Modeling squid axon K+ channel by a nucleation and growth kinetic mechanism

A kinetic model accounting for all salient features of the K⁺ channel of the squid giant axon, including the rising phase of the ON gating charge and the Cole-Moore effect, is provided. Upon accounting for a significant feature distinguishing K⁺, Na⁺ and Ca^2 + channels from channel-forming peptides modeled in our previous 2016 BBA paper, the nucleation-and-growth kinetic model developed therein is extended to simulate ON ionic and gating currents of the K⁺ channel of the squid giant axon at different depolarization potentials by the use of only two free parameters. K⁺ channel opening is considered to proceed by progressive aggregation of single subunits, while they are moving their gating charge outward under depolarizing conditions within their tetrameric structure; K⁺ channel closing proceeds in the opposite direction, by repolarization-induced disaggregation of subunits, accompanied by inward movement of their gating charge.     

Effects of yohimbine on squid axons.

Yohimbine, an indolealkylamine alkaloid, reduces the amplitude of the sodium current in the squid giant axon. For doses that reduce sodium current amplitude by up to 50%, there is no significant change in the kinetics or in any of the voltage-dependent parameters associated with sodium channels. The effective equilibrium constant for yohimbine binding to the sodium channel is 3 x 10(-4) M. Repetitive depolarizing pulses increase the inhibition of squid axon sodium current by yohimbine. This use-dependent inhibition is enhanced by increasing the concentration of yohimbine, by increasing the frequency of pulsing, and by increasing the magnitude or the duration of depolarization. It is reduced by hyperpolarizing prepulses. This behavior can be explained by a model wherein yohimbine binds more readily to open sodium channels than to closed sodium channels and wherein the Hodgkin-Huxley kinetic parameters are modified by the binding of the drug. This type of model may also explain the tonic and use-dependent inhibition previously described by others for local anesthetics.     

Dynamics of aminopyridine block of potassium channels in squid axon membrane

Aminopyridines (2-AP, 3-AP, and 4-AP) selectively block K channels of squid axon membranes in a manner dependent upon the membrane potential and the duration and frequency of voltage clamp pulses. They are effective when applied to either the internal or the external membrane surface. The steady-state block of K channels by aminopyridines is more complete for low depolarizations, and is gradually relieved at higher depolarizations. The K current in the presence of aminopyridines rises more slowly than in control, the change being more conspicuous in 3-AP and 4-AP than in 2-AP. Repetitive pulsing relieves the block in a manner dependent upon the duration and interval of pulses. The recovery from block during a given test pulse is enhanced by increasing the duration of a conditioning depolarizing prepulse. The time constant for this recovery is in the range of 10-20 ms in 3-AP and 4-AP, and shorter in 2-AP. Twin pulse experiments with variable pulse intervals have revealed that the time course for re-establishment of block is much slower in 3-AP and 4-AP than in 2-AP. These results suggest that 2-AP interacts with the K channel more rapidly than 3-AP and 4-AP. The more rapid interaction of 2-AP with K channels is reflected in the kinetics of K current which is faster than that observed in 3-AP or 4-AP, and in the pattern of frequency-dependent block which is different from that in 3-AP or 4-AP. The experimental observations are not satisfactorily described by alterations of Hodgkin-Huxley n-type gating units. Rather, the data are consistent with a simple binding scheme incorporating no changes in gating kinetics which conceives of aminopyridine molecules binding to closed K channels and being released from open channels in a voltage-dependent manner.     

Response of delayed (K+) channels to the time-dependent clamping function in squid giant axon. I. Ascending ramps

Squid giant axons are voltage-clamped with ascending potential ramps whose slopes range from 0.5 mV/msec to 60 mV/msec and delayed (K+) currents are observed. The parametric current-voltage curves exhibit a delay period of minimal current followed by a rapid increase of current toward a final steady state. Both the initial delay and the slope of the subsequent rising phase increase with increasing ramp slope. When the Hodgkin-Huxley equations are used to generate theoretical current-voltage curves, the sharp difference between the delay and rising phases is muted and the ramp slope must be increased to produce an adequate representation of the data. A muted biphasic response is also observed when the current-voltage curves are generated using modified Hodgkin-Huxley parameters and a correction for K+ accumulation in the periaxonal space. These modified equations provide an accurate fit for step-potential clamp current data. Since the ramp experiments include all relevant clamping potentials, the experiments provide a sensitive test for kinetic models of K+ on flow in the delayed (K+) channels of squid giant axon.     

Potassium currents in the giant axon of the crab Carcinus maenas

Measurements were made of the kinetic and steady-state characteristics of the potassium conductance in the giant axon of the crab Carcinus maenas. These measurements were made in the presence of tetrodotoxin, using the feedback amplifier concept introduced by Dodge and Frankenhaeuser (J. Physiol, (London) 143:76-90). The conductance increase during depolarizing voltage-clamp pulses was analyzed assuming that two separate potassium channels exist in these axons. The first potassium channel exhibited activation and fast inactivation gating which could be fitted using the m3h, Hodgkin-Huxley formalism. The second potassium channel exhibited the standard n4 Hodgkin-Huxley kinetics. These two postulated channels are blocked by internal application of caesium, tetraethylammonium and sodium ions. External application of 4 amino-pyridine also blocks these channels.     

