Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue1.Jul2000:

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 fromTorpedoelectroplax, 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 con-ductance levels. Each pore has the same concon-ductance 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 pa-rameter 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 in-dependence/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).

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 indepen-dent and iindepen-dentically 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 re-duced 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.

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 f1gandO2ˆ f2gconsisting respectively of the open states with conductance levels 1 and 2, andC 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 homo-geneous Markov chainfX…t†g ˆ fX…t†; tP0ghaving transition rate matrixQˆ ‰qijŠof the form

…2:1†

wheredi is the sum of non-diagonal entries of the ith row, i.e. di ˆ

P

j6ˆiqij. 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 forQ 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, andQ₀₁ governs transitions from states of Cto O1. Such a process is often called an aggregated Markov chain. When the focus is on whether the channel is open, without regard to

the conductance level, the relevant partitioning is into two classes,OˆO1[O2of open states and Cas above. The corresponding partition of Qcan be written as

Qˆ QOO QOC Q_CO Q_CC

; …2:2†

whereQ_CCis the same asQ00used 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 asso-ciated discrete-time Markov chain, the jump chain, having transition (probability) matrixP with diagonal entries all zero and o€-diagonal entriespijˆqij=di; i6ˆj. Thus, for example, if the ClC-0

channel is in state 2, there is a probabilityp24ˆq24=…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 pˆ …p1;. . .;p6†

T

which can be obtained by solving the global balance equation(s)pT_{Qˆ0, where} 0 is a vector with all zero entries, together with P

ipiˆ1. When the process is reversible, an

alternative approach to the equilibrium distribution is to solve the detailed balance equations

piqij ˆpjqji for alli;jˆ1;2;. . .;6, together with

P

ipiˆ1. The equilibrium distribution is also

the limiting distribution in the sense that its components satisfypk ˆlimt!1PfX…t† ˆkg. Just as

the partition of the state space intoO1,O2andCinduces a corresponding partition (2.1) ofQ, so it also induces a partition of the equilibrium distribution as pT _{ˆ …p}T

1..

Transitions between states in the continuous-time Markov chain are described by the jump chain, which has transition matrixP. 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 expo-nential distribution with parameter di (cf. [5, (24)]). Since there is a single open state in O1, it

follows that the duration of a sojourn inO1 has an exponential distribution with meanl1 ˆ1=d1. Similarly, the duration of a sojourn in O2 has an exponential distribution with meanl2 ˆ1=d2. Because there is more than one state inC, the distribution of aCsojourn-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 classCwe need the probability distribution/_C that, when the chain is in equilibrium, a visit to the closed statesCbegins in the various states ofC. It follows, for example from [13, (3.2)], that

/_C ˆpT_OQ_OC=…pT_OQ_OC1†; …2:3†

where1 is a column vector with all elements unity. Similarly,/_OˆpT

CQCO=…pTCQCO1†. Note that

LetZA₁;A2;...;As be the random vector whose s components are the durations of consecutive so-journs of an aggregated Markov chain in classesA1;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 classesA1;A2;. . .;As is

AA=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 classAs, the chain must exit to another class

B di€erent from As, which happens according to the transition rates …ÿQAsAs†1ˆ

P

B6ˆAsQAsB1,

where the sum is over all setsB, di€erent fromAs, in some state±space partition (see [16, p. 64]).

For eachi, the chain moves fromAitoAi‡1with transition ratesQAiAi‡1, and the distribution of the duration of a sojourn in Ai is described by eQAiAiti.

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 variablest1;. . .;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

which hold for any (®nite) square matrixKthat 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 inCis given by

fC…t† ˆ/T_C exp…QCCt†…ÿQCC†1 …tP0†:

Hence, using (2.5), the mean duration of a closed sojourn inC is

E…TC† ˆ

P10ˆp13‡p15ˆ …q13‡q15†=d1 andP12ˆ1ÿP10ˆp12ˆq12=d1. Similarly,P20ˆp24ˆq24=d2and

P21ˆ1ÿP20ˆp21ˆq21=d2.

