On the theory of ion transport across the nerve membrane. II. Potassium ion kinetics and cooperativity (with x = 4)

Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional (2D) case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional (1D) case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the 2D case, the linear theory gives description of vortical cooperative motion.

Resolving linear and non-linear interactive kinetic mechanisms for ions in membrane channels

Thallous ion in gramicidin channels displays the anomalous molefraction effect and other behavior that suggests its permeationmechanism might be more complicated than the mechanisms for sodiumor potassium ion permeation. The permeation is modeled by eithermultistate first order kinetics where the number of states and therate constants are modified to fit the data or an ion displacementmechanism that requires higher order rate terms. Although the twoclasses of mechanism are difficult to distinguish usingcurrent-voltage data, the two classes give different responses toa modulated transmembrane potential with frequency comparable tothe rate constants for intrachannel ion transitions. Themultistate first order kinetics give currents only at themodulation frequency. Information is transmitted in the phase andamplitude of the observed current. The non-linear iondisplacement mechanism produces harmonic frequencies. A detailedspectral analysis then distinguishes the two classes of mechanismand provides a range of frequency and phase data that permitsdetermination of the appropriate rate constants.

Amplitude histograms can identify positively but not negatively coupled channels

We investigated the ability of amplitude distributions to determine if the gating of a pair of channels is coupled. These distributions are expressed as probability density amplitude histograms with peaks corresponding to zero, one, or two open channels. If the channels gate independently, the areas under these peaks (A, B, and C, respectively) determine the open probabilities of the two channels (p(1) and p(2)). Manivannan et al. (Biophys J 1994;61:216) showed that if Delta=B(2)/AC was less than 4 then the channel gating is coupled. We defined a similar parameter, D=(B(2)/4)-AC. If D<0 then channel gating is coupled. However, amplitude histograms with D0 are consistent with both independent and coupled gating. We further present a simple model in which channels are assumed to be identical and can be positively or negatively coupled. Here, amplitude histograms determine q=(B+2C)/2 (open probability of the coupled channels) and r=-D (the coupling parameter). Thus, positively coupled channels (r0) produce amplitude histograms with D<0 whereas negatively coupled channels (r<0) produce amplitude histograms with D0.

Voltage-independent gating transitions in squid axon potassium channels

We have investigated the actions of internal and external Zn2+ on squid axon K channel ionic and gating currents. As has been noted previously, application of Zn2+ to either membrane surface substantially slowed the activation of these channels with little or no change in deactivation. Internal Zn2+ (near 200-300 nM) slowed channel activation by up to sixfold over the range of membrane voltages from -30 to +50 mV. External Zn2+ (10 mM) produced an approximate twofold slowing of activation from -40 to +40 mV. We found that the changes in ionic current activation kinetics were accompanied by less than a twofold slowing of channel-gating currents in a narrow range of potentials near -30 mV. There was, at most, only a few percent reduction of charge movement associated with Zn2+ application. We conclude that these ions interact with channel components involved in weakly voltage-dependent conformational changes. Although there are some differences in detail, the general similarity of the actions of both internal and external Zn2+ on channel function suggests that the modified channel-gating step involves amino acids accessible to both the internal and external membrane surface.

Superposition properties of interacting ion channels

Quantitative analysis of patch clamp data is widely based on stochastic models of single-channel kinetics. Membrane patches often contain more than one active channel of a given type, and it is usually assumed that these behave independently in order to interpret the record and infer individual channel properties. However, recent studies suggest there are significant channel interactions in some systems. We examine a model of dependence in a system of two identical channels, each modeled by a continuous-time Markov chain in which specified transition rates are dependent on the conductance state of the other channel, changing instantaneously when the other channel opens or closes. Each channel then has, e.g., a closed time density that is conditional on the other channel being open or closed, these being identical under independence. We relate the two densities by a convolution function that embodies information about, and serves to quantify, dependence in the closed class. Distributions of observable (superposition) sojourn times are given in terms of these conditional densities. The behavior of two channel systems based on two- and three-state Markov models is examined by simulation. Optimized fitting of simulated data using reasonable parameters values and sample size indicates that both positive and negative cooperativity can be distinguished from independence.

