Stochastic modelling of a single ion channel: interdependence of burst length and number of openings per burst

Previous modelling of single channel behaviour based on Markov processes has been concerned mainly with means and marginal distributions of particular quantities. The present study derives the joint distribution, conditional distributions, and associated mean values for the burst length (T) and the number (N) of openings per burst in two simple three-state models in which bursting is possible, one for an agonist-only and one for a channel blocking mechanism. In both models the conditional mean burst length (E(T/N = r)) increases linearly as a function of the number of openings per burst, while the conditional mean number of openings per burst (E(N/T = x)) is a nonlinear strictly increasing function of burst length, which is asymptotically linear for large burst length. The asymptotic intercept for each model is shown to be less than, equal to, or greater than unity according as mean channel closed-time is less than, equal to, or greater than mean open-time. For parameter values typical of the nicotinic receptor, this intercept is less than unity for the agonist-only model and greater than unity for the blocking model. As a result of the dependence between the number of openings per burst and burst length, it is shown that experimental estimates of the unconditional mean number of openings per burst may be biased if bursts of only short duration are collected.     

Complex voltage-dependent behavior of single unliganded calcium-sensitive potassium channels

study and characterization of unliganded openings is of central significance for the elucidation of gating mechanisms for allosteric ligand-gated ion channels. Unliganded openings have been reported for many channel types, but their low open probability can make it difficult to study their kinetics in detail. Because the large conductance calcium-activated potassium channel mSlo is sensitive to both intracellular calcium and to membrane potential, we have been able to obtain stable unliganded single-channel recordings of mSlo with relatively high opening probability. We have found that the single-channel gating behavior of mSlo is complex, with multiple open and closed states, even when no ligand is present. Our results rule out a Monod-Wyman-Changeux allosteric mechanism with a central voltage-dependent concerted step, and they support the existence of quaternary states with less than the full number of voltage sensors activated, as has been suggested by previous work involving measurements of gating currents.     

Modulation of single cardiac sodium channels by DPI 201-106

Single sodium channel currents were analysed in cell attached patches from single ventricular cells of guinea pig hearts in the presence of a novel cardiotonic compound DPI 201-106. The mean single channel conductance of DPI-treated Na channels was not changed by DPI (20.8 +/- 4 pS, control, 3 patches; 21.3 +/- 1 pS with DPI, 5 mumol/1,3 patches). DPI voltage-dependently prolongs the cardiac sodium channel openings by removal of inactivation at potentials positive to -40 mV. At potentials negative to -40 mV a clustering of short openings at the very beginning of the depolarizing voltage steps can be observed causing a transient time course of the averaged currents. Long openings induced an extremely slow inactivation. Short openings, long openings and nulls appeared in groups referring to a modal gating behaviour of DPI-treated sodium channels. DPI-modified Na channels showed a monotonously prolonged mean open time with increased depolarizing voltage steps, e.g. the open state probability within a sweep was increased. However, the number of non-empty sweeps was decreased with the magnitude of the depolarizing steps, e.g. the probability of the channel being open as calculated from the averaged currents was voltage-dependently decreased by DPI (50% decrease at -50.7 +/- 9 9 mV, 3 patches). Short and long openings of DPI-modified channels could be separated by variation of the holding potential. The occurrence of long Na channel openings was much more suppressed by reducing the holding potential (half maximum inactivation at -112 +/- 8 mV, 4 patches) than that of short openings (half maximum inactivation at -88 +/- 8 mV, 4 patches). Otherwise, short living openings completely disappeared at potentials positive to -40 mV where the occurrence of long openings was favoured. The differential voltage dependence of blocking and activating effects of DPI on cardiac Na channels as well as the differential voltage dependence of the appearance of short and long openings refers to a modal gating behaviour of cardiac Na channels.     

Ion-channel gating mechanisms: model identification and parameter estimation from single channel recordings

Patch-clamp data may be analysed in terms of Markov process models of channel gating mechanisms. We present a maximum likelihood algorithm for estimation of gating parameters from records where only a single channel is present. Computer simulated data for three different models of agonist receptor gated channels are used to demonstrate the performance of the procedure. Full details of the implementation of the algorithm are given for an example gating mechanism. The effects of omission of brief openings and closings from the single-channel data on parameter estimation are explored. A strategy for discriminating between alternative possible gating models, based upon use of the Schwarz criterion, is described. Omission of brief events is shown not to lead to incorrect model identification, except in extreme circumstances. Finally, the algorithm is extended to include channel gating models exhibiting multiple conductance levels.     

