Data transformations for improved display and fitting of single-channel dwell time histograms.

A.L. Blatz and K.L. Magleby (1986a. J. Physiol. [Lond.]. 378:141-174) have demonstrated the usefulness of plotting histograms with a logarithmic time axis to display the distributions of dwell times from recordings of single ionic channels. We derive here the probability density function (pdf) corresponding to logarithmically binned histograms. Plotted on a logarithmic time scale the pdf is a peaked function with an invariant width; this and other properties of the pdf, coupled with a variance-stabilizing (square root) transformation for the ordinate, greatly simplify the interpretation and manual fitting of distributions containing multiple exponential components. We have also examined the statistical errors in estimation, by the maximum-likelihood method, of kinetic parameters from logarithmically binned data. Using binned data greatly accelerates the fitting procedure and introduces significant errors only for bins spaced more widely than 8-16 per decade.     

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.     

Identifying kinetic gating mechanisms for ion channels by using two-dimensional distributions of simulated dwell times.

Ion channels are integral membrane proteins that regulate ionic flux through cell membranes by opening and closing (or gating) their pores. The gating can be monitored by observing step changes in the current flowing through single channels. Analysis of the durations of the open and closed intervals and of the correlations among the interval durations can give insight into the gating mechanism. Although it is well known that the correlation information can be essential to distinguish among possible gating mechanisms, it has been difficult to use this information because it has not been possible to correct the predicted correlations for the distortion of the single-channel data because of filtering and noise. To overcome this limitation we present a method based on a comparison of simulated and experimental two-dimensional dwell-time distributions constructed by analysing simulated and experimental single-channel currents in an identical manner. The simulated currents incorporate the true effects of filtering and noise, the two-dimensional distributions retain the correlation information, and the identical analysis allows direct maximum-likelihood comparison of the simulated and experimental two-dimensional distributions. We show that the two-dimensional simulation method has a greatly increased ability to distinguish among models, compared with methods that use one-dimensional distributions.     

Measuring kinetics of complex single ion channel data using mean-variance histograms.

The measurement of single ion channel kinetics is difficult when those channels exhibit subconductance events. When the kinetics are fast, and when the current magnitudes are small, as is the case for Na+, Ca2+, and some K+ channels, these difficulties can lead to serious errors in the estimation of channel kinetics. I present here a method, based on the construction and analysis of mean-variance histograms, that can overcome these problems. A mean-variance histogram is constructed by calculating the mean current and the current variance within a brief "window" (a set of N consecutive data samples) superimposed on the digitized raw channel data. Systematic movement of this window over the data produces large numbers of mean-variance pairs which can be assembled into a two-dimensional histogram. Defined current levels (open, closed, or sublevel) appear in such plots as low variance regions. The total number of events in such low variance regions is estimated by curve fitting and plotted as a function of window width. This function decreases with the same time constants as the original dwell time probability distribution for each of the regions. The method can therefore be used: 1) to present a qualitative summary of the single channel data from which the signal-to-noise ratio, open channel noise, steadiness of the baseline, and number of conductance levels can be quickly determined; 2) to quantify the dwell time distribution in each of the levels exhibited. In this paper I present the analysis of a Na+ channel recording that had a number of complexities. The signal-to-noise ratio was only about 8 for the main open state, open channel noise, and fast flickers to other states were present, as were a substantial number of subconductance states. "Standard" half-amplitude threshold analysis of these data produce open and closed time histograms that were well fitted by the sum of two exponentials, but with apparently erroneous time constants, whereas the mean-variance histogram technique provided a more credible analysis of the open, closed, and subconductance times for the patch. I also show that the method produces accurate results on simulated data in a wide variety of conditions, whereas the half-amplitude method, when applied to complex simulated data shows the same errors as were apparent in the real data. The utility and the limitations of this new method are discussed.     

Nuclear transport of single molecules: dwell times at the nuclear pore complex

The mechanism by which macromolecules are selectively translocated through the nuclear pore complex (NPC) is still essentially unresolved. Single molecule methods can provide unique information on topographic properties and kinetic processes of asynchronous supramolecular assemblies with excellent spatial and time resolution. Here, single-molecule far-field fluorescence microscopy was applied to the NPC of permeabilized cells. The nucleoporin Nup358 could be localized at a distance of 70 nm from POM121-GFP along the NPC axis. Binding sites of NTF2, the transport receptor of RanGDP, were observed in cytoplasmic filaments and central framework, but not nucleoplasmic filaments of the NPC. The dwell times of NTF2 and transportin 1 at their NPC binding sites were 5.8 +/- 0.2 and 7.1 +/- 0.2 ms, respectively. Notably, the dwell times of these receptors were reduced upon binding to a specific transport substrate, suggesting that translocation is accelerated for loaded receptor molecules. Together with the known transport rates, our data suggest that nucleocytoplasmic transport occurs via multiple parallel pathways within single NPCs.     

Open channel block by gadolinium ion of the stretch-inactivated ion channel in mdx myotubes.

Currents flowing through single stretch-inactivated ion channels were recorded from cell-attached patches on myotubes from mdx mice. Adding micromolar concentrations of gadolinium to patch electrodes containing normal saline produced rapid transitions in the single-channel current between the fully open and closed states. The kinetics of the current fluctuations followed the predictions of a simple model of open channel block in which the transitions in the current arise from the entry and exit of Gd from the channel pore: histograms of the open and closed times were well fit with single exponentials, the blocking rate depended linearly on the concentration of gadolinium in the patch electrode, and the unblocking rate was independent of the concentration of gadolinium. Hyperpolarizing the patch increased the rate of unblocking (approximately e-fold per 85 mV), suggesting the charged blocking particle can exit the channel into the cell under the influence of the applied membrane field. The rate of blocking was rapid and was independent of the patch potential, consistent with the rate of ion entry into the pore being determined by its rate of diffusion in solution. When channel open probability was reduced by applying suction to the electrode, the blocking kinetics were independent of the extent of inactivation, suggesting that mechanosensitive gating does not modify the structure of the channel pore.     

