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Mathematical Biosciences 166 (2000) 23±44
www.elsevier.com/locate/mbs

Burst properties of a supergated double-barrelled chloride ion
channel
Yun Li a, Geo€rey F. Yeo b, Robin K. Milne a,*, Barry W. Madsen c,
Robert O. Edeson d
a

Department of Mathematics and Statistics, The University of Western Australia, Nedlands, Perth 6907, Australia
b
Mathematics and Statistics, DSE, Murdoch University, Murdoch, 6150, Australia
c
Department of Pharmacology, The University of Western Australia, Nedlands, Perth 6907, Australia
d
Department of Anaesthesia, Sir Charles Gairdner Hospital, Nedlands, Perth 6009, Australia
Received 18 November 1999; received in revised form 27 April 2000; accepted 15 May 2000

Abstract
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 a€ects 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 diculty of selecting an appropriate critical time in
de®ning a burst empirically. Ó 2000 Published by Elsevier Science Inc. All rights reserved.
Keywords: Markov models; Binomial testing; Channel dependence and independence; Fast and slow gating;
Multiconductance level burst theory; Simulation

*

Corresponding author. Tel.: +61-8 9380 3346; fax: +61-8 9380 1028.
E-mail address: milne@maths.uwa.edu.au (R.K. Milne).

0025-5564/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 2 6 - 2

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Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

1. Introduction
Membrane channels permeable to chloride ions have important roles in cellular function, such
as the regulation of epithelial ion transport, cell volume and excitability. The chloride channel
from the electric organ of Torpedo, ClC-0, is the prototype of a family of voltage-dependent
chloride channels found in many species [1]. Unlike most well-studied ion channels that have a
single-gated pore, ClC-0 has a `double-barrelled' structure consisting of two seemingly independent and identically gated pores. Ion ¯ux through each of these is controlled by a subgate together
with a `supergate' that can simultaneously block both pores [2,3]. ClC-0 is unusual also in that
permeant ion concentration a€ects gating, and activation of the subgates and supergate does not
change unidirectionally as a function of membrane potential; the subgates are activated by
membrane depolarisation and the supergate by hyperpolarisation [4]. At the usual membrane
potentials found in cells, the time scales of gating kinetics are often well separated, with the
subgates being fast (lifetimes of order several milliseconds) compared with the slower supergate
(of order 100 ms). In patch clamp experiments, this leads to channel openings occurring in
groupings known as bursts. These bursts have greater complexity than that seen in typical single
channel systems because they can exhibit two open conductance levels which are not a simple
superposition of independent components (a consequence of supergating). Existing theory on the

properties of bursts (e.g. [5]) has not generally been extended to such cases. Whilst it was unclear
for some time whether all gates acted independently of one another [6], more recent results appear
to support independence of the subgates [7,8]. However, many of the studies addressing this issue
have been based on tests for binomial distribution of level occupancy [2,3,9], and because these are
known to lack sensitivity under certain circumstances [10] results should be interpreted with
caution.
The aim of the present paper was to consider an appropriate Markov model for ClC-0, develop
theory for its bursting behaviour, and explore the use of derived properties in studying questions
of independence. Although the paper is focused on the chloride channel ClC-0, the models
considered and results obtained for these models may well be relevant for other types of channel.

2. Markov chain models
2.1. Background
Studies on ClC-0 [3,11] suggested a four state model was sucient to represent the fast and slow
gating seen in patch clamp data. A more general model having six states is shown in Fig. 1, the
four state scheme being a special case where states 4±6 are collapsed into one. Whereas this reduced four state model may well represent ClC-0 data under many circumstances, the unusual
feature of slow gating being activated by hyperpolarisation and fast gating by depolarisation
means that the clear separation in gating kinetics may become obscured at some membrane
potentials. For this reason we have chosen to study the more general case.
In this model O, C, o and c indicate that the supergate and subgates are open and closed,

respectively. Three observed conductance levels correspond to state 2 (two subgates and the supergate open, double conductance level), state 1 (either subgate and the supergate open, single

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

25

Fig. 1. Six state model for the ClC-0 channel, where O, C, o and c imply that the supergate and subgates are open
and closed, respectively. States 3±6 exhibit zero conductance, and states 1 and 2 unit and twice-unit conductance,
respectively.

conductance level) and states 3±6 (either the supergate closed or both subgates closed, or all three,
zero conductance level). Bursts of channel activity consist largely of alternating sojourns among
states 1, 2 and 3 as entry to any of states 4, 5 or 6 can usually be expected to terminate the burst.
The six states in Fig. 1 may be partitioned into three classes, O1 ˆ f1g and O2 ˆ f2g consisting
respectively of the open states with conductance levels 1 and 2, and C ˆ f3; 4; 5; 6g which is the
class of zero conductance states. (State labels for the open states were chosen to avoid confusion
with the notation for corresponding conductance levels.) This division into non-overlapping
classes of states, known as aggregation, is important in derivation of channel properties.
Throughout the paper, vectors and matrices appear in bold. Except where indicated otherwise,
all vectors are column vectors and T denotes transpose, this being often used to express row

vectors.
2.2. Basic theory
Suppose that the state of the channel is described by an irreducible continuous-time homogeneous Markov chain fX …t†g ˆ fX …t†; t P 0g having transition rate matrix Q ˆ ‰qij Š of the form

…2:1†

P
where di is the sum of non-diagonal entries of the ith row, i.e. di ˆ j6ˆi qij . The indicated zero
entries correspond to pairs of states between which direct transitions are not allowable under the
assumed model (Fig. 1). The partitioning shown for Q re¯ects the aggregation described above
(i.e. into levels 0, 1 and 2): for example, the submatrix Q00 governs transitions between states
within C, and Q01 governs transitions from states of C to O1 . Such a process is often called an
aggregated Markov chain. When the focus is on whether the channel is open, without regard to

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Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

the conductance level, the relevant partitioning is into two classes, O ˆ O1 [ O2 of open states and
C as above. The corresponding partition of Q can be written as



