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Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology

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  • Egger, H.
  • Fellner, K.
  • Pietschmann, J.-F.
  • Tang, B.Q.

Abstract

We consider the numerical solution of coupled volume-surface reaction-diffusion systems having a detailed balance equilibrium. Based on the conservation of mass, an appropriate quadratic entropy functional is identified and an entropy-entropy dissipation inequality is proven. This allows us to show exponential convergence to equilibrium by the entropy method. We then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes. Mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error that hold uniformly in time. Some numerical tests are presented to illustrate these theoretical results. The analysis and the numerical approximation are discussed in detail for a simple model problem. The basic arguments however apply also in a more general context. This is demonstrated by investigation of a particular volume-surface reaction-diffusion system arising as a mathematical model for asymmetric stem cell division.

Suggested Citation

  • Egger, H. & Fellner, K. & Pietschmann, J.-F. & Tang, B.Q., 2018. "Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 351-367.
    • Handle: RePEc:eee:apmaco:v:336:y:2018:i:c:p:351-367
    DOI: 10.1016/j.amc.2018.04.031
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    1. Ugo Mayor & Nicholas R. Guydosh & Christopher M. Johnson & J. Günter Grossmann & Satoshi Sato & Gouri S. Jas & Stefan M. V. Freund & Darwin O. V. Alonso & Valerie Daggett & Alan R. Fersht, 2003. "The complete folding pathway of a protein from nanoseconds to microseconds," Nature, Nature, vol. 421(6925), pages 863-867, February.
    2. Jörg Betschinger & Karl Mechtler & Juergen A. Knoblich, 2003. "The Par complex directs asymmetric cell division by phosphorylating the cytoskeletal protein Lgl," Nature, Nature, vol. 422(6929), pages 326-330, March.
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    Cited by:

    1. Dmitry Lukyanenko & Tatyana Yeleskina & Igor Prigorniy & Temur Isaev & Andrey Borzunov & Maxim Shishlenin, 2021. "Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay," Mathematics, MDPI, vol. 9(4), pages 1-12, February.
    2. Raul Argun & Alexandr Gorbachev & Natalia Levashova & Dmitry Lukyanenko, 2021. "Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduc," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    3. Max Winkler, 2020. "Error estimates for the finite element approximation of bilinear boundary control problems," Computational Optimization and Applications, Springer, vol. 76(1), pages 155-199, May.
    4. Raul Argun & Alexandr Gorbachev & Dmitry Lukyanenko & Maxim Shishlenin, 2021. "On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front," Mathematics, MDPI, vol. 9(22), pages 1-18, November.

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