AND FITTING OF SINGLE-CHANNEL DWELL

Transcription

1 DATA TRANSFORMATONS FOR MPROVED DSPLAY AND FTTNG OF SNGLE-CHANNEL DWELL TME HSTOGRAMS F. J. SGWORTH AND S. M. SNE Departent ofphysiology, Yale University School of Medicine, New Haven, Connecticut 65 ABSTRACr A. L. Blatz and K. L. Magleby (1986a. J. Physiol. [Lond.l. 378: ) have deonstrated the usefulness of plotting histogras with a logarithic tie axis to display the distributions of dwell ties fro recordings of single ionic channels. We derive here the probability density function (pdf) corresponding to logarithically binned histogras. Plotted on a logarithic tie scale the pdf is a peaked function with an invariant width; this and other properties of the pdf, coupled with a variance-stabilizing (square root) transforation for the ordinate, greatly siplify the interpretation and anual fitting of distributions containing ultiple exponential coponents. We have also exained the statistical errors in estiation, by the axiu-likelihood ethod, of kinetic paraeters fro logarithically binned data. Using binned data greatly accelerates the fitting procedure and introduces significant errors only for bins spaced ore widely than 8-16 per decade. NTRODUCTON The rates of known conforational transitions in proteins span any orders of agnitude, fro picoseconds to hours, reflecting the fact that transition rates depend exponentially on underlying energy differences. t is therefore natural to plot relaxation data on a logarithic tie scale, as was done by Austin et al. (1975) in their study of optical absorbance changes in yoglobin over a tie scale of 2,As to 1 ks. Blatz and Magleby (1986a, b) have recently shown the utility of a siilar log-log representation for dwell-tie distributions fro single-channel recordings where the tie constants are spread over several orders of agnitude. McManus et al. (1987) have also considered the bias introduced into estiates of fitted paraeters by finite sapling intervals and binning. We present here an alternative, direct display ethod for histogras of constant logarithic bin width and derive the corresponding probability density function (pdf). We also present an iproved procedure for axiu-likelihood estiation of kinetic paraeters fro binned data, and we report the results of siulations to test the perforance of this procedure. Fig. 1 A illustrates the liitations of the traditional linear histogra in displaying a dwell-tie distribution having two exponential coponents. One coponent is well resolved on the tie scale chosen for this plot; it coprises 7% of the events and has a tie constant of s. The reaining 3% of the events belong to a coponent with a -s tie constant which is hardly visible on this tie scale. The other parts of Fig. 1 show the sae siulated data (5,12 events) and corresponding theoretical curves plotted on logarithic tie scales. Part B shows the representation introduced by Blatz and Magleby (1986a). Here the probability density, estiated by dividing the nuber of events in each bin by the bin width, is plotted on a log-log scale along with a log-log plot of the theoretical pdf. Parts C and D show the new representations to be considered in this paper, where the nuber of events in each bin (there are equally spaced bins per decade) is plotted on a linear (C) or square-root ordinate (D). Superiposed on the histogras are the appropriately transfored, theoretical pdfs. Under the transforation to a logarithic abscissa, the pdf corresponding to each exponential coponent of the distribution is not onotonic but has a peak at the value of the tie constant. The unusual shape of the pdf can be understood fro the fact that the bins at very short ties are narrow and so collect few events, whereas at long ties the frequency of events decreases exponentially, uch ore quickly than the increase in bin width. Thus a axiu is to be expected in the vicinity of the tie constant of the distribution. DSPLAY OF DSTRBUTONS Linear Histogras For coparison with the theory for the logarithic histogras, we first review the theory of the traditional linear histogra (see for exaple Colquhoun and Hawkes, 1983). A set of bins is defined as having a width bt and starting values belonging to a sequence of ties t,. Entries are ade in the bins according to the dwell ties observed, such that the nuber of entries ni in the ith bin is the nuber of events having durations t that satisfy ti < t < ti + bt /87/12/47/8 BOPHYS. J. Biophysical Society - Volue 52 Deceber $2. 47