QOO QOC
;
…2:2†
Qˆ
QCO QCC
where QCC is the same as Q00 used in (2.1). Section 3.8 of [12] gives a brief introduction to the basic
theory of aggregated Markov chains. Further details can be found in various papers in the ion
channel literature [13±15] and in the probability literature [16,17]. The remainder of this section
summarizes aspects of the theory that are key to the developments in the present paper.
Transitions between states in this continuous-time Markov chain are determined by an associated discrete-time Markov chain, the jump chain, having transition (probability) matrix P with
diagonal entries all zero and o€-diagonal entries pij ˆ qij =di ; i 6ˆ j. Thus, for example, if the ClC-0
channel is in state 2, there is a probability p24 ˆ q24 =…q21 ‡ q24 † that its next transition is to state 4,
representing supergate closure. This is the probability of the channel moving directly from the
state with conductance level 2 to a state (which must be state 4) with conductance level 0.
Since the process fX …t†g has six states and is irreducible, it has an equilibrium distribution
the global balance equation(s) pT Q ˆ 0, where
p ˆ …p1 ; . . . ; p6 †T which can be obtained by solving P
0 is a vector with all zero entries, together with i pi ˆ 1. When the process is reversible, an

alternative approach to the equilibrium distribution
P is to solve the detailed balance equations
pi qij ˆ pj qji for all i; j ˆ 1; 2; . . . ; 6, together with i pi ˆ 1. The equilibrium distribution is also
the limiting distribution in the sense that its components satisfy pk ˆ limt!1 P fX …t† ˆ kg. Just as
the partition of the state space into O1 , O2 and C induces a corresponding partition (2.1) of Q, so it
. .
also induces a partition of the equilibrium distribution as pT ˆ …pT1 ..pT2 ..pT0 †. The partitioning
.
pT ˆ …pTO ..pTC † will also be used.
2.3. Sojourn-time distributions and means
Transitions between states in the continuous-time Markov chain are described by the jump
chain, which has transition matrix P. Conditional on the successive states visited, the durations of
the sojourns in these states are independent, with each sojourn-time in state i having an exponential distribution with parameter di (cf. [5, (24)]). Since there is a single open state in O1 , it
follows that the duration of a sojourn in O1 has an exponential distribution with mean l1 ˆ 1=d1 .
Similarly, the duration of a sojourn in O2 has an exponential distribution with mean l2 ˆ 1=d2 .
Because there is more than one state in C, the distribution of a C sojourn-time, i.e. the duration of
a sojourn in C, is not in general an exponential distribution (cf. [5, Section 3.3]) but a linear
combination (mixture) of four exponential distributions.
Throughout the remainder of this paper it is supposed that the underlying continuous-time
Markov chain is in equilibrium. Then, to determine the distribution of, for example, a sojourn-time

in the class C we need the probability distribution /C that, when the chain is in equilibrium, a visit
to the closed states C begins in the various states of C. It follows, for example from [13, (3.2)], that
/C ˆ pTO QOC =…pTO QOC 1†;
where 1 is a column vector with all elements unity. Similarly, /O ˆ
the vectors /C and /O are of length 4 and 2, respectively.

…2:3†
pTC QCO =…pTC QCO 1†.

Note that

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

27

Let Z A1 ;A2 ;...;As be the random vector whose s components are the durations of consecutive sojourns of an aggregated Markov chain in classes A1 ; A2 ; . . . ; As (with consecutive classes distinct) of
some state±space partition. Fredkin et al. [15, p. 209] (see also [18, Section 3]) show that the joint
probability density function of durations of consecutive sojourns of the aggregated chain in
classes A1 ; A2 ; . . . ; As is
…2:4†

fZ A1 ;A2 ;...;As …t1 ; . . . ; ts † ˆ /TA1 eQA1 A1 t1 QA1 A2 eQA2 A2 t2


QAsÿ1 As eQAs As ts …ÿQAs As †1;
P
j j
where t1 ; . . . ; ts P 0. Here eQAA t ˆ 1
jˆ0 t QAA =j! is the usual matrix exponential; see, for example,
[19, p. 169] or [5, Section 4.1.2]. The sequence of sojourns starts in class A1 according to the
stationary probabilities /A1 . After the ®nal sojourn, in class As , the chain must exit to another
class
P
B di€erent from As , which happens according to the transition rates …ÿQAs As †1 ˆ B6ˆAs QAs B 1,
where the sum is over all sets B, di€erent from As , in some state±space partition (see [16, p. 64]).
For each i, the chain moves from Ai to Ai‡1 with transition rates QAi Ai‡1 , and the distribution of the
duration of a sojourn in Ai is described by eQAi Ai ti .
The result (2.4) is fundamental in that many other (joint) densities, probabilities and moments
can be obtained easily from it, for example by integrating over an appropriate selection of the
variables t1 ; . . . ; ts . Using such an approach, it is possible to avoid Laplace transform methods in
favour of direct calculation based on densities. In such manipulations key results are
Z 1
tk eKt dt ˆ …ÿ1†k‡1 k!Kÿ…k‡1† …k ˆ 0; 1; 2; . . .†;
…2:5†

0

Z

1

eKt dt ˆ ÿeKu Kÿ1

…u P 0†;

…2:6†

u

which hold for any (®nite) square matrix K that has the real parts of all its (distinct) eigenvalues
(strictly) negative. See [16,20] for more details concerning these results. When the model is viewed
as a parametric statistical model the joint density (2.4) considered as a function of the parameters
is just the likelihood function based on observations on the successive class sojourn-times. Qin
et al. [21] used this likelihood function as a basis for estimation of transition rates in the
underlying continuous-time Markov model.

It follows from (2.4) that the probability density function (pdf) of the duration of a sojourn
in C is given by
fC …t† ˆ /TC exp…QCC t†…ÿQCC †1

…t P 0†:

Hence, using (2.5), the mean duration of a closed sojourn in C is
Z 1
E…TC † ˆ
tfC …t† dt ˆ /TC …ÿQÿ1
CC †1:
0

2.4. Transition probabilities between conductance levels
For i; j 2 f0; 1; 2g let Pij denote the probability of the channel moving from the ith to the jth
conductance level. Since there is a single state with conductance level 1, it follows that

28

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

P10 ˆ p13 ‡ p15 ˆ …q13 ‡ q15 †=d1 and P12 ˆ 1 ÿ P10 ˆ p12 ˆ q12 =d1 . Similarly, P20 ˆ p24 ˆ q24 =d2 and
P21 ˆ 1 ÿ P20 ˆ p21 ˆ q21 =d2 .
Expressions for the probabilities P01 and P02 are more dicult to determine since there are four
states in C. Consider the partition of Q as in (2.1), and take s ˆ 2, A1 ˆ C and A2 ˆ O1 in (2.4).
Then the density function of Z 01 , the vector whose components are the durations of two consecutive sojourns, in classes C and O1 respectively, is
fZ 01 …t1 ; t2 † ˆ /TC eQ00 t1 Q01 eQ11 t2 …ÿQ11 †1

