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

Nernst-Planck-Gaussian finite element modelling of Ca2+ electrodiffusion in amphibian striated muscle transverse tubule-sarcoplasmic reticular triadic junctional domains

Introduction

Intracellular Ca2+ signalling regulates membrane permeabilities, enzyme activity, and gene transcription amongst other functions. Large transmembrane Ca2+ electrochemical gradients and low diffusibility between cell compartments potentially generate short-lived, localised, high-[Ca2+] microdomains. The highest concentration domains likely form between closely apposed membranes, as at amphibian skeletal muscle transverse tubule-sarcoplasmic reticular (T-SR, triad) junctions.

Materials and methods

Finite element computational analysis characterised the formation and steady state and kinetic properties of the Ca2+ microdomains using established empirical physiological and anatomical values. It progressively incorporated Fick diffusion and Nernst-Planck electrodiffusion gradients, K+, Cl-, and Donnan protein, and calmodulin (CaM)-mediated Ca2+ buffering. It solved for temporal-spatial patterns of free and buffered Ca2+, Gaussian charge differences, and membrane potential changes, following Ca2+ release into the T-SR junction.

Results

Computational runs using established low and high Ca2+ diffusibility (D Ca2+) limits both showed that voltages arising from intracytosolic total [Ca2+] gradients and the counterions little affected microdomain formation, although elevated D Ca2+ reduced attained [Ca2+] and facilitated its kinetics. Contrastingly, adopting known cytosolic CaM concentrations and CaM-Ca2+ affinities markedly increased steady-state free ([Ca2+]free) and total ([Ca2+]), albeit slowing microdomain formation, all to extents reduced by high D Ca2+. However, both low and high D Ca2+ yielded predictions of similar, physiologically effective, [Ca2+-CaM]. This Ca2+ trapping by the relatively immobile CaM particularly increased [Ca2+] at the junction centre. [Ca2+]free, [Ca2+-CaM], [Ca2+], and microdomain kinetics all depended on both CaM-Ca2+ affinity and D Ca2+. These changes accompanied only small Gaussian (∼6 mV) and surface charge (∼1 mV) effects on tubular transmembrane potential at either D Ca2+.

Conclusion

These physical predictions of T-SR Ca2+ microdomain formation and properties are compatible with the microdomain roles in Ca2+ and Ca2+-CaM-mediated signalling but limited the effects on tubular transmembrane potentials. CaM emerges as a potential major regulator of both the kinetics and the extent of microdomain formation. These possible cellular Ca2+ signalling roles are discussed in relation to possible feedback modulation processes sensitive to the μM domain but not nM bulk cytosolic, [Ca2+]free, and [Ca2+-CaM], including ryanodine receptor-mediated SR Ca2+ release; Na+, K+, and Cl- channel-mediated membrane excitation and stabilisation; and Na+/Ca2+ exchange transport.

Reduced K+ build-up in t-tubules contributes to resistance of the diaphragm to myotonia

Patients with myotonia congenita suffer from slowed muscle relaxation caused by hyperexcitability. The diaphragm is only mildly affected in myotonia congenita; discovery of the mechanism underlying its resistance to myotonia could identify novel therapeutic targets. Intracellular recordings from two mouse models of myotonia congenita revealed the diaphragm had less myotonia than either the extensor digitorum longus (EDL) or the soleus muscles. A mechanism contributing to resistance of the diaphragm to myotonia was reduced depolarization of the interspike membrane potential during repetitive firing of action potentials, a process driven by build-up of K+ in small invaginations of muscle membrane known as t-tubules. We explored differences between diaphragm and EDL that might underlie reduction of K+ build-up in diaphragm t-tubules. Smaller size of diaphragm fibres, which promotes diffusion of K+ out of t-tubules, was identified as a contributor. Intracellular recording revealed slower repolarization of action potentials in diaphragm suggesting reduced Kv conductance. Higher resting membrane conductance was identified suggesting increased Kir conductance. Computer simulation found that a reduction of Kv conductance had little effect on K+ build-up whereas increased Kir conductance lessened build-up, although the effect was modest. Our data and computer simulation suggest opening of K+ channels during action potentials has little effect on K+ build-up whereas opening of K+ channels during the interspike interval slightly lessens K+ build-up. We conclude that activation of K+ channels may lessen myotonia by opposing depolarization to action potential threshold without worsening K+ build-up in t-tubules. KEY POINTS: In mouse models of the muscle disease myotonia congenita, the diaphragm has much less myotonia (muscle stiffness) than the extensor digitorum longus or soleus muscles. Identifying why the diaphragm is resistant to myotonia may help in developing novel therapy. We found the reason the diaphragm has less myotonia is that it is less prone to depolarization caused by K+ build-up in t-tubules during repetitive firing of action potentials. Smaller fibre size contributes to resistance to K+ build-up with differences in K+ currents playing little role. Our data suggest drugs that open K+ channels may be effective in treating myotonia as they may lessen excitability without worsening K+ build-up in t-tubules.

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.

Is post-hypertonic lysis of human red blood cells caused by excessive cell volume regulation?

