<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the <italic>relaxation time</italic> and a measure of <italic>invasiveness</italic>, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be <italic>S</italic>-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called <italic>LeVeque–Yee formula</italic> to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.</p></abstract>

The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.

Phase transitions of biological phenotypes by means of a prototypical PDE model

Communications in Applied and Industrial Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Phase transitions of biological phenotypes by means of a prototypical PDE model"}

The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal...

Phase transitions of biological phenotypes by means of a prototypical PDE model

Published Online: Feb 01, 2020

Page range: 1 - 17

Received: Sep 12, 2018

Accepted: Nov 13, 2019

DOI: https://doi.org/10.2478/caim-2020-0001

Keywords
phase transitions, reaction-diffusion systems, propagating fronts, finite difference schemes, wave speed approximations

© 2020 C. Mascia et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Phase transitions of biological phenotypes by means of a prototypical PDE model

Published Online: Feb 01, 2020

Page range: 1 - 17

Received: Sep 12, 2018

Accepted: Nov 13, 2019

DOI: https://doi.org/10.2478/caim-2020-0001

Keywordsphase transitions, reaction-diffusion systems, propagating fronts, finite difference schemes, wave speed approximations

© 2020 C. Mascia et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
phase transitions, reaction-diffusion systems, propagating fronts, finite difference schemes, wave speed approximations