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Amplitude histograms can identify positively but not negatively coupled channels
Journal of Neuroscience Methods 96 (2000) 105 – 111 www.elsevier.com/locate/jneumeth
Amplitude histograms can identify positively but not negatively coupled channels J.L. Kenyon a,*, R.J. Bauer b a
Department of Physiology and Cell Biology/MS 352, Uni6ersity of Ne6ada School of Medicine, Reno, NV 89557, USA b Department of Pharmacology, Xoma Corporation, 2910 Se6enth St., Berkeley, CA 94710, USA Received 5 July 1999; received in revised form 25 November 1999; accepted 27 November 1999
Abstract We investigated the ability of amplitude distributions to determine if the gating of a pair of channels is coupled. These distributions are expressed as probability density amplitude histograms with peaks corresponding to zero, one, or two open channels. If the channels gate independently, the areas under these peaks (A, B, and C, respectively) determine the open probabilities of the two channels (p1 and p2). Manivannan et al. (Biophys J 1994;61:216) showed that if D =B 2/AC was less than 4 then the channel gating is coupled. We defined a similar parameter, D= (B 2/4) −AC. If DB 0 then channel gating is coupled. However, amplitude histograms with D\0 are consistent with both independent and coupled gating. We further present a simple model in which channels are assumed to be identical and can be positively or negatively coupled. Here, amplitude histograms determine q= (B+ 2C)/2 (open probability of the coupled channels) and r= −D (the coupling parameter). Thus, positively coupled channels (r\ 0) produce amplitude histograms with DB0 whereas negatively coupled channels (rB 0) produce amplitude histograms with D\ 0. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Single channel; Independent; Cooperative; Analysis; Ryanodine receptor; Double-barreled channel
1. Introduction The possibility that ion channels might gate cooperatively has long been recognized. Although early studies found channels gating independently (Hill and Chen, 1971a,b; Neher et al., 1978; Sigworth, 1979, 1980), the possibility that channels could coordinate their activity has remained and several reports have described a variety of methods to test this issue. These studies found evidence for coupled gating of acetylcholine receptor channels (Schindler et al., 1984; Yeramian et al., 1986), sodium channels (Iwasa et al., 1986), gap junction channels (Manivannan et al., 1992; Veenstra et al., 1994), and ryanodine receptor channels (Marx et al., 1998). A common method used to detect coupling of channels has been the analysis of the distribution of current amplitudes, i.e. amplitude histograms, by comparing * Corresponding author. Tel.: +1-775-7841649; fax: + 1-7757846903. E-mail address: [email protected] (J.L. Kenyon)
the observed distributions with those predicted for independent channels. For example, Manivannan et al. (1992) defined a parameter D determined by the areas under the peaks of the amplitude histogram and showed that histograms where D B 4 were incompatible with the gating of independent channels. They also demonstrated that changes in the open probabilities of independent channels could not result in amplitude distributions with D B 4, i.e. mode shifting cannot cause independent channels to appear to be coupled. Our work was stimulated by the amplitude histograms presented in the report by Marx et al. (1998) that the accessory protein FKBP12 induced coupled gating by pairs of ryanodine receptor channels. These histograms led us to investigate the consequences of coupled gating on amplitude distributions and to clarify and extend the results of Manivannan et al. (1992). We further developed a simple model of coupled channel gating to facilitate interpretation of amplitude histograms. This modeling led us to the result that although amplitude histograms produced by negatively coupled channels are indistinguishable from those aris-
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ing from independent channels, amplitude histograms produced by positively coupled channels are distinct from those produced by independent channels. This result, combined with the inability of mode shifting to produce an amplitude histogram consistent with positive coupling, implies that positively coupled gating by a pair of channels can be established by the analysis of the distribution of current amplitudes.
2. Results We have limited our analysis to records containing pairs of channels for two reasons. First, this greatly simplifies the analysis and allows us to obtain a more detailed picture of channel behavior than is possible when more than two channels are present. Second, we were interested in the analysis of data obtained by Marx et al. (1998) from pairs of channels. We also note that gating of ‘double-barreled’ channels produces data similar to that produced by a pair of channels (Miller, 1982; Morier and Sauve´, 1994). Probability density amplitude histograms generated by a pair of channels (or pores) have three peaks. We designate the areas under these peaks as A, B, and C where A is the fraction of time zero channels are open, B is the fraction of time one channel is open, and C is the fraction of time two channels are open. Thus, A, B, and C are probabilities that sum to unity (A+ B +C= 1). For identical channels, the open probability can be readily obtained from the amplitude histogram by the relationship Po = (B+ 2C)/2 (cf. Eq. 2 of Morier and Sauve´, 1994). More generally, if the two channels gate independently, the probability density histogram can be used to determine individual open probabilities of the channels. If p1 is the open probability of one of the channels and p2 is the open probability of the other, then: A = (1−p1)(1− p2)
(1)
B = p1(1− p2)+ p2(1 −p1)
(2)
C =p1p2
(3)
Combining Eqs. (1)– (3) gives a quadratic equation with the solution: 2C+ B p1, p2 = 9 2 2
'
B2 − AC 4
(4)
Defining D =B /4−AC, it is clear that p1 and p2 are real numbers if and only if D ] 0. If D B0 then p1 and p2 are complex numbers and the histogram is inconsistent with the gating of two independent channels with real open probabilities. Similar results were obtained by Manivannan et al. (1992). Thus, one can establish if the gating of a pair of channels is coupled using a x2 statistic to compare the
ability of two models to predict the observed distribution of current amplitudes. If a model with D] 0 is less likely to predict the amplitude distribution than a model with DB 0, one can conclude that the gating of the channels is coupled. Alternatively, D] 0 is consistent with channels that gate independently. Note that, as we describe below, histograms with D\ 0 can also be generated by a specific form of coupled channels. Lastly, we point out that valid results require that the gating is stationary, and that the current observations used to make the amplitude histogram are not correlated in time (Draber et al., 1993; Bauer et al., 1996). The alternative to independent gating is coupled gating where the probabilities of opening and closing of each channel depend on the gating status of the other. A complete investigation of coupled gating can be performed by second order correlation or dwell time analysis of the kinetics of channel gating (Yeramian et al., 1986; Bauer et al., 1987; Fredkin and Rice, 1987). However, the techniques for these analyses are not trivial to implement and, in general, recordings of channel gating do not contain sufficient information to fully characterize the complex gating schemes expected (Bauer et al., 1987; Fredkin and Rice, 1987). For these reasons, it is common to limit the analysis of channel gating to the steady state distribution of current levels and to postulate that channels gate independently in order to obtain values of open probability. Another reason for this assumption is the lack of a conceptual framework that can be used to approach the problem. We have approached this last issue by developing a simple model wherein two channels are postulated to be identical with the possibility that their gating is coupled. Probability density amplitude histograms generated by a pair of channels have information that can be used to determine two parameters describing channel gating. In the situation described earlier (channels are independent and can have different open probabilities), the two parameters are the open probabilities p1 and p2 (cf. Bauer et al., 1996). If the channels are identical and can be coupled, then two convenient parameters are q, the open probability of the identical channels, and r, a coupling factor. The areas of the probability density amplitude histogram for one such model are: A= (1− q)2 + r