Human red blood cells (RBC) exposed to hypertonic media are subject to post-hypertonic lysis - an injury that only develops during resuspension to an isotonic medium. The nature of post-hypertonic lysis was previously hypothesized to be osmotic when cation leaks were observed, and salt loading was suggested as a cause of the cell swelling upon resuspension in an isotonic medium. However, it was problematic to account for the salt loading since the plasma membrane of human RBCs was considered impermeable to cations. In this study, the hypertonicity-related behavior of human RBCs is revisited within the framework of modern cell physiology, considering current knowledge on membrane ion transport mechanisms - an account still missing. It is recognized here that the hypertonic behavior of human RBCs is consistent with the acute regulatory volume increase (RVI) response - a healthy physiological reaction initiated by cells to regulate their volume by salt accumulation. It is shown by reviewing the published studies that human RBCs can increase cation conductance considerably by activating cell volume-regulated ion transport pathways inactive under normal isotonic conditions and thus facilitate salt loading. A simplified physiological model accounting for transmembrane ion fluxes and membrane voltage predicts the isotonic cell swelling associated with increased cation conductance, eventually reaching hemolytic volume. The proposed involvement of cell volume regulation mechanisms shows the potential to explain the complex nature of the osmotic response of human RBCs and other cells. Cryobiological implications, including mechanisms of cryoprotection, are discussed.

Electromechanical interactions between cell membrane and nuclear envelope: Beyond the standard Schwan's model of biological cells

We investigate little-appreciated features of the hierarchical core-shell (CS) models of the electrical, mechanical, and electromechanical interactions between the cell membrane (CM) and nuclear envelope (NE). We first consider a simple model of an individual cell based on a coupled resistor-capacitor (Schwan model (SM)) network and show that the CM, when exposed to ac electric fields, acts as a low pass filter while the NE acts as a wide and asymmetric bandpass filter. We provide a simplified calculation for characteristic time associated with the capacitive charging of the NE and parameterize its range of behavior. We furthermore observe several new features dealing with mechanical analogs of the SM based on elementary spring-damper combinations. The chief merit of these models is that they can predict creep compliance responses of an individual cell under static stress and their effective retardation time constants. Next, we use an alternative and a more accurate CS physical model solved by finite element simulations for which geometrical cell reshaping under electromechanical stress (electrodeformation (ED)) is included in a continuum approach with spatial resolution. We show that under an electric field excitation, the elongated nucleus scales differently compared to the electrodeformed cell.

Nernst-Planck-Gaussian modelling of electrodiffusional recovery from ephaptic excitation between mammalian cardiomyocytes

Introduction: In addition to gap junction conduction, recent reports implicate possible ephaptic coupling contributions to action potential (AP) propagation between successive adjacent cardiomyocytes. Here, AP generation in an active cell, withdraws Na+ from, creating a negative potential within, ephaptic spaces between the participating membranes, activating the initially quiescent neighbouring cardiomyocyte. However, sustainable ephaptic transmission requires subsequent complete recovery of the ephaptic charge difference. We explore physical contributions of passive electrodiffusive ion exchange with the remaining extracellular space to this recovery for the first time. Materials and Methods: Computational, finite element, analysis examined limiting, temporal and spatial, ephaptic [Na+], [Cl-], and the consequent Gaussian charge differences and membrane potential recovery patterns following a ΔV∼130 mV AP upstroke at physiological (37°C) temperatures. This incorporated Nernst-Planck formalisms into equations for the time-dependent spatial concentration gradient profiles. Results: Mammalian atrial, ventricular and purkinje cardiomyocyte ephaptic junctions were modelled by closely apposed circularly symmetric membranes, specific capacitance 1 μF cm-2, experimentally reported radii a = 8,000, 12,000 and 40,000 nm respectively and ephaptic axial distance w = 20 nm. This enclosed an ephaptic space containing principal ions initially at normal extracellular [Na+] = 153.1 mM and [Cl-] = 145.8 mM, respective diffusion coefficients D Na = 1.3 × 109 and D Cl = 2 × 109 nm2s-1. Stable, concordant computational solutions were confirmed exploring ≤1,600 nm mesh sizes and Δt≤0.08 ms stepsize intervals. The corresponding membrane voltage profile changes across the initially quiescent membrane were obtainable from computed, graphically represented a and w-dependent ionic concentration differences adapting Gauss's flux theorem. Further simulations explored biological variations in ephaptic dimensions, membrane anatomy, and diffusion restrictions within the ephaptic space. Atrial, ventricular and Purkinje cardiomyocytes gave 40, 180 and 2000 ms 99.9% recovery times, with 720 or 360 ms high limits from doubling ventricular radius or halving diffusion coefficient. Varying a, and D Na and D Cl markedly affected recovery time-courses with logarithmic and double-logarithmic relationships, Varying w exerted minimal effects. Conclusion: We thereby characterise the properties of, and through comparing atrial, ventricular and purkinje recovery times with interspecies in vivo background cardiac cycle duration data, (blue whale ∼2000, human∼90, Etruscan shrew, ∼40 ms) can determine physical limits to, electrodiffusive contributions to ephaptic recovery.