2 ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.4. r.- - rb-- L ,,t- x 35C 3C 25C A o~ -o 4 B X 2 c 15c > 2 c 5 L 4. l i il l 7J i o Tie (s) Tie (s) [r]rr~~~~~~~~~~~~qrd]j1j]jq?]~~~~~~~~~~~~~~~~~~~~~, ' 2 co \15 > Li 5 o [ E-A-f<. C Tie (s) c 15,_s X c 5 Lu -5 vf ~ ~~~~~~~~~~~~~~~~ l, Tie (s) FGURE 1 Four representations of a dwell-tie distribution with two exponential coponents. 5,12 rando nubers were generated according to a distribution with tie constants of s (7% of the events) and s (3%) and binned for display as histogras in the lower panel of each part of the figure. Superiposed are the theoretical probability density functions for each coponent (dashed curves) and their su (continuous curve). n each part of the figure the upper panel plots the absolute value of the deviation of the height of each bin fro the theoretical curve, with dashed curves showing the expectation value of the standard deviation for each bin. The upper panels were plotted with vertical expansion factors of 2.1, 5.4, 3.1, and 4.9, respectively. (A) Linear histogra. Events are collected into bins of 1 s width and plotted on a linear scale. The -s coponent has a very sall aplitude in this plot. (B) Log-log display with variable-width (logarithic) binning. The nuber of entries in each bin is divided by the bin width to obtain a probability density in events/s which is plotted on the ordinate. (C) Direct display of a logarithic histogra. Events are collected into bins of width Ax =.2, and superiposed on the histogra is the su of two functions as in Eq. 11. (D) Square-root ordinate display of a logarithic histogra as in C. Note that the scatter about the theoretical curve is constant throughout the display. -i n order to copare the histogra values ni with a kinetic theory, a pdf is coputed fro the theory. The pdf is defined as a liit of the probability that the rando open tie t falls in a range centered on t, f(t) =li Prob(t h/2 < t < t + h/2) h- ~~~h Thus if the bin width bt is ade sufficiently sall, the expected values of ni becoe proportional to the value off at the center of the bin, 48 ni N6tf(ti + bt/2), (1) where N is the total nuber of events entered into the histogra. Single Exponential Distribution. The probability distribution function is defined as the probability that a rando dwell tie t falls below a given value t, F(t) = Prob (t < t). A kinetic process involving a single transition step gives rise to dwell ties having an exponential distribution. The probability distribution function for such a process with a BOPHYSCAL JOURNAL VOLUME

3 ean dwell tie r is F(t) = 1 - exp (-t/r), (2) The probability density function (pdf) is the derivative of the distribution function, f(t) = d F(t) = dt - exp (-tr). (3) T Logarithic Histogras Histogras can also be constructed by choosing bins to have constant widths on the logarithic tie axis, as shown in Fig. 1 C. Such bins have constant relative width; for exaple, one bin in this figure ranges fro 1. to 1.26 s, while another bin ranges fro to 126 s. We assue that the logarithic x-axis arises fro the transforation x = n t. (4) Let the bin width on the x-axis have the (diensionless) value Ax, and let the lower liit of the leftost bin be x,. Then the ith bin will have the lower liit Xi = Xs + imx which corresponds to the actual dwell tie where ts the range of tie values ti = ts exp (i5x), = exp (x,). The ith bin therefore corresponds to ti <t <ti exp (bx). Blatz and Magleby (1986a) accuulated their data into such bins, however for displaying the data they corrected for the variable bin width by dividing each ni by the width bti of each bin. By Eq. 1 this result is seen to be proportional to the pdf, n (S) - Nf (ti + bti/2). (6) These values were plotted on log-log coordinates along with the theoretical pdf, which in Fig. 1 B is the su of two exponentials. The pdf for Logarithic Histogras Our approach is to display the logarithic histogras directly, without the variable-binwidth correction of Eq. 6. Such a histogra is shown in Fig. 1 C, where the ordinate is