A kinetic model for determining the consequences of electrogenic active transport in cardiac muscle

When active transport is electrogenic in a tissue that is continuously active, such as cardiac muscle, the active transport current is as important in the generation of the action potential as are the passive currents. A thermodynamically constrained kinetic model of electrogenic active transport of sodium and potassium ions has been developed in which the influences of voltage and chemical composition are explicitly defined. This model is coupled to a system of passive permeabilities, of the minimum degree of complexity, to simulate the integrated activity of active and passive ion transport in the generation of the cardiac action potential. Results of preliminary simulations indicate that electrogenic active transport provides a mechanism for slowly changing currents both within the time scale of an action potential as well as of many action potentials. The presence of active transport also complicates the interpretation of isotopic flux measurements and the separation of currents.     

A model for conformational coupling of membrane potential and proton translocation to ATP synthesis and to active transport.

Acceptance of a membrane potential and/or a proton gradient as a possible means of transmitting energy from oxidations to ATP synthesis rests in part on a satisfactory hypothesis for how the potential or proton gradient could drive ATP synthesis. Recognition that energy input may drive ATP synthesis by change in binding of reactants at the catalytic site has led to the suggestions presented in this paper. These are that in oxidative phosphorylation and photophosphorylation, the requisite conformational changes may be coupled to exposure of charged groups to different sides of the membrane. The cycle of charged group exposure or movement may be driven by the membrane potential or, through protonation and deprotonation, may be coupled to proton translocation across the membrane. Effects of proton gradient and membrane potential may be additive. Similar conformational coupling suggestions may explain proton translocation coupled to ATP cleavage and active transport of metabolites coupled to membrane potential, proton gradients of ATP cleavage.     

How different two almost identical action potentials can be: A model study on cardiac repolarization

Spatial heterogeneity in the properties of ion channels generates spatial dispersion of ventricular repolarization, which is modulated by gap junctional coupling. However, it is possible to simulate conditions in which local differences in excitation properties are electrophysiologically silent and only play a role in pathological states. We use a numerical procedure on the Luo-Rudy phase 1 model of the ventricular action potential (AP1) in order to find a modified set of model parameters which generates an action potential profile (AP2) almost identical to AP1. We show that, although the two waveforms elicited from resting conditions as a single AP are very similar and belong to membranes sharing similar passive electrical properties, the modified membrane generating AP2 is a weaker current source than the one generating AP1, has different sensitivity to up/down-regulation of ion channels and to extracellular potassium, and a different electrical restitution profile. We study electrotonic interaction of AP1- and AP2 - type membranes in cell pairs and in cable conduction, and find differences in source-sink properties which are masked in physiological conditions and become manifest during intercellular uncoupling or partial block of ion channels, leading to unidirectional block and spatial repolarization gradients. We provide contour plot representations that summarize differences and similarities. The present report characterizes an inverse problem in cardiac cells, and strengthen the recently emergent notion that a comprehensive characterization and validation of cell models and their components are necessary in order to correctly understand simulation results at higher levels of complexity.     

A model of active transport of ions in hepatocytes

A mathematical model of the transport of basic ions (K⁺, Na⁺, Cl⁻) across the hepatocyte membrane has been created using the previously constructed models of active ion transport in biomembranes. The dependence of the resting potential on extracellular ion concentration has been established. Using the model, it is possible to independently calculate the resting potential at the biomembrane and the concentrations of sodium, potassium, and chlorine ions in the cell. The calculated internal concentrations of the ions are in good agreement with the corresponding experimental values.     

An upper limit for the electrogenic Na-K pump contribution to maximum diastolic potential in feline cardiac Purkinje fibers in steady state

We have estimated an upper limit for the electrogenic contribution of the Na-K pump to diastolic transmembrane potential. We simultaneously monitored the maximum diastolic potential and the extracellular space potassium activity during exposure to a very high concentration of ouabain. Exposure to ouabain caused a depolarization of approximately 3 mV (n = 33 experiments) over 34 +/- 3 s (mean +/- standard error) prior to any change in extracellular K activity. In four experiments, we monitored intracellular sodium activity and observed it to rise with approximately the same temporal lag (delay = 26 +/- 7 s). We also measured relative membrane conductance in one series of experiments and observed it to decrease to 91 +/- 2% of its control value by the time extracellular space K began to rise. Following the initial increase in extracellular space K activity the subsequent membrane depolarization is shown to be accurately predicted solely from the measured increase in extracellular space K activity as calculated from the Goldman equation. Limitations of the method and possible interpretations of the data are discussed. We interpret this ouabain-induced depolarization that occurs prior to the rise in external K to be an upper limit to the Na-K pump's electrogenic contribution to steady-state membrane potential.     

Contributions of secondary active transport processes to membrane potentials

Summary Equations are developed to examine the effects of secondary active transport processes on the steady-state membrane potential of symmetrical cells. It is shown that, with suitable modifications, equations of the type developed by Goldman, Hodgkin and Katz may be derived to accommodate the contributions to the membrane potential of both electroneutral and electrogenic transporters. Where the membrane potential is function of the dominant medium ions (Na, K, and Cl), other contributions can come only from an electrogenic Na pump and from neutral co- and counter-transporters if, and only if, these involve the dominant ions. Experimental approaches to measure the parameters necessary to solve the equations developed here are discussed.     

