Journal o]"Neuroscience Methods, 42 (1992) 91 - 103

91

~:; 1992 Elsevier Science Publishers B.V. All rights reserved 0165-0270/92/$05.00

NSM 01350

Model invariant method for extracting single-channel mean open and closed times from heterogeneous multichannel records * S.V. Ramanan, S.F. F a n a n d P . R . B r i n k Department of Physiology and Biophysics, State Unit'ersity of New York. Stony Brook, NY l 1794-8661 (USA I (Received 5 September 1991) (Revised version received 7 January 1992) (Accepted l(I January 1992)

Key wonls: Channels; O p e n time; Closed time; Mean times; lnvariance; Heterogeneity We present a proof that the mean open (and closed) times of the individual channels in a multichannel record can be found in a model-independent fashion. As the results are model independent, they can be derived by assuming the simplest model for all the channels, namely that they all have the basic C L O S E D - - ' O P 1 2 N scheme. In particular, the method can be applied to patches where the channel population is heterogenous ~ith respect to open probability. Multichannel simulations are performed to test the limits of applicability of this method to restricted amounts of data. One conclusion is that increasing the number of channels does not substantially reduce the errors in estimating the mean times, in spite of the 'increased information" present. We also prove the general applicability of the algorithm of Fenwick et al. (1982) in estimating the mean times without knowledge of the number of channels present, and discuss its limitations. An illustration using experimental data is also given.

Introduction Extracting mean open and closed dwell times in a record containing only 1 channel is a modelindependent process. For records where more than one channel are simultaneously active, it is possible to extract the open probabilities of the individual channels, provided they are independent. This extraction is a model-independent method as these probabilities obey a simple multinomial distribution. Knowledge of the open probabilities effectively amounts to knowledge of the ratio of the mean open to mean closed dwell time fl)r the individual channels. If all the channels in such a multichannel record are identical and independent, we can use an algorithm proposed by Fenwick et al. (1982) to

* This work was supported by National Institutes of Health grant HL 31299.

Correspondence: S.V. Ramanan, Dept. of Physiology and Biophysics, State University of New York, Stony Brook, NY 11794-8661, USA.

extract the mean open time for these identical channels. This algorithm is distinguished by the fact that it does not require knowledge of the number of channels in the multichannel record. Horn and Lange (1983) proved that the mean open time yielded by this algorithm maximizes the likelihood function under certain restrictions. For stationary records, i.e., records obtained with the m e m b r a n e potential held constant, these restrictions require that the kinetic Markov scheme for these identical channels have only one open state which closes by the same path as it opens. In a recent article Dabrowski et al. (199t)) have found a method to discriminate between independent and cooperative behaviour for a patch containing channels with identical kinetic schemes (and thus identical open probabilities). A side product of this method is that it yields estimates of the mean open and closed times for identical channels with identical probabilities as well as the pdfs of the open and closed states. In what follows, we show that, for independent channels with Markov kinetics, the mean open (and closed) times for individual channels in a stationary multichannel record can be found in a

~2

model-independent fashion. This observation is true even if the channels in the patch have different open probabilities and, indeed, different kinetic models. The model independence of the method places the mean open and closed times on a par with the open probability as fundamental single-channel characteristics extractable from multichannel records. The method enables us to also prove the generality of the algorithm of Fenwick et al. (1982) in finding an averaged mean open time for an arbitrary multichannel patch without knowledge of the number of channels.

Theory Single channel Define the transition matrix Q and its partitions Qcc, Qco, Qoo and Qoc for a single channel, as conventionally defined (Fredkin et al., 1985). The term ' o p e n aggregate' is used to mean the collection of all the ' o p e n ' states, and similarly for the closed aggregate. This reference and Colquhoun and Hawkes (1981) provide an expression for the open dwell time density f~,(t): T

fo(t) =

uc WcQco exp(Qoot)Qocuc T Uc WcQcouc

where u c is a column vector of ones and W c is a diagonal matrix of the probabilities of the closed states. We then find for the mean open time % r,, = f t f o ( t ) dt T

-2

Uc WcQcoQoo Qocuc T

uc WcQcoUo T

-2

Uo WoQooQoo Qocuc T

Uo WoQooU o since the row probability vector u~W c + u~W o is a left eigenvector of the transition matrix Q with eigenvalue zero, i.e., uTWcQco + uoWoQoo V = o. Using conservation of probability, i.e., Qoouo + Qo~u c = 0, this reduces to T

/Ao Wo/A o To ~

T

Uo WoQooUo

Denoting the total probability of the open aggregate by Po = u "r o W,,u o, we find the basic: relationship, P,, -

-

-i

= -u,,WoOoouo.