Expressions for the probabilitiesP01andP02are more dicult to determine since there are four states inC. Consider the partition ofQ as in (2.1), and takesˆ2,A1 ˆC andA2 ˆO1 in (2.4). Then the density function of Z01, the vector whose components are the durations of two con-secutive sojourns, in classes Cand O1 respectively, is

fZ₀₁…t1;t2† ˆ/TCe

Q00t1_Q

01eQ11t2…ÿQ11†1 …t1;t2P0†: …2:7† Integrating out both variables,t1 and t2, yields

P01ˆ/TC…ÿQ

where the latter expression follows since, in the present case,O1has 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,

P02ˆ/T_C…ÿQ00ÿ1†Q02. Note that …ÿQ

ÿ1

00†Q01, 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 asG01(cf. [13, (1.25)]). In this notation,P01ˆ/TCG01 and P02ˆ/T_CG02.

2.5. Burst theory

Channel records often exhibit periods of activity, known as bursts, which are noticeably sep-arated 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 thantc. 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 densityfT;NO…t;n†of the numberNO

of open sojourns within the burst and the vector T ˆ …T…1†_;T…2†_{;. . .;T}…n†_{† of their successive}

is the probability distribution that a burst begins in the various open states. Note that the probability distribution w_O 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 pT_OQ_OCin both numerator and denominator could be replaced by/_C.) Observe that the (marginal) densities for the components ofT, the open times within the burst, are genuine p.d.f.s, but that the `density' forNO is its probability (mass) function.

Eq. (2.9) yields several further results. Integration over t1;t2;. . .;tn gives probabilities for the

numberNO of open sojourns within a burst as

PfNOˆng ˆw_OTRn_tcÿ1…I ÿRtc†1 …nˆ1;2;. . .†; …2:11† (2.11) is a form of matrix geometric distribution (cf. [13, (3.5)]). The mean numberE…NO†of open

sojourns within a burst is _E…NO† ˆwTO‰IÿRtcŠ

ÿ1

1. As the number NC of gaps within a burst is

equal toNOÿ1, the distribution ofNC follows from (2.11).

For any particular open time, sayT…k†_{, within a burst havingnopen sojourns, the joint density}

ofT…k† _{and N}

O follows by integrating over all the open times except thekth, yielding (as in [24])

fT…k†_;NO…t;n† ˆwT_ORk_tÿ1

within a burst having nopen sojourns is

E…T…k†_{;n† ˆw}T

Hence, by summing over all possible values ofkandn, it follows that the mean open time within a burst is

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 byT1 the total open time at conductance level 1 and similarly for T2. Then de®ne a normalised charge transfer W ˆT1‡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

a101ˆ …ÿQÿ111†Q10…IÿeQ00tc†…ÿQ

ÿ1

00†Q01: …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 approxi-mations 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 em-ployed in Section 5 to illustrate the reliability of the simple approximations.

In the remainder of the paper P~fg and ~_E…† are used to denote the simple approximations to Pfg 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. LetS denote the index of the level which starts a burst. Then the probabilities forSarePfSˆ1g ˆw₁

andPfS ˆ2g ˆw2, wherew1andw2are the components ofwO, de®ned in (2.10), and so given by

w₁ˆpT_OQ_OCeQCCtc_ÿQÿ1

CC†Q01 …p T

OQOCe

QCCtc

ÿQÿ_CC1†Q₀₁1‡p_OTQ_OCeQCCtc_ÿQÿ1

CC†Q021† …3:2† andw2ˆ1ÿw1. (If the strict requirement that a burst be preceded by a closed sojourn of greater thantc were ignored, then the probabilitiesw1 andw2 could be approximated by /1 and /2, the components of /_O, de®ned near (2.3).)

LetE denote the level which ends the burst. Suppose that N0 is the number of gaps within a burst, and thatN1 andN2 are respectively the numbers of open sojourns at conductance levels 1 and 2. Observe that N0‡N2ˆN1ÿ1 when S ˆ1 and Eˆ1; see, for example, Fig. 2 where

n0 ˆ5,n2 ˆ5 andn1 ˆ11. We give an analytical derivation of the joint distribution ofN1 andN2, followed by a probabilistic derivation of the marginal distribution ofN1.