…t1 ; t2 P 0†:

…2:7†

Integrating out both variables, t1 and t2 , yields
T
ÿ1
P01 ˆ /TC …ÿQÿ1
00 †Q 01 1 ˆ /C …ÿQ00 †Q 01 ;

…2:8†

where the latter expression follows since, in the present case, O1 has a single open state. (Note that
in (2.7), (2.8) and elsewhere in this paper, in order to preserve the mathematical structure of
formulae we will often retain matrix expressions, even when they do reduce to scalars.) Similarly,
ÿ1
†Q02 . Note that …ÿQÿ1
P02 ˆ /TC …ÿQ00
00 †Q 01 , which gives the probabilities of moving from
the various level 0 states to level 1 states, corresponds to what, in notation closer to that of
Colquhoun and Hawkes [13], can be written as G 01 (cf. [13, (1.25)]). In this notation, P01 ˆ /TC G 01
and P02 ˆ /TC G 02 .
2.5. Burst theory
Channel records often exhibit periods of activity, known as bursts, which are noticeably separated from other such periods. This occurs when the closed states divide clearly into classes of
short-lived and long-lived states [13]. Then a (theoretical) burst is a sequence of open sojourns
separated by visits to the short-lived closed states, with any pair of successive bursts separated by
a visit to the long-lived closed states. The burst length is the time from the start of the ®rst open
sojourn to the end of the last such sojourn in the burst. Generally, for models of ClC-0 state 3 is a
short-lived closed state and states 4±6 are long-lived closed states.
In practice, with experimental data, it cannot be observed whether a channel at level 0 is in a
short-lived or long-lived state. The usual approach [22,23] then starts with some speci®ed critical
time tc , and identi®es an (empirical) burst as any series of open sojourns separated by closed
sojourns having duration at most tc , with any pair of successive bursts separated by a closed
sojourn of duration greater than tc . Short sojourns in the class of closed states are usually called
gaps within the burst. Except brie¯y in Section 6 this empirical de®nition of a burst is used.
To determine properties of a burst, we ®rst derive the joint density fT;NO …t; n† of the number NO
of open sojourns within the burst and the vector T ˆ …T …1† ; T …2† ; . . . ; T …n† † of their successive durations. Since all the n ÿ 1 gaps within the burst must have duration at most tc , it follows from
(2.4) with s ˆ 2n ÿ 1, after appropriate integration over the closed-time variables and use of (2.5)
and (2.6), that
QOO t2
QOC


QCO eQOO tn QOC eQCC tc 1;
fT;NO …t; n† ˆ wTO eQOO t1 QOC …I ÿ eQCC tc †…ÿQÿ1
CC †Q CO e

…2:9†

where t1 ; . . . ; tn P 0 and
ÿ


ÿ1
†QCO = pTO QOC eQCC tc … ÿ Qÿ1
wO ˆ pTO QOC eQCC tc …ÿQCC
CC †Q CO 1

…2:10†

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

29

is the probability distribution that a burst begins in the various open states. Note that the
probability distribution wO rather than /O must be used in (2.9), because the closed sojourn
preceding the burst is constrained to be of duration at least tc . (However, in (2.10) the initial
pTO QOC in both numerator and denominator could be replaced by /C .) Observe that the (marginal)
densities for the components of T, the open times within the burst, are genuine p.d.f.s, but that the
`density' for NO is its probability (mass) function.
Eq. (2.9) yields several further results. Integration over t1 ; t2 ; . . . ; tn gives probabilities for the
number NO of open sojourns within a burst as
P fNO ˆ ng ˆ wTO Rnÿ1
tc …I ÿ Rtc †1

…n ˆ 1; 2; . . .†;

…2:11†

ÿ1
QCC tc
QCC tc
†Qÿ1
1 ˆ …I ÿ Rtc †1. The distribution
where Rtc ˆ Qÿ1
OO QOC …I ÿ e
CC QCO and ÿQOO QOC e
(2.11) is a form of matrix geometric distribution (cf. [13, (3.5)]). The mean number E…NO † of open
sojourns within a burst is E…NO † ˆ wTO ‰I ÿ Rtc Šÿ1 1. As the number NC of gaps within a burst is
equal to NO ÿ 1, the distribution of NC follows from (2.11).
For any particular open time, say T …k† , within a burst having n open sojourns, the joint density
of T …k† and NO follows by integrating over all the open times except the kth, yielding (as in [24])
nÿkÿ1
QOO t
fT …k† ;NO …t; n† ˆ wTO Rkÿ1
…I ÿ Rtc †1;
QOC …I ÿ eQCC tc †…ÿQÿ1
CC †QCO Rtc
tc e

…2:12†

where t P 0 and n ˆ 1; 2; . . . From (2.12), the mean E…T …k† ; n† …k ˆ 1; . . . ; n† of the kth open time
within a burst having n open sojourns is
ÿ1
nÿk
E…T …k† ; n† ˆ wTO Rkÿ1
tc …ÿQ CC †Rtc …I ÿ Rtc †1:

Hence, by summing over all possible values of k and n, it follows that the mean open time within a
burst is
E…TO † ˆ wTO ‰I ÿ Rtc Šÿ1 …ÿQÿ1
CC †1:

…2:13†

Although the result (2.13) gives the mean open time within a burst, because the open times in
ClC-0 are of two di€erent types, some at conductance level 1 and some at level 2, E…TO † cannot be
used to determine the mean total charge transfer over the burst. Within a burst, denote by T1 the
total open time at conductance level 1 and similarly for T2 . Then de®ne a normalised charge
transfer W ˆ T1 ‡ 2T2 , and
E…W † ˆ E…T1 † ‡ 2E…T2 †:

…2:14†

The main aim of the next section is to approximate E…W † by ®nding simple but often good
approximations for E…T1 † and E…T2 †.