(1)

TO

The ith entry of the column vector - Q o o u o is the sum of the transition rates from the ith ()pen state to all states not in the open aggregate. The ith entry of the row vector u T o W o is the equilibrium probability of the ith open state. Stated in words, then Eqn. 1 says that the number of openings per second ( -- Po/r o) is equal to the product of (a) the equilibrium probability of each open state and (b) its transition rate to exit the open aggregate, summed over all open states. Denoting an arbitrary open state by i, its equilibrium probability by p~, and its transition rate to exit the open aggregate by %, Eqn. I can be rewritten as t'~,

7o

~

PiYi

(2)

l ~ open

A similar formula holds for the closed aggregate.

Two channels Consider any kinetic scheme which has a number of aggregates greater than 2. Pick any one of the aggregates and call it the ' o p e n ' aggregate; lump all the other aggregates into a 'closed' aggregate. It is easily verified that the derivation above carries through unchanged for the 'open' aggregate. As Eqns. 1 and 2 relate only the properties of the 'open' aggregate, a corresponding result must be true for every aggregate. A more formal proof may be provided by using the notation of Fredkin and Rice (1986). Now consider 2 channels with transition matrices Q and Q'. We can construct a transition matrix for a system consisting of 2 such independent channels. Every state in this 2-channel scheme will be connected to other states only by transitions of a single channel, and with the same rate constant as that in the single-channel scheme. Such a composite scheme may be denoted by the symbol Q ® Q ' , which implies that every composite state is, in some sense, a 'product' of the

93 individual single-channel states. This scheme has 3 aggregates: (1) both channels closed, (2) one channel closed and the other open, and (3) both channels open. These aggregates are denoted by cc', co (co' and c ' o ) and 00', respectively. Using Eqn. 2 we find Poo '

- E

To°'

E piiv,i

(3)

i~o j~o'

The assumption of independent channels implies that (1) ~,~j = y~ + yj and (2) p,.j =PiPs. Using this, we find

Note that since Poo' = PoP° ', this quantity can be cancelled out of the equation. For the aggregate with only 1 channel open, we have

-E Toc

E PijYij + E

i~oj~c'

E p,jv,j

i~o'j~c

and as above, we can reduce this to

"~ Toc

P,,P~.,

--

+

+

Po,Pc

TO

+

--

.

(4)

Tc

The key point about Eqns. 3 and 4 is that they do not contain any reference to the individual kinetic models of the different channels. In particular, they could have been derived by assuming that the channels all have the simple C----'O kinetic scheme, thus showing the model independence of the method.

Arbitrary' number of channels The above procedure can be repeated for any collection of independent channels. The net result is that, as for 2 channels, the mean open and closed times can be found without knowledge of the kinetic schemes of the individual channels. We only present the results. Let there be n independent channels in a patch, each of these channels can be represented by a Markov scheme. Experimentally, the following are known. (1) Pi, the equilibrium probability of the ith level, i.e., that i channels are open, i = 0, 1 . . . . . n. (2) k,, the reciprocal of the m e a n occupancy time of the

ith level, i = 0, 1. . . . . n. The following can then be found in a model-independent fashion: (a) &, the equilibrium open probability and qj = 1 - P i , the equilibrium closed probability of the jth channel, j = 1, 2 . . . . . n. Maximum likelihood procedures for determining the ps in the case where all the channels are identical are given in Korn et al. (1981) and Sachs et al. (1982). Extension of this procedure when there are channels with differing open probabilities in a patch is discussed in the methods section. (b) a j, the reciprocal of the mean closed time, and /3i, the reciprocal of the mean open time for the jth channel, j = 1, 2 . . . . . n. These are specified by (n + 1) linear equations, one corresponding to each level. Consider the equation corresponding to the jth level. The right-hand side of the equation is Pjk i. The n left-hand side is the sum of ( j ) terms, each of which corresponds to a unique configuration of the n channels such that j of them are open. For example, if the first j channels are open and the remaining channels are closed, the term corresponding to this configuration is Pi

qi i=j+ 1

/3i + i

i

For an arbitrary configuration with j channels open, the corresponding term on the left-hand side would be the product of (a) the probability of the configuration and (b) a sum of n reciprocal mean times, one for each channel, this being/3 if the channel is open or ce if the channel is closed. For illustration, we write the 4 relevant equations when n = 3. Equation set 5

qlq2q3[cq + a 2 + a3] = Pokll P,q2q3[/3, + a2 + °e3] + q, P2q3[ ce, +/32 + oe3] + qlq2P3[Cel + Ce2+ /33] = Plkl PlP2q3[/31 +/32 +ee3] +qlP2P3[ ce, +/32 +/33] + PlqzP3[/31 + ce2 +/33] = P2k2 Pl P2P3[~S1 +/32 +/33] = P3k3 Note that eel//3 i =pJq~. These, along with the above, provide (2n + 1) for the 2n quantities oe

