Multivariate semi-Markov analysis of burst properties of multiconductance single ion channels

Patch clamp recordings from ion channels often show periods of repetitive activity, known as bursts, which are noticeably separated from each other by periods of inactivity. Depending on the type of channel, such recordings may exhibit (conductance) levels between the closed (zero) level and the fully open level. Properties of bursts are less subject to problems that arise from recording than are properties for individual sojourns at different levels, and study of bursting behaviour provides important information about the finer structure of the underlying channel gating process. For a general finite state space continuous-time Markov chain model allowing one or more nonzero conductance levels, the present paper establishes results about the semi-Markov structure of a single burst and of a sequence of bursts, and uses this in a unified approach to properties of both theoretical and empirical bursts. The distribution and moments of particular burst properties, including the total charge transfer, the total sojourn time and the total number of visits to specified conductance levels during a burst, are derived. Various extensions are also described.

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

Multivariate semi-Markov analysis of burst properties of multiconductance single ion channels

Patch clamp recordings from ion channels often show periods of repetitive activity, known as bursts, which are noticeably separated from each other by periods of inactivity. Depending on the type of channel, such recordings may exhibit (conductance) levels between the closed (zero) level and the fully open level. Properties of bursts are less subject to problems that arise from recording than are properties for individual sojourns at different levels, and study of bursting behaviour provides important information about the finer structure of the underlying channel gating process. For a general finite state space continuous-time Markov chain model allowing one or more nonzero conductance levels, the present paper establishes results about the semi-Markov structure of a single burst and of a sequence of bursts, and uses this in a unified approach to properties of both theoretical and empirical bursts. The distribution and moments of particular burst properties, including the total charge transfer, the total sojourn time and the total number of visits to specified conductance levels during a burst, are derived. Various extensions are also described.

Burst properties of a supergated double-barrelled chloride ion channel

The chloride selective channel from Torpedo electroplax, ClC-0, is the prototype of a large gene family of chloride channels that behave as functional dimers, with channel currents exhibiting two non-zero conductance levels. Each pore has the same conductance and is controlled by a subgate, and these have seemingly identical fast gating kinetics. However, in addition to the two subgates there is a single slower 'supergate' which simultaneously affects both channels. In the present paper, we consider a six state Markov model that is compatible with these observations and develop approximations as well as exact results for relevant properties of groupings of openings, known as bursts. Calculations with kinetic parameter values typical of ClC-0 suggest that even simple approximations can be quite accurate. Small deviations from the assumption of independence within the model lead to marked changes in certain predicted burst properties. This suggests that analysis of these properties may be helpful in assessing independence/non-independence of gating in this type of channel. Based on simulations of models of both independent and non-independent gating, tests using binomial distributions can lead to false conclusions in each situation. This is made more problematic by the difficulty of selecting an appropriate critical time in defining a burst empirically.

Markov modeling of allosteric drug effects on ion channels, with particular reference to neuronal nicotinic acetylcholine receptors

Allosteric mechanisms have been suggested for an increasing number of ion channel drug interactions, but often these ideas are not examined quantitatively through use of Markov models that would allow statistical estimation of proposed coupling effects. In this paper we illustrate, using properties relevant to the neuronal nicotinic acetylcholine receptor, how these models can be used to provide insight into the behavior of cooperative systems. Such models would then provide the basis for inferential studies with experimental data aimed at quantifying the magnitude of drug-induced changes on particular channel parameters. It is shown that even small changes in agonist binding affinity or channel gating are sufficient to produce biphasic modulatory drug effects in an allosteric model of nicotinic receptor activity.

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.

Modulatory drug action in an allosteric Markov model of ion channel behaviour: biphasic effects with access-limited binding to either a stimulatory or an inhibitory site

Concentration-dependent biphasic effects of drugs on ion channel activity have been reported in a variety of preparations, usually with stimulatory effects seen at low concentrations followed by increasingly dominant inhibition at higher levels. Such behaviour is often interpreted as evidence for the existence of separate modulatory drug binding sites. We demonstrate in this paper that it is possible for biphasic effects to be produced in an allosteric model of a ligand-activated ion channel, where diffusion-limited binding of the modulatory drug is restricted to either a stimulatory or an inhibitory site (but not both) because of steric overlap. The possibility of such an interaction mechanism should be kept in mind when interpreting experimental data if stoichiometric evidence from complementary techniques suggests that only one drug molecule is bound per receptor/ion channel complex.