## Abstract — In this paper a systems biology framework for generalized fault diagnosis in the caspase signaling network of biomolecules is studied. This novel method is capable of identifying critical molecules whose dysfunction can affect the network function detrimentally. The generalized vulnerabilities defined and computed in the paper quantify the role of molecules in a complex network. Impact of network input activities and multiple faults are studied as well. The results and methods are useful for quantitative analysis of functional impacts of individual or a group of molecules on the overall performance of molecular signaling networks.

## Keywords – molecular systems biology, molecular networks, cell signaling, fault diagnosis, vulnerability.

## Introduction

Analysis of molecular networks using a variety of engineering and computational tools and approaches has been an active area of research in molecular biology in recent years []. Molecular systems biology looks at the orchestrated function of individual components together. Despite some progress made in recent years, there is a desperate need for proper methods to quantify the function of molecules within a network, as well as their shares in possible malfunction of the network. Molecular fault diagnosis engineering is introduced in recent years [Error: Reference source not found],[ ] to find critical molecules whose dysfunction can have detrimental impacts on the network function. Possible applications to target discovery and drug development are discussed in [] and []. In this study the fault diagnosis approach introduced in [Error: Reference source not found] is generalized in a number of ways. In Sec. a new parameter is introduced which is basically the fault probability of each molecule, and allows network vulnerabilities to be parameterized. This is useful for analysis and data fitting. A method for computing generalized vulnerabilities is provided in Sec. . The impact of changes in input activities is investigated in Sec. , whereas Sec. is devoted to studying vulnerabilities of pairs of molecules (in [Error: Reference source not found] only vulnerabilities of single molecules are studied). In Sec. a multi level (ternary) fault diagnosis is proposed and its performance is investigated in Sec. Error: Reference source not found and Sec. . Some concluding remarks are provided in Sec. .

## Proposed Generalized Faulty Network Model

In this paper we develop concepts and methods using the experimentally verified caspase3 signaling network [], shown in (the method is applicable to other networks as well). It has three input molecules, insulin, EGF, TNF, some intermediate molecules, and one output, caspase3. Using the input-output relationships in Table 1 of [Error: Reference source not found], the network transition probability matrix M can be written as: The caspase3 network.* MERGEFORMAT ()Each element of the above matrix is a conditional transition probability of the form . For any given set of 0/1 values for the inputs shown in , this gives the probability of the output to be 0 or 1. Now we introduce a faulty network model for the caspase3 network, to analyze the impact of dysfunctional (faulty) molecules. In this model, the probability of a molecule in the network to be faulty is p, i. e. . When a molecule is faulty, its activity state does not change in response to its regulators, i. e., becomes independent of its regulators, and it is stuck at 0. By calculating the conditional probabilities specified in , the transition probability matrix for the caspase3 network can be constructed, depending on which molecule is faulty, as listed in -. These matrices are needed to compute the vulnerability of each molecule in the network.* MERGEFORMAT ()* MERGEFORMAT ()* MERGEFORMAT ()* MERGEFORMAT ()

## Generalized Vulnerabilities in Caspase Network

The vulnerability value of a molecule is the probability that the network fails, i. e., incorrect network output, if the molecule is dysfunctional [Error: Reference source not found]. Using the total probability theorem [], the vulnerability of the network to the dysfunction of each individual molecule can be written as:* MERGEFORMAT ()Since in this section all the eight input patterns are equally probable, vulnerability in can be simplified to:* MERGEFORMAT ()The above probabilities are elements of network transition probability matrices in -. Upon using those matrices, results in the following equations for the vulnerabilities of molecules in terms of the fault probability p:* MERGEFORMAT ()These vulnerability levels are plotted in for all the molecules. The caspase3 network shows different fault behavior depending on the faulty molecule. For example, when AKT is faulty, vulnerability rapidly increases with p. This shows the critical role of AKT in the network. On the other hand, some molecules such as EGFR or MEKK1ASK1 cause a small increase in vulnerability as p increases. This indicates their less critical role in the function of the network. Vulnerability versus the fault probability p in the caspase3 network.

## Impact of Inputs Activities on Output Activity and Network Vulnerabilities

## Output activity with no faulty molecule in the network

Each input probability (activity) represents the probability of a ligand binding. To study the impact of input probabilities, we start from the case where all the molecules are functional. Let be input probabilities (activities). Then based on the input-output relations in Table 1 [Error: Reference source not found] and for independent inputs we have* MERGEFORMAT ()The activity of caspase3, , is plotted in versus , when . The results in this figure are consistent with the experimental data in [], i. e., activation of caspase3 by TNF and the subsequent cell death are inversely related to the probability of EGF or insulin activation. Caspase3 activity in terms of TNF activity (activities of EGF and insulin are the same, and fixed at 0. 1, 0. 5 and 0. 9).