The joint probability thatN1 ˆn1,N2 ˆn2andEˆ1 conditional on the burst starting at level 1 within a burst ending at level 1, p12p21 is the probability of a transition of type 121, and Kis a normalising constant which can be chosen to ensure a proper (rather than a defective)

distribu-tion. The joint distribution in (3.3) follows by observing that, given S ˆ1, the event

fN1 ˆn1;N2 ˆn2;Eˆ1g occurs if and only ifn2 of then1ÿ1 transitions within the burst away from level 1 are of type 121 (probability…p12p21†n2), and the othern0 ˆn1ÿ1ÿn2 transitions are of type 101 (probabilityan0

101). Similarly level 2 andKis 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,

Consequently, summing overn2 in (3.3)±(3.6) using the binomial expansion, and then using the total probability formula, yields the ®rst line in

~ excluded from consideration. The distribution (3.7) is a (zero-)modi®ed geometric distribution (cf. [25, pp. 312±316]).

This can be achieved by using an argument based on conditional independence. For example, when n1P1, considering ®rst the two ways in which a burst may begin leads to a probability which is the ®rst term in the product de®ningx. The term…p12p21‡a101†n1ÿ

1

in (3.7) arises as the probability that there aren1ÿ1 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 ex-cursion. Finally, consideration of the two ways in which a burst may end yields a probability which is the second term in the product de®ningx.

From (3.7), it follows that the mean number of level 1 open sojourns within a burst can be approximated by

~

E…N1† ˆKx=…1ÿp12p21ÿa101†

2

: …3:8†

Hence, sinceT1can 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

~

E…T1† ˆ~E…N1†=d1: …3:9†

3.2. Approximation to the distribution of N2 and to E…T2†

Using (3.3) and the negative binomial formula

X1

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

~

The modi®ed geometric distribution (3.11) could have been obtained by the same kind of prob-abilistic reasoning as was used for (3.7). Here the termp21p12=…1ÿa101†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.

~

E…N2† ˆKf 1

ÿ p12p21

1ÿa101 ÿ2

…3:12†

and the mean level 2 open time can be approximated by

~

E…T2† ˆE~…N2†=d2: …3:13†

From (3.9) and (3.13) it follows that a simple approximation to the mean normalised charge transfer is given by_E~…W† ˆE~…T1† ‡2~E…T2†.

3.3. Improved approximations to_E…T1†, E…T2† andE…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 timetcincreases, the simple approximations toE…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 approxi-mations 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 nde®ned by

nˆ E…TO†

~

E…T1† ‡~E…T2†

: …3:14†

Then, after scaling byn, improved approximationsE…T1† ˆn~E…T1†andE…T2† ˆnE~…T2†toE…T1† and E…T2†, respectively are obtained. Furthermore, it follows from (2.14) that an improved ap-proximation for the mean normalised charge transfer within a burst is

E…W† ˆn‰~E…T1† ‡2~E…T2†Š: …3:15†

It should be noted that this way of obtaining improved approximations applies only to ex-pectations and not, for example, to variances. If viewed as a rescaling of the probabilities P~, rescaling usingn6ˆ1 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 ofN1,N2,

T1, T2 and forE…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 corre-sponding covariance and q…N1;N2† the correlation. ApproximationsV~…N1† and V~…N2† to V…N1† and V…N1† respectively, can be derived from (3.7) and (3.11) as

~ V…N1† ˆ

Kx‰1ÿKxÿ …p12p21‡a101†2Š

…1ÿp12p21ÿa101†4

; …3:16†

~ V…N2† ˆ

Kf‰1ÿKfÿ …p12p21=…1ÿa101†† 2

Š …1ÿp12p21=…1ÿa101††

Similarly, a covariance approximation C~…N1;N2†, can be obtained from

~

E…N1N2† ˆ

2Kwp12p21

…1ÿa101ÿp12p21† 3‡

K…w1p12b2‡w2p21b1‡2w2p21p12b2†

…1ÿa101ÿp12p21†

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 forT1 and T2, note that these random

variables can be represented as random sums (see Section 4.2) and hence

~

V…T1† ˆ …V~…N1† ‡~E…N1††=d12;V~…T2† ˆ …V~…N2† ‡E~…N2††=d22, and C~…T1;T2† ˆC~…N1;N2†=…d1d2†. It 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