(5)
B= 2q(1−q)− 2r
(6)
C=q 2 + r
(7)
The reasoning behind this formulation is discussed later. Substituting from Eq. (7) into Eq. (6) allows one to solve for q and r: q=
B+ 2C = Po 2
(8)
r = C− q 2 = C−
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111
B +2C 2
2
=−
2
B − AC = − D 4 (9)
Because B and C have values between 0 and 1, the solutions for q and r are real numbers for all histograms. Inspection of Eq. (8) shows that q is the average open probability of the two channels. The model described by Eqs. (5) – (9) can be visualized by considering a pair of channels that gate identically with variable coupling. With independent gating, r is zero and A, B, and C will be binomially distributed with q=p1 =p2. If the channels were to become coupled with r \ 0 then A, B, and C will no longer be binomially distributed as A and C increase at the expense of B. That is, there is an increasing tendency for the channels to be opened or closed together. We refer to this as ‘positive coupling’ (see also Iwasa et al., 1986). On the other hand, if the channels become coupled with r B 0, B will be increased at the expense of A and C. Here, the tendency is for the gating configuration of one channel to be the opposite of the other, i.e. if one of the channels is open the other channel will tend to be closed. We refer to this as ‘negative coupling’. Values of r are limited by the requirement that A, B, and C must be between 0 and 1. This can be illustrated by considering again a pair of channels with variable coupling. For example, if the open probability of the channels is q=0.5 and the channels are independent, B will equal 0.5. Clearly, values of r are restricted to between −0.25 and + 0.25 to maintain B in Eq. (6) between 0 and 1. More generally, for any value of q, r equals rmax when B =0 and A + C = 1. Thus, rmax = q− q 2. If r= rmax then the opening and closing of the channels are coupled perfectly such that there are no openings to the single channel level (A + C =1, B= 0). rmin is more complicated. For q B0.5, rmin = − (1− q)2 with C=0 and A+ B =1. For q \0.5, rmin = −q 2 with A= 0 and B + C = 1. Finally, for q = 0.5, rmin = − q 2 with A= C=0 and B=1, i.e. the two channels take turns opening such that one of them is open at all times. Thus, the empirical amplitudes A, B, and C can be interpreted in terms of two independently gating channels with open probabilities p1 and p2 (Eqs. (1) – (3)), or in terms of two identical coupled channels with a common open probability q and a coupling factor r (Eqs. (5)–(7)). Comparing the solution to p1 and p2 expressed in Eq. (4) with the solution to q and r expressed in Eqs. (8) and (9), we can relate the parameterization of the coupled model to the parameterization of the independent gating model: p1, p2 = q9 −r
(10)
From Eq. (10), it is clear that amplitude distributions generated by negatively coupled channels (r B 0) can
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also be described by a model in which the channels are independent with real values for p1 and p2. Hence, independent channels and negatively coupled channels cannot be distinguished by amplitude histograms. In contrast, if r\ 0 then p1 and p2 are complex. Thus, amplitude distributions generated by positively coupled channels are incompatible with the gating of independent channels. The advantage of the particular coupled gating model described in Eqs. (5)–(7) can now be recognized. In contrast to the parameter D described by Manivannan et al. (1992), which has no intuitive meaning with regard to channel gating, the relationship D= −r (Eq. (9)) provides a simple and direct link between the model parameter D and the coupling of channel gating. The impact of r on the amplitude distribution is illustrated in Fig. 1 where we have calculated amplitude histograms with constant q and different values of r. The data sets shown consist of 2048 current observations to achieve an experimentally realizable set of temporally uncorrelated observations based on our experience with Ca2 + -activated K+ channels (Bauer et al., 1996). In that work, records of approximately 5 min duration typically contained 1000–2000 uncorrelated observations of current amplitude. The numbers of observations assigned to each current level were determined using Eqs. (5)–(7) with q= 0.45 and r=0, − 0.1013, or 0.124. For example, the number of closed observations is 2048((1− 0.45)2 + r). The value of q was chosen because it allows relatively large values for both rmin and rmax. The values chosen for r are discussed below. The amplitudes of the currents were set to 0, 4, or 8 pA (for zero, one, or two channels open) plus Gaussian noise to obtain a standard deviation of the current of 0.7. These values were chosen to facilitate comparison of our model with the data presented by Marx et al. (1998). Note that neither the kinetics nor the stochastic nature of channel gating is incorporated in this approach. Rather, it produces exact (albeit noisy) distributions of temporally uncorrelated current observations. The data in Fig. 1A were obtained with r=0, i.e. the channels are not coupled. The probability density amplitude histogram constructed from these data is shown in Fig. 1D and is characterized by D= 0 and D=4. The distribution of the openings was described equally well by two independent channels with equal open probabilities (p1 = p2 = 0.45) or two identical channels with q= 0.45 and r=0 (smooth lines on top of amplitude histogram). The data in Fig. 1B were generated with the same value of q but with r= − 0.1013. This is 0.5 times rmin for q= 0.45 giving the channels moderate negative coupling. Comparing Fig. 1A and B, one sees that negative coupling increases the observations of one open channel at the expense of joint openings and joint closings. Panel 1E is the amplitude histogram of these
108
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111
data. It is fit equally well by an independent model with p1 = 0.13 and p2 = 0.77, and by a coupled model with q=0.45 and r= − 0.10. The data in Fig. 1C were generated with q=0.45 and r = 0.124. Here the positive coupling of the channels has reduced the observations of one open channel and increased the observations of joint openings and closings. The amplitude histogram constructed from these openings was readily fit by a coupled model with q =0.45 and r =0.126. However, it was difficult to fit the amplitude distribution using a model based on independent channels. The broken line in Fig. 1F was obtained by fixing the standard deviation of the central peak to 0.7. With this constraint, the best fit was obtained with p1 =p2 =0.42 and it is clear that this model provides a poor fit to the data. We have observed that fits of independent gating models to histograms characterized by D B 0 have p1 =p2 and that these values are near q, the real portion of the complex solution to Eqs. (1) – (3) [cf. Manivannan et al. (1992)].
