Qualitative networks: a symbolic approach to analyze biological signaling networks - PubMed (original) (raw)

Qualitative networks: a symbolic approach to analyze biological signaling networks

Marc A Schaub et al. BMC Syst Biol. 2007.

Abstract

Background: A central goal of Systems Biology is to model and analyze biological signaling pathways that interact with one another to form complex networks. Here we introduce Qualitative networks, an extension of Boolean networks. With this framework, we use formal verification methods to check whether a model is consistent with the laboratory experimental observations on which it is based. If the model does not conform to the data, we suggest a revised model and the new hypotheses are tested in-silico.

Results: We consider networks in which elements range over a small finite domain allowing more flexibility than Boolean values, and add target functions that allow to model a rich set of behaviors. We propose a symbolic algorithm for analyzing the steady state of these networks, allowing us to scale up to a system consisting of 144 elements and state spaces of approximately 10(86) states. We illustrate the usefulness of this approach through a model of the interaction between the Notch and the Wnt signaling pathways in mammalian skin, and its extensive analysis.

Conclusion: We introduce an approach for constructing computational models of biological systems that extends the framework of Boolean networks and uses formal verification methods for the analysis of the model. This approach can scale to multicellular models of complex pathways, and is therefore a useful tool for the analysis of complex biological systems. The hypotheses formulated during in-silico testing suggest new avenues to explore experimentally. Hence, this approach has the potential to efficiently complement experimental studies in biology.

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Figures

Figure 1

Figure 1

Iterative improvement of the model. Schematic view of the iterative improvement process used to build a model which is consistent with the experimental data. The verification process is represented in blue. The improvement of the model based on counter examples is represented in red. The outer improvement loop, based on laboratory experiments, is represented in green.

Figure 2

Figure 2

Model of the gal system in E. coli. A: View of the gal system in E. coli. The activator CRP activates both galETK and GalS. GalS inhibits galETK and negatively auto-regulates itself. GalS inh, which is a shortcut representing both β-D-galactose and D-fucose, cancels the inhibitory effect of Gals S on galETK. B: Evolution of the level of galETK when CRP is activated at time t = 0. The level are represented using relative values with respect to the stable state galETK St. The situation in which no GalS inh is not expressed is represented in blue. This corresponds to an I1-FFL. The situation in which GalS inh is present is represented in red. In this situation, the model behaves like with a simple activation. The response time, measured as the time needed to reach galETK St/2 is shorter in the absence of GalS inh.

Figure 3

Figure 3

Schematic view of the multicellular model. View of the different layers of the mammalian skin, and how they are represented in the multicellular model. Connections through the Wnt pathway are represented in red and connections through the Notch pathway are represented in blue. The level of Notch receptor of each cell is fixed, and the corresponding values are indicated in the cells. Both the dermis and the cornified layer (composed of dead cells) are not represented in the model. The required result for each cell in the case of wild type simulation is indicated below the cell. Cells of the basal layer are proliferating, while cells of the suprabasal layers are differentiated.

Figure 4

Figure 4

Visualization of a single cell. The components of the Wnt pathway are represented in red and the components of the Notch pathway in blue. The canonical Wnt pathway starts from the extracellular level of the short range signaling Wnt protein, which we represent by the Wnt_ext variable. This short range molecule binds to the Frizzled receptor, which leads to an increase in the intracellular level of DSH. We therefore represent the interaction between these molecules by an activation from Wnt_ext to Frizzled and then to DSH. The level of the scaffolding protein Axin is dependent on the level of DSH which acts as an inhibitor and Casein Kinase 1 α (represented by the variable Cask1α), which is assumed to be an activator. β-Catenin is phosphorylated by a complex composed, amongst others, by Axin. We separately represent the expression level of _β_-Catenin, β-Catenin exp. _β_-Catenin is activated by β_-Catenin_exp and inhibited by Axin. The family of downstream target genes of the canonical Wnt signaling pathway, (GT1), are transcriptionally activated by _β_-Catenin. The Notch pathway is based on the cell-cell interaction between the Notch receptor (whose level is represented by the variable Notch) and the ligands expressed on the membrane of the immediate neighboring cells (Ligand in). Since a successful binding between the receptor and the ligand leads to the cleavage of the intracellular part of Notch (Notch-IC) we use the minimum target function to model this interaction. We represent the level of the downstream target genes of Notch signaling as a single variable (GT2), which is activated by Notch-IC. Amongst these targets is the protein p21, which is activated by Notch-IC and inhibits Wnt. Wnt is an external variable of the cell. The downstream targets of Notch signaling activate the Jagged ligand.

Figure 5

Figure 5

Arbitrary pathways for performance evaluation. This cell contains arbitrary chosen pathways that have no biological meaning. The interactions between components in this example were chosen in order to obtain a complex transition function. The component A ext is connected to the component I of the neighbor cells. The component L ext is connected to the component H of the neighbor cells.

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