Specificity of ion channel inhibitors for the maxi cation channel in rye root plasma membranes

The pharmacology of the maxi cation channel in the plasma membraneof rye (Secale cereale L.) root cells was studied followingits incorporation into planar lipid bilayers. The channel wasinhibited by ruthenium red, diltiazem, verapamil, and quinineat micromolar concentrations and TEA⁺ at millimolar concentrations. Key words: Calcium (Ca²⁺, cation channel, inhibitors, planar lipid bilayer, plasma membrane     

Computed energy profiles for ion transport in the gramicidin A channel

Energy profiles have been established for the transport of Na⁺, K⁺ and Cs⁺ through the gramicidin A channel. In Urry’s head-to-headβ _3.3 ^6.3 left-handed helical dimer structure, using a refined methodology for the calculation of intra and intermolecular interactions. The computations show the important role, for the energy profile and the position of the possible binding site, of the flexible ethanolamine chain, whose significance was till now overlooked. The calculations indicate that the barriers at the entrance and at the center of the channel should be in the order Na⁺ XXX K⁺ XXX Cs⁺ but predict also that the energies of the binding sites should be the greatest for Na⁺ and, then, comparable for K⁺ and Cs⁺. The indications concerning the barriers are confirmed by experiment.     

Identification of structures for ion channel kinetic models

Markov models of ion channel dynamics have evolved as experimental advances have improved our understanding of channel function. Past studies have examined limited sets of various topologies for Markov models of channel dynamics. We present a systematic method for identification of all possible Markov model topologies using experimental data for two types of native voltage-gated ion channel currents: mouse atrial sodium currents and human left ventricular fast transient outward potassium currents. Successful models identified with this approach have certain characteristics in common, suggesting that aspects of the model topology are determined by the experimental data. Incorporating these channel models into cell and tissue simulations to assess model performance within protocols that were not used for training provided validation and further narrowing of the number of acceptable models. The success of this approach suggests a channel model creation pipeline may be feasible where the structure of the model is not specified a priori.     

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.     

Interactions of tricyclic antidepressants with a synaptic ion channel

The interactions of four tricyclic antidepressants, including two dibenzazepines, imipramine and desipramine, and two dibenzocyclo-heptenes, amitriptyline and nortryptiline, were studied with the ion channel associated with the nicotinic acetylcholine receptor from the electric organs of $\text{Torpedo ocellata}$. These drugs inhibited the binding of tritiated perhydrohistrionicotoxin and phencyclidine to sites on the ion channel. All four compounds interacted with the ion channel with approximately equal affinities, inhibition constants for the two sites ranging from 1.1 to 3.4 μM in the absence and from 0.16 to 0.75 μM in the presence of 1 μM acetylcholine. The affinities of the secondary amines were increased by acetylcholine to a greater degree than were those of the tertiary amine compounds.     

Physical and molecular basis of ion channel gating: Can electrostatic interactions close the ion channel?

We analyzed voltage-dependent ion channel structure and conformational changes corresponding to channel gating. During the gating, S4 segments, as well as other parts of the channel, undergo a set of conformational modifications. These changes are accompanied by complicated movements of positive charges that are mostly located in the S4 segments. These charges electrostatically interact with the ions passing through the channel. The interaction energy depends on the conformational state of the channel, i.e., on the mutual positions of the permeant ions and these charges. Analyzing and making energetical estimations, we propose a hypothesis: the closed state of the ion channel corresponds to the S4 position when electrostatic interaction between positively charged groups of the S4 segments and permeant ions is strong enough to close the pathway for these ions.     

Properties of the stochastic energization-relaxation channel model for vectorial ion transport

A model for the primary active transport by an ion pump protein is proposed. The model, the "energization-relaxation channel model," describes an ion pump as a multiion channel that undergoes stochastic transitions between two conformational states by external energy supply. When the potential profile along ion transport pathway is asymmetrical, a net ion flux is induced by the transitions. In this model, the coupling of the conformational change and ion transport is stochastic and loose. The model qualitatively reproduces known properties of active transport such as the effect of ion concentration gradient and membrane potential on the rate of transport and the inhibition of ion transport at high ion concentration. We further examined the effect of various parameters on the ion transport properties of this model. The efficiency of the coupling was almost 100% under some conditions.     

Vacuolar ion channels in the liverwort Marchantia polymorpha: influence of ion channel inhibitors

Planta - Potassium-permeable slow activating vacuolar channels (SV) and chloride-permeable channels in the vacuole of the liverwort Marchantia polymorpha were characterized in respect to calcium...     

Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events.

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc. are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and the shut times, are too short to be detected reliably. If a fixed dead-time tau is assumed then it is possible to define, as an approximation to what is actually observed, an 'extended opening' or e-opening which starts with an opening of duration at least tau followed by any number of openings and shuttings, all the shut times being shorter than tau; the e-opening ends when a shut time longer than tau occurs. A similar definition is used for e-shut times. The probability densities, f(t), of these extended times have previously been obtained as expressions which become progressively more complicated, and numerically unstable to compute, as t-->infinity. In this paper we present, for the two-state model, an alternative representation as an infinite series of which a small number of terms gives a very accurate approximation of f (t) for large t. For the general model we present an asymptotic representation as a mixture of exponentials which is accurate for all except quite small values of t. Some simple model-independent corrections for missed events are discussed in relationship to the exact solutions.