3. Discussion A number of studies have used amplitude histograms to distinguish between independent and coupled gating (Iwasa et al., 1986; Yeramian et al., 1986; Manivannan et al., 1992; Draber et al., 1993; Liu and Dilger, 1993; Veenstra et al., 1994; Marx et al., 1998). These studies recognized that some amplitude histograms were inconsistent with independent channels, implying that the gating of the channels was coupled. In our work, we
have shown that an amplitude histogram characterized by DB 0 implies that the channels are not independent. In addition, we clarified previous work by pointing out that histograms characterized by D] 0 are compatible with either independent channels or coupled channels. We have further presented a simple model with parameters q (Po of the coupled channels) and r (the coupling parameter) in which channels are assumed to be identical and can be positively or negatively coupled, with r= − D. Thus, positively coupled channels (r\0) produce amplitude histograms with DB 0 and vice versa. Analysis of amplitude histograms is not sufficient to identify negatively coupled channels and other information must be used to make this determination. For example, Iwasa et al. (1986) found an amplitude histogram generated by a pair of Na+ channels that was consistent with two independent channels with different open probabilities (i.e. D\ 0). However, they determined that one of the open probabilities in this scheme was significantly different from the open probabilities observed in experiments containing single channels. Accordingly, they concluded that the pair of channels was negatively coupled. More generally, this distinction might be made by a second order analysis of the kinetics of channel gating (Yeramian et al., 1986; Bauer et al., 1987; Draber et al., 1993; Liu and Dilger, 1993; Keleshian et al., 1994). Amplitude histograms characteristic of positively coupled channels can often be detected by an experienced eye. Their distinguishing feature, a middle peak that is ‘too small’, can be seen by comparing Fig.
Fig. 1. Effects of r on the distribution of current amplitudes. Panels A, B, and C each show 2048 independent current observations calculated with q=0.45 and differing r as described in the text. The observations are arranged arbitrarily from left to right with zero, one and two channels open. Panel A was obtained with r= 0. Panel B was obtained with r= − 0.1013. Panel C was obtained with r= 0.124. Histograms in D, E, and F are the probability density amplitude histograms constructed from the data in A, B, and C, respectively. The continuous smooth lines are fits by the model with two identical coupled channels (Eqs. (5)–(7)). The broken lines are fits by the model with two independent channels (Eqs. (1)–(3)). These curves overlap in panels D and E but are distinct in panel F.
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111
Fig. 2. Analysis of gating of a double-barreled chloride channel. Original data are from Table 1 of Morier and Sauve´ (1994). Po values are open probabilities determined by those authors. Values of q and r/rmax were calculated from the numbers of observations of current levels corresponding to the 0, O1, and O2 current levels. The hatched bar at the bottom of the graph denotes the voltage range over which the distribution of current amplitudes were not compatible with independent channels.
1D – F. In our model, this is as a result of the −2r term in Eq. (6). Attempts to fit such a histogram with Eqs. (1) – (3) (for independent channels) fail because adjustments of p1 and p2 can fit the middle peak or the outer peaks but cannot fit all three peaks. In contrast, the increased middle peak of negatively coupled channels is readily fit by independent models with p1 "p2. Manivannan et al. (1992) demonstrated that mode shifting by independent channels cannot cause D B 4 and they concluded that changes in open probability cannot make independent channels appear coupled. This important result merits clarification. Given two independent channels with open probabilities p1 and p2, if p1 and p2 change during a recording their measured values will be the time averaged open probabilities. Because all the p1 and p2 values are real in this case, the average values will also be real and the amplitude histogram will be characterized by D \0 (or D \4 as shown by Manivannan et al., 1992). Thus, mode shifting can produce an amplitude histogram compatible with negative coupling but cannot make a pair of independent channels appear to be positively coupled. This implies that the observation of an amplitude histogram with D B 0 (or DB 4) cannot be attributed to mode shifting and is strong evidence in favor of positively coupled channels. This issue is of particular relevance to the study of ryanodine receptor channels that undergo spontaneous changes in open probability (Percival et al., 1994). Two limitations of our results are that the derivation of D is limited to the analysis of pairs of channels and that our modeling of coupled channels is limited to pairs of identical channels. The detection of coupling when more than two channels are recorded has been considered previously in some detail (Manivannan et al., 1992; Draber et al., 1993; Liu and Dilger, 1993). However, it remains to be established if positive and negative coupling of more than two channels give rise to distinguishable amplitude histograms.