~

V…W† ˆV~…T1† ‡4V~…T2† ‡4C~…T1;T2†: …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 re-spective probabilities a101 given by (3.1), and

a102ˆ …ÿQÿ111†Q10…IÿeQ00tc†…ÿQÿ 1

00†Q02; …4:1†

a201ˆ …ÿQÿ221†Q20…IÿeQ00tc†…ÿQÿ 1

00†Q01; …4:2†

a202ˆ …ÿQÿ221†Q20…IÿeQ00tc†…ÿQÿ 1

00†Q02: …4:3†

To ®nd the distribution ofN1, the number of level 1 open sojourns within a burst, letN01be the number of events that are transitions of type 101 within a burst, and let N0i …iˆ2;. . .;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 ˆa101, a2 ˆa102a201=…1ÿa202†, a3 ˆa102p21=…1ÿa202†,

a4 ˆp12a201=…1ÿa202†anda5ˆp12p21=…1ÿa202†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. (In all such cases the intermediate sojourns at level 0 must be of duration at most tc.) Let Aˆ

P5

iˆ1ai be the sum of

these probabilities. ThusAis the probability that the system, given it is at level 1, returns to level 1 before the burst ends.

Observe thatN1 ˆ1‡P5_iˆ1N0iwhenSˆ1 andEˆ1; such a realization is illustrated in Fig. 3.

PfN01ˆn01;N02ˆn02;N03ˆn03;N04ˆn04;N1ˆn1;Eˆ1jS ˆ1g This expression follows by observing that the event on the left-hand side of the above equation occurs if and only ifn0i …iˆ1;2;3;4†of n1ÿ1 transitions within the burst are of the respective types 101, 10…202†01, 10…202†1 and 1…202†01, and the othern05ˆn1ÿ1ÿn01ÿn02ÿn03ÿn04 transitions are of type 1…202†1. These cases have respective probabilitiesan01

1 ,a

5 . The ®nal term in (4.4), b1, is the probability that the burst ends from level 1. Similar ex-pressions to (4.4) may be derived for other ways of starting and ending the burst.

4.1. Exact distributions ofN1 andN2

Using the multinomial formula and summing overn01, n02, n03and n04in (4.4), and then using the total probability formula, yields the ®rst line in

PfN1 ˆn1g ˆ sA starts at level 2 and eventually visits level 1, andv2b2is 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 forn1P1, the event fN1 ˆn1g occurs if and only if the burst starts at level 1, or starts at level 2 and eventually visits level 1, with probability…w₁‡w₂v1†, and then the system returns to level 1n1ÿ1 times, with probabilityAn1ÿ1_{, 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‡v2b2†. 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 thatPfN1 ˆ0g ˆ1ÿs, and PfN1 ˆn1jN1P1g ˆ …1ÿA†An1ÿ1, n1 ˆ1;2;. . .

The distribution (4.5) is another (zero-)modi®ed geometric distribution. Clearly, the approxi-mate distribution ofN1 in (3.7) is the special case of (4.5) with a102ˆa202ˆa202 ˆ0, i.e. where direct transitions between levels 0 and 2 within a burst are not allowed. Note that the same type of

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 b1ˆa202, b2 ˆa201a102=…1ÿa101†, b3 ˆ

a201p12=…1ÿa101†, b4ˆp21a102=…1ÿa101†, b5 ˆp21p12=…1ÿa101† and B be the sum of

bi …iˆ1;. . .;5†. SoBis 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

PfN2 ˆn2g ˆ Whena102 ˆa201 ˆa202ˆ0, the distribution (4.6) ofN2 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† ˆs=…1ÿA†

