This approach may also provide valuable insight into signaling networks. Sensitivity analysis methods are typically used in conjunction with differential equation models of molecular networks to determine the sensitivity of the concentrations of various molecules to variations in kinetic parameters and rate constants (38). To implement such sensitivity analysis methods in large signaling networks, the nominal values of a large number of kinetic parameters and concentration levels need to be determined first. The proposed Boolean vulnerability assessment framework, however, does not use these parameters and can identify vulnerable molecules with less detailed information.

In summary, this paper takes advantage of the concepts of electronic circuit fault diagnosis engineering to identify the vulnerable molecules that play crucial roles in the dysfunction of molecular networks. The vulnerable molecules identified in caspase3, p53, and CREB signaling networks are functionally related sets of molecules with physiological and pathological relevance to the specific function of each network. This indicates the usefulness of the proposed approach. Application of this technique can improve the physiological understanding of the functionality of biological systems, identify key regulatory components, and potentially identify important targets for drug discovery.

Materials and Methods

Constructing the networks

For the caspase3–FKHR network, we used the experimentally verified networks of Janes et al. (5, 6). For the p53 and CREB networks, we used the approach of Ma’ayan et al. (10), as follows. First the input nodes (ligands) and the output node (transcription factor) of interest were specified, as well as the intermediate molecules that transmit the input signals to the output; then the type of the interactions among the molecules (activatory or inhibitory) was determined with the use of the existing literature and databases (see Supplementary Materials for details).

Identification of feedback paths

We used the depth-first search (DFS) algorithm (39) to identify feedback paths in a network. This step is done after a digital circuit is created. First, the digital circuit model is converted into a graph model. Each node in the digital circuit is modeled as a node in the corresponding graph and connections between the logic gates are modeled as edges in the graph. The DFS traversal is applied to the graph model to identify the feedback loops (39). This traversal visits all nodes of the graph one by one in the topological order by traversing the paths in the graph. Whenever a node is revisited (visited twice) in a path, it indicates the existence of a loop in the graph. Then, the loop is broken by removing the edge and replacing the corresponding connection in the digital circuit with a flip-flop. A flip-flop is a one-bit digital logic memory unit. This process is repeated until no further loop can be identified, i.e., the DFS traversal of the modified graph can be concluded without revisiting any node.

Vulnerability analysis of the logic circuits

We present a methodology for reliability analysis (vulnerability analysis) of logic circuits and its application to biological signaling networks to calculate the vulnerability values for the molecules in the network.

We extract the EPP of the internal nodes, which is the probability that an erroneous value on that node propagates from the error site to system outputs and results in an observable error in the system. We developed an EPP computation approach, which improves the runtime of previous EPP methods by several orders of magnitude. The presented approach uses the signal probabilities (SP) of all nodes in the combinational part and then computes EPPs based on the topological structure of the logic circuit. The SP of a line l indicates the probability of l having logic value 1 (40). Experiments on benchmark circuits and comparison of the results with the fault injection method based on random simulation show the effectiveness and the accuracy of the presented approach (9).

We first extract the structural paths from the error sites to all reachable primary outputs and then traverse these paths to compute the propagation probability of the erroneous value to the reachable primary outputs or to the reachable flip-flops. Based on the error site, we categorize nets and gates in the circuit as follows. An “on-path” signal is a net on a path from the error site to a reachable output. An “on-path gate” is the gate with at least one on-path input. Finally, an “off-path” signal is a net that is not on-path and is an input of an on-path gate (fig. S5).

For calculation of error propagation probability as we traverse the paths, we use signal probability for off-path signals and use our propagation rules for on-path signals. SP calculation and estimation techniques have been presented previously (41). The problem statement can be described as follows:

Given the failure probability in node ni, calculate the probability of the propagation of this error to primary outputs (POs) or flip-flops (FFs) (system failure).

Errors can be directly propagated to a primary output and cause a system failure at the same clock cycle, or they can be propagated to flip-flops repeatedly, and finally manifest as errors at a primary output several clock cycles later.

First, consider a simple case when there is only one path from the error site to an output. As we traverse this path gate by gate, the error propagation probability from an on-path input of a gate to its output depends on the type of the gate and the signal probability of other off-path signals. In the example shown in fig. S6, the error propagation probability to the output of the gate C (AND gate) is the product of the probability of the output of gate A being 1 and the error probability at the PI (1 × 0.2 = 0.2). Similarly, EPP at the output of the gate D (OR gate) is calculated as 0.2 × (1 − SPB) = 0.2 × 0.7 = 0.14.