Permeable ions differentially affect gating kinetics and unitary conductance of L-type calcium channels

Although ion permeation and gating of L-type Ca(2+) channels are generally considered separate processes controlled by distinct components of the channel protein, ion selectivity can vary with the kinetic state. To test this possibility, we studied single-channel currents (cell-attached) of recombinant L-type channels (Ca(V)1.2, beta(2a), and alpha(2)delta) transiently expressed in tsA201 cells in the presence of the channel agonist BayK 8644 which promotes long channel openings (Mode 2 openings). We found that both the brief (Mode 1) and long (Mode 2) mean open times in the presence of Ca(2+) were relatively longer than those with Ba(2+). The unitary slope conductance with Ba(2+) was significantly larger (p<0.05) in Mode 2 openings than for brief Mode 1 openings, whereas the conductance with Ca(2+) did not vary with mode gating. Consequently, the gamma(Ba):gamma(Ca) ratio was greater for Mode 2 than Mode 1 openings. Our findings indicate that both ion permeation and gating kinetics of the L-type channel are differentially modulated by permeable ions. Ca(2+) binding to the L-type channel may stabilize the alteration of channel ion permeability mediated by gating kinetics, and thus, play a role in preventing excessive ion entry when the activation gating of the channel is promoted to the prolonged open state.     

Kinetics of unliganded acetylcholine receptor channel gating.

Open- and closed-state lifetimes of unliganded acetylcholine receptor channel activity were analyzed by the method of likelihood maximazation. For both open times and closed times, the best-fitting density is most often a sum of two exponentials. These multiple open states cannot depend on the number of receptor binding sites occupied since they are observed in the absence of ligand. The rate of spontaneous opening and the faster decay constant of closing increased as the membrane was hyperpolarized. The voltage dependence of the rate of spontaneous opening is stronger than that for curare-liganded channels. Evidence that the acetylcholine receptor channel can open spontaneously in the absence of ligand has been presented previously (Sanchez et al, 1983; Brehm et al, 1984; Jackson, 1984). To add to this evidence, alpha-bungarotoxin was added to the patch electrode, causing the frequency of openings to decay with time. The rate constant determined from this decay is similar to rate constants reported for the binding of iodinated alpha-bungarotoxin to the acetylcholine receptor. The frequency of unliganded channel opening has been estimated as 2 X 10(-3) s-1 per receptor. A comparison of carbamylcholine-liganded and spontaneous gating transition rates suggests that ligand binding increases the rate of opening by a factor of 1.4 X 10(7). Carbamylcholine binding increases the mean open time by a factor of 5. Thus, a cholinergic agonist activates the acetylcholine receptor by destabilizing the closed state. The liganded and unliganded channel gating rates were used to analyze the energetics of ligand activation of the acetylcholine receptor channel, and to relate the open channel dissociation constant to the closed channel dissociation constant.     

Kinetic properties of single sodium channels in rat heart and rat brain

Single Na channel currents were compared in ventricular myocytes and cortical neurons of neonatal rats using the gigaseal patch-clamp method to determine whether tissue-specific differences in gating can be detected at the single-channel level. Single-channel currents were recorded in cell-attached and excised membrane patches at test potentials of -70 to -20 mV and at 9-11 degrees C. In both cell-attached and excised patches brain Na channel mean open time progressively increased from less than 1 ms at -70 mV to approximately 2 ms at -20 mV. Near threshold, single openings with dispersed latencies were observed. By contrast, in cell-attached patches, heart Na channel mean open time peaked near -50 mV, was three times brain Na channel mean open time, and declined continuously to approximately 2 ms at -20 mV. Near threshold, openings occurred frequently usually as brief bursts lasting several milliseconds and rarely as prolonged bursts lasting tens of milliseconds. Unlike what occurs in brain tissue where excision did not change gating, in excised heart patches both the frequency of prolonged bursting and the mean open time of single units increased markedly. Brain and cardiac Na channels can therefore be distinguished on the basis of their mean open times and bursting characteristics.     