3. Burst properties: simple approximations
Gaps within a burst may be of several types. For a gap of type 101, i.e. a transition from level 1
to level 0 and back to level 1, the gap has duration at most tc , and so the conditional probability
that the channel moves from level 1 to level 0 for a sojourn at most tc in duration and then moves
back to level 1, given that it starts at level 1, is

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Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

Fig. 2. Realization of a burst starting and ending in level 1 and having no direct transitions between levels 0 and 2
within the burst (tc is the critical time used to identify the burst).
Q00 tc
†…ÿQÿ1
a101 ˆ …ÿQÿ1
00 †Q 01 :
11 †Q10 …I ÿ e

…3:1†

The other possible types of gaps, namely 102, 201 and 202, are likely to occur much less frequently
in ClC-0, as illustrated in the numerical example in Section 5 (see Table 1). Therefore, in order to
simplify the approach to burst properties, for the remainder of this section such possibilities are
ignored.
By restricting to bursts with gaps of type 101, as shown in Fig. 2, a simple approximation is
obtained for various probabilities and other properties associated with bursts. A consequence of
having ignored certain types of burst is that the `distributions' which result from such approximations may be defective (i.e. the probabilities do not add to 1).
An improved approximation applicable to expectations of burst properties is developed in
Section 3.3. In Section 4, the same type of probabilistic argument as used to derive the simple
approximations yields corresponding exact results for the marginal distributions. These are employed in Section 5 to illustrate the reliability of the simple approximations.
~ are used to denote the simple approximations
In the remainder of the paper P~f
g and E…
†
to P f
g and E…
†. Corresponding symbols are used for derived quantities such as variances and
covariances.
3.1. Approximation to the distribution of N1 and to E…T1 †
As stated in Section 2.3, the aggregated Markov chain is assumed to be in equilibrium. Let S
denote the index of the level which starts a burst. Then the probabilities for S are P fS ˆ 1g ˆ w1
and P fS ˆ 2g ˆ w2 , where w1 and w2 are the components of wO , de®ned in (2.10), and so given by
 T
QCC tc
T
QCC tc
w1 ˆ pTO QOC eQCC tc …ÿQÿ1
… ÿ Qÿ1
…3:2†
… ÿ Qÿ1
CC †Q02 1†
CC †Q 01 1 ‡ pO Q OC e
CC †Q 01 …pO QOC e

and w2 ˆ 1 ÿ w1 . (If the strict requirement that a burst be preceded by a closed sojourn of greater
than tc were ignored, then the probabilities w1 and w2 could be approximated by /1 and /2 , the
components of /O , de®ned near (2.3).)
Let E denote the level which ends the burst. Suppose that N0 is the number of gaps within a
burst, and that N1 and N2 are respectively the numbers of open sojourns at conductance levels 1
and 2. Observe that N0 ‡ N2 ˆ N1 ÿ 1 when S ˆ 1 and E ˆ 1; see, for example, Fig. 2 where
n0 ˆ 5, n2 ˆ 5 and n1 ˆ 11. We give an analytical derivation of the joint distribution of N1 and N2 ,
followed by a probabilistic derivation of the marginal distribution of N1 .

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

31

The joint probability that N1 ˆ n1 , N2 ˆ n2 and E ˆ 1 conditional on the burst starting at level 1
is approximated by
; N2 ˆ n2
; E ˆ 1 j S ˆ 1g
P~fN1 ˆ n1
n1 ÿ 1
1 ÿn2 ÿ1
…p12 p21 †n2 an101
b1
ˆK
n2

…0 6 n2 6 n1 ÿ 1†;

…3:3†

where a101 is given by (3.1), b1 ˆ …ÿQ11 †ÿ1 Q10 eQ00 tc 1 is the probability of a speci®ed sequence
within a burst ending at level 1, p12 p21 is the probability of a transition of type 121, and K is a
normalising constant which can be chosen to ensure a proper (rather than a defective) distribution. The joint distribution in (3.3) follows by observing that, given S ˆ 1, the event
fN1 ˆ n1 ; N2 ˆ n2 ; E ˆ 1g occurs if and only if n2 of the n1 ÿ 1 transitions within the burst away
n
from level 1 are of type 121 (probability …p12 p21 † 2 ), and the other n0 ˆ n1 ÿ 1 ÿ n2 transitions are
0
). Similarly
of type 101 (probability an101
P~fN1 ˆ n1
; N2 ˆ n2
; E ˆ 2 j S ˆ 1g
n ÿ 1 n1 ÿn2
a
ˆK 1
…p12 p21 †n2 ÿ1 p12 b2
n2 ÿ 1 101

…1 6 n2 6 n1 †;

…3:4†

ÿ1

where b2 ˆ …ÿQ22 † Q20 eQ00 tc 1 is the probability of a speci®ed sequence within a burst ending at
level 2 and K is as above. Furthermore, since a burst which starts at level 2 and cannot proceed
directly to level 0 must ®rst move to level 1 and thereafter develop as if it were a burst starting at
level 1,
; N2 ˆ n2
; E ˆ 1 j S ˆ 2g
P~fN1 ˆ n1
n1 ÿ 1
n1 ÿn2
p21 b1
…p12 p21 †n2 ÿ1 a101
ˆK
n2 ÿ 1

…1 6 n2 6 n1 †;

…3:5†

and
P~fN1 ˆ n1
; N2 ˆ n2
; E ˆ 2 j S ˆ 2g
n1 ÿ 1
1 ÿn2 ‡1
…p12 p21 †n2 ÿ2 an101
…p21 p12 †b2
ˆK
n2 ÿ 2

…2 6 n2 6 n1 ‡ 1†:

…3:6†

Consequently, summing over n2 in (3.3)±(3.6) using the binomial expansion, and then using the
total probability formula, yields the ®rst line in

n1 ÿ1
…n1 P 1†;
P~fN1 ˆ n1 g ˆ Kx…p12 p21 ‡ a101 †
…3:7†
Kw2 b2
…n1 ˆ 0†;
where x ˆ …w1 ‡ w2 p21 †…b1 ‡ p12 b2 † and K ˆ 1=fw2 b2 ‡ x…1 ÿ p12 p21 ÿ a101 †g. For the second line
in (3.7) the total probability formula is again used, after observing that P~fN1 ˆ 0 j S ˆ 1g ˆ 0 and
P~fN1 ˆ 0 j S ˆ 2g ˆ b2 , since direct transitions within the burst between levels 2 and 0 have been
excluded from consideration. The distribution (3.7) is a (zero-)modi®ed geometric distribution
(cf. [25, pp. 312±316]).
Our analytical derivation of (3.7), based on (3.3)±(3.6), allows also derivation of the joint
distribution of N1 and N2 . The structure of (3.7) and the expression for x suggest that, if only
marginal distributions are required, a direct probabilistic derivation of these should be possible.

