Statistical methods for model discrimination. Applications to gating kinetics and permeation of the acetylcholine receptor channel.

Methods are described for discrimination of models of the gating kinetics and permeation of single ionic channels. Both maximum likelihood and regression procedures are discussed. In simple situations, where models are nested, standard hypothesis tests can be used. More commonly, however, non-nested models are of interest, and several procedures are described for model discrimination in these cases, including Monte Carlo methods, which allow the comparison of models at significance levels of choice. As an illustration, the methods are applied to single-channel data from acetylcholine receptor channels.     

A physical model of potassium channel activation: from energy landscape to gating kinetics

We have developed a method for rapidly computing gating currents from a multiparticle ion channel model. Our approach is appropriate for energy landscapes that can be characterized by a network of well-defined activation pathways with barriers. To illustrate, we represented the gating apparatus of a channel subunit by an interacting pair of charged gating particles. Each particle underwent spatial diffusion along a bistable potential of mean force, with electrostatic forces coupling the two trajectories. After a step in membrane potential, relaxation of the smaller barrier charge led to a time-dependent reduction in the activation barrier of the principal gate charge. The resulting gating current exhibited a rising phase similar to that measured in voltage-dependent ion channels. Reduction of the two-dimensional diffusion landscape to a circular Markov model with four states accurately preserved the time course of gating currents on the slow timescale. A composite system containing four subunits leading to a concerted opening transition was used to fit a series of gating currents from the Shaker potassium channel. We end with a critique of the model with regard to current views on potassium channel structure.     

Gating current harmonics. IV. Dynamic properties of secondary activation kinetics in sodium channel gating.

The kinetics for sodium channel gating appear to involve three coupled processes: (a) "primary" activation, (b) "secondary" activation, and (c) inactivation. Gating current data obtained in dynamic steady states with sinusoidal voltage-clamp were analyzed to give further details about the secondary activation process in sodium channel gating. Unlike primary activation and inactivation, the secondary activation kinetics involve physical processes that become defined when the data are analyzed as a function of the sinusoid frequency in addition to mean membrane potential. The effects of these processes are described, and a physical interpretation is presented.     

Modification of sodium and potassium channel gating kinetics by ether and halothane

The effect of ether and halothane on the kinetics of sodium and potassium currents were investigated in the crayfish giant axon. Both general anesthetics produced a reversible, dose-dependent speeding up of sodium current inactivation at all membrane potentials, with no change in the phase of the currents. Double-pulse inactivation experiments with ether also showed faster inactivation, but the rate of recovery from inactivation at negative potentials was not affected. Ether shifted the midpoint of the steady-state fast inactivation curve in the hyperpolarizing direction and made the curve steeper. The activation of potassium currents was faster with ether present, with no change in the voltage dependence of steady-state potassium currents. Ether and halothane are known to perturb the structure of lipid bilayer membranes; the alterations in sodium and potassium channel gating kinetics are consistent with the hypothesis that the rates of the gating processes of the channels can be affected by the state of the lipids surrounding the channels, but a direct effect of ether and halothane on the protein part of the channels cannot be ruled out. Ether did not affect the capacitance of the axon membrane.     

Reptation theory of ion channel gating.

Reptation theory is a highly successful approach for describing polymer dynamics in entangled systems. In turn, this molecular process is the basis of viscoelasticity. We apply a modified version of reptation dynamics to develop an actual physical model of ion channel gating. We show that at times longer than microseconds these dynamics predict an alpha-helix-screw motion for the amphipathic protein segment that partially lines the channel pore. Such motion has been implicated in several molecular mechanics studies of both voltage-gated and transmitter-gated channels. The experimental probability density function (pdf) for this process follows t-3/2 which has been observed in several experimental systems. Reptation theory predicts that channel gating will occur on the millisecond time scale and this is consistent with experimental results from single-channel recording. We examine the consequences of reptation over random barriers and we show that, to first order, the pdf remains unchanged. In the case of a charged helix undergoing reptation in the presence of a transmembrane potential we show that the tail of the pdf will be exponential. We provide a list of practical experimental predictions to test the validity of this physical theory.     

Stationarity of sodium channel gating kinetics in excised patches from neuroblastoma N1E 115.

Na channel gating parameters in a number of preparations are translated along the voltage axis in excised patches compared to cell attached or whole cell recording. The aim of this study is to determine whether these changes in gating behavior continue over an extended period or, rather, develop rapidly on excision with stationary kinetics thereafter. Average currents were constructed from single-channel records from neuroblastoma N1E 115 at various times after excision, excluding the first 5 min, in eight inside-out excised patches. Single exponentials were fitted to the current decay of the average records, and the mean time constant for each patch was determined. Values were plotted as the percentage difference from these means for each patch against time from excision. Collected results show no obvious trend in values from 5 min to 2 h. Kinetics are stationary, and shifts in Na channel gating parameters along the voltage axis seen in excised as compared to whole cell configuration in neuroblastoma must be complete by the first few minutes after excision. Raising the internal Na concentration reduced the single channel current amplitude, confirming that these are Na channels.     

