Ion Channel Models Research Articles

Consideration of Kv ion channels as firstorder systems

Abstract This paper discusses possibilities to explain which Kv channels can be considered as first-order systems. This means that the dynamic behavior of these channels can be described with a first-order differential equation and are the most straightforward dynamical systems. For example, a typical first-order system is an RC circuit, and analogously, Kv channels can be considered with the dynamic behavior of first-order systems following RC circuits. This work represents a family of KV channels that can be described as first-order systems, providing an efficient method of ion channel modeling in computational electrophysiology.

Geometric slow–fast analysis of a hybrid pituitary cell model with stochastic ion channel dynamics

To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow–fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems and increasingly used for continuous, non-smooth dynamical systems. However, they are much less used for random dynamical systems, in particular for hybrid models with discrete, random dynamics. Here we propose a geometrical method that works directly with the hybrid system. We illustrate our approach through an application to a hybrid pituitary cell model in which the stochastic dynamics of very few active large-conductance potassium (BK) channels is coupled to a deterministic model of the other ion channels and calcium dynamics. To employ our geometric approach, we exploit the slow–fast structure of the model. The random fast subsystem is analyzed by considering discrete phase planes, corresponding to the discrete number of open BK channels, and stochastic events correspond to jumps between these planes. The evolution within each plane can be understood from nullclines and limit cycles, and the overall dynamics, e.g., whether the model produces a spike or a burst, is determined by the location at which the system jumps from one plane to another. Our approach is generally applicable to other scenarios to study discrete random dynamical systems defined by hybrid stochastic–deterministic models.

Empirical Quantification of Predictive Uncertainty Due to Model Discrepancy by Training with an Ensemble of Experimental Designs: An Application to Ion Channel Kinetics

When using mathematical models to make quantitative predictions for clinical or industrial use, it is important that predictions come with a reliable estimate of their accuracy (uncertainty quantification). Because models of complex biological systems are always large simplifications, model discrepancy arises—models fail to perfectly recapitulate the true data generating process. This presents a particular challenge for making accurate predictions, and especially for accurately quantifying uncertainty in these predictions. Experimentalists and modellers must choose which experimental procedures (protocols) are used to produce data used to train models. We propose to characterise uncertainty owing to model discrepancy with an ensemble of parameter sets, each of which results from training to data from a different protocol. The variability in predictions from this ensemble provides an empirical estimate of predictive uncertainty owing to model discrepancy, even for unseen protocols. We use the example of electrophysiology experiments that investigate the properties of hERG potassium channels. Here, ‘information-rich’ protocols allow mathematical models to be trained using numerous short experiments performed on the same cell. In this case, we simulate data with one model and fit it with a different (discrepant) one. For any individual experimental protocol, parameter estimates vary little under repeated samples from the assumed additive independent Gaussian noise model. Yet parameter sets arising from the same model applied to different experiments conflict—highlighting model discrepancy. Our methods will help select more suitable ion channel models for future studies, and will be widely applicable to a range of biological modelling problems.

Nonextensive realizations in interacting ion channels: Implications for mechano-electrical transducer mechanisms

We propose a theoretical model to investigate the thermodynamics of single and coupled two-state ion channels, associated with mechanoelectrical transduction (MET) and hair cell biophysics. The modeling was based on the Tsallis nonextensive statistical mechanics. The choice for a nonextensive framework in modeling ion channels is encouraged on the fact that we take into account the presence of interactions or long-range correlations in the dynamics of single and coupled ion channels. However, the basic assumptions that support Boltzmann–Gibbs statistics, traditionally used to model ion channel dynamics, state that the system is formed by independent or weakly interacting elements. Despite being well studied in many biological systems, the literature has not yet addressed the study of both entropy and mutual information related to isolated or physically interacting pairs of MET channels. Inspired by hair cell biophysics, we show how the presence of nonextensivity, or subadditivity and superadditivity modulates the nonextensive entropy and mutual information as functions of stereocilia displacements. We also observe that the magnitude of the interaction between the two channels, given by a nonextensive parameter, influences the amplitude of the nonextensive joint entropy and mutual information as functions of the hair cell displacements. Finally, we show how nonextensivity regulates the current versus displacement curve for a single and a pair of interacting two-state channels. The present findings shed light on the thermodynamic process involved in the molecular mechanisms of the auditory system.

