283
J. Phy8iol. (1977), 272, pp. 283-294 With 5 text-figurem Printed in Great Britain
MOVEMENT OF LABELLED DECAMETHONIUM IN MUSCLE FIBRES OF THE RAT
BY R. CASE, R. CREESE,* W. J. DIXON, F. J. MASSEY AND D. B. TAYLOR From the Departments of Pharmacology and of Public Health and Biomathematics, U.C.L.A. School of Medicine, Los Angeles, California 90024, U.S.A.
(Received 2 December 1976) SUMMARY
1. Tritium-labelled decamethonium accumulated in diaphragm muscles of the rat in vitro with a peak at the end-plate region and the distributions were fitted by Gaussian curves. 2. Prolonged wash in physiological saline (10 hr) produced some loss in radioactivity but no detectable spread of the labelled compound along the fibres, which indicated that the decamethonium was not in a mobile form. 3. Rats injected with labelled decamethonium showed radioactivity in the diaphragm muscles after 21 days. 4. A slow spread of the labelled compound along the fibres was detected, and from the widening of the Gaussian curves the apparent diffusion coefficient was 1 2 x 10-8 cm2 sec-', which is less than 1/500 of that in free solution. INTRODUCTION
Decamethonium and similar compounds are known to enter skeletal muscle at the end-plate region (Taylor & Nedergaard, 1965; Creese & Maclagan, 1970). These substances are thought to pass through channels opened by the depolarizing drugs (e.g. Cookson & Paton, 1969), and the entry can be used to indicate the change in permeability of the end-plate (Creese, Franklin, Humphrey & Mitchell, 1976). An attempt was made to measure the internal diffusion along the muscle fibres, and the mobility was found to be much lower than expected. A brief account has been given previously (Taylor, Dixon, Creese & Case, 1967). * Present address: Department of Physiology, St Mary's Hospital Medical School, London, W2 1PG.
284
R. CASE AND OTHERS METHODS
Albino rats of approximately 100 g were injected i.v. with [3H-methyl]decamethonium chloride (1.64 mg/kg) in a volume of 0 01 ml. (Creese, Taylor & Case, 1971). After an interval which varied between 2 hr and several days the animals were stunned and the left hemidiaphragm was removed, frozen, sliced, weighed, dissolved and counted by liquid scintillation methods (Creese et al. 1971). In other rats the hemidiaphragm was removed and suspended in physiological saline (Creese & Northover, 1961) at 38 'C. For measurements of outwardmovement of labelled drug (as in Fig. 2 below) the left and right hemidiaphragm was used alternately. The muscles were exposed to a solution containing labelled decamethonium and then passed through a succession of vials containing 10 ml. physiological saline. At the end of the run the vials were freeze-dried and scintillator was added for counting. The radioactivity which remained in the muscles and tendon was found by dissolving the tissue and counting. An estimate of the thickness was obtained from the other hemidiaphragm by means of a cork-borer, which was used to punch out a segment of the muscle (Creese, 1968). In experiments designed to detect the longitudal movement of labelled decamethonium in vitro (as in Fig. 3 below), each rat was used to provide two strips from the left diaphragm, an anterior (or medial) strip and a posterior (or lateral) strip. The diaphragms were mounted on holders and a crocodile clip was attached to the thread (see Creese & Northover, 1961). The muscles were exposed to labelled decamethonium for 2 hr and washed either for 30 min or for 10 hr in inactive saline. The solution was changed each min for the first five min, then every 5 min for 30 min and subsequently every 15 min. When the muscle was removed marks were made on the thread and holder so that the alignment could be maintained while the muscle was being frozen on solid carbon dioxide. The diaphragms were sliced and counted as before. RESULTS
Diffusion of decamethonium in extracellular spaces Fig. 1 shows results from a diaphragm muscle which was loaded by immersion in a solution containing labelled decamethonium (20 min at 60 AM) and then passed through a succession of vials of inactive solution every 1-5 min. The radioactivity in the vials was counted as described above and at the end of the wash the muscle was dissolved and also counted. The commutative counts have been plotted and show a double exponential curve with a rapid phase having a half-time of 1-3 min and a slower component with half-time of 29-4 min. Results from six diaphragms were averaged. The half-time for the fast fraction, obtained by methods used by Creese, Taylor & Tilton (1963), gave a mean value of 1-1 min (range 0-9-1-5). The thickness was 0-472 mm (mean of 6, range 0-42-0-70). The apparent diffusion coefficient from these values (Creese, 1968) was 2-4 x 10-6 cm? sec- in extracellular fluid, which can be compared with the value of 7-7 x 10-6 cm2 sec1 obtained in solution at 36 'C by conductance measurements (Brookes & Mackay, 1971). The apparent coefficient in the extracellular spaces is approximately 1/3
285 LABELLED DECAMETHONIUM of that in free solution. The extracellular space, estimated from the fast fraction, was 0-36 ml. g'1 (mean of 6, range 0.29-0-45), and with a specific gravity of 1-07 this gives the volume fraction of 0 34, which is similar to the value of 0 33 obtained with the use of labelled sodium (Creese, 1968). Fig. 2 shows a longer experiment. The diaphragm was immersed for 2 hr in labelled decamethonium (60 pUM), and washed for a further 101 hr, 40,000 T-
20,000 1- - \0 50,000.
