Analysis of potassium conductance in squid giant axons

Assuming a model of facilitated ionic transport across axonal membranes proposed by McIlroy (1975) and extended by McIlroy and Hahn (1978), it is shown that if the selectivity coefficient, π_K, of the potassium conducting system ?59 the permeabilityP _Ks, of the periaxonal barrier of the squid giant axon for K⁺ ions?(1.2±0.44)×10^?4 cm sec^?1 and the thickness of the periaxonal space ?477±168 Å. Using a value (10^?4 cm sec^?1) ofP _Ks in the foregoing range the experimental curves for the steady state membrane ionic conductance versus measured membrane potential difference (p.d.), ?, of Gilbert and Ehrenstein (1969) are corrected for the effect of accumulation of K⁺ in the periaxonal space. This correction is most marked for the axon immersed in a natural ionic environment, whose conductance curve is shifted ?70mV along the voltage axis in the hyperpolarization direction. By assuming that the physico-chemical connection between a depolarization of the axonal membrane and the consequent membrane conductance changes is a Wien dissociative effect of the membrane's electric field on a weak electrolyte situated in the axolemma, the position of the peaks of the corrected conductance versus ? curves can be identified with zero membrane electric field and hence with zero p.d.across the axolemma. A set of values for the double-layer p.d.s at the axonal membrane interfaces with the external electrolytes in the vicinity of the K⁺ conducting pores can therefore be deduced for the various external electrolytes employed by Gilbert and Ehrenstein. A model of these double-layer p.d.s in which the membrane interfaces are assumed to possess fixed monovalent negatively charged sites, at least in the neighbourhood of the K⁺ conducting pores, is constructed. It is shown that, using the previously deduced values for the doublelayer p.d.s, such a model has a consistent, physically realistic solution for the distance between the fixed charged sites and for the dissociation constants of these sites in their interaction with the ions of the extramembrane electrolytes.     

Significant potassium ion accumulation at the external surface of Myxicola giant axons

Potassium accumulation associated with outward membrane potassium current was investigated experimentally in Myxicola giant axon. During prolonged voltage-clamp pulses to positive transmembrane potentials, the K+ equilibrium potential may approach zero mV, suggesting massive K+ accumulation outside the axonal membrane to concentrations many-fold higher than those in the bathing medium. The potassium accumulation can be satisfactorily described by a three-compartment model, consisting of the nerve fiber, a restricted physiological periaxonal space and the bulk solution. The average thickness, theta, of the periaxonal space is calculated as 177 +/- 59 A, i.e., comparable to that in the squid, while the permeability coefficient of the external barrier, PKs, was calculated to be (1.4 +/- 0.4) X 10(-4) cm/s. These conclusions are well supported by morphological study.     

Modeling repetitive firing and bursting in a small unmyelinated nerve fiber.

The Hodgkin-Huxley equations, originally developed to describe the electrical events in the squid giant axon, have been modified to simulate the ionic and electrical events in a small unmyelinated nerve fiber. The modified equations incorporate an electrogenic sodium-potassium pump, finite intra-axonal volume, a periaxonal space, a calcium current, and calcium-dependent potassium conductance (GKCa). The model shows that adaptation can occur in two ways: increased Na-K pump activity because of periaxonal K accumulation or intra-axonal Na accumulation; or from an increase in (GKCa) caused by calcium accumulating within the axon. Bursting is an extension of adaptation and occurs when the sensitivity of the Na-K pump or (GKCa) to changes in ionic concentration is increased.     

The Effects of Several Alcohols on the Properties of the Squid Giant Axon

The effects of several alcohols on the resting potential, action potential, and voltage-clamp currents of the squid giant axon have been measured. All the alcohols employed are similar in that they depress maximum sodium conductance much more than maximum potassium conductance. Octyl alcohol differs from the others (C₂ through C₅) in that it has less tendency to depolarize the axon. Depolarization is always accompanied by a decrease of g_K near the resting potential, such that the ratio g_K/g_leak is decreased. Steady-state inactivation of the sodium ion current is unaffected by alcohols, as is membrane capacity. Resting membrane conductance is usually decreased by alcohols. The findings are discussed in relation to work on monomolecular films.     

An analysis of the direct effect of chloridazepoxide on mammalian cardiac tissues and crayfish and squid giant axons: possible basis of antiarrhythmic activity.