A model for conductance changes in the squid giant axon. II. Channel kinetics

The kinetics of the sodium and potassium channels in voltage clamped squid giant axon following a relaxation of the membrane subunits are examined and compared with the Hodgkin-Huxley equations. Mechanisms are suggested for the turn-off of the sodium conductance and a set of kinetic states are proposed for the potassium channel which are consistent with the experimental observations. Determination of the rate constants for relaxation of the surface subunits which triggers the subsequent changes within the independent channels provide information on the equilibrium constant and free energy for this process. The free energy is observed to approach zero as the depolarizing voltage of the clamp approaches $\text{E}{}_{\mathrm{<mtext>Na</mtext>}}$, the voltage for zero sodium current in voltage clamped axons. Analysis of the final rate constants in the kinetic sequence for potassium indicates a symmetry of the channel when it is in its steady-state configuration during clamp in the absence of external gradients.     

Single sodium channels from the squid giant axon

Since the work of A. L. Hodgkin and A. F. Huxley (1952. J. Physiol. [Lond.].117:500-544) the squid giant axon has been considered the classical preparation for the study of voltage-dependent sodium and potassium channels. In this preparation much data have been gathered on macroscopic and gating currents but no single sodium channel data have been available. This paper reports patch clamp recording of single sodium channel events from the cut-open squid axon. It is shown that the single channel conductance in the absence of external divalent ions is approximately 14 pS, similar to sodium channels recorded from other preparations, and that their kinetic properties are consistent with previous results on gating and macroscopic currents obtained from the perfused squid axon preparation.     

Potassium currents in the giant axon of the crabCarcinus maenas

Summary Measurements were made of the kinetic and steadystate characteristics of the potassium conductance in the giant axon of the crabCarcinus maenas. These measurements were made in the presence of tetrodotoxin, using the feedback amplifier concept introduced by Dodge and Frankenhaeuser (J. Physiol. (London) 143:76–90). The conductance increase during depolarizing voltage-clamp pulses was analyzed assuming that two separate potassium channels exist in these axons. The first potassium channel exhibited activation and fast inactivation gating which could be fitted using them ³ h, Hodgkin-Huxley formalism. The second potassium channel exhibited the standardn ⁴ Hodgkin-Huxley kinetics. These two postulated channels are blocked by internal application of caesium, tetraethylammonium and sodium ions. External application of 4 amino-pyridine also blocks these channels.     

Homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve

A homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve is derived. It is pointed out that a homomorphism can correct the Cole-Moore discrepancy in delay of conductance for voltage clamp data with initial hyperpolarization. The voltage dependence of the rate constants can also be removed. Curves are presented to compare the representation of the nerve conductances by the Hodgkin-Huxley equations and the new homomorphism.     

Nonlinear cable equations for axons. II. Computations and experiments with external current electrodes

We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold.     

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.     

Differential blockage of two types of potassium channels in the crab giant axon

Summary Measurements were made of the kinetic and steady-state characteristics of the potassium conductance in the giant axon of the crabsCarcinus maenas andCancer pagirus. The conductance increase during depolarizing voltage-clamp pulses was analyzed assuming that two separate types of potassium channels exist in these axons (M. E. Quinta-Ferreira, E. Rojas and N. Arispe,J. Membrane Biol. 66:171–181, 1982). It is shown here that, with small concentrations of conventional K⁺-channel blockers, it is possible to differentially inhibit these channels. The potassium channels with activation and fast inactivation gating (m³h, Hodgkin-Huxley kinetics) were blocked by external application of 4 amino-pyridine (4-AP). The potassium channels with standard gating (n⁴, Hodgkin-Huxley kinetics) were preferentially inhibited by externally applied tetraethylammonium (TEA). The differential blockage of the two types of potassium conductance changes suggests that they represent two different populations of potassium channels.It is further shown here that blocking the early transient conductance increase leads to the inhibition of the repetitive electrical activity induced by constant depolarizing current injection in fibers fromCardisoma guanhumi.     

Modeling squid axon Na+ channel by a nucleation and growth kinetic mechanism

A kinetic model accounting for all salient features of the Na⁺ channel of the squid giant axon is provided. The model furnishes explanations for the Cole-Moore-like effect, the rising phase of the ON gating current and the slow ‘intermediate component’ of its decaying phase, as well as the gating charge immobilization. Experimental ON ionic currents are semi-quantitatively simulated by the use of only three free parameters, upon assuming that the Na⁺ channel opening proceeds along with the stepwise aggregation of its four domains, while they are moving their gating charge outward under depolarizing conditions. The inactivation phase of the ON ionic current is interpreted by a progressive electrostatic attraction between the positively charged ‘hinged lid’ containing the hydrophobic IFM triad and its receptor inside the channel pore, as the stepwise outward movement of the S4 segments of the Na⁺ channel progressively increases the negative charge attracting the triad to its receptor. The Na⁺ channel closing is assumed to proceed by repolarization-induced disaggregation of its domains, accompanied by inward movement of their gating charge. The phenomenon of ‘gating charge immobilization’ can be explained by assuming that gradual structural changes of the receptor over the time course of depolarization strengthen the interaction between the IFM triad and its receptor, causing a slow release of the gating charge during the subsequent repolarization.     