109
The restriction of our modeling to pairs of identical channels is severe and it is important to note that q and r are based on this assumption. In particular, if the channels are not identical then determinations of these parameters are invalid. That being noted, we find that our coupled model provides a useful framework within which to think about the consequences of coupling between channels. Furthermore, double-barreled channels (Gadsby, 1996) and ryanodine receptor channels (Bers and Fill, 1998) are two classes of identical coupled channels of considerable interest and importance that qualify for analysis by our model. Double-barreled channels have two pores, each with an independent gate, plus a common gate that closes both pores (Miller, 1982). In terms of the model presented here, this common gate confers positive coupling to the channel by increasing the time both channels are closed. Our analysis of the amplitude distributions from the study of double-barreled chloride channels by Morier and Sauve´ (1994) is consistent with this suggestion. This analysis is summarized in Fig. 2 where we have plotted Po (tabulated by Morier and Sauve´), q, and the normalized coupling parameter r/rmax as functions of membrane potential. One sees that q and Po are identical (within roundoff) and increase as membrane potential is made more negative. The coupling term, r/rmax, averages 0.31 at positive potentials, where Morier and Sauve´ determined that the amplitude histograms were not consistent with independent channels, but drops to near 0 at negative potentials where the histograms are compatible with independent channels. Pairs of ryanodine receptor channels are also candidates for analysis with our model. Indeed, the data of Marx et al. (1998) were the immediate stimulus of our work. Briefly, these workers examined the gating of RyR1 channels in artificial lipid bilayers under conditions where the single channel current was 4 pA. Fig. 1E of their report is an amplitude histogram indicating the presence of two channels with openings to 4 and 8 pA. They analyzed this distribution by measuring the probability of both channels being open (P2 in their notation, C in our notation) then calculated the predicted probability of one channel being open (P1calc, Bcalc = 2 C·(1− C) in our notation) using Eqs. (2) and (3) with the assumption that p1 = p2 = Po. They concluded that Bcalc was not significantly different from the observed area and that the distribution was consistent with the gating of two independent channels. However, we find that the middle peak of the amplitude histogram is tantalizingly low suggesting that r is greater than zero and that the channels were partially coupled in this experiment. This could be tested using a x2 statistic to compare the ability of two models based on Eqs. (5)–(7) to predict the observed distribution. In one model, r would be constrained to r= 0 (i.e. the channels are uncoupled). In the other, r would be
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110
allowed to float. If the first model were less likely to predict the data one could conclude that the two RyR1 channels were partially coupled. Eqs. (5)–(7) are a member of a class of formulations written generally as: A = (1−q)2 + jr
(11)
B =2q(1−q)+ lr
(12)
2
C =q + kr
(13)
where the coefficients j, l, and k weight the impact of coupling on the opened and closed configurations of the channels and are constrained to j+ l+ k = 0. We have noted two important properties of Eqs. (5)–(7) (i.e. j=1, l= − 2, k = 1). First, this formulation can describe any amplitude distribution generated by the gating of a pair of channels. Second, this formulation gives simple relationships between q and open probability (Eq. (8)) and between r and channel coupling (Eq. (9)). The first property is shared by all formulations with j=k and some others, the second is apparently unique to Eqs. (5)–(7) (see below). Different formulations give rise to different mechanistic views of channel gating. For example, in formulations with j =k, coupling of the channels has the same effect on the probabilities of both channels being closed or open. Other formulations can be made with different mechanistic interpretations and we have considered two such formulations: j =0, l = −1, k =1 and j= 1, l= − 1, k = 0. In the first of these, coupling of the channels affects directly only the configurations of one and two opened channels whereas in the second the direct effects of coupling are limited to configurations of two closed and one opened channels. Both of these formulations will fit any histogram generated by the gating of a pair of channels with real values of q (between 0 and 1) and r. However, the significance of q and r differs from parameters obtained using Eqs. (5) – (7). For example, with j=1, l = −1, k =0 one obtains: q = C
(14)
and r= 2q(1− q)−B= 2 C − 2C −B
(15)
Thus, q is not the average open probability (compare Eqs. (8) and (14)) and r is not the coupling status of the channels (compare Eqs. (9) and (15)). Similar results are obtained solving the equations based on j=0, l = − 1, k =1. Indeed, Eqs. (5) – (7) are the only formulation we have found that provide simple relationships between the parameters q and r and channel coupling. Therefore, analysis of amplitude distributions using variations of Eqs. (11) – (13) generally will not provide a simple interpretation distinguishing mechanistic models of channel coupling. Such distinctions may be possible using a complete kinetic analysis of channel gating, cf.
(Yeramian et al., 1986; Bauer et al., 1987; Draber et al., 1993; Liu and Dilger, 1993; Keleshian et al., 1994). Where this level of understanding is not required, we find that analysis of steady state amplitude distributions will provide information about the coupling of pairs of ion channels.
Acknowledgements We thank Dr Grant Nicol and Dr Helen Mason for their help with the manuscript. Supported by a Grantin-Aid from the Western States Affiliate of the American Heart Association 98-257.
References Bauer RJ, Bowman BF, Kenyon JL. Theory of the kinetic analysis of patch-clamp data. Biophys J 1987;52:961 – 78. Bauer RJ, Carl A, Kapicka CL, Kenyon JL. Determination of channel open probabilities from multichannel data. J Neurosci Methods 1996;68:101 – 11. Bers DM, Fill M. Muscle contraction — coordinated feet and the dance of ryanodine receptors. Science 1998;281:790 – 1. Draber S, Schultze R, Hansen U-P. Cooperative behavior of K+ channels in the tonoplast of Chara corallina. Biophys J 1993;65:1553 – 9. Fredkin DR, Rice JA. Correlation functions of a function of a finite-state Markov process with application to channel kinetics. Math Biosci 1987;86:1 – 12. Gadsby DC. Two-bit anion channel really shapes up. Nature 1996;383:295 – 6. Hill TL, Chen Y-D. On the theory of ion transport across the nerve membrane. II. Potassium ion kinetics and cooperativity (with x = 4). Proc Natl Acad Sci USA 1971a;68:1711– 5. Hill TL, Chen Y-D. On the theory of ion transport across the nerve membrane. III. Potassium ion kinetics and cooperativity (with x = 4, 6, 9). Proc Natl Acad Sci USA 1971b;68:2488 – 92. Iwasa K, Ehrenstein G, Moran N, Jia M. Evidence for interactions between batrachotoxin-modified channels in hybrid neuroblastoma cells. Biophys J 1986;50:531 – 7. Keleshian AM, Yeo GF, Edeson RO, Madsen BW. Superposition properties of interacting ion channels. Biophys J 1994;67:634–40. Liu Y, Dilger JP. Application of the one- and two-dimensional Ising models to studies of cooperativity between ion channels. Biophys J 1993;64:26 – 35. Manivannan K, Ramanan SV, Mathias RT, Brink PR. Multichannel recordings from membranes which contain gap junctions. Biophys J 1992;61:216 – 27. Marx SO, Ondrias K, Marks AR. Coupled gating between individual skeletal muscle Ca2 + release channels (ryanodine receptors). Science 1998;281:818 – 21. Miller C. Open-state substructure of single chloride channels from Torpedo electroplax. Philos Trans R Soc Lond [Biol] 1982;299:401 – 11. Morier N, Sauve´ R. Analysis of a novel double-barreled anion channel from rat liver rough endoplasmic reticulum. Biophys J 1994;67:590 – 602. Neher E, Sakmann B, Steinbach JH. The extracellular patch clamp: a method for resolving currents through individual open channels in biological membranes. Pflu¨gers Arch 1978;375:219 – 28.
