Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating

Human sodium current voltage-dependence at physiological temperature measured by coupling a patch-clamp experiment to a mathematical model

Voltage-gated Na+ channels are crucial to action potential propagation in excitable tissues. Because of the high amplitude and rapid activation of the Na+ current, voltage-clamp measurements are very challenging and are usually performed at room temperature. In this study, we measured Na+ current voltage-dependence in stem cell-derived cardiomyocytes at physiological temperature. While the apparent activation and inactivation curves, measured as the dependence of current amplitude on voltage, fall within the range reported in previous studies, we identified a systematic error in our measurements. This error is caused by the deviation of the membrane potential from the command potential of the amplifier. We demonstrate that it is possible to account for this artifact using computer simulation of the patch-clamp experiment. We obtained surprising results through patch-clamp model optimization: a half-activation of -11.5 mV and a half-inactivation of -87 mV. Although the half-activation deviates from previous research, we demonstrate that this estimate reproduces the conduction velocity dependence on extracellular potassium concentration. KEY POINTS: Voltage-gated Na+ currents play a crucial role in excitable tissues including neurons, cardiac and skeletal muscle. Measurement of Na+ current is challenging because of its high amplitude and rapid kinetics, especially at physiological temperature. We have used the patch-clamp technique to measure human Na+ current voltage-dependence in human induced pluripotent stem cell-derived cardiomyocytes. The patch-clamp data were processed by optimization of the model accounting for voltage-clamp experiment artifacts, revealing a large difference between apparent parameters of Na+ current and the results of the optimization. We conclude that actual Na+ current activation is extremely depolarized in comparison to previous studies. The new Na+ current model provides a better understanding of action potential propagation; we demonstrate that it explains propagation in hyperkalaemic conditions.

Calibration of ionic and cellular cardiac electrophysiology models

Cardiac electrophysiology models are among the most mature and well-studied mathematical models of biological systems. This maturity is bringing new challenges as models are being used increasingly to make quantitative rather than qualitative predictions. As such, calibrating the parameters within ion current and action potential (AP) models to experimental data sets is a crucial step in constructing a predictive model. This review highlights some of the fundamental concepts in cardiac model calibration and is intended to be readily understood by computational and mathematical modelers working in other fields of biology. We discuss the classic and latest approaches to calibration in the electrophysiology field, at both the ion channel and cellular AP scales. We end with a discussion of the many challenges that work to date has raised and the need for reproducible descriptions of the calibration process to enable models to be recalibrated to new data sets and built upon for new studies. This article is categorized under: Analytical and Computational Methods > Computational Methods Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Four Ways to Fit an Ion Channel Model

Mathematical models of ionic currents are used to study the electrophysiology of the heart, brain, gut, and several other organs. Increasingly, these models are being used predictively in the clinic, for example, to predict the risks and results of genetic mutations, pharmacological treatments, or surgical procedures. These safety-critical applications depend on accurate characterization of the underlying ionic currents. Four different methods can be found in the literature to fit voltage-sensitive ion channel models to whole-cell current measurements: method 1, fitting model equations directly to time-constant, steady-state, and I-V summary curves; method 2, fitting by comparing simulated versions of these summary curves to their experimental counterparts; method 3, fitting to the current traces themselves from a range of protocols; and method 4, fitting to a single current trace from a short and rapidly fluctuating voltage-clamp protocol. We compare these methods using a set of experiments in which hERG1a current was measured in nine Chinese hamster ovary cells. In each cell, the same sequence of fitting protocols was applied, as well as an independent validation protocol. We show that methods 3 and 4 provide the best predictions on the independent validation set and that short, rapidly fluctuating protocols like that used in method 4 can replace much longer conventional protocols without loss of predictive ability. Although data for method 2 are most readily available from the literature, we find it performs poorly compared to methods 3 and 4 both in accuracy of predictions and computational efficiency. Our results demonstrate how novel experimental and computational approaches can improve the quality of model predictions in safety-critical applications.

