Simulating Quantitative Cellular Responses Using Asynchronous Threshold Boolean Network Ensembles
© Jack et al; licensee BioMed Central Ltd. 2011
Received: 20 December 2010
Accepted: 11 July 2011
Published: 11 July 2011
With increasing knowledge about the potential mechanisms underlying cellular functions, it is becoming feasible to predict the response of biological systems to genetic and environmental perturbations. Due to the lack of homogeneity in living tissues it is difficult to estimate the physiological effect of chemicals, including potential toxicity. Here we investigate a biologically motivated model for estimating tissue level responses by aggregating the behavior of a cell population. We assume that the molecular state of individual cells is independently governed by discrete non-deterministic signaling mechanisms. This results in noisy but highly reproducible aggregate level responses that are consistent with experimental data.
We developed an asynchronous threshold Boolean network simulation algorithm to model signal transduction in a single cell, and then used an ensemble of these models to estimate the aggregate response across a cell population. Using published data, we derived a putative crosstalk network involving growth factors and cytokines - i.e., Epidermal Growth Factor, Insulin, Insulin like Growth Factor Type 1, and Tumor Necrosis Factor α - to describe early signaling events in cell proliferation signal transduction. Reproducibility of the modeling technique across ensembles of Boolean networks representing cell populations is investigated. Furthermore, we compare our simulation results to experimental observations of hepatocytes reported in the literature.
A systematic analysis of the results following differential stimulation of this model by growth factors and cytokines suggests that: (a) using Boolean network ensembles with asynchronous updating provides biologically plausible noisy individual cellular responses with reproducible mean behavior for large cell populations, and (b) with sufficient data our model can estimate the response to different concentrations of extracellular ligands. Our results suggest that this approach is both quantitative, allowing statistical verification and calibration, and extensible, allowing modification and revision as guided by experimental evidence. The simulation methodology is part of the US EPA Virtual Liver, which is investigating the effects of everyday contaminants on living tissues. Future models will incorporate additional crosstalk surrounding proliferation as well as the putative effects of xenobiotics on these signaling cascades within hepatocytes.
Thousands of chemicals are used in commerce and evaluating their public health risk remains a challenging problem [1, 2]. Much of our knowledge about mechanisms of toxicity is based on evidence from in vivo animal studies and in vitro experiments, where we can begin to unravel some of the molecular signaling and transcriptional changes induced via chemical perturbation; however, there are three main issues in translating these findings to humans. First, it is often impractical to design experiments with sufficient power to detect the subtle effects of very low environmentally relevant exposures . Hence most observations about chemical effects are at relatively high concentrations that cannot be easily used to quantify the risk of long-term and low-dose exposure to complex mixtures of chemicals . Second, since the molecular response to chemicals is not always conserved between species , the effects observed in rodents cannot be directly extrapolated from rodents to humans without additional mechanistic insight . Third, toxicity is a tissue level phenomenon arising from the behaviors of heterogeneous cell populations. Understanding the complex signaling processes between these different cell types is crucial in determining toxicity potential. We are building a cell-based tissue model to estimate the quantitative population-level effects of chemical exposures [7, 8]. Here we describe an asynchronous threshold Boolean network (BN) approach to model signal transduction in individual cells and to estimate tissue level responses using an ensemble of BNs.
A BN describes a signaling network as a digital circuit in which logical elements (proteins or genes) are either 'ON' or 'OFF'. The temporal evolution of the signaling network is calculated using a set of Boolean functions (AND, OR, NOT) to model regulatory interactions. Since they offer a biologically relevant and computationally efficient formalism for analyzing the relationship between molecular network topology and function, BNs have been used extensively to simulate the behavior of cells based on their network activity. Genetic regulatory networks have been particularly amenable to this formalism due to the binary nature of gene activation . The availability of large-scale transcriptional profiles spurred more recent applications of deterministic  and probabilistic  BNs for reconstructing and simulating genetic regulatory networks. Additionally, BNs have been used for modeling the cell cycle [12–15]; cell proliferation [16–18]; apoptosis ; and cancer . BNs can be used to represent the binary activity of molecular species across cell populations (in vitro and in vivo). One of the limitations of BNs is that they cannot readily estimate continuous functional responses, i.e., quantitative dose-response, which are fundamentally important in pharmacology and toxicology.
Liver cancer is a frequent outcome in testing the long-term effects of chemicals in rodents  and often considered in regulatory decisions . Since the mechanisms of carcinogenesis are poorly understood, it is difficult to translate chemical effects from rodents to humans. Cancer is believed to arise due to a breakdown of the homeostatic processes that maintain balance between cell death and division . Some chemicals (called mutagens) can alter cell phenotypes by damaging DNA resulting in harmful mutations that can spur the activation of oncogenes. Nongenotoxic carcinogens, on the other hand, can act via insidious mechanisms that suppress apoptosis or to stimulate cell proliferation.