siply the nuber of events in each bin. For coparison with theory, we copute the appropriate pdf for the logarithic tie axis by first transforing the probability distribution function and then differentiating it to obtain the new pdf. Starting with the exponential distribution function (Eq. 2) and transforing according to Eq. 4 we obtain the distribution G(x), where G(x) = F[exp (x)]= 1 - exp [-exp (x - x)], (7) xo = ln(r), the logarith of the tie constant. The corresponding probability density function g(x) is obtained by differentiating G with respect to x, g(x) = exp [x - x- exp (x - xo)]. (8) f we define the "generic" pdf as g(z) = exp [z - exp (z)] then we can write g(x) siply as g(x) go(x xo). (9) = - The function g(x) has three properties that are useful for our purposes. First, fro Eq. 9 it is clear that a change in the underlying tie constant X results only in a shift of the function along the x-axis, rather than a change of scale. Second, the axiu value of g(x) occurs when x = xo, i.e., at the logarith of the tie constant, where the value of g is e-1. Third, this axiu value is independent of the tie constant, unlike the pdf in the linear histogra (Eq. 3) where the axiu value varies as 1 /r. For coparison of the histogra with a theoretical distribution, we find analogously to Eq. 1 that if the grand total nuber of events is N, the expectation for the nuber of events in the ith bin is, in the liit of sall Ax, ni Nbxg(xi S + dx/2). () f the distribution consists of a su of exponential coponents, the pdf takes the for g(x) = Ej j-1 ajgo(x - sj), (11) where sj is the logarith of the jth tie constant, and aj is the fraction of the total events represented by that coponent. The sooth curve in Fig. 1 C was coputed as in Eq. 11 as the su of two ters, and the resulting g(x) was scaled according to Eq. to allow direct coparison with the histogra values. Square-Root Ordinate For the evaluation of fits to experiental data it is useful to know the characteristics of the expected scatter of the experiental points. Let the nuber of entries n in a bin have the ean value no and variance a. Assuing that n follows Poisson statistics, the variance is equal to the ean, na = no. SGWORTH AND SNE proved Display and Fitting ofsingle-channel Histogras 49

4 Thus when a histogra is plotted with a linear ordinate, the scatter is larger in the higher bins, as deonstrated by the upper panels in Fig. 1, A and C where the deviations between the theoretical and experiental bin heights are copared. On the other hand, plotting the histogra with a logarithic ordinate, as in Fig. 1 B, results in the largest apparent scatter fro the bins containing few entries. For Poisson-distributed bin heights the scatter takes on a constant size with the following transforation: y = n"2, (12) where y is the plotted value, and n is the nuber of entries in a bin. To deonstrate this property, consider an error bar corresponding to one standard deviation for a bin having expectation value no. The error bar would extend fro no to no + n'/2, which by the transforation of Eq. 12 is plotted as a bar of length by= n/2 -(no + n1/2)1/2 Y - (1 + l/yo)'/21. Expanding the ter in brackets as a Taylor series in yo' and reaining the first-order ter we obtain u) ~Yo( 2yo) 2 This transforation therefore yields error bars that have a constant length, independent of no. This property is deonstrated in Fig. 1 D, where the upper panel shows the agnitude of the scatter along with the constant estiate for the standard deviation throughout the width of the histogra. FTTNG OF DSTRBUTONS Maxiu Likelihood Methods The fitting of experiental sets of dwell-tie easureents is typically done by axiizing the logarith of the likelihood with respect to the set of fitting paraeters, denoted here by. The fit is based on assuing a particular for of the probability distribution of the tj, usually a su of exponential ters: in this case represents the set of tie constants and coefficients of the exponential coponents. The likelihood is equal to the probability of obtaining a particular set of observed dwell ties tj, given the for of the distribution and the paraeters, and is proportional to the product over the N observations N Lik = ts (tjl O), j,- where f(tj) is the probability density