Interactions of cell volume, membrane potential, and membrane transport parameters

Mathematical modeling of intracellular osmolarity and cell volume stabilization: The Donnan effect and ion transport

The presence of impermeant molecules within a cell can lead to an increase in cell volume through the influx of water driven by osmosis. This phenomenon is known as the Donnan (or Gibbs-Donnan) effect. Animal cells actively transport ions to counteract the Donnan effect and regulate their volume, actively pumping Na+ out and K+ into their cytosol using the Na+/K+ ATPase (NKA) pump. The pump-leak equations (PLEs) are a system of algebraic-differential equations to model the membrane potential, ion (Na+, K+, and Cl-), and water flux across the cell membrane, which provide insight into how the combination of passive ions fluxes and active transport contribute to stabilizing cell volume. Our broad objective is to provide analytical insight into the PLEs through three lines of investigation: (1) we show that the provision of impermeant extracellular molecules can stabilize the volume of a passive cell; (2) we demonstrate that the mathematical form of the NKA pump is not as important as the stoichiometry for cell stabilization; and (3) we investigate the interaction between the NKA pump and cation-chloride co-transporters (CCCs) on cell stabilization, showing that NCC can destabilize a cell while NKCC and KCC can stabilize it. We incorporate extracellular impermeant molecules, NKA pump, and CCCs into the PLEs and derive the exact formula for the steady states in terms of all the parameters. This analytical expression enables us to easily explore the effect of each of the system parameters on the existence and stability of the steady states.

The Na+,K+-ATPase and its stoichiometric ratio: some thermodynamic speculations

 Almost seventy years after its discovery, the sodium-potassium adenosine triphosphatase (the sodium pump) located in the cell plasma membrane remains a source of novel mechanistic and physiologic findings. A noteworthy feature of this enzyme/transporter is its robust stoichiometric ratio under physiological conditions: it sequentially counter-transports three sodium ions and two potassium ions against their electrochemical potential gradients per each hydrolyzed ATP molecule. Here we summarize some present knowledge about the sodium pump and its physiological roles, and speculate whether energetic constraints may have played a role in the evolutionary selection of its characteristic stoichiometric ratio.

Mesenchymal Stem/Stromal Cells in Three-Dimensional Cell Culture: Ion Homeostasis and Ouabain-Induced Apoptosis

This study describes the changes in ion homeostasis of human endometrial mesenchymal stem/stromal cells (eMSCs) during the formation of three-dimensional (3D) cell structures (spheroids) and investigates the conditions for apoptosis induction in 3D eMSCs. Detached from the monolayer culture, (2D) eMSCs accumulate Na⁺ and have dissipated transmembrane ion gradients, while in compact spheroids, eMSCs restore the lower Na⁺ content and the high K/Na ratio characteristic of functionally active cells. Organized as spheroids, eMSCs are non-proliferating cells with an active Na/K pump and a lower K⁺ content per g cell protein, which is typical for quiescent cells and a mean lower water content (lower hydration) in 3D eMSCs. Further, eMSCs in spheroids were used to evaluate the role of K⁺ depletion and cellular signaling context in the induction of apoptosis. In both 2D and 3D eMSCs, treatment with ouabain (1 µM) results in inhibition of pump-mediated K⁺ uptake and severe K⁺ depletion as well as disruption of the mitochondrial membrane potential. In 3D eMSCs (but not in 2D eMSCs), ouabain initiates apoptosis via the mitochondrial pathway. It is concluded that, when blocking the Na/K pump, cardiac glycosides prime mitochondria to apoptosis, and whether a cell enters the apoptotic pathway depends on the cell-specific signaling context, which includes the type of apoptotic protein expressed.

Unidirectional fluxes of monovalent ions in human erythrocytes compared with lymphoid U937 cells: Transient processes after stopping the sodium pump and in response to osmotic challenge

Recently, we have developed software that allows, using a minimum of required experimental data, to find the characteristics of ion homeostasis and a list of all unidirectional fluxes of monovalent ions through the main pathways in the cell membrane both in a balanced state and during the transient processes. Our approach has been successfully validated in human proliferating lymphoid U937 cells during transient processes after stopping the Na/K pump by ouabain and for staurosporine-induced apoptosis. In present study, we used this approach to find the characteristics of ion homeostasis and the monovalent ion fluxes through the cell membrane of human erythrocytes in a resting state and during the transient processes after stopping the Na/K pump with ouabain and in response to osmotic challenge. Due to their physiological significance, erythrocytes remain the object of numerous studies, both experimental and computational methods. Calculations showed that, under physiological conditions, the K+ fluxes through electrodiffusion channels in the entire erythrocyte ion balance is small compared to the fluxes through the Na/K pump and cation-chloride cotransporters. The proposed computer program well predicts the dynamics of the erythrocyte ion balance disorders after stopping the Na/K pump with ouabain. In full accordance with predictions, transient processes in human erythrocytes are much slower than in proliferating cells such as lymphoid U937 cells. Comparison of real changes in the distribution of monovalent ions under osmotic challenge with the calculated ones indicates a change in the parameters of the ion transport pathways through the plasma membrane of erythrocytes in this case. The proposed approach may be useful in studying the mechanisms of various erythrocyte dysfunctions.

Monovalent ions and stress-induced senescence in human mesenchymal endometrial stem/stromal cells

Monovalent ions are involved in growth, proliferation, differentiation of cells as well as in their death. This work concerns the ion homeostasis during senescence induction in human mesenchymal endometrium stem/stromal cells (hMESCs): hMESCs subjected to oxidative stress (sublethal pulse of H₂O₂) enter the premature senescence accompanied by persistent DNA damage, irreversible cell cycle arrest, increased expression of the cell cycle inhibitors (p53, p21) cell hypertrophy, enhanced β-galactosidase activity. Using flame photometry to estimate K⁺, Na⁺ content and Rb⁺ (K⁺) fluxes we found that during the senescence development in stress-induced hMESCs, Na⁺/K⁺pump-mediated K⁺ fluxes are enhanced due to the increased Na⁺ content in senescent cells, while ouabain-resistant K⁺ fluxes remain unchanged. Senescence progression is accompanied by a peculiar decrease in the K⁺ content in cells from 800-900 to 500-600 µmol/g. Since cardiac glycosides are offered as selective agents for eliminating senescent cells, we investigated the effect of ouabain on ion homeostasis and viability of hMESCs and found that in both proliferating and senescent hMESCs, ouabain (1 nM-1 µM) inhibited pump-mediated K⁺ transport (ID₅₀ 5 × 10^-8 M), decreased cell K⁺/Na⁺ ratio to 0.1-0.2, however did not induce apoptosis. Comparison of the effect of ouabain on hMESCs with the literature data on the selective cytotoxic effect of cardiac glycosides on senescent or cancer cells suggests the ion pump blockade and intracellular K⁺ depletion should be synergized with target apoptotic signal to induce the cell death.