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111 Percival AL, Williams AJ, Kenyon JL, Grinsell MM, Airey JA, Sutko JL. Chicken skeletal muscle ryanodine receptor isoforms: ion channel properties. Biophys J 1994;67:1834–50. Schindler H, Spillecke F, Neumann E. Different channel properties of Torpedo acetylcholine receptor monomers and dimers reconstituted in planar membranes. Proc Natl Acad Sci USA 1984;81:6222 – 6. Sigworth FJ. The Cole-Moore effect: cooperativity among potassium channels? Biophys J 1979;25:196a–a (Abstract).
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Amplitude histograms can identify positively but not negatively coupled channels J.L. Kenyon a,*, R.J. Bauer b a
Department of Physiology and Cell Biology/MS 352, Uni6ersity of Ne6ada School of Medicine, Reno, NV 89557, USA b Department of Pharmacology, Xoma Corporation, 2910 Se6enth St., Berkeley, CA 94710, USA Received 5 July 1999; received in revised form 25 November 1999; accepted 27 November 1999
Abstract We investigated the ability of amplitude distributions to determine if the gating of a pair of channels is coupled. These distributions are expressed as probability density amplitude histograms with peaks corresponding to zero, one, or two open channels. If the channels gate independently, the areas under these peaks (A, B, and C, respectively) determine the open probabilities of the two channels (p1 and p2). Manivannan et al. (Biophys J 1994;61:216) showed that if D =B 2/AC was less than 4 then the channel gating is coupled. We defined a similar parameter, D= (B 2/4) −AC. If DB 0 then channel gating is coupled. However, amplitude histograms with D\0 are consistent with both independent and coupled gating. We further present a simple model in which channels are assumed to be identical and can be positively or negatively coupled. Here, amplitude histograms determine q= (B+ 2C)/2 (open probability of the coupled channels) and r= −D (the coupling parameter). Thus, positively coupled channels (r\ 0) produce amplitude histograms with DB0 whereas negatively coupled channels (rB 0) produce amplitude histograms with D\ 0. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Single channel; Independent; Cooperative; Analysis; Ryanodine receptor; Double-barreled channel
1. Introduction The possibility that ion channels might gate cooperatively has long been recognized. Although early studies found channels gating independently (Hill and Chen, 1971a,b; Neher et al., 1978; Sigworth, 1979, 1980), the possibility that channels could coordinate their activity has remained and several reports have described a variety of methods to test this issue. These studies found evidence for coupled gating of acetylcholine receptor channels (Schindler et al., 1984; Yeramian et al., 1986), sodium channels (Iwasa et al., 1986), gap junction channels (Manivannan et al., 1992; Veenstra et al., 1994), and ryanodine receptor channels (Marx et al., 1998). A common method used to detect coupling of channels has been the analysis of the distribution of current amplitudes, i.e. amplitude histograms, by comparing * Corresponding author. Tel.: +1-775-7841649; fax: + 1-7757846903. E-mail address: [email protected] (J.L. Kenyon)
the observed distributions with those predicted for independent channels. For example, Manivannan et al. (1992) defined a parameter D determined by the areas under the peaks of the amplitude histogram and showed that histograms where D B 4 were incompatible with the gating of independent channels. They also demonstrated that changes in the open probabilities of independent channels could not result in amplitude distributions with D B 4, i.e. mode shifting cannot cause independent channels to appear to be coupled. Our work was stimulated by the amplitude histograms presented in the report by Marx et al. (1998) that the accessory protein FKBP12 induced coupled gating by pairs of ryanodine receptor channels. These histograms led us to investigate the consequences of coupled gating on amplitude distributions and to clarify and extend the results of Manivannan et al. (1992). We further developed a simple model of coupled channel gating to facilitate interpretation of amplitude histograms. This modeling led us to the result that although amplitude histograms produced by negatively coupled channels are indistinguishable from those aris-
0165-0270/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 0 2 7 0 ( 9 9 ) 0 0 1 8 9 - 2
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ing from independent channels, amplitude histograms produced by positively coupled channels are distinct from those produced by independent channels. This result, combined with the inability of mode shifting to produce an amplitude histogram consistent with positive coupling, implies that positively coupled gating by a pair of channels can be established by the analysis of the distribution of current amplitudes.