## Output activity with one faulty molecule in the network

When there is one faulty molecule in the network, the probability of caspase3 being active can be calculated using the total probability theorem as follows:* MERGEFORMAT ()Note that can be obtained from the second column of M matrices in -. To see how the activity of a faulty molecule may affect the output activity as input activity changes, consider the case in which , the activity of the input TNF changes from 0 to 1, whereas similarly to [Error: Reference source not found], activities of the other two inputs, EGF and insulin, are fixed at . Using , the output activity can be written as* MERGEFORMAT ()Now we study output activity when based on , the faulty molecule has high vulnerability, AKT, or low/medium vulnerability, MEKK1ASK1, or low vulnerability, IKK. B. 1. The faulty molecule is highly vulnerable: When AKT is faulty with probability p, by replacing the conditional probabilities in with the second column of the M matrix in , the output activity can be written as* MERGEFORMAT ()Eq. is plotted in versus , for . As a reference, output activity when there is no faulty molecule is also plotted in using , which is . Since AKT is highly vulnerable, changes in its fault probability can significantly alter the output activity, compared to the case where it is not faulty molecule. This is biologically relevant since the activity of AKT has a positive correlation with the activity of caspase3 []. Caspase3 activity in terms of TNF activity q3, when AKT is faulty with different probabilities p. EGF & insulin activities are fixed at 0. 5. B. 2. The faulty molecule has low/medium or low vulnerability: When MEKK1ASK1 or IKK is faulty with probability p, by replacing conditional probabilities in with the second column of the M matrices in or , the output activity can be written as:* MERGEFORMAT ()* MERGEFORMAT ()Eq. and are plotted in and , respectively, versus , for . Since MEKK1ASK1 has low/medium vulnerability, we observe much less changes in the output activity in , compared to a highly vulnerable molecule such as AKT. This insensitivity is even more evident for a low vulnerable molecule such as IKK, which causes no change in the output activity when faulty (Figure 6). Caspase3 activity in terms of TNF activity q3, when MEKK1ASK1 is faulty with different probabilities p. EGF & insulin activities are fixed at 0. 5. Caspase3 activity in terms of TNF activity q3, when IKK is faulty with different probabilities p. EGF & insulin activities are fixed at 0. 5.

## Impact of input activities on vulnerabilities

To study the impact of input probabilities (activities) on the vulnerability levels of molecules in the network, we fix the activities of the two inputs EGF and insulin at [Error: Reference source not found]. For two different activity levels of the input TNF, , vulnerabilities of different molecules are plotted in and , respectively, in terms of the fault probability p. Changing the input probabilities may affect the vulnerability levels of some molecules. Vulnerability versus the fault probability p for all the molecules in the caspase3 network, while TNF activity is 0. 1. Vulnerability versus the fault probability p for all the molecules in the caspase3 network, while TNF activity is 0. 9. Comparing , and illustrate that changing activity of TNF does not effect on AKT vulnerability. However, when TNF activity decreases from 0. 9 to 0. 1, EGFR becomes less vulnerable. More specifically, we have and for the vulnerability of AKT and EGFR, respectively. Notice the dependence of EFGR’s vulnerability on .

## Vulnerabilities with Pairs of Faulty Molecules

To compute network vulnerabilities, so far we assumed there is only one faulty molecule in the network. Now we study the case where there are two faulty molecules simultaneously. Let and represent two simultaneously faulty molecules, then both of them stuck at 0 and therefore in a faulty case we have:* MERGEFORMAT ()Using and by constructing transition probability matrices for all the double faults, network vulnerabilities for all possible double faulty molecules are computed and listed in with equi-probable inputs and . Sorted joint vulnerabilities are provided in . A summary of is provided in , i. e., minimum, maximum and average vulnerabilities for each molecule, when jointly faulty with other molecules. The diagonal elements in are single fault vulnerabilities, obtained from with . We notice that average vulnerability in is the highest when AKT is jointly faulty with other molecules. This again shows the critical role of AKT in the network. To better understand the double fault vulnerabilities in , we consider some examples. In we have listed the network output, for all input patterns, when faulty molecules are AKT or p38 or MEKK1ASK1, as well as the pairs (AKT, p38) or (AKT, MEKK1ASK1). Note that faulty (incorrect) outputs are marked in italic bold. When AKT is sa0, the number of incorrect outputs is 6, which gives the sa0 vulnerability of .. Also when AKT is jointly faulty with p38, the number of incorrect outputs is 6, which gives the vulnerability of . This indicates the same vulnerability, when both AKT and p38 are faulty. On the other hand, when both AKT and MEKK1ASK1 are faulty, number of incorrect outputs in , out of 8 possibilities, becomes 4, which gives the vulnerability of , which is a slight decrease. So, depending on what pairs are dysfunctional, the vulnerability can change. It also seems that when a low vulnerable molecule such as p38 is jointly faulty with a highly vulnerable one like AKT, the pair’s vulnerability is greater than the original single low vulnerability.