Cation-Chloride Cotransporters, Na/K Pump, and Channels in Cell Water/Ionic Balance Regulation Under Hyperosmolar Conditions: In Silico and Experimental Studies of Opposite RVI and AVD Responses of U937 Cells to Hyperosmolar Media

Studying the transport of monovalent ions across the cell membrane in living cells is complicated by the strong interdependence of fluxes through parallel pathways and requires therefore computational analysis of the entire electrochemical system of the cell. Current paper shows how to calculate changes in the cell water balance and ion fluxes caused by changes in the membrane channels and transporters during a normal regulatory increase in cell volume in response to osmotic cell shrinkage (RVI) followed by a decrease in cell volume associated with apoptosis (AVD). Our recently developed software is used as a computational analysis tool and the established human lymphoid cells U937 are taken as an example of proliferating animal cells. It is found that, in contrast to countless statements in the literature that cell volume restoration requires the activation of certain ion channels and transporters, the cellular responses such as RVI and AVD can occur in an electrochemical system like U937 cells without any changes in the state of membrane channels or transporters. These responses depend on the types of chloride cotransporters in the membrane and differ in a hyperosmolar medium with additional sucrose and in a medium with additional NaCl. This finding is essential for the identification of the true changes in membrane channels and transporters responsible for RVI and AVD in living cells. It is determined which changes in membrane parameters predicted by computational analysis are consistent with experimental data obtained on living human lymphoid cells U937, Jurkat, and K562 and which are not. An essential part of the results is the developed software that allows researchers without programming experience to calculate the fluxes of monovalent ions via the main transmembrane pathways and electrochemical gradients that move ions across the membrane. The software is available for download. It is useful for studying the functional expression of the channels and transporters in living cells and understanding how the cell electrochemical system works.

Cation-Chloride Cotransporters, Na/K Pump, and Channels in Cell Water and Ion Regulation: In silico and Experimental Studies of the U937 Cells Under Stopping the Pump and During Regulatory Volume Decrease

Cation-coupled chloride cotransporters play a key role in generating the Cl^- electrochemical gradient on the cell membrane, which is important for regulation of many cellular processes. However, a quantitative analysis of the interplay between numerous membrane transporters and channels in maintaining cell ionic homeostasis is still undeveloped. Here, we demonstrate a recently developed approach on how to predict cell ionic homeostasis dynamics when stopping the sodium pump in human lymphoid cells U937. The results demonstrate the reliability of the approach and provide the first quantitative description of unidirectional monovalent ion fluxes through the plasma membrane of an animal cell, considering all the main types of cation-coupled chloride cotransporters operating in a system with the sodium pump and electroconductive K⁺, Na⁺, and Cl^- channels. The same approach was used to study ionic and water balance changes associated with regulatory volume decrease (RVD), a well-known cellular response underlying the adaptation of animal cells to a hypoosmolar environment. A computational analysis of cell as an electrochemical system demonstrates that RVD may happen without any changes in the properties of membrane transporters and channels due to time-dependent changes in electrochemical ion gradients. The proposed approach is applicable when studying truly active regulatory processes mediated by the intracellular signaling network. The developed software can be useful for calculation of the balance of the unidirectional fluxes of monovalent ions across the cell membrane of various cells under various conditions.

The importance of water and hydraulic pressure in cell dynamics

All mammalian cells live in the aqueous medium, yet for many cell biologists, water is a passive arena in which proteins are the leading players that carry out essential biological functions. Recent studies, as well as decades of previous work, have accumulated evidence to show that this is not the complete picture. Active fluxes of water and solutes of water can play essential roles during cell shape changes, cell motility and tissue function, and can generate significant mechanical forces. Moreover, the extracellular resistance to water flow, known as the hydraulic resistance, and external hydraulic pressures are important mechanical modulators of cell polarization and motility. For the cell to maintain a consistent chemical environment in the cytoplasm, there must exist an intricate molecular system that actively controls the cell water content as well as the cytoplasmic ionic content. This system is difficult to study and poorly understood, but ramifications of which may impact all aspects of cell biology from growth to metabolism to development. In this Review, we describe how mammalian cells maintain the cytoplasmic water content and how water flows across the cell surface to drive cell movement. The roles of mechanical forces and hydraulic pressure during water movement are explored.

On the Effects of Mechanical Stress of Biological Membranes in Modeling of Swelling Dynamics of Biological Systems

We highlight mechanical stretching and bending of membranes and the importance of membrane deformations in the analysis of swelling dynamics of biological systems, including cells and subcellular organelles. Membrane deformation upon swelling generates tensile stress and internal pressure, contributing to volume changes in biological systems. Therefore, in addition to physical (internal/external) and chemical factors, mechanical properties of the membranes should be considered in modeling analysis of cellular swelling. Here we describe an approach that considers mechanical properties of the membranes in the analysis of swelling dynamics of biological systems. This approach includes membrane bending and stretching deformations into the model, producing a more realistic description of swelling. We also discuss the effects of membrane stretching on swelling dynamics. We report that additional pressure generated by membrane bending is negligible, compared to pressures generated by membrane stretching, when both membrane surface area and volume are variable parameters. Note that bending deformations are reversible, while stretching deformation may be irreversible, leading to membrane disruption when they exceed a certain threshold level. Therefore, bending deformations need only be considered in reversible physiological swelling, whereas stretching deformations should also be considered in pathological irreversible swelling. Thus, the currently proposed approach may be used to develop a detailed biophysical model describing the transition from physiological to pathological swelling mode.

