The interpretation of membrane current voltage relations: a Nernst-Planck analysis

Atomic-level simulation of current-voltage relationships in single-file ion channels

The difficulty in characterizing ion conduction through membrane channels at the level of individual permeation events has made it challenging to elucidate the mechanistic principles underpinning this fundamental physiological process. Using long, all-atom simulations enabled by special-purpose hardware, we studied K⁺ permeation across the K_V1.2/2.1 voltage-gated potassium channel. At experimentally accessible voltages, which include the physiological range, the simulated permeation rate was substantially lower than the experimentally observed rate. The current–voltage relationship was also nonlinear but became linear at much higher voltages. We observed permeation consistent with a “knock-on” mechanism at all voltages. At high voltages, the permeation rate was in accordance with our previously reported K_V1.2 pore-only simulations, after the simulated voltages from the previous study were recalculated using the correct method, new insight into which is provided here. Including the voltage-sensing domains in the simulated channel brought the linear current–voltage regime closer to the experimentally accessible voltages. The simulated permeation rate, however, still underestimated the experimental rate, because formation of the knock-on intermediate occurred too infrequently. Reducing the interaction strength between the ion and the selectivity filter did not increase conductance. In complementary simulations of gramicidin A, similar changes in interaction strength did increase the observed permeation rate. Permeation nevertheless remained substantially below the experimental value, largely because of infrequent ion recruitment into the pore lumen. Despite the need to apply large voltages to simulate the permeation process, the apparent voltage insensitivity of the permeation mechanism suggests that the direct simulation of permeation at the single-ion level can provide fundamental physiological insight into ion channel function. Notably, our simulations suggest that the knock-on permeation mechanisms in K_V1.2 and KcsA may be different.

Thermal noise due to surface-charge effects within the Debye layer of endogenous structures in dendrites

An assumption commonly used in cable theory is revised by taking into account electrical amplification due to intracellular capacitive effects in passive dendritic cables. A generalized cable equation for a cylindrical volume representation of a dendritic segment is derived from Maxwell's equations under assumptions: (i) the electric-field polarization is restricted longitudinally along the cable length; (ii) extracellular isopotentiality; (iii) quasielectrostatic conditions; and (iv) homogeneous medium with constant conductivity and permittivity. The generalized cable equation is identical to Barenblatt's equation arising in the theory of infiltration in fissured strata with a known analytical solution expressed in terms of a definite integral involving a modified Bessel function and the solution to a linear one-dimensional classical cable equation. Its solution is used to determine the impact of thermal noise on voltage attenuation with distance at any particular time. A regular perturbation expansion for the membrane potential about the linear one-dimensional classical cable equation solution is derived in terms of a Green's function in order to describe the dynamics of free charge within the Debye layer of endogenous structures in passive dendritic cables. The asymptotic value of the first perturbative term is explicitly evaluated for small values of time to predict how the slowly fluctuating (in submillisecond range) electric field attributed to intracellular capacitive effects alters the amplitude of the membrane potential. It was found that capacitive effects are almost negligible for cables with electrotonic lengths L>0.5 , contributes up to 10% of the signal for cables with electrotonic lengths in the range between 0.25<L<0.5 , and dominates the membrane potential for electrotonically short cables (L<0.2) . These results show that electrotonically short dendritic cables with both ends sealed are prone to significant neurobiological thermal noise due to intracellular capacitive effects. The presence of significant thermal noise weakens the assumption of intracellular isopotentiality when approximating dendrites with compartments.

Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na(+) and K(+). In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Phi) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na(+) activation (micro) and a state-dependent inactivation (eta) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na(+) channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.

Reversal potential of the calcium current in bull-frog atrial myocytes

1. Voltage clamp recordings of the calcium current (ICa) in single myocytes which were enzymatically isolated from bull-frog atrium show that a genuine reversal of the current flowing through Ca2+ channels can be recorded (ef. Reuter & Scholz, 1977; Lee & Tsien, 1982, 1984; Campbell, Giles & Shibata, 1988c). In normal 2.5 mM [Ca2+]0 Ringer solution this apparent reversal potential (Erev) is near +50 mV, a value well below the predicted thermodynamic Ca2+ equilibrium potential (ECa). 2. None the less, Erev shifts with variations in extracellular divalent ion concentrations (Ca2+, Sr2+ and Ba2+) according to the predictions of a Nernstian divalent cation electrode, i.e. approximately 29 mV per 10-fold change in the external concentration of divalent ion. 3. The existing theoretical analysis of this Erev has been extended in order to clarify its interpretation with regard to the selectivity characteristics of ICa. 4. The apparent reversal potential is analysed using a form of the constant field equation which has been modified to include (i) simultaneous monovalent and divalent cation movements and (ii) the presence of a surface potential (V'). This equation can be solved to yield an explicit expression for Erev. The effects of V' on apparent permeability ratios for the Ca2+ channel Erev are demonstrated. 5. In combination, our experimental results and calculations suggest that: (i) previous estimates of V' which were used to describe permeability (P) ratios of Ca2+ channels in various cardiac preparations may be in error, (ii) in normal [Ca2+]o the PNa/PCa ratio is very small, and (iii) PCa/PK must be greater than 1000. An analysis of the relative selectivity of the channel for divalent cations compared to K+ shows that PCa greater than PSr greater than PBa, assuming that PK remains the same after the divalent substitutions. 6. The Ca2+ channel in bull-frog atrial cells is thus much more selective for Ca2+ ions than had previously been estimated; in particular, inward flow of monovalent cations (e.g. Na+) through these channels does not contribute significantly to the observed ICa. The physiological implications of this high selectivity for Ca2+ ions are discussed.