In the general case in which “reconvergent paths” (one signal directly or indirectly drives more than one input of a logic gate) might exist, the propagation probability from the error site to the output of the reconvergent gate depends not only on the type of the gate and the signal probabilities of the off-path signals, but also on the polarities of the propagated error on the on-path signals. In the presence of errors, the status of each signal can be expressed with four values:

1. 0: no error is propagated to this signal line and the signal has an error-free value of 0.

2. 1: no error is propagated to this signal line and it has a logic value of 1.

3. a: the signal has an erroneous value with the same polarity as the original erroneous value at the error site (denoted by a).

4. ā: the signal has an erroneous value, but the erroneous value has an opposite polarity compared to the erroneous value at the error site (denoted by ā).

Based on this four-value logic, we can redefine propagation rules for each logic gate. These probabilities, denoted by Pa(Ui), Pā(Ui), P1(Ui), and P0(Ui), are explained as follows:

5. Pa(Ui) and Pā(Ui) are defined as the probability of the output of node Ui being a and ā, respectively. In other words, Pa(Ui) is the probability that the erroneous value is propagated from the error site to Ui with an even number of inversions, whereas Pā(Ui) is the similar propagation probability with an odd number of inversions.

6. P1(Ui) and P0(Ui) are defined as the probability of the output of node Ui being 1 and 0, respectively. In these cases, the error is blocked and not propagated.

Note that for on-path signals, Pa(Ui) + Pā(Ui) + P1(Ui) + P0(Ui) = 1, and for off-path signals, P1(Uj) + P0(Uj) = 1. Because we have considered the polarity of error effect propagation, this will take care of reconvergent points. The error propagation calculation rules for elementary gates are shown in table S4. To illustrate how to use the propagation rules for reconvergent paths, consider the example shown in fig. S7. In this example, the error propagation probability from the output of gate A to PO is calculated. Assume that the gate A becomes erroneous. So, initially, we set Pa(A) = 1, Pā(A) = 0, P1(A) = 0, and P0(A) = 0. Then, we propagate these probabilities through gates D, E, and H. As an example, we do the following steps to compute the error propagation probability of the erroneous value to the output of gate H.

P0(H) = P0(C) × P0(D) × P0(G) = 0.7 × 0.8 × 0.3 = 0.168

Pa(H) = (P0(C) + Pa(C)) × (P0(D) + Pa(D)) × (P0(G) + Pa(G))

P0(H) = (0.7) × (0.2 + 0.8) × (0.3)−0.168 = 0.042

Pā (H) = (P0(C) + Pā(C)) × (P0(D) + Pā(D)) × (P0(G) + Pā(G))

P0(H) = (0.7) × (0.8) × (0.7 + 0.3)−0.168 = 0.392

P1(H) = 1 − (0.168 + 0.042 + 0.392) = 0.398

P(H) = 0.042(a) + 0.392(ā) + 0.168(0) + 0.398(1)

Finally, the EPP of gate A to outputs can be computed as:

EPPA→PO = [Pa(H) + Pā (H)] = (0.042 + 0.392) = 0.434

In general, the following algorithm shows how we can extract and then traverse all paths from a given error site to all reachable outputs and how we apply the propagation probability rules as we traverse the paths.

The main algorithm

For every node, ni, do:

1. Path construction: Extract all on-path signals (and gates) from ni to every reachable primary output POj and/or flip-flop FFk. This is achieved with the forward depth-first search (DFS) algorithms (39).

2. Ordering: Prioritize signals on these paths based on their distance level with the use of the topological sorting algorithm (39). Topological sort of a directed acyclic graph is an ordered list of the vertices such that if there is an edge (u, v) in the graph, then u appears before v in the list.

3. Propagation probabilities computation: Traverse the paths in the topological order and apply propagation rules to compute the probability for each on-path node based on propagation probability rules (table S4).

Using the above formulation, we can compute error propagation probabilities from an arbitrary error site to any flip-flop, primary output, or both in just one pass starting from the error site to reachable output. As a result, the complexity of this approach is linear to the size of the circuit (number of logic gates). In other words, the exponential path enumeration problem is not observed in this algorithm. After the EPP of each node to all outputs is computed, the overall EPP of an arbitrary node A to all primary outputs at the first clock cycle can be computed as:

EPPc=1(A)=1i=1k1EPPAPOi

where k is the number of primary outputs. Note that EPPc = 1(A) computes the EPP of node A at the first clock cycle. The error, however, can be captured in flip-flops and propagated to primary outputs in the next clock cycles. To compute EPP of a node in the next clock cycles (c >1), the same error propagation rules can be repeated in the next cycles. In multiple cycle simulations (c >1), flip-flops are considered as possible error sites. Using the error propagation rules, EPPc = 2(A), EPPc = 3(A), EPPc = 4(A) are computed accordingly. Our experiments show that EPPc = x(A) for x > 4 becomes close to 0 such that we can ignore these probability values without sacrificing any accuracy. Finally, we compute the overall EPP of a node A according to the following equation:

