Estimation of ion channel kinetics from fluctuations of macroscopic currents

For single channel recordings, the maximum likelihood estimation (MLE) of kinetic rates and conductance is well established. A direct extrapolation of this method to macroscopic currents is computationally prohibitive: it scales as a power of the number of channels. An approximated MLE that ignored the local time correlation of the data has been shown to provide estimates of the kinetic parameters. In this article, an improved approximated MLE that takes into account the local time correlation is proposed. This method estimates the channel kinetics using both the time course and the random fluctuations of the macroscopic current generated by a homogeneous population of ion channels under white noise. It allows arbitrary kinetic models and stimulation protocols. The application of the proposed algorithm to simulated data from a simple three-state model on nonstationary conditions showed reliable estimates of all the kinetic constants, the conductance and the number of channels, and reliable values for the standard error of those estimates. Compared to the previous approximated MLE, it reduces by a factor of 10 the amount of data needed to secure a given accuracy and it can even determine the kinetic rates in macroscopic stationary conditions.     

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

Mathematical and kinetic modeling of biofilm reactor based on ant colony optimization

Inverse estimation of model parameters via mathematical modeling route, known as inverse modeling (IM), is an attractive alternative approach to the experimental methods. This approach makes use of efficient optimization techniques in the course of solution of an inverse problem with the aid of measured data. In this study, a novel optimization method based on ant colony optimization (ACO), denoted by ACO-IM, is presented for inverse estimation of kinetic and film thickness parameters of biofilm models that describe an experimental fixed bed anaerobic reactor. The proposed optimization method for parameter estimation emulates the fact that ants are capable of finding the shortest path from a food source to their nest by depositing a trial of pheromone during their walk. The efficacy of the ACO-IM for numerical estimation of bio-kinetic parameters is demonstrated through its application for the anaerobic treatment of industry wastewater in a fixed bed biofilm process. The results explain the rigorousness of mathematical models, the form of kinetic and film thickness models and the type of packing to be used with the biofilm process for accurate determination of kinetic and film thickness parameters so as to ensure reliable predictive performance of the biofilm reactor models.     

Quantitative analysis of tetrapentylammonium-induced blockade of open N-methyl-D-aspartate channels

The blockade of open N-methyl-d-aspartate (NMDA) channels by tetrapentylammonium (TPentA) in acutely isolated rat hippocampal neurons was studied using whole-cell patch-clamp techniques. TPentA prevented the closure of the NMDA channel following what is known as the foot-in-the-door mechanism. Hooked tail currents appearing after termination of the agonist (aspartate) and TPentA coapplication were analyzed quantitatively according to the corresponding sequential kinetic model. Studies of the hooked tail current amplitude and the degree of the stationary current inhibition dependence on the blocker concentration led to a new method for estimation of fast foot-in-the-door blocker binding/unbinding rate constants. The application of this method to the NMDA channel blockade by TPentA allowed finding the values of its binding (1.48 microM(-1)s(-1)) and unbinding (14 s(-1)) rate constants. An analysis of the dependence of the electric charge carried during the hooked tail current on the blocker concentration led to a new method for estimation of the maximum NMDA channel open probability, P(0). The value of P(0) found in experiments with TPentA was 0.04.     

Poisson sampling-based inference for single ion channel data with time interval omission.

Patch-clamp recording allows investigations of the gating kinetics of single ion channels. Statistical analysis of kinetic data can enhance our understanding of channel gating at a molecular level. Experimental channel records suffer from time interval omission, i.e. failure to detect brief channel openings and closings. It is important to incorporate this phenomenon into statistical analyses of ion channel data. When time interval omission is ignored, the method of maximum likelihood can usually be used to estimate gating parameters from a single channel record. However, it is far more difficult to apply this method when time interval omission is incorporated. We present an alternative approach to parameter estimation based on Poisson sampling. A simulated homogeneous Poisson process is superimposed onto the channel record and inference is based on the numbers of points in successive open and closed sojourns, rather than on the sojourn times themselves. We describe the method for the two-state Markov model C<-->O, although it is applicable to more general models. Computer-simulated data are used to demonstrate the efficacy of the method. Modifications of the method are discussed briefly.     

Superposition properties of independent ion channels

Membrane patches usually contain several ion channels of a given type. However, most of the stochastic modelling on which data analysis (in particular, estimation of kinetic constants) is currently based, relates to a single channel rather than to multiple channels. Attempts to circumvent this problem experimentally by recording under conditions where channel activity is low are restrictive and can introduce bias; moreover, possibly important information on how multichannel systems behave will be missed. We have extended existing theory to multichannel systems by applying results from point process theory to derive some distributional properties of the various types of sojourn time that occur when a given number of channels are open in a system containing a specified number of independent channels in equilibrium. Separate development of properties of a single channel and the superposition of several such independent channels simplifies the presentation of known results and extensions. To illustrate the general theory, particular attention is given to the types of sojourn time that occur in a two channel system; detailed expressions are presented for a selection of models, both Markov and non-Markov.     