10,000
0
"0
40,000
5,000 0
30,000- - o
0 0
C
en *-)
Co
cJ
2,000
l
I
'O
20,000]
0 1 2 3 4 5 6
,do
10,000I
0
10
I
20
l
30
Min Fig. 1. Outward movement of labelled decamethonium in rat diaphragm (uptake 20 min in 60 FM). Ordinate shows counts remaining. After a short time the curve is described by the sum of two exponential, with halftimes of 1-3 min and 29 min. Inset shows the fast fraction.
the saline being changed as for Fig. 1 for the first 30 min and thence at intervals of 15 min. The outward movement consists of a rapid fraction as above, plus a slower fraction with a half-time of 19 min, and a much slower fraction with a half-time of 8-9 hr. Attempted measurement of longitudinal diffusion in vitro Fig. 3 shows the distribution of labelled decamethonium in experiments similar to that shown in Fig. 2. The design was symmetrical, and the two strips of muscle were obtained from the left diaphragm of each rat. The muscles were exposed to labelled decamethonium (60 /M) for 2 hr and then washed either for 30 min or for 101 hr, as described above. The muscles were frozen on solid carbon dioxide, sliced and counted.
R. CASE AND OTHERS
286 1-2
100,000
0
T 10 o 0) E o 0 Cu
80,000 0
E 07
o 0
0
°Q
-
-c
E
o50,000 O°
0.
0)~~~~~~~~~~~~~~~~~~0 ~ ~ ~ ~ 0) ~ ~~
~
~
~
~
~
0
~~0
aQ'^O"
- 30,000
"
03 _
0-3~~~~~~~~~~~~~~~~~~~~~~-
0
.>
-
6 8 10 Hr Fig. 2. Long washout following uptake of 2 hr at 60 saM. Abscissa shows labelled decamethonium which remains, expressed as a clearance, being ,ul. mg-" or (counts min-1mg-')/(counts min-1,al.-1 of radio-active solution). From 2-5 to 10-5 hr there is a very slow fraction with half-time of 8-9 hr. 4
2
A
B
8000
6
6000
E
.~4000-
E C
0m
2000 0
0
2
4
6
8
2 10 12 0 mm from tendon
4
6
8
10
Fig. 3. Distribution of labelled decamethonium in diaphragm muscles following exposure for 2 hr in 60 /SM with washout (A) 30 min and (B) 10-5 hr. Ordinate gives counts min-'mg-", abscissa is distance in mm from tendon. After prolonged wash the radioactivity in (B) is smaller. Gauss curves fitted to the histograms show little difference in the S.D. which is 0-99 mm for A and 1-10 mm for B.