Novel kinetics in the sodium conductance system predicted by the aggregation model of channel gating.

Special voltage-clamp pulse protocols are given that make differential predictions for the kinetics of models based on a simple sequential, simple cyclic, and an aggregation scheme. Detailed kinetic time-courses for the discriminating pulse protocols are numerically derived from the differential equation system that describes the aggregation model.     

Estimating transition rates in aggregated Markov models of ion channel gating with loops and with nearly equal dwell times

A typical task in the application of aggregated Markov models to ion channel data is the estimation of the transition rates between the states. Realistic models for ion channel data often have one or more loops. We show that the transition rates of a model with loops are not identifiable if the model has either equal open or closed dwell times. This non-identifiability of the transition rates also has an effect on the estimation of the transition rates for models which are not subject to the constraint of either equal open or closed dwell times. If a model with loops has nearly equal dwell times, the Hessian matrix of its likelihood function will be ill-conditioned and the standard deviations of the estimated transition rates become extraordinarily large for a number of data points which are typically recorded in experiments.     

Applications of nonequilibrium response spectroscopy to the study of channel gating. Experimental design and optimization

A novel experimental technique known as non-equilibrium response spectroscopy (NRS) based on ion channel responses to rapidly fluctuating voltage waveforms was recently described (Millonas & Hanck, 1998a). It was demonstrated that such responses can be affected by subtle details of the kinetics that are otherwise invisible when conventional stepped pulses are applied. As a consequence, the kinetics can be probed in a much more sensitive way by supplementing conventional techniques with measurements of the responses to more complex voltage waveforms. In this paper we provide an analysis of the problem of the design and optimization of such waveforms. We introduce some methods for determination of the parametric uncertainty of a class of kinetic models for a particular data set. The parametric uncertainty allows for a characterization of the amount of kinetic information acquired through a set of experiments which can in turn be used to design new experiments that increase this information. We revisit the application of dichotomous noise (Millonas & Hanck, 1998a, b), and further consider applications of a more general class of continuous wavelet -based waveforms. A controlled illustration of these methods is provided by making use of a simplified "toy" model for the potassium channel kinetics.     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Mechanism underlying slow kinetics of the OFF gating current in Shaker potassium channel

Based on the structure of the KcsA potassium channel, the Shaker K+ channel is thought to have, near the middle of the membrane, a cavity that can be occupied by a permeant or a blocking cation. We have studied the interaction between cations in the cavity and the activation gate of the channel, using a set of monovalent cations together with Shaker mutants that modify the structure of the cavity. Our results show that reducing the size of the side chain at position 470 makes it possible for the mutant channel, unlike native Shaker, to close with tetraethylammonium (TEA+) or the long-chain TEA-derivative C10+ trapped inside the channel. Neither I470 mutants nor Shaker can close when N-methyl-glucamine (NMG+) is in the channel, even though this ion is smaller than C10+. Apparently, the carbohydrate side chain of NMG+ prevents gate closing. Gating currents recorded from Shaker and I470C were measured in the presence of different intracellular cations to further analyze the interaction of cations with the gate. Our results suggest that the cavity in Shaker is so small that even permeant cations like Rb+ or Cs+ must leave the cavity before the channel gate can close.     

Nicotinic receptor fourth transmembrane domain: hydrogen bonding by conserved threonine contributes to channel gating kinetics

The fourth transmembrane domain (M4) of the nicotinic acetylcholine receptor (AChR) contributes to the kinetics of activation, yet its close association with the lipid bilayer makes it the outermost of the transmembrane domains. To investigate mechanistic and structural contributions of M4 to AChR activation, we systematically mutated alphaT422, a conserved residue that has been labeled by hydrophobic probes, and evaluated changes in rate constants underlying ACh binding and channel gating steps. Aromatic and nonpolar mutations of alphaT422 selectively affect the channel gating step, slowing the rate of opening two- to sevenfold, and speeding the rate of closing four- to ninefold. Additionally, kinetic modeling shows a second doubly liganded open state for aromatic and nonpolar mutations. In contrast, serine and asparagine mutations of alphaT422 largely preserve the kinetics of the wild-type AChR. Thus, rapid and efficient gating of the AChR channel depends on a hydrogen bond involving the side chain at position 422 of the M4 transmembrane domain.     

