# American Institute of Mathematical Sciences

June  2020, 40(6): 3813-3836. doi: 10.3934/dcds.2020160

## Turing type instability in a diffusion model with mass transport on the boundary

 1 Department of Applied Mathemarics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan 2 Department of Mathematical and Life Science, Hiroshima University, 1-3-1 Kagamiyama Higashi-Hiroshima, 739-8526, Japan

* Corresponding author: Kunimochi Sakamoto

Received  January 2019 Revised  January 2020 Published  March 2020

Fund Project: The first author was partially supported by JSPS KAKENHI Grant JP18H01139 and the second author was partially supported by JSPS KAKENHI Grant JP19K03564

Some reaction-diffusion models describing the cell polarity are proposed, where the system has two independent variables standing for the concentration of proteins in the membrane and the cytosol respectively. In this article we deal with such a polarity model consisting of one equation on a unit sphere and the other one in the ball inside the sphere. The two equations are coupled through a nonlinear boundary condition and the total mass is conserved. We investigate the linearized stability of a constant steady state and provide conditions under which a Turing type instability takes place, namely, the constant state is stable against spatially uniform perturbations on the sphere for all choices of diffusion rates, while unstable against nonuniform perturbations on the sphere as the diffusion coefficient of the equation on the sphere becomes small relative to the one in the ball.

Citation: Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
##### References:
 [1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8. [2] A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049. [3] A. Anma and K. Sakamoto, Turing type mechanisms for linear diffusion systems under nondiagonal Robin boundary conditions, SIAM J. Math. Anal., 45 (2013), 3611-3628.  doi: 10.1137/130908270. [4] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim., 25 (2004), 321-348.  doi: 10.1081/NFA-120039655. [5] J.-L. Chern, Y. Morita, and T.-T. Shieh, Asymptotic behavior of equilibrium states of reaction-diffusion systems with mass conservation, J. Differential Equations, 264 (2018), 550–574. doi: 10.1016/j.jde.2017.09.015. [6] R. Courant and D. Hilbert, Methods of Mathematical Physics, John Wiley & Sons, Inc., New York, 1989. [7] R. Diegmiller, H. Montanelli, C. B. Muratov and S. Y. Shvartsman, Spherical caps in cell polarization, Biophysical J., 115 (2018), 1-5.  doi: 10.1016/j.bpj.2018.05.033. [8] N. W. Goehring, C. Hoege, S. W. Grill and A. A. Hyman, PAR proteins diffuse freely across the anterior-posterior boundary in polarized C. elegans embryos, J. Cell. Biol., 193 (2011), 583-594.  doi: 10.1083/jcb.201011094. [9] N. W. Goehring, D. Chowdhury, A. A. Hyman and S. W. Grill, FRAP analysis of membrane-associated proteins: Lateral diffusion and membrane-cytoplasmic exchange, Biophys. J., 99 (2010), 2443-2452.  doi: 10.1016/j.bpj.2010.08.033. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. [11] S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.015203. [12] S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021. [13] S. Jimbo and Y. Morita, Nonlocal eigenvalue problems arising in a generalized phase-field-type system, Jpn J. Ind. Appl. Math., 34 (2017), 555-584.  