Stochastic models for systems of interacting ion channels.

We consider a variety of Markov based models for systems of ion channels exhibiting dependence between channels. It is shown how many useful properties which may be calculated for an aggregated single-channel model, or a system of independent channels, can be extended to various types of interacting channel systems. Key structure and results from the theory of aggregated Markov processes are summarized in a convenient matrix form. These are then applied to the superposition of independent and dependent channels, including a patch of channels in a random environment, and a system of channels with spatial interactions. Calculations based on the resultant matrix expressions and intensity arguments can be implemented straightforwardly in a matrix-oriented package such as Matlab. The role of reversibility is also studied. A number of examples illustrate the strengths of the methods and enable numerical comparisons between the different types of systems.

Ion channel gating and time interval omission: statistical inference for a two-state Markov model.

Statistical inference is considered for a two-state Markov model of a single ion channel, when time interval omission is incorporated. A simple method of obtaining confidence sets for the mean open and closed sojourn times for the underlying single channel, based on the method-of-moments estimators, is presented. Time interval omission induces non-identifiability, in that the method-of-moments usually leads to two distinct estimates of the mean open and closed sojourn times, one corresponding to the true values and the other being an artefact of time interval omission. A new method of overcoming such non-identifiability on the basis of one single channel record is described. The methodology is illustrated by a numerical example.

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.

Single ion channel models incorporating aggregation and time interval omission.

We present a general theoretical framework, incorporating both aggregation of states into classes and time interval omission, for stochastic modeling of the dynamic aspects of single channel behavior. Our semi-Markov models subsume the standard continuous-time Markov models, diffusion models and fractal models. In particular our models allow for quite general distributions of state sojourn times and arbitrary correlations between successive sojourn times. Another key feature is the invariance of our framework with respect to time interval omission: that is, properties of the aggregated process incorporating time interval omission can be derived directly from corresponding properties of the process without it. Even in the special case when the underlying process is Markov, this leads to considerable clarification of the effects of time interval omission. Among the properties considered are equilibrium behavior, sojourn time distributions and their moments, and auto-correlation and cross-correlation functions. The theory is motivated by ion channel mechanisms drawn from the literature, and illustrated by numerical examples based on these.

Single-channel data and missed events: analysis of a two-state Markov model.

Patch-clamp recording permits investigation of the gating kinetics of single ion channels. Careful statistical analysis of kinetic data can yield clues as to the molecular events underlying channel gating. However, it is important that such analysis should take full account of the limitations that arise from the finite time resolution of patch-clamp recording techniques. Single-ion-channel data are generally interpreted in terms of Markov process models of channel gating mechanisms. Experimental channel records suffer from time interval omission, i.e. failure to detect brief channel openings and closings. This leads to an identifiability problem when analysing single-channel data, i.e. different gating mechanisms provide equally convincing descriptions of the same experimental data. We consider a two-state Markov model of receptor-channel gating in which the channel opening rate is proportional to the agonist concentration, C in equilibrium with OA. By using computer-simulated data, the approximate likelihood of the data is maximized to yield parameter estimates for the model. At a single agonist concentration there is an identifiability problem in that two pairs of parameter estimates are obtained. The 'true' parameter estimates cannot be distinguished from the 'false' ones. By considering data corresponding to a range of agonist concentrations one may identify the 'true' parameter estimates as those that do not change as the agonist concentration is increased. Alternatively, one may identify the 'true' parameter estimates directly by maximizing a global likelihood, the latter being obtained by simultaneous consideration of data obtained at several different agonist concentrations.(ABSTRACT TRUNCATED AT 250 WORDS)

Markov, fractal, diffusion, and related models of ion channel gating. A comparison with experimental data from two ion channels.

The gating kinetics of single-ion channels are generally modeled in terms of Markov processes with relatively small numbers of channel states. More recently, fractal (Liebovitch et al. 1987. Math. Biosci. 84:37-68) and diffusion (Millhauser et al. 1988. Proc. Natl. Acad. Sci. USA. 85:1502-1507) models of channel gating have been proposed. These models propose the existence of many similar conformational substrates of the channel protein, all of which contribute to the observed gating kinetics. It is important to determine whether or not Markov models provide the most accurate description of channel kinetics if progress is to be made in understanding the molecular events of channel gating. In this study six alternative classes of gating model are tested against experimental single-channel data. The single-channel data employed are from (a) delayed rectifier K+ channels of NG 108-15 cells and (b) locust muscle glutamate receptor channels. The models tested are (a) Markov, (b) fractal, (c) one-dimensional diffusion, (d) three-dimensional diffusion, (e) stretched exponential, and (f) expo-exponential. The models are compared by fitting the predicted distributions of channel open and closed times to those observed experimentally. The models are ranked in order of goodness-of-fit using a boot-strap resampling procedure. The results suggest that Markov models provide a markedly better description of the observed open and closed time distributions for both types of channel. This provides justification for the continued use of Markov models to explore channel gating mechanisms.

A note on single-channel autocorrelation functions.

It is shown for time-reversible ion channel gating mechanisms that the sojourn time autocorrelation functions are necessarily nonnegative, decreasing, and convex. It is also shown, again for time-reversible mechanisms, that the lagged moments of all orders are determined by the first three moments. The application of these results to the statistical analysis of single-channel kinetics is briefly discussed.

Temporal clustering of ion channel openings incorporating time interval omission.

Ion channel gating mechanisms can be satisfactorily modelled by a time-reversible continuous-time Markov chain on a finite state space. The complete process is not observable, but rather the state space is partitioned into 'open' and 'closed' states corresponding to the receptor channel being open or closed, and it is only possible to observe whether the process is in an open or a closed state. Previous studies of locust muscle glutamate receptor channels have revealed single channel openings to be highly clustered in time. This clustering can be described by the ratio of the variance var N(t) to the mean E[N(t)] of the number of channel openings in a time interval of length t. In this paper we obtain expressions for (formula; see text) for the above aggregated Markov process. Applications of these expressions to a model for the locust muscle glutamate receptor channel show this aspect of the model to be reasonably consistent with experimental data. In practice very short sojourns in either the open or closed states will fail to be detected, a phenomenon known as time interval omission. Using a semi-Markov approach, we outline a general theoretical framework for analysing dynamic properties of aggregated Markov processes incorporating time interval omission. We illustrate the applicability of this framework by using it to find limt----infinity [[var N(t)]/E[N(t)]] theoretically, when time interval omission is incorporated. This allows us to study the robustness of limt----infinity [[var N(t)]/E[N(t)]] to time interval omission, as a measure of temporal clustering.

A survey of the problems and needs of homeless consumers of acute psychiatric services.

Efforts to alter the repeated use costly hospital-based psychiatric services and the underuse of community mental health services by some homeless adults have yielded few successes. To better understand these failed efforts, the authors interviewed 112 self-identified homeless recidivists in San Francisco on several demographic variables, the problems they face in living in the community, and the resources they feel they need to be able to remain in the community. Most striking among the findings was the low priority respondents accorded to the psychiatric and social services currently offered by community mental health agencies; instead the respondents often blamed their inability to avoid readmissions on their lack of basic resources for survival. The authors discuss the relevance of the findings for mental health research, funding policies, and programming.