A scheme to account for the effects of Rb+ and K+ on inward rectifier K channels of bovine artery endothelial cells

An electrochemical gating model is presented to account for the effects described in the companion paper by M. R. Silver, M. S. Shapiro, and T. E. DeCoursey (1994. Journal of General Physiology, 103:519-548) of Rb+ and Rb+/K+ mixtures on the kinetics and voltage dependence of an inwardly rectifying (IR) K+ channel. The model proposes that both Rb+ and K+ act as allosteric modulators of an intrinsically voltage dependent isomerization between open and closed states. Occupancy of binding sites on the outside of the channel promotes channel opening and stabilizes the open state. Rb+ binds to separate sites within the pore and plugs IR channels. Occupancy of the pore by Rb+ can modify the rates of isomerization and the affinity of the allosteric sites for activator ions. The model also incorporates the proposed triple-barreled nature of the IR channel (Matsuda, H., 1988. Journal of Physiology. 397:237-258.) by proposing that plugging of the channel is a cooperative process involving a single site in each of the three bores, 80% of the way through the membrane field. Interaction between bores during plugging and permeation is consistent with correlated flux models of the properties of the IR channel. Parallel bores multiply the number allosteric sites associated with the macromolecular channel and allow for steep voltage dependence without compromising the parallel shift of the half-activation potential with reversal potential. Our model proposes at least six and possibly 12 such allosteric binding sites for activator ions. We derive algebraic relations that permit derivation of parameters that define simple versions of our model from the data of Silver et al. (1994). Numerical simulations based on those parameters closely reproduce that data. The model reproduces the RS+ induced slowing of IR kinetics and the negative shift of the relation between the half-activation voltage (V1/2) and reversal potential when channel plugging is associated with (a) a slowing of the isomerization rates; (b) an increase in the affinity of allosteric sites on closed channels that promote opening; and (c) a decrease in the affinity of sites on open channels that slow closing. Rb+ also slows closing at positive potentials where open channel blockade is unlikely. Allowing Rb+ to be 1.5 times more potent than K+ as an activator in the model can account for this effect and improves the match between the predicted and observed relation between the Rb+ to K+ mole fraction and the opening rate at V1/2.(ABSTRACT TRUNCATED AT 400 WORDS)

Shaker potassium channel gating. II: Transitions in the activation pathway

Voltage-dependent gating behavior of Shaker potassium channels without N-type inactivation (ShB delta 6-46) expressed in Xenopus oocytes was studied. The voltage dependence of the steady-state open probability indicated that the activation process involves the movement of the equivalent of 12-16 electronic charges across the membrane. The sigmoidal kinetics of the activation process, which is maintained at depolarized voltages up to at least +100 mV indicate the presence of at least five sequential conformational changes before opening. The voltage dependence of the gating charge movement suggested that each elementary transition involves 3.5 electronic charges. The voltage dependence of the forward opening rate, as estimated by the single-channel first latency distribution, the final phase of the macroscopic ionic current activation, the ionic current reactivation and the ON gating current time course, showed movement of the equivalent of 0.3 to 0.5 electronic charges were associated with a large number of the activation transitions. The equivalent charge movement of 1.1 electronic charges was associated with the closing conformational change. The results were generally consistent with models involving a number of independent and identical transitions with a major exception that the first closing transition is slower than expected as indicated by tail current and OFF gating charge measurements.

