Abstract
Living cells are characterized by their capacity to maintain a stable steady state. For instance, cells are able to conserve their volume, internal ionic composition and electrical potential difference across the plasma membrane within values compatible with the overall cell functions. The dynamics of these cellular variables is described by complex integrated models of membrane transport. Some clues for the understanding of the processes involved in global cellular homeostasis may be obtained by the study of the local stability properties of some partial cellular processes. As an example of this approach, I perform, in this study, the neighborhood stability analysis of some elementary integrated models of membrane transport. In essence, the models describe the rate of change of the intracellular concentration of a ligand subject to active and passive transport across the plasma membrane of an ideal cell. The ligand can be ionic or nonionic, and it can affect the cell volume or the plasma membrane potential. The fundamental finding of this study is that, within the physiological range, the steady states are asymptotically stable. This basic property is a necessary consequence of the general forms of the expressions employed to describe the active and passive fluxes of the transported ligand.
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References
Aihara, K. and G. Matsumoto (1982). Temporally coherent organization and instabilities in squid giant axons. J. Theor. Biol. 95, 697–720.
Baumgarten, C. M. and J. J. Feher (1995). Osmosis and the regulation of the cell volume, in Cell Physiology. Source Book, N. Sperelakis (Ed.), New York: Academic Press, pp. 180–211.
Blumenthal, R., J.-P. Changeaux and R. Lefever (1970). Membrane excitability and dissipative instabilities. J. Membr. Biol. 2, 351–374.
Cristina, E. and J. A. Hernández (2000). An elementary kinetic model of energy coupling in biological membranes. Biochim. Biophys. Acta 1460, 276–290.
Csonka, L. N. and A. D. Hanson (1991). Prokariotic osmoregulation: genetics and physiology. Annu. Rev. Microbiol. 15, 569–606.
Diamond, J. M. (1982). Transcellular cross-talk between epithelial cell membranes. Nature 300, 683–685.
Falciatore, A., M. R. d’Alcalà, P. Croot and C. Bowler (2000). Perception of environmental signals by a marine diatom. Science 288, 2363–2366.
Goldbeter, A. (1991). Models for oscillations and excitability in biochemical systems, in Biological Kinetics, L. A. Segel (Ed.), Cambridge, London: Cambridge University Press, pp. 107–154.
Goldman, D. E. (1943). Potential, impedance and rectification in membranes. J. Gen. Physiol. 27, 37–60.
Hansen, U.-P., D. Gradmann, D. Sanders and C. L. Slayman (1981). Interpretation of current-voltage relationships for “active” ion transport systems: I. Steady-state reaction-kinetic analysis of class-I mechanisms. J. Membr. Biol. 63, 165–190.
Hassard, B. (1978). Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axons. J. Theor. Biol. 71, 401–420.
Hernández, J. A. and E. Cristina (1998). Modeling cell volume regulation in nonexcitable cells: the roles of the Na+ pump and of cotransport systems. Am. J. Physiol. 275, C1067–C1080.
Hernández, J. A. and S. Chifflet (2000). Electrogenic properties of the sodium pump in a dynamic model of membrane transport. J. Membr. Biol. 176, 41–52.
Hernández, J. A., J. Fischbarg and L. S. Liebovitch (1989). Kinetic model of the effects of electrogenic enzymes on the membrane potential. J. Theor. Biol. 137, 113–125.
Hill, T. L. (1977). Free Energy Transduction in Biology, New York: Academic Press, pp. 1–56.
Hodgkin, A. L. and B. Katz (1949). The effect of sodium ions on the electrical activity of the giant axon of the squid. J. Physiol. Lond. 108, 37–77.
Jakobsson, E. (1980). Interactions of cell volume, membrane potential, and membrane transport parameters. Am. J. Physiol. 238, C196–C206.
Katchalsky, A. and R. Spangler (1968). Dynamics of membrane processes. Quart. Rev. Biophys. I, 127–175.