Two stable steady states in the Hodgkin-Huxley axons.

Two stable steady states were found in the numerical solution of the Hodgkin-Huxley equations for the intact squid axon bathed in potassium-rich sea water with an externally applied inward current. Under the conditions the two stable steady-states exist, the Hodgkin-Huxley equations have a complex bifurcation structure including, in addition to the two stable steady-states, a stable limit cycle, two unstable equilibrium points, and one asymptotically stable equilibrium point. It was also concluded that two stable steady states can appear in the Hodgkin-Huxley axons when the leak current is comparable to the currents through the Na and K channels.     

Slow Changes of Potassium Permeability in the Squid Giant Axon

A slow potassium inactivation i.e. decrease of conductance when the inside of the membrane is made more positive with respect to the outside, has been observed for the squid axon. The conductance-potential curve is sigmoid shaped, and the ratio between maximum and minimum potassium conductance is at least 3. The time constant for the change of potassium conductance with potential is independent of the concentration of potassium in the external solution, but dependent upon potential and temperature. At 9 degrees C and at the normal sea water resting potential, the time constant is 11 sec. For lower temperature or more depolarizing potentials, the time constant is greater. The inactivation can be described by modifying the Hodgkin-Huxley equation for potassium current, using one additional parameter. The modified equation is similar in form to the Hodgkin-Huxley equation for sodium current, suggesting that the mechanism for the passive transport of potassium through the axon membrane is similar to that for sodium.     

The Cole-Moore Effect: Still Unexplained?

In the first issue, on the first page of the Biophysical Journal in 1960, Cole and Moore provided the first confirmation of the Hodgkin and Huxley formulation of the sodium and potassium conductances that underlie the action potential. In addition, working with the squid giant axon, Cole and Moore noted that strong hyperpolarization preceding a depolarizing voltage-clamp pulse delayed the rise of the potassium conductance: once started, the time course of the rise was always the same but after significant hyperpolarization there was a long lag before the rise began. This phenomenon has come to be known as the Cole-Moore effect. Their article examines and disproves the hypothesis that the lag reflects the time required to refill the membrane with potassium ions after the ions are swept out of the membrane into the axoplasm by hyperpolarization. The work by Cole and Moore indirectly supports the idea of a membrane channel for potassium conductance. However, the mechanism of the Cole-Moore effect remains a mystery even now, buried in the structure of the potassium channel, which was completely unknown at the time.     

Pharmacological and kinetic analysis of K channel gating currents

We have measured gating currents from the squid giant axon using solutions that preserve functional K channels and with experimental conditions that minimize Na channel contributions to these currents. Two pharmacological agents were used to identify a component of gating current that is associated with K channels. Low concentrations of internal Zn2+ that considerably slow K channel ionic currents with no effect on Na channel currents altered the component of gating current associated with K channels. At low concentrations (10-50 microM) the small, organic, dipolar molecule phloretin has several reported specific effects on K channels: it reduces K channel conductance, shifts the relationship between channel conductance and membrane voltage (Vm) to more positive potentials, and reduces the voltage dependence of the conductance-Vm relation. The K channel gating charge movements were altered in an analogous manner by 10 microM phloretin. We also measured the dominant time constants of the K channel ionic and gating currents. These time constants were similar over part of the accessible voltage range, but at potentials between -40 and 0 mV the gating current time constants were two to three times faster than the corresponding ionic current values. These features of K channel function can be reproduced by a simple kinetic model in which the channel is considered to consist of two, two-state, nonidentical subunits.     

A physical model of sodium channel gating

Most current models of membrane ion channel gating are abstract compartmental models consisting of many undefined states connected by rate constants arbitrarily assigned to fit the known kinetics. In this paper is described a model with states that are defined in terms of physically plausible real systems which is capable of describing accurately most of the static and dynamic properties measured for the sodium channel of the squid axon. The model has two components. The Q-system consists of charges and dipoles that can move in response to an electric field applied across the membrane. It would contain and may compose the gating charge that is known to transfer prior to channel opening. The N-system consists of a charged group or dipole that is constrained to move only in the plane of the membrane and thus does not interact directly with the trans-membrane electric field but can interact electrostatically with the Q-system. The N-system has only two states, its resting state (channel closed) and its excited state (channel open) and its response time is very short in comparison with that of the Q-system. On depolarizing the membrane the the N-system will not make a transition to its open state until a critical amount of Q-charge transfer has occurred. Using only four adjustable parameters that are fully determined by fitting the equilibrium properties of the model to those of the sodium channel in the squid axon, the model is then able to describe with some accuracy the kinetics of channel opening and closing and includes the Cole and Moore delay. In addition to these predictions of the behaviour of assemblies of channels the model predicts some of the individual channel properties measured by patch clamp techniques.