Nonequilibrium response of a voltage gated sodium ion channel and biophysical characterization of dynamic hysteresis

Here we have studied the dynamic as well as the non-equilibrium thermodynamic response properties of voltage-gated Na-ion channel. Using sinusoidally oscillating external voltage protocol we have both kinetically and energetically studied the non-equilibrium steady state properties of dynamic hysteresis in details. We have introduced a method of estimating the work done associated with the dynamic memory due to a cycle of oscillating voltage. We have quantitatively characterised the loop area of ionic current which gives information about the work done to sustain the dynamic memory only for ion conduction, while the loop area of total entropy production rate gives the estimate of work done for overall gating dynamics. The maximum dynamic memory of Na-channel not only depends on the frequency and amplitude but it also depends sensitively on the mean of the oscillating voltage and here we have shown how the system optimize the dynamic memory itself in the biophysical range of field parameters. The relation between the average ionic current with increasing frequency corresponds to the nature of the average dissipative work done at steady state. It is also important to understand that the utilization of the energy from the external field can not be directly obtained only from the measurement of ionic current but also requires nonequilibrium thermodynamic study.

Probing kinetic drug binding mechanism in voltage-gated sodium ion channel: open state versus inactive state blockers

The kinetics and nonequilibrium thermodynamics of open state and inactive state drug binding mechanisms have been studied here using different voltage protocols in sodium ion channel. We have found that for constant voltage protocol, open state block is more efficient in blocking ionic current than inactive state block. Kinetic effect comes through peak current for mexiletine as an open state blocker and in the tail part for lidocaine as an inactive state blocker. Although the inactivation of sodium channel is a free energy driven process, however, the two different kinds of drug affect the inactivation process in a different way as seen from thermodynamic analysis. In presence of open state drug block, the process initially for a long time remains entropy driven and then becomes free energy driven. However in presence of inactive state block, the process remains entirely entropy driven until the equilibrium is attained. For oscillating voltage protocol, the inactive state blocking is more efficient in damping the oscillation of ionic current. From the pulse train analysis it is found that inactive state blocking is less effective in restoring normal repolarisation and blocks peak ionic current. Pulse train protocol also shows that all the inactive states behave differently as one inactive state responds instantly to the test pulse in an opposite manner from the other two states.

Dynamical characterization of inactivation path in voltage-gated Na(+) ion channel by non-equilibrium response spectroscopy

Inactivation path of voltage gated sodium channel has been studied here under various voltage protocols as it is the main governing factor for the periodic occurrence and shape of the action potential. These voltage protocols actually serve as non-equilibrium response spectroscopic tools to study the ion channel in non-equilibrium environment. In contrast to a lot of effort in finding the crystal structure based molecular mechanism of closed-state(CSI) and open-state inactivation(OSI); here our approach is to understand the dynamical characterization of inactivation. The kinetic flux as well as energetic contribution of the closed and open- state inactivation path is compared here for voltage protocols, namely constant, pulsed and oscillating. The non-equilibrium thermodynamic quantities used in response to these voltage protocols serve as improved characterization tools for theoretical understanding which not only agrees with the previously known kinetic measurements but also predict the energetically optimum processes to sustain the auto-regulatory mechanism of action potential and the consequent inactivation steps needed. The time dependent voltage pattern governs the population of the conformational states which when couple with characteristic rate parameters, the CSI and OSI selectivity arise dynamically to control the inactivation path. Using constant, pulsed and continuous oscillating voltage protocols we have shown that during depolarization the OSI path is more favored path of inactivation however, in the hyper-polarized situation the CSI is favored. It is also shown that the re-factorisation of inactivated sodium channel to resting state occurs via CSI path. Here we have shown how the subtle energetic and entropic cost due to the change in the depolarization magnitude determines the optimum path of inactivation. It is shown that an efficient CSI and OSI dynamical profile in principle can characterize the open-state drug blocking phenomena.