ESTIMATION OF THE KINETIC PARAMETERS OF UNIDIRECTIONAL TRANSPORT ACROSS THE BLOOD-BRAIN BARRIER

Abstract— The unidirectional transport of metabolic substrates from blood to brain may be defined in terms of Michaelis-Menten saturable ( K m, V _max) and non-saturable ( K _d) components of influx. Various computation procedures have been previously reported to estimate the kinetic parameters when an intracarotid injection technique is used. Transformations of the influx data which allow linear plots to obtain estimates were compared with estimates obtained directly from a best fit on a least means squares criterion for both experimental and simulated data. Large discrepancies were apparent between the various estimates of the kinetic parameters when an equal weight was given to transformed data. For pyruvate (21-day-old rats), K _m, values varied between 1.02 and 6.25 mM and V _max varied between 0.68 and 2.30 μmol g⁻¹ min⁻¹. The estimates were almost equivalent when pyruvate data was re-analysed using a weighting scheme based on the finding that the absolute value of the S.D. of influx increased in proportion to influx. It is recommended that estimates of kinetic parameters be obtained by an iterative, non-linear least squares method to fit appropriately weighted data directly.     

Opening rate of acetylcholine receptor channels.

The nicotinic acetylcholine (ACh) receptor is responsible for rapid conversion of chemical signals to electrical signals at the neuromuscular junction. Because the receptor and its ion channel are components of a single transmembrane protein, the time between ACh binding and channel opening can be minimized. To determine just how quickly the channel opens, we made rapid (100-400 microseconds) applications of 0.1-10 mM ACh to outside-out, multichannel membrane patches from BC3H-1 cells, while measuring the onset of current flow through the channels at 11 degrees C. Onset time is steeply dependent upon ACh concentration when channel activation is limited by binding of ACh (0.1-1 mM). At +50 mV, the 20-80% onset time reaches a plateau near 110 microseconds above 5 mM ACh as channel opening becomes rate limiting. Thus, we calculate the opening rate, beta = 12/ms, without reference to specific channel activation schemes. At -50 mV, the combination of a rapid, voltage-dependent block of channels by ACh with a finite solution exchange time distorts onset. To determine opening rate at -50 mV, we determine the kinetic parameters of block from "steady-state" current and noise analyses, assume a sequential model of channel activation/block, and numerically simulate current responses to rapid perfusion of ACh. Using this approach, we find beta = 15/ms. In contrast to the channel closing rate, the opening rate is relatively insensitive to voltage.     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Rapid ligand-regulated gating kinetics of single inositol 1,4,5-trisphosphate receptor Ca2+ release channels

The ubiquitous inositol 1,4,5-trisphosphate receptor (InsP(3)R) intracellular Ca(2+) release channel is engaged by thousands of plasma membrane receptors to generate Ca(2+) signals in all cells. Understanding how complex Ca(2+) signals are generated has been hindered by a lack of information on the kinetic responses of the channel to its primary ligands, InsP(3) and Ca(2+), which activate and inhibit channel gating. Here, we describe the kinetic responses of single InsP(3)R channels in native endoplasmic reticulum membrane to rapid ligand concentration changes with millisecond resolution, using a new patch-clamp configuration. The kinetics of channel activation and deactivation showed novel Ca(2+) regulation and unexpected ligand cooperativity. The kinetics of Ca(2+)-mediated channel inhibition showed the single-channel bases for fundamental Ca(2+) release events and Ca(2+) release refractory periods. These results provide new insights into the channel regulatory mechanisms that contribute to complex spatial and temporal features of intracellular Ca(2+) signals.     

Estimation of kinetic parameters in a structured yeast model using regularisation.

In this work, a procedure for estimating kinetic parameters in biochemically structured models was developed. The approach is applicable when the structure of a kinetic model has been set up and the kinetic parameters should be estimated. The procedure consists of five steps. First, initial values were found in or calculated from literature. Hereafter using sensitivity analysis the most sensitive parameters were identified. In the third step physiological knowledge was combined with the parameter sensitivities to manually tune the most sensitive parameters. In step four, a global optimisation routine was applied for simultaneous estimation of the most sensitive parameters identified during the sensitivity analysis. Regularisation was included in the simultaneous estimation to reduce the effect of insensitive parameters. Finally, confidence intervals for the estimated parameters were calculated. This parameter estimation approach was demonstrated on a biochemically structured yeast model containing 11 reactions and 37 kinetic constants as a case study.     