287 LABELLED DECAMETHONIUM The wash out period was 30 min for Fig. 3A and 10 hr for Fig. 3B. The radioactivity in Fig. 3 B is diminished, but Gauss curves fitted to the histograms gave a standard deviation which was close to 1 mm in each case. In the series the S.D. for 30 min wash was 1-06 mm (mean of 8, range 0.75-1.42), while for muscles washed for 101 hr the S.D. was 1-10 (mean of 8, range 0.87-1.27). There is no significant difference in these results. 300 r-
200 H E
cI E C
0
100 H
U
0 0
2
4
6
1H A
8 10 mm
0 2 from
4
6
8 10
0
2
tendon
6 8 10 12
4
C B
Fig. 4. Distribution of labelled decamethonium in diaphragm muscles injected with labelled decamethonium and removed after intervals of (A) 2 hr, (B) 6 days and (C) 10 days. There is a reduction in radioactivity with time, and also a widening of the Gaussian curves fitted to the histograms. The
S.D.
is indicated beneath each
muscle, and
was
0-81
mm
for
(A),
and 1-35 mm for (B) and 1-82 mm for (C).
Longitudinal diffusion of decamethonium in vivo It was clear from Fig. 3 and similar experiments that the movements of decamethonium in the fibre was very slow. Fig. 4 shows the results of experiments in which labelled decamethonium was injected into rats in vivo (1.64 mg/kg). In Fig. 4A the left diaphragm was removed after 2 hr, while Fig. 4B and Fig. 4C show the distribution in diaphragm removed after 6 days and 10 days respectively. Gaussian curves have been fitted to the histograms by a method which is described below (see Appendix). In Fig. 4 there is a fall in radioactivity with time, and also a progressive widening of the Gauss curves, shown by the computed values of the
R. CASE AND OTHERS 288 standard deviation which are shown for each of the curves. The results for the whole series are listed in Table 1. If the widening of the Gaussian curves is attributed to a process with diffusion-like kinetics then the increase in variance should be linear with time and
0-12-0o2= 2D't,
(1)
TAiRZ 1. Labelled decamethonium in vivo
No. of muscles 12 6 6 12
Time 2hr 2 days 6 days 12 days
Standard deviation (mm) 0802 + 0140 0987 + 0O197 1*352 + 0*237 1-643 + 0253
Total counts (min- mg-L) 651 655 446 619
Ratio 058 072 0-69 071
Mean values of o, the computed standard deviation from histograms similar to those of Fig. 4, are listed together with the S.D. of the group. Mean values of the total counts min-' mg-' are also shown. The last column gives the fraction of the total count which is alloted to the Gauss curve, obtained as described in the text.
3H .1 .1
.1
11
.1
2 CN
E E 0)
., .1
>
.1
1
I
.1
0
I
-
O
2
I
6
4
8
J 10
Days
Fig. 5. Plot of variance against time in days. The points show (mean value of the standard deviation)', obtained from histograms similar to those of Fig. 4. The limits show the 95 % confidence values. The interrupted line is the regression, obtained as described in the text.
LABELLED DECAMETHONIUM ass where ro- is the S.D. at zero time and o the S.D. after an interval t; D' is an apparent diffusion coefficient (e.g. Jacobs, 1935). In Fig. 5 the variance in mmx is plotted against time in days. The points give values of (mean S.D.)., and the limits give the estimated 95 % confidence values. There is a progressive increase of variance with time, and the interrupted line gives the regression where the slope is 0-2055 mm2 day-1, and hence D' is 1-2 x 1o-8 cm sec-1 with 95% limits 1-1-13 x 10-8 cm2 sec-1 (see Appendix). The deviation ao at zero time is 0 79 mm. The histograms shown in Fig. 4 are representative of muscles removed at various times after injection. A fall in the total radioactivity occurred during the first few days, but there was little change between the sixth and the tenth days apart from the widening of the Gaussian curves. It was found in other experiments that it was possible to detect radioactivity in diaphragms removed 21 days after injection. The total radioactivity in the individual muscles was variable, and this was partly attributed to variations in the dose which was injected. The fraction of total radioactivity attributable to the Gaussian curve is also shown in Table 1. This was 0-58 after 2 hr and later became nearly constant at approximately 0-71. It can be seen in Fig. 4A that the values at the end of the muscles at 2 hr are much larger than values in Fig. 4 B and Fig. 4C, and the change in the ratio could be attributed to loss of non-specific component with a comparatively rapid turnover. After 2 days the ratio is nearly constant, and no difference is then detectable in the rate of exchange of the
different components. DISCUSSION