function evaluated at tj with the particular set of paraeters. Because the likelihood typically takes on very sall values, nuerical 5 (13) evaluation of its logarith is preferable. n practice the log likelihood is also corrected for the absence of very short and very long intervals that are issed due to experiental liitations. With this correction the log likelihood is given, within a fixed constant, by (Colquhoun and Sigworth, 1983) L(O) = 2 n [f (tjo)/p(tnn, taxl)], (14) where the tj are the N experientally observed dwell ties, and Atini taxo) =-- (15) is the probability that dwell ties fall within the range of experientally easurable ties characterized by ti. and t.,, coputed fro the probability distribution with paraeters. Binned Maxiu Likelihood The likelihood for binned data is the probability that a set of data results in a particular set of bin occupancies ni. The log likelihood fro binned data can be calculated as k JF(ti+1) - F(t,J )l L(13) = p(ts, tko) (16) where F(tiO) is the probability distribution evaluated at the lower bound t, of the ith bin using the paraeter values, and p(t,, tk) is, as before, the probability that the experiental dwell ties fall within the range of the k bins in the histogra (Eq. 15), p(ts, tko) = F(t,) - F(tkO) Notice that for the binned data we use the probability distribution function F(t) in the calculations, rather than the probability density (pdf) as in Eq. 14. We evaluate F(t) at each bin edge and take the difference; this gives the probability that an event falls in the bin. Taking this difference is equivalent to integrating the pdf over the width of the bin. Maxiu likelihood estiation fro binned data is equivalent to the unbinned estiation as in Eq. 15 in the liit of sall bin width. Fro Eqs. 1, 14, and 16 it can be shown that: Lb(O) - L(o) Prob k E ni n (5ti), i- (17) where bti is the width of the ith bin; equality holds in the liit of sall bti. Since in the fitting proble the ni and bt, are constants and only the paraeters are varied, the particular that axiizes L will also axiize Lb. For a su of exponential coponents the probability distribution function takes the for F(tjO) = 1 - (tnin --- t < taxlo)l j-1 aj exp (-t/rt), (18) BOPHYSCAL JOURNAL VOLUME

5 where consists of the entire set of the coefficients aj and tie constants Tj. Each a represents the fraction of the total nuber of events contained in thejth coponent, and these coefficients su to unity, z7= 1 j_- so that the set of paraeters has 2-1 independent eleents. Maxiu-likelihood fitting of the binned data then consists of finding the set of paraeters that axiizes Lb(O). STATSTCAL ERRORS N FTTNG BNNED DATA Sine and Steinbach (1986) and McManus et al. (1987) have considered two kinds of systeatic error in estiating kinetic paraeters that arise fro discrete sapling and binning of data. First, there is an error in the estiates of brief exponential coponents when the event durations are easured as discrete ultiples of a sapling interval. n our laboratory we use the 5%-threshold-crossing technique for estiating event durations, but we interpolate the experiental current trace data in the vicinity of threshold crossings (Colquhoun and Sigworth, 1983). The resulting dwell-tie estiates are not quantized, so that this source of systeatic error does not arise. (t should however be noted that there reains a ore subtle proble of rando errors in dwell-tie estiates that arise fro noise in the current trace. This sort of error was considered in Colquhoun and Sigworth, 1983, and is expected to be sall.) Second, McManus et al. (1987) have pointed out a "binning error" that occurs when bin heights are copared with theoretical probability density values. This systeatic error can be understood fro the approxiations (Eqs. 1 and ) that relate the bin heights to the pdf values at the bin centers: the approxiations becoe poor when the bin width increases to becoe coparable to the tie constants in the pdf. We have presented here a siple solution to this proble, naely to use the probability distribution function, rather than the probability density, in the axiu likelihood evaluation (Eq. 16). Having dealt with these two sources of error, we wanted to see how wide the bins could be ade without introducing other errors into