A numerical approach to ion channel modelling using whole-cell voltage-clamp recordings and a genetic algorithm

The activity of trans-membrane proteins such as ion channels is the essence of neuronal transmission. The currently most accurate method for determining ion channel kinetic mechanisms is single-channel recording and analysis. Yet, the limitations and complexities in interpreting single-channel recordings discourage many physiologists from using them. Here we show that a genetic search algorithm in combination with a gradient descent algorithm can be used to fit whole-cell voltage-clamp data to kinetic models with a high degree of accuracy. Previously, ion channel stimulation traces were analyzed one at a time, the results of these analyses being combined to produce a picture of channel kinetics. Here the entire set of traces from all stimulation protocols are analysed simultaneously. The algorithm was initially tested on simulated current traces produced by several Hodgkin-Huxley–like and Markov chain models of voltage-gated potassium and sodium channels. Currents were also produced by simulating levels of noise expected from actual patch recordings. Finally, the algorithm was used for finding the kinetic parameters of several voltage-gated sodium and potassium channels models by matching its results to data recorded from layer 5 pyramidal neurons of the rat cortex in the nucleated outside-out patch configuration. The minimization scheme gives electrophysiologists a tool for reproducing and simulating voltage-gated ion channel kinetics at the cellular level.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.     

Mechanism of blocking K+-channels with tetraethylammonium

Using the patch-voltage-clamp method action of tetraethylammonium on the fast (30 pS) and slow K+ channels was investigated. The slow K+ channels were presented by two types: with whole (30 pS) and decreased (20 pS) conductance. In all cases tetraethylammonium decreased the current magnitude and modified the channel kinetic parameters. Apparent blocking constants determined from the current decreasing are as 8-50 and 4-12 mM for the slow K+ channels with whole and decreased conductance, respectively, and 0.05-0.08 mM--for the fast K+ channel. The potential dependency of the blocking constants correlates with that of the channel conductance. Probability of the channel open state for the slow K+ channels decreases, and that for the fast K+ channel increases under application of tetraethylammonium. It is concluded that there are two sites of tetraethylammonium binding: the first site is into the channel pore, and the second one--into the regulatory centre responsible for the channel kinetic behaviour. Blocking of general conductance of the slow channels is accompanied by proportional decrease of the channel substate conductances without change of their number and cooperatively. Block of the fast K+ channel occurs without change of the channel elementary conductance but with decrease of the number of the channel substates and reversible violation of the channel transition cooperativity. The data are discussed from the point of the hypothesis on the channel clustery organization.     

Modeling Facilitative Sugar Transporters: Transitions Between Single and Double Ligand Occupancy of Multiconformational Channel Models Explain Anomalous Kinetics

The four-state simple carrier model (SCM) is employed to describe ligand translocation by diverse passive membrane transporters. However, its application to systems like facilitative sugar transporters (GLUTs) is controversial: unidirectional fluxes under zero-trans and equilibrium-exchange experimental conditions fit a SCM, but flux data from infinite-cis and infinite-trans experiments appear not to fit the same SCM. More complex kinetic models have been proposed to explain this ``anomalous' behavior of GLUTs, but none of them accounts for all the experimental findings. We propose an alternative model in which GLUTs are channels subject to conformational transitions, and further assume that the results from zero-trans and equilibrium-exchange experiments as well as trans-effects corresponds to a single-occupancy channel regime, whereas the results from the infinite-cis and infinite-trans experiments correspond to a regime including higher channel occupancies. We test the plausibility of this hypothesis by studying a kinetic model of a two-site channel with two conformational states. In each state, the channel can bind the ligand from only one of the compartments. Under single-occupancy, for conditions corresponding to zero-trans and equilibrium-exchange experiments, the model behaves as a SCM capable of exhibiting trans-stimulations. For a regime including higher degrees of occupancy and infinite-cis and infinite-trans conditions, the same channel model can exhibit a behavior qualitatively similar to a SCM, albeit with kinetic parameters different from those for the single-occupancy regime. Numerical results obtained with our model are consistent with available experimental data on facilitative glucose transport across erythrocyte membranes. Hence, if GLUTs are multiconformational channels, their particular kinetic properties can result from transitions between single and double channel occupancies. Received: 12 April 1995/Revised: 28 August 1995     

Temperature dependence of kinetic parameters of single potential dependent K+ channels in mollusc neurons. Similarity in the action of potential and temperature]

