Robert O. Edeson

Evidence for Cooperativity Between Nicotinic Acetylcholine Receptors in Patch Clamp Records

It is often assumed that ion channels in cell membrane patches gate independently. However, in the present study nicotinic receptor patch clamp data obtained in cell-attached mode from embryonic chick myotubes suggest that the distribution of steady-state probabilities for conductance multiples arising from concurrent channel openings may not be binomial. In patches where up to four active channels were observed, the probabilities of two or more concurrent openings were greater than expected, suggesting positive cooperativity. For the case of two active channels, we extended the analysis by assuming that 1) individual receptors (not necessarily identical) could be modeled by a five-state (three closed and two open) continuous-time Markov process with equal agonist binding affinity at two recognition sites, and 2) cooperativity between channels could occur through instantaneous changes in specific transition rates in one channel following a change in conductance state of the neighboring channel. This allowed calculation of open and closed sojourn time density functions for either channel conditional on the neighboring channel being open or closed. Simulation studies of two channel systems, with channels being either independent or cooperative, nonidentical or identical, supported the discriminatory power of the optimization algorithm. The experimental results suggested that individual acetylcholine receptors were kinetically identical and that the open state of one channel increased the probability of opening of its neighbor.

Statistical inference from single channel records: two-state Markov model with limited time resolution

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little consideration. This paper describes techniques of statistical inference, in particular likelihood methods, suitable for Markov models incorporating limited time resolution by means of a discrete detection limit. To simplify the analysis, attention is restricted to two-state models, although the methods have more general applicability. Non-uniqueness of the mean open-time and mean closed-time estimators obtained by moment methods based on single exponential approximations to the apparent open-time and apparent closed-time distributions has been reported. The present study clarifies and extends this previous work by proving that, for such approximations, the likelihood equations as well as the moment equations (usually) have multiple solutions. Such non-uniqueness corresponds to non-identifiability of the statistical model for the apparent quantities. By contrast, higher-order approximations yield theoretically identifiable models. Likelihood-based estimation procedures are developed for both single exponential and bi-exponential approximations. The methods and results are illustrated by numerical examples based on literature and simulated data, with consideration given to empirical distributions and model control, likelihood plots, and point estimation and confidence regions.

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.

Stochastic Modelling of a Single Ion Channel: An Alternating Renewal Approach with Application to Limited Time Resolution

Stochastic models of ion channels have been based largely on Markov theory where individual states and transition rates must be specified, and sojourn-time densities for each state are constrained to be exponential. This study presents an approach based on random-sum methods and alternating-renewal theory, allowing individual states to be grouped into classes provided the successive sojourn times in a given class are independent and identically distributed. Under these conditions Markov models form a special case. The utility of the approach is illustrated by considering the effects of limited time resolution (modelled by using a discrete detection limit, ξ) on the properties of observable events, with emphasis on the observed open-time (ξ-open-time). The cumulants and Laplace transform for a ξ-open-time are derived for a range of Markov and non-Markov models; several useful approximations to the ξ-open-time density function are presented. Numerical studies show that the effects of limited time resolution can be extreme, and also highlight the relative importance of the various model parameters. The theory could form a basis for future inferential studies in which parameter estimation takes account of limited time resolution in single channel records. Appendixes include relevant results concerning random sums and a discussion of the role of exponential distributions in Markov models.

Verapamil, neuromuscular transmission and the nicotinic receptor

The effect of verapamil on neuromuscular transmission was studied in the frog by analysing ionophoretic endplate current (iEPC) trains and the growth and decay phases of miniature endplate currents (mepcs). In addition, single channel data on the interaction of verapamil with the nicotinic acetylcholine receptor were obtained from cultured embryonic chick skeletal muscle cells. Verapamil caused both open and closed channel blockade in the iEPC trains. Mepc amplitude was decreased at low micromolar concentrations, and at higher concentrations there was also accelerated mepc decay indicating open channel blockade. The latter effect could not be explained by a sequential channel occlusion mechanism. Analysis of the mepc rising phase showed that low micromolar concentrations of the drug decreased the pool of receptors which could be activated. Single channel data confirmed the specific interaction of verapamil with the nicotinic receptor, showing closed channel blockade at low concentrations, and at higher levels the shortening of open channel lifetime. It is suggested that both forms of blockade may involve desensitization processes.