32

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

This can be achieved by using an argument based on conditional independence. For example,
when n1 P 1, considering ®rst the two ways in which a burst may begin leads to a probability
which is the ®rst term in the product de®ning x. The term …p12 p21 ‡ a101 †n1 ÿ1 in (3.7) arises as the
probability that there are n1 ÿ 1 excursions away from the level 1 in a situation where there are
only two possibilities (121, or 101 where the sojourn time at level 0 is at most tc ) for each excursion. Finally, consideration of the two ways in which a burst may end yields a probability
which is the second term in the product de®ning x.
From (3.7), it follows that the mean number of level 1 open sojourns within a burst can be
approximated by
~ 1 † ˆ Kx=…1 ÿ p12 p21 ÿ a101 †2 :
E…N

…3:8†

Hence, since T1 can be represented as a random sum (see Section 4.2), the mean level 1 open time
E…T1 † within a burst can be approximated by
~ 1 † ˆ E…N
~ 1 †=d1 :
E…T

…3:9†

3.2. Approximation to the distribution of N2 and to E…T2 †
Using (3.3) and the negative binomial formula

1 
 a r
X
r‡iÿ1 r i
ab ˆ
i
1ÿb
iˆ0
yields
P~fN2 ˆ n2 ; E ˆ 1 j S ˆ 1g ˆ K

b1
1 ÿ a101



p21 p12
1 ÿ a101

n 2

…n2 P 0†:

…3:10†

Analogous expressions can be derived, in a manner similar to (3.3)±(3.6), for the corresponding
probabilities when the burst is started or ended di€erently. Then it follows, again using the total
probability formula as in Section 3.1, that

n2 ÿ1
…n2 P 1†;
…3:11†
P~fN2 ˆ n2 g ˆ Kf…p21 p12 =…1 ÿ a101 ††
Kw1 b1 =…1 ÿ a101 †
…n2 ˆ 0†;
where
fˆ



w1 p12
‡ w2
1 ÿ a101




p21 b1
‡ b2 :
1 ÿ a101

The modi®ed geometric distribution (3.11) could have been obtained by the same kind of probabilistic reasoning as was used for (3.7). Here the term p21 p12 =…1 ÿ a101 † arises as the probability of
an excursion away from level 2; this excursion must go to level 1 and return from this level, but in
between may have any number of sojourns, each of duration at most tc , at level 0.
From (3.7), the mean number of level 2 open sojourns within a burst can be approximated
by

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44


ÿ2
p12 p21
~
E…N2 † ˆ Kf 1 ÿ
1 ÿ a101

33

…3:12†

and the mean level 2 open time can be approximated by
~ 2 † ˆ E…N
~ 2 †=d2 :
E…T

…3:13†

From (3.9) and (3.13) it follows that a simple approximation to the mean normalised charge
~ 2 †.
~ † ˆ E…T
~ 1 † ‡ 2E…T
transfer is given by E…W
3.3. Improved approximations to E…T1 †, E…T2 † and E…W †
As mentioned at the beginning of this section, direct transitions between conductance levels 2
and 0 within a burst are omitted. Since the number of such omitted transitions increase as the
critical time tc increases, the simple approximations to E…N1 †, E…T1 †, E…N2 † and E…T2 † are likely to
become less e€ective for larger tc , as illustrated in Table 2. Nevertheless, these simple approximations can be modi®ed to make them less sensitive to changes in tc .
For this model, the exact mean open sojourn E…TO † may be calculated and is given by (2.13).
Thus we can introduce a scaling factor n de®ned by
nˆ

E…TO †
:
~
~ 2†
E…T1 † ‡ E…T

…3:14†

~ 1 † and E …T2 † ˆ nE…T
~ 2 † to E…T1 †
Then, after scaling by n, improved approximations E …T1 † ˆ nE…T
and E…T2 †, respectively are obtained. Furthermore, it follows from (2.14) that an improved approximation for the mean normalised charge transfer within a burst is
~ 1 † ‡ 2E…T
~ 2 †Š:
E …W † ˆ n‰E…T

…3:15†

It should be noted that this way of obtaining improved approximations applies only to expectations and not, for example, to variances. If viewed as a rescaling of the probabilities P~,
rescaling using n 6ˆ 1 would result in a defective distribution.
That the above approximations are often good for estimating the mean normalised charge
transfer E…W † is shown in Section 5, after derivation of exact results for the distributions of N1 , N2 ,
T1 , T2 and for E…W † in Section 4.
3.4. Approximate variances, covariances and correlations
Let V …N1 † and V …N1 † denote the variances of N1 and N2 , respectively, C…N1 ; N2 † the corresponding covariance and q…N1 ; N2 † the correlation. Approximations V~ …N1 † and V~ …N2 † to V …N1 †
and V …N1 † respectively, can be derived from (3.7) and (3.11) as
2

Kx‰1 ÿ Kx ÿ …p12 p21 ‡ a101 † Š
V~ …N1 † ˆ
;
4
…1 ÿ p12 p21 ÿ a101 †

…3:16†

Kf‰1 ÿ Kf ÿ …p12 p21 =…1 ÿ a101 ††2 Š
V~ …N2 † ˆ
:
…1 ÿ p12 p21 =…1 ÿ a101 ††4

…3:17†

34

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

~ 1 ; N2 †, can be obtained from
Similarly, a covariance approximation C…N
~ 1 N2 † ˆ
E…N

2Kwp12 p21
3

…1 ÿ a101 ÿ p12 p21 †

‡

K…w1 p12 b2 ‡ w2 p21 b1 ‡ 2w2 p21 p12 b2 †
…1 ÿ a101 ÿ p12 p21 †2

;

…3:18†

which follows from (3.3)±(3.6). The associated approximate correlation q~…N1 ; N2 † can then be
written down.
In order to obtain corresponding approximate results for T1 and T2 , note that these random
variables can be represented as random sums (see Section 4.2) and hence
~ 1 ††=d 2 ; V~ …T2 † ˆ …V~ …N2 † ‡ E…N
~ 2 ††=d 2 , and C…T
~ 1 ; T2 † ˆ C…N
~ 1 ; N2 †=…d1 d2 †. It
V~ …T1 † ˆ …V~ …N1 † ‡ E…N
1
2
~
~
follows that the absolute value of q…T1 ; T2 † is always less than q…N1 ; N2 †, as d1 and d2 cancel. An
approximation to V …W † can be obtained as
~ 1 ; T2 †:
V~ …W † ˆ V~ …T1 † ‡ 4V~ …T2 † ‡ 4C…T