2. Results We have limited our analysis to records containing pairs of channels for two reasons. First, this greatly simplifies the analysis and allows us to obtain a more detailed picture of channel behavior than is possible when more than two channels are present. Second, we were interested in the analysis of data obtained by Marx et al. (1998) from pairs of channels. We also note that gating of ‘double-barreled’ channels produces data similar to that produced by a pair of channels (Miller, 1982; Morier and Sauve´, 1994). Probability density amplitude histograms generated by a pair of channels (or pores) have three peaks. We designate the areas under these peaks as A, B, and C where A is the fraction of time zero channels are open, B is the fraction of time one channel is open, and C is the fraction of time two channels are open. Thus, A, B, and C are probabilities that sum to unity (A+ B +C= 1). For identical channels, the open probability can be readily obtained from the amplitude histogram by the relationship Po = (B+ 2C)/2 (cf. Eq. 2 of Morier and Sauve´, 1994). More generally, if the two channels gate independently, the probability density histogram can be used to determine individual open probabilities of the channels. If p1 is the open probability of one of the channels and p2 is the open probability of the other, then: A = (1−p1)(1− p2)
(1)
B = p1(1− p2)+ p2(1 −p1)
(2)
C =p1p2
(3)
Combining Eqs. (1)– (3) gives a quadratic equation with the solution: 2C+ B p1, p2 = 9 2 2
'
B2 − AC 4
(4)
Defining D =B /4−AC, it is clear that p1 and p2 are real numbers if and only if D ] 0. If D B0 then p1 and p2 are complex numbers and the histogram is inconsistent with the gating of two independent channels with real open probabilities. Similar results were obtained by Manivannan et al. (1992). Thus, one can establish if the gating of a pair of channels is coupled using a x2 statistic to compare the
ability of two models to predict the observed distribution of current amplitudes. If a model with D] 0 is less likely to predict the amplitude distribution than a model with DB 0, one can conclude that the gating of the channels is coupled. Alternatively, D] 0 is consistent with channels that gate independently. Note that, as we describe below, histograms with D\ 0 can also be generated by a specific form of coupled channels. Lastly, we point out that valid results require that the gating is stationary, and that the current observations used to make the amplitude histogram are not correlated in time (Draber et al., 1993; Bauer et al., 1996). The alternative to independent gating is coupled gating where the probabilities of opening and closing of each channel depend on the gating status of the other. A complete investigation of coupled gating can be performed by second order correlation or dwell time analysis of the kinetics of channel gating (Yeramian et al., 1986; Bauer et al., 1987; Fredkin and Rice, 1987). However, the techniques for these analyses are not trivial to implement and, in general, recordings of channel gating do not contain sufficient information to fully characterize the complex gating schemes expected (Bauer et al., 1987; Fredkin and Rice, 1987). For these reasons, it is common to limit the analysis of channel gating to the steady state distribution of current levels and to postulate that channels gate independently in order to obtain values of open probability. Another reason for this assumption is the lack of a conceptual framework that can be used to approach the problem. We have approached this last issue by developing a simple model wherein two channels are postulated to be identical with the possibility that their gating is coupled. Probability density amplitude histograms generated by a pair of channels have information that can be used to determine two parameters describing channel gating. In the situation described earlier (channels are independent and can have different open probabilities), the two parameters are the open probabilities p1 and p2 (cf. Bauer et al., 1996). If the channels are identical and can be coupled, then two convenient parameters are q, the open probability of the identical channels, and r, a coupling factor. The areas of the probability density amplitude histogram for one such model are: A= (1− q)2 + r
(5)
B= 2q(1−q)− 2r
(6)
C=q 2 + r
(7)
The reasoning behind this formulation is discussed later. Substituting from Eq. (7) into Eq. (6) allows one to solve for q and r: q=
B+ 2C = Po 2
(8)
r = C− q 2 = C−
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111
B +2C 2
2
=−
2
B − AC = − D 4 (9)
Because B and C have values between 0 and 1, the solutions for q and r are real numbers for all histograms. Inspection of Eq. (8) shows that q is the average open probability of the two channels. The model described by Eqs. (5) – (9) can be visualized by considering a pair of channels that gate identically with variable coupling. With independent gating, r is zero and A, B, and C will be binomially distributed with q=p1 =p2. If the channels were to become coupled with r \ 0 then A, B, and C will no longer be binomially distributed as A and C increase at the expense of B. That is, there is an increasing tendency for the channels to be opened or closed together. We refer to this as ‘positive coupling’ (see also Iwasa et al., 1986). On the other hand, if the channels become coupled with r B 0, B will be increased at the expense of A and C. Here, the tendency is for the gating configuration of one channel to be the opposite of the other, i.e. if one of the channels is open the other channel will tend to be closed. We refer to this as ‘negative coupling’. Values of r are limited by the requirement that A, B, and C must be between 0 and 1. This can be illustrated by considering again a pair of channels with variable coupling. For example, if the open probability of the channels is q=0.5 and the channels are independent, B will equal 0.5. Clearly, values of r are restricted to between −0.25 and + 0.25 to maintain B in Eq. (6) between 0 and 1. More generally, for any value of q, r equals rmax when B =0 and A + C = 1. Thus, rmax = q− q 2. If r= rmax then the opening and closing of the channels are coupled perfectly such that there are no openings to the single channel level (A + C =1, B= 0). rmin is more complicated. For q B0.5, rmin = − (1− q)2 with C=0 and A+ B =1. For q \0.5, rmin = −q 2 with A= 0 and B + C = 1. Finally, for q = 0.5, rmin = − q 2 with A= C=0 and B=1, i.e. the two channels take turns opening such that one of them is open at all times. Thus, the empirical amplitudes A, B, and C can be interpreted in terms of two independently gating channels with open probabilities p1 and p2 (Eqs. (1) – (3)), or in terms of two identical coupled channels with a common open probability q and a coupling factor r (Eqs. (5)–(7)). Comparing the solution to p1 and p2 expressed in Eq. (4) with the solution to q and r expressed in Eqs. (8) and (9), we can relate the parameterization of the coupled model to the parameterization of the independent gating model: p1, p2 = q9 −r
(10)
From Eq. (10), it is clear that amplitude distributions generated by negatively coupled channels (r B 0) can
