Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells

Abstract Calcium is one of the most important intracellular messengers, which occurs in the cytosol and the endoplasmic reticulum of animal cells. While most calcium dynamics models either do not account properly for the fact that the endoplasmic reticulum constitutes a microstructure of the cell or are infeasible by resolving the fine structure very explicitly, Goel et al. [15] derived an effective macroscopic model by formal homogenization. In this paper, this approach is made rigorous using periodic homogenization techniques to upscale the nonlinear coupled system of reaction–diffusion equations and, moreover, the appropriate scaling of the interfacial exchange term is taken into consideration.

[1]  James P. Keener,et al.  Mathematical physiology , 1998 .

[2]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[3]  Alain Damlamian,et al.  Two-Scale Convergence On Periodic Surfaces And Applications , 1995 .

[4]  W. Krassowska,et al.  Effective boundary conditions for syncytial tissues , 1994, IEEE Transactions on Biomedical Engineering.

[5]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  Malte A. Peter,et al.  Multiscale Modelling of Chemical Degradation Mechanisms in Porous Media with Evolving Microstructure , 2009, Multiscale Model. Simul..

[8]  Jesús Ildefonso Díaz Díaz,et al.  EFFECTIVE CHEMICAL PROCESSES IN POROUS MEDIA , 2003 .

[9]  N. Bakhvalov,et al.  Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials , 1989 .

[10]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[11]  E. S. Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[12]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[13]  V. Marchenko,et al.  Homogenization of Partial Differential Equations , 2005 .

[14]  Doina Cioranescu,et al.  Homogénéisation du problème de Neumann non homogène dans des ouverts perfores , 1988 .

[15]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[16]  Maria Neuss-Radu,et al.  Some extensions of two-scale convergence , 1996 .

[17]  M. Böhm,et al.  A note on the existence of extension operators for Sobolev spaces on periodic domains , 2014 .

[18]  Grégoire Allaire,et al.  Multiscale convergence and reiterated homogenisation , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  Malte A. Peter,et al.  Coupled reaction–diffusion processes inducing an evolution of the microstructure: Analysis and homogenization , 2009 .

[20]  Daniele Andreucci,et al.  On a hierarchy of models for electrical conduction in biological tissues , 2006 .

[21]  Peter Wall,et al.  Two-scale convergence , 2002 .

[22]  P. Donato,et al.  An introduction to homogenization , 2000 .

[23]  Sara Monsurrò,et al.  Homogenization of a two-component composite with interfacial thermal barrier , 2003 .

[24]  Pranay Goel,et al.  Homogenization of the Cell Cytoplasm: The Calcium Bidomain Equations , 2006, Multiscale Model. Simul..

[25]  C. Conca,et al.  On the homogenization of a transmission problem arising in chemistry , 2004 .

[26]  Michael Böhm,et al.  Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium , 2008 .

[27]  HOMOGENIZATION OF SEMILINEAR PARABOLIC EQUATIONS IN PERFORATED DOMAINS , 2004 .

[28]  A. Holmbom.  Homogenization of parabolic equations an alternative approach and some corrector-type results , 1997 .

[29]  G. Burton.  Sobolev Spaces , 2013 .

[30]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[31]  P. Bisegna,et al.  Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues , 2004 .

[32]  W. Krassowska,et al.  Homogenization of syncytial tissues. , 1993, Critical reviews in biomedical engineering.