…3:19†

4. Burst properties: exact results
Consider now a burst which may have within it direct transitions between levels 2 and 0. In this
case the burst may contain gaps of any of the four types 101, 102, 201 and 202, which have respective probabilities a101 given by (3.1), and
Q00 tc
a102 ˆ …ÿQÿ1
†…ÿQÿ1
00 †Q 02 ;
11 †Q10 …I ÿ e

…4:1†

Q00 tc
†…ÿQÿ1
a201 ˆ …ÿQÿ1
00 †Q 01 ;
22 †Q20 …I ÿ e

…4:2†

Q00 tc
†…ÿQÿ1
a202 ˆ …ÿQÿ1
00 †Q 02 :
22 †Q20 …I ÿ e

…4:3†

To ®nd the distribution of N1 , the number of level 1 open sojourns within a burst, let N01 be the
number of events that are transitions of type 101 within a burst, and let N0i …i ˆ 2; . . . ; 5† be the



number of events that are transitions of the respective types 10…202† 01, 10…202† 1, 1…202† 01 and
1…202† 1 within a burst. Here …202† denotes either a single sojourn at level 2 or possibly repeated
¯uctuations between levels 2 and 0 (starting and ending at level 2), with all sojourns at level 0
having duration at most tc . Let a1 ˆ a101 , a2 ˆ a102 a201 =…1 ÿ a202 †, a3 ˆ a102 p21 =…1 ÿ a202 †,
a4 ˆ p12 a201 =…1 ÿ a202 † and a5 ˆ p12 p21 =…1 ÿ a202 † be the respective probabilities of the above types
of transitions. Here, for example, a3 arises from an initial transition of type 102 followed by any
number of transitions of type 202 and a ®nal transition back to level 1. P
(In all such cases the
5
intermediate sojourns at level 0 must be of duration at most tc .) Let A ˆ iˆ1 ai be the sum of
these probabilities. Thus A is the probability that the system, given it is at level 1, returns to level 1
before the burst ends. P
Observe that N1 ˆ 1 ‡ 5iˆ1 N0i when S ˆ 1 and E ˆ 1; such a realization is illustrated in Fig. 3.
Joint probabilities for N01 , N02 , N03 , N04 and E ˆ 1 conditional on the burst starting at level 1 are
given by

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

35

Fig. 3. Realization of a burst starting and ending in level 1, with all possible transitions allowable. Here n01 ˆ 6,
n02 ˆ 1, n03 ˆ 2, n04 ˆ 2, n05 ˆ 3, n1 ˆ 15 and n2 ˆ 16.

P fN01 ˆ n01 ; N02 ˆ n02 ; N03 ˆ n03 ; N04 ˆ n04 ; N1 ˆ n1 ; E ˆ 1 j S ˆ 1g
n1 ÿ 1
ˆ
an01 an02 an03 an04 an1 ÿ1ÿn01 ÿn02 ÿn03 ÿn04 b1
n01 ; n02 ; n03 ; n04 ; n05 1 2 3 4 5

…4:4†

…n0i P 0; i ˆ 1; . . . ; 5 and n01 ‡


‡ n05 ˆ n1 ÿ 1†, which is a modi®ed multinomial distribution.
This expression follows by observing that the event on the left-hand side of the above equation
occurs if and only if n0i …i ˆ 1; 2; 3; 4† of n1 ÿ 1 transitions within the burst are of the respective



types 101, 10…202† 01, 10…202† 1 and 1…202† 01, and the other n05 ˆ n1 ÿ 1 ÿ n01 ÿ n02 ÿ n03 ÿ n04

transitions are of type 1…202† 1. These cases have respective probabilities an101 , an202 , an303 and an404 and
an505 . The ®nal term in (4.4), b1 , is the probability that the burst ends from level 1. Similar expressions to (4.4) may be derived for other ways of starting and ending the burst.
4.1. Exact distributions of N1 and N2
Using the multinomial formula and summing over n01 , n02 , n03 and n04 in (4.4), and then using
the total probability formula, yields the ®rst line in
 n ÿ1
sA 1
…n1 P 1†;
P fN1 ˆ n1 g ˆ
…4:5†
w2 b2 =…1 ÿ a202 † …n1 ˆ 0†:
Here s ˆ …w1 ‡ w2 v1 †…b1 ‡ v2 b2 †, where w1 and w2 are the components of wO given by (3.2), v1 ˆ
…p21 ‡ a201 †=…1 ÿ a202 † and v2 ˆ …p12 ‡ a102 †=…1 ÿ a202 †. Clearly, w2 v1 is the probability that a burst
starts at level 2 and eventually visits level 1, and v2 b2 is probability that a burst, presently at level
1, does not return to level 1 and ends at level 2. Hence the ®rst line in (4.5) shows that for n1 P 1,
the event fN1 ˆ n1 g occurs if and only if the burst starts at level 1, or starts at level 2 and
eventually visits level 1, with probability …w1 ‡ w2 v1 †, and then the system returns to level 1 n1 ÿ 1
times, with probability An1 ÿ1 , and then the burst ends from level 1 or does not return to level 1
before the burst ends from level 2, with probability …b1 ‡ v2 b2 †. For the second line in (4.5) the
total probability formula is again used, after observing that, in this case, the burst must start at
level 2 and end from this level, possibly after repeated ¯uctuations between levels 0 and 2. Note
that P fN1 ˆ 0g ˆ 1 ÿ s, and P fN1 ˆ n1 j N1 P 1g ˆ …1 ÿ A†An1 ÿ1 , n1 ˆ 1; 2; . . .
The distribution (4.5) is another (zero-)modi®ed geometric distribution. Clearly, the approximate distribution of N1 in (3.7) is the special case of (4.5) with a102 ˆ a202 ˆ a202 ˆ 0, i.e. where
direct transitions between levels 0 and 2 within a burst are not allowed. Note that the same type of