Asymptotic behaviors of aggregated Markov processes

AbstractFinite state Markov processes and their aggregated Markov processes have been extensively studied, especially in ion channel modeling and reliability modeling. In reliability field, the asymptotic behaviors of repairable systems modeled by both processes have been paid much attention to. For a Markov process, it is well-known that limiting measures such as availability and transition probability do not depend on the initial state of the process. However, for an aggregated Markov process, it is difficult to directly know whether this conclusion holds true or not from the limiting measure formulas expressed by the Laplace transforms. In this paper, four limiting measures expressed by Laplace transforms are proved to be independent of the initial state through Tauber’s theorem. The proof is presented under the assumption that the rank of transition rate matrix is one less than the dimension of state space for the Markov process, which includes the case that all states communicate with each other. Some numerical examples and discussions based on these are presented to illustrate the results directly and to show future related research topics. Finally, the conclusion of the paper is given.

A Poisson-Nernst-Planck single ion channel model and its effective finite element solver

A single ion channel is a membrane protein with an ion selectivity filter that allows only a single species of ions (such as potassium ions) to pass through in the “open” state. Its selectivity filter also naturally separates a solvent domain into an intracellular domain and an extracellular domain. Such biological and geometrical characteristics of a single ion channel are novelly adopted in the construction of a new kind of dielectric continuum ion channel model, called the Poisson-Nernst-Planck single ion channel (PNPSIC) model, in this paper. An effective PNPSIC finite element solver is then developed and implemented as a software package workable for a single ion channel with a three-dimensional X-ray crystallographic molecular structure and a mixture of multiple ionic species. Numerical results for a potassium channel confirm the convergence and efficiency of the PNPSIC finite element solver and demonstrate the high performance of the software package. Moreover, the PNPSIC model is applied to the calculation of electric current and validated by biophysical experimental data.

NeuroML-DB: Sharing and characterizing data-driven neuroscience models described in NeuroML.

As researchers develop computational models of neural systems with increasing sophistication and scale, it is often the case that fully de novo model development is impractical and inefficient. Thus arises a critical need to quickly find, evaluate, re-use, and build upon models and model components developed by other researchers. We introduce the NeuroML Database (NeuroML-DB.org), which has been developed to address this need and to complement other model sharing resources. NeuroML-DB stores over 1,500 previously published models of ion channels, cells, and networks that have been translated to the modular NeuroML model description language. The database also provides reciprocal links to other neuroscience model databases (ModelDB, Open Source Brain) as well as access to the original model publications (PubMed). These links along with Neuroscience Information Framework (NIF) search functionality provide deep integration with other neuroscience community modeling resources and greatly facilitate the task of finding suitable models for reuse. Serving as an intermediate language, NeuroML and its tooling ecosystem enable efficient translation of models to other popular simulator formats. The modular nature also enables efficient analysis of a large number of models and inspection of their properties. Search capabilities of the database, together with web-based, programmable online interfaces, allow the community of researchers to rapidly assess stored model electrophysiology, morphology, and computational complexity properties. We use these capabilities to perform a database-scale analysis of neuron and ion channel models and describe a novel tetrahedral structure formed by cell model clusters in the space of model properties and features. This analysis provides further information about model similarity to enrich database search.

On the crucial features of a single‐file transport model for ion channels

AbstractIt has long been accepted that the multiple‐ion single‐file transport model is appropriate for many kinds of ion channels. However, most of the purely theoretical works in this field did not capture all of the important features of realistic systems. Nowadays, large‐scale atomic‐level simulations are more feasible. The discrepancy between theories, simulations, and experiments are getting obvious, enabling people to carefully examine the missing parts of the theoretical models and methods. In this work, it is attempted to find out the essential features that such kinds of models should possess, such that the physical properties of an ion channel be adequately reflected.

Molecular simulations of state-specific drug interactions with multiple cardiac ion channels to reveal mechanisms of arrhythmogenesis.

Unintended drug block of cardiac ion channels remains a major problem in drug development. The voltage-gated potassium channel KV11.1 also known as hERG is a major drug anti-target binding a diverse set of small molecule drugs that potently reduce the critical repolarizing current IKr. Many drugs that bind the hERG channel promote deadly arrythmias while some hERG blockers present significantly lower proarrhythmic risk. Two hypotheses were proposed to elucidate this discrepancy: (1) preferential drug binding to the inactivated state of the hERG channel confers greater proarrhythmic risk and (2) simultaneous drug binding to other cardiac ion channels can ameliorate the risk associated with hERG channel block. Here, we present a state-specific molecular modeling assessment of drug binding to different conformations of the hERG and voltage-gated sodium NaV1.5 and calcium CaV1.2 channels. We have developed structural models of these cardiac ion channels in open and inactivated conformations and performed all-atom molecular dynamics (MD) simulations to validate structural stabilities and assess ion conduction. Ligand docking was then performed using Site Identification by Ligand Competitive Saturation (SILCS), a pre-compute ensemble molecular docking technique. SILCS allows us to perform a high-throughput assessment of ligand binding affinities using molecular fragment energy maps derived from MD simulations of state-specific ion channel models. Bayesian machine learning was used to provide improved correlation of SILCS-computed affinities with experimental data. Using SILCS multi-ligand docking we also estimated interactions of drugs with sex hormones in the hERG channel pore to assess a potential molecular mechanism for an increased proarrhythmia risk in females. We aim to use SILCS computed state-specific drug affinity data to inform multi-scale functional kinetic models of cardiac electrophysiology to estimate emergent drug effects on the cardiac action potential and heart rhythm.