It has been suggested that nongenotoxic carcinogens may increase hepatocyte proliferation by perturbing the crosstalk network regulated by growth factors and cytokines . Crosstalk refers to the sequence of protein regulation activated by any one growth factor or cytokine ligand overlapping with the sequences of other ligands, which allows for complex behavior. The presence of crosstalk allows a cell to behave as a multiplexer, integrating multiple signals to select from many possible outcomes, such as cell cycle initiation and progression.
A number of computational models have been proposed for simulating cell proliferation [12–18, 25], however, BNs have not been extensively used in modeling chemical induced toxicity or in hepatic biology. In order for a chemical to produce a chronic or acute tissue level effect, there must be some level of perturbed protein activity in the signal transduction of one or more cells. We are evaluating BNs for modeling early molecular signaling events in hepatocytes that lead to proliferative changes, which are key events in carcinogenesis. Hence, our initial objective is to model some of the normal cues, i.e. homeostatic processes, that can stimulate healthy, quiescent hepatocytes (G0) into entering the cell cycle (G1).
Technological advancements such as flow cytometry and high content screening have made it possible to measure protein levels with single cell resolution. Experimental observations suggest that protein levels within cells may exhibit a switch-like 'all' or 'nothing' ('ON' or 'OFF') response - for example, p53 response to DNA damage [26, 27], TNFα stimulation [28, 29], MAPK signaling events , and drug treatment . These types of observations serve as a foundation for the hypothesis that a Boolean representation is sufficient for describing the molecular multiplicities of individual cells in a simulation framework. Next, we assume that the aggregate activity of molecules across a population of hundreds, thousands or millions of cells can be used to estimate tissue level responses.
Our work is based on two extensions of asynchronous BNs, which employ a non-deterministic updating scheme. First, we use threshold functions to calculate the activation of each protein in our model. This technique has been applied to other systems [32, 33], and it provides a simple representation and adjustable parameter for investigating the interactions between signaling molecules. Second, we model an ensemble of BNs to simulate the quantitative responses of thousands of cells. As such, we can estimate dose dependent responses of cell populations. We defined the topology of the BN semi-automatically using structured information about canonical signaling network from a public pathway repository. Here we describe our initial results on the reproducibility of asynchronous threshold Boolean network ensembles and their potential utility for estimating quantitative time- and concentration-dependent biological responses.
Cell Signaling Network Construction
Pathway Information for Building Crosstalk Model
Epidermal Growth Factor (EGF)
Insulin-like Growth Factor Type 1 (IGF-1)
Tumor Necrosis Factor alpha (TNFα)
Merge Four Pathways
Subgraph: all paths from receptors to c-Jun or c-Fos
Final network: remove with in degree = 0
Simulating individual cellular responses
We used the biochemical interaction network in Figure 1 to describe the response of an individual hepatocyte to the growth factors (EGF, IGF-1 and INS) and the inflammatory cytokine (TNFα). In order to simulate the dynamics of signal transduction, we translated the biochemical interaction network into a threshold BN. As in a traditional BN approach, we assumed that: (a) proteins in the network are described by one of two states, active (ON) or inactive (OFF) and, (b) interactions result in either the activation or inhibition of output proteins by input proteins. Our approach deviates from traditional BNs in three important ways. First, we replace the logical operators with an integer summation function that incorporates an activation threshold. This allows us to adjust the activating potential of each protein in the network. Second, we employ a nondeterministic, asynchronous updating scheme (see Methods), which can simulate biological 'noise' observed in protein signaling cascades. Third, we provide a method for using Hill functions for calibrating the probability of activation for proteins in the network, which can be calibrated with concentration-response data.
Simulating cell population responses
In order to estimate the quantitative response to treatments, we assume that cell populations can be modeled as an ensemble of asynchronous BNs. This allows us to estimate the dynamic response across a simulated biological sample as the aggregate activity of each protein across thousands of BNs (see Methods). Hence, an ensemble of BNs can be considered a simulated 'replicate' as illustrated in Figure 2(B). We investigated the response of an ensemble of 1000 asynchronous BNs to treatment with INS (including the BN depicted in Figure 2(A) until all BNs reached a steady state. The resulting aggregate activity profiles of IRS, c-Jun, c-Fos, and ERK1/2 are shown in Figure 2b (right panel). These trends captured by the simulated BN ensembles appear similar to experimental data from molecular assays performed on in vivo and in vitro replicates (which contain a large number of cells). While this requires additional quantitative and mechanistic evaluation, it is important to note that such continuous protein activity profiles cannot be generated using traditional BNs. Before further evaluation with experimental data we analyzed the reproducibility of our approach with respect to the network depicted in Figure 1.