36

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

direct probabilistic argument as was indicated for (3.7) can be used to derive (4.5), and could
replace the above analytical arguments.
Next, we derive the distribution of N2 . Let b1 ˆ a202 , b2 ˆ a201 a102 =…1 ÿ a101 †, b3 ˆ
a201 p12 =…1 ÿ a101 †, b4 ˆ p21 a102 =…1 ÿ a101 †, b5 ˆ p21 p12 =…1 ÿ a101 † and B be the sum of
bi …i ˆ 1; . . . ; 5†. So B is the probability that the system, given that it is at level 2, returns to level 2
before the burst ends. By symmetry in (4.5), it follows immediately that by interchanging 1 and 2
 n ÿ1
…n2 P 1†;
jB 2
…4:6†
P fN2 ˆ n2 g ˆ
w1 b1 =…1 ÿ a101 † …n2 ˆ 0†;
where j ˆ …w1 v02 ‡ w2 †…v01 b1 ‡ b2 †, v01 ˆ …p21 ‡ p201 †=…1 ÿ a101 † and v02 ˆ …p12 ‡ a102 †=…1 ÿ a101 †.
When a102 ˆ a201 ˆ a202 ˆ 0, the distribution (4.6) of N2 collapses to the approximate distribution
given by (3.11).
It follows from (4.5) and (4.6) that the mean numbers E…N1 † and E…N2 † of level 1 and level 2
open sojourns within a burst and their variances are
E…N1 † ˆ s=…1 ÿ A†2

and V …N1 † ˆ s…1 ÿ A2 ÿ s†=…1 ÿ A†4 ;

E…N2 † ˆ j=…1 ÿ B†2

and V …N2 † ˆ j…1 ÿ B2 ÿ j†=…1 ÿ B†4 :

4.2. Exact distributions of T1 and T2
To derive the exact distribution of T1 , the (total) level 1 open time within a burst, observe that
T1 ˆ 0 if and only if N1 ˆ 0, and hence P …T1 ˆ 0† ˆ P …N1 ˆ 0†, and that when N1 P 1
T1 ˆ

N1
X

U …i† ;

iˆ1

…i†

where U is the duration of the ith level 1 opening within the burst. Thus, conditional on N1 P 1,
T1 is sum of a geometrically distributed number N1 of independent and identically distributed
random variables U …1† ; U …2† ; . . . ; each having an exponential distribution with mean 1=d1 . It follows (see [26, Appendix]) that T1 itself has an exponential distribution, with mean
E…T1 j N1 P 1† ˆ 1=…d1 …1 ÿ A††:
Taking account of both cases, the cumulative distribution function of T1 is
P …T1 6 t† ˆ

w2 b2
s
…1 ÿ eÿd1 …1ÿA†t †
‡
…1 ÿ a202 † 1 ÿ A

…t P 0†:

The exact distribution of T1 is thus a (zero-)modi®ed exponential distribution.
The mean and variance of the level 1 open time within a burst are
E…T1 † ˆ E…N1 †=d1 ˆ s=…d1 …1 ÿ A†2 †
and
V …T1 † ˆ

s…2 ÿ 2A ÿ s†
4

d12 …1 ÿ A†

:

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

37

By symmetry, similar results can be written down for the exact distribution of T2 . In particular, the
mean and variance of the level 2 open time within a burst are
E…T2 † ˆ E…N2 †=d2 ˆ j=…d2 …1 ÿ B†2 †
and
V …T2 † ˆ

j…2 ÿ 2B ÿ j†
4

d22 …1 ÿ B†

:

Therefore, the exact mean normalised charge transfer E…W † within a burst can obtained as
E…W † ˆ E…T1 † ‡ 2E…T2 † ˆ s=…d1 …1 ÿ A†2 † ‡ 2j=…d2 …1 ÿ B†2 †:

…4:7†

An expression for the exact variance V …W † ˆ V …T1 † ‡ 4V …T2 † ‡ 4C…T1 ; T2 † of the normalised
charge transfer W is not immediately available since the above methods do not yield the (exact)
covariance C…T1 ; T2 † of T1 and T2 . However, the approximate covariance obtained in Section 3.4
may be a reasonable approximation to the exact value in many cases, and this approach is used in
~ 1 ; T2 †. It
the numerical examples given in the next section, where V # …W † ˆ V …T1 † ‡ 4V …T2 † ‡ 4C…T
is likely that T1 and T2 are rather highly correlated, as large (small) values of T1 and T2 would most
often occur with long (short) bursts.

5. Numerical results
Given transition rates for the Markov model, burst properties can be studied numerically, and
comparisons made between the approximate and exact results. In addition, simulations assist in
understanding behaviour, and provide data for testing analytic techniques. The transition rates
((5.1), units of msÿ1 ) used for this exercise, implying time constants for the subgates and supergate
of the order of 10 and 100 ms, respectively, were chosen to re¯ect the kinetic separation often
found experimentally for the situation where the supergate and subgates act independently (as can
be seen by substituting values from (5.1) into Fig. 1). Bursts are then reasonably well distinguished
from comparatively long interburst intervals caused by supergate closure. Typically, analysis of
experimental data has proceeded by excluding the (assumed) interburst sojourns and considering
only events within bursts [3,27].

…5:1†

Fig. 4(a) shows a 10 second segment of a simulated recording based on the six state Markov
model with transition rates given in (5.1). There appears to be bursting behaviour, for example,
between about 7.3 and 8.3 s; this is shown more clearly in (b) and (c) at higher time resolution.

38

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

Fig. 4. A 10 second segment of simulated ClC-0 activity based on the six state Markov model with transition rate
matrix given by (5.1) is shown in (a). Higher time resolution features of segments of bursting behaviour are shown in
(b) and (c).

Between 7.3 and 7.7 s there are several open sojourns; whether these are judged to belong to the
same or di€erent bursts would depend on the choice of tc . Although the empirical de®nition of a
burst must be used in dealing with data from patch clamp recordings, in a simulation it is possible
to `observe' which visits to level 0 involve only the short-lived closed state 3 and which involve
supergate closure (a sojourn in one or more of states 4±6); hence properties of both theoretical and
empirical bursts may be investigated and compared.
The numerical values presented in Table 1 for the probabilities a101 , a102 , a201 and a202 of the
four types of gaps within a burst, show that, for all values of tc , the assumption made in Section 3
concerning the simple approximations, namely that transitions of types 102, 201 and 202 within
bursts are rare compared with a transition of type 101 (and so could be ignored), is reasonable.
(The probability of a transition of type 121 is p12 p21 ˆ 0:2976 for all values of tc , and so is not
shown in the table.) Values are presented also for the probabilities of some other events associated
with bursts. Because a burst tends to last longer as tc increases, the respective probabilities b1 and
b2 of a burst ending at level 1 and 2 naturally decrease. The exact probabilities that a burst starts
at level 2 increase with tc ; these values are all greater than the simple approximation /2 ˆ 0:0222
(which does not depend on tc ), as the approximation does not restrict the burst to start after a
closed sojourn of duration greater than tc . Also shown in Table 1 are values of the normalizing
constant K (which is needed for the simple approximations), and the scaling constant n (for the
improved approximations).
Exact and approximate results for some important properties of bursts as tc increases from 10
to 100 ms are given in Table 2, which again is based on the particular transition rates in (5.1). The
mean numbers, E…N1 † and E…N2 †, of visits within a burst to the respective levels 1 and 2, the mean