~4

and /3. The consistency equation for this overdetermined set is

E t'~k,= E P~kg i odd

(6)

i even

This says that the number of openings must be equal to the number of closures. It is worthwhile reemphasizing that the simple structure of Eqn. set 5 results only because the method for extracting the m e a n and closed times is a model-independent one. Thus the equations for deriving these from the experimental data cannot be more complicated than that produced by assuming the simplest C---~O model for all the channels; this results in simple coefficients from a multinomial formula arising as prefactors of the linear combinations of the a s and /3s. From these (2n + 1) equations, we can use singular value decomposition (SVD) (Press et al. 1988) to find the values of the mean open and mean closed dwell times. This method is the one preferred in finding the solution of linear equation sets such as Eqn. 5 above. Thus in Eqs. 5, there are only 2n variables to be solved for, whilst there are (2n + 1) equations. As noted in Eqn. 6, these equations are consistent, but use of SVD enables us to proceed in the solution without actually inserting Eqn. 6 to eliminate one of the Eqn. set 5. Such a brute-force insertion would also remove the symmetry of the formulation. A more pertinent reason for using SVD is that if some of the channels are identical, then the Eqn. set 5 for the mean times is singular, since there are fewer than 2n variables to be solved for. A further complication is that Eqn. set 5 cannot distinguish between e.g., the following 2 cases: (a) 2 channels both with open probability p = q = 0.5, but the first channel has 1 / a = 1//3 = 0.5 and the other has 1 / a = 1//3 = 1.5; (b) 2 channels both with p = q = 0.5 and both with l/c~ = 1//3 = 1 s, i.e., the mean for the 2 channels in case (a). In this case SVD yields the 'closest' solution in the sense of least norm for the difference vector. Such a solution has the property that, e.g., in the above scenario, case (b) would be preferred, i.e., all channels with the same probability would also have the same mean times (see also Dabrowski et al., 1990).

We check for consistency of this 'closest' solution by verifying that the values for the mean open and closed times obtained do indced givc the correct open probabilities. Any discrepancy here could mean that the channels in the patch are not independent. Finally, if the mean open closed or open times given by SVD arc negative, and thus physically impossible, then this could be another indicator that the channels in the patch are not independent.

The algorithm of Fenwick et al. We finally prove the extensibility of the algorithm of Fenwick et al. (1982) to a general multichannel record. This algorithm says that the quantity /3, which is the total number of closing (or opening) transitions, divided by the total time that all the channels are open, is an estimator of the averaged inverse mean open time. In the notation above, this is the equation

~_

i:o

(7)

~iPi i-1

The numerator may be verified, on inspection, to be equal to ~';=lPi/3iPi. The denominator can be shown by induction to be equal to ~=lPi.~ is thus a weighted average of the mean open time of the various channels in the patch, the weighting factor being the open probability. If all the channels have identical open probability, then =/3, the inverse mean open time for any channel, as expected.

Methods Similation of any single channel model is done by the method of Clay and DeFelice (1983). The single-channel records are then composed into a multichannel record, from which the occupation probabilities (Pi) and the mean occupancy time ( 1 / k i) are found for each level i. Maximum likelihood procedures are then used to find the most probable number of channels in this simulated record. This is described below.

95

We consider first the case of identical channels. Denote the number of channels actually seen in the simulation by Nsee,. Let N~hut and N,,pen denote the number of channels which never opened and never shut during the time of the simulation, respectively. The most likely open probability /3 for these channels is given by the formula N,ccl~

/3 = Y'. (i + N.pen)Pi/Uto,,

(8)

i- 0

where Ntot = N~e, + N~hut + Nop~, is the total number of channels assumed to be present (Korn et al. 1981; Patlak and Horn, 1982). The log likelihood ( L L ) for this case is N, .....

LL ct ~, P, log

[ ( N t o t ] p N ....i

i == (I

i + Nooen J

×(l-p)i+N

...... [,

N (9)

where the term m square brackets is the theoretical probability of the ith level. N~hut and Nopen are varied to find the values at which L L is maximized. Consider the case where there are 2 types of channels numbering N 1 and N 2 , respectively (see also Glasbey and Martin, 1990). For a given N,,p~ and Nshut, we must also have Ntot = N 1 + N 2. The most likely probabilities/31 and/32 for the 2 types of channels are related by the formula (see the appendix for a proof of the generalized equality), Nseen

N,/3,+N2/32=

(i+Nopen)Pi"

E i

(10)

