Creating Computational Models of Ion Channel Dynamics.

Markov models are widely used to represent ion channel protein configurations as different states in the model's topology. Such models allow for dynamic simulation of ion channel kinetics through the simulated application of voltage potentials across a cell membrane. In this chapter, we present a general method for creating Markov models of ion channel kinetics using computational optimization alongside a fully featured example model of a cardiac potassium channel. Our methods cover designing training protocols, iteratively testing potential model topologies for structure identification, creation of algorithms for model simulation, as well as methods for assessing the quality of fit for a finalized model.

Analysis of pressure-activated Piezo1 open and subconductance states at a single channel level.

Mechanically activated Piezo1 channels undergo transitions from closed to open-state in response to pressure and other mechanical stimuli. However, the molecular details of these mechanosensitive gating transitions are unknown. Here, we used cell-attached pressure-clamp recordings to acquire single channel data at steady-state conditions (where inactivation has settled down), at various pressures and voltages. Importantly, we identify and analyze subconductance states of the channel which were not reported before. Pressure-dependent activation of Piezo1 increases the occupancy of open and subconductance state at the expense of decreased occupancy of shut-states. No significant change in the mean open time of subconductance states was observed with increasing negative pipette pressure or with varying voltages (ranging from -40 to -100 mV). Using Markov-chain modeling, we identified a minimal four-states kinetic scheme, which recapitulates essential characteristics of the single channel data, including that of the subconductance level. This study advances our understanding of Piezo1-gating mechanism in response to discrete stimuli (such as pressure and voltage) and paves the path to develop cellular and tissue level models to predict Piezo1 function in various cell types.

Exploration of the Parameter Space of an Ion Channel Kinetic Model by a Markov-Chain-Based Methodology.

In this work, we propose a methodology based on Monte Carlo Markov chains to explore the parameter space of kinetic models for ion channels. The methodology allows the detection of potential parameter sets of a model that are compatible with experimentally obtained whole-cell currents, which could remain hidden when methods focus on obtaining the parameters that provide the best fit. To show its implementation and utility, we considered a four-state kinetic model proposed in the literature to describe the activation of the voltage-gated proton channel (Hv1), Biophysical Journal, 2014, 107, 1564. In that work, a set of values for the rate transitions that describe the channel kinetics at different intracellular H+ concentration (pHi) were obtained by the Simplex method. With our approach, we find that, in fact, there is more than one parameter set for each pHi, which renders the same open probability temporal course within the experimental error margin for all of the considered voltages. The large differences that we obtained for the values of some rate constants among the different solutions show that there is more than one possible interpretation of this channel behavior as a function of pHi. We also simulated a proposed new experimental condition where it is possible to observe that different sets of parameters yield different results. Our study highlights the importance of a comprehensive analysis of parameter space in kinetic models and the utility of the proposed methodology for finding potential solutions.

High-throughput assessment identifying major platelet Ca2+ entry pathways via tyrosine kinase-linked and G protein-coupled receptors.

In platelets, elevated cytosolic Ca2+ is a crucial second messenger, involved in most functional responses, including shape change, secretion, aggregation and procoagulant activity. The platelet Ca2+ response consists of Ca2+ mobilization from endoplasmic reticulum stores, complemented with store-operated or receptor-operated Ca2+ entry pathways. Several channels can contribute to the Ca2+ entry, but their relative contribution is unclear upon stimulation of ITAM-linked receptors such as glycoprotein VI (GPVI) and G-protein coupled receptors such as the protease-activated receptors (PAR) for thrombin. We employed a 96-well plate high-throughput assay with Fura-2-loaded human platelets to perform parallel [Ca2+]i measurements in the presence of EGTA or CaCl2. Per agonist condition, this resulted in sets of EGTA, CaCl2 and Ca2+ entry ratio curves, defined by six parameters, reflecting different Ca2+ ion fluxes. We report that threshold stimulation of GPVI or PAR, with a variable contribution of secondary mediators, induces a maximal Ca2+ entry ratio of 3-7. Strikingly, in combination with Ca2+-ATPase inhibition by thapsigargin, the maximal Ca2+ entry ratio increased to 400 (GPVI) or 40 (PAR), pointing to a strong receptor-dependent enhancement of store-operated Ca2+ entry. By pharmacological blockage of specific Ca2+ channels in platelets, we found that, regardless of GPVI or PAR stimulation, the Ca2+ entry ratio was strongest affected by inhibition of ORAI1 (2-APB, Synta66) > Na+/Ca2+ exchange (NCE) > P2×1 (only initial). In contrast, inhibition of TRPC6, Piezo1/2 or STIM1 was without effect. Together, these data reveal ORAI1 and NCE as dominating Ca2+ carriers regulating GPVI- and PAR-induced Ca2+ entry in human platelets.

