Boolean modeling and fault diagnosis in oxidative stress response
- Sriram Sridharan1,
- Ritwik Layek1,
- Aniruddha Datta1Email author and
- Jijayanagaram Venkatraj2
https://doi.org/10.1186/1471-2164-13-S6-S4
© Sridharan et al.; licensee BioMed Central Ltd. 2012
Published: 26 October 2012
Abstract
Background
Oxidative stress is a consequence of normal and abnormal cellular metabolism and is linked to the development of human diseases. The effective functioning of the pathway responding to oxidative stress protects the cellular DNA against oxidative damage; conversely the failure of the oxidative stress response mechanism can induce aberrant cellular behavior leading to diseases such as neurodegenerative disorders and cancer. Thus, understanding the normal signaling present in oxidative stress response pathways and determining possible signaling alterations leading to disease could provide us with useful pointers for therapeutic purposes. Using knowledge of oxidative stress response pathways from the literature, we developed a Boolean network model whose simulated behavior is consistent with earlier experimental observations from the literature. Concatenating the oxidative stress response pathways with the PI 3-Kinase-Akt pathway, the oxidative stress is linked to the phenotype of apoptosis, once again through a Boolean network model. Furthermore, we present an approach for pinpointing possible fault locations by using temporal variations in the oxidative stress input and observing the resulting deviations in the apoptotic signature from the normally predicted pathway. Such an approach could potentially form the basis for designing more effective combination therapies against complex diseases such as cancer.
Results
In this paper, we have developed a Boolean network model for the oxidative stress response. This model was developed based on pathway information from the current literature pertaining to oxidative stress. Where applicable, the behaviour predicted by the model is in agreement with experimental observations from the published literature. We have also linked the oxidative stress response to the phenomenon of apoptosis via the PI 3k/Akt pathway.
Conclusions
It is our hope that some of the additional predictions here, such as those pertaining to the oscillatory behaviour of certain genes in the presence of oxidative stress, will be experimentally validated in the near future. Of course, it should be pointed out that the theoretical procedure presented here for pinpointing fault locations in a biological network with feedback will need to be further simplified before it can be even considered for practical biological validation.
Keywords
Introduction
The control of gene expression in eukaryotic organisms is achieved via multivariate interactions between different biological molecules such as proteins and DNA [1]. Consequently, in recent years, various genetic regulatory network modeling approaches such as differential equations and their discrete-time counterparts, Bayesian networks, Boolean networks (BNs) and their probabilistic generalizations, the so-called probabilistic Boolean networks (PBNs) [2] have been proposed for capturing the holistic behavior of the relevant genes. Some of these approaches such as differential equations involve finer models and require a lot of data for inference while others such as Boolean networks yield coarse models with lower data requirements for model inference. On the other hand, historically biologists have focused on experimentally establishing marginal cause-effect relationships between different pairs of genes, which when concatenated together leads to what is known as pathway information. Biological pathways are used by biologists to represent complex interactions occurring at the molecular level inside living cells [3]. Pathway diagrams describe how the biological molecules interact to achieve their biological function in the presence of appropriate stimuli [4]. At a very simple level, biological pathways represent the graphical interactions between different molecules. However, as already noted, the pathways give only a marginal picture of the regulations (up-regulation or down-regulation) of the different genes/RNAs/proteins by other genes/RNAs/proteins.
The complexity of biological signaling and the prevelance of prior information in the form of pathway knowledge demand that genetic regulatory network models consistent with pathway information be developed. Motivated by this, we developed an approach to generate Boolean network models consistent with given pathway information and applied it to studying the p53-mediated DNA damage stress response [5]. In addition, we used a signaling diagram of the MAP-Kinase pathways to predict possible location(s) of the single signaling breakdowns, based on the cancer-causing breakdown signature [6]. Moreover, we also made theoretical predictions of the efficacy of different combination therapies involving six anti-cancer drugs, which we plan to validate in the near future.
In this paper, we first develop a Boolean network model consistent with oxidative stress response pathway information from the biological literature. Thereafter this model is linked with the PI 3k/Akt pathway and the composite model is used to pinpoint the possible fault locations based on the observed deviations in the apoptotic signature over different time windows. The paper is organized as follows. Section contains a brief general description of Stress Response Pathways while Section presents a discussion specific to the case of oxidative stress. The Boolean network model for oxidative stress response is developed in Section. The role of mitochondria as the site in a cell where the oxidative stress is generated is discussed in Section. In Section, we develop an integrated network linking oxidative stress response to the phenomenon of apoptosis via the PI 3k/Akt pathways. Section presents an approach for pinpointing fault locations in the integrated network by observing the apoptotic signature in response to certain test stress input sequences. Finally, Section contains some concluding remarks.