Chlordiazepoxide has been found to be antiarrhythmic $\text{in vivo}$. The purpose of the present investigation is to identify the mechanism(s) of such antiarrhythmic activity. In canine heart, chlordiazepoxide effectively depressed the enhanced repetitive discharges in subendocardial Purkinje fibers surviving acute myocardial infarction. Chlordiazepoxide altered the action potential characteristics of Purkinje fiber by shortening the APD₅₀, APD₁₀₀ and effective refractory period with little effect on the resting membrane potential. The maximal rate of upstroke (dv/dt) was significantly reduced only at 1 × 10^?4M and above in Purkinje fibers and the membrane response curve was consistently shifted to the right by chlordiazepoxide. The ventricular muscle was little affected by chlordiazepoxide except for the shortened APD₅₀ and reduced dv/dt. Chlordiazepoxide exerted nerve blocking potency comparable to lidocaine in the crayfish giant axon. Voltage-clamp experiments in squid axon showed that chlordiazepoxide suppressed both components of membrane current, the transient inward sodium current being diminished far greater than the steady-state potassium current. These results demonstrate a direct action on cardiac and axonal membranes which may be partially responsible for the antiarrhythmic activity of this agent.     

Macromolecular rate theory explains the temperature dependence of membrane conductance kinetics

The factor Q₁₀ is used in neuroscience to adjust reaction rates of voltage-activated membrane conductances to different temperatures and is widely assumed to be constant. By performing an analysis of published data of the reaction rates of sodium, potassium, and calcium membrane conductances, we demonstrate that 1) Q₁₀ is temperature dependent, 2) this relationship is similar across conductances, and 3) there is a strong effect at low temperatures (<15°C). We show that macromolecular rate theory (MMRT) explains this temperature dependency. MMRT predicts the existence of optimal temperatures at which reaction rates decrease as temperature increases, a phenomenon that we also found in the published data sets. We tested the consequences of using MMRT-adjusted reaction rates in the Hodgkin-Huxley model of the squid’s giant axon. The MMRT-adjusted model reproduces the temperature dependence of the rising and falling times of the action potential. Furthermore, the model also reproduces these properties for different squid species that live in different climates. In a second example, we compare spiking patterns of biophysical models based on human pyramidal neurons from the Allen Cell Types database at room and physiological temperatures. The original models, calibrated at 34°C, failed to generate realistic spikes at room temperature in more than half of the tested models, while the MMRT produces realistic spiking in all conditions. In another example, we show that using the MMRT correction in hippocampal pyramidal cell models results in 100% differences in voltage responses. Finally, we show that the shape of the Q₁₀ function results in systematic errors in predicting reaction rates. We propose that the optimal temperature could be a thermodynamical barrier to avoid over excitation in neurons. While this study is centered on membrane conductances, our results have important consequences for all biochemical reactions involved in cell signaling.     

Potassium Ion Current in the Squid Giant Axon: Dynamic Characteristic

Measurements of the potassium current in the squid axon membrane have been made, after changes of the membrane potential to the sodium potential of Hodgkin and Huxley (HH), from near the resting potential, from depolarizations of various durations and amplitudes, and from hyperpolarizations of up to 150 mv. The potassium currents I given by I = I_∞ {1 - exp [- (t + t₀)/τ]}²⁵, where t₀ is determined by the initial conditions, represent the new data and approximate the HH functions in the regions for which they are adequate. A corresponding modification for the sodium current does not appear necessary. The results support the HH assumptions of the independence of the potassium and sodium currents, the dependence of the potassium current upon a single parameter determined by the membrane potential, and the expression of this parameter by a first order differential equation, and, although the results drastically modify the analytical expressions, they very considerably extend the range of apparent validity of these assumptions. The delay in the potassium current after severe hyperpolarization is used to estimate a potassium ion mobility in the membrane as 10^-5 of its value in aqueous solutions.     