How Merkel cells transduce mechanical stimuli: A biophysical model of Merkel cells

Merkel cells combine with Aβ afferents, producing slowly adapting type 1(SA1) responses to mechanical stimuli. However, how Merkel cells transduce mechanical stimuli into neural signals to Aβ afferents is still unclear. Here we develop a biophysical model of Merkel cells for mechanical transduction by incorporating main ingredients such as Ca2+ and K+ voltage-gated channels, Piezo2 channels, internal Ca2+ stores, neurotransmitters release, and cell deformation. We first validate our model with several experiments. Then we reveal that Ca2+ and K+ channels on the plasma membrane shape the depolarization of membrane potentials, further regulating the Ca2+ transients in the cells. We also show that Ca2+ channels on the plasma membrane mainly inspire the Ca2+ transients, while internal Ca2+ stores mainly maintain the Ca2+ transients. Moreover, we show that though Piezo2 channels are rapidly adapting mechanical-sensitive channels, they are sufficient to inspire sustained Ca2+ transients in Merkel cells, which further induce the release of neurotransmitters for tens of seconds. Thus our work provides a model that captures the membrane potentials and Ca2+ transients features of Merkel cells and partly explains how Merkel cells transduce the mechanical stimuli by Piezo2 channels.

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.

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.

The Donnan-dominated resting state of skeletal muscle fibers contributes to resilience and longevity in dystrophic fibers

Duchenne muscular dystrophy (DMD) is an X-linked dystrophin-minus muscle-wasting disease. Ion homeostasis in skeletal muscle fibers underperforms as DMD progresses. But though DMD renders these excitable cells intolerant of exertion, sodium overloaded, depolarized, and spontaneously contractile, they can survive for several decades. We show computationally that underpinning this longevity is a strikingly frugal, robust Pump-Leak/Donnan (P-L/D) ion homeostatic process. Unlike neurons, which operate with a costly “Pump-Leak–dominated” ion homeostatic steady state, skeletal muscle fibers operate with a low-cost “Donnan-dominated” ion homeostatic steady state that combines a large chloride permeability with an exceptionally small sodium permeability. Simultaneously, this combination keeps fiber excitability low and minimizes pump expenditures. As mechanically active, long-lived multinucleate cells, skeletal muscle fibers have evolved to handle overexertion, sarcolemmal tears, ischemic bouts, etc.; the frugality of their Donnan dominated steady state lets them maintain the outsized pump reserves that make them resilient during these inevitable transient emergencies. Here, P-L/D model variants challenged with DMD-type insult/injury (low pump-strength, overstimulation, leaky Nav and cation channels) show how chronic “nonosmotic” sodium overload (observed in DMD patients) develops. Profoundly severe DMD ion homeostatic insult/injury causes spontaneous firing (and, consequently, unwanted excitation–contraction coupling) that elicits cytotoxic swelling. Therefore, boosting operational pump-strength and/or diminishing sodium and cation channel leaks should help extend DMD fiber longevity.

Zinc accelerates respiratory burst termination in human PMN

The respiratory burst of phagocytes is essential for human survival. Innate immune defence against pathogens relies strongly on reactive oxygen species (ROS) production by the NADPH oxidase (NOX2). ROS kill pathogens while the translocation of electrons across the plasma membrane via NOX2 depolarizes the cell. Simultaneously, protons are released into the cytosol. Here, we compare freshly isolated human polymorphonuclear leukocytes (PMN) to the granulocytes-like cell line PLB 985. We are recording ROS production while inhibiting the charge compensating and pH regulating voltage-gated proton channel (H_V1). The data suggests that human PMN and the PLB 985 generate ROS via a general mechanism, consistent of NOX2 and H_V1. Additionally, we advanced a mathematical model based on the biophysical properties of NOX2 and H_V1. Our results strongly suggest the essential interconnection of H_V1 and NOX2 during the respiratory burst of phagocytes. Zinc chelation during the time course of the experiments postulates that zinc leads to an irreversible termination of the respiratory burst over time. Flow cytometry shows cell death triggered by high zinc concentrations and PMA. Our data might help to elucidate the complex interaction of proteins during the respiratory burst and contribute to decipher its termination.

Molecular basis of ventricular arrhythmogenicity in a Pgc-1α deficient murine model

Mitochondrial dysfunction underlying metabolic disorders such as obesity and diabetes mellitus is strongly associated with cardiac arrhythmias. Murine Pgc-1α-/- hearts replicate disrupted mitochondrial function and model the associated pro-arrhythmic electrophysiological abnormalities. Quantitative PCR, western blotting and histological analysis were used to investigate the molecular basis of the electrophysiological changes associated with mitochondrial dysfunction. qPCR analysis implicated downregulation of genes related to Na+-K+ ATPase activity (Atp1b1), surface Ca2+ entry (Cacna1c), action potential repolarisation (Kcnn1), autonomic function (Adra1d, Adcy4, Pde4d, Prkar2a), and morphological properties (Myh6, Tbx3) in murine Pgc-1α-/- ventricles. Western blotting revealed reduced NaV1.5 but normal Cx43 expression. Histological analysis revealed increased tissue fibrosis in the Pgc-1α-/- ventricles. These present findings identify altered transcription amongst a strategically selected set of genes established as encoding proteins involved in cardiac electrophysiological activation and therefore potentially involved in alterations in ventricular activation and Ca2+ homeostasis in arrhythmic substrate associated with Pgc-1α deficiency. They complement and complete previous studies examining such expression characteristics in the atria and ventricles of Pgc-1 deficient murine hearts.