Analyzing single-molecule time series via nonparametric Bayesian inference

The ability to measure the properties of proteins at the single-molecule level offers an unparalleled glimpse into biological systems at the molecular scale. The interpretation of single-molecule time series has often been rooted in statistical mechanics and the theory of Markov processes. While existing analysis methods have been useful, they are not without significant limitations including problems of model selection and parameter nonidentifiability. To address these challenges, we introduce the use of nonparametric Bayesian inference for the analysis of single-molecule time series. These methods provide a flexible way to extract structure from data instead of assuming models beforehand. We demonstrate these methods with applications to several diverse settings in single-molecule biophysics. This approach provides a well-constrained and rigorously grounded method for determining the number of biophysical states underlying single-molecule data.

Dynamic memory of a single voltage-gated potassium ion channel: A stochastic nonequilibrium thermodynamic analysis

In this work, we have studied the stochastic response of a single voltage-gated potassium ion channel to a periodic external voltage that keeps the system out-of-equilibrium. The system exhibits memory, resulting from time-dependent driving, that is reflected in terms of dynamic hysteresis in the current-voltage characteristics. The hysteresis loop area has a maximum at some intermediate voltage frequency and disappears in the limits of low and high frequencies. However, the (average) dissipation at long-time limit increases and finally goes to saturation with rising frequency. This raises the question: how diminishing hysteresis can be associated with growing dissipation? To answer this, we have studied the nonequilibrium thermodynamics of the system and analyzed different thermodynamic functions which also exhibit hysteresis. Interestingly, by applying a temporal symmetry analysis in the high-frequency limit, we have analytically shown that hysteresis in some of the periodic responses of the system does not vanish. On the contrary, the rates of free energy and internal energy change of the system as well as the rate of dissipative work done on the system show growing hysteresis with frequency. Hence, although the current-voltage hysteresis disappears in the high-frequency limit, the memory of the ion channel is manifested through its specific nonequilibrium thermodynamic responses.

Computational tools to investigate genetic cardiac channelopathies

The aim of this perspective article is to share with the community of ion channel scientists our thoughts and expectations regarding the increasing role that computational tools will play in the future of our field. The opinions and comments detailed here are the result of a 3-day long international exploratory workshop that took place in October 2013 and that was supported by the Swiss National Science Foundation.

On the estimation of cooperativity in ion channel kinetics: activation free energy and kinetic mechanism of Shaker K+ channel

In this paper, we have explored generic criteria of cooperative behavior in ion channel kinetics treating it on the same footing with multistate receptor-ligand binding in a compact theoretical framework. We have shown that the characterization of cooperativity of ion channels in terms of the Hill coefficient violates the standard Hill criteria defined for allosteric cooperativity of ligand binding. To resolve the issue, an alternative measure of cooperativity is proposed here in terms of the cooperativity index that sets a unified criteria for both the systems. More importantly, for ion channel this index can be very useful to describe the cooperative kinetics as it can be readily determined from the experimentally measured ionic current combined with theoretical modelling. We have analyzed the correlation between the voltage value and slope of the voltage-activation curve at the half-activation point and consequently determined the standard free energy of activation of the ion channel using two well-established mechanisms of cooperativity, namely, Koshland-Nemethy-Filmer (KNF) and Monod-Wyman-Changeux (MWC) models. Comparison of the theoretical results for both the models with appropriate experimental data of mutational perturbation of Shaker K(+) channel supports the experimental fact that the KNF model is more suitable to describe the cooperative behavior of this class of ion channels, whereas the performance of the MWC model is unsatisfactory. We have also estimated the mechanistic performance through standard free energy of channel activation for both the models and proposed a possible functional disadvantage in the MWC scheme.