Frequency entrainment of squid axon membrane

Summary Sinusoidally varying stimulating currents were applied to space-clamped squid giant axon membranes in a double sucrose gap apparatus. Stimulus parameters varied were peak-to-peak current amplitude, frequency, and DC offset bias. In response to these stimuli, the membranes produced action potentials in varying patterns, according to variation of input stimulus parameters. For some stimulus parameters the output patterns were stable and obviously periodic with the periods being simple multiples of the input period; for other stimulus parameters no obvious periodicity was manifest in the output. The experimental results were compared with simulations using a computer model which was modified in several ways from the Hodgkin-Huxley model to make it more representative of our preparation. The model takes into account K⁺ accumulation in the periaxonal space, features of Na⁺ inactivation which are anomalous to the Hodgkin-Huxley model, sucrose gap hyperpolarization current, and membrane current noise. Many aspects of the experiments are successfully simulated but some are not, possibly because some very slow process present in the preparation is not included in the model.     

Potassium ion accumulation in a periaxonal space and its effect on the measurement of membrane potassium ion conductance

Summary Potassium currents of various durations were obtained from squid giant axons voltage-clamped in artificial seawater solutions containing sufficient tetrodotoxin to block the sodium conductance completely. From instantaneous potassium current-voltage relations, the reversal potentials immediately at the end of these currents were determined. On the basis of these reversal potential measurements, the potassium ion concentration gradient across the membrane was shown to decrease as the potassium current duration increased. The kinetics of this change was shown to vary monotonically with the potassium ion efflux across the membrane estimated from the integral over time of the potassium current divided by the Faraday, and to be independent of both the external sodium ion concentration and the presence or absence of membrane series resistance compensation. It was assumed that during outward potassium current flow, potassium ions accumulated in a periaxonal space bounded by the membrane and an external diffusion barrier. A model system was used to describe this accumulation as a continuous function of the membrane currents. On this basis, the mean periaxonal space thickness and the permeability of the external barrier to K⁺ were found to be 357 Å and 3.21×10^–4 cm/sec, respectively. In hyperosmotic seawater, the value of the space thickness increased significantly even though the potassium currents were not changed significantly. Values of the resistance in series with the membrane were calculated from the values of the permeability of the external barrier and these values were shown to be roughly equivalent to series resistance values determined by current clamp measurements. Membrane potassium ion conductances were determined as a function of time and voltage. When these were determined from data corrected for the potassium current reversal potential changes, larger maximal potassium conductances were obtained than were obtained using a constant reversal potential. In addition, the potassium conductance turn-on with time at a variety of membrane potentials was shown to be slower when potassium conductance values were obtained using a variable reversal potential than when using a constant reversal potential.     

Effects of Replacement of External Sodium Chloride with Sucrose on Membrane Currents of the Squid Giant Axon

It was observed that a reduction of the sodium chloride concentration in the external solution bathing a squid giant axon by replacement with sucrose resulted in marked decreases in the peak inward and steady-state outward currents through the axon membrane following a step decrease in membrane potential. These effects are quantitatively acounted for by the increase in series resistance resulting from the decreased conductivity of the sea water and the assumption that the sodium current obeys a relation of the form I = k₁C₁ - k₂C₂ where C₁, C₂ are internal and external ion activities and k₁, k₂ are independent of concentration. It is concluded that the potassium ion current is independent of the sodium concentration. That the inward current is carried by sodium ions has been confirmed. The electrical potential (or barrier height) profile in the membrane which drives sodium ions appears to be independent of sodium ion concentration or current. A specific effect of the sucrose on hyperpolarizing currents was observed and noted but not investigated in detail.     

Protein release from the internal surface of the squid giant axon membrane during excitation and potassium depolarization

The proteins in the perfusate collected from intracellularly perfused squid giant axons were analyzed after being labeled with radioactive ¹²⁵I-labeled Bolton-Hunter reagent. The rate of protein release into the perfusate was found to be increased by the following electrophysiological manipulations of the axons: (1) repetitive electrical stimulation at 60 Hz in axons perfused with normal potassium fluoride-containing solution or at 0.125 Hz in axons perfused with tetraethylammonium containing solution, (2) perfusion with 4-amino-pyridine solution which induces spontaneous electrical activity in the axon, and (3) depolarization of the axon induced by raising the external potassium concentration. Sodium dodecyl sulfate polyacrylamide gel electrophoresis of the proteins released under these conditions yielded molecular weight profiles different from those of the extruded axoplasmic proteins. These observations indicate that there exists, in close association with the axonal membrane, a particular group of proteins, the solubility of which is readily affected by changes in the state of the membrane.     