doi: 10.1007/s13160-017-0254-z. [14] E. Latos, Y. Morita and T. Suzuki, Stability and spectral comparison of a reaction-diffusion system with mass conservation, J. Dynam. Differential Equations, 30 (2018), 823-844.  doi: 10.1007/s10884-018-9650-6. [15] E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039. [16] S. S. Lee and T. Shibata, Self-organisation and advective transport in the cell polarity formation for asymmetric cell division, J. Theoret. Biol., 382 (2015), 1-14.  doi: 10.1016/j.jtbi.2015.06.032. [17] H. Levine and W.-J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E (3), 72 (2005), 5pp. doi: 10.1103/PhysRevE.72.061912. [18] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅰ, Nonlinear Anal., 83 (2013), 91-101.  doi: 10.1016/j.na.2012.10.012. [19] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅱ, J. Math. Anal. Appl., 433 (2016), 642-680.  doi: 10.1016/j.jmaa.2015.07.053. [20] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅲ, Nonlinear Anal. Real World Appl., 31 (2016), 88-99.  doi: 10.1016/j.nonrwa.2016.01.007. [21] T. Mori, K. Kuto, T. Tsujikawa, M. Nagayama and S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, 10th AIMS Conference. Suppl., 2015,861–877. doi: 10.3934/proc.2015.0861. [22] T. Mori, K. Kuto, T. Tsujikawa and S. Yotsutani, Exact multiplicity of stationary limiting problem of a cell polarization model, Discrete Contin. Dyn. Syst., 36 (2016), 5627-5655.  doi: 10.3934/dcds.2016047. [23] Y. Mori, Y. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from bistable reaction-diffusion system, Biophys. J., 94 (2008), 3684-3697.  doi: 10.1529/biophysj.107.120824. [24] Y. Mori, Y. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization, SIAM J. Appl. Math., 71 (2011), 1401-1427.  doi: 10.1137/10079118X. [25] Y. Morita, Spectrum comparison for a conserved reaction-diffusion system with a variational property, J. Appl. Anal. Comput., 2 (2012), 57-71. [26] Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007. [27] Y. Morita and K. Sakamoto, A diffusion model for cell polarization with interactions on the membrane, Jpn J. Ind. Appl. Math., 35 (2018), 261-276.  doi: 10.1007/s13160-017-0290-8. [28] Y. Morita and N. Shinjo, Reaction-diffusion models with a conservation law and pattern formation, Josai Mathmatical Monographs, 9 (2016), 177-190. [29] I. L. Novak, F. Gao, Y.-S. Choi, D. Resasco, J. C. Schaff and B. M. Slepchenko, Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology, J. Comput. Phys., 226 (2007), 1271-1290.  doi: 10.1016/j.jcp.2007.05.025. [30] M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108. [31] A. Rätz and M. Röger, Turing instability in a mathematical model for signaling networks, J. Math. Biol., 65 (2012), 1215-1244.  doi: 10.1007/s00285-011-0495-4. [32] A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, Nonlinearity, 27 (2014), 1805-1828.  doi: 10.1088/0951-7715/27/8/1805. [33] V. Sharma and J. Morgan, Global existence of solutions to reaction-diffusion systems with mass transport type boundary conditions, SIAM J. Math. Anal., 48 (2016), 4202-4240.  doi: 10.1137/15M1015145.

