Abstract
We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anguige, K., Röger, M.: Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells. J. Math. Anal. Appl. 448, 213–244 (2017)
Aubin, T.: Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252. Springer, New York (1982)
Berestycki, H., Coulon, A.-C., Roquejoffre, J.-M., Rossi, L.: The effect of a line with nonlocal diffusion on Fisher–KPP propagation. Math. Models Methods Appl. Sci. 25, 2519–2562 (2015)
Bothe, D., Köhne, M., Maier, S., Saal, J.: Global strong solutions for a class of heterogeneous catalysis models. J. Math. Anal. Appl. 445, 677–709 (2017)
Chipot, M.: Elements of Nonlinear Analysis. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2000)
Day, C. A., Kenworthy, A. K.: Tracking microdomain dynamics in cell membranes. In: Biochimica et biophysica acta 1788, pp. 245–253 (2009)
Elliott, C.M., Ranner, T.L.: Finite element analysis for a coupled bulk–surface partial differential equation. IMA J. Numer. Anal. 33, 377–402 (2013)
Elliott, C.M., Ranner, T., Venkataraman, C.: Coupled bulk–surface free boundary problems arising from a mathematical model of receptor–ligand dynamics. SIAM J. Math. Anal. 49, 360–397 (2017)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Fan, J., Sammalkorpi, M., Haataja, M.: Formation and regulation of lipid microdomains in cell membranes: theory, modeling, and speculation. FEBS Lett. 584, 1678–1684 (2010)
Fellner, K., Latos, E., Tang, B.Q.: Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling. Ann. Inst. H. Poincarè Anal. Non Linèaire 35, 643–673 (2018)
Fellner, K., Rosenberger, S., Tang, B.Q.: Quasi-steady-state approximation and numerical simulation for a volume–surface reaction–diffusion system. Commun. Math. Sci. 14, 1553–1580 (2016)
Fellner, K., Tang B.Q.: Entropy methods and convergence to equilibrium for volume-surface reaction-diffusion systems. In: Gonçalves, P., Soares, A. (eds.) From Particle Systems to Partial Differential Equations. PSPDE 2015. Springer Proceedings in Mathematics & Statistics, vol. 209. Springer, Cham, 153–176 (2017)
Friedmann, E., Neumann, R., Rannacher, R., et al.: Well-posedness of a linear spatio-temporal model of the JAK2/STAT5 signaling pathway. Commun. Math. Anal. 15, 76–102 (2013)
Garcke, H., Kampmann, J., Rätz, A., Röger, M.: A coupled surface–Cahn–Hilliard bulkdiffusion system modeling lipid raft formation in cell membranes. Math. Models Methods Appl. Sci 26, 1149–1189 (2016)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Glitzky, A., Mielke, A.: A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces. Z. Angew. Math. Phys. 64, 29–52 (2013)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)
Hale, J.K., Sakamoto, K.: Shadow systems and attractors in reaction–diffusion equations. Appl. Anal. Int. J. 32, 287–303 (1989)
Hausberg, S.: Mathematical analysis of a spatially coupled reaction–diffusion system for signaling networks in biological cells. Ph.D. thesis. Technische Universität Dortmund (2016)
Kavallaris, N.I., Suzuki, T.: On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system. Nonlinearity 30, 1734–1761 (2017)
Keener, J.P.: Activators and inhibitors in pattern formation. Stud. Appl. Math. 59, 1–23 (1978)
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)
Levine, H., Rappel, W.-J.: Membrane-bound Turing patterns. Phys. Rev. E 72, 061912 (2005)
Li, F., Ni, W.-M.: On the global existence and finite time blow-up of shadow systems. J. Differ. Equ. 247, 1762–1776 (2009)
Madzvamuse, A., Chung, A.H.W., Venkataraman, C.: Stability analysis and simulations of coupled bulk–surface reaction–diffusion systems. Proc. A. 471, 20140546 (2015)
Marciniak-Czochra, A., Härting, S., Karch, G., Suzuki, K.: Dynamical spike solutions in a nonlocal model of pattern formation. Nonlinearity 31, 1757 (2018)
Marciniak-Czochra, A., Mikelić, A.: Shadow limit for parabolic-ODE systems through a cut-off argument. In: Rad Hrvatske akademije znanosti i umjetnosti: Matemati.cke znanosti, pp. 99–116 (2017)
Mielke, A.: Thermomechanical modeling of energy–reaction–diffusion systems, including bulk–interface interactions. Discrete Contin. Dyn. Syst. Ser. S 6, 479–499 (2013)
Muller, N., Piel, M., Calvez, V., Voituriez, R., Gonçalves-Sá, J., Guo, C.-L., Jiang, X., Murray, A., Meunier, N.: A predictive model for yeast cell polarization in pheromone gradients. PLoS Comput. Biol. 12, e1004795 (2016)
Nelson, W.J.: Adaptation of core mechanisms to generate cell polarity. Nature 422, 766–774 (2003)
Novak, I.L., Gao, F., Choi, Y.-S., Resasco, D., Schaff, J.C., Slepchenko, B.M.: Diffusion on a curved surface coupled to diffusion in the volume: application to cell biology. J. Comput. Phys. 226, 1271–1290 (2007)
Pierre, M.: Global existence in reaction–diffusion systems with control of mass: a survey. Milan J. Math. 78, 417–455 (2010)
Rätz, A., Röger, M.: Turing instabilities in a mathematical model for signaling networks. J. Math. Biol. 65, 1215–1244 (2012)
Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153, 2nd edn. Birkhäuser/Springer, Basel (2013)
Rätz, A., Röger, M.: Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks. Nonlinearity 27, 1805 (2014)
Sharma, V., Morgan, J.: Global existence of solutions to reaction–diffusion systems with mass transport type boundary conditions. SIAM J. Math. Anal. 48, 4202–4240 (2016)
Sharma, V., Morgan, J.: Uniform bounds for solutions to volume–surface reaction–diffusion systems. Differ. Integral Equ. 30, 423–442 (2017)
Teigen, K.E., Li, X., Lowengrub, J., Wang, F., Voigt, A.: A diffusion–interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun. Math. Sci. 7, 1009–1037 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were partially funded by DFG under contract RO 3850/2-1.
Rights and permissions
About this article
Cite this article
Hausberg, S., Röger, M. Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system. Nonlinear Differ. Equ. Appl. 25, 17 (2018). https://doi.org/10.1007/s00030-018-0508-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-018-0508-8