Influencia de la curvatura en la formación de patrones: el mecanismo de Turing en el círculo
Resumen
En este trabajo estudiamos la influencia de la curvatura del medio sobre la formaci´on de patrones mediante el mecanismo de inestabilidad de Turing generada por difusi´on. Para analizar el efecto de la curvatura consideramos la variedad curva cerrada m´as simple, una circunferencia. Presentamos el operador de Laplace-Beltrami, que es la generalizaci´on del Laplaciano sobre variedades, con el fin de resolver la ecuaci´on de difusi´on y describir las modificaciones que induce la curvatura en la funci´on de distribuci´on y en el camino cuadr´atico medio. Como un caso particular de estudio empleamos el modelo de Gierer-Meinhardt. Se muestra que el rango de modos inestables donde ser´a posible hallar los patrones espacio-temporales depende del radio y, por lo tanto, de la curvatura del c´ırculo espec´ıfico.
Descargas
Citas
Amazon, J. J., Goh, S. L., and Feigenson, G. W. (2013). Competition between line tension and curvature stabilizes modulated phase patterns on the surface of giant unilamellar vesicles: A simulation study. Phys. Rev. E., 87:022708.
Arfken, G. and Weber, H., editors (2005). Mathematical Methods for Physicists. Elsevier Academic Press.
Ball, P. (2015). Forging patterns and making waves from biology to geology: a commentary on Turing (1952) the chemical basis of morphogenesis. Phil. Trans. R. Soc. B, 370:20140218.
Brigatti, E., N´u˜nez-L´opez, M., and Oliva, M. (2011). Analysis of a spatial Lotka-Volterra model with a finite range predator-prey interaction. Eur. Phys. J. B, 81:321.
Brinkmann, F., Mercker, M., Richter, T., and Marciniak-Czochra, A. (2018). Post-Turing tissue pattern formation: Advent of mechanochemistry. PLOS Computational Biology, 14:e1006259.
Capasso, V. and Wilson, R. (1997). Analysis of a reaction-di usion system modeling man–environment–man epidemics. SIAM J. Appl. Math., 57:327–346.
Castro-Villarreal, P., Villada-Balbuena, A., M´endez-Alcaraz, J. M., Casta˜neda-Priego, R., and Estrada-Jim´enez, S (2014). A brownian dynamics algorithm for colloids in curved manifolds. J. Chem. Phys., 140:214115.
Chac´on-Acosta, G., N´u˜nez-L´opez, M., and Pineda, I. (2020). Turing instability conditions in confined systems with an effective position-dependent diffusion coefficient. J. Chem. Phys., 152.
Chaplain, M., Ganesh, M., and Graham, I. (2001). Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. Journal of Mathematical, 42.
Crank, J. (1980). The Mathematics of Diffusion. Oxford University Press. do Carmo, M., editor (1976). Di erential Geometry of curves and surfaces. Prentice Hall, Inc., New Jersey.
Fuselier, E. and Wright, G. (2013). A high-order kernel method for diffusion and reaction-di usion equations on surfaces. J. Sci. Comput., 56:535.
Fuseya, Y., Katsuno, H., Behnia, K., and Kapitulnik, A. (2021). Nanoscale Turing patterns in a bismuth monolayer. Nat. Phys., 17:1031–1036.
Gatenby, R. A. and Gawlinski, E. (1996). A reaction-diffusion model of cancer invasion. Cancer Research, 56:5745 -5753.
Getzin, S., Yizhaq, H., Bell, B., Erickson, T. E., Postle, A. C., Katra, I., Tzuk, O., Zelnik, Y. R., Wiegand, K., Wiegand, T., and Meron, E. (2016). Discovery of fairy circles in Australia supports self-organization theory. PNAS, 113:3551–3556.
Gierer, A. and Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12:30–39.
Gjorgjieva, J. and Jacobsen, J. (2007). Turing patterns on growing spheres: the exponential case. DCDS Supplements, 2007(Special):436–445.