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also be described by a model in which the channels are independent with real values for p1 and p2. Hence, independent channels and negatively coupled channels cannot be distinguished by amplitude histograms. In contrast, if r\ 0 then p1 and p2 are complex. Thus, amplitude distributions generated by positively coupled channels are incompatible with the gating of independent channels. The advantage of the particular coupled gating model described in Eqs. (5)–(7) can now be recognized. In contrast to the parameter D described by Manivannan et al. (1992), which has no intuitive meaning with regard to channel gating, the relationship D= −r (Eq. (9)) provides a simple and direct link between the model parameter D and the coupling of channel gating. The impact of r on the amplitude distribution is illustrated in Fig. 1 where we have calculated amplitude histograms with constant q and different values of r. The data sets shown consist of 2048 current observations to achieve an experimentally realizable set of temporally uncorrelated observations based on our experience with Ca2 + -activated K+ channels (Bauer et al., 1996). In that work, records of approximately 5 min duration typically contained 1000–2000 uncorrelated observations of current amplitude. The numbers of observations assigned to each current level were determined using Eqs. (5)–(7) with q= 0.45 and r=0, − 0.1013, or 0.124. For example, the number of closed observations is 2048((1− 0.45)2 + r). The value of q was chosen because it allows relatively large values for both rmin and rmax. The values chosen for r are discussed below. The amplitudes of the currents were set to 0, 4, or 8 pA (for zero, one, or two channels open) plus Gaussian noise to obtain a standard deviation of the current of 0.7. These values were chosen to facilitate comparison of our model with the data presented by Marx et al. (1998). Note that neither the kinetics nor the stochastic nature of channel gating is incorporated in this approach. Rather, it produces exact (albeit noisy) distributions of temporally uncorrelated current observations. The data in Fig. 1A were obtained with r=0, i.e. the channels are not coupled. The probability density amplitude histogram constructed from these data is shown in Fig. 1D and is characterized by D= 0 and D=4. The distribution of the openings was described equally well by two independent channels with equal open probabilities (p1 = p2 = 0.45) or two identical channels with q= 0.45 and r=0 (smooth lines on top of amplitude histogram). The data in Fig. 1B were generated with the same value of q but with r= − 0.1013. This is 0.5 times rmin for q= 0.45 giving the channels moderate negative coupling. Comparing Fig. 1A and B, one sees that negative coupling increases the observations of one open channel at the expense of joint openings and joint closings. Panel 1E is the amplitude histogram of these
108
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data. It is fit equally well by an independent model with p1 = 0.13 and p2 = 0.77, and by a coupled model with q=0.45 and r= − 0.10. The data in Fig. 1C were generated with q=0.45 and r = 0.124. Here the positive coupling of the channels has reduced the observations of one open channel and increased the observations of joint openings and closings. The amplitude histogram constructed from these openings was readily fit by a coupled model with q =0.45 and r =0.126. However, it was difficult to fit the amplitude distribution using a model based on independent channels. The broken line in Fig. 1F was obtained by fixing the standard deviation of the central peak to 0.7. With this constraint, the best fit was obtained with p1 =p2 =0.42 and it is clear that this model provides a poor fit to the data. We have observed that fits of independent gating models to histograms characterized by D B 0 have p1 =p2 and that these values are near q, the real portion of the complex solution to Eqs. (1) – (3) [cf. Manivannan et al. (1992)].
3. Discussion A number of studies have used amplitude histograms to distinguish between independent and coupled gating (Iwasa et al., 1986; Yeramian et al., 1986; Manivannan et al., 1992; Draber et al., 1993; Liu and Dilger, 1993; Veenstra et al., 1994; Marx et al., 1998). These studies recognized that some amplitude histograms were inconsistent with independent channels, implying that the gating of the channels was coupled. In our work, we
have shown that an amplitude histogram characterized by DB 0 implies that the channels are not independent. In addition, we clarified previous work by pointing out that histograms characterized by D] 0 are compatible with either independent channels or coupled channels. We have further presented a simple model with parameters q (Po of the coupled channels) and r (the coupling parameter) in which channels are assumed to be identical and can be positively or negatively coupled, with r= − D. Thus, positively coupled channels (r\0) produce amplitude histograms with DB 0 and vice versa. Analysis of amplitude histograms is not sufficient to identify negatively coupled channels and other information must be used to make this determination. For example, Iwasa et al. (1986) found an amplitude histogram generated by a pair of Na+ channels that was consistent with two independent channels with different open probabilities (i.e. D\ 0). However, they determined that one of the open probabilities in this scheme was significantly different from the open probabilities observed in experiments containing single channels. Accordingly, they concluded that the pair of channels was negatively coupled. More generally, this distinction might be made by a second order analysis of the kinetics of channel gating (Yeramian et al., 1986; Bauer et al., 1987; Draber et al., 1993; Liu and Dilger, 1993; Keleshian et al., 1994). Amplitude histograms characteristic of positively coupled channels can often be detected by an experienced eye. Their distinguishing feature, a middle peak that is ‘too small’, can be seen by comparing Fig.
Fig. 1. Effects of r on the distribution of current amplitudes. Panels A, B, and C each show 2048 independent current observations calculated with q=0.45 and differing r as described in the text. The observations are arranged arbitrarily from left to right with zero, one and two channels open. Panel A was obtained with r= 0. Panel B was obtained with r= − 0.1013. Panel C was obtained with r= 0.124. Histograms in D, E, and F are the probability density amplitude histograms constructed from the data in A, B, and C, respectively. The continuous smooth lines are fits by the model with two identical coupled channels (Eqs. (5)–(7)). The broken lines are fits by the model with two independent channels (Eqs. (1)–(3)). These curves overlap in panels D and E but are distinct in panel F.
J.L. Kenyon, R.J. Bauer / Journal of Neuroscience Methods 96 (2000) 105–111
Fig. 2. Analysis of gating of a double-barreled chloride channel. Original data are from Table 1 of Morier and Sauve´ (1994). Po values are open probabilities determined by those authors. Values of q and r/rmax were calculated from the numbers of observations of current levels corresponding to the 0, O1, and O2 current levels. The hatched bar at the bottom of the graph denotes the voltage range over which the distribution of current amplitudes were not compatible with independent channels.