2D-dwell-time analysis with simulations of ion-channel gating using high-performance computing.

Single-channel patch-clamp recordings allow observing the action of a single protein complex in real time and hence the deduction of the underlying conformational changes in the ion-channel protein. Commonly, recordings are modeled using hidden Markov chains, connecting open and closed states in the experimental data with protein conformations. The rates between states denote transition probabilities that could be modified by membrane voltage or ligand binding. Modeling algorithms have to deal with limited recording bandwidth and a very noisy background. It was previously shown that the fit of two-dimensional (2D)-dwell-time histograms with simulations is very robust in that regard. Errors introduced by the low-pass filter or noise cancel out to a certain degree when comparing experimental and simulated data. In addition, the topology of models (that is, the chain of open and closed states) could be inferred from 2D-histograms. However, the 2D-fit was never applied to its full potential. A major reason may be the extremely time-consuming and often unreliable fitting process, due to the stochastic variability in the simulations. We have now solved these issues by introducing a message-passing interface (MPI) allowing massive parallel computing on a high-performance computing (HPC) cluster and obtaining ensemble solutions. With ensembles, we have demonstrated how important ranked solutions are for difficult tasks related to a noisy background, fast gating events beyond the corner frequency of the low-pass filter, and topology estimation of the underlying Markov model. Finally, we have shown that, by combining the objective function of the 2D-fit with the deviation of the current amplitude distributions, automatic determination of the current level of the conducting state is possible, even with an apparent current reduction due to low-pass filtering. Making use of an HPC cluster, the power of 2D-dwell-time analysis can be used to its fullest with minor input of the experimenter.

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.

A549 in-silico 1.0: A first computational model to simulate cell cycle dependent ion current modulation in the human lung adenocarcinoma.

Lung cancer is still a leading cause of death worldwide. In recent years, knowledge has been obtained of the mechanisms modulating ion channel kinetics and thus of cell bioelectric properties, which is promising for oncological biomarkers and targets. The complex interplay of channel expression and its consequences on malignant processes, however, is still insufficiently understood. We here introduce the first approach of an in-silico whole-cell ion current model of a cancer cell, in particular of the A549 human lung adenocarcinoma, including the main functionally expressed ion channels in the plasma membrane as so far known. This hidden Markov-based model represents the electrophysiology behind proliferation of the A549 cell, describing its rhythmic oscillation of the membrane potential able to trigger the transition between cell cycle phases, and it predicts membrane potential changes over the cell cycle provoked by targeted ion channel modulation. This first A549 in-silico cell model opens up a deeper insight and understanding of possible ion channel interactions in tumor development and progression, and is a valuable tool for simulating altered ion channel function in lung cancer electrophysiology.

Memory in Ion Channel Kinetics.