Using many different voltage protocols to characterise discrepancy in mathematical ion channel models.

The Kv11.1 protein encoded by the hERG gene forms the primary subunit of a voltage-sensitive ion channel responsible for IKr in cardiomyocytes. Mathematical models of the macroscopic current are fitted to data from patch-clamp experiments - in which the whole-cell current is recorded in response to an applied voltage protocol. These models are a valuable tool for understanding the proarrhythmic effects of drugs and mutations, as they can provide accurate, quantitative predictions of the macroscopic current. In recent years, information rich protocols (that do not rely on the channels returning to steady state between voltage pulses) were designed to calibrate mathematical IKr models using short, dense time series. These protocols have been adapted for use on high-throughput automated patch clamp machines in CHO cells overexpressing hERG at either room or physiological temperatures. There are many choices of such protocols to apply during a patch-clamp experiment. We designed twelve protocols under various criteria, all of which should provide enough information to fit simple models of IKr uniquely. However, the parameters in imperfect models are forced to make different compromises to fit data from different protocols. The results highlight the impact of model discrepancy (the difference between models and reality) on parameter estimation and prediction. In particular, we show that the parameter estimates obtained depend on the chosen protocol, and how parameters fitted to protocols which highlight a significant amount of model discrepancy can lead to poor predictions. These results motivate the need for new experimental design methods that account for model discrepancy and aid in model selection. Furthermore, we suggest cross-validation as a tool to help evaluate the trustworthiness of model predictions, which may help identify good models of IKr as well as other macroscopic currents.

Cell-Free Synthesis Goes Electric: Dual Optical and Electronic Biosensor via Direct Channel Integration into a Supported Membrane Electrode.

Assembling transmembrane proteins on organic electronic materials is one promising approach to couple biological functions to electrical readouts. A biosensing device produced in such a way would enable both the monitoring and regulation of physiological processes and the development of new analytical tools to identify drug targets and new protein functionalities. While transmembrane proteins can be interfaced with bioelectronics through supported lipid bilayers (SLBs), incorporating functional and oriented transmembrane proteins into these structures remains challenging. Here, we demonstrate that cell-free expression systems allow for the one-step integration of an ion channel into SLBs assembled on an organic conducting polymer, poly(3,4-ethylenedioxythiophene) polystyrenesulfonate (PEDOT:PSS). Using the large conductance mechanosensitive channel (MscL) as a model ion channel, we demonstrate that MscL adopts the correct orientation, remains mobile in the SLB, and is active on the polyelectrolyte surface using optical and electrical readouts. This work serves as an important illustration of a rapidly assembled bioelectronic platform with a diverse array of downstream applications, including electrochemical sensing, physiological regulation, and screening of transmembrane protein modulators.

Nonionic, Cleavable Surfactant for Top-Down Proteomics.

Nonionic surfactants are often used as general reagents for cell lysis enabling protein extraction, stabilization, and purification under nondenaturing conditions for downstream analysis in structural biology. However, the presence of surfactants in the sample matrix often has a deleterious effect on electrospray ionization (ESI)-mass spectrometry (MS) analysis of proteins and complexes. Here, we report a nonionic, cleavable surfactant, n-decyl-disulfide-β-D-maltoside (DSSM), for top-down proteomics. DSSM was designed to mimic the properties of one of the most common surfactants used in structural biology, n-dodecyl-β-D-maltoside (DDM), but contains a disulfide bond that allows for facile cleavage and surfactant removal before or during MS analysis. We have shown that DSSM is compatible with direct electrospray ionization (ESI)-MS analysis and reversed-phase liquid chromatography (RPLC)-MS analysis of proteins and protein complexes. We have demonstrated that DSSM can facilitate top-down proteomic characterization of membrane proteins such as a model ion channel protein and a G protein-coupled receptor as well as endogenous proteins from cell lysates for the determination of sequence variations and posttranslational modifications (PTMs). Conceivably, DSSM could serve as a general replacement for DDM in proteomic experiments and structural biology studies.