Reproducibility of Protein Activity Profiles
We systematically evaluated the reproducibility of the asynchronous BN ensembles of the model shown in Figure 1 by analyzing their response to different treatment conditions. For each treatment condition, we simulated 100 replicates with 1000 cells per replicate (i.e., 100,000 cells per treatment condition). Each treatment condition is defined by combinatorial stimulation of the four extracellular ligand cues: (i) EGF, TNFα, IGF-1, and INS combined; (ii) TNFα and INS; (iii) TNFα and IGF-1; (iv) TNFα and EGF; (v) EGF only; (vi) IGF-1 only; (vii) INS only; (viii) TNFα only; (ix) and no active extracellular ligands (the control group). We assumed that each cell is exposed to enough ligand in order to activate a sufficient number of receptors for signal propagation. Hence, for each of the simulated treatment conditions 100% of the cells receive stimulation. Moreover, following the logic of Boolean abstraction of protein concentration, we assumed that the ligand is switched 'ON' in every cell upon initialization.
In Figure 4(B), we show the distribution of the CV for the steady state protein activities across all treatment groups as a box and whisker plot. Similarly, Figure 4(C) shows the reproducibility across the proteins for each treatment group. Whereas the heatmap of Figure 4(A) shows information on the CV per protein per treatment condition, the plots in Figures 4(B) and 4(C) visualize the overall behavior of the model across each treatment condition and protein, respectively. We found that treatment conditions (iii), (vi), (viii) and (ix) - IGF-1 and TNFα, IGF-1 only, TNFα only, and the untreated control group - were the least reproducible in comparison to all other treatment conditions. For the control group, a possible explanation for the reproducibility result is that the median activity of proteins in the control group is very low. As a result, signaling molecules other than ligands and receptors have a very low probability of being active at initialization, in order to simulate a background level of hepatocyte proliferation. This very low mean value has the effect of inflating the CV. In the case of IGF-1 or TNFα, the low reproducibility could be due to the inclusion of fewer reactions than the two other growth factors (EGF and INS). Hence, stimulating with IGF-1 or TNFα may not sufficiently stimulate the individual BNs for the entire ensemble to synchronize in response to treatment. Similar logic governs the model reproducibility following combined stimulation with IGF-1 and TNFα. These results also help to illuminate the sensitivity of our simulation approach to the topology of the signaling network. Importantly, the key endpoint of the model, AP-1 formation, is very reproducible across all treatment conditions.
Comparison with Experimental Data
We used experimental data on primary hepatocytes in culture  for a preliminary evaluation of our simulation approach and putative crosstalk model. In this experiment, rat primary hepatocytes were treated with varying concentrations of EGF and/or TNFα, and then the proportion of cells entering S phase (DNA Synthesis) was measured using Bromodeoxyuridine (5'-bromo-2'-deoxyuridine, BrdU). Although we do not explicitly model S phase in our network, the formation of the AP-1 transcription factor complex is believed to precede S phase in cell cycle progression. Hence, we assumed the formation of AP-1 to be a potential surrogate of S phase and, therefore, correlated with BrdU incorporation. We adjusted the probabilities of activation for proteins in our network in order to closely approximate the levels of BrdU incorporation in the absence of any treatment (control group). Further details on the calibration of the model are described in the Methods section.
We simulated the effects of different treatments on AP-1 formation.
BNs have been used extensively to model the dynamics of molecular signaling and genetic regulatory networks. Because they require the discretization of molecular activity levels, however, a BN cannot be readily used to model the continuous concentration- and time-dependent effects of treatments. To address this issue we extended traditional BNs in three ways. First, we model the molecular response of an individual cell using a BN. Hence, we assumed that the average activity of signaling molecules in individual cells exhibits a switch-like (ON or OFF) response. Although this may not always be the case, we believe it is more biologically plausible than a binary representation of tissue level molecular activities. Second, we assumed that cellular response to stimulation is nondeterministic. It has been suggested synchronous updating schemes for simulating BNs can produce spurious attractors that disappear in the presence of noise [43, 44]. An asynchronous updating scheme, on the other hand, allows for variability (or noise) in cellular responses: two identical cells with the same initial configurations respond differentially to the same stimulus over time. Although asynchronous updating is not new in BNs , we define a probability of activation for each molecular species that is amenable to calibration and evaluation using experimental data. Third, we simulate an ensemble of asynchronous BNs to estimate the aggregate activity of each molecular species across cell populations. Our results show that this approach produces continuous responses similar to experimental observations from tissues. We believe this opens up new possibilities for estimating quantitative dose- and time-dependent responses in toxicology and in disease progression using knowledge of molecular mechanisms.
For this work, we used this simulation methodology to analyze the dynamics of a specific biochemical interaction network, which was constructed to investigate early molecular events surrounding hepatocellular proliferation. This is important because sustained cell proliferation is one of the key events in the progression of liver cancer. We find that (a) our extension of BNs yields highly reproducible results that have variability consonant with biological data and (b) our pathway-driven preliminary cytokine and growth factor protein signaling network is concordant with experimental observations on DNA synthesis in hepatocytes.