The effect of temperature on veratridine action in squid giant axons

Resting membrane potential and intracellular sodium and potassium concentrations were determined at 5 and 21°C in normal and veratridine-treated axons of the squid Doryteuthis plei. 300 μM veratridine produced an increase in the intracellular sodium concentration, which changed from 52 to 284 mM in 10 min of exposure at 21°C, and from 76 to 260 mM at 5°C. Under the same treatment the intracellular potassium concentration changed from 357 to 221 mM (21°C) and from 334 to 194 mM (5°C). All the changes could be prevented by adding 1 μM tetrodotoxin. Veratridine (30, 100 and 300 μM) increased the resting sodium permeability of the giant axon, and the effect was greater at 21°C. The affinity of the membrane for veratridine increases when the nerves are cooled, the three concentrations tested produce maximum activation of the sodium channels at 5°C. But only the higher two concentrations are saturating at 21°C.     

Effects of narcotics on the giant axon of the squid

—Levorphanol (10^-3 M) reversibly blocked conduction in the giant axon of the squid and axons from the walking legs of spider crab and lobster. Similar concentrations of levallorphan and dextrorphan blocked conduction in the squid giant axon. Under the same experimental condition morphine caused an approximately 40 per cent decrease in spike height. Levorphanol did not affect the resting potential or resistance of the squid axon. Spermidine, spermine and dinitrophenol had little or no direct effect on the action potential nor did they alter the potency of levorphanol. Concentrations of levorphanol as low as 5 × 10^-5 M blocked repetitive or spontaneous activity in the squid axon induced by decreasing the divalent cations in the medium. After exposure to tritiated levorphanol, the axoplasm and envelope of the squid axon accumulated up to 500 per cent of the concentration of tritium found in the external medium, dependent on time of exposure, and other variables. At pH 6 the levels of penetration were 33-50% of those found at pH 8, which correlates with our observation that levorphanol is about 33 % as potent in blocking the action potential at pH 6. The penetrability of levorphanol was not affected by spermidine, dinitrophenol or cottonmouth moccasin venom. Levorphanol did not alter the penetration of [C¹⁴]acetylcholine nor did it render the squid axon sensitive to it. The block of axonal conduction by compounds of the morphine series is discussed both as to possible mechanisms and significance.     

An analysis of conductance changes in squid axon

The membrane of the squid axon is considered on the basis of a pore model in which the distribution of the pore sizes strongly favors K⁺ transfer when there is no potential. Electrical asymmetry causes non-penetrating ions on the membrane capacitor to exert a mechanical force on both membrane surfaces and this force results in a deformation of the membrane pore system such that it assumes a distribution of sizes favoring the ions exerting mechanical force. The ions involved appear to be Ca⁺⁺ on the outside of the membrane and isethionate^-, (i^-) on the inside; as Ca⁺⁺ is equivalent in size to Na⁺, the charged membrane is potentially able to transfer Na⁺, when the ions deforming the membrane pore distribution are removed. A depolarization of the membrane leads to an opening of pores that will allow Na⁺ penetration and a release of the membrane from deformation. The pores revert to the zero-potential pore size distribution hence the Na permeability change is a transient. Calculation shows that the potassium conductance vs. displacement of membrane potential curve for the squid axon and the "inactivation" function, h, can be obtained directly from the assumed membrane distortion without the introduction of arbitrary parameters. The sodium conductance, because it is a transient, requires assumptions about the time constants with which ions unblock pores at the outside and the inside of the membrane.     

A fit to nerve membrane rectification curves with a double-dipole-layer membrane model

Following Wei's suggestion that nerve stimulation and conduction properties are due to dipole layers at the two membrane surfaces (Wei, 1969), we have done steady-state electro-diffusion calculations in the constant field approximation for a simple double-dipole-layer model. We are thereby able to quantitatively fit the recent potassium iso-osmotic rectification curves of Gilbert and Ehrenstein for the squid giant axon membrane. For the squid axon membrane in a natural ion environment, only the outside dipole layer is present in the fit to the data.     

Non-Linear Current-Potential Relations in an Axon Membrane

The membrane current density, I_m, in the squid giant axon has been calculated from the measured external current applied to the axon, I_o, by the equation See PDF for Equation where V_m is the membrane potential under the current electrode and r₁ and r₂ are the external and internal longitudinal resistances. The original derivation of this equation included in one step an assumption of a linear relation between I_m and V_m. It is shown that the same equation can be obtained without this restricting assumption.     

