Abstract
A continuum model of diffusion-coupled cells that more accurately reflects the presence of low-permeability gap junctions between cells is analyzed. It is shown by a multi-scale analysis that to lowest order the slow evolution of the mean concentration is described by the usual ordinary differential equations for a discrete model. Furthermore, stable non-uniform steady solutions are shown to exist in the continuum model of a one component system, whereas this is impossible for the standard reaction-diffusion model of this system. It is also shown how to average the equations in this continuum model to obtain a system of reaction-diffusion equations with constant coefficients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allgower, E.: On a discretization of 368-01. Proc. Conf. Roy. Irish Acad. Miller, J., (ed.) New York: Academic Press 1975
Ashkenazi, M., Othmer, H. G.: Spatial patterns in coupled biochemical oscillators. J. Math. Biol. 5, 305–350 (1978)
Bensoussan, A., Lions, J. L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Amsterdam: North-Holland 1978
Bunow, B., Kernevez, J. P., Joly, G., Thomas, D.: Pattern formation by reaction-diffusion instabilities: Application to morphogenesis in Drosophila. J. Theoret. Biol. 84, 629–649 (1980)
Casten, R. G., Holland, D. J.: Instability results for reaction-diffusion equations with Neumann boundary conditions. J. Diff. Eqns. 27, 266–273 (1978)
Chafee, N.: Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Diff. Eqns. 18, 111–134 (1975)
Courant, R., Hubert, D.: Methods of mathematical physics. New York: Interscience 1953
Crick, F.: Diffusion in embryogenesis. Nature 225, 420–422 (1971)
Feldman, J., Gilula, N. B., Pitts, J. D.: Intercellular junctions and synapses. London: Chapman and Hall 1978
Fife, P., Peletier, L. A.: Clines induced by variable selection and migration. MRC Technical Summary, June 1980
Flagg-Newton, J., Simpson, I., Loewenstein, W. R.: Permeability of the cell-to-cell membrane channels in mammalian cell junction. Science 205, 404–407 (1979)
Griep, E. B., Revel, J. P.: Gap junctions in development. In: De Mello, W. C. (ed.) Intercellular communication. New York: Plenum Press 1977
Heinrich, R., Rapoport, S. M., Rapoport, T. A.: Metabolic regulation and mathematical models. Prog. Biophys. Molec. Biol. 32, 1–82 (1977)
Hodgkin, A. L.: Exchange and diffusion of tracer in a bundle of cardiac fibers. Appendix to: Weidmann, S.: The diffusion of radiopotassium across intercalated disks of mammalian cardiac muscle. J. Physiol. 187, 323–342 (1966)
Imanaga, I.: Cell-to-cell diffusion of procion yellow in sheep and calf purkinje fibers. J. Mem. Biol. 16, 381–388 (1974)
Jürgens, H., Peitgen, H. O., Saupe, D.: Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems. In: Robinson, S. M. (ed.) Analysis and computation of fixed points. New York: Academic Press 1980
Levin, S. A.: Non-uniform stable solutions to reaction-diffusion equations: Applications to ecological pattern formation. In: Haken, H. (ed.) Proc. Intl. Symp. on Synergetics. Springer 1979
Levitt, D. G.: Kinetics of diffusion and convection in 3.2 Å pores. Biophys. J. 13, 186–206 (1973)
Loewenstein, W. R., Rose, B.: Calcium in (junctional) intercellular communication and a thought on its behavior in intracellular communication. Ann. N. Y. Acad. Sc. 307, 285–307 (1978)
Matano, H.: Asymptotic behavior and stability of solutions of semi-linear diffusion equations. Publ. Res. Inst. Math. Sci. (Kyoto) 15, 401–454 (1979)
Nicolis, G., Prigogine, I.: Self-organization in nonequilibrium systems. New York: J. Wiley and Sons 1977
Othmer, H. G., Aldridge, J. A.: The effects of cell density and metabolite flux on cellular dynamics. J. Math. Biol. 5, 169–200 (1978)
Othmer, H. G., Pate, E.: Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc. Natl. Acad. Sci. (USA) 77, 4180–4184 (1980)
Weingart, R.: The permeability to tetraethylammonium ions of the surface membrane and the intercalcated disks of sheep and calf myocardium. J. Physiol. 240, 741–762 (1974)
Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theoret. Biol. 25, 1–47 (1969)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Othmer, H.G. A continuum model for coupled cells. J. Math. Biology 17, 351–369 (1983). https://doi.org/10.1007/BF00276521
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276521