We investigated the effects of protein deletion from the network. In Additional file 1, we show a heatmap of these results. From this analysis, we believe our network is robust to the deletion of single proteins. Additionally, this information helps illustrate important signaling nodes in the network. For example, ERK1/2 and JNK appear to play important roles, since the removal of these changes the steady state values of other signaling molecules. To our surprise, the removal of IRS activity did not have a large effect on the signaling processes relative to other experimental conditions, even though the signaling molecule has a high connectivity in the graph.
A number of formalisms have been used to model the dynamics of eukaryotic cell cycle initiation/progression. Tyson and coauthors used ordinary differential equations (ODEs) to describe key cell division in frog oocytes . Zielinski and colleagues used fuzzy Boolean logic to simulate receptor mediated crosstalk preceding cell proliferation in SKOV3 human epithelial cell line . Similar fuzzy models have been proposed by others . One advantage common to both techniques - i.e., ODEs and fuzzy logic - is that they can represent continuous or multivalued treatment concentrations. While these methods are powerful, they predict the population behavior of molecular species without emphasizing individual cellular protein activity. Our objective is to model the heterogeneous response of cell populations in order to estimate histological effects due to different treatments, which necessitates a modeling paradigm with individual cell clarity.
In order to help unravel the mechanisms of toxicity, we are compelled to investigate a simulation framework with a strong emphasis on network topology and reduced parameter space. ODE-based methods involve rate constants which are often difficult to quantify. Testing perturbations to systems of ODEs is not always straightforward with limited data. With BN dynamics, we lose some resolution of time but significantly reduce the number of parameters. We believe our technique is amenable to high throughput modeling of perturbations over a diverse chemical landscape, where the calibration of parameters can be limited due to scarce data on large numbers of environmental chemicals. We hope the modeling framework proposed will be useful in testing chemical perturbations from HTS data to generalized networks on signaling events assessed at the tissue level.
A comparable modeling framework was proposed  which also considered populations of networks. There are several differences in the modeling approaches. First, we use a binary representation of protein activity in a single cell while  used a ternary description to capture the level of protein expression relative to a population average under normal conditions. An ON/OFF representation of protein activity may not always be sufficient (e.g., caspase 8 activity) but it is generally consistent with single cell level observations (e.g. flow cytometry or high-content screening). The ternary representation used by  is based on western blot data on individual proteins with a comparison between treatment and control groups. While a mathematical transformation could relate one approach to another, the two techniques use a different abstraction for describing single cell biology.
Second, there is a subtle distinction between the abstraction of population level behavior between the two approaches. The authors of  calculated the average behavior of proteins across a set of BN whereas we use summation. This allows us to compare dose-dependent differences in potency and efficacy between treatments. Furthermore, it also enables the quantitative evaluation of population level "up-regulation" or "down-regulation" between treatment and control groups without using a ternary representation.
Third, the authors of  evaluate the effects of knockouts by maintaining some of the proteins in at 'control' or 'below control'. On the other hand, we consider the effects of dose-dependent perturbations in protein activities (e.g. extracellular ligands or intracellular signaling molecules) by using Hill functions to define probability of activation for certain protein(s) across the cell population. To our knowledge, no one has used this approach to incorporate concentration-response data in a BN modeling framework. The ability to reproduce and predict concentration-response data is essential for toxicological applications, bridging data from toxicity studies with systems biology to anticipate adverse outcomes.
We did not consider edge weights as an adjustable parameter for the model. All edges are weighted equally (set to 1.0). Modifying the edge weights would change the dynamics of the simulation. For example, in , the authors use a sigmoid function of the weighted sum to determine the probability per node in the propagation of the signal. Their technique offers a unique method for additional stochasticity to a threshold modeling framework. Modifying edge weights would require careful consideration of the updating scheme (describe in Methods), which would be affected if the edge weights were allowed to vary throughout simulation.
For the crosstalk network model investigated in this paper, randomly increasing the threshold of a single molecule from an integer to a noninteger value has little effect on the steady state levels of AP-1 (results not shown). These changes can effect the activity profile of the proteins over time - allowing sustained (threshold = 0.0) or transient activity (0.0 < threshold < 1.0) after activation, or recapitulating protein lability and potential loss of signal. For example,in the case of EGF stimulation, the receptor has negative feedback encoded in the model. Therefore, increasing the thresholds of proteins downstream of EGFR to noninteger values affects the overall activity profiles of the molecules across time (results not shown). Increasing thresholds by integer values can have the same effect as node deletion when the number of activating (input) molecules is equal to the threshold. In general, increasing the threshold increases the required input value for signal propagation.
There are some important limitations in our approach. First, the model presented in this work did not consider any communication between the cells (autocrine or paracrine signals), or between the cells and the extracellular matrix. Contact inhibition and matrix attachment, and cell communication are important factors in cell cycle progression and we are considering their role in a cell-based model of liver tissues . Second, we also recognize that the biochemical interaction network we used in this work is incomplete. Our primary goal was to develop a flexible modeling approach that can incrementally accommodate additional mechanistic information as it becomes available.
