Mathematical modeling of ion homeostasis & cell volume stabilization: impact of ion transporters, impermeant molecules, & Donnan effect

The pump-leak mechanism (PLM) first, described by Tosteson and Hoffman (1960), demonstrates how the activity of the Na+ − K+ ATPase (NKA) can counteract the osmotic influx of water stimulated by the presence of impermeant intracellular molecules. We derive analytical solutions for the steady state ion concentrations, voltage, and volume of a cell, by including impermeant extracellular molecules, variable impermeant charge, and Cation-Chloride Co-transporters (CCC). We demonstrate that impermeant extracellular molecules could stabilize a cell without NKA activity but argue that it is unlikely to play a significant role in vivo. Significantly we have shown that the precise form of the NKA is unimportant for determining the steady state in PLMs. We have derived an analytical expression for the steady state of the PLM with one of the Cation-Chloride Co-transporters, either KCC, NCC, or NKCC, active. Notably, we have demonstrated that NCC at high pump rates can destabilize cells, which could account for the rarity of this co-transporter. In addition, we show that the reversal of any of the CCCs is unlikely. Importantly, we link the thermodynamics of the NKA to the PLM to show that there is a natural limit to the energy utilized by the PLM that prevents futile cycles. We show that the average charge on the intracellular impermeant molecules influences ion distributions but has no impact on energy utilization. Our study shows that analytical mathematical solutions from physically well-grounded models provide insight into ion transport systems that could only be obtained from numerical simulations with great difficulty. Significance Statement The regulation of cell volume is fundamental to the stability of all tissue. Animal cells regulate their volume by actively pumping sodium and potassium ions, preventing the water’s osmotic influx from blowing up the cell. Based on the physical laws that determine ion and water fluxes, we derive equations that allow one to predict how pump rates and ion conductances combine to stabilize cell volume. The action of the sodium pump consumes about 30% of a cell’s energy budget, and we demonstrate the rate of ion pumping is constrained so that cells do not consume excessive energy. Our work also demonstrates the power of closed-form mathematical equations in characterizing such pump-leak systems.