Mott Memristors and Neuronal Ion Channels: A Qualitative Analysis

Memristive devices have earned increased recognition over the past decade, owing to their application in memory and neuromorphic systems. Recently, memristive devices have found a propitious role in Spiking Neural Networks. This is partly attributed to an emerging family of memristive devices that are volatile. Volatile memristors, especially Mott memristors, have been used to implement Leaky Integrate and Fire neurons along with neuron circuits with richer dynamics, referred to as neuristors. The Mott memristor is often compared to voltage-gated ion channels in the Hodgkin-Huxley neuron model. However, voltage-gated ion channels in the Hodgkin-Huxley model do not exhibit voltage hysteresis in their quasi-static I-V characteristics, a critical property of Mott memristors. We discuss the association between Mott memristors based on joule heating and the Hodgkin-Huxley voltage-gated ion channels, and we prove that they are qualitatively dissimilar. Consequently, we propose a hypothetical potassium current-gated ion channel that exhibits voltage hysteresis. We simulate a Mott memristor Pearson-Anson oscillator and a current-gated ion channel oscillator. We use the nullcline plots and bifurcation diagrams of the circuits to demonstrate that both circuits are qualitatively similar. We further show that both circuits produce oscillations through a supercritical Andronov-Hopf bifurcation and have a similar oscillation-generation mechanism. We conclude that for the Mott memristor to share qualitative similarity to a neuronal ion channel, the channel must be current-gated rather than voltage-gated. Our aim is to supplement the understanding of how hardware neurons relate to their biological counterparts, which is a crucial first step towards implementing better brain-inspired systems.

A general pattern of non-spiking neuron dynamics under the effect of potassium and calcium channel modifications.

Electrical activity of excitable cells results from ion exchanges through cell membranes, so that genetic or epigenetic changes in genes encoding ion channels are likely to affect neuronal electrical signaling throughout the brain. There is a large literature on the effect of variations in ion channels on the dynamics of spiking neurons that represent the main type of neurons found in the vertebrate nervous systems. Nevertheless, non-spiking neurons are also ubiquitous in many nervous tissues and play a critical role in the processing of some sensory systems. To our knowledge, however, how conductance variations affect the dynamics of non-spiking neurons has never been assessed. Based on experimental observations reported in the biological literature and on mathematical considerations, we first propose a phenotypic classification of non-spiking neurons. Then, we determine a general pattern of the phenotypic evolution of non-spiking neurons as a function of changes in calcium and potassium conductances. Furthermore, we study the homeostatic compensatory mechanisms of ion channels in a well-posed non-spiking retinal cone model. We show that there is a restricted range of ion conductance values for which the behavior and phenotype of the neuron are maintained. Finally, we discuss the implications of the phenotypic changes of individual cells at the level of neuronal network functioning of the C. elegans worm and the retina, which are two non-spiking nervous tissues composed of neurons with various phenotypes.

The Kv10.1 voltage-gated ion channel modeling based on control system theory

Abstract In this paper, we present a control system theory based modeling approach for the voltage-gated ion-channel Kv10.1 at different temperatures. Kv10.1 is expressed in the central nervous system and the brain of healthy humans. The study of the activity of this ion channel is important, because its activity is associated with the occurrence of cancer in different organs or tissues. According to systems and control theory, the voltage-gated channel Kv10.1 was assumed to be a linear time-invariant system and, as such, exhibits dynamic behavior. The experimental results show that Kv10.1 operates as a first-order model based on the input voltage step protocol and the measured macroscopic ion current output.

An efficient finite element iterative method for solving a nonuniform size modified Poisson-Boltzmann ion channel model

In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution of Model 1 collecting all the singular points and Models 2 and 3 much easier to solve numerically than the original nuSMPBIC model. A damped two-block iterative method is then presented to solve Model 3, along with a novel modified Newton iterative scheme for solving each related nonlinear algebraic system. To this end, an effective nuSMPBIC finite element solver is derived and then implemented as a program package that works for an ion channel protein with a three-dimensional molecular structure and a mixture of multiple ionic species. Numerical results for a voltage-dependent anion channel (VDAC) in a mixture of four ionic species demonstrate a fast convergence rate of the damped two-block iterative method, the high performance of the software package, and the importance of considering nonuniform ion sizes. Moreover, the nuSMPBIC model is validated by the anion selectivity property of VDAC.

Response of a neuronal network computational model to infrared neural stimulation.