## Ternary fault Diagnosis:

Previously we considered two activity states of molecules, i. e., active or inactive states. Now we propose a ternary activity model, where a molecule could be either active, partially active or inactive, represented by 1, 1/2 and 0, respectively. This modeling approach is proposed because there are cases where it might be needed to have multiple states for a molecule [],[]. For example, phosphorylation of AKT at Thr-308 is required for its activity, whereas the second phosphorylation at Ser-473 can make the molecule more active[]. Therefore now we develop a ternary fault diagnosis method. The proposed multi-level (ternary) fault diagnosis method: We explain the proposed approach by an example on the caspase3 network (). To include different levels of activity, we consider inactive, partially active and active states for each molecule, i. e., 0, 1/2 and 1, respectively. Also we use the same definition as [],[] for the ternary logic function to define the input –output relationships which is shown in Error: Reference source not found. Using these input-output relationships, the network transition probability matrix M for ternary model can be computed (Eq )Now we introduce a faulty network model for the caspase3 network, to analyze the impact of dysfunctional (faulty) molecules. Similar to binary model in ternary model, the probability of a molecule in the network to be faulty is p, i. e. . When a molecule is faulty, its activity state does not change in response to its regulators, i. e., becomes independent of its regulators, and it is stuck at 0. Also by calculating the conditional probabilities, the transition probability matrix for the caspase3 network can be constructed, depending on which molecule is faulty, as listed in Eq. -. These matrices are needed to compute the vulnerability of each molecule in the network. Using the above matrices, similarly to the binary case we have derived formulas for the vulnerability of each molecule with three activity states when all the 27 input patterns are equally probable. The results are shown in the following equations:* MERGEFORMAT ()These vulnerability levels are plotted in for all the molecules. Vulnerability versus the fault probability p in ternary model of the caspase3 networkAlso for comparing binary case with ternary case some results for both cases are shown in . Vulnerability versus the fault probability p in the caspase3 network when there is a faulty molecule in the network: binary versus ternary modelingWe observe that the ternary model has more precision in categorizing the function of molecules according to their vulnerability values. For example, EGFR and MEKK1ASK1 were predicted previously to have the same fault behavior (), whereas now they behave differently (). Another interesting observation is that still AKT has the highest vulnerability level. Vulnerabilities of the rest of the molecules have changed slightly. Since they all fall below the lowest line in , they are not shown in , to keep the figure easier to read.

## Impact of Input Activities on Vulnerabilities in Ternaty Model

To see the effect of input activity on vulnerability, in and vulnerability versus fault probability for different TNF activity are plotted base on which molecule is faulty. In and the input activities are. andrespectively . In both figure AKT is the highest vulnerable molecule. Also when AKT is faulty, although in binary case the vulnerability does not depends on TNF activity, in ternary model vulnerability depends on TNF activity and it is increasing by increasing TNF activity. In addition in where the activity of TNF is 0. 5, EGFR and MEKK1ASK1 have the same vulnerability. However, when TNF activity decreases to 0. 1, EGFR becomes more vulnerable than MEKK1ASK1, as shown in . On the other hand, increase of TNF activity to 0. 9 in renders EGFR less vulnerable. However in binary model, the order of vulnerability based on which molecule is faulty does not change by changing TNF activity, which means ternary model is more accurate in showing the effect of input activity in vulnerability. Vulnerability versus the fault probability p for all the molecules in the caspase3 network, while TNF activity is 0. 1 (ternary model)Vulnerability versus the fault probability p for all the molecules in the caspase3 network, while TNF activity is 0. 9 (ternary model)