0

The L L is again given by Eqn. 9, but with the appropriate formula for the theoretical probability. We use Brent's method (Press et al. 1988) to search for/31 which maximizes the L L for given N~hut, Nooe~ and N I. This method is useful in searching for minima of one dimensional continuous functions. It requires that the solutions be 'bracketed', i.e., that the solutions lie in a predetermined and prespecified range. Since we are maximizing for probabilities that can lie only between () and 1, this method which almost 'surely' finds a maximum (through a combination of

parabolic interpolation and golden-sections) is quite suitable. Thus N l is varied for given Nt,,t, and Nshut and Noven are varied to find the values of Nope., N~hut and N 1 at which LL is maximized. It should be noted that, for fixed Nope,,, N~hut and N l, the L L in the 2-channel-type case may (and often does) have an additional local maximum in additional to the global maximum in /31. The 2 maxima have been found in all cases to lie on opposing sides of the value of /5 predicted to maximize the L L assuming only identical channels. This information must be used to bracket the maximums for Brent's method, i.e., search first in the bracketed range [0, /3] and then in the range [/3, 1] and compare the two maxima in LL obtained to find the true global maximum. We moreover assume that the L L has no local maxima when Nopc,1, N s h u t and N l are varied. We do not consider more than 2 types of channels in our simulations. Theoretically, these cases can be handled similar to the case with only 2 types discussed above, except for the replacement of Brent's method by a suitable multidimensional maximization algorithm (e.g., the method of Nelder and Mead described in Press et al., 1988). It should be noted that additional local maxima, as discussed above, are likely to be present. Once the best/3s have been found by maximizing the LL, the mean open and closed times for the individual channels can be found by a direct application of the singular value algorithm to the equations in the theory section (for example, Eqn. set 5 for the case when the number of channels is 3). In plotting the results, we calculate the mean and SD of the inverse mean open and closed times/3 and a from 100 simulations. We also plot the mean and SD of the number of channels which maximize the LL over all simulations.

Estimation of errors The right-hand side of Eqn. 5 is the quantity

Pik, which can be rewritten as N J T , where N i is the number of events at level i, and the T is the total time of recording. With the usual assumption that the error in measuring N~ is of order }."Ni , we can scale all the Eqns. 5 appropriately by this

~h

error factor; such a scaling is necessary for the SVD algorithm to work. A side product of SVD then is that it yields a variance-covariance matrix for the estimated parameters c~ and/3. As pointed out in the results section, this estimate of the variances of the parameters is of somewhat dubious value as the channel kinetics may have correlations. These correlations may cause substantials bias in the mean times experimentally measured for the various levels, and thus in the final estimates. For a single channel, the following estimates for the p a r a m e t e r s a and /3 found in a recording time T can be explicitly calculated for the simple C---~O scheme (Manivannan et al., 1992). These are given approximately by A a / o ~ - : a + b / o ~ ; A/3//3 = a + b//3; where

a= ~

+/3

; b=~/(a+~)/2T

Given the estimates for a and /3 from SVD, we use these formulas to estimate errors. As these are calculated for a scheme with no correlations, the actual errors are probably 2 or 3 times as large; so the calculated errors serve only as very approximate estimates.

Results

Identical channels Fig. 1 shows the results of simulations on the basic model C----"O. All simulations are 60-s long, and a, the inverse mean closed time, is fixed at 10/s. In all figures, we plot the ratio of the means and the SDs of the p a r a m e t e r s estimated from the simulations to their theoretical values. The parameters plotted are a , / 3 and N, where N is the number of channels found by maximum LL. The display of the p a r a m e t e r s for any abscissa are staggered, ce(O) to the l e f t , / 3 ( x ) in the center and N ( [ ] ) to the right, to avoid overlaying the error bars. In all cases discussed in the results section, the inverse mean open times estimated by the algorithm of Fenwick et al. differed insignificantly from the corresponding value of 13

estimated from SVD. Fig. la shows the results when the number of channels is fixed at 4, and /3 is varied. When /3 = 0.1, the number of e v e n t s / channel for the 60 s is ~ 6 and the estimate for/3 is significantly different from its expected value. This deviation seems to be correlated with an underestimate of the number of channels present (see discussion of Fig. l b - d below). As the figure is symmetrical for a and /3 around /3 = I0, except for the number of e p i s o d e s / c h a n n e l , it may bc noted that it is essential to have ~ 100 e p i s o d e s / channel to obtain reasonable estimates for the inverse times. It may also be noted that the error in the estimates of the number of channels increases when either the open probability p is high ( > 0.9) or low ( < 0.1). Such error in estimating the number of channels increases the error in ~e (when p is low) or/3 (when p is high). Fig. lb, c and d show the cases where /3 is fixed at 1, 10 and 100/s, respectively, and the number of channels is varied from 2 to 9. Note, for purposes of comparison, that the scale of the y-axis in Fig. le differs from the other figures. Note in all cases that the errors do not decrease significantly with increasing channel number, in spite of the increasing information. This is because the LL best estimate of the number of channels can differ from the actual number for large number of channels. This difference seem to compensate for the 'extra' information from more channels. Even in the case when ~e =/3 = 10, where the error seems to decrease for moderate channel number, this error starts to increase again when this number is greater than 8. It is also worth noting again that c~ is estimated well when the open probability is high ( ~ 0.9, Fig. lb) while /3 can be found with little error when the open probability is low ( ~ 0.1, Fig. ld). When the open probability is equal to 0.5, both a and /3 can be found to within 10% (Fig. lc). Of Fig. l b - d , the worst errors appear in Fig. lb, when /3 = 1 and there are ~ 60 e v e n t s / c h a n n e l s for the 60 s of simulation. As Fig. l b and ld are complementary with regards to oe and /3, we see that the error approximately halves when the n u m b e r of episodes increases from ~ 6 0 / c h a n n e l (Fig. lb) to - 6 0 0 / c h a n n e l (Fig. ld). Comparing Fig. lc and d, we further note that an accurate knowl-