Intracellular K+ and water content in human blood lymphocytes during transition from quiescence to proliferation

Many evidence shows that K⁺ ions are required for cell proliferation, however, changes in intracellular K⁺ concentration during transition of cells from quiescence to cycling are insufficiently studied. Here, we show using flame emission assay that a long-term increase in cell K⁺ content per g cell protein is a mandatory factor for transition of quiescent human peripheral blood lymphocytes (PBL) to proliferation induced by phytohemagglutinin, phorbol ester with ionomycin, and anti-CD3 antibodies with interleukin-2 (IL-2). The long-term increase in K⁺ content is associated with IL-2-dependent stage of PBL activation and accompanies the growth of small lymphocytes and their transformation into blasts. Inhibition of PBL proliferation with drugs specific for different steps of G0/G1/S transit prevented both blast-transformation and an increase in K⁺ content per cell protein. Determination of the water content in cells by measuring the density of cells in the Percoll gradient showed that, unlike the K⁺ content, the concentration of K⁺ in cell water remains unchanged, since water and K⁺ change in parallel. Correlation of proliferation with high cell K⁺ and water content has been confirmed by the data obtained in comparative study of PBL and permanently cycling Jurkat cells. Our data suggest that K⁺ is important for successful proliferation as the main intracellular ion that participates in regulation of cell water content during cell transition from quiescence to proliferation. We concluded that high K⁺ content in cells and the associated high water content is a characteristic feature of proliferating cells.

A Tool for Computation of Changes in Na+, K+, Cl- Channels and Transporters Due to Apoptosis by Data on Cell Ion and Water Content Alteration

Monovalent ions are involved in a vast array of cellular processes. Their movement across the cell membrane is regulated by numerous channels and transporters. Identification of the pathways responsible for redistribution of ions and cell water in living cells is hampered by their strong interdependence. This difficulty can be overcome by computational analysis of the whole cell flux balance. Our previous computational studies were concerned with monovalent ion fluxes in cells under the conditions of balanced ion distribution or during transition processes after stopping the Na+/K+ pump. Here we analyze a more complex case-redistribution of ions during cell apoptosis when the parameters keep changing during the process. New experimental data for staurosporine-induced apoptosis of human lymphoma cells U937 have been obtained: the time course of changes in cellular K+, Na+, Cl-, and water content, as well as Rb+ fluxes as a marker of the Na/K pump activity. Using a newly developed computational tool, we found that alteration of ion and water balance was associated with a 55% decrease in the Na+/K+-ATPase rate coefficient over a 4-h period, with a time-dependent increase in potassium channel permeability, and a decrease in sodium channel permeability. The early decrease in [Cl-]i and cell volume were associated with an ~5-fold increase in chloride channel permeability. The developed approach and the presented executable file can be used to identify the channels and transporters responsible for alterations of cell ion and water balance not only during apoptosis but in other physiological scenarios.

The logic of ionic homeostasis: Cations are for voltage, but not for volume

Neuronal activity is associated with transmembrane ionic redistribution, which can lead to an osmotic imbalance. Accordingly, activity-dependent changes of the membrane potential are sometimes accompanied by changes in intracellular and/or extracellular volume. Experimental data that include distributions of ions and volume during neuronal activity are rare and rather inconsistent partly due to the technical difficulty of performing such measurements. However, progress in understanding the interrelations among ions, voltage and volume has been achieved recently by computational modelling, particularly “charge-difference” modelling. In this work a charge-difference computational model was used for further understanding of the specific roles for cations and anions. Our simulations show that without anion conductances the transmembrane movements of cations are always osmotically balanced, regardless of the stoichiometry of the pump or the ratio of Na⁺ and K⁺ conductances. Yet any changes in cation conductance or pump activity are associated with changes of the membrane potential, even when a hypothetically electroneutral pump is used in calculations and K⁺ and Na⁺ conductances are equal. On the other hand, when a Cl^- conductance is present, the only way to keep the Cl^-equilibrium potential in accordance with the changed membrane potential is to adjust cell volume. Importantly, this voltage-evoked Cl^--dependent volume change does not affect intracellular cation concentrations or the amount of energy that is necessary to support the system. Taking other factors into consideration (i.e. the presence of internal impermeant poly-anions, the activity of cation-Cl^- cotransporters, and the buildup of intra- and extracellular osmolytes, both charged and electroneutral) adds complexity, but does not change the main principles.

We have developed software that calculates membrane potential and cell volume that result from redistribution of principal ions (K⁺, Na⁺, and Cl^-) during normal cellular activity and experimental manipulations. Calculations in the model are done by an iterative charge-difference method that makes few assumptions about governing equations. Most of the features that were considered to be important for volume and voltage regulation were incorporated in the model, including the unique capability to perform calculations with different values of transmembrane water permeability. We have used the program to reexamine interactions between ionic fluxes, membrane potential, and cell volume and found that there was a previously unappreciated difference in the way that the distribution of cations and anions affect the cell. Na⁺ and K⁺, which are distributed unevenly across the membrane by the Na⁺/K⁺-ATPase, are primarily responsible for the membrane potential, but, contrary to popular belief, do not directly participate in volume regulation. On the other hand, the Cl^- conductance determines the extent of volume changes, because Cl^- has to follow the changes of membrane potential, which inevitably leads to changes in cell volume. The software is available to download and use for other investigations.

Evolution of our understanding of cell volume regulation by the pump-leak mechanism

Kay and Blaustein recount the history of the pump-leak mechanism, which stabilizes cell volume by pumping sodium ions out of cells.

All animal cells are surrounded by a flexible plasma membrane that is permeable to water and to small ions. Cells thus face a fundamental problem: the considerable tension that their membranes would experience if the osmotic influx of water, driven by the presence of impermeant intracellular ions, was left unopposed. The pivotal study that described the cell’s remedy for this impending osmotic catastrophe—the “pump-leak mechanism” (PLM)—was published in the Journal of General Physiology by Tosteson and Hoffman in 1960. Their work revealed how the sodium pump stabilizes cell volume by eliminating the osmotic gradient. Here we describe the mechanistic basis of the PLM, trace the history of its discovery, and place it into the context of our current understanding.

Proliferation-related changes in K+ content in human mesenchymal stem cells

Intracellular monovalent ions have been shown to be important for cell proliferation, however, mechanisms through which ions regulate cell proliferation is not well understood. Ion transporters may be implicated in the intracellular signaling: Na⁺ and Cl⁻ participate in regulation of intracellular pH, transmembrane potential, Ca²⁺ homeostasis. Recently, it is has been suggested that K⁺ may be involved in “the pluripotency signaling network”. Our study has been focused on the relations between K⁺ transport and stem cell proliferation. We compared monovalent cation transport in human mesenchymal stem cells (hMSCs) at different passages and at low and high densities of culture as well as during stress-induced cell cycle arrest and revealed a decline in K⁺ content per cell protein which was associated with accumulation of G1 cells in population and accompanied cell proliferation slowing. It is suggested that cell K⁺ may be important for successful cell proliferation as the main intracellular ion that participates in regulation of cell volume during cell cycle progression. It is proposed that cell K⁺ content as related to cell protein is a physiological marker of stem cell proliferation and may be used as an informative test for assessing the functional status of stem cells in vitro.