Ion channels are transport proteins present in the lipid bilayers of biological membranes. They are involved in many physiological processes, such as the generation of nerve impulses, hormonal secretion, and heartbeat. Conformational changes in the ion channel-forming protein allow the opening or closing of pores to control the ionic flux through the cell membranes. The opening and closing of the ion channel have been classically treated as a random kinetic process, known as a Markov process. Here the time the channel remains in a given state is assumed to be independent of the condition it had in the previous state. More recently, however, several studies have shown that this process is not random but a deterministic one, where both the open and closed dwell-times and the ionic current flowing through the channel are history-dependent. This property is called long memory or long-range correlation. However, there is still much controversy regarding how this memory originates, which region of the channel is responsible for this property, and which models could best reproduce the memory effect. In this article, we provide a review of what is, where it is, its possible origin, and the mathematical methods used to analyze the long-term memory present in the kinetic process of ion channels.

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.

Prediction and Dissipation in Nonequilibrium Molecular Sensors: Conditionally Markovian Channels Driven by Memoryful Environments.

Biological sensors must often predict their input while operating under metabolic constraints. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is challenging. This arises partly from the functional constraints being at cross purposes and partly since quantifying the prediction performance of even in silico sensors can require prohibitively long simulations, especially when highly complex environments drive sensors out of equilibrium. To circumvent these difficulties, we develop new expressions for the prediction accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to complex, correlated (unifilar hidden semi-Markov) environmental inputs in nonequilibrium steady state. Predictive metrics include the instantaneous memory and the total predictable information (the mutual information between present sensor state and input future), while dissipation metrics include power extracted from the environment and the nonpredictive information rate. Success in deriving these formulae relies on identifying the environment's causal states, the input's minimal sufficient statistics for prediction. Using these formulae, we study large random channels and the simplest nontrivial biological sensor model-that of a Hill molecule, characterized by the number of ligands that bind simultaneously-the sensor's cooperativity. We find that the seemingly impoverished Hill molecule can capture an order of magnitude more predictable information than large random channels.

Cardiac safety assessment with motion field imaging analysis of human iPS cell-derived cardiomyocytes is improved by an integrated evaluation with cardiac ion channel profiling.

We validated a motion field imaging (MFI) assay with human induced pluripotent stem cell-derived cardiomyocytes (hiPS-CMs) as a model to assess multiple cardiac liabilities by comparing the guinea-pig Langendorff heart with hiPS-CMs using 4 reference compounds and 9 internal compounds. We investigated repolarization duration, beating rate (BR), conduction speed, contractility, and inhibitory profile of three cardiac ion channels: hERG, Cav1.2, and Nav1.5. For repolarization, the contraction-relaxation duration (CRDc) of hiPS-CMs was generally consistent with the QTc interval of Langendorff heart. However, 2 internal compounds shortened CRDc despite QTc prolongation in Langendorff heart. Cardiac ion channel profiling revealed that hiPS-CMs could not be used to detect QTc prolongation when the value of Cav1.2 IC50 / hERG IC50 for a compound was between 1 and 10, whereas hiPS-CMs showed responses largely consistent with Langendorff heart when Cav1.2 IC50 / hERG IC50 was below 1 or above 10. The accuracy of hiPS-CMs for the BR was not high, mainly because the BR of hiPS-CMs was increased by an inhibition of Cav1.2. The hiPS-CMs were highly sensitive to conduction speed and contractility, able to detect QRS widening caused by Nav1.5-inhibition, as well as decreased LVdP/dtmax caused by the inhibition of Cav1.2 and/or Nav1.5. In conclusion, the MFI assay with hiPS-CMs would be useful for evaluating multiple cardiac liabilities. The ion channel profile helps to interpret the results of MFI assay and correctly evaluate cardiac risks. Therefore, an integrated cardiac safety assessment with MFI and ion channel profiling is recommended.

Moving average filtering with deconvolution (MAD) for hidden Markov model with filtering and correlated noise.

Ion channel data recorded using the patch clamp technique are low-pass filtered to remove high-frequency noise. Almanjahie et al. (Eur Biophys J 44:545-556, 2015) based statistical analysis of such data on a hidden Markov model (HMM) with a moving average adjustment for the filter but without correlated noise, and used the EM algorithm for parameter estimation. In this paper, we extend their model to include correlated noise, using signal processing methods and deconvolution to pre-whiten the noise. The resulting data can be modelled as a standard HMM and parameter estimates are again obtained using the EM algorithm. We evaluate this approach using simulated data and also apply it to real data obtained from the mechanosensitive channel of large conductance (MscL) in Escherichia coli. Estimates of mean conductances are comparable to literature values. The key advantages of this method are that it is much simpler and computationally considerably more efficient than currently used HMM methods that include filtering and correlated noise.

