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Modeling TGF-β signaling pathway in epithelial-mesenchymal transition
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16. Incidentally, we are now in a position to clarify why the families of fixed points (i), (ii), (iii) are not compatible with the selected initial data. First, let us note that TGF-β and R get progressively consumed at exactly the same pace, see the first two equations of system (7), as they are both implicated in the creation of the R* species. However, at t = 0, the quota of free receptors R is significantly larger than the number of injected TGF-β molecules (R(t = 0) = 1nM vs. TGF-β(t = 0) = 0.113nM). As a consequence, and because R and TGF-β obey to an identical kinetic, it is the TGF-β that vanish first, reaching its asymptotic state TGF-β = 0 when a residual quota of R is still present. From here on, the receptors R cannot be mutated in R* any longer and are therefore indefinitely frozen to the asymptotic value R(t = 0) − TGF-β(t = 0), which is positive and different from zero. Both solutions (i) and (ii) are therefore to be rejected because they require R = 0. A similar reasoning can be invoked to exclude solution (iii). This latter would in fact imply Sc = 0. However, as it can be readily appreciated by inspection of system (7), the rate of loss of both R* and Sc is governed by the same term, namely −kp[R*][Sc]. The maximum amount of bound receptors R* is equal to the number of TGF-β(t = 0) molecules, while the Sc elements at time t = 0 are definitely many more (121.1nM). Moreover, the population of Sc gets also re-integrated, via a source term controlled by the reaction rate kexp, which acts as long as Sn is different from zero. Based on the above, we can therefore conclude that Sc molecules are still present when R* becomes zero. From here on the Sc cannot decrease any longer and stay frozen to the value that they have eventually attained when the condition R* = 0 is met. Hence, solution (iii) cannot apply to the scrutinized setting, as it assumes the asymptotic condition Sc = 0.
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The epithelial-mesenchymal transition (EMT) consists in a morphological change in epithelial cells characterized by the loss of the cell adhesion and the acquisition of mesenchymal phenotype. This process plays a crucial role in the embryonic development and in regulating the tissue homeostasis in the adult, but it proves also fundamental for the development of cancermetastasis. Experimental evidences have shown that the EMT depends on the TGF-β signaling pathway, which in turn regulates the transcriptional cellular activity. In this work, a dynamical model of the TGF-β pathway is proposed and calibrated versus existing experimental data on lung cancer A549 cells. The analysis combines Bayesian Markov Chain Monte Carlo (MCMC) and standard Ordinary Differential Equations (ODEs) techniques to interpolate the gene expression data via an iterative adjustments of the parameters involved. The kinetic of the Smad proteins phosphorylation, as predicted within the model is found in excellent agreement with available experiments, an observation that confirms the adequacy of the proposed mathematical picture.
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