Quantitative Model for Ion Transport and Cytoplasm Conductivity of Chinese Hamster Ovary Cells

In mammalian cells cytoplasm ion concentrations and hence cytoplasm conductivity is an important indicator of their physiological state. Changes in the cytoplasm conductivity has been associated with physiological changes such as progression of cancer and apoptosis. In this work, a model that predicts the effects of physiological changes in ion transport on the cytoplasm conductivity of Chinese hamster ovary (CHO) cells is demonstrated. We determined CHO-specific model parameters, Na⁺/K⁺ ATPase pumps and ion channels densities, using a flux assay approach. The obtained sodium (P_Na), potassium (P_K) and chloride (P_Cl) permeability and Na⁺/K⁺ ATPase pump density were estimated to be 5.6 × 10^-8 cm/s, 5.6 × 10^-8 cm/s, 3.2 × 10^-7 cm/s and 2.56 × 10^-11 mol/cm², respectively. The model was tested by comparing the model predictions with the experimentally determined temporal changes in the cytoplasm conductivity of Na⁺/K⁺ ATPase pump inhibited CHO cells. Cells' Na⁺/K⁺ ATPase pumps were inhibited using 5 mM Ouabain and the temporal behavior of their cytoplasm conductivity was measured using dielectrophoresis cytometry. The measured results are in close agreement with the model-calculated values. This model will provide insight on the effects of processes such as apoptosis or external media ion concentration on the cytoplasm conductivity of mammalian cells.

Calibration and characterization of intracellular Asante Potassium Green probes, APG-2 and APG-4

The response of fluorescent ion probes to ions is affected by intracellular environment. To properly calibrate them, intracellular and extracellular concentrations of the measured ion must be made equal. In the first, computational, part of this work, we show, using the example of potassium, that the two requirements for ion equilibration are complete dissipation of membrane potential and high membrane permeability for both potassium and sodium. In the second part, we tested the ability of various ionophores to achieve potassium equilibration in Jurkat and U937 cells and found a combination of valinomycin, nigericin, gramicidin and ouabain to be the most effective. In the third part, we applied this protocol to two potassium probes, APG-4 and APG-2. APG-4 shows good sensitivity to potassium but its fluorescence is sensitive to cell volume. Because ionophores cause cell swelling, calibration buffers had to be supplemented with 50 mM sucrose to keep cell volume constant. With these precautions taken, the average potassium concentrations in U937 and Jurkat cells were measured at 132 mM and 118 mM, respectively. The other tested probe, APG-2, is nonselective for cations; this is, however, a potentially useful property because the sum [K⁺] + [Na⁺] determines the amount of intracellular water.

How Cells Can Control Their Size by Pumping Ions

The ability of all cells to set and regulate their size is a fundamental aspect of cellular physiology. It has been known for sometime but not widely so, that size stability in animal cells is dependent upon the operation of the sodium pump, through the so-called pump-leak mechanism (Tosteson and Hoffman, 1960). Impermeant molecules in cells establish an unstable osmotic condition, the Donnan effect, which is counteracted by the operation of the sodium pump, creating an asymmetry in the distribution of Na⁺ and K⁺ staving off water inundation. In this paper, which is in part a tutorial, I show how to model quantitatively the ion and water fluxes in a cell that determine the cell volume and membrane potential. The movement of water and ions is constrained by both osmotic and charge balance, and is driven by ion and voltage gradients and active ion transport. Transforming these constraints and forces into a set of coupled differential equations allows us to model how the ion distributions, volume and voltage change with time. I introduce an analytical solution to these equations that clarifies the influence of ion conductances, pump rates and water permeability in this multidimensional system. I show that the number of impermeant ions (x) and their average charge have a powerful influence on the distribution of ions and voltage in a cell. Moreover, I demonstrate that in a cell where the operation of active ion transport eliminates an osmotic gradient, the size of the cell is directly proportional to x. In addition, I use graphics to reveal how the physico-chemical constraints and chemical forces interact with one another in apportioning ions inside the cell. The form of model used here is applicable to all membrane systems, including mitochondria and bacteria, and I show how pumps other than the sodium pump can be used to stabilize cells. Cell biologists may think of electrophysiology as the exclusive domain of neuroscience, however the electrical effects of ion fluxes need to become an intimate part of cell biology if we are to understand a fundamental process like cell size regulation.

Unidirectional Flux Balance of Monovalent Ions in Cells with Na/Na and Li/Na Exchange: Experimental and Computational Studies on Lymphoid U937 Cells

Monovalent ion traffic across the cell membrane occurs via various pathways. Evaluation of individual fluxes in whole cell is hampered by their strong interdependence. This difficulty can be overcome by computational analysis of the whole cell flux balance. However, the previous computational studies disregarded ion movement of the self-exchange type. We have taken this exchange into account. The developed software allows determination of unidirectional fluxes of all monovalent ions via the major pathways both under the balanced state and during transient processes. We show how the problem of finding the rate coefficients can be solved by measurement of monovalent ion concentrations and some of the fluxes. Interdependence of fluxes due to the mandatory conditions of electroneutrality and osmotic balance and due to specific effects can be discriminated, enabling one to identify specific changes in ion transfer machinery under varied conditions. To test the effectiveness of the developed approach we made use of the fact that Li/Na exchange is known to be an analogue of the coupled Na/Na exchange. Thus, we compared the predicted and experimental data obtained on U937 cells under varied Li+ concentrations and following inhibition of the sodium pump with ouabain. We found that the coupled Na/Na exchange in U937 cells comprises a significant portion of the entire Na+ turnover. The data showed that the loading of the sodium pump by Li/Na exchange involved in the secondary active Li+ transport at 1-10 mM external Li+ is small. This result may be extrapolated to similar Li+ and Na+ flux relationships in erythrocytes and other cells in patients treated with Li+ in therapeutic doses. The developed computational approach is applicable for studying various cells and can be useful in education for demonstrating the effects of individual transporters and channels on ion gradients, cell water content and membrane potential.