1D – F. In our model, this is as a result of the −2r term in Eq. (6). Attempts to fit such a histogram with Eqs. (1) – (3) (for independent channels) fail because adjustments of p1 and p2 can fit the middle peak or the outer peaks but cannot fit all three peaks. In contrast, the increased middle peak of negatively coupled channels is readily fit by independent models with p1 "p2. Manivannan et al. (1992) demonstrated that mode shifting by independent channels cannot cause D B 4 and they concluded that changes in open probability cannot make independent channels appear coupled. This important result merits clarification. Given two independent channels with open probabilities p1 and p2, if p1 and p2 change during a recording their measured values will be the time averaged open probabilities. Because all the p1 and p2 values are real in this case, the average values will also be real and the amplitude histogram will be characterized by D \0 (or D \4 as shown by Manivannan et al., 1992). Thus, mode shifting can produce an amplitude histogram compatible with negative coupling but cannot make a pair of independent channels appear to be positively coupled. This implies that the observation of an amplitude histogram with D B 0 (or DB 4) cannot be attributed to mode shifting and is strong evidence in favor of positively coupled channels. This issue is of particular relevance to the study of ryanodine receptor channels that undergo spontaneous changes in open probability (Percival et al., 1994). Two limitations of our results are that the derivation of D is limited to the analysis of pairs of channels and that our modeling of coupled channels is limited to pairs of identical channels. The detection of coupling when more than two channels are recorded has been considered previously in some detail (Manivannan et al., 1992; Draber et al., 1993; Liu and Dilger, 1993). However, it remains to be established if positive and negative coupling of more than two channels give rise to distinguishable amplitude histograms.
109
The restriction of our modeling to pairs of identical channels is severe and it is important to note that q and r are based on this assumption. In particular, if the channels are not identical then determinations of these parameters are invalid. That being noted, we find that our coupled model provides a useful framework within which to think about the consequences of coupling between channels. Furthermore, double-barreled channels (Gadsby, 1996) and ryanodine receptor channels (Bers and Fill, 1998) are two classes of identical coupled channels of considerable interest and importance that qualify for analysis by our model. Double-barreled channels have two pores, each with an independent gate, plus a common gate that closes both pores (Miller, 1982). In terms of the model presented here, this common gate confers positive coupling to the channel by increasing the time both channels are closed. Our analysis of the amplitude distributions from the study of double-barreled chloride channels by Morier and Sauve´ (1994) is consistent with this suggestion. This analysis is summarized in Fig. 2 where we have plotted Po (tabulated by Morier and Sauve´), q, and the normalized coupling parameter r/rmax as functions of membrane potential. One sees that q and Po are identical (within roundoff) and increase as membrane potential is made more negative. The coupling term, r/rmax, averages 0.31 at positive potentials, where Morier and Sauve´ determined that the amplitude histograms were not consistent with independent channels, but drops to near 0 at negative potentials where the histograms are compatible with independent channels. Pairs of ryanodine receptor channels are also candidates for analysis with our model. Indeed, the data of Marx et al. (1998) were the immediate stimulus of our work. Briefly, these workers examined the gating of RyR1 channels in artificial lipid bilayers under conditions where the single channel current was 4 pA. Fig. 1E of their report is an amplitude histogram indicating the presence of two channels with openings to 4 and 8 pA. They analyzed this distribution by measuring the probability of both channels being open (P2 in their notation, C in our notation) then calculated the predicted probability of one channel being open (P1calc, Bcalc = 2 C·(1− C) in our notation) using Eqs. (2) and (3) with the assumption that p1 = p2 = Po. They concluded that Bcalc was not significantly different from the observed area and that the distribution was consistent with the gating of two independent channels. However, we find that the middle peak of the amplitude histogram is tantalizingly low suggesting that r is greater than zero and that the channels were partially coupled in this experiment. This could be tested using a x2 statistic to compare the ability of two models based on Eqs. (5)–(7) to predict the observed distribution. In one model, r would be constrained to r= 0 (i.e. the channels are uncoupled). In the other, r would be
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allowed to float. If the first model were less likely to predict the data one could conclude that the two RyR1 channels were partially coupled. Eqs. (5)–(7) are a member of a class of formulations written generally as: A = (1−q)2 + jr
(11)
B =2q(1−q)+ lr
(12)
2
C =q + kr
(13)
where the coefficients j, l, and k weight the impact of coupling on the opened and closed configurations of the channels and are constrained to j+ l+ k = 0. We have noted two important properties of Eqs. (5)–(7) (i.e. j=1, l= − 2, k = 1). First, this formulation can describe any amplitude distribution generated by the gating of a pair of channels. Second, this formulation gives simple relationships between q and open probability (Eq. (8)) and between r and channel coupling (Eq. (9)). The first property is shared by all formulations with j=k and some others, the second is apparently unique to Eqs. (5)–(7) (see below). Different formulations give rise to different mechanistic views of channel gating. For example, in formulations with j =k, coupling of the channels has the same effect on the probabilities of both channels being closed or open. Other formulations can be made with different mechanistic interpretations and we have considered two such formulations: j =0, l = −1, k =1 and j= 1, l= − 1, k = 0. In the first of these, coupling of the channels affects directly only the configurations of one and two opened channels whereas in the second the direct effects of coupling are limited to configurations of two closed and one opened channels. Both of these formulations will fit any histogram generated by the gating of a pair of channels with real values of q (between 0 and 1) and r. However, the significance of q and r differs from parameters obtained using Eqs. (5) – (7). For example, with j=1, l = −1, k =0 one obtains: q = C
(14)
and r= 2q(1− q)−B= 2 C − 2C −B
(15)
Thus, q is not the average open probability (compare Eqs. (8) and (14)) and r is not the coupling status of the channels (compare Eqs. (9) and (15)). Similar results are obtained solving the equations based on j=0, l = − 1, k =1. Indeed, Eqs. (5) – (7) are the only formulation we have found that provide simple relationships between the parameters q and r and channel coupling. Therefore, analysis of amplitude distributions using variations of Eqs. (11) – (13) generally will not provide a simple interpretation distinguishing mechanistic models of channel coupling. Such distinctions may be possible using a complete kinetic analysis of channel gating, cf.
(Yeramian et al., 1986; Bauer et al., 1987; Draber et al., 1993; Liu and Dilger, 1993; Keleshian et al., 1994). Where this level of understanding is not required, we find that analysis of steady state amplitude distributions will provide information about the coupling of pairs of ion channels.
Acknowledgements We thank Dr Grant Nicol and Dr Helen Mason for their help with the manuscript. Supported by a Grantin-Aid from the Western States Affiliate of the American Heart Association 98-257.
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