A viscoplastic model for the active component in cardiac muscle

The HMK model (Hunter et al. in Prog Biophys Mol Biol 69:289–331, 1998) proposes mechanobiological equations for the influence of intracellular calcium concentration \(\hbox {Ca}_\mathrm{i}\) on the evolution of bound calcium concentration \(\hbox {Ca}_\mathrm{b}\) and the tropomyosin kinetics parameter z, which model processes in the active component of the tension in cardiac muscle. The inelastic response due to actin-myosin crossbridge kinetics is modeled in the HMK model with a function Q that depends on the history of the rate of total stretch of the muscle fiber. Here, an alternative model is proposed which models the active component of the muscle fiber as a viscoplastic material. In particular, an evolution equation is proposed for the elastic stretch \(\lambda _\mathrm{a} \) in the active component. Specific forms of the constitutive equations are proposed and used to match experimental data. The proposed viscoplastic formulation allows for separate modeling of three processes: the high rate deactivation of crossbridges causing rapid reduction in active tension; the high but lower rate reactivation of crossbridges causing recovery of active tension; and the low rate relaxation effects characterizing the Hill model of muscles.     

K-Na transport: the Brewer model replaces the active transport model

In the 1940's several experimental observations were made regarding the K+ and Na+ content of chilled and restored red blood cells. As a consequence, the concept of active transport was developed. Brewer, a physicist, developed a model for membrane transport based on the electrical properties of double bonds in the ground and excited states. Of particular importance is the membrane double bond P = O. This model was largely formulated from isotope concentration studies using mass spectroscopy, photospectrometry and the nature of malignant cells. In this study, it is shown that the Brewer model completely explains the experimental results which led to the concept of active transport. In addition, it also explains the results of some adjunct experiments.     

A two-current model for the dynamics of cardiac membrane

In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon—subcritical alternans—that does not occur for the commonly used exponential map.     

3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study

The number of mathematical models of cardiac cellular excitability is rapidly growing, and compact graphical representations of their properties can make new acquisitions available for a broader range of scientists in cardiac field. Particularly, the intrinsic over-determination of the model equations systems when fitted only to action potential (AP) waveform and the fact that they are frequently tuned on data covering only a relatively narrow range of dynamic conditions, often lead modellers to compare very similar AP profiles, which underlie though quite different excitable properties. In this study I discuss a novel compact 3D representation of the cardiac cellular AP, where the third dimension represents the instantaneous current–voltage profile of the membrane, measured as repolarization proceeds. Measurements of this type have been used previously for in vivo experiments, and are adopted here iteratively at a very high time, voltage, current-resolution on (i) the same human ventricular model, endowed with two different parameters sets which generate the same AP waveform, and on (ii) three different models of the same human ventricular cell type. In these 3D representations, the AP waveforms lie at the intersection between instantaneous time–voltage–current surfaces and the zero-current plane. Different surfaces can share the same intersection and therefore the same AP; in these cases, the morphology of the current surface provides a compact view of important differences within corresponding repolarization dynamics.Refractory period, supernormal excitability window, and extent of repolarization reserve can be visualized at once. Two pivotal dynamical properties can be precisely assessed, i.e. all-or-nothing repolarization window and membrane resistance during recovery. I discuss differences in these properties among the membranes under study, and show relevant implications for cardiac cellular repolarization.     

Model of active transport of ions in cardiac cell

A model of the active transport of ions in a cardiac muscle cell, which takes into account the active transport of Na⁺, K⁺, Ca²⁺, Mg²⁺, HCO₃⁻ and Cl⁻ ions, has been constructed. The model allows independent calculations of the resting potential at the biomembrane and concentrations of basic ions (sodium, potassium, chlorine, magnesium and calcium) in a cell. For the analysis of transport processes in cardiac cell hierarchical algorithm “one ion-one transport system” was offered. The dependence of the resting potential on concentrations of the ions outside a cell has been established. It was shown, that ions of calcium and magnesium, despite their rather small concentration, play an essential role in maintenance of resting potential in cardiac cell. The calculated internal concentrations of ions are in good agreement with the corresponding experimental values.     

Coupling ratio of the Na-K pump in the lobster cardiac ganglion

The electrogenic Na-K pump coupling ratio in the large neurons of the lobster cardiac ganglion was determined by two different electrophysiological techniques. A graphical analysis plotting exp(EmF/RT) vs. [K]o after the pump was blocked by ouabain was used to determine values for [K]i, PNa/PK, and the pump coupling ratio. These measurements were made 4-8 h after the cells were penetrated with microelectrodes, and thus represent non-Na-loaded steady state values. The value obtained for the pump coupling ratio under these conditions was 1.44 +/- 0.06 (n = 9) or close to 3 Na for 2 K. The second technique used to measure the coupling ratio was to iontophoretically inject Na ions into the neuron. Neurons were penetrated with three microelectrodes, two of which were filled with 2 M Na-citrate; the third electrode contained either 2 M K-citrate or 3 M KCl. By passing current between the Na salt-containing electrodes, Na was injected into the cell soma. The injection system was calibrated by injecting 24Na-citrate into counting vials from representative microelectrodes (calculated 24Na transport = 0.92). By knowing the Na load injected into the cells, and by measuring the time-current area produced by the Na activation of the Na-K pump, the coupling ratio was calculated to be 1.54 +/- 0.05 (n = 19), which is not significantly different from the value obtained by the first method. This value represents a Na-loaded experimental situation. When Na was removed from the external bathing solution, the coupling ratio shifted to 2 Na to 1 K (2.0 +/- 0.07, n = 4). These results suggest that the pump normally operates with a 3:2 ratio both in steady state and under Na load but that in the absence of external Na, it can operate with less than a full complement (2) of K on the external surface of the pump.