Stress response pathways
General scheme of stress response pathways. This figure explains the general flow of information in stress response pathways.
Oxidative stress response pathways
Nrf2 Activation. Explains how Nrf2 is activated and how it is able to neutralize the free radicals.
Nrf2 Deactivation. Explains how after neutralizing free radicals Nrf2 is transported back to cytoplasm from mitochondria.
One of the byproducts of normal metabolism is the production of a large number of free radicals. Oxidative stress is caused by the production of free radicals in quantities beyond those that can be handled by the cellular antioxidant system. Indeed, oxidative stress has been implicated in the development of many age-related diseases, including neurodegenerative ones, such as Alzheimer's and Parkinson's, and in aging itself. In addition, excess free radicals react with the nucleotides in the DNA resulting in mutations in the long run. Although there are cellular mechanisms to sense and repair the oxidative DNA damage, mutations can accumulate over a period of time and result in a major disease like cancer. In the next section, we develop a Boolean network model for oxidative stress response pathways. This network will be later utilized to analyze different failure modes that can supress apoptosis and possibly lead to cancer.
Boolean network modeling of oxidative stress response pathways
Before proceeding to the actual modeling of the specific oxidative stress response pathways, we first formally define the general terms 'pathway' and 'Boolean Network' following the detailed development in [5]. Given two genes/proteins A and B and binary values a, b ∈ {0, 1}, we define the term pathway segment to mean that if gene/protein A assumes the value a then gene/protein B transitions to b in no more than t subsequent time steps. A pathway is defined to be a sequence of pathway segments of the form .
A Boolean Network (BN), ϒ = (V, F ), on n genes is defined by a set of nodes/genes V = {x1, ..., x n }, x i ∈ {0, 1}, i = 1, ..., n, and a list F = (f1, ..., f n ), of Boolean functions, f i : {0, 1} n → {0, 1}, i = 1, ..., n [18]. The expression of each gene is quantized to two levels, and each node x i represents the state/expression of the gene i, where x i = 0 means that gene i is OFF and x i = 1 means that gene i is ON. The function f i is called the predictor function for gene i. Updating the states of all genes in ϒ is done synchronously at every time step according to their predictor functions. At time t, the network state is given by x(t) = (x1(t), x2 (t), ..., x n (t)), which is also called the gene activity profile (GAP) of the network.
Oxidative Stress Response Pathways. The major pathways involved in oxidative stress response.
Karnaugh Maps for Deriving the Oxidative Stress Response Boolean Network. K-map simplification for all the elements involved in the system.
From the pathways described above and using the Karnaugh-map reduction techniques, the Boolean update equations for each node of the network are deduced. Some logical reasoning has been used for determining the equations: 1) the maximum number of predictors for updating a variable is fixed to be 3; 2) Small Maf Protein is assumed to be ubiquitously expressed and the pathway given by Eqn.(10) only increases the concentration of SMP, which in conjunction with Eqn.(9), binds to ARE and down-regulates the antioxidant gene; 3) a gene being turned on implies that the corresponding protein is being produced although, in reality, this is not necessarily true; and 4) in the case of a conflict in the Karnaugh map, biological knowledge has been used to assign either a 0 or a 1. This last point is demonstrated by a specific example. For instance, in the case of ARE, the entry shown with a grey circle around it says that when both Bach 1 and Nrf 2 are upregulated and antioxidant gene is downregulated, then at the next time step antioxidant gene will be upregulated. The biological explanation for such an update is that it corresponds to the situation where, in the presence of Stress, Nrf 2 has been activated and is relocating to the nucleus while the inhibitor Bach 1 is simultaneously relocating to the cytoplasm prior to the activation of antioxidant gene at the next time step. Such intuitive reasoning has been used to model the system here. One might use a different reasoning which could lead to a different set of update equations. However, since we are concerned only about the final steady-state behavior, such reasoning can be justified as long as the overall system behavior, defined by the update equations, matches the steady-state. As an example, the final update equation for ARE is derived as follows. In the K-maps, the ones are grouped up in pairs of 2,4,8 and so on and each group should have at least one variable staying constant. So for this case there are two groups whose equations correspond to and . The final update equation for ARE is the sum of these two equations. Please refer to Additional file 1 for some additional details. Indeed, by working with different sets of update equations, we determined that all biologically plausible ones led to the same/similar attractor behavior. From the set of possible Boolean networks we chose the ones that appealed most to our biological understanding and the resulting update equations are given below:
Equivalent Boolean Network for Oxidative Stress Response. Boolean network model for oxidative stress response based on the equations derived using K-maps.
