Discrete-state theory in nerve impulse modelling.

Hodgkin-Huxley models have been the standard for describing ionic current kinetics. However, many single channel behaviors cannot be described using traditional Hodgkin-Huxley models; they can be described by expanding the Hodgkin-Huxley models to have multiple resting and inactivated states. The model, based on charge translocation between a finite number of discrete Markovian states, is a biophysical kinetic model, according to current generalizations of channel structure, capable of reproducing channel behavior. The elaboration of the model is based on the Markov process. This type of model assumes that each channel has a discrete number of states that are connected by a kinetic diagram that defines the allowable transitions between these states and the rates at which these transitions occur. The application of the model presented here leads to results in accordance with the experimental data regarding the shape and characteristics of the nerve impulse registered along the nerve fibre. Unlike the traditional Hodgkin-Huxley models, the model based on the Markov processes has the advantage of removing the empirical equations, simplifying the computation of the membrane potential and revealing the single-channel variables. The average behavior is obtained by the repetition in one channel of the same stimulus, a number of times equal to the number of channels, which means that the macroscopic variables are predictable by the repetition, a certain number of times, of the same observations in a single channel.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

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.     

Estimating kinetic constants from single channel data.

The process underlying the opening and closing of ionic channels in biological or artificial lipid membranes can be modeled kinetically as a time-homogeneous Markov chain. The elements of the chain are kinetic states that can be either open or closed. A maximum likelihood procedure is described for estimating the transition rates between these states from single channel data. The method has been implemented for linear kinetic schemes of fewer than six states, and is suitable for nonstationary data in which one or more independent channels are functioning simultaneously. It also provides standard errors for all estimates of rate constants and permits testing of smoothly parameterized subhypotheses of a general model. We have illustrated our approach by analysis of single channel data simulated on a computer and have described a procedure for analysis of experimental data.     

Gating kinetics of potassium channel in rat dorsal root ganglion neurons analyzed with fractal model

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and kinetic rate constants connecting these states are constant. To study the gating kinetics of voltage-dependent K(+) channel in rat dorsal root ganglion neurons, K(+) channel current were recorded using cell-attached patch-clamp technique. The K(+) channel characteristic of kinetics were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimension D for the closed times and for the open times depend on the pipette potential. For the open and closed times of kinetic setpoint, it was found dependent on the applied pipette potential, which indicated that the ion channel gating kinetics had nonlinear kinetic properties. Thus, the open and closed durations, which had the voltage dependence of the gating of this ion channel, were well described by the fractal model.     

Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and the kinetic rate constants connecting these states are constant. In the alternative fractal model the spontaneous fluctuations of the channel protein at many different time scales are represented by a kinetic rate constant k = At1-D, where A is the kinetic setpoint and D the fractal dimension. Single-channel currents were recorded at 146 mM external K+ from an inwardly rectifying, 120 pS, K+ selective, voltage-sensitive channel in cultured mouse hippocampal neurons. The kinetics of these channels were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimensions were approximately 2 for the closed times and approximately 1 for the open times and did not depend on voltage. For both the open and closed times the logarithm of the kinetic setpoint was found to be proportional to the applied voltage, which indicates that the gating of this channel involves the net inward movement of approximately one negative charge when this channel opens. Thus, the open and closed times and the voltage dependence of the gating of this channel are well described by the fractal model.     

Characterization of the voltage-dependent properties of a volume- sensitive anion conductance

Outwardly rectified, swelling-activated anion conductances have been described in numerous cell types. The major functional variable observed amongst these conductances is the extent and rate of depolarization-induced inactivation. In general, the conductances can be divided into two broad classes, those that show rapid inactivation in response to strong depolarization and those that show little or no voltage dependence. The swelling-activated anion conductance in rat C6 glioma cells is inactivated nearly completely by membrane depolarization above +90 mV and reactivated by membrane hyperpolarization. The kinetics of inactivation and reactivation are fit by single and double exponentials, respectively. Voltage-dependent behavior is well described by a simple linear kinetic model in which the channel exists in an open or one of three inactivated states. pH- induced changes in voltage-dependent gating suggest that the voltage sensor contains critical basic amino acid residues. Extracellular ATP blocks the channel in a voltage-dependent manner. The block is sensitive to the direction of net Cl- movement and increases open channel noise indicating that ATP interacts with the channel pore. Blockage of the channel with ATP dramatically slows depolarization- induced inactivation.     

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states

BACKGROUND: The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. MAIN CONTRIBUTIONS: We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.     

Fractal and Markov behavior in ion channel kinetics

Kinetic analysis of ion channel recordings attempts to distinguish the number and lifetimes of channel molecular states. Most kinetic analysis assumes that the lifetime of each state is independent of previous channel history, so that open and closed durations are Markov processes whose probability densities are sums of exponential decays. An alternative approach assumes that channel molecules have many configurtions with widely varying lifetimes. Rates of opening and closing then vary with the time scale of observation, leading to fractal kinetics. We have examined kinetic behavior in two types of channels from human and avian fibroblasts, using a maximum likehood method to test the dependence of rates on observational time scale. For both channels, openings showed mixed fractal and Markov behavior, while closings gave mainly fractal kinetics.     