Description of modal gating of the cardiac calcium release channel in planar lipid membranes.

Single channel activity of the cardiac ryanodine-sensitive calcium-release channel in planar lipid membranes was studied in order to elucidate the calcium-dependent mechanism of its steady-state behavior. The single channel kinetics, observed with Cs+ as the charge carrier at different activating (cis) Ca2+ concentrations in the absence of ATP and Mg2+, were similar to earlier reports and were extended by analysis of channel modal behavior. The channel displayed three episodic levels of open probability defining three gating modes: H (high activity), L (low activity), and I (no activity). The large difference in open probabilities between the two active modes resulted from different bursting patterns and different proportions of two distinct channel open states. I-mode was without openings and can be regarded as the inactivated mode of the channel; L-mode was composed of short and sparse openings; and H-mode openings were longer and grouped into bursts. Modal gating may explain calcium-release channel adaptation (as transient prevalence of H-mode after Ca2+ binding) and the inhibitory effects of drugs (as stabilization of mode I), and it provides a basis for understanding the regulation of calcium release.     

Voltage-dependent modulation of single N-Type Ca2+ channel kinetics by receptor agonists in IMR32 cells.

The voltage-dependent inhibition of single N-type Ca(2+) channels by noradrenaline (NA) and the delta-opioid agonist D-Pen(2)-D-Pen (5)-enkephalin (DPDPE) was investigated in cell-attached patches of human neuroblastoma IMR32 cells with 100 mM Ba(2+) and 5 microM nifedipine to block L-type channels. In 70% of patches, addition of 20 microM NA + 1 microM DPDPE delayed markedly the first channel openings, causing a four- to fivefold increase of the first latency at +20 mV. The two agonists or NA alone decreased also by 35% the open probability (P(o)), prolonged partially the mean closed time, and increased the number of null sweeps. In contrast, NA + DPDPE had little action on the single-channel conductance (19 versus 19.2 pS) and minor effects on the mean open time. Similarly to macroscopic Ba(2+) currents, the ensemble currents were fast activating at control but slowly activating and depressed with the two agonists. Inhibition of single N-type channels was effectively removed (facilitated) by short and large depolarizations. Facilitatory pre-pulses increased P(o) significantly and decreased fourfold the first latency. Ensemble currents were small and slowly activating before pre-pulses and became threefold larger and fast decaying after facilitation. Our data suggest that slowdown of Ca(2+) channel activation by transmitters is mostly due to delayed transitions from a modified to a normal (facilitated) gating mode. This single-channel gating modulation could be well simulated by a Monte Carlo method using previously proposed kinetic models predicting marked prolongation of first channel openings.     

A note on correlations in single ion channel records

General expressions are derived for the correlation coefficients between the length of an opening and that of the nth subsequent opening for a single ion channel. Analogous results are given for the correlation between shut times, and between an open time and subsequent shut times. An alternative derivation of the results of Fredkin et al. (in Proc. Berkeley Conf. in honor of Neyman & Kiefer, vol. 1, pp. 269-289 (1985] is given, and their results are extended to the case where openings occur in bursts. Expressions are given for the correlation between the first and nth opening in a burst, between the lengths of bursts, and between the number of openings per burst. Each of these sorts of correlation can give information about the connections that exist between the various states of the system; interpretations of the correlations are discussed. Expressions are derived for the distributions of the nth open time, shut time, burst length, etc. following the application of a perturbation (e.g. a voltage jump or a concentration jump). It is shown that these distributions will all be the same (namely the equilibrium distribution) only in the case where the openings, burst lengths, etc. are not correlated. Certain reaction schemes predict a component in the distribution of the number of openings per burst that has a unit mean (i.e. a component of isolated single openings). For some schemes this component is predicted to have zero amplitude, in principle, whereas in others it may be quite prominent. The presence or absence of this component can give information about the way in which the various states of the system are connected. The interpretation in terms of mechanism is discussed.     