Quantitative model analysis of the resting membrane potential in insect skeletal muscle: Implications for low temperature tolerance

Abiotic stressors, such as cold exposure, can depolarize insect cells substantially causing cold coma and cell death. During cold exposure, insect skeletal muscle depolarization occurs through a 2-stage process. Firstly, short-term cold exposure reduces the activity of electrogenic ion pumps, which depolarize insect muscle markedly. Secondly, during long-term cold exposure, extracellular ion homeostasis is disrupted causing further depolarization. Consequently, many cold hardy insects improve membrane potential stability during cold exposure through adaptations that secure maintenance of ion homeostasis during cold exposure. Less is known about the adaptations permitting cold hardy insects to maintain membrane potential stability during the initial phase of cold exposure, before ion balance is disrupted. To address this problem it is critical to understand the membrane components (channels and transporters) that determine the membrane potential and to examine this question the present study constructed a mathematical "charge difference" model of the insect muscle membrane potential. This model was parameterized with known literature values for ion permeabilities, ion concentrations and membrane capacitance and the model was then further developed by comparing model predictions against empirical measurements following pharmacological inhibitors of the Na⁺/K⁺ ATPase, Cl^- channels and symporters. Subsequently, we compared simulated and recorded membrane potentials at 0 and 31 °C and at 10-50 mM extracellular [K⁺] to examine if the model could describe membrane potentials during the perturbations occurring during cold exposure. Our results confirm the importance of both Na⁺/K⁺ ATPase activity and ion-selective Na⁺, K⁺ and Cl^- channels, but the model also highlights that additional electroneutral flux of Na⁺ and K⁺ is needed to describe how membrane potentials respond to temperature and [K⁺] in insect muscle. While considerable further work is still needed, we argue that this "charge difference" model can be used to generate testable hypotheses of how insects can preserve membrane polarization in the face of stressful cold exposure.

A Reverse-Osmosis Model of Apoptotic Shrinkage

The standard theory of apoptotic volume decrease (AVD) posits activation of potassium and/or chloride channels, causing an efflux of ions and osmotic loss of water. However, in view of the multitude of possible channels that are known to support apoptosis, a model based on specific signaling to a channel presents certain problems. We propose another mechanism of apoptotic dehydration based on cytoskeletal compression. As is well known, cytoskeleton is not strong enough to expel a substantial amount of water against an osmotic gradient. It is possible, however, that an increase in intracellular pressure may cause an initial small efflux of water, and that will create a small concentration gradient of ions, favoring their exit. If the channels are open, some ions will exit the cell, relieving the osmotic gradient; in this way, the process will be able to continue. Calculations confirm the possibility of such a mechanism. An increase in membrane permeability for water or ions may also result in dehydration if accompanied even by a constant cytoskeletal pressure. We review the molecular processes that may lead to apoptotic dehydration in the context of this model.

Chloride dynamics alter the input-output properties of neurons

Fast synaptic inhibition is a critical determinant of neuronal output, with subcellular targeting of synaptic inhibition able to exert different transformations of the neuronal input-output function. At the receptor level, synaptic inhibition is primarily mediated by chloride-permeable Type A GABA receptors. Consequently, dynamics in the neuronal chloride concentration can alter the functional properties of inhibitory synapses. How differences in the spatial targeting of inhibitory synapses interact with intracellular chloride dynamics to modulate the input-output function of neurons is not well understood. To address this, we developed computational models of multi-compartment neurons that incorporate experimentally parametrised mechanisms to account for neuronal chloride influx, diffusion, and extrusion. We found that synaptic input (either excitatory, inhibitory, or both) can lead to subcellular variations in chloride concentration, despite a uniform distribution of chloride extrusion mechanisms. Accounting for chloride changes resulted in substantial alterations in the neuronal input-output function. This was particularly the case for peripherally targeted dendritic inhibition where dynamic chloride compromised the ability of inhibition to offset neuronal input-output curves. Our simulations revealed that progressive changes in chloride concentration mean that the neuronal input-output function is not static but varies significantly as a function of the duration of synaptic drive. Finally, we found that the observed effects of dynamic chloride on neuronal output were mediated by changes in the dendritic reversal potential for GABA. Our findings provide a framework for understanding the computational effects of chloride dynamics on dendritically targeted synaptic inhibition.

A Theoretical Approach for the Electrochemical Characterization of Ciliary Epithelium

The ciliary epithelium (CE) is the primary site of aqueous humor (AH) production, which results from the combined action of ultrafiltration and ionic secretion. Modulation of ionic secretion is a fundamental target for drug therapy in glaucoma, and therefore it is important to identify the main factors contributing to it. As several ion transporters have been hypothesized as relevant players in CE physiology, we propose a theoretical approach to complement experimental methods in characterizing their role in the electrochemical and fluid-dynamical conditions of CE. As a first step, we compare two model configurations that differ by (i) types of transporters included for ion exchange across the epithelial membrane, and by (i) presence or absence of the intracellular production of carbonic acid mediated by the carbonic anhydrase enzyme. The proposed model configurations do not include neurohumoral mechanisms such as P2Y receptor-dependent, cAMP, or calcium-dependent pathways, which occur in the ciliary epithelium bilayer and influence the activity of ion transporters, pumps, and channels present in the cell membrane. Results suggest that one of the two configurations predicts sodium and potassium intracellular concentrations and transmembrane potential much more accurately than the other. Because of its quantitative prediction power, the proposed theoretical approach may help relate phenomena at the cellular scale, that cannot be accessed clinically, with phenomena occurring at the scale of the whole eye, for which clinical assessment is feasible.