97

edge of the number of channels also reduces the error for a given number of episodes significantly. Fig. 2a shows a plot identical to Fig. 1, but for a model of the ACh receptor (Colquhoun and Hawkes, 1981).

be compared

to Fig. ld, where

the open

and the number

of events/channel

Again,

that

we

find

while the error

C - 26 . C 1.9X 104 O . 1 (I 4 i000

¢=I0

The number of e v e n t s / c h a n n e l for the 12 s is ~ 600. As the open probability is ~ 5% this can

/3 is e s t i m a t e d

in the estimate

p ~ 10%

is a l s o

well,

of a seems

to be

c o r r e l a t e d t o a n t h e e r r o r in t h e e s t i m a t e n u r n D c r or c l l a n n c l 5 p r c ~ c n t ,

C--O

a=lO p=lO

N:4

~ 600.

quite

of the

C--O

4 8 o

3

o

J= -=

-=

tO

2 vl

I

il

/ J &

0

........

,

10"2

........

I0"1

,

100

........

,

......

~a

'"

........

'

102

'

103

''

0.9

.....

3

104

| of idmka cMmm

a=lO p=s

2 o

C--O

==10 p=lO0

.~

V~ 2

3

4 5 6 7 ~1of ide~ic'a chome~

C--O

2

i -g

8

9

10

~ 0

2

TI

TI

i-i v l

3

4 5 6 7 t of ide~ica ¢ h o ~

8

TT

i z

9

Fig. 1. a: semilog plot where the total number of channels is fixed at 4. Each channel is considered as a 2-state model (C----"O), with opening rate constant a fixed at 10/s, while the closing rate constant /3 is varied from 0.1 to 100/s. All simulations are 60-s long, and each point describes the mean and SD of various parameters over 100 simulations. The ratio of simulated to theoretical inverse mean closed time (~, ), inverse m e a n open time (/3, × ), and n u m b e r of channels ([]) are plotted vs. log(/3). See the explanation in the text for the deviation in the simulated to theoretical ratio of channel number and /3 from the expected value of 1 when /3 is 0.1/s. For all points the SDs are plotted as well. For convenience the ratioed values of a, /3 and channel number are staggered so that the error bars (SD) can be distinguished for any point on the abscissa, b: simple c l o s e d - o p e n (C-~-O) channel model. In the case shown the inverse m e a n closed time (o~) is 1 0 / s and the inverse mean open time (/3) is 1/s. All 3 normalized parameters ( a , / 3 and channel number) are plotted vs. the n u m b e r of identical channels, which is varied from 2 to 9. The values are again staggered about the abscissa values to allow easy determination of the error bars. In this case no significant deviation from the expected value of 1 for all parameters occurs regardless of the n u m b e r of channels, c and d: variants of (b) where c~ is fixed at 1 0 / s while /3 (inverse mean open time) is varied such that the a//3 is 1 in (c) and 0.1 in (d).

Fig. 2b is a plot similar to Fig. 2a for a model for the maxi-K channel with [Ca]~ = 0.5 ~zM (Magleby and Pallotta, 1983, Eqn. 1 and Table I, column 2) for 25 s of simulation.

a = 49.84

C --- C --- 0

1]= 1000 2

Model for the

a

AcHreceptor

I

o 6011

1811

o

120 J 1811

39511 I 322

O O The n u m b e r of e v e n t s / c h a n n e l is ~ 3750. This can be compared to Fig. lc ( p = 0.5 in both) where the number of e v e n t s / c h a n n e l is 300. In spite of the fact that the number of e v e n t s / channel is greater by a factor of ~ 10 in Fig. 2b, the errors in Figs. 2b and l c are approximately the same. We attribute this to the fact that the model of the maxi-K channel has intrinsic correlations (see e.g., Fredkin et al., 1985; McManus and Magleby, 1988) i.e., the length of successive openings (and closures) are not independent. Such correlations may cause the channel to go into a ' m o d e ' , i.e., a subset of the state space, for a prolonged time. As the channel does not visit all the states available in a short recording time, the fluctuations in the mean times are greater than in a model where such correlations or modes do not exist as in Fig. lc. The existence of a rarely visited closed state might be the reason for the differences in the errors of c~ and /3 in Fig. 2b. As the best estimate of the number of channels occurs when p = 0.5, a reasonable experimental protocol for a patch with identical voltage-dependent channels would to be to set the m e m b r a n e voltage such that p = 0.5, and use this estimate for other potentials where the open probability takes on other values. If, however, the patch contains various types of channels whose open probability has different voltage dependencies, there are additional complications discussed below. In some cases, setting the m e m b r a n e potential such that p = 0.5 for one type of channel may even lead to more serious errors than for other probabilities.