Shaker potassium channel gating. III: Evaluation of kinetic models for activation

Predictions of different classes of gating models involving identical conformational changes in each of four subunits were compared to the gating behavior of Shaker potassium channels without N-type inactivation. Each model was tested to see if it could simulate the voltage dependence of the steady state open probability, and the kinetics of the single-channel currents, macroscopic ionic currents and macroscopic gating currents using a single set of parameters. Activation schemes based upon four identical single-step activation processes were found to be incompatible with the experimental results, as were those involving a concerted, opening transition. A model where the opening of the channel requires two conformational changes in each of the four subunits can adequately account for the steady state and kinetic behavior of the channel. In this model, the gating in each subunit is independent except for a stabilization of the open state when all four subunits are activated, and an unstable closed conformation that the channel enters after opening. A small amount of negative cooperativity between the subunits must be added to account quantitatively for the dependence of the activation time course on holding voltage.

Application of the one- and two-dimensional Ising models to studies of cooperativity between ion channels

The Ising model of statistical physics provides a framework for studying systems of protomers in which nearest neighbors interact with each other. In this article, the Ising model is applied to the study of cooperative phenomena between ligand-gated ion channels. Expressions for the mean open channel probability, rho o, and the variance, sigma 2, are derived from the grand partition function. In the one-dimensional Ising model, interactions between neighboring open channels give rise to a sigmoidal rho o versus concentration curve and a nonquadratic relationship between sigma 2 and rho o. Positive cooperativity increases the slope at the midpoint of the rho o versus concentration curve, shifts the apparent binding affinity to lower concentrations, and increases the variance for a given rho o. Negative cooperativity has the opposite effects. Strong negative cooperativity results in a bimodal sigma 2 versus rho o curve. The slope of the rho o versus concentration curve increases linearly with the number of binding sites on a protomer, but the sigma 2 versus rho o relationship is independent of the number of ligand binding sites. Thus, the sigma 2 versus rho o curve provides unambiguous information about channel interactions. In the two-dimensional Ising model, rho o and sigma 2 are calculated numerically from a series expansion of the grand partition function appropriate for weak interactions. Virtually all of the features exhibited by the one-dimensional model are qualitatively present in the two-dimensional model. These models are also applicable to voltage-gated ion channels.

Cooperative denaturation kinetics of homogeneous polymers

The kinetics of denaturation of a homogeneous, helical biopolymer with nearest neighbor interactions are described, using a kinetic Ising model in which the configuration of its neighbors dictates the transition probability for a single residue in the chain. The actual kinetics that are simulated using Monte Carlo techniques are compared with the results of analytical kinetic equations for the fraction of helix, (s), generated using the mean-field approximation. This mean-field rate equation is expanded as a hierarchy of terms that characterize the nature of rate constants for interacting systems. The first term in the expansion is first order in (s) and varies linearly with the interaction energy. Subsequent rate terms involve higher powers of (s) and demonstrate the need for nonlinear equations in systems with larger interaction energies. Both the simulations and the mean-field approximation show an intrinsic induction period for the single-step kinetic process. They also yield an apparent first-order rate constant that changes as the reaction proceeds. However, only the simulated kinetics yield ordered regions of chain and a nonzero, nearest-neighbor correlation function.