Krause, A. L., Burton, A. M., Fadai, N. T., and Gorder, R. A. V. (2018). Emergent structures in reaction-advection-diffusion systems on a sphere. Phys. Rev. E, 97:042215.
Krause, A. L., Ellis, M. A., and Gorder, R. A. V. (2019). Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds. Bull. Math. Biol., 81:759–799.
Le´on-Velasco, D. A. and Chac´on-Acosta, G. (2021). Full finite element scheme for reaction-diffusion systems on embedded curved surfaces in R3. Adv. Math. Phys., page 8898484.
Liaw, S., Yang, C., Liu, R., and Hong, J. (2001). Turing model for the patterns of lady beetles. Phys. Rev. E, 64:041909.
Liu, Y. J., Zhu, L., Wang, A., and Wang, B. (2011). Dynamical behavior of an epidemic model. Braz. J. Phys., 41:304–308.
Mimar, S., Juane, M. M., Park, J., Mu˜nuzuri, A. P., and Ghoshal, G. (2019). Turing patterns mediated by network topology in homogeneous active systems. Phys. Rev. E, 99:062303.
Murray, J. D., editor (2003). Mathematical Biology II: Spatial Models and Biomedical Applications. Third Edition, Springer-Verlag, Berlin Heidelberg.
Nampoothiri, S. (2016). Stability of patterns on thin curved surfaces. Phys. Rev. E, 94:022403.
N´u˜nez-L´opez, M., Chac´on-Acosta, G., and Santiago, J. A. (2017). Diffusiondriven instability on a curved surface: Spherical case revisited. Braz. J Phys, 47:231–238.
Plaza, R., S´anchez-Gardu˜no, F., Padilla, P., Barrio, R., and Maini, P. (2004). The effect of growth and curvature on pattern formation. J. Dyn. Diff. Eqs., 16:1093.
Ramakrishnan, N., Kumar, P. S., and Radhakrishnan, R. (2014). Mesoscale computational studies of membrane bilayer remodeling by curvatureinducing proteins. Phys. Rep., 543:1–60.
Sens, P. and Turner, M. S. (2011). Microphase separation in nonequilibrium biomembranes. Phys. Rev. Lett., 106:238101.
Stoop, N., Lagrange, R., Terwagne, D., Reis, P., and Dunkel, J. (2015). Curvature-induced symmetry breaking determines elastic surface patterns. Nature Materials, 14:337–342.
S´anchez-Gardu˜no, F., Krause, A. L., Castillo, J. A., and Padilla, P. (2019). Turing-Hopf patterns on growing domains: the torus and the sphere. J. Theor. Biol., 481:136–150.
Turing, A. M. (1952). The chemical basis of morphogenesis. Philos. Trans. Roy. Soc., 237:37–72.
Turing, A. M. (1954). Part iii: A solution of the equations morphogenetical for the case of spherical symmetry.
van Meer, G., Voelker, D. R., and Feigenson, G. W. (2008). Membrane lipids: where they are and how they behave. Nat. Rev. Mol. Cell Biol., 9:112–124.
Vandin, G., Marenduzzo, D., Giryachev, A. B., and Orlandini, E. (2016). Correction: Curvature-driven positioning of Turing patterns in phase-separating curved membranes. Soft Matt., 12:3888–3896.
Varea, C., Arag´on, J., and Barrio, R. (1999). Turing patterns on a sphere. Phys. Rev. E, 60:4588.
Webb, G. F. (1981). A reaction-diffusion model for a deterministic diffusive epidemic. J. Math. Anal. Appl., 84:150–161.
Zhu, L., Yuan, H., Wu, K., Wang, X., Liu, G., Sun, J., Liao, X., and Chen, X. (2021). Curvature-controlled delamination patterns of thin films on spherical substrates. iScience, 24:102616.