## Vulnerabilities with Pairs of Faulty Molecules in Ternary Model

Similar to section for binary case here we consider pairs of faulty molecules in ternary model. The results are shown and summarized in Error: Reference source not found-As expected we see high value for vulnerability when one of the two faulty molecules is AKT. Based on the results for the ternary model and comparing them with binary results, an initial conclusion is that one can start with the less complex active/inactive fault diagnosis approach, to analyze the malfunction of signaling networks. This will assist in identifying the molecules whose dysfunction does not contribute much to the network failure (low vulnerability). Afterwards, if we need to understand, with more precision, the role of molecules with high vulnerabilities, then we will try a ternary model, at the cost of higher computational complexity. We anticipate that some predictions of ternary fault diagnosis will be the same as the binary one, specifically for molecules which were identified to have low vulnerabilities previously. However, it is likely that molecules diagnosed with medium or high vulnerabilities in the binary method may have their vulnerabilities changed in the ternary method. By increasing the number of activity levels from two to three, the complexity of the model increases. However, based on our results, the predictive power of the model does not necessarily increase proportionally. So, to have more predictive power, perhaps we do not need to develop a ternary model for the entire network. Instead, we can start with the binary method, to get a preliminary picture of the vulnerability levels of different molecules. According to our studies on the CREB and many other large networks, typically many molecules have very low vulnerabilities [Error: Reference source not found]. So, we can then focus on building a hybrid model where a small set of molecules with high vulnerabilities have three activity levels (This is conceptually similar to the binary network of Schlatter et al. [] where only few nodes have three states. Of course there is no fault analysis conducted by Schlatter et al.[]). This hybrid model is less complex than a fully ternary model and is likely to have more predictive power.

## Conclusion

In this paper a generalized fault analysis method is developed and applied to the caspase network. We have analyzed the network under different assumptions and conditions. The biological interpretations of some of the findings are consistent with the previously published experimental data. Biological relevance of some other predictions made here needs to be investigated via experimentation. The method is also applicable to large molecular networks.

## Supporting Information

Network Vulnerabilities for All Pairs of Faulty Molecules For binary model, vulnerabilities for all possible double faulty molecules when the inputs are equi-probable and . are computed and listed. (The diagonal elements are single fault vulnerabilities with ). Sorted . For binary model, sorted joint vulnerabilities for all possible double faulty molecules are provided. Minumum, Maximum and Average Vulnerabilties for each Faulty Molecule. Minimum, maximum and average vulnerabilities for each molecule, when jointly faulty with other molecules (for binary model) are listed, Network Output for Some Single and Double Faulty Molecules. For binary model, The network output, for all input patterns, when faulty molecules are AKT or p38 or MEKK1ASK1, as well as the pairs (AKT, p38) or (AKT, MEKK1ASK1) are listed. Note that faulty (incorrect) outputs are marked in italic bold. Error: Reference source not foundError: Reference source not found. The input – output relationships in ternary model are defined. Error: Reference source not foundError: Reference source not found. For ternary model, vulnerabilities for all possible double faulty molecules when the inputs are equi-probable and . are computed and listed in . (The diagonal elements are single fault vulnerabilities with ). Error: Reference source not foundError: Reference source not found. For ternary model, sorted joint vulnerabilities for all possible double faulty molecules are provided.(Double Stuck-at-0 Fault Is More Probable). Minimum, maximum and average vulnerabilities for each molecule, when jointly faulty with other molecules (for ternary model) are listed. Eq. . The network transition probability matrix M for ternary model when there is no faulty molecule in the network.* MERGEFORMAT (S1)Each element of the above matrix is a conditional transition probability of the form . For any given set of 0/½/1 values for the inputs, this gives the probability of the output to be 0, 1/2 or 1. Eq. . The network transition probability matrix M for ternary model when the faulty molecule is AKT.* MERGEFORMAT (S2)Eq. . The network transition probability matrix M for ternary model when the faulty molecule is EGFR.* MERGEFORMAT (S3)Eq. . The network transition probability matrix M for ternary model when the faulty molecule is MEKK1ASK1.* MERGEFORMAT (S4)Eq. . The network transition probability matrix M for ternary model when the faulty molecule is caspase8, ERK or MEK.* MERGEFORMAT (S5)Eq.. The network transition probability matrix M for ternary model when the faulty molecule is IRS1.* MERGEFORMAT (S6)Eq. . The network transition probability matrix M for ternary model when the faulty molecule is IKK, ComplexI, ComplexII, NFκB, JNK1, MK2, cFLIPL, MKK3, MKK7 or p38.* MERGEFORMAT (S7)