Characterization of the pace-maker current kinetics in calf Purkinje fibres

Kinetics of the cardiac pace-maker current (if) were studied using high K+, low Na+ solutions under conditions where the current time course could be dissected from other components. Activation of if during relatively large negative pulses is S-shaped, and is approximated by an exponential function of time to the third power. Less-pronounced S-shaped activation occurs at potentials close to the middle of the activation curve (near -70/-80 mV). Here, allowing for the presence of a very slow component, the power required to fit the current activation approaches 1. The comparison between current activation and deactivation at the same potentials shows that although deactivation can be approximated by a single exponential, the two processes have a quite different time dependence, and this difference depends on the membrane potential. This behaviour is not compatible with Hodgkin-Huxley kinetics. While near the half-activation range the current decays with an apparently single exponential time course, at more positive potentials the current deactivation becomes sigmoidal. At least the third power of an exponential is required to fit its time course at potentials positive to about -40 mV. These data imply that both open and closed states correspond to several distinct channel configurations. The 'delay' in the current onset during a hyperpolarization is decreased by applying large, short hyperpolarizations before activation. Suitable pre-pulse durations and/or amplitudes can reduce the subsequent current activation to a single exponential. Records with and without a pre-pulse do not always superimpose. After the activation 'delay' has been removed by a suitable hyperpolarization preceding an activating pulse, the time course of its recovery can be studied by applying depolarizations of given amplitude and variable duration. The time course of the delay recovery does not seem to be linked to the time course of current deactivation recorded at the same voltage. Reduction of the activation 'delay' by conditioning pre-hyperpolarizations does not affect current decay during a subsequent depolarizing pulse. The current decay appears to depend only on the current amplitude reached before a deactivating pulse is applied. This, and the evidence in the preceding paragraph, suggest that the delay recovery and the current deactivation are independent processes. A reaction scheme is proposed, which has been developed on the basis of the experimentally determined kinetic properties of if. The channel model is composed of five gating subunits of three different types, not all independent in their movements.(ABSTRACT TRUNCATED AT 400 WORDS)

Delayed activation of the cardiac pacemaker current and its dependence on conditioning pre-hyperpolarizations

The activation time course of the pacemaker current if in Purkinje fibres during hyperpolarizations is s-shaped, which requires kinetics more complex than first order. Large, short hyperpolarizations preceding a fixed activating test pulse cause the current trace to shift to the left on the time-axis, as if a delay in if activation were removed, and eventually lead to a current trace which is a simple exponential and activates with no delay during a test pulse. Current traces before and after a relatively short and small prepulse will superimpose after a time translation. Further increasing prepulse amplitude and/or duration gives current traces that cannot be made to superimpose with the test current trace. The channel opening can be described by first order kinetics only if a delay in current activation is introduced.

Potassium ion currents in the crayfish giant axon. Dynamic characteristics

The kinetics of the voltage-sensitive potassium channel in crayfish axon have been examined. The conductance increase after a step depolarization from rest can be described by a first-order kinetic process raised to the third power. When conditioning voltage levels preceded the test pulse, the steady-state conductance was found to be independent of initial conditions. Depolarizing conditioning voltages in general allowed superposition of test voltage potassium currents by a shift along the time axis. Hyperpolarizing conditioning voltages produced a delay in onset of conductance during the test pulse and changed the kinetics so that superposition was not possible. The delay increased during the hyperpolarization with a first-order lag having a time constant in the range of 1.5-3 ms. Return to the resting level caused recovery from the delayed state to follow a single exponential decay with a time constant of 1.9-2.2 ms. The steady state delay vs. voltage curves were not saturated at potentials as negative as -180 mV.

The Cole-Moore effect in nodal membrane of the frog Rana ridibunda: evidence for fast and slow potassium channels

The K conductance (gK) kinetics were studied in voltage-clamped frog nodes (Rana ridibunda) in double-pulse experiments. The Cole-Moore translation for gK--t curves associated with different initial potentials (E) was only observed with a small percentage of fibers. The absence of the translation was found to be caused by the involvement of an additional, slow, gK component. This component cannot be attributed to a multiple-state performance of the k channel. It can only be accounted for by a separate, slow K channel, the fast channel being the same as the n4 K channel in R. pipiens. The slow K channel is characterized by weaker sensitivity to TEA, smaller density, weaker potential (E) dependence, and somewhat more negative E range of activation than the fast K channel. According to characteristics of the slow K system, three types of fibers were found. In Type I fibers (most numerous) the slow K channel behaves as and n4 HH channel. In Type II fibers (the second largest group found) the slow K channel obeys the HH kinetics within a certain E range only; beyond this range the exponential decline of the slow gK component is preceded by an E-dependent delay, its kinetics after the delay being the same as those in Type I fibers. In Type III fibers (rare) the slow K channel is lacking, and it is only in these fibers that the Cole-Moore translation of the measured gK--t curves can be observed directly. The physiological role of the fast and slow K channel in amphibian nerves is briefly discussed.