Kinetics of (+)‐tubocurarine blockade at the neuromuscular junction

1 Although (+)‐tubocurarine (Tc) is classically considered to be a competitive antagonist at the neuromuscular junction, kinetic details of the interaction remain unclear. 2 We studied the competitive action of Tc on the nicotinic receptor at the frog neuromuscular junction using a quantitative analysis of the generation phase of miniature endplate currents (m.e.p.cs) recorded in Ringer solution (20°C) under voltage clamp (–90 mV) in the absence or presence of 1–5 μm Tc. Under control conditions four neurotransmission parameters were estimated by non‐linear regression using a mathematical model of synaptic transmission incorporating transmitter release, diffusion, hydrolysis, receptor binding and channel gating. These parameters were then used in a further regression to estimate binding rate constants for Tc at the same endplate. Allowance was made for open channel block by Tc, which under the conditions of this study was only a small component of total blockade. 3 The results suggest that Tc binds to the two agonist recognition sites on the nicotinic receptor with equal affinity (stoichiometric KDs of 2.2 and 8.8 μm), and that most of the functional blockade at concentrations up to 5 μm is due to occupancy of only one site. 4 The association rate constant for Tc binding to sites on the nicotinic acetylcholine receptor appears to be very fast (k+ D = 8.9× 108 m−1 s−1) and comparable to that for acetylcholine (ACh). 5 In the brief time during which an m.e.p.c. is generated (≅ 200 μs), reversal of Tc blockade by transiently high concentrations of ACh seems to be kinetically limited.

Model Properties Underlying Non-Identifiability in Single Channel Inference

Models of ion channel kinetics subserve inferential methods applied to patch clamp data. For Markov models the density function of a sojourn time in a class of states is a mixture of exponentials. Determination of kinetic parameters from density functions may be complicated by non-uniqueness of solutions. This non-identifiability is investigated analytically for a class of two states, assuming detailed balance; relations between model properties, observable density parameters, and non-uniqueness are presented. The results are further developed in terms of similarity transform methods. Additional information provided by joint distributions is discussed. An example is given where identifiability of a model can be demonstrated explicitly. Attention is drawn to instances where the number of components in a density function may be misleading when used to infer the number of underlying states.

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.

Distribution of exponentiality in miniature endplate current decay

Exponential functions are commonly used to describe miniature endplate current (MEPC) decay; under control conditions a monoexponential is usually regarded as sufficient, whereas in the presence of some drugs a biexponential may be necessary. Using an automated fitting procedure which estimated exponential parameters and the period of decay from peak to baseline, a unimodal distribution of curvature was found for control MEPCs recorded in frog sartorius muscle. The majority of MEPCs were of monoexponential form, while the remainder were biexponential with either less or greater curvature than expected for a simple exponential (hypoexponential or hyperexponential, respectively). The proportion of MEPCs in each of the 3 groups was constant for a given endplate but varied between endplates. A possible explanation for this phenomenon could be differences in synaptic geometry within and between endplates. The increased curvature of individual MEPCs in the hyperexponential group was analyzed by assuming a sequential model for agonist blockade or desensitization, and calculating closing and reopening rate constants. These rate constants were altered by procaine and verapamil (100 microM) in a manner consistent with blockade of the acetylcholine receptor by enhancement of agonist-induced desensitization.

On the use of time constants as estimates of mean lifetimes in single channel data analysis

For Markov models of single channel kinetics, a sojourn time in a class of states has a density function which is usually a linear combination of exponential densities. There are many instances in the single channel literature where the time constants of exponentials fitted to sojourn time data have been used as estimated mean sojourn times in individual states, though the two may be very different. In the present study the nature and magnitude of this difference in the case of a two state class is illustrated analytically and numerically. The time constants should be viewed at best as approximations, possibly poor, to the estimated mean sojourn times. Estimates of kinetic parameters cannot in general be obtained explicitly from the fitted parameters of the density alone. However, this is shown to be possible in some special cases and enables direct estimation of, for example, the channel opening rate constant β (or an upper limit to the estimate of β in the case of multiple channels) in standard sequential three or four state models of nicotinic receptor kinetics, using only the fitted parameters of the closed-time density.