The Boolean State Transition Diagram when the Stress input is 0. The state transition diagram for the Boolean network with no stress on the system. This gives us an idea of the attractor states of the system.
The Boolean State Transition Diagram when the Stress input is 1. The state transition diagram for the Boolean network with stress on the system. This gives us an idea of the attractor states of the system.
It is clear from the preceding discussion that some kind of oscillatory behavior of the genes will be observed when the external Stress input equals 1. On the other hand, when the Stress input equals 0, the system will rest in only one state meaning that there will be no oscillation.
Time domain simulation results
Time response behaviour of the system in Fig.4. Time response simulation of the Boolean network to observe oscillations of the proteins in the system.
Mitochondria and free radical generation
Stages of Oxidative Phosphorylation producing free radicals. Explains Krebs cycle and how and where free radicals are produced in the mitochondria.
Even though it has been long recognized that increased ROS production in mitochondria leads to genetic instability and progression of cancer, there remain several unanswered questions regarding the complex signalling capacity of this organelle [27]. The DNA is highly susceptible to free radical attacks. Free radicals can break DNA strands or delete bases. These mutations can prove to be carcinogenic. It has been estimated that more than 10,000 hits of oxidative stress occur each day. So it is important to tackle these free radicals at the source of their generation, which is why the mitochondria is also a very rich source of anti-oxidants. Although cellular mechanisms can tackle this stress, damage accumulates with age. At present altered energy metabolism is considered to be an additional hall mark of cancer progression [28] and these metabolic pathways have been investigated as targets for cancer therapy. In this paper, we will specifically focus on the PI 3k/Akt pathway which is one such pathway and is described in the following section.
An integrated network for oxidative stress response and apoptosis
Pathway Diagram of Oxidative Stress along with PI3k/Akt. Inclusion of PI3k/Akt pathways along with oxidative stress pathways and study how they can lead to aberrant be-haviour in cells.
Boolean Network modeling of Fig. 11. A Boolean network model of the network along with PI3k/Akt pathways.
Classification of faults in the integrated network
In the integrated pathway diagram of Figure 11, the two genes namely Bad and Bcl 2 are instrumental in deciding the fate of the cell. The preferred status of the two genes, when oxidative stress is not being neutralized, are 1 and 0 respectively since it corresponds to the situation where the pro-apoptotic factor is turned ON and the anti-apoptotic factor is turned OFF. Although a deviation from this state may not signal that the cell is turning cancerous, there is a higher possiblity of the cell exhibiting aberrant behaviour.
Depending on the final resting status of these two genes, one may be able to characterize the degree of invasiveness of the disease especially if it is being caused by apoptosis supression. Once it has been determined that a cell is exhibiting abberant behavior, one would like to pinpoint the location of the fault/error so that the necessary therapeutic intervention(s) can be applied. Since the digital circuit model of Figure 12 uses logic gates, it should be possible to use the fault detection techniques from the Digital Logic literature [37, 38] to pinpoint the fault locations. This will be carried out in this section. An important difference between the results obtained in Layek et al. [6] for pinpointing the fault locations in the MAPKinase pathways and the results to be presented here is that the digital circuit in Figure 12 involves feedback and its behaviour is, therefore, much more complicated to analyze. However, it should be pointed out that the simpler fault pinpointing methodology presented in Layek et al. [6] is much more amenable to biological validation via appropriately designed experiments while the same cannot be said about the results to be presented here. Indeed, the results to be presented here show that the pinpointing of the fault locations is theoretically possible even in this case, although the biological feasibility of the methods required is open to question.
We note that the faults in a digital circuit are mainly of two types [37]:
-
Stuck-at Faults: As the name implies, this is a fault where a particular line l is stuck at a particular value α ∈ {0, 1}, denoted by line l,s-a-α (s-a-α means stuck-at-α). This means that the value at that line is always going to be α regardless of the inputs coming in. This can be thought of as something similar to a mutation in a gene, where a particular gene is either permanently turned ON or OFF.
-
Bridging Faults: This is the type of fault where new interconnections are introduced among elements of the network. This can be thought of as new pathways being created in the cell. This type of fault is not considered in the current paper due to the lack of biological knowledge about new pathways being introduced.