The conductance of sodium channels under conditions of reduced current at the node of Ranvier

1. The single sodium channel conductance gamma and the number of channels N were estimated from fluctuation analysis in voltage-clamped nodes of Ranvier under conditions that decreased the size of the sodium current. 2. Reduction of the sodium current by depolarizing prepulses had no effect on gamma, and, in cases where it could be determined, had no significant effect on N. Partial block of the sodium conductance with tetrodotoxin and saxitoxin also did not affect gamma significantly, but reduced N. 3. gamma was reduced to about 40% of the control value at -5 mV when the pH of the external solution was reduced to 5.0. The pH dependence of gamma is consistent with the theories of Woodhull and of Drouin & Neumcke. 4. The differing effects of prepulses, toxins and pH are interpreted in view of the different time scales of channel inactivation or block under these conditions. 5. The nearly unchanged gamma with prepulses and partial toxin block provides further evidence for the absence of interactions among sodium channels.

The use of ascorbate as a probe of opioid receptor structure: evidence for two independent mechanisms of receptor destruction by ascorbate

Ascorbate (20mM) pretreatment of brain membrane suspensions at 37 degrees produced a rapid irreversible loss of specific opioid binding. There was no reduction in specific 3H-halo-peridol binding. Ascorbate induced loss of opioid binding under these experimental conditions was not blocked by low concentrations of EDTA or MN++. In contrast, the slowly developing loss of opioid binding during exposure to 1 mM ascorbate at 23 degrees was completely inhibited by 10(-5)M EDTA or Mn++. At 37 degrees, D-isoascorbate, and several other reducing agents (glutathione, dithiothreitol, cysteine) produced a loss of opioid binding similar to that seen with ascorbate. It is concluded that 1 mM ascorbate at 23 degrees, and 20 mM ascorbate at 37 degrees, destroy opioid binding sites by two independent mechanisms. Lipid peroxidation is implicated at low ascorbate concentrations; a reductive process appears to be responsible for the ascorbate induced loss of binding at higher concentrations.

Conditioning hyperpolarization-induced delays in the potassium channels of myelinated nerve

Hyperpolarizing conditioning pulses delay the onset of potassium channel current in voltage-clamped myelinated nerve fibers. Both the development of and recovery from this conditioning are approximately exponential functions of time: the time constants are functions of the conditioning voltage. The delay is larger and develops faster for more hyperpolarized conditioning pulses. The magnitude of the delay (but not the rate of development or recovery) depends upon the test potential-small test depolarizations produce larger delays than large depolarizations. The currents with and without the conditioning pulse cannot be made to superimpose by a simple time translation.

Cole-Moore effect in the frog node

Potassium currents were recorded from the voltage-clamped frog node (Rana esculenta) during various test pulses that followed hyperpolarizing prepulses of different amplitudes and durations. Both the delay in potassium current onset and the shape of the current trace as a function of time were found to be a function of prepulse parameters. This finding is different from the current trace superposition described by Cole and Moore for a specific test pulse, sodium equilibrium potential in the squid giant axon. The Cole-Moore effect, which was found here only under a specific set of conditions, thus may be a special case rather than the general property of the membrane. The implication of these findings to the various excitable membrane potassium channel models, which are based on the Cole-Moore effect, is discussed.