Wavelet-based protocols for ion channel electrophysiology

BACKGROUND

Fluctuation-induced phenomena caused by both random and deterministic stimuli have been previously studied in a variety of contexts. They are based on the interplay between the spectro-temporal patterns of the signal and the kinetics of the system it is applied to. The aim of this study was to develop a method for designing fluctuating inputs into nonlinear system which would elicit the most desired system output and to implement the method to studies of ion channels.

RESULTS

We describe an algorithm based on constructing the input as a superposition of wavelets and optimizing it according to a selected cost functional. The algorithm is applied to ion channel electrophysiology where the input is the fluctuating voltage delivered through a patch-clamp experimental apparatus and the output is the whole-cell ionic current. The algorithm is optimized to aid selection of Markov models of the gating kinetics of the voltage-gated Shaker K+ channel and tested by comparison of numerically obtained ionic currents predicted by different models with experimental data obtained from the Shaker K+ channels. Other applications and optimization criteria are also suggested.

CONCLUSION

The method described in this paper can be useful in development and testing of models of ion channel gating kinetics, developing voltage inputs that optimize certain nonequilibrium phenomena in ion channels, such as the kinetic focusing, and potentially has applications to other fields.

Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel

Here we have studied the nonequilibrium thermodynamic response of a voltage-gated Shaker potassium ion channel using a stochastic master equation. For a constant external voltage, the system reaches equilibrium indicated by the vanishing total entropy production rate, whereas for oscillating voltage the current and entropy production rates show dynamic hysteretic behavior. Here we have shown quantitatively that although the hysteresis loop area vanishes in low and high frequency domains of the external voltage, they are thermodynamically distinguishable. In the very low frequency domain, the system remains close to equilibrium, whereas at high frequencies it goes to a nonequilibrium steady state (NESS) associated with a finite value of dissipation function. At NESS, the efficiency of the ion conduction can also be related with the nonlinear dependence of the dissipation function on the power of the external field. Another intriguing aspect is that, at the high frequency limit, the total entropy production rate oscillates at NESS with half of the time period of the external voltage.

A novel frequency analysis method for assessing K(ir)2.1 and Na (v)1.5 currents

Voltage clamping is an important tool for measuring individual currents from an electrically active cell. However, it is difficult to isolate individual currents without pharmacological or voltage inhibition. Herein, we present a technique that involves inserting a noise function into a standard voltage step protocol, which allows one to characterize the unique frequency response of an ion channel at different step potentials. Specifically, we compute the fast Fourier transform for a family of current traces at different step potentials for the inward rectifying potassium channel, K_ir2.1, and the channel encoding the cardiac fast sodium current, Na_v1.5. Each individual frequency magnitude, as a function of voltage step, is correlated to the peak current produced by each channel. The correlation coefficient vs. frequency relationship reveals that these two channels are associated with some unique frequencies with high absolute correlation. The individual IV relationship can then be recreated using only the unique frequencies with magnitudes of high absolute correlation. Thus, this study demonstrates that ion channels may exhibit unique frequency responses.

Recapitulation of an ion channel IV curve using frequency components

INTRODUCTION

Presently, there are no established methods to measure multiple ion channel types simultaneously and decompose the measured current into portions attributable to each channel type. This study demonstrates how impedance spectroscopy may be used to identify specific frequencies that highly correlate with the steady state current amplitude measured during voltage clamp experiments. The method involves inserting a noise function containing specific frequencies into the voltage step protocol. In the work presented, a model cell is used to demonstrate that no high correlations are introduced by the voltage clamp circuitry, and also that the noise function itself does not introduce any high correlations when no ion channels are present. This validation is necessary before the technique can be applied to preparations containing ion channels. The purpose of the protocol presented is to demonstrate how to characterize the frequency response of a single ion channel type to a noise function. Once specific frequencies have been identified in an individual channel type, they can be used to reproduce the steady state current voltage (IV) curve. Frequencies that highly correlate with one channel type and minimally correlate with other channel types may then be used to estimate the current contribution of multiple channel types measured simultaneously.

