Proc. R. Soc. Lond. B. 199, 231-262 (1977)
The theoretical background is presented for (a) the relaxation
towards equilibrium of drug-induced membrane currents, and (b)
the fluctuations of membrane current about its equilibrium value
that originate in the opening and closing of membrane ion
channels. General expressions are given that relate the
relaxation current, autocovariance function, spectral density
function, fluctuation variance and mean open channel lifetime to
the rate constants and single channel conductances for any theory
of drug action based on the law of mass action. The question of
how much can be validly inferred from experimental spectra that
appear to have only one component is discussed. The equations are
illustrated by their application to some simple theories of drug
action that are currently under consideration.
[get pdf]
Phil. Trans. R. Soc. Lond. B300, 1-59 (1982)
Characteristics of observed bursts of single channel openings were derived recently for two particular ion channel mechanisms. In this paper these methods are generalised so that the observable characteristics of bursts can be calculated directly for any mechanism that has transition probabilities that are independent of time as long as the process is at equilibrium or is maintained in a steady state by an energy supply.
General expressions are given for the distributions of the open time, the number of openings per burst, the total open time per burst, the gaps within and between bursts, and so on.
With the aid of these general results a single computer program can be written that will provide numerical values for such distributions for any postulated mechanism, given only the transition rates between the various states.
The results are illustrated by a numerical example of mechanism in which two agonist molecules can bind sequentially, and either singly or doubly occupied receptor ion channels may open.
The analogous theory is also given for the case
where bursts of channel openings are grouped into clusters; many
of the results bear a close analogy with those found for simple
bursts.
[get pdf]
INTRODUCTION
1. BASIC ASSUMPTIONS AND DEFINITIONS
(a) Basic definitions 5
(b) Some possible mechanisms 7
(i) A simple agonist mechanism 7
(ii) A simple open ion channel block mechanism 8
(iii) A more complex agonist mechanism 9
(c) Theoretical background 9
(i) General theory 9
(ii) Numerical evaluation of results 11
(iii) Another approach 12
(d) Definition of bursts in practice 13
2. SOME STANDARD METHODS 13
(a) Probability of occurrence of a specified sequence of events 13
(b) The duration of a specified sequence of events 14
3. THE ANALYSIS OF BURSTS 15
(a) The start of a burst 15
(i) Some special cases 16
(b) The end of a burst 16
(c) The number of openings per burst 17
(i) The case of a single open state 18
(ii) Distributions conditional on starting state 18
(iii) Simple examples 18
(d) The burst length 18
(i) Distributions conditional on the starting state 19
(ii) The case of a single open state 19
(e) The total open time per burst 20
(i) A simple example 20
(ii) The case of a single open state 20
(f) The total shut time per burst 21
(i) The case of a single state in B 21
(ii) Simple example 23
(g) The length of individual openings 23
(i) Distribution of the kth opening in a burst with r openings 24
(ii) Distribution of the kth opening in a burst (regardless of r) 25
(iii) Overall distribution of the length of an opening 25
(h) Shut periods (gaps) within bursts 26
(i) Distribution of the kth gap in a burst with r openings 27
(ii) Distribution of the kth gap in a burst (regardless of r) 27
(iii) Overall distribution of gaps within bursts 28
(i) Shut periods between bursts 28
(i) Another view of the initial vector f _{b} 30
(j) The distribution of all shut times 30
(i) Inferences about bursts from the distribution of all shut times30
4. A NUMERICAL EXAMPLE 31
(a) The mechanism and its parameter values 31
(i) Equiplibrium state occupancies 33
(ii) The Q and II matrices 33
(iii) The predicted noise spectrum 34
(b) The start of a burst 34
(c) The number of openings per burst 34
(d) The burst length 35
(e) The total open time per burst 35
(f) The total shut time per burst 36
(g) The length of individual openings 36
(h) The length of gaps within a burst 38
(i) The gaps between bursts 39
(j) The distribution of all shut times 39
(i) Inferences about bursts from the distribution of all shut times 40