Prediction of Ion Channels and their Types from Protein Sequences: Comprehensive Review and Comparative Assessment.

BACKGROUND: Ion channels are a large and growing protein family. Many of them are associated with diseases, and consequently, they are targets for over 700 drugs. Discovery of new ion channels is facilitated with computational methods that predict ion channels and their types from protein sequences. However, these methods were never comprehensively compared and evaluated. OBJECTIVE: We offer first-of-its-kind comprehensive survey of the sequence-based predictors of ion channels. We describe eight predictors that include five methods that predict ion channels, their types, and four classes of the voltage-gated channels. We also develop and use a new benchmark dataset to perform comparative empirical analysis of the three currently available predictors. RESULTS: While several methods that rely on different designs were published, only a few of them are currently available and offer a broad scope of predictions. Support and availability after publication should be required when new methods are considered for publication. Empirical analysis shows strong performance for the prediction of ion channels and modest performance for the prediction of ion channel types and voltage-gated channel classes. We identify a substantial weakness of current methods that cannot accurately predict ion channels that are categorized into multiple classes/types. CONCLUSION: Several predictors of ion channels are available to the end users. They offer practical levels of predictive quality. Methods that rely on a larger and more diverse set of predictive inputs (such as PSIONplus) are more accurate. New tools that address multi-label prediction of ion channels should be developed.

Spatiotemporal relationships defining the adaptive gating of the bacterial mechanosensitive channel MscS.

Adaptive desensitization and inactivation are common properties of most ion channels and receptors. The mechanosensitive channel of small conductance MscS, which serves as a low-threshold osmolyte release valve in most bacteria, inactivates not from the open, but from the resting state under moderate tensions. This mechanism enables the channel to respond differently to slow tension ramps versus abruptly applied stimuli. In this work, we present a reconstruction of the energy landscape for MscS transitions based on patch current kinetics recorded under special pressure protocols. The data are analyzed with a three-state continuous time Markov model, where the tension-dependent transition rates are governed by Arrhenius-type relations. The analysis provides assignments to the intrinsic opening, closing, inactivation, and recovery rates as well as their tension dependencies. These parameters, which define the spatial (areal) distances between the energy wells and the positions of barriers, describe the tension-dependent distribution of the channel population between the three states and predict the experimentally observed dynamic pulse and ramp responses. Our solution also provides an analytic expression for the area of the inactivated state in terms of two experimentally accessible parameters: the tension at which inactivation probability is maximized, γ*, and the midpoint tension for activation, γ0.5. The analysis initially performed on Escherichia coli MscS shows its applicability to the recently characterized MscS homolog from Pseudomonas aeruginosa. Inactivation appears to be a common property of low-threshold MscS channels, which mediate proper termination of the osmotic permeability response and contribute to the environmental fitness of bacteria.

In silico assessment of the effects of various compounds in MEA/hiPSC-CM assays: Modeling and numerical simulations.

We propose a mathematical approach for the analysis of drugs effects on the electrical activity of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) based on multi-electrode array (MEA) experiments. Our goal is to produce an in silico tool able to simulate drugs action in MEA/hiPSC-CM assays. The mathematical model takes into account the geometry of the MEA and the electrodes' properties. The electrical activity of the stem cells at the ion-channel level is governed by a system of ordinary differential equations (ODEs). The ODEs are coupled to the bidomain equations, describing the propagation of the electrical wave in the stem cells preparation. The field potential (FP) measured by the MEA is modeled by the extracellular potential of the bidomain equations. First, we propose a strategy allowing us to generate a field potential in good agreement with the experimental data. We show that we are able to reproduce realistic field potentials by introducing different scenarios of heterogeneity in the action potential. This heterogeneity reflects the differentiation atria/ventricles and the age of the cells. Second, we introduce a drug/ion channels interaction based on a pore block model. We conduct different simulations for five drugs (mexiletine, dofetilide, bepridil, ivabradine and BayK). We compare the simulation results with the field potential collected from experimental measurements. Different biomarkers computed on the FP are considered, including depolarization amplitude, repolarization delay, repolarization amplitude and depolarization-repolarization segment. The simulation results show that the model reflect properly the main effects of these drugs on the FP.