Gating of the ATP-sensitive K+ channel by a pore-lining phenylalanine residue

ATP-sensitive K(+) (K(ATP)) channels are gated by intracellular ATP, proton and phospholipids. The pore-forming Kir6.2 subunit has all essential machineries for channel gating by these ligands. It is known that channel gating involves the inner helix bundle of crossing in which a phenylalanine residue (Phe168) is found in the TM2 at the narrowest region of the ion-conduction pathway in the Kir6.2. Here we present evidence that Phe168-Kir6.2 functions as an ATP- and proton-activated gate via steric hindrance and hydrophobic interactions. Site-specific mutations of Phe168 to a small amino acid resulted in losses of the ATP- and proton-dependent gating, whereas the channel gating was well maintained after mutation to a bulky tryptophan, supporting the steric hindrance effect. The steric hindrance effect, though necessary, was insufficient for the gating, as mutating Phe168 to a bulky hydrophilic residue severely compromised the channel gating. Single-channel kinetics of the F168W mutant resembled the wild-type channel. Small residues increased P(open), and displayed long-lasting closures and long-lasting openings. Kinetic modeling showed that these resulted from stabilization of the channel to open and long-lived closed states, suggesting that a bulky and hydrophobic residue may lower the energy barrier for the switch between channel openings and closures. Thus, it is likely that the Phe168 acts as not only a steric hindrance gate but also potentially a facilitator of gating transitions in the Kir6.2 channel.     

Cardiac Na currents and the inactivating, reopening, and waiting properties of single cardiac Na channels

Tetrodotoxin (TTX)-sensitive Na currents were examined in single dissociated ventricular myocytes from neonatal rats. Single channel and whole cell currents were measured using the patch-clamp method. The channel density was calculated as 2/micron 2, which agreed with our usual finding of four channels per membrane patch. At 20 degrees C, the single channel conductance was 20 pS. The open time distributions were fit by a single-exponential function with a mean open time of approximately 1.0 ms at membrane potentials from -60 to -40 mV. Averaged single channel and whole cell currents were similar when scaled and showed both fast and slow rates of inactivation. The inactivation and activation gating shifted quickly to hyperpolarized potentials for channels in cell-attached as well as excised patches, whereas a much slower shift occurred in whole cells. Slowly inactivating currents were present in both whole cell and single channel current measurements at potentials as positive as -40 mV. In whole cell measurements, the potential range could be extended, and slow inactivation was present at potentials as positive as -10 mV. The curves relating steady state activation and inactivation to membrane potential had very little overlap, and slow inactivation occurred at potentials that were positive to the overlap. Slow inactivation is in this way distinguishable from the overlap or window current, and the slowly inactivating current may contribute to the plateau of the rat cardiac action potential. On rare occasions, a second set of Na channels having a smaller unit conductance and briefer duration was observed. However, a separate set of threshold channels, as described by Gilly and Armstrong (1984. Nature [Lond.]. 309:448), was not found. For the commonly observed Na channels, the number of openings in some samples far exceeded the number of channels per patch and the latencies to first opening or waiting times were not sufficiently dispersed to account for the slowly inactivating currents: the slow inactivation was produced by channel reopening. A general model was developed to predict the number of openings in each sample. Models in which the number of openings per sample was due to a dispersion of waiting times combined with a rapid transition from an open to an absorbing inactivated state were unsatisfactory and a model that was more consistent with the results was identified.     

Coupling interactions between voltage sensors of the sodium channel as revealed by site-specific measurements

The voltage-sensing S4 segments in the sodium channel undergo conformational rearrangements in response to changes in the electric field. However, it remains unclear whether these structures move independently or in a coordinated manner. Previously, site-directed fluorescence measurements were shown to track S4 transitions in each of the four domains. Here, using a similar technique, we provide direct evidence of coupling interactions between voltage sensors in the sodium channel. Pairwise interactions between S4s were evaluated by comparing site-specific conformational changes in the presence and absence of a gating perturbation in a distal domain. Reciprocity of effect, a fundamental property of thermodynamically coupled systems, was measured by generating converse mutants. The magnitude of a local gating perturbation induced by a remote S4 mutation depends on the coupling strength and the relative equilibrium positions of the two voltage sensors. In general, our data indicates that the movement of all four voltage sensors in the sodium channel are coupled to a varying extent. Moreover, a gating perturbation in S4-DI has the largest effect on the activation of S4-DIV and vice versa, demonstrating an energetic linkage between S4-DI and S4-DIV. This result suggests a physical mechanism by which the activation and inactivation process may be coupled in voltage-gated sodium channels. In addition, we propose that cooperative interactions between voltage sensors may be the mechanistic basis for the fast activation kinetics of the sodium channel.     