Redox reagents and divalent cations alter the kinetics of cystic fibrosis transmembrane conductance regulator channel gating.

Gating of the cystic fibrosis Cl(-) channel requires hydrolysis of ATP by its nucleotide binding folds, but how this process controls the kinetics of channel gating is poorly understood. In the present work we show that the kinetics of channel gating and presumably the rate of ATP hydrolysis depends on the species of divalent cation present and the oxidation state of the protein. With Ca(2+) as the dominant divalent cation instead of Mg(2+), the open burst duration of the channel is increased approximately 20-fold, and this change is reversible upon washout of Ca(2+). In contrast, "soft" divalent cations such as Cd(2+) interact covalently with cystic fibrosis transmembrane conductance regulator (CFTR). These metals decrease both opening and closing rates of the channel, and the effects are not reversed by washout. Oxidation of CFTR channels with a variety of oxidants resulted in a similar slowing of channel gating. In contrast, reducing agents had the opposite effect, increasing both opening and closing rates of the channel. In cell-attached patches, CFTR channels exhibit both oxidized and reduced types of gating, raising the possibility that regulation of the redox state of the channel may be a physiological mode of control of CFTR channel activity.     

Kinetics of unliganded acetylcholine receptor channel gating.

Open- and closed-state lifetimes of unliganded acetylcholine receptor channel activity were analyzed by the method of likelihood maximazation. For both open times and closed times, the best-fitting density is most often a sum of two exponentials. These multiple open states cannot depend on the number of receptor binding sites occupied since they are observed in the absence of ligand. The rate of spontaneous opening and the faster decay constant of closing increased as the membrane was hyperpolarized. The voltage dependence of the rate of spontaneous opening is stronger than that for curare-liganded channels. Evidence that the acetylcholine receptor channel can open spontaneously in the absence of ligand has been presented previously (Sanchez et al, 1983; Brehm et al, 1984; Jackson, 1984). To add to this evidence, alpha-bungarotoxin was added to the patch electrode, causing the frequency of openings to decay with time. The rate constant determined from this decay is similar to rate constants reported for the binding of iodinated alpha-bungarotoxin to the acetylcholine receptor. The frequency of unliganded channel opening has been estimated as 2 X 10(-3) s-1 per receptor. A comparison of carbamylcholine-liganded and spontaneous gating transition rates suggests that ligand binding increases the rate of opening by a factor of 1.4 X 10(7). Carbamylcholine binding increases the mean open time by a factor of 5. Thus, a cholinergic agonist activates the acetylcholine receptor by destabilizing the closed state. The liganded and unliganded channel gating rates were used to analyze the energetics of ligand activation of the acetylcholine receptor channel, and to relate the open channel dissociation constant to the closed channel dissociation constant.     

Kinetic analysis of channel gating. Application to the cholinergic receptor channel and the chloride channel from Torpedo californica.

Identification of the minimum number of ways in which open and closed states communicate is a crucial step in defining the gating kinetics of multistate channels. We used certain correlation functions to extract information about the pathways connecting the open and closed states of the cation channel of the purified nicotinic acetylcholine receptor and of the chloride channel of Torpedo californica electroplax membranes. Single channel currents were recorded from planar lipid bilayers containing the membrane channel proteins under investigation. The correlation functions are conveniently computed from single channel current records and yield information on E, the minimum number of entry/exit states into the open or closed aggregates. E gives a lower limit on the numbers of transition pathways between open and closed states. For the acetylcholine receptor, the autocorrelation analysis shows that there are at least two entry/exit states through which the open and closed aggregates communicate. The chloride channel fluctuates between three conductance substates, here indentified as C, M, and H for closed, intermediate, and high conductance, respectively. Correlation analysis shows that E is greater than or equal to 2 for the M aggregate, indicating that there are at least two distinct entry/exit states in the M aggregate. In contrast, there is no evidence for the existence of more than one entry/exit state in the C or H aggregates. Thus, these correlation functions provide a simple and general strategy to extract information on channel gating kinetics.     

The use of pressure-jump relaxation kinetics to study protein folding landscapes

Pressure-jump induced relaxation kinetics can be used to study both protein unfolding and refolding. These processes can be initiated by upward and downward pressure-jumps of amplitudes of a few 10 to 100 MPa, with a dead-time on the order of milliseconds. In many cases, the relaxation times can be easily determined when the pressure cell is connected to a spectroscopic detection device, such as a spectrofluorimeter. Adiabatic heating or cooling can be limited by small pressure-jump amplitudes and a special design of the sample cell. Here, we discuss the application of this method to four proteins: 33-kDa and 23-kDa proteins from photo-system II, a variant of the green fluorescent protein, and a fluorescent variant of ribonuclease A. The thermodynamically predicted equivalency of upward and downward pressure-jump induced protein relaxation kinetics for typical two-state folders was observed for the 33-kDa protein, only. In contrast, the three other proteins showed significantly different kinetics for pressure-jumps in opposite directions. These results cannot be explained by sequential reaction schemes. Instead, they are in line with a more complex free energy landscape involving multiple pathways.     

