Logical model specification aided by model- checking techniques: application to the mammalian cell cycle regulation

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Logical modeling of the mammalian cell cycle

Proper understanding of the behavior of complex biological regulatory networks requires the integration of heterogeneous data into predictive mathematical models. Logical modeling focuses on qualitative data and offers a flexible framework to delineate the main dynamical properties of such networks. However, formal analysis faces a combinatorial explosion as the number of regulatory components and interactions increases. Here, we show how model-checking techniques can be used to verify sophisticated dynamical properties resulting from model regulatory structure. We demonstrate the power of this approach through the updating of a model of the molecular network controlling mammalian cell cycle. We use model-checking to progressively refine this model in order to fit recent experimental observations. The resulting model accounts for the sequential activation of cyclins, the role of Skp2, and emphasizes a multifunctional role for the cell cycle inhibitor Rb

Mechanistic epithelial model, simulation.

<p>The simulation starts with a naive configuration (i.e. all cells are undifferentiated) upon which Grk, Dpp, and Mid input levels are applied. All components are updated synchronously except dpERK and the integration variables (not shown), which are always updated earlier (see text). From left to right are depicted the successive states of each component in the epithelium (gene expression patterns), before Grk extinction. The right panels show the components' states after Grk extinction. Color intensities are used for multi-valued components (Grk, Rho and EGRF, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003527#pcbi-1003527-g003" target="_blank">Figure 3B</a>). Red frames denote pattern changes.</p

Mechanistic epithelium model, clonal analyses.

<p>(<b>A</b>) Loss-of-function clones. (<b>B</b>) Gain-of-function clones. Before Grk extinction (top row) and after Grk extinction (lower row). Row organization and color codes as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003527#pcbi-1003527-g007" target="_blank">Figure 7</a>. In the Br GOF case, the oscillatory attractor obtained before Grk extinction is due to the synchronous simulation scheme (see text). Here, we show the most consistent pattern of the two stable states resulting from the Grk removal.</p

Phenomenological model: rules and result.

<p>(<b>A</b>) Regulatory graph: the model links three distinct follicle cell fates, Operculum, Floor and Roof, to a combination of input components EGF, BMP, <i>anterior</i>, and Roof_adj. Oval nodes are Boolean (0 or 1) and the rectangular node (EGF) is associated to a multi-valued variable, which here takes values between 0 and 2 (absent, intermediate and high level). EGF directly influences the position of the three domains on the dorsal-ventral axis. BMP establishes the anterior border of the roof, while <i>anterior</i> defines the anterior competence region. Roof_adj is an input variable accounting for the differentiated state of neighboring cells. Green and red edges denote positive and negative effects, respectively. The edge in purple denotes a dual effect, i.e. activating or repressing, depending on the level of its source. (<b>B</b>) Logical functions driving the dynamics of the model: Each rule specifies under which conditions the variable evolves to value 1 (otherwise, the variable tends to 0). The condition of the presence of EGF is simply denoted as “EGF”, and “EGF:1” or “EGF:2” whenever distinction between levels is required. Logical connectors are: & for a conjunction (and), | for a disjunction (or) and ! for a negation (not). (<b>C</b>) Epithelial model: left, patterns for the inputs EGF (yellow), BMP (purple) and <i>anterior</i> (pink) as used during the simulation. Right, final cell fates are shown in green (operculum), blue (floor) and red (roof).</p

Overview of oogenesis in <i>Drosophila melanogaster</i>.

<p>(<b>A</b>) Schematic of an ovariole. Egg chambers, displayed at progressively later stages from anterior (left) to posterior (right), are formed in the germarium, and consist of three main cell types: nurse cells and the oocyte, both germ line, enveloped by a layer of somatic follicle cells (FC). After stage 9, the FCs have remodeled to form a columnar epithelium over the oocyte, and a squamous epithelium over the nurse cells. (<b>B–B′</b>) At early stages, ligand Gurken (Grk; in yellow) co-localizes with the oocyte nucleus to the posterior pole of the oocyte. It signals to EGFR in the overlying FC, activating the EGF pathway in a posterior-anterior gradient. (<b>C–C′</b>) After oocyte repolarization, Grk and the oocyte nucleus are located at the dorsal-anterior cortex of the oocyte. The EGF pathway is locally activated in overlying FC. (<b>D–D′</b>) Dpp ligand produced in the anterior FC establishes a steep anterior-posterior gradient of BMP signaling activity in the columnar FC. (<b>E–E″</b>) The appendage primordia are defined at stage 10 and consist, on either side of the midline, of two groups of cells, roof and floor. The eggshell deposited between the oocyte (Oo) and the follicle cells (FC) contains the operculum (OP), the micropyle (MP), and two dorsal appendages (DA); and is constituted by the vitelline membrane (VM), the inner chorionic layer (ICL), an endochorion (EnC) and an exochorion (ExC) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003527#pcbi.1003527-Marmion1" target="_blank">[31]</a>.</p