Here, it is appropriate to mention that the biological relevance of each of these two types of faults has been discussed in Layek et al. [6].
The digital circuit in Figure 12 has feedback (shown in bold lines) and is, therefore, a sequential circuit. To detect a fault in a sequential circuit we need a test sequence. Let T be a test sequence and let R(q,T) be the response of the fault-free sequential system N starting in the intial state q. Now let the faulty sequential circuit be denoted by N f where f is the fault. Let us denote by R f (q f ,T) the response of N f to T starting in the initial state q f . A test sequence T detects a fault f iff (if and only if or equivalently this condition is both necessary and sufficient) for every possible pair of initial states q and q f , the output sequences R(q,T) and R f (q f ,T) are different for some specified vector t i ∈ T. The output being observed is the status of [Bad, Bcl 2].
Block Diagram Representation of Fig.12. A simple description of the system showing clearly the feedback lines in the system.
The purpose of this procedure is to pinpoint the possible locations for the fault f in N f , given the output sequence of Bad and Bcl 2 for the normal and faulty circuits. It is assumed that we have no knowledge about the initial status of any of the genes. Knowledge of the initial status of the internal states is important as all future computations are based on these values. The Homing sequence is an initial input sequence that brings the network to a known internal state. So, once the Homing Sequence is given to N and N f , N will come to a known internal state. Note that a similar claim cannot be made about N f as the fault f is not known apriori. For the circuit in Figure 13, a possible Homing sequence is [0 0 0 0 0 0 0 0], which brings the internal state of the system to [0 1 0]. This means that if the Stress input is zero for eight time steps, then at the end of that period, the internal state of the system becomes [0 1 0], regardless of the initial status of any of the genes in the network. A reason for choosing this Homing sequence is that it implies that no input needs to be given to the system and it evolves to the indicated internal state. In future when we are trying to validate these results experimentally this will be of immense help. If we refer back to Figure 7, we see that regardless of the initial state, within four time steps the trajectory reaches the state ('010010') where ARE = 0 and Keap 1 = 1. This is consistent with the conclusion that we are getting from the Homing sequence here with the only difference that a slightly longer sequence is required here as the state transition diagram has a higher cardinality than that in Figure 7.
Fault Detection using Time-Frame Expansion. Fault detection in the boolean(digital) network using time-frame expansion method.
Test Sequences for detecting single stuck-at-faults. The test sequence which can be given to system to find out single stuck-at-faults based on output signature.
Concluding remarks
In this paper, we have developed a Boolean network model for the oxidative stress response. This model was developed based on pathway information from the current literature pertaining to oxidative stress. Where applicable, the behaviour predicted by the model is in agreement with experimental observations from the published literature. It is our hope that some of the additional predictions here, such as those pertaining to the oscillatory behaviour of certain genes in the presence of oxidative stress, will be experimentally validated in the near future.
We have also linked the oxidative stress response to the phenomenon of apoptosis via the PI 3k/Akt pathway. An integrated model based on collectively considering the PI 3k/Akt pathways and the oxidative stress response pathways was developed and then used to pinpoint possible fault locations based on the Bad-Bcl 2 apoptotic signatures in response to 'test' oxidative stress inputs. The approach used to achieve this differs significantly from the earlier results in Layek et al. [6] since the Boolean network of this paper has feedback. The approaches used here and in Layek et al. [6] could potentially have a significant effect on cancer therapy in the future as pinpointing the possible fault location(s) in cancer could permit the choice of the appropriate combination of drugs (such as kinase inhibitors) for maximum therapeutic effectiveness. Of course, it should be pointed out that the theoretical procedure presented here for pinpointing fault locations in a biological network with feedback will need to be further simplified before it can be even considered for practical biological validation.
Declarations
Acknowledgements
Based on “Modelling oxidative stress response pathways”, by Sriram Sridharan, Ritwik Layek, Aniruddha Datta and Jijayanagaram Venkatraj which appeared in Genomic Signal Processing and Statistics (GENSIPS), 2011 IEEE International Workshop on. © 2011 IEEE [39].
This work was supported in part by the National Science Foundation under Grants ECCS-0701531 and and ECCS-1068628 and in part by the J. W. Runyon, Jr. '35 Professorship II Endowment Funds at Texas A & M University.
This article has been published as part of BMC Genomics Volume 13 Supplement 6, 2012: Selected articles from the IEEE International Workshop on Genomic Signal Processing and Statistics (GENSIPS) 2011. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/13/S6.
Authors’ Affiliations
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