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

C. Mascia 1 , P. Moschetta 1  and C. Simeoni 2
  • 1 Department of Mathematics G. Castelnuovo, Italy
  • 2 Laboratoire J.A. Dieudonné, France


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.

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  • 1. D. Yao, C. Dai and S. Peng, Mechanism of the Mesenchymal-Epithelial Transition and its relationship with metastatic tumor formation, Molecular Cancer Research, vol. 9, no. 12, pp. 1608–1620, 2011.

  • 2. J.P. Thiery and J.P. Sleeman, Complex networks orchestrate epithelial-mesenchymal transitions, Nature Reviews Molecular Cell Biology, vol. 7, no. 2, pp. 131–142, 2006.

  • 3. P.C. Davies, L. Demetrius and J.A. Tuszynki, Cancer as dynamical phase transition, Theoretical Biology and Medical Modelling, vol. 8, pp. 1–30, 2011.

  • 4. M. Mojtahedi, A. Skupin, J. Zhou, I.G. Castaño, R.Y.Y. Leong-Quong, H. Chang, K. Trachana, A. Giuliani and S. Huang, Cell fate decision as high-dimensional critical state, PLOS Biology, vol. 14, no. 12, p. e2000640, 2016.

  • 5. J. Xu, S. Lamouille and R. Derynck, TGF-β-induced epithelial to mesenchymal transition, Cell Research, vol. 19, no. 2, pp. 156–172, 2009.

  • 6. M. Bertolaso, Philosophy of cancer. A dynamic and relational view. History, Philosophy and Theory of the Life Sciences 18, Springer Science+Business Media Dordrecht, Springer Netherlands, 2016.

  • 7. C. Simeoni, S. Dinicola, A. Cucina, C. Mascia and M. Bizzarri, Systems Biology approach and Mathematical Modeling for analyzing phase-space switch during Epithelial-Mesenchymal Transition, in Systems Biology (M. Bizzarri, ed.), Methods in Molecular Biology 1702, pp. 95–123, Springer Protocols+Business Media LLC, 2018.

  • 8. M. Bizzarri, A. Cucina, F. Conti and F. D’Anselmi, Beyond the oncogene paradigm: understanding complexity in carcinogenesis, Acta Biotheoretica, vol. 56, no. 3, pp. 173–196, 2008.

  • 9. G. Barrière, P. Fici, G. Gallerani, F. Fabbri and M. Rigaud, Epithelial Mesenchymal Transition: a double-edged sword, Clinical and Translational Medicine, vol. 4, no. 14, pp. 1–6, 2015.

  • 10. M.W. Green and B.D. Sleeman, On FitzHugh’s nerve axon equations, Journal of Mathematical Biology, vol. 1, pp. 153–163, 1974.

  • 11. M.A. Jones, B. Song and D.M. Thomas, Controling wound healing through debridement, Mathematical and Computer Modelling, vol. 40, pp. 1057–1064, 2004.

  • 12. R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki, The Hill equation and the origin of quantitative pharmacology, Archive for History of Exact Sciences, vol. 66, pp. 427–438, 2012.

  • 13. J. Monod, J. Wyman and J.P. Changeux, On the nature of allosteric transitions: a plausible model, Journal of Molecular Biology, vol. 12, pp. 88–118, 1965.

  • 14. J.N. Weiss, The Hill equation revisited: uses and misuses, The FASEB Journal, vol. 11, no. 11, pp. 835–841, 1997.

  • 15. R.I. Masel, Principles of adsorption and reaction on solid surfaces, Wiley Series in Chemical Engineering 3, John Wiley & Sons, 1996.

  • 16. C.S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, vol. 91, no. 7, pp. 385–398, 1959.

  • 17. P.C. Fife and J.B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, vol. 65, pp. 335–361, 1977.

  • 18. J.A. Sherratt and B.P. Marchant, Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation, IMA Journal of Applied Mathematics, vol. 56, pp. 289–302, 1996.

  • 19. A. Quarteroni, Numerical models for differential problems. Third edition. Modeling, Simulation and Applications 16, Springer, Cham, 2017.

  • 20. R.J. LeVeque and H.C. Yee, A study of numerical methods for hyperbolic conservation laws with sti source terms, Journal of Computational Physics, vol. 86, no. 1, pp. 187–210, 1990.

  • 21. C. Lattanzio, C. Mascia, R.G. Plaza and C. Simeoni, Kinetic schemes for assessing stability of traveling fronts for the Allen-Cahn equation with relaxation, Applied Numerical Mathematics, vol. 141, pp. 234–247, 2019.

  • 22. P. Moschetta and C. Simeoni, Numerical investigation of the Gatenby-Gawlinski model for acid-mediated tumour invasion, Rendiconti di Matematica e delle sue Applicazioni, pp. 1–31, online first, 2019.

  • 23. R.J. LeVeque, Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.

  • 24. B.H. Gilding and R. Kersner, Travelling waves in nonlinear diffusion-convection reaction. Progress in Nonlinear Differential Equations and Their Applications 60, Springer Basel AG, 2004.


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