The quantal gating charge of sodium channel inactivation

Using a very low noise voltage clamp technique it has been possible to record from the squid giant axon a slow component of gating current (I _g ) during the inactivation phase of the macroscopic sodium current (I _Na ) which was hitherto buried in the baseline noise. In order to examine whether this slowI _g contains gating charge that originates from transitions between the open (O) and the inactivated (I) states, which would indicate a true voltage dependence of inactivation, or whether other transitions contribute charge to slowI _g , a new model independent analysis termed isochronic plot analysis has been developed. From a direct correlation ofI _g and the time derivative of the sodium conductance dg _Na/d the condition when only O-I transitions occur is detected. Then the ratio of the two signals is constant and a straight line appears in an isochronic plot ofI _g vs. dg _Na/d . Its slope does not depend on voltage or time and corresponds to the quantal gating charge of the O-I transition (q _h ) divided by the single channel ionic conductance (). This condition was found at voltages above – 10 mV up to + 40 mV and a figure of 1.21e ^– was obtained forq _h at temperatures of 5 and 15°C. At lower voltages additional charge from other transitions, e.g. closed to open, is displaced during macroscopic inactivation. This means that conventional Eyring rate analysis of the inactivation time constant _h is only valid above – 10 mV and here the figure forq _h was confirmed also from this analysis. It is further shown that most of the present controversies surrounding the voltage dependence of inactivation can be clarified. The validity of the isochronic plot analysis has been confirmed using simulated gating and ionic currents.Abbreviations I _g gating current - I _Na sodium ionic current - g _Na macroscopic sodium conductance - single channel conductance - C, O, I closed, open, inactivated state occupancy of channels - g _h quantal charge displaced in a single O-I transition of Na channel - e ^– equivalent electron charge - h index referring to inactivation process - S _l limiting slope in isochronic plot see Eq.(3) - fractional distance, see Fig. 4 and (4, 5) - TMA tetramethylammonium - TTX tetrodotoxin - Tris tris(hydroxymethyl)aminomethane - HEPES N-2-hydroxyethylpiperazine-N-2-ethanesulfonic acid     

Diffusion model in ion channel gating. Extension to agonist-activated ion channels.

Previously, we described a model which treats ion channel gating as a discrete diffusion problem. In the case of agonist-activated channels at high agonist concentration, the model predicts that the closed lifetime probability density function from single channel recording approximates a power law with an exponent of -3/2 (Millhauser, G. L., E. E. Salpeter, and R. E. Oswald. 1988a. Proc. Natl. Acad. Sci. USA. 85: 1503-1507). This prediction is consistent with distributions derived from a number of ligand-gated channels at high agonist concentration (Millhauser, G. L., E. E. Salpeter, and R. E. Oswald. 1988b. Biophys. J. 54: 1165-1168.) but does not describe the behavior of ion channels at low activator concentrations. We examine here an extension of this model to include an agonist binding step. This extended model is consistent with the closed time distributions generated from the BC3H-1 nicotinic acetylcholine receptor for agonist concentrations varying over three orders of magnitude.     

A negatively charged residue in the outer mouth of rat sodium channel determines the gating kinetics of the channel

Previous studies using combined techniques of site-directed mutagenesis and electrophysiology of voltage-gated Na(+) channels have demonstrated that there are significant overlaps in the regions that are important for the two fundamental properties of the channels, namely gating and permeation. We have previously shown that a pore-lining residue, W402 in S5-S6 region (P loop) in domain I of the micro1 skeletal muscle Na(+) channel, was important in the gating of the channel. Here, we determined the role of an adjacent pore-lining negatively charged residue (E403) in channel gating. Charge neutralization or substitution with positively charged side chain at this position resulted in a marked delay in the rate of recovery from slow inactivation. Indeed, the fast inactivation process appeared intact. Restoration of the negatively charged side chain with a sulfhydryl modifier, MTS-ethylsulfonate, resulted in a reactivation profile from a slow-inactivated state, which was indistinguishable from that of the wild-type channels. We propose an additional functional role for the negatively charged residue. Assuming no major changes in the pore structure induced by the mutations, the negatively charged residue E403 may work in concert with other pore regions during recovery from slow inactivation of the channel. Our data represent the first report indicating the role of negative charge in the slow inactivation of the voltage-gated Na(+) channel.