the estiation of paraeters by the binned axiu likelihood ethod. The process of collecting events into bins inevitably reoves soe inforation fro the data. One therefore expects that at larger bin widths the paraeters should show additional scatter that reflects the loss of inforation. To characterize these errors we used the axiu likelihood technique to estiate paraeters fro groups of synthetic data sets. Each data set was created using the sae values of tie constants and nubers of events but were generated with different rando nubers. Fro the sets of estiated paraeters so obtained, we coputed the standard deviation and ean values for coparison with the starting paraeters used in constructing the data sets. The process of axiu-likelihood estiation was repeated using densities of bins ranging fro 2 to 64 bins per decade (corresponding to Ax = 1.15 to.36). n soe cases we also subjected the data sets to the direct log-likelihood evaluation (involving no binning; Eq. 14), to test the dependence of the errors on the binning procedure. n these siulations we used a variety of starting paraeters and different nubers of events ranging fro N = 1,24 to 25,6 in order to pose both well-conditioned and illconditioned fitting probles. METHODS Synthetic data in the for of exponentially distributed rando nubers were obtained by taking the logarith of the output of the FORTRAN- 77 RND function on our PDP-1 1/73 coputer (Digital Equipent Corp., Marlboro, MA). This RND function uses a 32-bit algorith and has a period uch longer than the size of our data sets. The resulting rando nubers were scaled and pooled according to the tie constants and aplitudes of the distribution to be siulated, and synthetic dwell ties saller than t, = -5 s were eliinated fro the data set to siulate a detection liit. The events were collected into bins with widths given by Eq. (4) with t, = 's and Ax = 2.33/, where is the nuber of bins per decade. The log likelihood was coputed according to Eq. 16 for binned data, or according to Eq. 14 for the unbinned axiu-likelihood (ML) estiation. The coputations used 32-bit floating-point arithetic and the floating-point buffer operations of BASC-23. A siplex search procedure (Caceci and Cacheris, 1984) was used to axiize the log likelihood; for a typical three-coponent fit (five paraeters) the roughly 2 iterations required 3 in for the unbinned ML estiation fro 2,48 data points, but only 2 in when binning was used (16 bins/decade). n each group of fitting operations the ean,u; and standard deviation o, was coputed for each of the fitted paraeters. As an overall easure of the fitting errors, root-ean-square values of the noralized standard deviation a'and bias,u' were coputed fro the np paraeters as np i-1 vi np i (- vi where v1 is the theoretical value of the ith paraeter, i.e., the value that was used in generating the synthetic data. RESULTS Effect of Bin Density on the Precision of Fitting. We perfored binned and unbinned axiu-likelihood fitting on data sets fro eight different distributions containing two or three exponential coponents. For each of these underlying distributions we used two different nubers of events N. The greatest sensitivity to the effects of binning appeared, as ight be expected, in the "illconditioned" probles where N was relatively sall, and where the tie constants were closely spaced or the aplitudes of coponents were sall. Three distributions SGWORTH AND SNE proved Display and Fitting ofsingle-channel Histogras 51

6 of this kind are illustrated in the upper panels of Fig. 2, and the dependence of the rs errors on bin width is shown in the lower panels. The fitting proble illustrated in Fig. 2 A was the ost difficult one we tried. The distribution had tie constants of.2, 1, and s and aplitudes of.2,.1, and.7, respectively. This proble exeplifies a coon situation in which a coponent of interediate tie constant is "buried" between larger coponents; here the 1-s coponent represented only % of the events, and is hardly visible in a plot of the coposite distribution (Fig. 2 A, top panel). This coponent was also very difficult to fit: in the data sets of 2,56 events, even the best binned fitting operation (at 16 bins/decade) yielded estiates of this tie constant with a scatter (SD) of ±.58 s, and the aplitude estiates had the ean and SD of.12 ±.41. The noralized, overall errors in fitting this distribution at a bin density of 16/decade were o' =.31 and,t' =.13. n this distribution, and in general, the quality of fitting was surprisingly insensitive to the density of bins used. The scatter a' (plotted as squares with solid lines in the lower panel of Fig. 2 A) declined with increasing bin density up to 8-16 bins/decade, beyond which it reained essentially constant. The bias paraeter,' (squares with dashed lines) continued to decrease as the bin density increased to 64 bins/decade; however, since the bias values were usually sall copared with o', changes in these values reflect only sall effects on the quality of fits. This is because the expected standard error E' of the paraeters in a single fit depends on the su of the squares of a' and,u', f' (1,s2 + ao2)1/2. When the nuber of events was increased -fold to N = 25,6, the precision of the fitted paraeters iproved by roughly a factor of 3. The sae general trends were apparent in the dependence of a' ( triangles with solid lines) and,u' (triangles with dashed lines) in Fig. 2 A, except that the bias effects due to binning were ore proinent in this case because the large nuber of events akes the statistical errors saller. Fig. 2 B shows the corresponding results for another three-coponent distribution that posed better-conditioned fitting probles. The coponents had tie constants of 1, 3, and s and relative aplitudes.3,.4, and.3, respectively. Again, the quality of fitting was only weakly dependent on bin density with no significant increase in the precision of fitting occurring beyond 16 bins/decade. Fig. 2 C shows the results for two fitting probles c c U) -4-, cv a) LLJ Log duration A (n Q1) LU % % t,,,,, % l Log duration B U) cv LU " % % l O Log duration C 3 L L. ci3 L) a) _ s - - E, x, ML Bins per decade 'A 31 L 1 cv a) 3 Lvi Q) 1..3 \~~~ _ A. % -- A -"-- - so Z7z ML Bins per decade / 3 L- a) cv c3 a)1 L..3 G -- _ ML Bins per decade FGURE 2 Dependence of fitting errors on the density of bins, as estiated fro fitting groups of synthetic data sets. The rs scatter a' (sybols with solid lines) and bias,s' (sybols with dashed lines) are plotted as a function of bin density, given as bins per decade; ML indicates the results of unbinned axiu-likelihood estiation. The upper panels illustrate the distributions, plotted as in Fig. 1 C and Eq n A, the distribution consisted of three exponential coponents with tie constants.2, 1., and s and aplitudes (i.e., fractions of total events).2,. 1, and.7, respectively. The squares represent paraeters fro fits to 2,56 data points, while the triangles were fro fits to 25,6 data points. (Only, rather than, unbinned ML fits were perfored for 25,6 points.) (B) Fits to a distribution with tie constants of 1, 5, and 25 s and aplitudes.3,.4, and.3. The squares are fro 2,56 data points, the triangles fro a total of 25,6 data points. (C) Fits to distributions with two coponents of equal aplitude. Squares, 1,24 points with tie constants of 1 and 3 s. Triangles, 1,24 points with tie constants of 1 and s..7 tr b 52 BOPHYSCAL JOURNAL VOLUME

7 involving 1,24 events consisting of two exponential coponents. The squares represent errors in fitting a distribution having the two tie constants separated by only a factor of 3 (shown in the top panel of Fig. 3 C); the triangles are fro a distribution with tie constants separated by a factor of. n each case there is rearkably little difference in the fitting errors between a bin density of 2/decade and the highest density used, 64/ decade. zzzzzzzzzzzzzzz U L Coparison with Classical Maxiu-Likelihood Fitting The binned axiu-likelihood fitting, which required one to two orders of agnitude less coputer tie, nevertheless gave coparable results to the classical, unbinned axiu likelihood technique (indicated by ML in each part of Fig. 2). Asyptotically the binned and unbinned techniques are equivalent (Eq. 17) but they have different sensitivities to roundoff errors and therefore are expected to behave differently in practice. The largest difference between binned and unbinned fitting was seen in the runs of Fig. 2 A. The a' and,u' values for the ML run with N = 25,6 are less