Keener, J. and J. Sneyd (1998). Mathematical Physiology, New York: Springer, pp. 33–73.
Lang, F., G. L. Busch, M. Ritter, H. Volkl, S. Waldegger, E. Gulbins and D. Haussinger (1998). Functional significance of cell volume regulatory mechanisms. Physiol. Rev. 78, 247–306.
Läuger, P. (1991). Electrogenic Ion Pumps, Sunderland, MA: Sinauer Associates, pp. 3–14.
Lemieux, D. R., F. A. Roberge and P. Savard (1990). A model study of the contribution of active Na-K transport to membrane repolarization in cardiac cells. J. Theor. Biol. 142, 1–33.
May, R. M. (1974). Stability and Complexity in Model Ecosystems, Princeton, NJ: Princeton University Press, pp. 13–36.
Minton, A. P. (1983). The effect of volume occupancy upon the thermodynamic activity of proteins: some biochemical consequences. Mol. Cell. Biochem. 55, 119–140.
Mizraji, E., L. Acerenza and J. A. Hernández (1988). Time delays in metabolic control systems. Biosystems 22, 11–17.
Murray, J. D. (1989). Mathematical Biology, Berlin: Springer, pp. 36–62.
Olivotto, M., A. Arcangeli, M. Carlà and E. Wanke (1996). Electric fields at the plasma membrane level: a neglected element in the mechanisms of cell signalling. Bioessays 18, 495–504.
Parker, J. C. (1993). In defense of cell volume? Am. J. Physiol. 2650, C1191–C1200.
Post, R. L. and P. C. Jolly (1957). The linkage of sodium, potassium, and ammonium active transport across the human erythrocyte membrane. Biochim. Biophys. Acta 25, 118–128.
Reuss, L. and C. U. Cotton (1994). Volume regulation in epithelia: transcellular transport and cross-talk, in Cellular and Molecular Physiology of Cell Volume Regulation, K. Strange (Ed.), Boca Raton, FL: CRC Press, pp. 31–47.
Ricard, J. and G. Noat (1984). Enzyme reactions at the surface of living cells: II. Destabilization in the membranes and conduction of signals. J. Theor. Biol. 109, 571–580.
Schultz, S. G. (1981). Homocellular regulatory mechanisms in sodium-transporting epithelia: avoidance of extinction by “flush-through”. Am. J. Physiol. 242, F579–F590.
Segel, L. A. (1987). Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge, London: Cambridge University Press, pp. 36–50.
Stein, W. D. (1990). Channels, Carriers and Pumps. An Introduction to Membrane Transport, New York: Academic Press, pp. 271–310.
Tosteson, D. C. and J. F. Hoffman (1960). Regulation of cell volume by active cation transport in high and low potassium sheep red cells. J. Gen. Physiol. 44, 169–194.
Weinstein, A. M. (1995). A kinetically defined Na+/H+ antiporter within a mathematical model of the rat proximal tubule. J. Gen. Physiol. 105, 617–641.
Weinstein, A. M. (1997). Dynamics of cellular homeostasis: recovery time for a perturbation from equilibrium. Bull. Math. Biol. 59, 451–481.
Weiss, T. F. (1996). Cellular Biophysics. Transport, Vol. 1, Cambridge, MA: MIT Press, pp. 571–643.
Yancey, P. H., M. E. Clark, S. C. Hand, R. D. Bowlus and G. N. Somero (1982). Living with water stress: evolution of osmolyte systems. Science 217, 1214–1222.
Zeuthen, T. (1996). Molecular Mechanisms of Water Transport, Heidelberg: Springer, pp. 1–10.
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Hernández, J.A. Stability properties of elementary dynamic models of membrane transport. Bull. Math. Biol. 65, 175–197 (2003). https://doi.org/10.1006/bulm.2002.0325
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DOI: https://doi.org/10.1006/bulm.2002.0325