4.2. Exact distributions ofT1 andT2

To derive the exact distribution ofT1, the (total) level 1 open time within a burst, observe that

T1ˆ0 if and only if N1 ˆ0, and henceP…T1ˆ0† ˆP…N1ˆ0†, and that whenN1P1

whereU…i†_{is the duration of theith level 1 opening within the burst. Thus, conditional onN}

1P1,

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=d}

1. It fol-lows (see [26, Appendix]) that T1 itself has an exponential distribution, with mean

E…T1jN1P1† ˆ1=…d1…1ÿA††:

Taking account of both cases, the cumulative distribution function of T1 is

P…T16t† ˆ

The exact distribution ofT1 is thus a (zero-)modi®ed exponential distribution. The mean and variance of the level 1 open time within a burst are

By symmetry, similar results can be written down for the exact distribution ofT2. In particular, the mean and variance of the level 2 open time within a burst are

E…T2† ˆE…N2†=d2 ˆj=…d2…1ÿB†2† and

V…T2† ˆ

j…2ÿ2Bÿj†

d2

2…1ÿB† 4 :

Therefore, the exact mean normalised charge transfer_E…W† within a burst can obtained as

E…W† ˆE…T1† ‡2E…T2† ˆs=…d1…1ÿA†2† ‡2j=…d2…1ÿB†2†: …4:7†

An expression for the exact variance V…W† ˆV…T1† ‡4V…T2† ‡4C…T1;T2† of the normalised charge transferW is not immediately available since the above methods do not yield the (exact) covarianceC…T1;T2† ofT1 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 the numerical examples given in the next section, whereV#_{W† ˆV…T}

1† ‡4V…T2† ‡4C~…T1;T2†. It is likely thatT1 andT2 are rather highly correlated, as large (small) values ofT1andT2would 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†

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 oftc. 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 probabilitiesa101, a102, a201 and a202 of the four types of gaps within a burst, show that, for all values oftc, 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 p12p21ˆ0:2976 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 astcincreases, the respective probabilitiesb1and

b₂ of a burst ending at level 1 and 2 naturally decrease. The exact probabilities that a burst starts at level 2 increase withtc; these values are all greater than the simple approximation/2 ˆ0:0222 (which does not depend on tc), as the approximation does not restrict the burst to start after a closed sojourn of duration greater thantc. Also shown in Table 1 are values of the normalizing constant K(which is needed for the simple approximations), and the scaling constantn (for the improved approximations).

Exact and approximate results for some important properties of bursts astcincreases 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†andE…N2†, of visits within a burst to the respective levels 1 and 2, the mean

times, _E…T1† and E…T2†, spent at these levels within a burst, and the mean normalized charge transfer, E…W†, all increase with tc, as might be expected. The correlation between T1 and T2 is substantial and also increases with tc, as does the standard deviation r#…W† ˆ



V#_W† p

. The simple approximations are good for reasonable choices oftc, 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 oftc; there is a similar accuracy for the means of N1, N2,T1 and T2.

The non-independent case was examined by increasing bothq12andq42either 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†andE…T2†, as functions of tc, for the independence model (based on (5.1)) and for the above two models ex-hibiting dependence. This enables useful comparisons between the predicted burst behaviour

Table 1

Probabilities associated with bursts for given critical times (in milliseconds)a

tc a101 a102 a201 a202 b1 b2 w2 n K

10 0.38 0.0006 0.0019 0.0022 0.30 0.044 0.045 1.004 1.006

20 0.52 0.0015 0.0046 0.0029 0.17 0.040 0.072 1.020 1.022

30 0.57 0.0025 0.0075 0.0034 0.12 0.037 0.093 1.045 1.046

40 0.59 0.0034 0.0103 0.0037 0.10 0.034 0.103 1.074 1.075

50 0.60 0.0043 0.0130 0.0041 0.08 0.031 0.108 1.106 1.106

80 0.62 0.0065 0.0197 0.0049 0.06 0.023 0.111 1.215 1.212

100 0.63 0.0076 0.0233 0.0054 0.05 0.019 0.111 1.304 1.301

a

Values based on the six state Markov model with transition rates given in (5.1). The probabilitya101, given by (3.1), is

the probability that a channel at level 1 moves to level 0 for a duration of at mosttc and then returns to level 1.