39

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44
Table 1
Probabilities associated with bursts for given critical times (in milliseconds)a
tc

a101

a102

a201

a202

b1

b2

w2

n

K

10
20
30
40
50
80
100

0.38
0.52
0.57
0.59
0.60
0.62
0.63

0.0006
0.0015
0.0025
0.0034
0.0043
0.0065
0.0076

0.0019
0.0046
0.0075
0.0103
0.0130
0.0197
0.0233

0.0022
0.0029
0.0034
0.0037
0.0041
0.0049
0.0054

0.30
0.17
0.12
0.10
0.08
0.06
0.05

0.044
0.040
0.037
0.034
0.031
0.023
0.019

0.045
0.072
0.093
0.103
0.108
0.111
0.111

1.004
1.020
1.045
1.074
1.106
1.215
1.304

1.006
1.022
1.046
1.075
1.106
1.212
1.301

a

Values based on the six state Markov model with transition rates given in (5.1). The probability a101 , given by (3.1), is
the probability that a channel at level 1 moves to level 0 for a duration of at most tc and then returns to level 1.
Correspondingly, a102 , a201 and a202 are de®ned in (4.1)±(4.3). The closing probabilities b1 and b2 and the starting
probability w2 are de®ned in the text of Section 3.1, and the scaling factors n and K are given at (3.14) and after (3.7),
respectively.

Table 2
Burst properties and their approximations for given critical times (in milliseconds)a
~ 2†
~ †
~ 1†
tc
E…T
E…T1 †
E…T
E…T2 †
q~…T1 ; T2 †
E…W
E …W †
10
20
30
40
50
80
100

19.6
33.7
46.1
54.8
61.1
76.1
86.1

19.7
34.4
48.1
58.8
67.4
91.9
111.3

4.9
8.3
11.3
13.4
14.9
18.3
20.5

4.9
8.6
12.0
14.7
16.8
23.0
27.8

0.55
0.65
0.71
0.74
0.76
0.80
0.81

29.3
50.3
68.7
81.6
90.8
112.7
127.1

29.5
51.3
71.7
87.6
100.5
136.9
165.8

E…W †

r# …W †

29.6
51.6
72.1
88.1
101.1
137.8
167.0

31.6
52.2
70.3
83.0
92.2
114.0
128.5

a

Values based on the Markov model with transition rates given in (5.1). E…T1 † and E…T2 † are the mean times spent at
~ 1 † and E…T
~ 2 † are approximations of E…T1 † and E…T2 †; q~…T1 ; T2 † is the approximated
levels 1 and 2 within a burst, and E…T
correlation between T1 and T2 given in Section 3.4; E…W † and r# …W † are the mean and the standard deviation of the
~ † and E …W † are the simple approximation and improved approximation to the mean
normalized charge transfer; E…W
normalized charge transfer E…W †.

times, E…T1 † and E…T2 †, spent at these levels within a burst, and the mean normalized charge
T1 and T2 is
transfer, E…W †, all increase with tc , as might be expected. The correlation between
p
substantial and also increases with tc , as does the standard deviation r# …W † ˆ V # …W †. The
simple approximations are good for reasonable choices of tc , indicating their possible usefulness in
similar but more complex situations where exact results might not be obtainable. The improved
approximation given in (3.15) to the mean normalized charge transfer is extremely precise for any
value of tc ; there is a similar accuracy for the means of N1 , N2 , T1 and T2 .
The non-independent case was examined by increasing both q12 and q42 either two- or four-fold.
This preserved microscopic reversibility in the model while introducing positive cooperativity into
gating such that: (i) with the supergate open, opening of one subgate increases the probability that
the second will open, and (ii) with the supergate shut, opening of both subgates increases the
probability of the supergate opening. Fig. 5 shows plots of E…W †, E…N1 †, E…N2 †, E…T1 † and E…T2 †, as
functions of tc , for the independence model (based on (5.1)) and for the above two models exhibiting dependence. This enables useful comparisons between the predicted burst behaviour

40

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

Fig. 5. Predicted burst behaviour as a function of tc (in milliseconds). In each plot the solid curve (±±) represents the
independence model based on transition rates given in (5.1). The dotted (




) and dashed (- - - -) curves show the
e€ects of increasing both q12 and q42 either two- or four-fold, respectively.

under these models. In particular, comparison of Fig. 5(b) and (d) emphasizes that, although
E…N1 † increases faster with tc in the four-fold dependent case than in the other cases, this same
behaviour is not seen in E…T1 † because of (3.9) and the fact that d1 increases to 0.21 and 0.26,
respectively in the two- and four-fold dependent cases. (Similar behaviour is not seen with E…N2 †
and E…T2 †, because d2 does not change.) As positive cooperativity (through q12 and q42 ) increases,
the mean normalized charge transfer also increases, for all choices of tc . If q12 and q42 were both
decreased in the same ratio (to maintain reversibility) the mean normalized charge transfer would
also decrease.

6. Binomial testing for independence of gating
Methods applying binomial distributions to level occupancy data have often been used to assess
whether the individual pores in multi-barrelled channels such as ClC-0 [3,9,28] or other channels

Y. Li et al. / Mathematical Biosciences 166 (2000) 23±44

41

[27,29] gate independently. Given the limitations of such methods [10], we examined whether
dependence resulting from small changes to the parameter set (5.1) used above would be detected.
From simulated data of 20 000 sojourns using (5.1) and deleting sojourns in states 4±6, we
generated a continuous sequence of sojourns in states 1±3 corresponding to idealized bursting
periods when the supergate is open. Ten samples of size n ˆ 500 (as in [27]) were then randomly
selected from this sequence of sojourn-times, and the observed conductance levels compared with
predictions from a binomial distribution, using the chi-squared test. In all instances the null
hypothesis of independent channel gating was correctly accepted. The non-independent case was
examined, as described in Section 5, by increasing both q12 and q42 either two- or four-fold. For
the two-fold increase in q12 and q42 the null hypothesis was rejected at the 5% signi®cance level in
all 10 tests, whereas with the four-fold chan