The BN used in this work was constructed semi-automatically from a molecular network topology, which was defined with little manual intervention using curated information on pathways. In future work we will evaluate the effects of additional pathways in the crosstalk network, for example, the interleukin protein family as well as the behavior of mito-inhibitors like Transforming Growth Factor Beta (TGF-β). For this work, we did not investigate alternative hypotheses on the signaling mechanisms of individual hepatocytes. The model development in  provides a method for investigating signaling differences between cell types. Understanding the signaling differences between cell types, such as, kinetic differences, is undoubtedly important in extrapolating cell line in vitro data to acute/chronic in vivo responses. Leveraging information from alternate pathway databases [49–51] should increase the descriptive power of our model, and is critical to decipher the role of molecular crosstalk in cellular responses to endogenous ligands and xenobiotics.
AP-1 is among several transcription factors important for cell cycle initiation/progression. In the current model we did not include gene expression regulated by the activation of transcription factors. Therefore, we chose AP-1 formation as the surrogate for downstream events, since it promotes cell cycle progression through increased expression of proteins, such as cyclin D1 , and antagonizes the function other molecules, such as p53 and p21 . As we continue to develop this model, we can incorporate the activity of additional important transcription factors, such as Nuclear Factor kappa B (NF-κB) and Forkhead Box (FOX) proteins. Finally, most BN based approaches cannot directly relate simulation time steps to physiologic time intervals. This is an open problem in BN simulation and our approach is not immune to this issue, but we hope to address this in future work.
An advantage of asynchronous BN ensembles is that they can be simulated very rapidly. A single simulated treatment group (1000 cells/replicate) can be executed on a computer in minutes, while some of the more complex simulations, such as the reproducibility investigation (Figure 4) can be simulated on the scale of hours. However, since each cell is initialized/simulated to steady state individually, the approach is amenable to parallelization. We believe this efficiency will allow us to simultaneously investigate the role of molecular network topology using background knowledge on quantitative tissue level responses from experimental data.
Systems biology approaches are vital for efficiently analyzing the effects of environmental contaminants on living tissues in order to evaluate the potential risk to public health. We developed asynchronous threshold Boolean network ensembles for translating mechanisms to in vitro and in vivo observations on molecular activity. The reproducibility of our modeling framework was evaluated by systematically analyzing the variability of our predictions across multiple simulations. We also showed that populations of hepatocytes can be simulated in this manner to predict experimentally observed quantitative responses. We believe that ensembles of Boolean networks can allow us to probe deeper mechanistic questions about the mode of action for chronic liver injury. We are testing this modeling approach as part of a broader computational and experimental effort aimed at estimating the putative effects of xenobiotics on the human liver by integrating chemical concentration, molecular pathways, cellular responses, and the role of cell-cell communication.
Threshold Boolean Network
Furthermore, we let and , and let n be the number of proteins in the graph. Additionally, we store the binary vector, , of the state (active or inactive) of every vertex at time τ. The state of a protein is dependent on the states of its predecessors. Therefore, we define a vector, , representing the threshold of activation for each vertex, a biologically inspired variable guiding the interplay predecessor vertices and protein activation. For the model, all thresholds were set to 0.0 with the exception of AP-1 formation, which is set to θAP-1∈ (1,2). This modification to the AP-1 threshold reflects the underlying biochemistry in that both c-Jun and c-Fos must be active for the activation of the AP-1.
Finally, we define the vector , which represents the probability of activation for each protein. For most models/proteins, there is a basal level of activity. We assume that individual BNs can have different protein activity profiles upon initialization (τ = 0), which allows for biological variability across the cell population.
Temporal Evolution of the Boolean Network
We used the BrdU concentration response data  to estimate the parameters for Equation 8. In order to use the Nelder-Mead algorithm to make a maximum likelihood estimation of Vmax, K A and n, we assumed that EGF activity corresponds to AP-1 activation. Although the network was encoded with negative feedback for the EGF receptor, representing the internalization and ubiquitination of this receptor, we make this assumption based on the simulation results with φ EGF = 1.0 (Figure 5a). The maximum likelihood estimates we found are n = 0.7, K A = 5.9ng/ml and Vmax = .28 (probability of activation).
Simulating Populations of Cells
Similarly the steady state activity of a protein across the ensemble is denoted as v T .
Implementation and Input Formats
The simulator is implemented in Python and the network model was produced interactively using Cytoscape. The molecular interaction network topology is defined in the Cytoscape exported format and protein information as well as quantitative parameters can be defined in the node attributes file (e.g., protein name, probabilities of activation, and threshold). The Python code as well as the network model are included (see Additional file 2 and Additional file 3).
This work was completed at the National Center for Computational Toxicology at the US Environmental Protection Agency. We would like to thank R. Woodrow Setzer and David Reif of the EPA and Julio Saez-Rodriguez of European Bioinformatics Insititute for useful conversations and critical review.