precise because we perfored only of the 3-h fitting runs to copute the; however, both ML runs appeared to have slightly lower a' values. n the other fitting probles tested, as in Fig. 2, B and C, essentially no difference was seen between unbinned fits and binned fits with densities of 16/decade or greater, and in soe cases the ML fitting actually showed higher bias values, which is to be expected fro the possible roundoff errors in accuulating the large su (Eq. 14) that runs over the total nuber of events. We conclude fro these and other siulations that the binned fitting perfors coparably to the unbinned axiu-likelihood estiation, and that relatively coarse binning can be used without introducing substantial errors. Little iproveent in perforance is seen fro bin densities above 16/decade, and acceptable behavior is obtained at 8 bins/decade. ndeed, for two-coponent distributions as in Fig. 2 C the very coarse binning of 2/decade ay be acceptable. We have standardized on a density of bins/decade, which is the density shown in the logarithic histogras of Figs. 1 and 3. DSCUSSON We have presented here a coparison of ethods for displaying dwell-tie histogras and have derived the for of the corresponding theoretical functions for the coparison of the histogras with exponential distributions. As the ost useful representation of dwell-tie data we suggest the display of histogras with a logarithic tie axis and a square-root vertical axis, as in Fig. 1 D. This display has a nuber of advantages over the traditional linear histogra display, including the properties that (a) exponential coponents appear as peaked func "-" 4 2 U 35C 3C 25C 2C.s 15C 1 C (n Ca 5 1 6C 14C 12C 1OC 8 * i i """""""""""" Tie (s) -J 3 FGURE 3 Histogras and fitted probability density functions fro the data of Sine and Steinbach (1987). Recordings fro cell-attached patches on BC3H-1 cells were ade at 11 C and -7 V patch ebrane potential. (A) Open tie distribution with 6 ji acetylcholine in the pipette, fro 2,9 observed events. The fit predicts a total of 2,12 events, 9% of which are contained in a 16-,gs coponent, and 91% in a coponent with a tie constant of 15 s. (B) Closed-tie histogra at 2 AM ACh, fro 4,65 observed events; sae data as in Fig. 4 of Sine and Steinbach. Highlighted as solid curves, the equal sized coponents with tie constants of 15.6 and 2.1 s are associated with the effective channel opening rate and the agonist dissociation rate of the singly liganded receptor, respectively. The estiated tie constants and fractions of events are: 1.32 s,.9; 113 s,.36; 15.6 s,.8; 2.1 s,.8; 25,us,.13; 33 js,.66. (C) Closed-tie histogra at 6 jm ACh, fro the sae recording as in A. The activation closures which were seen as two coponents at 2,M ACh appear here as a single coponent (solid curve). The shift of the coponent is interpreted as resulting fro an increase of the effective channel opening rate, in which the tie constant decreases fro 15.6 to 2.9 s. The other coponent, associated with agonist dissociation, is predicted to take on a uch saller aplitude at 6,M, and is apparently not visible here. The estiated nuber of events is 5,951; tie constants and relative fractions of events are: 3.7 s,.7; 38 s,.17; 38 s,.33; 2.6 s,.17; 29 s,.12; 31 A s,.66. For a detailed description of the interpretation of the various coponents in the histogras, see Sine and Steinbach (1987). A 6 pm ACH SGWORTH AND SNE proved Display and Fitting ofsingle-channel Histogras 53

8 tions in the display; (b) the position of the peak in each coponent corresponds directly to its tie constant; (c) the height of each peak corresponds to the total nuber of events in that coponent; and (d) the expected statistical scatter in the data is of constant agnitude throughout this type of display. We have also considered soe issues in the fitting of distributions to logarithically binned data. As Blatz and Magleby (1986a) and McManus et al. (1987) have pointed out, the binning of data can greatly accelerate the estiation of paraeters using the axiu-likelihood ethod. McManus et al. (1987) found that bin densities as low as 25/decade are acceptable, but we conclude fro siulations and fro using an iproved likelihood evaluation ethod that even lower bin densities (i.e., wider bins) can