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.

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.

Biological interactions of biocompatible and water-dispersed MoS2 nanosheets with bacteria and human cells

Two dimensional materials beyond graphene such as MoS2 and WS2 are novel and interesting class of materials whose unique physico-chemical properties can be exploited in applications ranging from leading edge nanoelectronics to the frontiers between biomedicine and biotechnology. To unravel the potential of TMD crystals in biomedicine, control over their production through green and scalable routes in biocompatible solvents is critically important. Furthermore, considering multiple applications of eco-friendly 2D dispersions and their potential impact onto live matter, their toxicity and antimicrobial activity still remain an open issue. Herein, we focus on the current demands of 2D TMDs and produce high-quality, few-layered and defect-free MoS2 nanosheets, exfoliated and dispersed in pure water, stabilized up to three weeks. Hence, we studied the impact of this material on human cells by investigating its interactions with three cell lines: two tumoral, MCF7 (breast cancer) and U937 (leukemia), and one normal, HaCaT (epithelium). We observed novel and intriguing results, exhibiting evident cytotoxic effect induced in the tumor cell lines, absent in the normal cells in the tested conditions. The antibacterial action of MoS2 nanosheets is then investigated against a very dangerous gram negative bacterium, such as two types of Salmonellas: ATCC 14028 and wild-type Salmonella typhimurium. Additionally, concentration and layer-dependent modulation of cytotoxic effect is found both on human cells and Salmonellas.

Biophysical models reveal the relative importance of transporter proteins and impermeant anions in chloride homeostasis

Fast synaptic inhibition in the nervous system depends on the transmembrane flux of Cl^- ions based on the neuronal Cl^- driving force. Established theories regarding the determinants of Cl^- driving force have recently been questioned. Here, we present biophysical models of Cl^- homeostasis using the pump-leak model. Using numerical and novel analytic solutions, we demonstrate that the Na⁺/K⁺-ATPase, ion conductances, impermeant anions, electrodiffusion, water fluxes and cation-chloride cotransporters (CCCs) play roles in setting the Cl^- driving force. Our models, together with experimental validation, show that while impermeant anions can contribute to setting [Cl^-]_i in neurons, they have a negligible effect on the driving force for Cl^- locally and cell-wide. In contrast, we demonstrate that CCCs are well-suited for modulating Cl^- driving force and hence inhibitory signaling in neurons. Our findings reconcile recent experimental findings and provide a framework for understanding the interplay of different chloride regulatory processes in neurons.

Cells called neurons in the brain communicate by triggering or inhibiting electrical activity in other neurons. To inhibit electrical activity, a signal from one neuron usually triggers specific receptors on the second neuron to open, which allows particles called chloride ions to flow into or out of the neuron.

The force that moves chloride ions (the so-called ‘chloride driving force’) depends on two main factors. Firstly, chloride ions, like other particles, tend to move from an area where they are plentiful to areas where they are less abundant. Secondly, chloride ions are negatively charged and are therefore attracted to areas where the net charge (determined by the mix of positively and negatively charged particles) is more positive than their current position.

It was previously believed that a group of proteins known as CCCs, which transport chloride ions and positive ions together across the membranes surrounding cells, sets the chloride driving force. However, it has recently been suggested that negatively charged ions that are unable to cross the membrane (or ‘impermeant anions’ for short) may set the driving force instead by contributing to the net charge across the membrane. Düsterwald et al. used a computational model of the neuron to explore these two possibilities.

In the simulations, altering the activity of the CCCs led to big changes in the chloride driving force. Changing the levels of impermeant anions altered the volume of cells, but did not drive changes in the chloride driving force. This was because the flow of chloride ions across the membrane led to a compensatory change in the net charge across the membrane.

Düsterwald et al. then used an experimental technique called patch-clamping in mice and rats to confirm the model’s predictions. Defects in controlling the chloride driving force in brain cells have been linked with epilepsy, stroke and other neurological diseases. Therefore, a better knowledge of these mechanisms may in future help to identify the best targets for drugs to treat such conditions.

Role of the low-selective organic anion transport in regulation of osmotic balance of renal collecting duct principal cells under hypo-osmotic conditions

In the course of adaptation of the rat kidney collecting duct cells to hypo-osmotic medium, the organic anion transporter inhibitor probenecid reduced significantly the regulatory cell volume decrease in response to a hypotonic shock. Both probenecid and hypotonic shock delayed significantly the entry into a cell of the fluorescent dye calcein, which exists as anion at neutral pH. Thus, the organic osmolyte transport plays an important role in the regulatory decrease of the principal cell volume under the hypo-osmotic conditions.

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.