Regulation of ion permeabilities of isolated rat liver cells by external calcium concentration and temperature

Regulation of ion transport through the plasma membrane was studied on single cell suspensions of hepatocytes, obtained after perfusion of rat liver with collagenase/hyaluronidase solution. Steady-state intracellular K and Na contents were shown to be markedly dependent on external Ca concentration and temperature, the sum of both ion concentrations remaining nearly constant. In contrast, steady-state intracellular chloride content was found to be independent of external Ca concentration, but dependent on temperature. Using the constant field relations, the passive permeabilities PK and PCl for potassium and chloride, respectively, were derived from the experimental data. At temperatures at and above 37 degrees C, with increasing external Ca concentration, PK, exhibits a sharp decrease at about 10(-4)M. In contrast, PCl at 37 degrees C was found to be independent of Ca concentration within experimental error. Earth alkali ions other than Ca, show marked but different effects on PK if compared at equal concentrations. Preincubation of the cells with cholesterol leads to a broadening of the dependence of PK on external Ca concentration. The above results, as well as those on the dependence of PK on external Ca concentration obtained by other authors, could be quantitatively described by a theoretical model of the plasma membrane proposed earlier. This model postulates regulatory binding sites, which cooperatively undergo a cation exchange of divalent cations by K+ ions from the external medium if the cation composition of the latter is altered.

An analysis of the dose-response curve at voltage-clamped frog-endplates

The agonist concentration--endplate conductance relation was examined for a number of agonists (such as carbachol, alkyl trimethylammonium salts, choline and decamethonium). The endplate current evoked varied as Imax [a/(a + K)]2, where a is the agonist concentration and Imax and K are agonist-specific parameters. This finding suggests that the endplate receptor has 2 equivalent subunits which bind agonist approximately non-cooperatively. The liganded subunits then switch to an active conformation with a probability that depends on the nature of the agonist. Both subunits must adopt the active conformation for the channel to open, but the transitions of the subunits could be either independent or concerted.

Statistical mechanics applied to cooperative ligand binding to proteins

By using the lattice statistical argument, we have shown that for a protein whose subunits have the same number of neighbors, the three parameters (K(AB), K(BB), and K(S)K(t)) in the sequential theory formulated by Koshland, Nemethy, and Filmer [Biochemistry (1966) 5, 365] can be reduced to two parameters. One of the parameters, Z, measures the strength of the subunit interactions and is related to the apparent free energy of interaction (DeltaF degrees I) by Z = exp (-DeltaF degrees I/2mkT), where m is the number of neighbors in a subunit and kT has the usual meaning. In addition, we relate Wyman's allosteric binding potential [Advan. Protein Chem. (1964) 19, 223] to the canonical partition function of the McMillan-Mayer theory [J. Chem. Phys. (1945) 13, 276]. An explicit form relating the apparent free energy of interaction and the Hill coefficient is given for an allosteric protein that has nonequivalent and independent ligand-binding sites. The present formulation can be used to account for a number of recent experimental results on hemoglobins.

Fluctuations and noise in kinetic systems. Application to K+ channels in the squid axon

We consider the equilibrium or steady-state noise power density spectrum in the quantity N = Sigma(x) (i=0)a(i)N(i) for an ensemble of independent and equivalent systems each of which can exist in the discrete set of states i = 0, 1, ..., x. N(i) is the number of systems of the ensemble in state i and the a(i)'s are constants. There is a transition rate constant alpha(ij) for an arbitrary transition i --> j; the kinetic equations are linear. There are possible applications to enzyme and biochemical kinetics generally, to membrane transport, muscle contraction, binding on macromolecules, etc. In each case, noise measurements would provide information about the kinetic scheme. The particular application considered here is to K(+) channels or gates (one channel = one system) in the squid axon membrane: a(i)g(K) is the K(+) conductance of a channel in state i and the kinetic scheme is of the Hodgkin-Huxley type (HH). Here we allow an arbitrary set of a(i)'s. This is a generalization of our treatment of K(+) channel noise in an earlier paper. The theory is discussed and some calculations made using Fishman's recent experimental results on K(+) channel noise as a guide. Preliminary indications are that the HH choice of a(i)'s may be oversimplified and that a(0) congruent with 0, a(1) not equal a(0), a(x) not equal a(x-1). Quite possibly the a(i)'s increase from a(0) to a(x), though the early a(i)'s must be relatively small to give the observed induction behavior in g(K)(t). An increase in equal steps is unsatisfactory because this is essentially HH with x = 1 (no induction). More refined experiments may modify these tentative conclusions. In any case, it appears from Fishman's work that noise measurements will probably be very useful in distinguishing between rival models of K(+) channels.