Anions Govern Cell Volume: A Case Study of Relative Astrocytic and Neuronal Swelling in Spreading Depolarization

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little about the dynamics of cell and tissue swelling, and the differential changes in the volumes of neurons and glia during pathological states such as spreading depolarizations (SD) under ischemic and non–ischemic conditions, and epileptic seizures. By combining the Hodgkin–Huxley type spiking dynamics, dynamic ion concentrations, and simultaneous neuronal and astroglial volume changes into a comprehensive model, we elucidate why glial cells swell more than neurons in SD and the special case of anoxic depolarization (AD), and explore the relative contributions of the two cell types to tissue swelling. Our results demonstrate that anion channels, particularly Cl⁻, are intrinsically connected to cell swelling and blocking these currents prevents changes in cell volume. The model is based on a simple and physiologically realistic description. We introduce model extensions that are either derived purely from first physical principles of electroneutrality, osmosis, and conservation of particles, or by a phenomenological combination of these principles and known physiological facts. This work provides insights into numerous studies related to neuronal and glial volume changes in SD that otherwise seem contradictory, and is broadly applicable to swelling in other cell types and conditions.

A Thermodynamic Model of Monovalent Cation Homeostasis in the Yeast Saccharomyces cerevisiae

Cationic and heavy metal toxicity is involved in a substantial number of diseases in mammals and crop plants. Therefore, the understanding of tightly regulated transporter activities, as well as conceiving the interplay of regulatory mechanisms, is of substantial interest. A generalized thermodynamic description is developed for the complex interplay of the plasma membrane ion transporters, membrane potential and the consumption of energy for maintaining and restoring specific intracellular cation concentrations. This concept is applied to the homeostasis of cation concentrations in the yeast cells of S. cerevisiae. The thermodynamic approach allows to model passive ion fluxes driven by the electrochemical potential differences, but also primary or secondary active transport processes driven by the inter- play of different ions (symport, antiport) or by ATP consumption (ATPases). The model-confronted with experimental data-reproduces the experimentally observed potassium and proton fluxes induced by the external stimuli KCl and glucose. The estimated phenomenological constants combine kinetic parameters and transport coefficients. These are in good agreement with the biological understanding of the transporters thus providing a better understanding of the control exerted by the coupled fluxes. The model predicts the flux of additional ion species, like e.g. chloride, as a potential candidate for counterbalancing positive charges. Furthermore, the effect of a second KCl stimulus is simulated, predicting a reduced cellular response for cells that were first exposed to a high KCl stimulus compared to cells pretreated with a mild KCl stimulus. By describing the generalized forces that are responsible for a given flow, the model provides information and suggestions for new experiments. Furthermore, it can be extended to other systems such as e.g. Candida albicans, or selected plant cells.

The Balance of Fluid and Osmotic Pressures across Active Biological Membranes with Application to the Corneal Endothelium

The movement of fluid and solutes across biological membranes facilitates the transport of nutrients for living organisms and maintains the fluid and osmotic pressures in biological systems. Understanding the pressure balances across membranes is crucial for studying fluid and electrolyte homeostasis in living systems, and is an area of active research. In this study, a set of enhanced Kedem-Katchalsky (KK) equations is proposed to describe fluxes of water and solutes across biological membranes, and is applied to analyze the relationship between fluid and osmotic pressures, accounting for active transport mechanisms that propel substances against their concentration gradients and for fixed charges that alter ionic distributions in separated environments. The equilibrium analysis demonstrates that the proposed theory recovers the Donnan osmotic pressure and can predict the correct fluid pressure difference across membranes, a result which cannot be achieved by existing KK theories due to the neglect of fixed charges. The steady-state analysis on active membranes suggests a new pressure mechanism which balances the fluid pressure together with the osmotic pressure. The source of this pressure arises from active ionic fluxes and from interactions between solvent and solutes in membrane transport. We apply the proposed theory to study the transendothelial fluid pressure in the in vivo cornea, which is a crucial factor maintaining the hydration and transparency of the tissue. The results show the importance of the proposed pressure mechanism in mediating stromal fluid pressure and provide a new interpretation of the pressure modulation mechanism in the in vivo cornea.

Monitoring the dielectric response of single cells following mitochondrial adenosine triphosphate synthase inhibition by oligomycin using a dielectrophoretic cytometer

One of the main uses of adenosine triphosphate (ATP) within mammalian cells is powering the Na(+)/K(+) ATPase pumps used to maintain ion concentrations within the cell. Since ion concentrations determine the cytoplasm conductivity, ATP concentration is expected to play a key role in controlling the cytoplasm conductivity. The two major ATP production pathways within cells are via glycolysis within the cytoplasm and via the electron transport chain within the mitochondria. In this work, a differential detector combined with dielectrophoretic (DEP) translation in a microfluidic channel was employed to observe single cell changes in the cytoplasm conductivity. The DEP response was made sensitive to changes in cytoplasm conductivity by measuring DEP response versus media conductivity and using double shell models to choose appropriate frequencies and media conductivity. Dielectric response of Chinese hamster ovary (CHO) cells was monitored following inhibition of the mitochondria ATP production by treatment with oligomycin. We show that in CHO cells following exposure to oligomycin (8 μg/ml) the cytoplasm conductivity drops, with the majority of the change occurring within 50 min. This work demonstrates that dielectric effects due to changes in ATP production can be observed at the single cell level.

Possible causes of apoptotic volume decrease: an attempt at quantitative review

Cell shrinkage and dehydration are essential characteristics of apoptosis, and loss of as much as half of the initial cell volume is not uncommon. This phenomenon is usually explained by efflux of K+and Cl−. We reexamine this hypothesis on the basis of the available data for ion concentrations and the requirements for osmotic equilibrium and electroneutrality. In addition to ion loss, we discuss the possible impacts of several other processes: efflux of low-molecular-weight osmolytes, acidification of the cytosol, effects of water channels and pumps, heterogeneity of intracellular water, and dissociation of apoptotic bodies. We conclude that most mammalian cells are theoretically capable of reducing their volume by 15–20% through ion loss or a decrease in cytosolic pH, although, in reality, the contribution of these mechanisms to apoptotic shrinkage may be smaller. Transitions between osmotically active and inactive water pools might influence cell volume as well; these mechanisms are poorly understood but are amenable to experimental study. Dissociation of apoptotic bodies is a separate mechanism of volume reduction and should be monitored closely; this can be best achieved by measurement of intracellular water, rather than cell volume.