Two types of channels As stated in the methods section, we restrict our consideration to cases where only 2 different

!

2

3

4

5

6

7

8

9

10

t of ACh channels

1.1 . . . . . a = 295.9

C --- C --- C

= 307.5 ~ = I

Model for

b

0 0 the ~xi-Kchonnel.

x

-7 o

1.0

1 0.9 2

3

4

5

6

7

8

9

1O

t 0f Jl~i-K cl~nnet~ Fig. 2. a: simulated to theoretical ratio of inverse mean closed and mean open times (a and /3) and channel number for a well known 3-state model of the A C h channel (Colquhoun and Hawkes, 1981) as the n u m b e r of channels in the simulation is varied. Inverse mean closed time (c~) of each channel was 49.84/s and inverse m e a n ()pen time (/3) was 1000/s, i.e., the mean closed time is ~ 20 ms and the m e a n open time is 1 ms. b: same plot as shown in (a) but for a 5-state model of the maxi-K channel (Magleby and Pallotta, 1983). Again, all parameter ratios are plotted vs. the n u m b e r of maxi-K channels present, a is 295.9/s and /3 is 307.5/s, i.e., the mean closed time and mean open time are 3.38 ms and 3.25 ms, respectively.

types of channels are simultaneously present in a patch. Fig. l b - d give 3 prototypes, namely channels based on a C---~O model, with different probabilities of = 0.1, 0.5, and - 0 . 9 , respectively. We assume patches where there is 1 channel of one of these prototypes (called type I), and a varying number (1-8) of another of these proto-

9q

types (called type II). This gives 6 possibilities, some of which are illustrated in Figs. 3 and 4. The top panel of Fig. 3 shows the parameters c~, /3 and N for the sole channel of type I in the patch. The corresponding bottom panel shows the same parameters for the other type of channel (type II), the number of which is present varies from 1 to 8. All the parameters are scaled

I

I

1111°f T ~ I d ~

1

to their respective theoretical values, which arc printed in the graph. For purposes of comparison, note that the scale of the y axis may differ from graph to graph. Severe deviations from the theoretical values occur in Fig. 3. Here variable number of type II channels with p = 0.5 are paired with one type I channel with p -- 0.1 in Fig. 3a and 0.9 in Fig. 3b.

I

10

7

o 1- 10

C--O

pl,,m

6

9

I

I

I

i

I

a1=10 pl. 1

C---O

0 2.'. 10

(2 - - 0

8 7

5 4

5 3

2

tm

I 0

2

2

I 0

4

~2., 10

C --0

I~-10

0

0

0

1

2

3

6 # of TypeI chneb

8

1

2

3

4 5 tl of Tl~e Id~m~b

Fig. 3. Results from simulations of patches containing 2 channel types. In all patches, the number of type I channels is fixed at 1, while the number of type II channels is varied. A closed-open (C---'O) model is used for both types. The inverse mean closed and open times for the single type I channel are a 1 = 10/s and /31 = 100/s, and every type II channel introduced has a z = 10/s and /32 = 100/s. The top panel shows the normalized parameters (simulated to theoretical ratios of a, /3 and channel number) for the single type I channel, and the bottom panel shows the normalized parameters for the type I1 channel. The x axis in the top panel serves as a reminder that there is only one type I channel in all the simulations, while the x axis in the bottom panel shows the increasing number of type II channels. Note that the simulated to theoretical ratio for channel number in the top panel (type I) deviates significantly from 1 as the number of type II channels is increased. With a large number of type !I channels in the patch the ability to accurately find the number of type I channels ( = 1) becomes impossible (see text for explanation). As before, squares are channel number, diamonds are a and :x: s are/3. SD is plotted as error bars.

I{)()

3

I ¢=1= tO ps. lO

,

~ C~O

I/ i

i

o

e2=~

C--O

~-B -~

2

iJ.T

fat~laumn Fig. 4. Conditions similar to those of Fig. 3, except here we have a ~ = 10/s and/3 ~ = 10/s for the single channel of type I and a 2= 10/s and /32= 100/s for all type II channels present, i.e., the parameters for the type I and type II channels are exchanged from the ones in Fig. 3a. The upper panel is a plot of the 3 normalized parameters for the single type 1 channel, and the lower panel is a plot of the 3 normalized parameters for type II channels vs. the total number of type 1I channels. In this case, the normalized parameters for type I and II channels seem to be good indicators of channel number and behavior.