一个改进的马尔科夫模型

离子通道是细胞膜里的大分子孔道,是跨越细胞膜里的蛋白质大分子,是神经、肌肉等细胞膜兴奋性的基础。人体细胞均具有完成特殊功能的离子通道,构建离子通道,尤其其门控行为的动力学模型,对于研究离子通道的相关课题具有重要意义。离子通道的开关反映了蛋白质构象变化的动力学过程。本文介绍了细胞膜离子通道的基本内容和几种常用模型,并根据Markov链对离子通道门控行为的一个二态（闭、开）模型的Markov过程进行了改进,得到了包含失活状态的离子通道门控行为的Markov模型。     

Opening mechanism of a cyclic nucleotide-gated channel based on analysis of single channels locked in each liganded state.

Cyclic nucleotide-gated channels contain four subunits, each with a binding site for cGMP or cAMP in the cytoplasmic COOH-terminal domain. Previous studies of the kinetic mechanism of activation have been hampered by the complication that ligands are continuously binding and unbinding at each of these sites. Thus, even at the single channel level, it has been difficult to distinguish changes in behavior that arise from a channel with a fixed number of ligands bound from those that occur upon the binding and unbinding of ligands. For example, it is often assumed that complex behaviors like multiple conductance levels and bursting occur only as a consequence of changes in the number of bound ligands. We have overcome these ambiguities by covalently tethering one ligand at a time to single rod cyclic nucleotide-gated channels (Ruiz, ML., and J.W. Karpen. 1997. Nature. 389:389-392). We find that with a fixed number of ligands locked in place the channel freely moves between three conductance states and undergoes bursting behavior. Furthermore, a thorough kinetic analysis of channels locked in doubly, triply, and fully liganded states reveals more than one kinetically distinguishable state at each conductance level. Thus, even when the channel contains a fixed number of bound ligands, it can assume at least nine distinct states. Such complex behavior is inconsistent with simple concerted or sequential allosteric models. The data at each level of liganding can be successfully described by the same connected state model (with different rate constants), suggesting that the channel undergoes the same set of conformational changes regardless of the number of bound ligands. A general allosteric model, which postulates one conformational change per subunit in both the absence and presence of ligand, comes close to providing enough kinetically distinct states. We propose an extension of this model, in which more than one conformational change per subunit can occur during the process of channel activation.     

The kinetic and physical basis of K(ATP) channel gating: toward a unified molecular understanding

K(ATP) channels can be formed from Kir6.2 subunits with or without SUR1. The open-state stability of K(ATP) channels can be increased or reduced by mutations throughout the Kir6.2 subunit, and is increased by application of PIP(2) to the cytoplasmic membrane. Increase of open-state stability is manifested as an increase in the channel open probability in the absence of ATP (Po(zero)) and a correlated decrease in sensitivity to inhibition by ATP. Single channel lifetime analyses were performed on wild-type and I154C mutant channels expressed with, and without, SUR1. Channel kinetics include a single, invariant, open duration; an invariant, brief, closed duration; and longer closed events consisting of a "mixture of exponentials," which are prolonged in ATP and shortened after PIP(2) treatment. The steady-state and kinetic data cannot be accounted for by assuming that ATP binds to the channel and causes a gate to close. Rather, we show that they can be explained by models that assume the following regarding the gating behavior: 1) the channel undergoes ATP-insensitive transitions from the open state to a short closed state (C(f)) and to a longer-lived closed state (C(0)); 2) the C(0) state is destabilized in the presence of SUR1; and 3) ATP can access this C(0) state, stabilizing it and thereby inhibiting macroscopic currents. The effect of PIP(2) and mutations that stabilize the open state is then to shift the equilibrium of the "critical transition" from the open state to the ATP-accessible C(0) state toward the O state, reducing accessibility of the C(0) state, and hence reducing ATP sensitivity.     

Enhanced Closed-state Inactivation in a Mutant Shaker K+ Channel

Many mutations that shift the voltage dependence of activation in Shaker channels cause a parallel shift of inactivation. The I2 mutation (L382I in the Shaker B sequence) is an exception, causing a 45 mV activation shift with only a 9 mV shift of inactivation midpoint relative to the wildtype (WT) channel. We compare the behavior of WT and I2 Shaker 29-4 channels in macropatch recordings from Xenopus oocytes. The behavior of WT channels can be described by both simple and detailed kinetic models which assume that inactivation proceeds only from the open state. The behavior of I2 channels requires that they inactivate from closed states as well, a property characteristic of voltage-gated sodium channels. A detailed ``multiple-state inactivation' model is presented that describes both activation and inactivation of I2 channels. The results are consistent with the view that residue L382 is associated with the receptor for the inactivation particles in Shaker channels. Received: 16 December 1996/Revised: 5 February 1997     

Time course of TEA(+)-induced anomalous rectification in squid giant axons

Changes in the voltage clamp currents of squid giant axons wrought by low axoplasmic TEA+ (tetraethylammonium chloride) concentrations (0.3 mM and above) are described. They are: (a) For positive steps from the resting potential in sea water, the K+ current increases, decreases, then increases, instead of increasing monotonically. (b) For positive steps from the resting potential in 440 mM external K+, the current has an exponentially decaying component, whose decay rate increases with axoplasmic [TEA+]. The control currents increase monotonically. (c) For negative steps from the resting potential in 440 mM external K+, the current record has a peak followed by a decay that is slow relative to the control. The control record decreases monotonically. Qualitatively these findings can be described by a simple kinetic model, from which, with one assumption, it is possible to calculate the rate at which K+ ions move through the K+ channels. An interesting conclusion from (c) is that the channels cannot be closed by the normal voltage-sensitive mechanism (described by Hodgkin and Huxley) until they are free of TEA+.     