Infrared neural stimulation (INS), as a novel form of neuromodulation, allows modulating the activity of nerve cells through thermally induced capacitive currents and thermal sensitivity ion channels. However, fundamental questions remain about the exact mechanism of INS and how the photothermal effect influences the neural response. Computational neural modeling can provide a powerful methodology for understanding the law of action of INS. We developed a temperature-dependent model of ion channels and membrane capacitance based on the photothermal effect to quantify the effect of INS on the direct response of individual neurons and neuronal networks. The neurons were connected through excitatory and inhibitory synapses and constituted a complex neuronal network model. Our results showed that a slight increase in temperature promoted the neuronal spikes and enhanced network activity, whereas the ultra-temperature inhibited neuronal activity. This biophysically based simulation illustrated the optical dose-dependent biphasic cell response with capacitive current as the core change condition. The computational model provided a new sight to elucidate mechanisms and inform parameter selection of INS.

Spiral wave drift under optical feedback in cardiac tissue.

Spiral waves occur in various types of excitable media and their dynamics determine the spatial excitation patterns. An important type of spiral wave dynamics is drift, as it can control the position of a spiral wave or eliminate a spiral wave by forcing it to the boundary. In theoretical and experimental studies of the Belousov-Zhabotinsky reaction, it was shown that the most direct way to induce the controlled drift of spiral waves is by application of an external electric field. Mathematically such drift occurs due to the onset of additional gradient terms in the Laplacian operator describing excitable media. However, this approach does not work for cardiac excitable tissue, where an external electric field does not result in gradient terms. In this paper, we propose a method of how to induce a directed linear drift of spiral waves in cardiac tissue, which can be realized as an optical feedback control in tissue where photosensitive ion channels are expressed. We illustrate our method by using the FitzHugh-Nagumo model for cardiac tissue and the generic model of photosensitive ion channels. We show that our method works for continuous and discrete light sources and can effectively move spiral waves in cardiac tissue, or eliminate them by collisions with the boundary or with another spiral wave. We finally implement our method by using a biophysically motivated photosensitive ion channel model included to the Luo-Rudy model for cardiac cells and show that the proposed feedback control also induces directed linear drift of spiral waves in a wide range of light intensities.

Ion channel model reduction using manifold boundaries.

Mathematical models of voltage-gated ion channels are used in basic research, industrial and clinical settings. These models range in complexity, but typically contain numerous variables representing the proportion of channels in a given state, and parameters describing the voltage-dependent rates of transition between states. An open problem is selecting the appropriate degree of complexity and structure for an ion channel model given data availability. Here, we simplify a model of the cardiac human Ether-à-go-go related gene (hERG) potassium ion channel, which carries cardiac IKr, using the manifold boundary approximation method (MBAM). The MBAM approximates high-dimensional model-output manifolds by reduced models describing their boundaries, resulting in models with fewer parameters (and often variables). We produced a series of models of reducing complexity starting from an established five-state hERG model with 15 parameters. Models with up to three fewer states and eight fewer parameters were shown to retain much of the predictive capability of the full model and were validated using experimental hERG1a data collected in HEK293 cells at 37°C. The method provides a way to simplify complex models of ion channels that improves parameter identifiability and will aid in future model development.

The 70-year search for the voltage-sensing mechanism of ion channels.

This retrospective on the voltage‐sensing mechanisms and gating models of ion channels begins in 1952 with the charged gating particles postulated by Hodgkin and Huxley, viewed as charges moving across the membrane and controlling its permeability to Na+ and K+ ions. Hodgkin and Huxley postulated that their movement should generate small and fast capacitive currents, which were recorded 20 years later as gating currents. In the early 1980s, several voltage‐dependent channels were cloned and found to share a common architecture: four homologous domains or subunits, each displaying six transmembrane α‐helical segments, with the fourth segment (S4) displaying four to seven positive charges invariably separated by two non‐charged residues. This immediately suggested that this segment was serving as the voltage sensor of the channel (the molecular counterpart of the charged gating particle postulated by Hodgkin and Huxley) and led to the development of the sliding helix model. Twenty years later, the X‐ray crystallographic structures of many voltage‐dependent channels allowed investigation of their gating by molecular dynamics. Further understanding of how channels gate will benefit greatly from the acquisition of high‐resolution structures of each of their relevant functional or structural states. This will allow the application of molecular dynamics and other approaches. It will also be key to investigate the energetics of channel gating, permitting an understanding of the physical and molecular determinants of gating. The use of multiscale hierarchical approaches might finally prove to be a rewarding strategy to overcome the limits of the various single approaches to the study of channel gating.