Murine Electrophysiological Models of Cardiac Arrhythmogenesis

Cardiac arrhythmias can follow disruption of the normal cellular electrophysiological processes underlying excitable activity and their tissue propagation as coherent wavefronts from the primary sinoatrial node pacemaker, through the atria, conducting structures and ventricular myocardium. These physiological events are driven by interacting, voltage-dependent, processes of activation, inactivation, and recovery in the ion channels present in cardiomyocyte membranes. Generation and conduction of these events are further modulated by intracellular Ca2+ homeostasis, and metabolic and structural change. This review describes experimental studies on murine models for known clinical arrhythmic conditions in which these mechanisms were modified by genetic, physiological, or pharmacological manipulation. These exemplars yielded molecular, physiological, and structural phenotypes often directly translatable to their corresponding clinical conditions, which could be investigated at the molecular, cellular, tissue, organ, and whole animal levels. Arrhythmogenesis could be explored during normal pacing activity, regular stimulation, following imposed extra-stimuli, or during progressively incremented steady pacing frequencies. Arrhythmic substrate was identified with temporal and spatial functional heterogeneities predisposing to reentrant excitation phenomena. These could arise from abnormalities in cardiac pacing function, tissue electrical connectivity, and cellular excitation and recovery. Triggering events during or following recovery from action potential excitation could thereby lead to sustained arrhythmia. These surface membrane processes were modified by alterations in cellular Ca2+ homeostasis and energetics, as well as cellular and tissue structural change. Study of murine systems thus offers major insights into both our understanding of normal cardiac activity and its propagation, and their relationship to mechanisms generating clinical arrhythmias.

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.

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.

Protective effect of Lai Fu Cheng Qi decoction on severe acute pancreatitis-induced myocardial injury in a rat model

The aim of the present study was to evaluate the effects of Lai Fu Cheng Qi decoction on myocardial injury in rats with severe acute pancreatitis (SAP). In total, 30 rats were randomly divided into sham, SAP and decoction treatment groups. SAP was induced by a retrograde pancreatic duct injection of 5% sodium taurocholate in the SAP and decoction treatment groups. Rats in decoction treatment group also received intragastric administration of Lai Fu Cheng Qi decoction. The serum levels of creatine kinase isoenzyme (CK-MB) and lactate dehydrogenase (LDH) were detected using an automatic biochemical analyzer. In addition, myocardial Na⁺-K⁺-ATPase activity was analyzed using a spectrophotometric method and the mitochondrial membrane potential was measured by flow cytometry. Myocardial apoptosis was assessed using a TUNEL assay and pathological changes to the heart and pancreas were detected by hematoxylin and eosin staining. Compared with the rats in the sham group, rats in the SAP and decoction treatment groups exhibited significantly higher levels of serum CK-MB and LDH, apoptosis index and pathological scores, and had significantly lower levels of Na⁺-K⁺-ATPase activity and mitochondrial membrane potential. However, when compared with the SAP group, the serum levels of CK-MB and LDH, the pathological scores of the pancreas and heart, and the myocardial cell apoptosis index in the decoction treatment group were significantly lower. Furthermore, the Na⁺-K⁺-ATPase activity and mitochondrial membrane potential were significantly increased in the decoction treatment group when compared with the SAP group. Therefore, Lai Fu Cheng Qi decoction was shown to exert a protective effect on myocardial injury induced by SAP in rats.

The determinants of transverse tubular volume in resting skeletal muscle

The transverse tubular (t)-system of skeletal muscle couples sarcolemmal electrical excitation with contraction deep within the fibre. Exercise, pathology and the composition of the extracellular fluid (ECF) can alter t-system volume (t-volume). T-volume changes are thought to contribute to fatigue, rhabdomyolysis and disruption of excitation–contraction coupling. However, mechanisms that underlie t-volume changes are poorly understood. A multicompartment, history-independent computer model of rat skeletal muscle was developed to define the minimum conditions for t-volume stability. It was found that the t-system tends to swell due to net ionic fluxes from the ECF across the access resistance. However, a stable t-volume is possible when this is offset by a net efflux from the t-system to the cell and thence to the ECF, forming a net ion cycle ECF→t-system→sarcoplasm→ECF that ultimately depends on Na⁺/K⁺-ATPase activity. Membrane properties that maximize this circuit flux decrease t-volume, including P_Na(t) > P_Na(s), P_K(t) < P_K(s) and N_(t) < N_(s) [P, permeability; N, Na⁺/K⁺-ATPase density; (t), t-system membrane; (s), sarcolemma]. Hydrostatic pressures, fixed charges and/or osmoles in the t-system can influence the magnitude of t-volume changes that result from alterations in this circuit flux. Using a parameter set derived from literature values where possible, this novel theory of t-volume was tested against data from previous experiments where t-volume was measured during manipulations of ECF composition. Predicted t-volume changes correlated satisfactorily. The present work provides a robust, unifying theoretical framework for understanding the determinants of t-volume.

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.

Functional consequences of NKCC2 splice isoforms: insights from a Xenopus oocyte model

The Na(+)-K(+)-2Cl(-) cotransporter NKCC2 is exclusively expressed in the renal thick ascending limb (TAL), where it exists as three main splice isoforms, NKCC2B, NKCC2A, and NKCC2F, with the latter two predominating. NKCC2A is expressed in both medullary and cortical TAL, but NKCC2F localizes to the medullary TAL. The biochemical characteristics of the isoforms have been extensively studied by ion uptake studies in Xenopus oocytes, but the functional consequences of alternative splicing remain unclear. We developed a charge-difference model of an NKCC2-transfected oocyte. The model closely recapitulated existing data from ion-uptake experiments. This allowed the reconciliation of different apparent Km values reported by various groups, which have hitherto either been attributed to species differences or remained unexplained. Instead, simulations showed that apparent Na(+) and Cl(-) dependencies are influenced by the ambient K(+) or Rb(+) bath concentrations, which differed between experimental protocols. At steady state, under bath conditions similar to the outer medulla, NKCC2F mediated greater Na(+) reabsorption than NKCC2A. Furthermore, Na(+) reabsorption by the NKCC2F-transfected oocyte was more energy efficient, as quantified by J NKCC/J Pump. Both the increased Na(+) reabsorption and the increased efficiency were eroded as osmolarity decreased toward levels observed in the cortical TAL. This supports the hypothesis that the NKCC2F is a medullary specialization of NKCC2 and demonstrates the utility of modeling in analyzing the functional implications of ion uptake data at physiologically relevant steady states.