Resolvability of fluorescence lifetime distributions using phase fluorometry.

The analysis of the fluorescence decay using discrete exponential components assumes that a small number of species is present. In the absence of a definite kinetic model or when a large number of species is present, the exponential analysis underestimates the uncertainty of the recovered lifetime values. A different approach to determine the lifetime of a population of molecules is the use of probability density functions and lifetime distributions. Fluorescence decay data from continuous distributions of exponentially decaying components were generated. Different magnitudes of error were added to the data to simulate experimental conditions. The resolvability of the distributional model was studied by fitting the simulated data to one and two exponentials. The maximum width of symmetric distributions (uniform, gaussian, and lorentzian), which cannot be distinguished from single and double exponential fits for statistical errors of 1 and 0.1%, were determined. The width limits are determined by the statistical error of the data. It is also shown that, in the frequency domain, the discrete exponential analysis does not uniformly weights all the components of a distribution. This systematic error is less important when probability and distribution functions are used to recover the decay. Finally, it is shown that real lifetime distributions can be proved using multimodal probability density functions. In the companion paper that follows we propose a physical approach, which provides lifetime distribution functions for the tryptophan decay in proteins. In the third companion paper (Alcala, J.R., E. Gratton, and F.J. Prendergast, 1987, Biophys. J., in press) we use the distribution functions obtained to fit data from the fluorescence decay of single tryptophan proteins.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Model of calcium channel inactivation: a qualitative analysis

A simple model of calcium channel inactivation has been developed, based on the accumulation of calcium ions at the inner mouth of the channel and on their binding to a receptor which inactivates the channel. A qualitative analysis has shown that upon an appropriate choice of parameters corresponding to the cell structure and to kinetic properties of its components, the calcium dependent inactivation and that assumed to be voltage dependent can both be emulated. The model suggests that the supposed variety of calcium channels might be explained by quantitative differences in nonlinear interactions of the channels with other cell components.     

Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating.

We describe a new electrophysiological technique called nonequilibrium response spectroscopy, which involves application of rapidly fluctuating (as high as 14 kHz) large-amplitude voltage clamp waveforms to ion channels. As a consequence of the irreversible (in the sense of Carnot) exchange of energy between the fluctuating field and the channel protein, the gating response is exquisitely sensitive to features of the kinetics that are difficult or impossible to adequately resolve by means of traditional stepped potential protocols. Here we focus on the application of dichotomous (telegraph) noise voltage fluctuations, a broadband Markovian colored noise that fluctuates between two values. Because Markov kinetic models of channel gating can be embedded within higher-dimensional Markov models that take into account the effects of the voltage fluctuations, many features of the response of the channels can be calculated algebraically. This makes dichotomous noise and its generalizations uniquely suitable for model selection and kinetic analysis. Although we describe its application to macroscopic ionic current measurements, the nonequilibrium response method can also be applied to gating and single channel current recording techniques. We show how data from the human cardiac isoform (hH1a) of the Na+ channel expressed in mammalian cells can be acquired and analyzed, and how these data reveal hidden aspects of the molecular kinetics that are not revealed by conventional methods.     

Thermodynamically irreversible gating of ryanodine receptors in situ revealed by stereotyped duration of release in Ca(2+) sparks

For a single or a group of Markov channels gating reversibly, distributions of open and closed times should be the sum of positively weighted decaying exponentials. Violation of this microscopic reversibility has been demonstrated previously on a number of occasions at the single channel level, and has been attributed to possible channel coupling to external sources of free energy. Here we show that distribution of durations of Ca(2+) release underlying Ca(2+) sparks in intact cardiac myocytes exhibits a prominent mode at approximately 8 ms. Analysis of the cycle time for repetitive sparks at hyperactive sites revealed no intervals briefer than approximately 35 ms and a mode at approximately 90 ms. These results indicate that, regardless of whether Ca(2+) sparks are single-channel or multi-channel in origin, they are generated by thermodynamically irreversible stochastic processes. In contrast, data from planar lipid bilayer experiments were consistent with reversible gating of RyR under asymmetric cis (4 microM) and trans Ca(2+) (10 mM), suggesting that the irreversibility for Ca(2+) spark genesis may reside at a supramolecular level. Modeling suggests that Ca(2+)-induced Ca(2+) release among adjacent RyRs may couple the external energy derived from Ca(2+) gradients across the SR to RyR gating in situ, and drive the irreversible generation of Ca(2+) sparks.