EPP(A)=EPPc=1(A)+ (1EPPc=1(A))×EPPc=2(A)+(1EPPc=1(A))× (1EPPc=2(A))×EPPc=3(A)+(1EPPc=1(A))× (1EPPc=2(A))×(1EPPc=3(A))×EPPc=4(A)

To apply our vulnerability analysis methodology for the biological circuits of interest, we first need to convert their set of Boolean equations (tables S1, S2, and S3) into sequential circuits, without any combinational feedback (the original set of equations may contain combinational feedbacks). For this purpose, the feedback paths of these circuits are extracted with the DFS algorithm (39). Then, appropriate flip-flops are inserted at each feedback path, identified by backward edges in the DFS, to convert the original circuit to a sequential circuit (figs. S2 and S4). Because the Boolean equations of table S1 do not have any combinational feedback, no flip-flop is inserted in the corresponding combinational circuit (Fig. 2B). Finally, the vulnerability analysis algorithm is applied to the circuit to extract the SP and EPP values for all the nodes.

Developing the EPP method for the toy molecular network

As an example, we explain the EPP computation method for the digital electronic circuit (Fig. 1C) of the toy molecular network (Fig. 1A). First, we need to obtain the signal probabilities (SP) of all nodes. Staring with input SPs of 0.5, this yields the following:

SP(A) = 0.5, SP(B) = 0.5

SP(D) = SP(A) = 0.5

SP(C) = 1 − SP(B) = 0.5

SP(E) = 1 − [(1 − SP(B)) × (1 − SP(D))] = 0.75

SP(C + D) = 1 − [(1 − SP(C)) × (1 − SP(D))] = 0.75

SP(E′) = 1 − SP(E) = 0.25

SP(F) = SP(E′) × SP(C + D) = 0.185

SP(G) = 1 − [(1 − SP(F)) × (1 − SP(E))] = 0.796875

Once we have the signal probability values of all nodes in the circuit, we can apply the error propagation probability rules (table S4) to the logic gates in the circuit, starting from input toward the output, to obtain EPP values for the output.

The topological order of this circuit is as follows:

A, B: level 0

C, D: level 1

E: level 2

F: level 3

G: level 4

Given an error in input A (dysfunction in the corresponding molecule), the probability that this error affects the output (causing the system to fail) is obtained as follows:

Because A is erroneous, Pa(A) = 1, Pā(A) = 0, P1(A) = 0, and P0(A) = 0.

Pa(B) = 0, Pā(B) = 0, P1(B) = 0.5, P0(B) = 0.5

Pa(C) = 0, Pā(C) = 0, P1(C) = 0.5, P0(C) = 0.5

Pa(D) = Pa(A) = 1, Pā(D) = Pā(A) = 0, P1(D) = P1(A) = 0, P0(D) = P0(A) = 0

P0(E) = P0(B) × P0(D) = 0

Pa(E) = [0.5 × 1] − 0 = 0.5,

Pā(E) = 0,

P1(E) = 1 − (0 + 0.5 + 0) = 0.5,

Pa(E′) = 0, Pā(E′) = 0.5, P1(E′) = 0, P0(E′) = 0.5

P0(C + D) = P0(C) × P0(D) = 0

Pa(C + D) = [0.5 × 1] − 0 = 0.5,

Pā(C + D) = 0,

P1(C + D) = 1 − (0 + 0.5 + 0) = 0.5,

P1(F) = P1(C + D) × P1(E′) = 0,

Pa(F) = [0 × 1] − 0 = 0,

Pā(F) = [0.5 × 0.5] − 0 = 0.25,

P0(F) = 1 − (0 + 0 + 0.25) = 0.75,

P0(G) = P0(E) × P0(F) = 0

Pa(G) = [0.5 × 0.75] − 0 = 0.375,

Pā(G) = 0,

P1(G) = 1 − (0 + 0.375 + 0) = 0.625.

The probability of error propagation from A to output G is calculated as EPP(A→G) = Pa(G) + Pā(G) = 0.375.

Similarly, the error propagation probabilities from all other nodes to the output can be computed and the signals can be ranked based on their EPPs.

Primary cortical culture

Primary cortical cultures were prepared from brains of embryonic day 17 to 18 Sprague–Dawley rats (Charles River). After trituration of cortical sections with a glass pipette, 2 × 105 to 4 × 105 neurons were plated on a coverslip (diameter, 12 mm) precoated with poly-D-lysine (BD Biocoat). For biochemical analysis, primary cortical cells were plated in 35-mm dishes precoated with poly-D-lysine (~3 × 106 neurons per dish). Neurons were grown in Neurobasal medium supplemented with 0.5 mM L-glutamine, B27 (2%), and N2 (1%) supplements.