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##### References:
 [1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8. [2] A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049. [3] A. Anma and K. Sakamoto, Turing type mechanisms for linear diffusion systems under nondiagonal Robin boundary conditions, SIAM J. Math. Anal., 45 (2013), 3611-3628.  doi: 10.1137/130908270. [4] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim., 25 (2004), 321-348.  doi: 10.1081/NFA-120039655. [5] J.-L. Chern, Y. Morita, and T.-T. Shieh, Asymptotic behavior of equilibrium states of reaction-diffusion systems with mass conservation, J. Differential Equations, 264 (2018), 550–574. doi: 10.1016/j.jde.2017.09.015. [6] R. Courant and D. Hilbert, Methods of Mathematical Physics, John Wiley & Sons, Inc., New York, 1989. [7] R. Diegmiller, H. Montanelli, C. B. Muratov and S. Y. Shvartsman, Spherical caps in cell polarization, Biophysical J., 115 (2018), 1-5.  doi: 10.1016/j.bpj.2018.05.033. [8] N. W. Goehring, C. Hoege, S. W. Grill and A. A. Hyman, PAR proteins diffuse freely across the anterior-posterior boundary in polarized C. elegans embryos, J. Cell. Biol., 193 (2011), 583-594.  doi: 10.1083/jcb.201011094. [9] N. W. Goehring, D. Chowdhury, A. A. Hyman and S. W. Grill, FRAP analysis of membrane-associated proteins: Lateral diffusion and membrane-cytoplasmic exchange, Biophys. J., 99 (2010), 2443-2452.  doi: 10.1016/j.bpj.2010.08.033. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. [11] S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.015203. [12] S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021. [13] S. Jimbo and Y. Morita, Nonlocal eigenvalue problems arising in a generalized phase-field-type system, Jpn J. Ind. Appl. Math., 34 (2017), 555-584.  doi: 10.1007/s13160-017-0254-z. [14] E. Latos, Y. Morita and T. Suzuki, Stability and spectral comparison of a reaction-diffusion system with mass conservation, J. Dynam. Differential Equations, 30 (2018), 823-844.  doi: 10.1007/s10884-018-9650-6. [15] E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039. [16] S. S. Lee and T. Shibata, Self-organisation and advective transport in the cell polarity formation for asymmetric cell division, J. Theoret. Biol., 382 (2015), 1-14.  doi: 10.1016/j.jtbi.2015.06.032. [17] H. Levine and W.-J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E (3), 72 (2005), 5pp. doi: 10.1103/PhysRevE.72.061912. [18] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅰ, Nonlinear Anal., 83 (2013), 91-101.  doi: 10.1016/j.na.2012.10.012. [19] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅱ, J. Math. Anal. Appl., 433 (2016), 642-680.  doi: 10.1016/j.jmaa.2015.07.053. [20] M. Małogrosz, A model of morphogen transport in the presence of glypicans Ⅲ, Nonlinear Anal. Real World Appl., 31 (2016), 88-99.  doi: 10.1016/j.nonrwa.2016.01.007. [21] T. Mori, K. Kuto, T. Tsujikawa, M. Nagayama and S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, 10th AIMS Conference. Suppl., 2015,861–877. doi: 10.3934/proc.2015.0861. [22] T. Mori, K. Kuto, T. Tsujikawa and S. Yotsutani, Exact multiplicity of stationary limiting problem of a cell polarization model, Discrete Contin. Dyn. Syst., 36 (2016), 5627-5655.  doi: 10.3934/dcds.2016047. [23] Y. Mori, Y. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from bistable reaction-diffusion system, Biophys. J., 94 (2008), 3684-3697.  doi: 10.1529/biophysj.107.120824. [24] Y. Mori, Y. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization, SIAM J. Appl. Math., 71 (2011), 1401-1427.  doi: 10.1137/10079118X. [25] Y. Morita, Spectrum comparison for a conserved reaction-diffusion system with a variational property, J. Appl. Anal. Comput., 2 (2012), 57-71. [26] Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007. [27] Y. Morita and K. Sakamoto, A diffusion model for cell polarization with interactions on the membrane, Jpn J. Ind. Appl. Math., 35 (2018), 261-276.  doi: 10.1007/s13160-017-0290-8. [28] Y. Morita and N. Shinjo, Reaction-diffusion models with a conservation law and pattern formation, Josai Mathmatical Monographs, 9 (2016), 177-190. [29] I. L. Novak, F. Gao, Y.-S. Choi, D. Resasco, J. C. Schaff and B. M. Slepchenko, Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology, J. Comput. Phys., 226 (2007), 1271-1290.  doi: 10.1016/j.jcp.2007.05.025. [30] M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108. [31] A. Rätz and M. Röger, Turing instability in a mathematical model for signaling networks, J. Math. Biol., 65 (2012), 1215-1244.  doi: 10.1007/s00285-011-0495-4. [32] A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, Nonlinearity, 27 (2014), 1805-1828.  doi: 10.1088/0951-7715/27/8/1805. [33] V. Sharma and J. Morgan, Global existence of solutions to reaction-diffusion systems with mass transport type boundary conditions, SIAM J. Math. Anal., 48 (2016), 4202-4240.  doi: 10.1137/15M1015145.
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