Correspondingly, a102, a201 anda202 are de®ned in (4.1)±(4.3). The closing probabilities b1 andb2 and the starting

probabilityw₂ are de®ned in the text of Section 3.1, and the scaling factorsnandKare given at (3.14) and after (3.7), respectively.

Table 2

Burst properties and their approximations for given critical times (in milliseconds)a

tc ~E…T1† E…T1† E~…T2† E…T2† q…~T1;T2† ~E…W† E…W† E…W† r#…W†

10 19.6 19.7 4.9 4.9 0.55 29.3 29.5 29.6 31.6

20 33.7 34.4 8.3 8.6 0.65 50.3 51.3 51.6 52.2

30 46.1 48.1 11.3 12.0 0.71 68.7 71.7 72.1 70.3

40 54.8 58.8 13.4 14.7 0.74 81.6 87.6 88.1 83.0

50 61.1 67.4 14.9 16.8 0.76 90.8 100.5 101.1 92.2

80 76.1 91.9 18.3 23.0 0.80 112.7 136.9 137.8 114.0

100 86.1 111.3 20.5 27.8 0.81 127.1 165.8 167.0 128.5

a

Values based on the Markov model with transition rates given in (5.1).E…T1†andE…T2†are the mean times spent at

levels 1 and 2 within a burst, and~E…T1†and~E…T2†are approximations ofE…T1†andE…T2†;q…~T1;T2†is the approximated

correlation betweenT1 andT2 given in Section 3.4;E…W†andr#…W†are the mean and the standard deviation of the

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† andE…T2†, becaused2 does not change.) As positive cooperativity (throughq12andq42) increases, the mean normalized charge transfer also increases, for all choices oftc. Ifq12 andq42were 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

Fig. 5. Predicted burst behaviour as a function oftc(in milliseconds). In each plot the solid curve (±±) represents the

[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 sizenˆ500 (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 bothq12and q42 either two- or four-fold. For the two-fold increase inq12andq42the null hypothesis was rejected at the 5% signi®cance level in all 10 tests, whereas with the four-fold change the same conclusion was obtained even at the 0.1% signi®cance level. These results, based on (5.1), indicate that moderate deviations from inde-pendence have a very high probability of being detected when interburst intervals are excluded, provided sojourns in states 1±3are correctly identi®ed.

Experimentally, however, this identi®cation cannot be guaranteed because zero conductance events could include (short) sojourns in states 4±6 as well as state 3. This is why, in practice, a critical time (tc) is used to classify closed times into short intraburst and longer interburst closed times, accepting that for any given tc there will be some incorrect assignments. It might be ex-pected that these classi®cation errors, depending on their magnitude, could compromise the bi-nomial test for independence of subgates. Obviously, the greater the di€erence between fast and slow gating time constants, the easier this approach to classi®cation should be. Some principles for choosing optimaltcvalues in single pore channels having two exponential components in the closed-time distribution are presented in [23, pp. 535,536], but the situation for the present model is more complex.

Table 3 for the independence model shows that there is an optimal range fortcif false rejection of independence is to be avoided; for the transition rates given in (5.1) this would appear to be 37± 45 ms. Much below this leveltcis too short and some sojourns in closed state 3 may be regarded as interburst intervals and excluded from the data for binomial testing. This leads to a de®cit in level 0 sojourns and increases the likelihood that the binomial test will be found incorrectly signi®cant. By contrast, much above tcˆ45 ms sojourns in states 4±6 are increasingly categorized as be-longing within bursts, resulting in a surplus of level 0 events and false rejection of the null hy-pothesis that individual pores behave independently. If the very low weight inverse eigenvalues for the closed distribution in Table 4 given ass1ands2 are discounted, it is perhaps reasonable to see whytcmust lie between about 10 and 100 ms for the independence model (Case I). However, as increasing levels of dependence are introduced into the model (shown as two- and four-fold in-creases inq12andq42), the largest inverse eigenvalue (s4 in Table 4) steadily decreases from 105 to 67 ms whiles3 remains almost unchanged. An experimenter observing these distributions would probably choose a similar (though more likely smaller)tcfor Cases II and III compared to Case I. Table 3 shows that extreme values of tc will, of course, lead to signi®cant results due to false inclusion and exclusion of data, but the precise range where the correct conclusion is made concerning dependence is dicult to gauge. Indeed for each of the dependent cases there aretc values when the null hypothesis would be falsely accepted at the 5% signi®cance level.