On the theory of ion transport across the nerve membrane, VII. Cooperativity between channels of a large square lattice

The exact kinetics of a 10 x 10 Ising system (periodic boundary conditions) with two-state channels arranged in a square lattice was studied by computer simulation. With all three values of the cooperativity parameter used, no induction in the K(+)-current curve was obtained. This confirms one of our previous conclusions concerning K(+) channels in the squid axon membrane: models with interacting channels arranged in a twodimensional lattice (Adam's model) are apparently excluded. Other topics included: equilibrium properties; short-range pair correlation functions; phase transition.

On the theory of ion transport across the nerve membrane. V. Two models for the Cole-Moore K + hyperpolarization delay

Two illustrative molecular models, designed to explain the Cole-Moore K(+) hyperpolarization delay, are proposed and analyzed. Both introduce a process supplementary to the usual Hodgkin-Huxley (HH) one for a K(+) channel. In both cases the new process becomes involved as a consequence of the conditioning hyperpolarization of the membrane and would account for the observed delay time in the K(+) current after depolarization to near ENa. The first model uses adsorption or desorption of phospholipid molecules on the surface of the assumed protein K(+) channel or gate. The second model involves the translocation of the charged subunits of the channel in the hyperpolarizing electric field.

On the theory of ion transport across the nerve membrane. IV. Noise from the open-close kinetics of K + channels

The theoretical power density spectrum G(f) of fluctuations in the steady-state squid axon K(+) current in the 10(3) Hz region has been derived assuming that these are fluctuations in the number of open K(+) channels in the Hodgkin-Huxley (HH) model. Various modifications of the HH model were also studied. The results were negative in all cases when compared with experiment. This confirms the generally held view that the observed G(f) approximately 1/f is associated primarily with K(+) current through open K(+) channels and not with the open-close kinetics of these channels.

Kinetics of small Ising systems: deviations from internal equilibrium in a tetrahedral model

Computer modeling is used to derive some of the exact kinetic properties of a small Ising system.

On the theory of ion transport across the nerve membrane. VI. Free energy and activation free energies of conformational change

Empirical functions, such as n(infinity)(V) and tau(n)(V) (of the Hodgkin-Huxley type), can be recast in terms of more fundamental functions F(V) (related to a conformational free energy change) and theta(V) (related to the corresponding free energies of activation). Examples of F(V) and theta(V) are given, for squid and frog node. F(V) is essentially a quadratic function of V. The possible molecular origin, for protein-like subunits, of the linear (e.g., net charge) and quadratic (e.g., polarizability) terms in F(V) is discussed. The F(V), theta(V) kind of analysis leads rather automatically to a simple explanation of the well-known approximate coincidence in location (V value) of the maximum in tau(n)(V) (time constant) and the steeply rising part of n(infinity)(V) (also m, 1 - h).

On the theory of ion transport across the nerve membrane. 3. Potassium ion kinetics and cooperativity (with x=4,6,9)

The calculations of Part II of this series have been extended to square (x = 4) and octahedral (x = 6) arrangements of subunits in a potassium channel (or gate). The conclusion is the same as before: experimental induction and superposition properties of gK(t), on depolarization, seem to rule out any significant degree of interaction or cooperativity between the (protein?) subunits of K(+) channel. Calculations for x = 4, 6, and 9 have also been made for a square lattice of interacting channels (periodic boundary conditions). Because of apparent rapid convergence with x, it seems fairly safe to conclude that this model is unsatisfactory. There is some difficulty with superposition but the principal shortcoming is a failure to produce induction behavior. Aggregation models for the K(+) channel are also discussed briefly here. They, too, appear rather unpromising (for the same reasons as seem to exclude conformational cooperativity within a channel).