Abstract
Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.
Similar content being viewed by others
References
Beck, J. S., R. Laprade and J.-Y. Lapointe (1994). Coupling between transepithelial Na transport and basolateral K conductance in renal proximal tubule. Am. J. Physiol. 266, F517–F527.
Dawson, D. C. and N. W. Richards (1990). Basolateral K conductance: role in regulation of NaCl absorption and secretion. Am. J. Physiol. 259, C181–C195.
Grasset, E., P. Gunter-Smith and S. G. Schultz (1983). Effects of Na-coupled alanine transport on intracellular K activities and the K conductance of the basolateral membranes of Necturus small intestine. J. Memb. Biol. 71, 89–94.
Gunter-Smith, P. J., E. Grasset and S. G. Schultz (1982). Sodium-coupled amino acid and sugar transport by Necturus small intestine. An equivalent electrical circuit analysis of a rheogenic co-transport system. J. Memb. Biol. 66, 25–39.
Hernandez, J. A. (2003). Stability properties of elementary dynamic models of membrane transport. Bull. Math. Biol. 65, 175–197.
Hewer, G. and C. Kenney (1988). The sensitivity of the stable Lyapunov equation. SIAM J. Control Optimization 26, 321–344.
Kirk, K. L., J. A. Schafer and D. R. DiBona (1987). Cell volume regulation in rabbit proximal straight tubule perfused in vitro. Am. J. Physiol. 252, F933–F942.
Lang, F., G. L. Busch, M. Ritter, H. Volkl, S. Waldegger, E. Gulbins and D. Haussinger (1998a). Functional significance of cell volume regulatory mechanisms. Physiol. Rev. 78, 247–306.
Lang, F., G. L. Busch and H. Volkl (1998b). The diversity of volume regulatory mechanisms. Cell. Physiol. Biochem. 8, 1–45.
Lapointe, J.-Y. and M. Duplain (1991). Regulation of basolateral membrane potential after stimulation of Na+ transport in proximal tubules. J. Membr. Biol. 120, 165–172.
Schultz, S. G. (1981). Homocellular regulatorymechanisms in sodium-transporting epithelia: avoidance of extinction by ‘flush-through’. Am. J. Physiol. 241, F579–F590.
Schultz, S. G. (1992). Membrane cross-talk in sodium-absorbing epithelial cells, in The Kidney. Physiology and Pathophysiology, D. W. Seldin and G. Giebisch (Eds), New York: Raven Press, pp. 287–299 (Chapter 11).
Trentelman, H. L., A. A. Stoorvogel and M. Hautus (2001). Control Theory for Linear Systems, London: Springer, pp. 1–389.
Tsuchiya, K., W. Wang, G. Giebisch and P. A. Welling (1992). ATP is a coupling modulator of parallel Na, K-ATPase-K-channel activity in the renal proximal tubule. Proc. Natl Acad. Sci. 89, 6418–6422.
Weinstein, A. M. (1992). Chloride transport in a mathematical model of the rat proximal tubule. Am. J. Physiol. 263, F784–F798.
Weinstein, A. M. (1994). Ammonia transport in a mathematical model of rat proximal tubule. Am. J. Physiol. 267, F237–F248.
Weinstein, A. M. (1996). Coupling of entry to exit by peritubular K+-permeability in a mathematical model of the rat proximal tubule. Am. J. Physiol. 271, F158–F168.
Weinstein, A. M. (1997). Dynamics of cellular homeostasis: recovery time for a perturbation from equilibrium. Bull. Math. Biol. 59, 451–481.
Weinstein, A. M. (1999). Modeling epithelial cell homeostasis: steady-state analysis. Bull. Math. Biol. 61, 1065–1091.
Weinstein, A. M. (2002). Assessing homeostatic properties of epithelial cell models: application to kidney proximal tubule, in Membrane Transport and Renal Physiology, H. E. Layton and A. M. Weinstein (Eds), The IMA volumes in Mathematics and its Applications 129, New York: Springer, pp. 119–140.
Welling, P. A. and M. A. Linshaw (1988). Importance of anion in hypotonic volume regulation of rabbit proximal straight tubule. Am. J. Physiol. 255, F853–F860.
Wonham, W. M. (1985). Linear Multivariable Control: A Geometric Approach, New York: Springer, pp. 1–334.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Weinstein, A.M. Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms. Bull. Math. Biol. 66, 1201–1240 (2004). https://doi.org/10.1016/j.bulm.2003.12.002
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1016/j.bulm.2003.12.002