be used with little degradation in the quality of paraeter estiation. t should be noted that our concern here has been to optially display and fit distributions of observed dwell ties. We have not considered here the corrections necessary for interpreting these distributions and coparing the with theory in cases where a significant nuber of events are not resolved. Strategies for aking such corrections have been discussed in the literature (Roux and Sauve, 1985; Blatz and Magleby, 1986a). Fig. 3 deonstrates the logarithic display and the bin fitting techniques applied to dwell ties recorded fro acetylcholine receptor channels in BC3H-1 cells. The data are taken fro the study of the agonist concentration dependence of closed and open durations described by Sine and Steinbach (1987). Siply plotting the dwell ties on the logarithic tie scale reveals qualitative features of the distributions. The open tie histogra (Fig. 3 A) consists of a ajor coponent with a peak near 15 s plus a sall coponent with a uch briefer ean duration. The closed tie distributions (parts B and C of the figure) span five decades of tie. Raising the agonist concentration fro 2 to 6,gM causes a general shift in the closed tie distribution toward shorter durations. The logarithic and square-root transforations cobine to show the nuber of coponents required to fit each histogra. n both histogras a partially resolved shoulder is seen at short durations, and five ore peaks are visible spaced about a decade apart. Thus the logarithic abscissa and the square root ordinate ake qualitative features of the data iediately apparent. This kind of display also serves as a starting point for quantitative analysis of the dwell-tie histogras. A reasonable description of the histogra can be obtained fro a anual fit to the su of exponentials. The resulting areas and tie constants then becoe seed values for axiu likelihood fitting. Because the histogra entries are collected into only about 6 bins, the axiu likelihood estiation of paraeters requires little coputer tie. Starting fro a anual fit of the six coponents, the fits in Fig. 3, B and C, converged in < in using our BASC- 23 progra. A further advantage of this display ethod is that the quality of the fit can be readily judged because the scatter is expected to have a constant aplitude across the histogra. For exaple, deletion of one of the six closed duration coponents would leave undefined a highly significant population of dwells. We thank Dr. C. Methfessel for his FORTRAN translation of the siplex routine. This work was supported by grant nuber NS-2151 fro the National nstitutes of Health. Received for publication 27 March 1987 and in final for 12 July REFERENCES Austin, R. H., K. W. Beeson, L. Eisenstein, H. Frauenfelder, and. C. Gunsalus Dynaics of ligand binding to yoglobin. Biocheistry. 14: Blatz, A. L., and K. L. Magleby. 1986a. Quantitative description of three odes of activity of fast chloride channels fro rat skeletal uscle. J. Physiol. (Lond.). 378: Blatz, A. L. and K. L. Magleby. 1986b. Correcting single channel data for issed events. Biophys. J. 49: Caceci, M. S., and W. P. Cacheris Fitting curves to data: the siplex algorith is the answer. Byte. May 1984: Colquhoun, D., and A. G. Hawkes The principles of the stochastic interpretation of ion-channel echaniss. n Single Channel Recording. B. Sakann and E. Neher, editors. Plenu Publishing Corp., New York Colquhoun, D., and Sigworth Fitting and statistical analysis of single-channel records. n Single Channel Recording. B. Sakann and E. Neher, editors. Plenu Publishing Corp., New York McManus,. B., A. L. Blatz, and K. L. Magleby Sapling, log, binning, fitting, and plotting durations of open and shut intervals fro single channels. Pfluegers Arch. Eur. J. Physiol. n press. Roux, B., and R. Sauv A general solution to the tie interval oission proble applied to single channel analysis. Biophys. J. 48: Sine, S. M., and J. H. Steinbach Activation of acetylcholine receptors on clonal aalian BC3H- 1 cells by low concentrations of agonist. J. Physiol. (Lond.). 373: Sine, S. M., and J. H. Steinbach Activation of acetylcholine receptors on clonal aalian BC3H- 1 cells by high concentrations of agonist. J. Physiol. (Lond). 385: BOPHYSCAL JOURNAL VOLUME