Modeling the neuroprotective role of enhanced astrocyte mitochondrial metabolism during stroke

A mathematical model that integrates the dynamics of cell membrane potential, ion homeostasis, cell volume, mitochondrial ATP production, mitochondrial and endoplasmic reticulum Ca(2+) handling, IP3 production, and GTP-binding protein-coupled receptor signaling was developed. Simulations with this model support recent experimental data showing a protective effect of stimulating an astrocytic GTP-binding protein-coupled receptor (P2Y1Rs) following cerebral ischemic stroke. The model was analyzed to better understand the mathematical behavior of the equations and to provide insights into the underlying biological data. This approach yielded explicit formulas determining how changes in IP3-mediated Ca(2+) release, under varying conditions of oxygen and the energy substrate pyruvate, affected mitochondrial ATP production, and was utilized to predict rate-limiting variables in P2Y1R-enhanced astrocyte protection after cerebral ischemic stroke.

Steady-state solutions of cell volume in a cardiac myocyte model elaborated for membrane excitation, ion homeostasis and Ca2+ dynamics

The cell volume continuously changes in response to varying physiological conditions, and mechanisms underlying volume regulation have been investigated in both experimental and theoretical studies. Here, general formulations concerning cell volume change are presented in the context of developing a comprehensive cell model which takes Ca(2+) dynamics into account. Explicit formulas for charge conservation and steady-state volumes of the cytosol and endoplasmic reticulum (ER) are derived in terms of membrane potential, amount of ions, Ca(2+)-bound buffer molecules, and initial cellular conditions. The formulations were applied to a ventricular myocyte model which has plasma-membrane Ca(2+) currents with dynamic gating mechanisms, Ca(2+)-buffering reactions with diffusive and non-diffusive buffer proteins, and Ca(2+) uptake into or release from the sarcoplasmic reticulum (SR) accompanied by compensatory cationic or anionic currents through the SR membrane. Time-dependent volume changes in cardiac myocytes induced by varying extracellular osmolarity or by action potential generation were successfully simulated by the novel formulations. Through application of bifurcation analysis, the existence and uniqueness of steady-state solutions of the cell volume were validated, and contributions of individual ion channels and transporters to the steady-state volume were systematically analyzed. The new formulas are consistent with previous fundamental theory derived from simple models of minimum compositions. The new formulations may be useful for examination of the relationship between cell function and volume change in other cell types.

THE ROLE OF MEMBRANE ELECTROSTATICS IN THE REGULATION OF CELL VOLUME AND ION CONCENTRATIONS

We present a model to study how membrane surface negative charges can affect the electro-osmotic regulation properties of a cell. This model is based on the cellular analog proposed by Jakobsson, which includes passive and active ion transports; we further introduce the effect of membrane surface charges, using a generalized formulation of the Gouy–Chapman theory. We derive a system of nonlinear differential-algebraic equations (DAEs) which describes the dynamics of the cellular analog. The system admits a unique asymptotically stable stationary state, in which the Na-pump rate, which is crucial for electro-osmotic regulation, is inversely related to the Ca2+level in the extracellular milieu; numerical integration shows that this apparent inhibition of the Na-pump by external Ca2+results from a decrease in the electrostatic field produced by surface charges at the external side of the membrane. Furthermore, the degree of stability of the stationary state dramatically depends on the amount of negative charges on the membrane; a maximal stability is obtained for densities around - e /500 Å2, where the Na-pump is maximally activated by an increase in the Na content of the cytoplasm.

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.

Balance of unidirectional monovalent ion fluxes in cells undergoing apoptosis: why does Na+/K+ pump suppression not cause cell swelling?

Cells dying according to the apoptotic program, unlike cells dying via an unprogrammed mode, are able to avoid swelling and osmotic bursting with membrane disruption.There are indications that apoptosis is accompanied by suppression of the Na+/K+ pump and changes in the K+ and Cl− channels. It remains unclear how ion fluxes through individual ion pathways are integrated so as to induce loss of intracellular ions and concomitant apoptotic volume decrease. A decrease in activity of the sodium pump during apoptosis should cause cell swelling rather than shrinkage. We have made the first systemic analysis of the monovalent ion flux balance in apoptotic cells. Experimental data were obtained for human U937 cells treated with staurosporine for 4–5 h, which is known to induce apoptosis. The data include cellular Cl− content and fluxes, K+, Na+, water content and ouabain-sensitive and -resistant Rb+ fluxes.Unidirectional monovalent ion fluxeswere calculated using these data and a cell model comprising the double Donnan system with the Na+/K+ pump, Cl−, K+, Na+ channels, the Na+–K+–2Cl−cotransporter (NKCC), the Na+–Cl− cotransporter (NC), and the equivalent Cl−/Cl− exchange.Apoptotic cell shrinkage was found to be caused, depending on conditions, either by an increase in the integral channel permeability of membrane for K+ or by suppression of the pump coupledwith a decrease in the integral channel permeability of membrane for Na+. The decrease in the channel permeability of membrane for Na+ plays a crucial role in cell dehydration in apoptosis accompanied by suppression of the pump. Supplemental Table S1 is given for easy calculating flux balance under specified conditions.

Peritubular membrane potential in kidney proximal tubular cells of spontaneously hypertensive rats

Peritubular membrane potential in kidney proximal tubular cells of spontaneously hypertensive rats (SHR-Okamoto strain adult rats) was measured with conventional 3 mol KCl microelectrodes, in vivo. Peritubular cell membrane potential was not different in SHR (-66.5 ± 0.7 mV) as compared with normotensive control Wistar rats (-67.5 ± 1.2 mV). To test the effects of possible altered sodium membrane transport in SHR on proximal tubule peritubular membrane potential, we allowed SHR and control rats to drink 1% NaCl for two weeks. Again, proximal tubule peritubular membrane potential was not different in SHR on 1% NaCl (-67.0 ± 1.0 mV) as compared with control rats on 1% NaCl (-64.7 ± 1.3 mV). From these results we concluded that peritubular membrane potential in kidney proximal tubular cells of SHR was not different from normotensive Wistar control rats, and if some alteration of sodium transport in kidney proximal tubular cells of SHR could exist, that was not possible to evaluate from the measurements of peritubular membrane potential in kidney proximal tubular cells.