Nonmodal gating of cardiac calcium channels as revealed by dihydropyridines

The hypothesis that dihydropyridine (DHP)-sensitive calcium channels have three distinct modes of gating has been examined. The major prediction is that the relative frequencies among modes depend on DHP concentration while the kinetics within a mode do not. We tested this by studying whole-cell and single-channel calcium currents in neonatal rat and adult guinea pig cardiac myocytes in different concentrations of several DHPs. In the absence of DHPs calcium currents declined with time but the kinetics, which are the focus of this study, were unchanged. Open-time frequency distributions had insignificant numbers of prolonged openings and were well fit by single tau's. Agonist DHP stereoisomers produced concentration-dependent changes in whole-cell tail current tau's. The frequency distribution of single calcium channel current open times became biexponential and the tau's were concentration dependent. The average number of openings per trace of channels with customary open times increased with increases in DHP concentration. Latencies to first opening for the customary openings and for prolonged openings were shorter in the presence of DHPs. A second larger conductance is another important feature of DHP-bound single calcium channels. Thus DHPs not only caused prolonged openings; they produced numerous changes in the kinetics of customary openings and increased channel conductance. It follows that these effects of DHPs do not support the hypothesis of modal gating of calcium channels. The mode model is not the only model excluded by the results; models in which DHPs are allowed to act only or mainly on open states are excluded, as are models in which the effects are restricted to inactivated states. We suggest a different type of model in which cooperative binding of DHPs at two sites produces the essential changes in kinetics and conductance.     

Gating of the ATP-sensitive K channel by a pore-lining phenylalanine residue

ATP-sensitive K⁺ (K_ATP) channels are gated by intracellular ATP, proton and phospholipids. The pore-forming Kir6.2 subunit has all essential machineries for channel gating by these ligands. It is known that channel gating involves the inner helix bundle of crossing in which a phenylalanine residue (Phe168) is found in the TM2 at the narrowest region of the ion-conduction pathway in the Kir6.2. Here we present evidence that Phe168-Kir6.2 functions as an ATP- and proton-activated gate via steric hindrance and hydrophobic interactions. Site-specific mutations of Phe168 to a small amino acid resulted in losses of the ATP- and proton-dependent gating, whereas the channel gating was well maintained after mutation to a bulky tryptophan, supporting the steric hindrance effect. The steric hindrance effect, though necessary, was insufficient for the gating, as mutating Phe168 to a bulky hydrophilic residue severely compromised the channel gating. Single-channel kinetics of the F168W mutant resembled the wild-type channel. Small residues increased P_open, and displayed long-lasting closures and long-lasting openings. Kinetic modeling showed that these resulted from stabilization of the channel to open and long-lived closed states, suggesting that a bulky and hydrophobic residue may lower the energy barrier for the switch between channel openings and closures. Thus, it is likely that the Phe168 acts as not only a steric hindrance gate but also potentially a facilitator of gating transitions in the Kir6.2 channel.     

Stochastic simulation of calcium microdomains in the vicinity of an L-type calcium channel

We study numerically the local dynamics of the intracellular calcium concentration in the vicinity of a voltage- and calcium-dependent plasma membrane L-type calcium channel. To account for the low number of Ca²⁺ ions and buffer molecules present in sub-femtoliter volumes, we use an exact stochastic simulation algorithm including diffusion. We present a novel, unified simulation method that implements reaction-diffusion events of Ca²⁺ ions and buffer molecules, stochastic ion channel gating and channel conductance as a multivariate Markov process. For fixed-voltage dynamics, e.g. under voltage-clamp conditions, it is shown that voltage-sensitive channel-gating steps can be incorporated exactly. We compare multi- and single-voxel geometries and show that the single-voxel approach leads to almost identical first- and second-order moments, at much lower computation time. Numerical examples illustrate the variability in local Ca²⁺ fluctuations as induced by bursts of channel openings in response to membrane depolarisations. Finally, by introducing calmodulin as a link, it is shown how this variability is passed on to downstream signalling pathways. The method may prove useful to study calcium microdomains and calcium-regulated processes triggered by membrane depolarisations as evoked by, e.g., viral channel-forming proteins during virus-host cell interactions.     

