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 & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
References:
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H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

[2]

A. AnmaK. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049.  Google Scholar

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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.  Google Scholar

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R. DiegmillerH. MontanelliC. 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.  Google Scholar

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N. W. GoehringC. HoegeS. 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.  Google Scholar

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N. W. GoehringD. ChowdhuryA. 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.  Google Scholar

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E. LatosY. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. MoriK. KutoT. 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.  Google Scholar

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Y. MoriY. 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.  Google Scholar

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Y. MoriY. 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.  Google Scholar

[25]

Y. Morita, Spectrum comparison for a conserved reaction-diffusion system with a variational property, J. Appl. Anal. Comput., 2 (2012), 57-71.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[28]

Y. Morita and N. Shinjo, Reaction-diffusion models with a conservation law and pattern formation, Josai Mathmatical Monographs, 9 (2016), 177-190.   Google Scholar

[29]

I. L. NovakF. GaoY.-S. ChoiD. ResascoJ. 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.  Google Scholar

[30]

M. OtsujiS. IshiharaC. CoK. KaibuchiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. AnmaK. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[7]

R. DiegmillerH. MontanelliC. 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.  Google Scholar

[8]

N. W. GoehringC. HoegeS. 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.  Google Scholar

[9]

N. W. GoehringD. ChowdhuryA. 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.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. LatosY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Levine and W.-J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E (3), 72 (2005), 5pp. doi: 10.1103/PhysRevE.72.061912.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

T. MoriK. KutoT. 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.  Google Scholar

[23]

Y. MoriY. 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.  Google Scholar

[24]

Y. MoriY. 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.  Google Scholar

[25]

Y. Morita, Spectrum comparison for a conserved reaction-diffusion system with a variational property, J. Appl. Anal. Comput., 2 (2012), 57-71.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. Morita and N. Shinjo, Reaction-diffusion models with a conservation law and pattern formation, Josai Mathmatical Monographs, 9 (2016), 177-190.   Google Scholar

[29]

I. L. NovakF. GaoY.-S. ChoiD. ResascoJ. 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.  Google Scholar

[30]

M. OtsujiS. IshiharaC. CoK. KaibuchiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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