The views expressed in this article are those of the authors and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.
- Andersen ME, Krewski D: Toxicity Testing in the 21st Century: Bringing the Vision to Life. Toxicological Sciences. 2009, 107: 324-330.View ArticlePubMedGoogle Scholar
- Holsapple MP, Afshari CA, Lehman-McKeeman LD: Forum Series: The "Vision" for Toxicity Testing in the 21st Century: Promises and Conundrums. Toxicological Sciences. 2009, 107: 307-308.View ArticlePubMedGoogle Scholar
- White RH, Cote I, Zeise L, Fox M, Dominici F, Burke TA, White PD, Hattis DB, Samet JM: State-of-the-Science Workshop Report: Issues and Approaches in Low-Dose-Response Extrapolation for Environmental Health Risk Assessment. Environ Health Perspect. 2008, 117:Google Scholar
- De Rosa C, El-Masri H, Pohl H, Cibulas W, Mumtaz M: Implications of Chemical Mixtures in Public Health Practice. Journal of Toxicology and Environmental Health Part B: Critical Reviews. 2004, 7: 339-350. 10.1080/10937400490498075.View ArticlePubMedGoogle Scholar
- Lawrence JW, Li Y, Chen S, DeLuca JG, Berger JP, Umbenhauer DR, Moller DE, Zhou G: Differential Gene Regulation in Human Versus Rodent Hepatocytes by Peroxisome Proliferator-activated Receptor (PPAR) α. Journal of Biological Chemistry. 2001, 276: 31521-31527. 10.1074/jbc.M103306200View ArticlePubMedGoogle Scholar
- Cohen SM, Lawson TA: Rodent bladder tumors do not always predict for humans. Cancer Letters. 1995, 93: 9-16. 10.1016/0304-3835(95)03785-UView ArticlePubMedGoogle Scholar
- Wambaugh J, Shah I: Simulating Microdosimetry in a Virtual Hepatic Lobule. PLoS Comput Biol. 2010, 6: e1000756- 10.1371/journal.pcbi.1000756PubMed CentralView ArticlePubMedGoogle Scholar
- Shah I, Wambaugh J: Virtual Tissues in Toxicology. Journal of Toxicology and Environmental Health Part B: Critical Reviews. 2010, 13: 314-328. 10.1080/10937404.2010.483948.View ArticleGoogle Scholar
- Kauffman S: Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology. 1969, 22: 437-467. 10.1016/0022-5193(69)90015-0View ArticlePubMedGoogle Scholar
- Zoltan S, Shoudan L: Modeling the normal and neoplastic cell cycle with 'realistic Boolean genetic networks': Their application for understanding carcinogenesis and assessing therapeutic strategies. Book Modeling the normal and neoplastic cell cycle with 'realistic Boolean genetic networks': Their application for understanding carcinogenesis and assessing therapeutic strategies. 1998, City: World Scientific Publishing,Google Scholar
- Shmulevich I, Dougherty ER, Kim S, Zhang W: Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002, 18: 261-274. 10.1093/bioinformatics/18.2.261View ArticlePubMedGoogle Scholar
- Sánchez L, Thieffry D: A Logical Analysis of the Drosophila Gap-gene System. Journal of Theoretical Biology. 2001, 211: 115-141. 10.1006/jtbi.2001.2335View ArticlePubMedGoogle Scholar
- Davidich MI, Bornholdt S: Boolean Network Model Predicts Cell Cycle Sequence of Fission Yeast. PLoS ONE. 2008, 3: e1672- 10.1371/journal.pone.0001672PubMed CentralView ArticlePubMedGoogle Scholar
- Li F, Long T, Lu Y, Ouyang Q, Tang C: The yeast cell-cycle network is robustly designed. Proceedings of the National Academy of Sciences of the United States of America. 2004, 101: 4781-4786. 10.1073/pnas.0305937101PubMed CentralView ArticlePubMedGoogle Scholar
- Faure A, Naldi A, Chaouiya C, Thieffry D: Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics. 2006, 22: e124-131. 10.1093/bioinformatics/btl210View ArticlePubMedGoogle Scholar
- Saez-Rodriguez J, Alexopoulos LG, Epperlein J, Samaga R, Lauffenburger DA, Klamt S, Sorger PK: Discrete logic modelling as a means to link protein signalling networks with functional analysis of mammalian signal transduction. Mol Syst Biol. 2009, 5:Google Scholar
- Samaga R, Saez-Rodriguez J, Alexopoulos LG, Sorger PK, Klamt S: The Logic of EGFR/ErbB Signaling: Theoretical Properties and Analysis of High-Throughput Data. PLoS Comput Biol. 2009, 5: e1000438- 10.1371/journal.pcbi.1000438PubMed CentralView ArticlePubMedGoogle Scholar