METHODS

Voltage clamp measurements were performed on a model cell using a standard voltage step protocol (-150 to +50 mV, 5mV steps). Noise functions containing equal magnitudes of 1-15 kHz frequencies (zero to peak amplitudes: 50 or 100mV) were inserted into each voltage step. The real component of the Fast Fourier transform (FFT) of the output signal was calculated with and without noise for each step potential. The magnitude of each frequency as a function of voltage step was correlated with the current amplitude at the corresponding voltages.

RESULTS AND CONCLUSIONS

In the absence of noise (control), magnitudes of all frequencies except the DC component correlated poorly (|R|<0.5) with the IV curve, whereas the DC component had a correlation coefficient greater than 0.999 in all measurements. The quality of correlation between individual frequencies and the IV curve did not change when a noise function was added to the voltage step protocol. Likewise, increasing the amplitude of the noise function also did not increase the correlation. Control measurements demonstrate that the voltage clamp circuitry by itself does not cause any frequencies above 0 Hz to highly correlate with the steady-state IV curve. Likewise, measurements in the presence of the noise function demonstrate that the noise function does not cause any frequencies above 0 Hz to correlate with the steady-state IV curve when no ion channels are present. Based on this verification, the method can now be applied to preparations containing a single ion channel type with the intent of identifying frequencies whose amplitudes correlate specifically with that channel type.

A single-cell model of phase-driven control of ventricular fibrillation frequency

The mechanisms controlling the rotation frequency of functional reentry in ventricular fibrillation (VF) are poorly understood. It has been previously shown that Ba2+ at concentrations up to 50 mumol/L slows the rotation frequency in the intact guinea pig (GP) heart, suggesting a role of the inward rectifier current (I(K1)) in the mechanism governing the VF response to Ba2+. Given that other biological (e.g., sinoatrial node) and artificial systems display phase-locking behavior, we hypothesized that the mechanism for controlling the rotation frequency of a rotor by I(K1) blockade is phase-driven, i.e., the phase shift between transmembrane current and voltage remains constant at varying levels of I(K1) blockade. We measured whole-cell admittance in isolated GP myocytes and in transfected human embryonic kidney (HEK) cells stably expressing Kir 2.1 and 2.3 channels. The admittance phase, i.e., the phase difference between current and voltage, was plotted versus the frequency in control conditions and at 10 or 50 micromol/L Ba2+ (in GP heart cells) or 1 mM Ba2+ (in HEK cells). The horizontal distance between plots was called the "frequency shift in a single cell" and analyzed. The frequency shift in a single cell was -14.14 +/- 5.71 Hz (n = 14) at 10 microM Ba2+ and -18.51 +/- 4.00 Hz (n = 10) at 50 microM Ba2+, p < 0.05. The values perfectly matched the Ba2+-induced reduction of VF frequency observed previously in GP heart. A similar relationship was found in the computer simulations. The phase of Ba2+-sensitive admittance in GP cells was -2.65 +/- 0.32 rad at 10 Hz and -2.79 +/- 0.26 rad at 30 Hz. In HEK cells, the phase of Ba2+-sensitive admittance was 3.09 +/- 0.03 rad at 10 Hz and 3.00 +/- 0.17 rad at 30 Hz. We have developed a biological single-cell model of rotation-frequency control. The results show that although rotation frequency changes as a result of I(K1) blockade, the phase difference between transmembrane current and transmembrane voltage remains constant, enabling us to quantitatively predict the change of VF frequency resulting from I(K1) blockade, based on single-cell measurement.