Regulatory volume decrease of rat kidney principal cells after successive hypo-osmotic shocks

Outer Medullary Collecting Duct (OMCD) principal cells are exposed to significant changes of the extracellular osmolarity and thus the analysis of their regulatory volume decrease (RVD) function is of great importance in order to avoid cell membrane rupture and subsequent death. In this paper we provide a sub-second temporal analysis of RVD events occurring after two successive hypo-osmotic challenges in rat kidney OMCD principal cells. We performed experimental cell volume measurements and created a mathematical model based on our experimental results. As a consequence of RVD the cell expels part of intracellular osmolytes and reduces the permeability of the plasma membrane to water. The next osmotic challenge does not cause significant RVD if it occurs within a minute after the primary shock. In such a case the cell reacts as an ideal osmometer. Through our model we provide the basis for further detailed studies on RVD dynamical modeling.

Extracellular magnesium and calcium reduce myotonia in ClC-1 inhibited rat muscle

Loss-of-function mutations in the ClC-1 Cl(-) channel trigger skeletal muscle hyperexcitability in myotonia congenita. For reasons that remain unclear, the severity of the myotonic symptoms can vary markedly even among patients with identical ClC-1 mutations, and may become exacerbated during pregnancy and with diuretic treatment. Since both these conditions are associated with hypomagnesemia and hypocalcemia, we explored whether extracellular Mg(2+) and Ca(2+) ([Mg(2+)]o and [Ca(2+)]o) can affect myotonia. Experimental myotonia was induced in isolated rat muscles by ClC-1 inhibition and effects of [Mg(2+)]o or [Ca(2+)]o on myotonic contractions were determined. Both cations dampened myotonia within their physiological concentration ranges. Thus, myotonic contractile activity was 6-fold larger at 0.3 than at 1.2 mM [Mg(2+)]o and 82-fold larger at 0.3 than at 1.27 mM [Ca(2+)]o. In intracellular recordings of action potentials, the threshold for action potential excitation was raised by 4-6 mV when [Mg(2+)]o was elevated from 0.6 to 3 mM, compatible with an increase in the depolarization of the membrane potential necessary to activate the Na(+) channels. Supporting this notion, mathematical simulations showed that myotonia went from appearing with normal Cl(-) channel function to disappearing in the absence of Cl(-) channel function when Na(+) channel activation was depolarized by 6 mV. In conclusion, variation in serum Mg(2+) and Ca(2+) may contribute to phenotypic variation in myotonia congenita patients.

Regulation of cell volume by glycosaminoglycans

Cell volume is regulated by a delicate balance between ion distribution across the plasma membrane and the osmotic properties of intra- and extracellular components. Using a fluorescent calcein indicator, we analysed the effects of glycosaminoglycans on the cell volume of hyaluronan producing fibroblasts and hyaluronan deficient HEK cells over a time period of 30 h. Exogenous glycosaminoglycans induced cell blebbing after 2 min and swelling of fibroblasts to about 110% of untreated cell volume at low concentrations which decreased at higher concentrations. HEK cells did not show cell blebbing and responded by shrinking to 65% of untreated cell volume. Heparin induced swelling of both fibroblasts and HEK cells. Hyaluronidase treatment or inhibition of hyaluronan export led to cell shrinkage indicating that the hyaluronan coat maintained fibroblasts in a swollen state. These observations were explained by the combined action of the Donnan effect and molecular crowding.

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.

Relationships between resting conductances, excitability, and t-system ionic homeostasis in skeletal muscle

Activation of skeletal muscle fibers requires rapid sarcolemmal action potential (AP) conduction to ensure uniform excitation along the fiber length, as well as successful tubular excitation to initiate excitation–contraction coupling. In our companion paper in this issue, Pedersen et al. (2011. J. Gen. Physiol. doi:10.1085/jgp.201010510) quantify, for subthreshold stimuli, the influence upon both surface conduction velocity and tubular (t)-system excitation of the large changes in resting membrane conductance (G_M) that occur during repetitive AP firing. The present work extends the analysis by developing a multi-compartment modification of the charge–difference model of Fraser and Huang to provide a quantitative description of the conduction velocity of actively propagated APs; the influence of voltage-gated ion channels within the t-system; the influence of t-system APs on ionic homeostasis within the t-system; the influence of t-system ion concentration changes on membrane potentials; and the influence of Phase I and Phase II G_M changes on these relationships. Passive conduction properties of the novel model agreed with established linear circuit analysis and previous experimental results, while key simulations of AP firing were tested against focused experimental microelectrode measurements of membrane potential. This study thereby first quantified the effects of the t-system luminal resistance and voltage-gated Na⁺ channel density on surface AP propagation and the resultant electrical response of the t-system. Second, it demonstrated the influence of G_M changes during repetitive AP firing upon surface and t-system excitability. Third, it showed that significant K⁺ accumulation occurs within the t-system during repetitive AP firing and produces a baseline depolarization of the surface membrane potential. Finally, it indicated that G_M changes during repetitive AP firing significantly influence both t-system K⁺ accumulation and its influence on the resting membrane potential. Thus, the present study emerges with a quantitative description of the changes in membrane potential, excitability, and t-system ionic homeostasis that occur during repetitive AP firing in skeletal muscle.