Protein extraction and immunoblot analysis

Primary neurons were cultured as described. Cells were homogenized in ice-cold lysate buffer (0.25 M tris, pH 7.5) containing protease inhibitors (Protease Inhibitor Cocktail tablets, Boehringer Mannheim) and phosphatase inhibitors (Phosphatase Inhibitor Cocktails I and II, Sigma) and lysed through three cycles of freezing (in liquid nitrogen) and thawing (in a 37°C water bath). Protein concentration was measured by Bio-Rad’s protein assay and spectrometry at 595°A. Equal amounts of total protein were loaded on 4 to 12% gradient bis–tris gels, separated by the NuPAGE system (Invitrogen), and transferred onto a nitrocellulose membrane. The membrane was probed with primary and secondary antibodies and signals were detected by chemiluminescence followed by autoradiography. The following antibodies were used: anti-CREB antibody (Cell Signaling, 1:1000), anti-phospho Ser133 CREB antibody (Cell Signaling, 1:1000), anti–P/Q-type calcium channel antibody (Chemicon, 1:1000), and anti-actin antibody (Sigma, 1:1000).

Immunofluorescence studies

Cortical neurons were grown on coverslips, fixed for 10 min in phosphate-buffered saline (PBS) plus 3.7% formaldehyde, and permeabilized for 2 min with cold acetone. Coverslips were coated with 100 μl of primary antibody diluted in PBS (anti-phospho Ser133 CREB, Cell Signaling, 1:200; anti–Map-2 antibody, Upstate Biotechnology, 1:250). Coverslips were washed three times and labeled with Alexa 568 anti-rabbit antibody (1:500), Alexa 488 anti-mouse antibody (1:500), and the nuclear marker DRAQ-5 (1:10,000).

Analysis of CREB regulation by P/Q-type calcium channels or Gαi

Primary cortical neurons were cultured as described. We followed the methods of Dolmetsch et al. (42) to analyze the effect of calcium channel blockers on CREB activity. We characterized the signaling properties of the endogenous P/Q-type calcium channels by monitoring the endogenous activity and total protein levels of CREB. To minimize the effect of other elements of calcium or G-protein signaling after neuronal depolarization, all experiments were performed without depolarizing the neurons. Previous time-course studies have shown that short-term Ser133 phosphorylations of CREB are transient events and prolonged Ser133 phosphorylation (more than 40 min) is required for transcriptional activity (4244). Therefore, in these experiments, we analyzed the endogenous Ser133 phosphorylation and total CREB protein abundance after at least 2 hours treatment with agonist or antagonists of P/Q-type calcium channels and 2 hours treatment with agonist or antagonist of Gαi. ω-Agatoxin IVA (Calbiochem), pertussis toxin (Sigma), and Mas-7 (Calbiochem) were dissolved in the appropriate vehicle and added to the medium. Following the treatment, cortical neurons were harvested, lysed, and subjected to Western blot analysis as described.

Adenoviral gene knockdown

We used AdenoSilence RNA interference viral vector kit (Millipore, cat. no. GAL10021) to target the expression of P/Q-type calcium channels in primary neuronal cultures. This kit consisted of three unique adenoviral shRNA constructs targeting the transcript variants 1 and 2 of the alpha 1A subunit of P/Q-type calcium channels (CACNA1A). Following the kit instruction, 60 × 106 viral particle units from the crude virus of each construct were added to 50 μl of complete medium and added to 35-mm dishes of primary neuronal cultures. Cells were harvested 48 hours after viral transduction, protein was extracted, and the lysates were subjected to Western blot analysis.

Acknowledgments

We thank H. Asadi (Northeastern University, Boston, MA) for helping us with the analysis of digital circuits. We also thank A. Ma’ayan and R. Iyengar (Mount Sinai School of Medicine, New York, NY) for useful comments regarding the use of their SAVI software tool.

Supplementary Materials

www.sciencesignaling.org/cgi/content/full/1/42/ra10/DC1

Methods

Fig. S1. Intermolecular interactions of the p53 network.

Fig. S2. The digital electronic p53 circuit.

Fig. S3. Intermolecular interactions of the CREB network.

Fig. S4. The digital electronic CREB circuit.

Fig. S5. A typical path between an erroneous node to primary outputs and flip-flops.

Fig. S6. A simple path between an erroneous input to a primary output.

Fig. S7. Applying error propagation rules for a reconverging path.

Table S1. Logic equations of the caspase3–FKHR network.

Table S2. Logic equations of the p53 network.

Table S3. Logic equations of the CREB network.

Table S4. Computing probability at the output of a gate in terms of its inputs.

References

References and Notes

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