First-principles modeling of fluid and solute exchange in the human during normal and hemodialysis conditions

A first-principles computer model of fluid and solute exchange under both physiological and hemodialysis condition is presented. The whole system has been modeled and simulated under the MODELICA integrated environment, which uses a hierarchical modeling strategy. The model performance has been analyzed by simulation in the light of existing hypothesis and physiological data used here for validation purposes. The results obtained provide a physiological interpretative key to patient's hemodynamic behavior during hemodialysis.

Volumetric response of vertebrate hepatocytes challenged by osmotic gradients: a theoretical approach

In this study we use a theoretical approach to study the volumetric response of goldfish hepatocytes challenged by osmotic gradients and compared it with that of hepatocytes from another teleost (the trout) and a mammal (the rat). Particular focus was given to the multiple non-linear interactions of transport systems enabling hypotonically challenged cells to trigger a compensatory response known as volume regulatory decrease or RVD. For this purpose we employed a mathematical model which describes the rates of change of the intracellular concentrations of main diffusible ions, of the cell volume, and of the membrane potential. The model was fitted to experimental data on the kinetics of volume change of hepatocytes challenged by anisotonic media. In trout and rat hepatocytes, experimental results had shown that hypotonic cell swelling was followed by RVD, whereas goldfish cells swelled with no concomitant RVD (M.V. Espelt et al., 2003, J. Exp. Biol. 206, 513-522). A comparison between data predicted by the model and that obtained experimentally suggests that in trout and rat hepatocytes hypotonicity activates a sensor element and this, in turn, activates an otherwise silent efflux of KCl - whose kinetics could be successfully predicted - thereby leading to volume down-regulation. In contrast, with regard to the absence of RVD in goldfish hepatocytes the model proposed suggests that either a sensor element triggering RVD is absent or that the effector mechanism (the loss of KCl) remains inactive under the conditions employed. In line with this, we recently found that extracellular nucleotides may be required to induce RVD in these cells, indicating that our model could indeed lead to useful predictions.

Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E(m)) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E(m) due to its influence upon transmembrane ion fluxes and cell volume (V(c)). These interrelationships between E(m), V(c) and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman-Hodgkin-Katz equation and the Gibbs-Donnan equilibrium, to more recent "current-summing" models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the "charge-difference" modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E(m), V(c) and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E(m), V(c) and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E(m), V(c) and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.

A macrophage cell model for pH and volume regulation

A whole-cell model of a macrophage (mphi) is developed to simulate pH and volume regulation during a NH4Cl prepulse challenge. The cell is assumed spherical, with a plasma membrane that separates the cytosolic and extracellular bathing media. The membrane contains background currents for Na+, K+ and Cl-, a Na(+)-K+ pump, a V-type H(+)-extruder (V-ATPase), and a leak pathway for NH4+. Cell volume is controlled by instantaneous osmotic balance between cytosolic and extracellular osmolytes. Simulations reveal that the mphi model can mimic alterations in measured pH(i) and cell volume (Vol(i)) data during and after delivery of an ammonia prepulse, which induces an acid load within the cell. Our analysis indicates that there are substantial problems in quantifying transporter-mediated H+ efflux solely from experimental observations of pH(i) recovery, as is commonly done in practice. Problems stemming from the separation of effects arise, since there is residual NH4+ dissociation to H+ inside the mphi during pH(i) recovery, as well as, proton extrusion via the V-ATPase. The core assumption of conventional measurement techniques used to estimate the H+ extrusion current (I(H)) is that the recovery phase is solely dependent on transporter-mediated H+ extrusion. However, our model predictions suggest that there are major problems in using this approach, due to the complex interactions between I(H), NH3/NH4+ buffering and NH3/NH4+ efflux during the active acid extrusion phase. That is, the conventional buffer capacity-based I(H) estimation must also take into account the perturbation that a prepulse challenge brings to the cytoplasmic acid buffer itself. The importance of this whole-cell model of mphipH(i) and volume regulation lies in its potential for extension to the characterization of several other types of non-excitable cells, such as the microglia (brain macrophage) and the T-lymphocyte.

A mathematical model of ion movements in grey matter during a stroke

The development of cytotoxic oedema during a stroke consists in cell swelling and shrinking of the extracellular space. This phenomenon is triggered by ion movements through voltage-gated channels, exchangers and pumps. During ischaemia, sodium, calcium and chloride enter the neurons whereas potassium and glutamate are expelled out of the cells. A mathematical model is proposed to represent the long-term dynamics of membrane potentials, cell volumes and ionic concentrations in intracellular and extracellular spaces during a stroke and to study the influence of each ionic current on cell swelling. The model relies on electrophysiological mechanisms and takes into account the behaviour of two types of cells: neurons and also astrocytes known to play a key role in the excitotoxic process in grey matter. The results obtained when a severe or a moderate ischaemia is simulated are consistent with those observed in the in vitro and in vivo experiments. As this model appears to be robust, it is used to perform illustrative simulations aimed at studying the effect of some channel blockers on cell swelling. This approach may help to explore new therapeutic strategies in order to reduce stroke damage.

Ischemia-Induced Norepinephrine Release, but not Norepinephrine-Derived Free Radicals, Contributes to Myocardial Ischemia - Reperfusion Injury

BACKGROUND

Norepinephrine (NE)-derived free radicals may contribute to myocyte injury after ischemia -reperfusion, so the influence of sympathetic denervation on myocardial ischemia - reperfusion injury was investigated in the present study.

METHODS AND RESULTS

Cardiac sympathetic denervation was produced in Wistar rats by a solution of 10% phenol 1 week before ischemia. Atenolol (0.5 mg/kg) was intravenously administered 10 min before the coronary occlusion. The left coronary artery was occluded for 30 min and thereafter reperfused. Cardiac interstitial fluid was collected by a microdialysis probe and free radicals in dialysate were determined by electron paramagnetic resonance (EPR) spin trapping, using 5,5-dimethyl-1-pyrroline-N-oxide as a spin trap. The ratio of infarct size to the ischemic area at risk (I/R) was decreased in both the phenol and atenolol groups compared with control (28.5+/-11.3, 31.8+/-10.7 vs 50.6+/-14.7%, p<0.05). During the coronary occlusion, concentrations of interstitial NE increased markedly in the control and atenolol groups, but was unchanged in the phenol group. EPR signal intensity (relative value to internal standard) was maximal at 1 h after reperfusion and was similar in the phenol and control groups (0.32+/-0.15 vs 0.45+/-0.19).