Control of cell volume in skeletal muscle

Regulation of cell volume is a fundamental property of all animal cells and is of particular importance in skeletal muscle where exercise is associated with a wide range of cellular changes that would be expected to influence cell volume. These complex electrical, metabolic and osmotic changes, however, make rigorous study of the consequences of individual factors on muscle volume difficult despite their likely importance during exercise. Recent charge-difference modelling of cell volume distinguishes three major aspects to processes underlying cell volume control: (i) determination by intracellular impermeant solute; (ii) maintenance by metabolically dependent processes directly balancing passive solute and water fluxes that would otherwise cause cell swelling under the influence of intracellular membrane-impermeant solutes; and (iii) volume regulation often involving reversible short-term transmembrane solute transport processes correcting cell volumes towards their normal baselines in response to imposed discrete perturbations. This review covers, in turn, the main predictions from such quantitative analysis and the experimental consequences of comparable alterations in extracellular pH, lactate concentration, membrane potential and extracellular tonicity. The effects of such alterations in the extracellular environment in resting amphibian muscles are then used to reproduce the intracellular changes that occur in each case in exercising muscle. The relative contributions of these various factors to the control of cell volume in resting and exercising skeletal muscle are thus described.

Extracellular charge adsorption influences intracellular electrochemical homeostasis in amphibian skeletal muscle

The membrane potential measured by intracellular electrodes, E_m, is the sum of the transmembrane potential difference (E₁) between inner and outer cell membrane surfaces and a smaller potential difference (E₂) between a volume containing fixed charges on or near the outer membrane surface and the bulk extracellular space. This study investigates the influence of E₂ upon transmembrane ion fluxes, and hence cellular electrochemical homeostasis, using an integrative approach that combines computational and experimental methods. First, analytic equations were developed to calculate the influence of charges constrained within a three-dimensional glycocalyceal matrix enveloping the cell membrane outer surface upon local electrical potentials and ion concentrations. Electron microscopy confirmed predictions of these equations that extracellular charge adsorption influences glycocalyceal volume. Second, the novel analytic glycocalyx formulation was incorporated into the charge-difference cellular model of Fraser and Huang to simulate the influence of extracellular fixed charges upon intracellular ionic homeostasis. Experimental measurements of E_m supported the resulting predictions that an increased magnitude of extracellular fixed charge increases net transmembrane ionic leak currents, resulting in either a compensatory increase in Na⁺/K⁺-ATPase activity, or, in cells with reduced Na⁺/K⁺-ATPase activity, a partial dissipation of transmembrane ionic gradients and depolarization of E_m.

A mathematical model of fatigue in skeletal muscle force contraction

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation-contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation-contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

Membrane potentials in Rana temporaria muscle fibres in strongly hypertonic solutions

Conventional microelectrode methods were used to measure variations in resting membrane potentials, E(m), of intact amphibian skeletal muscle fibres over a wide range of increased extracellular tonicities produced by inclusion of varying extracellular concentrations of sucrose. Moderate increases in extracellular tonicity to up to 2.6x normal (2.6tau) under Cl(-) free conditions produced negative shifts in E(m) that followed expectations for the K(+) Nernst equation (E(K)) applied to a perfect osmometer containing a conserved intracellular K(+) content despite any accompanying cell volume change. In contrast, E(m) remained stable in fibres studied in otherwise similar Cl(-) containing solutions, consistent with E(m) stabilization despite negative shifts in E(K) through inward cation-Cl(-) co-transport activity. Short exposures to higher tonicities (>3tau) similarly produced negative shifts in E(m) in Cl(-) free but not Cl(-) containing solutions. However, prolonged exposures to solutions of >3tau caused gradual net positive changes in E (m) in both Cl(-) containing and Cl(-) free solutions suggesting that these changes were independent of cation-Cl(-) transport. Indeed, there was no evidence of cation-Cl(-) co-transport activity in strongly hypertonic solutions despite its predicted energetic favourability, suggesting its possible regulation by E (m) in muscle. Additional findings implicated a failure to maintain greatly increased transmembrane [K(+)] gradients in these E(m) changes. Thus: (1) halving or doubling [K(+)](e) produced negative or positive shifts in E(m), respectively in isotonic or moderately hypertonic (<2.7tau), but not strongly hypertonic (>3tau) solutions; (2) subsequent restoration of isotonic extracellular conditions produced further positive changes in E(m) consistent with a dilution of the depleted [K(+)](i) by fibres regaining their original resting volumes; (3) quantitative modelling similarly predicted a gradual net efflux of K(+) as the balance between active and passive [K(+)] fluxes altered due to increased transmembrane [K(+)] gradients in hypertonic and low [K(+)](e) solutions. However, the observed positive changes in E(m) in the most strongly hypertonic solutions eventually exceeded these predictions suggesting additional limitations on Na(+)/K(+)-ATPase activity in strongly hypertonic solutions.

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.