4.1. Canonical Wnt Pathway (CWP)

Parameter identification/tuning in system biology usually involves an iterative process to develop a model that is capable of replicating a set of in vitro/in vivo experiments, which in turn reinforces the model’s reliability, while providing a comprehensive understanding of complex biological systems. While some parameters can be directly obtained from published experimental data, others need to be inferred and finally tested by means of experimental assays. The ARVC model herein devised uses both literature information/data and knowledge of the domain experts. Our model focuses on the link between ARVC and two well-characterized signaling pathways: the Wnt/β-catenin and RhoA-ROCK. In the first step of our model, we focused on the canonical Wnt pathway only (Figure 9), for whose development we extended a previously published model based on the prediction of β-catenin regulation by APC and Axin [32].

Scheme of reactions for modeling canonical Wnt signaling.

Reactions from 1 to 19 refer to the original model (Lee et al.). Reactions numbered from 20 to 23 in the blue box represent an extended version of Lee’s model. Reaction 20 describes the inhibition of “adipogenic mRNA” by β-catenin/TCF complex; reaction 21 refers to the regulatory loop of TCF synthesis; Reaction 22 shows the binding of PG to TCF, and Reaction 23 describes the activation of “adipogenic mRNA” by PG/TCF complex.

Figure 9 shows 19 reactions described in Lee’s model: each reaction refers to a specific biochemical event (e.g., protein binding/unbinding, activation/inactivation, transcriptional regulation). Many of the numerical values of the input quantities were experimentally extracted from Xenopus eggs data, while others were deduced from the literature, and the remaining ones were estimated on the basis of the output (degradation of β-catenin), according to the experimental data. The model developed by Lee et al. includes both ordinary differential equations and algebraic ones. Values of the parameters and initial reaction conditions and species present in the original model were adapted in this work. To extend Lee’s original model, we introduced two main mechanisms: (i) the interaction between PG and the β-catenin degradation complex; and (ii) the direct competition of PG with β-catenin for the binding to TCF. The details on the ODE system, all the species and reactions of the WCP-extended model are reported in System A and Tables S1–S3 of the Supplementary Materials, respectively. The mechanism of competition between PG and β-catenin is not yet completely understood, and therefore in our model we considered two different hypotheses: (1) the two proteins only compete to form a heterodimeric complex with TCF; and (2) there is also a competition of the complexes β-catenin/TCF and PG/TCF at the transcriptional level. To test the latter hypothesis, we introduced a generic “adipogenic mRNA” species, which is responsible for the adipogenic program. In basal conditions, the Wnt/β-catenin pathway is active and the complex β-catenin/TCF inhibits the transcription of adipogenic genes. Another crucial difference with respect to Lee’s model is the introduction of a regulatory feedback of TCF expression; in the original model, the concentration of TCF is assumed constant (15nM), while in our model the TCF synthesis is regulated through a positive feedback mediated by β-catenin/TCF dimer. This is an important point, since the Wnt pathway undergoes both positive and negative regulation by the induction of TCF/LEF and Axin synthesis, respectively [14]. Therefore, to investigate the effects of such positive and negative regulation, we implemented our model by introducing a “TCF encoding RNA” species, responsible for TCF production. The mathematical description of such transcriptional mechanisms has been extrapolated from another model available in literature [35]. The negative feedback introduced in our system involves the rate of Axin synthesis, whose induction depends on the availability of β-catenin and β-catenin/TCF complex [36]. In the lack of detailed biochemical information, the kinetics of the binding/unbinding of PG/TCF and the interaction between PG and β-catenin degradation complexes are assumed to be identical to those of β-catenin. The kinetics of “adipogenic mRNA” transcription was assumed to follow a classical biochemical mechanism of two transcriptional factors, competing for the same binding sites to the gene promoter (Figure 10) [37].

Competition of β-catenin and PG for TCF/LEF binding and relative activity. (A) β-catenin and PG compete for the binding to TCF/LEF; (B) when β-catenin/TCF binds to the promoter the transcription of adipogenic genes is inhibited; and (C) PG/TCF complex binds to the promoter of adipogenic genes and mediates their transcription.