CONCLUSIONS

Cardiac denervation protected myocyte against ischemia-reperfusion injury through decreasing direct NE toxicity, but not through decreasing NE-derived free radicals.

The Na/K pump, Cl ion, and osmotic stabilization of cells

An equation for membrane voltage is derived that takes into account the electrogenicity of the Na/K pump and is valid dynamically, as well as in the steady state. This equation is incorporated into a model for the osmotic stabilization of cells. The results emphasize the role of the pump and membrane voltage in lowering internal Cl(-) concentration, thus making osmotic room for vital substances that must be sequestered in the cell.

The influence of plasma membrane electrostatic properties on the stability of cell ionic composition

An electro-osmotic model is developed to examine the influence of plasma membrane superficial charges on the regulation of cell ionic composition. Assuming membrane osmotic equilibrium, the ion distribution predicted by Gouy-Chapman-Grahame (GCG) theory is introduced into ion transport equations, which include a kinetic model of the Na/K-ATPase based on the stimulation of this ion pump by internal Na(+) ions. The algebro-differential equation system describing dynamics of the cell model has a unique resting state, stable with respect to finite-sized perturbations of various types. Negative charges on the membrane are found to greatly enhance relaxation toward steady state following these perturbations. We show that this heightened stability stems from electrostatic interactions at the inner membrane side that shift resting state coordinates along the sigmoidal activation curve of the sodium pump, thereby increasing the pump sensitivity to internal Na(+) fluctuations. The accuracy of electrostatic potential description with GCG theory is proved using an alternate formalism, based on irreversible thermodynamics, which shows that pressure contribution to ion potential energy is negligible in electrostatic double layers formed at the surfaces of biological membranes. We discuss implications of the results regarding a reliable operation of ionic process coupled to the transmembrane electrochemical gradient of Na(+) ions.

Osmotic forces and gap junctions in spreading depression: a computational model

In a computational model of spreading depression (SD), ionic movement through a neuronal syncytium of cells connected by gap junctions is described electrodiffusively. Simulations predict that SD will not occur unless cells are allowed to expand in response to osmotic pressure gradients and K+ is allowed to move through gap junctions. SD waves of [K+]out approximately 25 to approximately 60 mM moving at approximately 2 to approximately 18 mm/min are predicted over the range of parametric values reported in gray matter, with extracellular space decreasing up to approximately 50%. Predicted waveform shape is qualitatively similar to laboratory reports. The delayed-rectifier, NMDA, BK, and Na+ currents are predicted to facilitate SD, while SK and A-type K+ currents and glial activity impede SD. These predictions are consonant with recent findings that gap junction poisons block SD and support the theories that cytosolic diffusion via gap junctions and osmotic forces are important mechanisms underlying SD.

An elementary kinetic model of energy coupling in biological membranes

The purpose of this work is to contribute to the understanding of the fundamental kinetic properties of the processes of energy coupling in biological membranes. For this, we consider a model of a microorganism that, in its plasma membrane, expresses two electrogenic enzymes (E(1) and E(2)) transporting the same monovalent cation C and electrodiffusive paths for C and for a monovalent anion A. E(1) (E(2)) couples transport C to the reaction S(1)<-->P(1) (S(2)<-->P(2)). We developed a mathematical model that describes the rate of change of the electrical potential difference across the membrane, of the internal concentrations of C and A, and of the concentrations of S(2) and P(2). The enzymes are incorporated via two-state kinetic models; the passive ionic fluxes are represented by classical formulations of electrodiffusion. The microorganism volume is maintained constant by accessory regulatory devices. The model is utilized for stationary and dynamic studies for the case of bacteria employing the electrochemical gradient of Na(+) as energetic intermediate. Among other conclusions, the results show that the membrane potential represents the relevant kinetic intermediate for the overall coupling between the energy donor reaction S(1)<-->P(1) and the synthesis of S(2).

Transport of fluid and solutes in the body I. Formulation of a mathematical model

A compartmental model of short-term whole body fluid, protein, and ion distribution and transport is formulated. The model comprises four compartments: a vascular and an interstitial compartment, each with an embedded cellular compartment. The present paper discusses the assumptions on which the model is based and describes the equations that make up the model. Fluid and protein transport parameters from a previously validated model as well as ionic exchange parameters from the literature or from statistical estimation [see companion paper: C. C. Gyenge, B. D. Bowen, R. K. Reed, and J. L. Bert. Am. J. Physiol. 277 (Heart Circ. Physiol. 46): H1228-H1240, 1999] are used in formulating the model. The dynamic model has the ability to simulate 1) transport across the capillary membrane of fluid, proteins, and small ions and their distribution between the vascular and interstitial compartments; 2) the changes in extracellular osmolarity; 3) the distribution and transport of water and ions associated with each of the cellular compartments; 4) the cellular transmembrane potential; and 5) the changes of volume in the four fluid compartments. The validation and testing of the proposed model against available experimental data are presented in the companion paper.

Activation of KATP channels by Na/K pump in isolated cardiac myocytes and giant membrane patches

Strophanthidin inhibits KATP channels in 2,4-dinitrophenol-poisoned heart cells (). The current study shows that the Na/K pump interacts with KATP current (IK-ATP) via submembrane ATP depletion in isolated giant membrane patches and in nonpoisoned guinea pig cardiac cells in whole-cell configuration. IK-ATP was inhibited by ATP, glibenclamide, or intracellular Cs+. Na/K pump inactivation by substitution of cytoplasmic Na+ for Li+ or N-methylglucamine decreased both IK-ATP by 1/3 (1 mM ATP, zero calcium), and IC50 of ATP for IK-ATP (0.3 +/- 0.1 mM) by 2/5. The Na+/Li+ replacement had no effect on IK-ATP at low pump activity ([ATP] </= 0.1 mM or 100 microM ouabain) or when IK-ATP was completely inhibited by 10 mM ATP. In whole-cell configuration, ouabain inhibited up to 60% of inwardly rectifying IK-ATP at 1 mM ATP in the pipette but not at 10 mM ATP and 10 mM phosphocreatine when IK-ATP was always blocked. However, mathematical simulation of giant-patch experiments revealed that only 20% of ATP depletion may be attributed to the ATP concentration gradient in the bulk solution, and the remaining 80% probably occurs in the submembrane space.