We conclude that in analysis of experimental ClC-0 data requiring application of a critical time

erroneous conclusions regarding independence of the subgates. Recording noise and limited time resolution would further complicate such inference. By focusing on particular properties of bursts in double-barrelled channels, such as the number of open sojourns at levels 1 and 2, mean so-journ-times at each level and mean normalized charge transfer, the present theory may assist investigation of subgate independence. (This would complement direct maximum likelihood es-timation of individual transition rates from full data sets.) The considerable sensitivity of many burst properties, as shown in Fig. 5, to the existence and degree of dependence in the channel illustrates the promise of such an approach.

Note that the closed-time distribution has three faster components with weightswi and means si …iˆ1;2;3†, and one slower component with weightw4 and mean s4. Observe that the latter is close to the mean duration of a closed sojourn which includes closure of the supergate. For all three models considered here, s3 is approximately 9.1, suggesting that, in the present case, this

Table 3

Assessment of the independence assumptiona

tc Independence model Dependence models

Two-fold increase Four-fold increase

5 10

10 10 0

15 10 4 9

20 6 0 10

25 5 4

30 3 7

35 2 9

37 0 10

40 0 10

45 0

80 2

100 4

a

The independence assumption was examined using simulated data and chi-squared tests based on the binomial dis-tribution. The table gives, for speci®ed critical times (tcin milliseconds), the number of rejections of the null hypothesis

of independence in 10 tests based on a 5% signi®cance level. For each condition, 20 000 sojourns were generated by simulation using (5.1), or (5.1) with the indicated increases inq12andq42, closed sojourns greater thantcwere classi®ed

as interburst intervals and excluded, and 10 samples each of size 500 were then randomly selected from the remaining data.

Table 4

Inverse eigenvalues (si) and weights (wi as a percentage) for closed-time distributionsa

Case Model s1 w1 s2 w2 s3 w3 s4 w4

I Independence 3.2 0:07 6.2 2:1 9.1 77:0 105 20:7

II Dependence

(two-fold)

3.2 0:04 6.1 2:6 9.1 74:9 94 22:6

III Dependence

(four-fold)

2.6 0:05 6.2 4:6 9.1 69:3 67 26:0

a

Values based on transition rates given in (5.1) for the independence model, which is shown as Case I; in Cases II and III,q12andq42in (5.1) were multiplied by two and four, respectively. Time constantss(inverse eigenvalues) are given in

component may correspond to gaps within a burst (because it is approximately the inverse of

d3ˆ0:11), and the other three components contribute to the long-shut period between bursts. Such interpretations are not generally straightforward.

From the above analysis it is clear that correct interpretation of binomial tests for independence requires careful choice of the critical timetc, for which there is no simple criterion. Three methods have been proposed for selecting a value oftcthat lies between two components of the closed-time distribution. For the numerical examples based on the transition rates given in (5.1) and their modi®cations which introduce dependence, the procedure of minimizing the total number of misclassi®ed sojourns gives better results than the other methods (see [23, pp. 535,536]). The optimum values oftcfor the three situations (independence, two- and four-fold changes) are about 37.5, 35.7 and 31.5 ms, respectively. The other two methods give substantially smaller values for

tc. The general selection problems arising where at least one of the closed classes has several states will be addressed in more detail elsewhere.

Acknowledgements

Support was received from the Australian Research Council (Grant A69801864). We thank the referees for their helpful comments.

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