The ion channel inverse problem: neuroinformatics meets biophysics

Ion channels are the building blocks of the information processing capability of neurons: any realistic computational model of a neuron must include reliable and effective ion channel components. Sophisticated statistical and computational tools have been developed to study the ion channel structure-function relationship, but this work is rarely incorporated into the models used for single neurons or small networks. The disjunction is partly a matter of convention. Structure-function studies typically use a single Markov model for the whole channel whereas until recently whole-cell modeling software has focused on serial, independent, two-state subunits that can be represented by the Hodgkin-Huxley equations. More fundamentally, there is a difference in purpose that prevents models being easily reused. Biophysical models are typically developed to study one particular aspect of channel gating in detail, whereas neural modelers require broad coverage of the entire range of channel behavior that is often best achieved with approximate representations that omit structural features that cannot be adequately constrained. To bridge the gap so that more recent channel data can be used in neural models requires new computational infrastructure for bringing together diverse sources of data to arrive at best-fit models for whole-cell modeling. We review the current state of channel modeling and explore the developments needed for its conclusions to be integrated into whole-cell modeling.

Sodium Channel Inactivation: Molecular Determinants and Modulation

Voltage-gated sodium channels open (activate) when the membrane is depolarized and close on repolarization (deactivate) but also on continuing depolarization by a process termed inactivation, which leaves the channel refractory, i.e., unable to open again for a period of time. In the “classical” fast inactivation, this time is of the millisecond range, but it can last much longer (up to seconds) in a different slow type of inactivation. These two types of inactivation have different mechanisms located in different parts of the channel molecule: the fast inactivation at the cytoplasmic pore opening which can be closed by a hinged lid, the slow inactivation in other parts involving conformational changes of the pore. Fast inactivation is highly vulnerable and affected by many chemical agents, toxins, and proteolytic enzymes but also by the presence of β-subunits of the channel molecule. Systematic studies of these modulating factors and of the effects of point mutations (experimental and in hereditary diseases) in the channel molecule have yielded a fairly consistent picture of the molecular background of fast inactivation, which for the slow inactivation is still lacking.

Maximum likelihood estimation of ion channel kinetics from macroscopic currents

We describe a maximum likelihood method for direct estimation of rate constants from macroscopic ion channel data for kinetic models of arbitrary size and topology. The number of channels in the preparation, and the mean and standard deviation of the unitary current can be estimated, and a priori constraints can be imposed on rate constants. The method allows for arbitrary stimulation protocols, including stimuli with finite rise time, trains of ligand or voltage steps, and global fitting across different experimental conditions. The initial state occupancies can be optimized from the fit kinetics. Utilizing arbitrary stimulation protocols and using the mean and the variance of the current reduce or eliminate problems of model identifiability (Kienker, 1989). The algorithm is faster than a recent method that uses the full autocovariance matrix (Celentano and Hawkes, 2004), in part due to the analytical calculation of the likelihood gradients. We tested the method with simulated data and with real macroscopic currents from acetylcholine receptors, elicited in response to brief pulses of carbachol. Given appropriate stimulation protocols, our method chose a reasonable model size and topology.

Optimal-sensitivity analysis of ion channel gating kinetics

We propose a new approach to analysis of kinetic models for ion channel gating, based on application of fluctuating voltages through a voltage clamp, in addition to conventional techniques. We show that the channel kinetics can be probed in a much more sensitive way, leading to more efficient model selection and more reliable estimates of model parameters. We use wavelet transform as an analytic tool for fluctuating currents and parametric dispersion plots as a measure of model compatibility with experimental data.

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.

Escape from a fluctuating system: a master equation and trapping approach

We present a general solution for the mean exit time in a system with on-site fluctuations between two configurations described by a master equation. The coupled configurations represent a spatially discretized version of an escape over a fluctuating barrier [C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)], and passage through modulating channels. Based on the general properties of the mean exit time, we obtain a simple solution for a coupled "birth" and "death" case that exhibits resonant activation. Within this exactly solvable model we derive analytically the optimal fluctuating rate, which is sensitive to the initial condition and scales as 1/n, where n is the system size. Our approach unifies a number of escape problems and points towards the generality of resonant activation.