5. THE ANALYSIS OF CLUSTERS OF BURSTS 41
(a) Basic definitions 41
(b) Two approaches to clusters 42
(i) Changing subscripts 42
(ii) Definition of new transition matrices 42
(c) The start and end of a cluster 43
(d) The number of openings per cluster 44
(e) The number of bursts per cluster 44
(f) The number of openings per burst 44
(g) The cluster length 45
(h) The total open time per cluster 45
(i) The total shut time per cluster 46
(j) The length of individual openings 46
(i) The vth opening in a cluster with w openings 46
(ii) The kth opening in a burst with r openings, that is the mth
burst in a cluster of n bursts 47
(iii)The kth opening in a burst with r openings 47
(iv) The overall distribution of open lifetime 47
(k) All Gaps within a cluster 48
(i) the vth gap in acluster with w openings 48
(ii) Overall distribution of gaps within a cluster 48
(l) Gaps between bursts within a cluster 48
(i) The mth gap between bursts in a cluster with n bursts 48
(ii) Overall distribution of gaps between bursts within clusters 48
(m) Gaps between clusters 49
(n) All shut times 49
(o) The bursts length 49
(i) the mth burst in a cluster with n bursts 49
(ii) The overall distribution of burst length 49
(p) Other overall distributions of burst characteristics 50
(q) The total burst time per cluster 50
(r) The total gap between burst time per cluster 50
6. DISCUSSION
(a) Inferences from the number of components 51
(b) Inferences from the time spent in a single state or set of states 52
(c) Inferences from the number of openings per burst 53
(d) Inferences from the total open time per burst 53
7. APPENDIX 1. SOME USEFUL RESULTS FOR ANALYSIS OF BURSTS
(a) Miscellaneous results 54
(b) The relationship between f _{o} and f _{b} 54
(c) Inversion of partitioned matrix 55
(i) Application to the distribution of burst length 56
8. APPENDIX 2. SOME USEFUL RESULTS FOR ANALYSIS OF CLUSTERS
(a) Miscellaneous results 56
(b) The relation between the various initial vectors 57
REFERENCES 58<
[get pdf]
J Physiol (Lond) 1985 Dec;369:501-557
The fine structure of ion-channel
activations by junctional nicotinic receptors in adult frog
muscle fibres has been investigated. The agonists used were
acetylcholine (ACh), carbachol (CCh), suberyldicholine (SubCh)
and decan-1,10-dicarboxylic acid dicholine ester (DecCh).
Individual activations (bursts) were interrupted by short closed
periods; the distribution of their durations showed a major fast
component ('short gaps') and a minor slower component
('intermediate gaps'). The mean duration of both short and
intermediate gaps was dependent on the nature of the agonist. For
short gaps the mean durations (microseconds) were: ACh, 20;
SubCh, 43; DecCh, 71; CCh, 13. The mean number of short gaps per
burst were: ACh, 1.9; SubCh, 4.1; DecCh, 2.0. The mean number of
short gaps per burst, and the mean number per unit open time,
were dependent on the nature of the agonist, but showed little
dependence on agonist concentration or membrane potential for
ACh, SubCh and DecCh. The short gaps in CCh increased in
frequency with agonist concentration and were mainly produced by
channel blockages by CCh itself. Partially open channels
(subconductance states) were clearly resolved rarely (0.4% of
gaps within bursts) but regularly. Conductances of 18% (most
commonly) and 71% of the main value were found. However, most
short gaps were probably full closures. The distribution of burst
lengths had two components. The faster component represented
mainly isolated short openings that were much more common at low
agonist concentrations. The slower component represented bursts
of longer openings. Except at very low concentrations more than
85% of activations were of this type, which corresponds to the
'channel lifetime' found by noise analysis. The frequency of
channel openings increased slightly with hyperpolarization. The
short gaps during activations were little affected when (a) the
[H^{+}]_{o} or [Ca^{2+}]_{o} were
reduced to 1/10th of normal, (b) when extracellular Ca^{2+}
was replaced by Mg^{2+}, (c) when the [Cl^{-}]_{i
}was raised or (d) when, in one experiment on an isolated
inside-out patch, the normal intracellular constituents were
replaced by KCl. Reduction of [Ca^{2+}]_{O} to
1/10 of normal increased the single-channel conductance by 50%,
and considerably increased the number of intermediate gaps. No
temporal asymmetry was detectable in the bursts of openings.