A s t h e n u m b e r o f t y p e II c h a n n e l s with p = 0.5 increases, t h e e x t r e m e c l o s e d a n d o p e n states a r e r a r e l y visited, as all the p r o b a b i l i t y is c o n c e n t r a t e d in t h e states w h e r e t h e n u m b e r o f o p e n c h a n n e l s - - - t h e n u m b e r o f c l o s e d c h a n n e l s . In such cases the L L a l g o r i t h m s e e m s to find a

m a x i m u m by i n c r e a s i n g the n u m b e r o f c h a n n e l s with high or low o p e n probabilities. This causes a c o r r e s p o n d i n g increase in e r r o r in the e s t i m a t e s for the m e a n times. As a n t i c i p a t e d in the previous section, if the m e m b r a n e voltage is set such that p = 0.5 for 1 type of c h a n n e l , and t h e r e arc also c h a n n e l s with low or high p r o b a b i l i t y simult a n e o u s l y p r e s e n t , this could cause skewness in the various estimates. N o t e that this severe i n c r e a s e in skewness with increasing n u m b e r o f c h a n n e l s d o e s not h a p p e n in the case c o m p l e m e n t a r y to Fig. 3a as shown in Fig. 4. H e r e the c h a n n e l type with p = 0.5 is now the sole c h a n n e l of type I and is p a i r e d with variable n u m b e r of c h a n n e l s of type 1I with p --0.l. In b o t h figures, a l t h o u g h the e r r o r increases with i n c r e a s i n g c h a n n e l n u m b e r , the d e v i a t i o n of the m e a n from the t h e o r e t i c a l value is much less t h a n the c o r r e s p o n d i n g o n e s in Fig. 3. In t h e o t h e r cases w h e r e t h e r e is a m i x t u r e of c h a n n e l s with p -~ 0.1 a n d p -~ 0.9, the predictions o f the L L a n d S V D a l g o r i t h m s c o m p a r e favorably with the t h e o r e t i c a l values. In all cases, the e r r o r in e s t i m a t i n g t h e p a r a m e t e r s for the u n i q u e c h a n n e l in the p a t c h s e e m s to i n c r e a s e linearly with the total n u m b e r o f channels. T h e best e s t i m a t e in all t h e figures is seen to occur when t h e r e is only 1 c h a n n e l of e a c h type in the patch. In all cases, d a t a not shown i n d i c a t e s that the m a x i m u m e r r o r is r e d u c e d by a factor of 2 for an i n c r e a s e in the n u m b e r o f e p i s o d e s / c h a n n e l by ~ 10, as was also n o t e d in the discussion o f Fig. 1. As an illustration of the m o d e l i n d e p e n d e n c e of the m e t h o d , we c o n s i d e r a s i m u l a t i o n w h e r e the p a t c h c o n t a i n s a m i x t u r e of A C h c h a n n e l s (Fig. 2a) a n d m a x i - K c h a n n e l s (Fig. 2b). T h e total n u m b e r o f c h a n n e l s is fixed at 5, a n d the n u m b e r o f m a x i - K c h a n n e l s is i n c r e a s e d from 1 to 4. C o r r e s p o n d i n g l y , the n u m b e r o f A C h c h a n n e l s d e c r e a s e s from 4 to 1. N o t e that the m a x i - K c h a n n e l has ~ 150 e p i s o d e s / s / c h a n n e l while the A C h c h a n n e l has ~ 50 e p i s o d e s / s / c h a n n e l . T h u s as the n u m b e r of m a x i - K c h a n n e l s increases, t h e total n u m b e r o f e p i s o d e s in the 10 s o f s i m u l a t i o n also increases. W i t h t h e p a r a m e t e r s c h o s e n the A C h c h a n n e l has p ~ 0.05 while for the m a x i - K c h a n n e l , the o p e n p r o b a b i l i t y p = 0.5.

lOI t of ACh ch0nne~ :3 I

a I = 49.84

3 (large number of channels with p = 0.5) and Fig. 4 (large number of channels with p = 0.1). However, the method presented is able to extract the mean times for both types of channels, despite the difference in the actual model for the 2 types.

2 I

C--C--O

pi = 1000

) Conclusion

= 295. 9

c -- ,c -

!

= 307.48

~)

0

x o o

.,Y

i

1

2 3 t of Maxi-K c~nnels

4

5

Fig. 5. Another variant of Figs. 3 and 4, except that the patch now contains a mixture of ACh channels (3-state model) and Maxi-K channels (5-state model). The top panel is a plot of the 3 normalized parameters for the 3-state model (ACh channel) plotted vs. the number of ACh channels ill the simulation. The lower panel represents a plot of the 3 normalized parameters for the maxi-K channel (5-state model) vs. the number of maxi-K channels present. Note that the total number of channels, ACh and maxi-K together, remains constant and is equal to 5 in all cases shown. Thus one goes from a simulation with 1 ACh channel and 4 maxi-K channels (lef-hand side) to a simulation with 4 ACh channels and 1 maxi-K channel (right-hand side). The mean inverse times a and /3 [or the 2 channel types are the same as those given in Fig. 2a and b. Reasonable estimates of all parameters are obtained in all cases.