Using the patch-voltage-clamp method on excised membrane fragments from molluscan neurones temperature dependences of kinetic parameters of the fast and slow K(+)-channels were investigated in the temperature range 1 to 40 degrees C. Temperature dependences of probability of the channel open state (P0) for the slow and fast K(+)-channels are, generally, opposite, that is P0 increases for the slow channel and decreases for the fast channel with temperature. Similar dependences characterize durations of single channel open intervals (tau 0) and burst durations (t(p)). Durations of interburst and interpulse intervals (respectively, t(i) and tau) decrease for the slow channel and increase, in contrast, for the fast channel with temperature. For the channels of both types temperature dependences of P0 (as for other parameters) are essentially nonmonotonous. There are two local extrema, at least: for the slow K(+)-channel-maximum at 15 degrees C (minimum for the fast channel) and minimum at 20-25 degrees C (maximum for the fast channel). In some cases the number of local extrema may be greater than two. Some similarity in the action of temperature and membrane potential on the kinetic parameters was observed. For the slow K(+)-channel P0, tau 0 and t p increase with temperature and membrane potential. For the fast channel these parameters decrease at the same conditions. Moreover, for the channels of both types temperature dependences of the kinetic parameters are slightly pronounced at the potentials where potential dependences of the parameters are least. As a whole, temperature measurements showed that there are, possibly, several points of structural transitions (similar to phase transitions) in the temperature range 0 to 40 degrees C. Primarily, the kinetic parameters are determined by these transitions.     

Amplification of electromagnetic signals by ion channels

Cells may respond to the exposure of low-frequency electromagnetic fields with changes in cell division, ion influx, chemical reaction rates, etc. The chain of events leading to such responses is difficult to study, mainly because of extremely small energies associated with low-frequency fields, usually much smaller than the thermal noise level. However, the presence of stochastic systems (for instance, ion channels) provides a basis for signal amplification, and could therefore, despite the low signal-to-noise ratio of the primary response, lead to the transmission of weak signals along the signaling pathways of cells. We have explored this possibility for an ion channel model, and we present a theory, based on the formalism of stochastically driven processes, that relates the time averages of the ion channel currents to the amplitude and frequency of the applied signal. It is concluded from this theory that the signal-to-noise ratio increases with the number of channels, the magnitude of the rate constants, and the frequency response of the intracellular sensing system (for instance, a calcium oscillator). The amplification properties of the stochastic system are further deduced from numerical simulations carried out on the model, which consists of multiple identical two-state channels, and the behavior for different parameters is examined. Numerical estimates of the parameters show that under optimum conditions, even very weak low-frequency electromagnetic signals (<100 Hz and down to 100 microT) may be detected in a cellular system with a large number of ion channels.     

Three-dimensional differentiation of photo-autotrophic biofilm constituents by multi-channel laser scanning microscopy (single-photon and two-photon excitation)

A simple microscopic method to three-dimensionally differentiate between various members in photo-autotrophic biofilm systems is described. By dual-channel single-photon (confocal) and two-photon laser scanning microscopy, the signals in the red and far red channels as well as their combination can be simultaneously recorded. The method takes advantage of the autofluorescent signal of cyanobacteria-recorded in the red and far red channel and the autofluorescent signal of the green algae-recorded in the far red channel only. The differentiation is based on the specific pigment composition of cyanobacteria and green algae in combination with the appropriate filter settings for detection of the autofluorescent emission signals. The method allows the non-destructive, three-dimensional examination of fully hydrated interfacial microbial communities at high resolution as well as the clear separation between autofluorescent signals of cyanobacteria and green algae. Furthermore, there is a third option to record additional signals simultaneously such as nucleic acid stained bacteria, bacteria labeled with phylogenetic probes or glycoconjugates stained by using lectins. With state of the art laser scanning microscopes, even a fourth channel is available for recording yet another parameter, e.g. in the reflection (single-photon only) or fluorescence (single- and two-photon) mode. Thus the approach represents a convenient tool to study multiple parameters of complex photo-autotrophic biofilm systems.     

Kinetic parameters of single ion channels and stationary conductivities of phytotoxin modified lipid bilayers

As shown earlier, phytotoxins produced by Pseudomonas syringae pv. syringae form ion channels of "small" and "large" conductance when incorporated into planar lipid membranes. The multilevel conductance is due to cluster organization of the channels (Kaulin et al., 1998; Gurnev et al., 2002). In this study the kinetic parameters of syringomycin E (SRE) and syringostatin A (SSA) channels in negatively charged bilayer lipid membranes were estimated. The average time of open state of the small channels (t(s)(open)) did not depend on transmembrane voltage (in the range of +/- 200 mV). The channel characteristics differed between two phytotoxins: the t(s)(open) for the SRE-channels was much larger than that for SSA-channels. An energetic diagram with two non-conducting states illustrating the formation of the small channel is proposed to explain the voltage independence of the kinetic parameters. The probability for synchronous functioning of small channels with SSA was higher than that with SRE. To analyse the role of the clusters in the biological activities of SRE and SSA, we estimated the cluster contribution to a net transmembrane currents to be 60 and 90%, respectively.