Dynamic Histogram Analysis To Determine Free Energies and Rates from Biased Simulations.

We present an algorithm to calculate free energies and rates from molecular simulations on biased potential energy surfaces. As input, it uses the accumulated times spent in each state or bin of a histogram and counts of transitions between them. Optimal unbiased equilibrium free energies for each of the states/bins are then obtained by maximizing the likelihood of a master equation (i.e., first-order kinetic rate model). The resulting free energies also determine the optimal rate coefficients for transitions between the states or bins on the biased potentials. Unbiased rates can be estimated, e.g., by imposing a linear free energy condition in the likelihood maximization. The resulting "dynamic histogram analysis method extended to detailed balance" (DHAMed) builds on the DHAM method. It is also closely related to the transition-based reweighting analysis method (TRAM) and the discrete TRAM (dTRAM). However, in the continuous-time formulation of DHAMed, the detailed balance constraints are more easily accounted for, resulting in compact expressions amenable to efficient numerical treatment. DHAMed produces accurate free energies in cases where the common weighted-histogram analysis method (WHAM) for umbrella sampling fails because of slow dynamics within the windows. Even in the limit of completely uncorrelated data, where WHAM is optimal in the maximum-likelihood sense, DHAMed results are nearly indistinguishable. We illustrate DHAMed with applications to ion channel conduction, RNA duplex formation, α-helix folding, and rate calculations from accelerated molecular dynamics. DHAMed can also be used to construct Markov state models from biased or replica-exchange molecular dynamics simulations. By using binless WHAM formulated as a numerical minimization problem, the bias factors for the individual states can be determined efficiently in a preprocessing step and, if needed, optimized globally afterward.

Ion Channel Conformation and Oligomerization Assessment by Site-Directed Spin Labeling and Pulsed-EPR.

Mechanosensitive (MS) ion channels are multimeric integral membrane proteins that respond to increased lipid bilayer tension by opening their nonselective pores to release solutes and relieve increased cytoplasmic pressure. These systems undergo major conformational changes during gating and the elucidation of their mechanism requires a deep understanding of the interplay between lipids and proteins. Lipids are responsible for transmitting lateral tension to MS channels and therefore play a key role in obtaining a molecular-detail model for mechanosensation. Site-directed spin labeling combined with electron paramagnetic resonance (EPR) spectroscopy is a powerful spectroscopic tool in the study of proteins. The main bottleneck for its use relates to challenges associated with successful isolation of the protein of interest, introduction of paramagnetic labels on desired sites, and access to specialized instrumentation and expertise. The design of sophisticated experiments, which combine a variety of existing EPR methodologies to address a diversity of specific questions, require knowledge of the limitations and strengths, characteristic of each particular EPR method. This chapter is using the MS ion channels as paradigms and focuses on the application of different EPR techniques to ion channels, in order to investigate oligomerization, conformation, and the effect of lipids on their regulation. The methodology we followed, from the initial strategic selection of mutants and sample preparation, including protein purification, spin labeling, reconstitution into lipid mimics to the complete set-up of the pulsed-EPR experiments, is described in detail.

Comparison of different in vitro cell models for the assessment of pesticide-induced dopaminergic neurotoxicity.