- Wu M, Yang X, Chan C: A Dynamic Analysis of IRS-PKR Signaling in Liver Cells: A Discrete Modeling Approach. PLoS ONE. 2009, 4: e8040- 10.1371/journal.pone.0008040PubMed CentralView ArticlePubMedGoogle Scholar
- Philippi N, Walter D, Schlatter R, Ferreira K, Ederer M, Sawodny O, Timmer J, Borner C, Dandekar T: Modeling system states in liver cells: Survival, apoptosis and their modifications in response to viral infection. BMC Systems Biology. 2009, 3: 97- 10.1186/1752-0509-3-97PubMed CentralView ArticlePubMedGoogle Scholar
- Zhang R, Shah MV, Yang J, Nyland SB, Liu X, Yun JK, Albert R, Loughran TP: Network model of survival signaling in large granular lymphocyte leukemia. Proceedings of the National Academy of Sciences. 2008, 105: 16308-16313. 10.1073/pnas.0806447105.View ArticleGoogle Scholar
- Martin MT, Judson RS, Reif DM, Kavlock RJ, Dix DJ: Profiling Chemicals Based on Chronic Toxicity Results from the U.S. EPA ToxRef Database. Book Profiling Chemicals Based on Chronic Toxicity Results from the US EPA ToxRef Database. 2009, 117: 392-399. City: National Institute of Environmental Health Sciences,Google Scholar
- Chhabra RS, Bucher JR, Wolfe M, Portier C: Toxicity characterization of environmental chemicals by the US National Toxicology Program: an overview. International Journal of Hygiene and Environmental Health. 2003, 206: 437-445. 10.1078/1438-4639-00240View ArticlePubMedGoogle Scholar
- Hanahan D, Weinberg RA: The Hallmarks of Cancer. Cell. 2000, 100: 57-70. 10.1016/S0092-8674(00)81683-9View ArticlePubMedGoogle Scholar
- Butterworth BE: Consideration of both genotoxic and nongenotoxic mechanisms in predicting carcinogenic potential. Mutation Research/Reviews in Genetic Toxicology. 1990, 239: 117-132. 10.1016/0165-1110(90)90033-8.View ArticleGoogle Scholar
- Csikász-Nagy A, Battogtokh D, Chen KC, Novák B, Tyson JJ: Analysis of a Generic Model of Eukaryotic Cell-Cycle Regulation. 2006, 90: 4361-4379.Google Scholar
- Lahav G, Rosenfeld N, Sigal A, Geva-Zatorsky N, Levine AJ, Elowitz MB, Alon U: Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat Genet. 2004, 36: 147-150. 10.1038/ng1293View ArticlePubMedGoogle Scholar
- Batchelor E, Loewer A, Lahav G: The ups and downs of p53: understanding protein dynamics in single cells. Nat Rev Cancer. 2009, 9: 371-377. 10.1038/nrc2604PubMed CentralView ArticlePubMedGoogle Scholar
- Tay S, Hughey JJ, Lee TK, Lipniacki T, Quake SR, Covert MW: Single-cell NF-[kgr]B dynamics reveal digital activation and analogue information processing. Nature. 2010, 466: 267-271. 10.1038/nature09145PubMed CentralView ArticlePubMedGoogle Scholar
- Spencer SL, Gaudet S, Albeck JG, Burke JM, Sorger PK: Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis. Nature. 2009, 459: 428-432. 10.1038/nature08012PubMed CentralView ArticlePubMedGoogle Scholar
- Luke N, Devito M, Portier C, El-Masri H: Employing a Mechanistic Model for the MAPK Pathway to Examine the Impact of Cellular All or None Behavior on Overall Tissue Response. Dose Response. 2010, 8: 347-367. 10.2203/dose-response.09-017.LukePubMed CentralView ArticlePubMedGoogle Scholar
- Cohen AA, Geva-Zatorsky N, Eden E, Frenkel-Morgenstern M, Issaeva I, Sigal A, Milo R, Cohen-Saidon C, Liron Y, Kam Z, et al.: Dynamic Proteomics of Individual Cancer Cells in Response to a Drug. Science. 2008, 322: 1511-1516. 10.1126/science.1160165View ArticlePubMedGoogle Scholar
- Derrida B: Dynamical phase transition in nonsymmetric spin glasses. Journal of Physics A: Mathematical and General. 1987, 20: L721-10.1088/0305-4470/20/11/009.View ArticleGoogle Scholar
- Bornholdt S: Boolean network models of cellular regulation: prospects and limitations. Journal of The Royal Society Interface. 2008, 5: S85-S94. 10.1098/rsif.2008.0132.focus.PubMed CentralView ArticleGoogle Scholar
- Robert J, Lefkowitz SKS: IGF-1 Receptor Signaling through betat-Arrestin. Sci Signal. 2005,Google Scholar
- Schlessinger J: Epidermal Growth Factor Receptor Pathway. Book Epidermal Growth Factor Receptor Pathway. 2005, City,Google Scholar