The results are plotted in Fig. 5, which may be compared to Figs. 3 and 4. Thus as the number of maxi-K channels increases, the errors and deviations from the means increases for all parameters, in spite of the increased total episodes. This corresponds to results discussed previously in Fig.

A model-independent method for estimating the mean open and closed times for individual channels in a multichannel patch is useful both in cases where the channel population is homogeneous or heterogeneous. Theoretically, these mean times can be found by solving a set of linear equations, provided the number of channels in the patch and their open probabilities are known. Finding these last, however, is a non-linear process which involves use of likelihood techniques. Any improvement in this process would substantially reduce the errors in estimating the mean times from a limited amount of data. By way of applicability to actual data, Fig. 6 illustrates a histogram from an excised patch of the 100 pS channel from the septum of the earthworm giant nerve fiber under conditions similar to those described in Brink and Fan (1989). The actual Cs concentration was 170 mM on both sides. There are 5 channels seen during the 5 min of recording; the total number of openings during the entire record is 7262. Experimental parameters for the various levels are given in Table I. LL procedures applied to the idealized record gives a maximum L L with 5 channels, 1 channel with p = 0.91 (type I) and 4 channels with p - - 0 . 0 8 8 (type II). The fit with these probabilities is also shown in Fig. 6 in dotted lines and is seen to be reasonably good. Use of the SVD algorithm gave a 1=35.18_+ 1.5 s- i and / 3 1 = 3 . 4 4 + 0 . 4 s i for the inverse mean closed and open times for the single type I channel, and c~ = 7.42 +_ 0.7 s 1 and /32=77.31_+ 1.5 s 1 for the 4 type II channels. We again note, as pointed out in the methods section that the actual estimates are probably a few times larger than the ones given here if there are any correlations in the single-channel kinetics.

11!2 "FABLE 1 EXPERIMENTAL PARAMETERS C H A N N E L R E C O R D IN FIG. 6

FOR

THE

MULTI-

Level

Probability

Events

Meantime (ms)

0 [ 2 3 4 5

0.061731 0.652794 0.251068 0.031383 0.002825 11.000200

1 111 5 7111 5971 1 548 175 19

14.8 30.5 11.2 5.4 4.3 2.8

It has been noted in the literature that the algorithm of Fenwick et al. (1982) yields an estimate for the mean open time without knowledge of the number of channels in the patch (for identical channels). It should be noted that this is

true only if the current level with all channels closed or all channels open is observable. If, as in the case of a large number of channels with p = 0.5, these levels have low probabilities and thus may not be seen in a limited time of recording, the above algorithm has to be used in cot]junction with a maximum LL algorithm which estimates the most likely number of channels which never opened or closed during the time of observation. In our simulations, we have ignored the issue of dead time or instrument bandwidth. Such a finite dead time introduces systematic errors in the m e a s u r e m e n t of fast rate constants. For single channels, various strategies to correct for limited bandwidth have been noted (Blatz and Magleby, 1986; Crouzy and Sigworth, 1990). These

6BB81 48B8]

248e

-4

*16

+4

+12

.211

.24

+28

*Z8

+32

+28

tall

pA

Fig. 6. Amplitude histogram of the current distribution from a multichannel detached patch from the s e p t u m of earthworm giant nerve fibre. The holding potential was + 4 8 INV. The vertical axis is the n u m b e r of digitized points and the horizontal axis the current. The dotted line is the fit of the histogram using gaussians, assuming 1 channel with open probability of 0:91 and 4 channels with open probability of 0.088. The raw data was filtered at 0.5 KHz and digitized at 360/zs. The SD of the thermal noise was 0.81 p A and the SD of the extra noise due each open channel was 0.57 pA. The peaks in the gaussians used in fitting the histogram were 0, 6,58, 12.8 18.8, 24.9 and 30.6 pA. The insert is an expansion of the amplitude histogram from 16 pA to 36 pA. It is shown primarily to illustrate the presence of a 5th channel within the patch.

103 s t r a t e g i e s a r e q u i t e c o m p l e x in t h e i r a p p l i c a t i o n . W c h a v e c h o s e n n o t t o a d d r e s s t h i s i s s u e h e r e . as the procedures to correct simulations to include l i m i t e d d e a d t i m e in m u l t i c h a n n e l r e c o r d s w o u l d be computationally extremely time-consuming.

Appendix

Consider an assortment of N channels, with open probabilities p, and corresponding closed p r o b a b i l i t i e s qr = 1 -- pr, 1