Dynamics of cellular homeostasis: Recovery time for a perturbation from equilibrium

In the collecting ductin vivo, the principal cell encounters a wide range in luminal flow rate and luminal concentration of NaCl. As a consequence, there are substantial variations in the transcellular fluxes of Na⁺ and Cl⁻, conditions which would be expected to perturb cell volume and cytosolic concentrations. Several control mechanisms have been identified which can potentially blunt these perturbations, and these entail cellular regulation of the luminal membrane Na⁺ channel and peritubular membrane K⁺ and Cl⁻ channels. To illustrate the impact of these regulated channels, a mathematical model of the principal cell of the rat cortical collecting duct has been developed, in which ion channel permeabilities are either constant or regulated. In comparison to the model with fixed permeabilities, the model with regulated channels demonstrates enhanced cellular homeostasis following steady-state variation in luminal NaCl. However, in the transient response to a cytosolic perturbation, the difference in recovery time between the models is small. An approximate analysis is presented which casts these models as dynamical systems with constant coefficients. Despite the presence of regulated ion channels, concordance of each model with its linear approximation is verified for experimentally meaningful perturbations from the reference condition. Solution of a Lyapunov equation for each linear system yields a matrix whose application to a perturbation permits explicit estimation of the time to recovery. Comparison of these solution matrices for regulated and non-regulated cells confirms the similarity of the dynamic response of the two models. These calculations suggest that enhanced homeostasis by regulated channels may be protective, without necessarily hastening recovery from cellular perturbations.