- White MF: Insulin Signaling Pathway. Sci Signal. 2005,Google Scholar
- David V, Goeddel GC: Tumor Necrosis Factor Pathway. Sci Signal. 2005,Google Scholar
- Shannon P, Markiel A, Ozier O, Baliga N, Wang J, Ramage D, Amin N, Schwikowski B, Ideker T: Cytoscape: A Software Environment for Integrated Models of Biomolecular Interaction Networks. Genome Research. 2003, 13: 2498-2504. 10.1101/gr.1239303PubMed CentralView ArticlePubMedGoogle Scholar
- Ayala J, Boustead J, Chapman S, Svitek C, Oeser J, Robey R, O'Brien R: Insulin-mediated activation of activator protein-1 through the mitogen-activated protein kinase pathway stimulates collagenase-1 gene transcription in the MES 13 mesangial cell line. J Mol Endocrinol. 2004, 33: 263-280. 10.1677/jme.0.0330263View ArticlePubMedGoogle Scholar
- Newman JRS, Ghaemmaghami S, Ihmels J, Breslow DK, Noble M, DeRisi JL, Weissman JS: Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nature. 2006, 441: 840-846. 10.1038/nature04785View ArticlePubMedGoogle Scholar
- Chevalier S, Macdonald N, Roberts R: Induction of DNA replication by peroxisome proliferators is independent of both tumour necrosis factor (alpha) priming and EGF-receptor tyrosine kinase activity. J Cell Sci. 1999, 112: 4785-4791.PubMedGoogle Scholar
- Nadori F, Lardeux B, Rahmani M, Bringuier A, Durand-Schneider A, Bernuau D: Presence of distinct AP-1 dimers in normal and transformed rat hepatocytes under basal conditions and after epidermal growth factor stimulation. Hepatology. 1997, 26: 1477-1483. 10.1002/hep.510260614View ArticlePubMedGoogle Scholar
- Greil F, Drossel B: Dynamics of Critical Kauffman Networks under Asynchronous Stochastic Update. Physical Review Letters. 2005, 95: 048701-View ArticlePubMedGoogle Scholar
- Klemm K, Bornholdt S: Stable and unstable attractors in Boolean networks. Physical Review E. 2005, 72: 055101-View ArticleGoogle Scholar
- Zwolak JW, Tyson JJ, Watson LT: Parameter Estimation for a Mathematical Model of the Cell Cycle in Frog Eggs. Journal of Computational Biology. 2005, 12: 48-63. 10.1089/cmb.2005.12.48View ArticlePubMedGoogle Scholar
- Zielinski R, Przytycki P, Zheng J, Zhang D, Przytycka T, Capala J: The crosstalk between EGF, IGF, and Insulin cell signaling pathways - computational and experimental analysis. BMC Systems Biology. 2009, 3: 88- 10.1186/1752-0509-3-88PubMed CentralView ArticlePubMedGoogle Scholar
- Huang Z, Hahn J: Fuzzy modeling of signal transduction networks. Chemical Engineering Science. 2009, 64: 2044-2056. 10.1016/j.ces.2009.01.041.View ArticleGoogle Scholar
- Rotroff D, Beam A, Dix D, Farmer A, Freeman K, Houck K, Judson R, LeCluyse E, Martin M, Reif D, Ferguson S: Xenobiotic-Metabolizing Enzyme and Transporter Gene Expression in Primary Cultures of Human Hepatocytes Modulated by Toxcast Chemicals. Journal of Toxicology and Environmental Health, Part B: Critical Reviews. 2010, 13: 329-346. 10.1080/10937404.2010.483949.View ArticleGoogle Scholar
- Kanehisa M, Goto S, Furumichi M, Tanabe M, Hirakawa M: KEGG for representation and analysis of molecular networks involving diseases and drugs. Nucleic Acids Research. 2010, 38: D355-D360. 10.1093/nar/gkp896PubMed CentralView ArticlePubMedGoogle Scholar
- Joshi-Tope G, Gillespie M, Vastrik I, D'Eustachio P, Schmidt E, de Bono B, Jassal B, Gopinath GR, Wu GR, Matthews L, et al.: Reactome: a knowledgebase of biological pathways. Nucleic Acids Research. 2005, 33: D428-D432.PubMed CentralView ArticlePubMedGoogle Scholar
- Schaefer CF, Anthony K, Krupa S, Buchoff J, Day M, Hannay T, Buetow KH: PID: the Pathway Interaction Database. Nucleic Acids Research. 2009, 37: D674-D679. 10.1093/nar/gkn653PubMed CentralView ArticlePubMedGoogle Scholar
- Bassingthwaighte J, Liebovitch LS, West BJ: Fractal Physiology. 1994, Oxford University Press, New York,View ArticleGoogle Scholar
- Hunt CA, Yan L, Ropella G, Park S, Sheikh-Bahael S: The multiscale in silico liver. Journal of Critical Care. 2007, 22: 348-349. 10.1016/j.jcrc.2007.10.031.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.