Positive correlations were found between the lengths of
successive apparent open times at low SubCh concentrations, but
no correlations between burst lengths were detectable. The
component of brief openings behaves, at low concentrations, as
though it originates from openings of singly occupied channels.
[get pdf]
Colquhoun, D., Hawkes, A.G. (1987) Proc. R. Soc. Lond. B 230,15-52.
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.
[get pdf]
[see Colquhoun, D., Hawkes, A. G., Merlushkin, A.
& Edmonds, B. (1997)
for a more complete description oj single channels after a jump]
[get pdf]
Philosophical Transactions of the Royal Society London A 354, 2555-2590.
[This is Mathematical and Physical Sciences series]
The openings and shuttings of individual ion channel molecules
can be modelled in terms of an underlying Markov process with
discrete states in continuous time. In practice, some of the open
times, and/or shut times, are too short to be detected reliably,
making the durations of some of these intervals appear to be
longer than they really are. Under certain assumptions about how
this happens, the probability densities of these apparent times
have previously been obtained. It has been shown that the ability
to distinguish between alternative postulated reaction mechanisms
can be greatly improved by considering bivariate distributions.
In this paper we obtain joint distributions, and hence
conditional distributions, of adjacent apparent open and shut
times. Numerical examples illustrate what insight these
conditional distributions may provide about the underlying
mechanisms. Bivariate distributions are readily generalized to
multivariate distributions which enable the likelihood for an
entire single-channel recording to be computed, and hence
efficient maximum likelihood estimates for the mechanism’s
rate constants can be obtained. Numerical examples of such
fitting are given.
[get pdf]
Philosophical Transactions of the Royal Society London A. 355, 1743-1786 (1997)
[This is Mathematical and Physical Sciences series]
Experiments are often performed to study the behaviour of a single ion channel in response to a perturbation produced by a step change (‘jump’) in a variable that influences its equilibrium position, for example a voltage jump or jump in agonist concentration. It is also common to apply a rectangular pulse (consisting of an on jump followed by an off jump); for example brief concentration pulses are used to mimic synaptic transmission.
Assuming a general Markov mechanism for channel dynamics, we
obtain theoretical probability distributions of observable
characteristics that describe the non-stationary behaviour of
single ion channels which are subject to a jump, or to a pulse of
finite duration. These characteristics are such things as open
times, shut times, first latency, burst length and length of
activation. We concentrate particularly on jumps to or from a
zero level of agonist, which necessitates some modification to
the usual arguments to cope with having some absorbing sets of
states. Where possible, we include results which make allowance
for the phenomenon of time interval omission, whereby some short
intervals may be missed due to imperfect resolution of the
recording method. A numerical example is studied in detail.
[get pdf]
1. Introduction and background 1744
(a) Introduction 1744
(b) The ideal case 1745
(c) The case of limited time resolution 1747
2. Single channels following a jump to zero agonist concentration 1748
(a) The fraction of channels that fail to open after t = 0 1749
(b) Distribution of the number of openings after t = 0 1750
(c) Distributions of the lengths of openings 1752
(d) Distributions of the lengths of shut times 1753
(e) The length of the burst 1755
(f) The length of the entire activation 1756
(g) The macroscopic time course 1757
(h) Distribution of total open time per burst 1757
3. Response of single channel to a pulse of agonist 1758
(a) Recording from the end of the pulse 1758
(b) Recording from the start of the pulse 1758
4. Allowing for time interval omission 1764
(a) Recording from the start of the pulse 1764
(b) Recording from the end of the pulse 1766
(c) Response to a single jump 1767
5. A numerical example 1769
(a) Channel behaviour after a jump to zero concentration 1770
(b) Response to a pulse of agonist - recording from the end of the pulse 1776
(c) Single jump from zero concentration 1778
(d) Response to a pulse: recording from the beginning of the pulse 1780
6. Discussion 1782
Appendix A 1783
References 1585
[get pdf]
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