A quantitative analysis of cell volume and resting potential determination and regulation in excitable cells

Abstract

This paper quantifies recent experimental results through a general physical description of the mechanisms that might control two fundamental cellular parameters, resting potential (E m ) and cell volume (V c ), thereby clarifying the complex relationships between them. E m was determined directly from a charge difference (CD) equation involving total intracellular ionic charge and membrane capacitance (C m ). This avoided the equilibrium condition dE m /dt = 0 required in determinations of E m by previous work based on the Goldman-Hodgkin-Katz equation and its derivatives and thus permitted precise calculation of E m even under non-equilibrium conditions. It could accurately model the influence upon E m of changes in C m or V c and of membrane transport processes such as the Na + -K + -ATPase and ion cotransport. Given a stable and adequate membrane Na + -K + -ATPase density (N ), V c and E m both converged to unique steady-state values even from sharply divergent initial intracellular ionic concentrations. For any constant set of transmembrane ion permeabilities, this set point of V c was then determined by the intracellular membrane-impermeant solute content (X − i ) and its mean charge valency (z X ), while in contrast, the set point of E m was determined solely by z X . Independent changes in membrane Na + (P Na ) or K + permeabilities (P K ) or activation of cation-chloride cotransporters could perturb V c and E m but subsequent reversal of such changes permitted full recovery of both V c and E m to the original set points. Proportionate changes in P Na , P K and N , or changes in Cl − permeability (P Cl ) instead conserved steady-state V c and E m but altered their rates of relaxation following any discrete perturbation. P Cl additionally determined the relative effect of cotransporter activity on V c and E m , in agreement with recent experimental results. In contrast, changes in X i − produced by introduction of a finite permeability term to X − (P X ) that did not alter z X caused sustained changes in V c that were independent of E m and that persisted when P X returned to zero. Where such fluxes also altered the effective z X they additionally altered the steady state E m . This offers a basis for the suggested roles of amino acid fluxes in long-term volume regulatory processes in a variety of excitable tissues.

Quantitative estimation of transmembrane ion transport in rat renal collecting duct principal cells

“…These models were first introduced by Tosteson and Hoffman (Tosteson and Hoffman 1960). Subsequent development of such models was described by several authors (Jakobsson 1980; Hernandez and Cristina 1998;Fraser and Huang 2004). Various models of this type differ by the equations that govern transmembrane ion fluxes, and by the number of intracellular osmolytes and mechanisms of transmembrane transport considered (Keener and Sneyd 1998).…”

“…The mathematical model used in the present work is similar to that proposed in Fraser and Huang (2004), Ilyaskin et al (2011), Zarogiannis et al (2013). The model includes a set of ordinary differential equations, which describe the changes of the cell volume V and intracellular amounts of K + , Na + , and Cl -(n Na , n K , n Cl ) due to the transmembrane fluxes of ions and water through their respective channels and the Na/K-pump (Fig.…”

“…E m was calculated at each time step using one of the two different approaches. In the first one, called the charge difference (CD) method, E m was expressed directly through the membrane capacitance (C m ) and the total intracellular charge as discussed by Fraser and Huang previously (Fraser and Huang 2004):…”

“…The amount of organic anions (n X ) was chosen so that the calculated stationary cell volume V was close to the size of OMCD cells (~1 × 10 -9 cm 3 ). The mean charge of organic anions z was set equal to -1.5 according to assumptions made by Fraser (Fraser and Huang 2004) to balance steady-state intracellular cation concentrations typical for OMCD cells (Gifford et al 1990). …”

“…The permeability parameters of the model are coefficients in equations of electrodiffusion (P Na , P K , P Cl ) and surface density of Na/KATPase (N pump ). Despite the fact that other free parameters such as z (mean organic osmolyte valence) and n X (intracellular amount of organic anions) have an influence on the values of steady state characteristics (Fraser and Huang 2004), the investigation of the their impact on the model behavior was out of the scope of the current work and thus their values were kept constant (see "Methods of mathematical modeling" in Materials and Methods section).…”