Biomedical and (neuro) toxicity research on (neuro) degenerative diseases still relies strongly on animal models. However, the use of laboratory animals is often undesirable for both ethical and technical reasons. Current in vitro research thus largely relies on tumor derived- or immortalized cell lines. Notably, the suitability of cell lines for studying neurodegeneration is determined by their intrinsic properties. We therefore characterized PC12, SH-SY5Y, MES23.5 and N27 cells with respect to the presence of functional membrane ion channels and receptors as well as for the effects of five known neurotoxic pesticides on cytotoxicity, oxidative stress and parameters of intracellular calcium homeostasis using a combined alamar Blue/CFDA assay, a H2DCFDA assay and single cell fluorescent (Fura-2) calcium imaging, respectively. Although all pesticides demonstrated a certain level of functional neurotoxicity in the different cell lines, our results also demonstrate considerable differences in intrinsic properties and pesticide-induced effects between the cell lines. This clearly indicates that care should be taken when interpreting (neuro)toxicity data as the chosen cell model may greatly influence the outcome.

Bayesian Statistical Inference in Ion-Channel Models with Exact Missed Event Correction.

The stochastic behavior of single ion channels is most often described as an aggregated continuous-time Markov process with discrete states. For ligand-gated channels each state can represent a different conformation of the channel protein or a different number of bound ligands. Single-channel recordings show only whether the channel is open or shut: states of equal conductance are aggregated, so transitions between them have to be inferred indirectly. The requirement to filter noise from the raw signal further complicates the modeling process, as it limits the time resolution of the data. The consequence of the reduced bandwidth is that openings or shuttings that are shorter than the resolution cannot be observed; these are known as missed events. Postulated models fitted using filtered data must therefore explicitly account for missed events to avoid bias in the estimation of rate parameters and therefore assess parameter identifiability accurately. In this article, we present the first, to our knowledge, Bayesian modeling of ion-channels with exact missed events correction. Bayesian analysis represents uncertain knowledge of the true value of model parameters by considering these parameters as random variables. This allows us to gain a full appreciation of parameter identifiability and uncertainty when estimating values for model parameters. However, Bayesian inference is particularly challenging in this context as the correction for missed events increases the computational complexity of the model likelihood. Nonetheless, we successfully implemented a two-step Markov chain Monte Carlo method that we called "BICME", which performs Bayesian inference in models of realistic complexity. The method is demonstrated on synthetic and real single-channel data from muscle nicotinic acetylcholine channels. We show that parameter uncertainty can be characterized more accurately than with maximum-likelihood methods. Our code for performing inference in these ion channel models is publicly available.

Comparison between Hodgkin-Huxley and Markov formulations of cardiac ion channels.

When simulating the macroscopic current flowing through cardiac ion channels, two mathematical formalisms can be adopted: the Hodgkin-Huxley model (HHM) formulation, which describes openings and closings of channel 'gates', or the Markov model (MM) formulation, based on channel 'state' transitions. The latter was first used in 1995 to simulate the effects of mutations in ionic currents and, since then, its use has been extended to wild-type channels also. While the MMs better describe the actual behavior of ion channels, they are mathematically more complex than HHMs in terms of parameter estimation and identifiability and are computationally much more demanding, which can dramatically increase computational time in large-scale (e.g. whole heart) simulations. We hypothesize that a HHM formulation obtained from classical patch-clamp protocols in wild-type and mutant ion channels can be used to correctly simulate cardiac action potentials and their static and dynamic properties. To validate our hypothesis, we selected two pivotal cardiac ionic currents (the rapid delayed rectifier K(+) current, IKr, and the inward Na(+) current, INa) and formulated HHMs for both wild-type and mutant channels (LQT2-linked T474I mutation for IKr and LQT3-linked ΔKPQ mutation for INa). Action potentials were then simulated using the MM and HHM versions of the currents, and the action potential waveforms, biomarkers and action potential duration rate dependence properties were compared in control conditions and in the presence of physiological variability. While small differences between ionic currents were found between the two models (correlation coefficient ρ>0.92), the simulations yielded almost identical action potentials (ρ>0.99), suggesting that HHMs may also be valid to simulate the effects of mutations affecting IKr and INa on the action potential.