Surmounting barriers in ionic channels

Biological cells are defined by the membrane that shields their vital molecules from the environment. The lipid bilayer of the membrane is an effective dielectric shield (Parsegian, 1969; Andersen, 1978; Honiget al.1986), preventing penetration by charged molecules: the lipid presents a large electrostatic energy barrier because it cannot neutralize the charge of solute molecules nearly as well as water. This energy barrier inhibits the permeation of solutes with local charge, even metabolites. Of course, metabolites do enter cells, and so physiologists have suspected (for a very long time, Hille, 1984, ch. 8) that the membrane shield is pierced by aqueous channels, through which solutes diffuse (with their local charge substantially neutralized) as they cross the membrane. These aqueous pores now have molecular reality (e.g. Nodaet al.1984; Miller, 1986). Each is formed by a specialized protein, integral to membranes, perhaps shaped like a thick-walled pipe, called ionic channels. Channels control the movement of many important molecules in and out of cells by the ‘gating’ mechanism that controls their opening and closing and by the selective properties of their open channel.

The arrhythmogenic current ITI in the absence of electrogenic sodium-calcium exchange in sheep cardiac Purkinje fibres

Sheep cardiac Purkinje fibres were voltage clamped with a two-microelectrode technique. Under conditions that are known to elevate intracellular calcium (0 mM-external potassium), membrane currents were examined. In the above conditions, a brief depolarizing pulse leads to an oscillatory inward current (ITI) which peaks at about 300 ms after the repolarization. An after-contraction is also observed, the peak of which occurs about 80 ms after the peak of ITI. This result is in accord with the results of Kass, Lederer, Tsien & Weingart (1978a). We replaced external sodium with an isotonic CaCl2 solution to remove the sodium-calcium exchange mechanism as a possible current carrier for ITI. In the steady state under these conditions an oscillatory membrane current and after-contraction are seen following repolarization. This current was identified as ITI on the basis of its temporal relation to both the repolarization step and the after-contraction. In isotonic CaCl2, ITI has a reversal potential of -37 mV. Because of this fact ITI cannot be explained by an electrogenic sodium-calcium exchange mechanism alone. The reversal potential suggests that ITI arises from a channel which is permeable to both potassium and calcium. Fluctuations of membrane current and of tension were recorded in the steady state at different holding potentials. Power spectral analysis showed that the current fluctuations were at a minimum at a holding potential of -37 mV. Tension fluctuations were, however, relatively constant over the range of membrane potentials examined (-17 to -70 mV). The peak power of the current fluctuations occurred at about 1.5 Hz (at a holding potential of -70 mV). This peak shifted towards higher frequencies with increasing depolarization. A similar shift in frequency was observed for the tension fluctuations. Cross-correlations between membrane current and tension were calculated for various steady membrane potentials. This analysis shows that the current fluctuations are associated with the tension fluctuations, each with a principal period of about 0.5 s. This analysis also shows that at potentials more negative than the reversal potential of ITI, increasing tension is associated with increasing inward current and that the tension fluctuations follow current fluctuations by about 70 ms. At potentials positive to the reversal potential of ITI, increasing tension was associated with increasing outward current. This analysis therefore indicates that the fluctuations in membrane current reverse at a potential similar to the reversal potential of ITI.(ABSTRACT TRUNCATED AT 400 WORDS)

Relations between chord and slope conductances and equivalent electromotive forces

Nonlinear current-voltage relations for ion movement across biological membranes have been observed and significantly complicate the interpretation of electrical measurements on these transport processes. To enable analysis of the electrical measurements two formalisms have evolved, chord and slope, by which equivalent conductances and electromotive forces (emfs) can be obtained. Because, in the presence of nonlinear relations between current and voltage, the chord conductances and emfs are generally not equal to their slope counterparts, it is imperative that they not be intermixed (8). However, when the functional relationship between the current and voltage is known, such as the Goldman-Hodgkin-Katz (GHK) flux equation, it becomes possible to compare the voltage dependencies of these parameters and examine interrelationships between them. In this communication analytical expressions are derived for the chord and slope conductances and emfs for transport of a single ionic species that obeys the GHK flux equation. Using these expressions, it is possible to convert electrical equivalent circuit parameters derived for one formalism to electrical equivalent parameters of the other formalism. Therefore data obtained using either formalism can be used to obtain values for intracellular activity and membrane permeability to the transported ion. Parallel analyses can be applied to other models of ion transport.

An experimental comparison between the continuum and single jump descriptions of nonactin-mediated potassium transport through black lipid membranes

The continuum and single jump treatments of ion transport through black lipid membranes predict experimentally distinguishable results, even when the same mechanistic assumptions are made and the same potential-distance profile is used. On the basis of steady-state current-voltage curves for nonactin-mediated transport of potassium ions, we find that the continuum model describes the data accurately, whereas the single jump model fails to do so, for all cases investigated in which capacitance measurements indicate that the membrane thickness varies little with applied potential.

The surprising heart: a review of recent progress in cardiac electrophysiology

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