GluN1-specific redox effects on the kinetic mechanism of NMDA receptor activation

NMDA receptors are glutamate-activated ion channel complexes central to the functioning of the mammalian nervous system. Opening of the NMDA receptor ion channel pore is initiated by agonist-induced conformational changes in the extracellular ligand-binding domain (LBD) but the dynamic mechanism of this process remains unresolved. We studied how a disulfide bond in the obligatory GluN1 subunit—the sole site of redox modulation in NMDA receptors—controls this activation gating mechanism. This disulfide bond is located in the hinge region of the LBD, and presumably constrains agonist-induced cleft closure of the clamshell-like LBD. Elimination of this bond, by either DTT-mediated reduction or mutagenesis, enhances gating efficiency such that pore opening now occurs with higher frequency and longer duration. The most prominent effect was to shift opening modes to long duration openings reminiscent of a high P_o gating mode that the NMDA receptor exhibits under ambient oxidizing conditions. In terms of preopen gating steps, elimination of this bond has effects only on the fast gating step consistent with this step being GluN1-specific and reflecting GluN1 gating movements immediately before channel opening. Overall, our results suggest that the dynamics of the GluN1 LBD have strong effects on late pore opening steps including regulating the duration of pore opening. This redox-mediated gating modulation could be an underlying mechanism of NMDA receptor malfunction in redox-dependent disease states and presents a potential target of pharmacologic action.     

A unified view of cystic fibrosis transmembrane conductance regulator (CFTR) gating: combining the allosterism of a ligand-gated channel with the enzymatic activity of an ATP-binding cassette (ABC) transporter

The cystic fibrosis transmembrane conductance regulator (CFTR) is a unique ion channel in that its gating is coupled to an intrinsic enzymatic activity (ATP hydrolysis). This enzymatic activity derives from the evolutionary origin of CFTR as an ATP-binding cassette transporter. CFTR gating is distinct from that of a typical ligand-gated channel because its ligand (ATP) is usually consumed during the gating cycle. However, recent findings indicate that CFTR gating exhibits allosteric properties that are common to conventional ligand-gated channels (e.g. unliganded openings and constitutive mutations). Here, we provide a unified view of CFTR gating that combines the allosterism of a ligand-gated channel with its unique enzymatic activity.     

Activation of the Drosophila TRP and TRPL channels requires both Ca2+ and protein dephosphorylation

The Transient Receptor Potential (TRP) proteins constitute a large and diverse family of channel proteins, which is conserved through evolution. TRP channel proteins have critical functions in many tissues and cell types, but their gating mechanism is an enigma. In the present study patch-clamp whole-cell recordings was applied to measure the TRP- and TRP-like (TRPL)-dependent currents in isolated Drosophila ommatidia. Also, voltage responses to light and to metabolic stress were recorded from the eye in vivo. We report new insight into the gating of the Drosophila light-sensitive TRP and TRPL channels, by which both Ca2+ and protein dephosphorylation are required for channel activation. ATP depletion or inhibition of protein kinase C activated the TRP channels, while photo-release of caged ATP or application of phorbol ester antagonized channels openings in the dark. Furthermore, Mg(2+)-dependent stable phosphorylation event by ATPgammaS or protein phosphatase inhibition by calyculin A abolished activation of the TRP and TRPL channels. While a high reduction of cellular Ca2+ abolished channel activation, subsequent application of Ca2+ combined with ATP depletion induced a robust dark current that was reminiscent of light responses. The results suggest that the combined action of Ca2+ and protein dephosphorylation activate the TRP and TRPL channels, while protein phosphorylation by PKC antagonized channels openings.     

Hodgkin-Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.