A Single-File Model for Potassium Transport in Squid Giant Axon: Simulation of Potassium Currents at Normal Ionic Concentrations

A physical model for potassium transport in squid giant axon is proposed. The model is designed to explain the empirical data given by the Hodgkin-Huxley model and related experiments. It is assumed that K⁺ moves across the axon membrane by single-file diffusion through narrow pores. In the model a pore has three negatively charged sites that can be occupied alternatively by K⁺ or by a gating particle, GP⁺⁺, coming from the external surface. GP⁺⁺ is considered to be part of the membrane rather than a diffusible component of the surrounding solutions. A high activation barrier for GP⁺⁺ is supposed at the inner membrane border so that it cannot change over to the internal surface. Therefore potassium diffusion can be blocked by GP⁺⁺ penetrating into the pores. This mechanism controls the dynamic behaviour of the model. The time-dependent probabilities of the pore states are described by a system of differential equations. The rate constants in these equations depend on the ionic concentrations, the membrane voltage, and the electrostatic interaction between ions in a single pore. Detailed computational tests for normal composition of external and internal solutions show that the model agrees remarkably well with the stationary and dynamic behaviour of the Hodgkin-Huxley model. However, the hyperpolarization delay is not reproduced. A structural modification, concerning this delay and the way in which GP⁺⁺ is attached to the membrane, is proposed, and the qualitative behavior of the model at varied external and internal concentrations is discussed.     

Potassium ion accumulation slows the closing rate of potassium channels in squid axons.

Potassium ion accumulation in the periaxonal space between squid axonal membrane and the Schwann cell surrounding the axon slows the rate of potassium channel closing to a degree that is consistent with the effect on channel closing of an equivalent change in the bulk external potassium concentration. The alteration of channel gating is independent of membrane potential, V, for V less than or equal to -60 mV, which suggests that the effect is mediated at a site on the outer surface of the membrane, rather than a site within the channel.     

K+ accumulation in the space between giant axon and Schwann cell in the squid Alloteuthis. Effects of changes in osmolarity.

In a train of impulses in squid giant axon, accumulation of extracellular potassium causes successive afterhyperpolarizations to be progressively less negative. In Loligo, Frankenhaeuser and Hodgkin had satisfactorily accounted for the characteristics of this effect with a model in which the axon is surrounded by a space, width theta, and a barrier of permeability P. In axons isolated from Alloteuthis, we found that the model fitted the observations quite well. Superfusing the axon with hypotonic artificial seawater (ASW) caused theta and P to decrease, and, conversely, hypertonic ASW caused them to increase: this would be the case if both the space and the pathway through the barrier were extracellular. In some cases, in normal ASW, the afterhyperpolarizations in a train decreased very little, less than 0.7 mV. In these extreme cases, theta was estimated to be 190 nm and P to be 7 x 10(-4) cm s-1, both several times the values of 30 nm and 6 x 10(-5) cm s-1 estimated by Frankenhaeuser and Hodgkin. We suggest that in vivo the periaxonal space may be considerably wider than that seen in conventionally fixed squid tissue.     

Temperature Characteristics of Excitation in Space-Clamped Squid Axons

Temperature characteristics of excitability in the squid giant axon were measured for the space-clamped axon with the double sucrose gap technique. Threshold strength-duration curves were obtained for square wave current pulses from 10 µsec to 10 msec and at temperatures from 5°C to 35°C. The threshold change of potential, at which an action potential separated from a subthreshold response, averaged 17 mv at 20°C with a Q¹⁰ of 1.15. The average threshold current density at rheobase was 12 µa/cm² at 20°C with a Q¹⁰ of 2.35 compared to 2.3 obtained previously. At short times the threshold charge was 1.5·10^-8 coul/cm². This was relatively independent of temperature and occasionally showed a minimum in the temperature range. At intermediate times and all temperatures the threshold currents were less than for both the single time constant model and the two factor excitation process as developed by Hill. FitzHugh has made computer investigations of the effect of temperature on the excitation of the squid axon membrane as represented by the Hodgkin-Huxley equations. These are in general in good agreement with our experimental results.