Decamethonium has been found to induce the entry of labelled sodium in the end-plate region (Creese & Mitchell, 1975), and labelled decamethonium also enters the fibres (Creese & Maclagan, 1970). The compound appears to become internally bound, for 10 hr of wash in the present study produced no change in the distribution apart from a slow loss of radioactivity to the outside solution. This is consistent with the autoradiographic findings (Creese & Maclagan, 1970) in which no change was detected in the distribution of radioactivity in single fibres over a period of 2 hr. Labelled calcium also enters at the junctional region and becomes attached to internal structures so that it is not easily removed by subsequent washing (Evans, 1974). The initial distribution of decamethonium in whole muscles seen in Fig. 4A has a S.D. of 0-8 mm, and this may be due chiefly to the scatter of end-plates in the various layers of the diaphragm. The normal curves shown in Fig. 4 slowly widen with time, and this indicates a very sluggish longitudinal movement with an apparent diffusion coefficient which is less than 1/500 of that in free solution. In 10 days the standard deviation more than doubles, and this is not likely to be produced solely by growth of the muscle. Simple diffusion through narrow
R. CASE AND OTHERS 290 tubules or endothelial reticulum is also unlikely to explain this slow rate, and Endo (1966) found that a dye could be cleared from the tubule spaces within a couple of minutes. The slow spread in Fig. 4 could be attributed to random movements along a series of adsorption sites, which would produce a gradual increase in the scatter. In Fig. 5 the slope gives variance/ time and is proportional to an apparent diffusion coefficient with dimensions L2T-1. In rat peroneus muscle the distribution of labelled decamethonium three minutes after injection was found by autoradiography to be maximal at the end-plate region, but the radioactivity extended for several hundred micra along the fibres (Creese & Maclagan, 1970). Similar results have been obtained in cat muscle (Creese & Maclagan, 1976). The simplest explanation is in terms of extra-junctional receptors, for the distribution found by autoradiography would not be produced if the drug entered only at the end-plate with diffusion coefficient as small as 1-2 x 10-8 cm2 sec-1. The existence of extra-junctional receptors is needed to explain the effects produced by iontophoretic application of acetylcholine in rat diaphragm (Miledi 1960, 1962) and also in mouse omohyoideus muscle (Dreyer, Muller, Peper & Sterz, 1976). These results differ from those in some frog muscles (Dreyer & Peper, 1974; Kuffler & Yashikami, 1975). From Fig. 5 the initial distribution of radioactivity has a S.D. of 0*79 mm, and a similar value is seen in Fig. 4A. This must be mainly due to the scatter of individual end-plates in the muscle. The irregular zig-zag appearance of end-plates in the diaphragm can be seen in the photographs published by Hebb, Krnjevi6 & Silver (1964) and England (1970), and the distribution may be further distorted when the diaphragm is mounted in holders and frozen. Figs. 4 and 5 show a steady spread of radioactivity, and from the increase of variance with time the internal diffusivity has been estimated by means of eqn. (1). During the process of diffusion, the distribution in the whole muscle will be the algebraic sum produced by individual fibres whose end-plates have themselves a considerable scatter. The sum of a number of variates when the origins are not aligned has been considered in connexioan with the central limit theorem of statistics, and Thomasian (1969) points out that if one variate is drastically greater than all others then 'the distribution of the sum of errors might well be about the same as the distribution of the one large error'. The estimation of the diffusion coefficient is thus not feasible by this method for short runs, and in practice the experiment was allowed to proceed until the width of the Gaussian curve as shown by the standard deviation had more than doubled. This research was supported by U.S. Public Health Service Grant NB-00738.
LABELLED DECAMETHONIUM
291
APPENDIX
Fitting of Gau8sian curves to distribution of labelled decamethonium The histograms shown in Figs. 3 and 4 were fitted to normal curves of the form y = b+l exp{-(xX _)2/2o4}, (2) where y is counts min-' mg-', x is distance in mm from the tendon, and b, 1, Ad and o- are constants. This is a Gauss curve plus the term b which is the value of the non-junctional uptake at the ends of the muscle. This term is in general unknown, and because of this additional constant it was not possible to fit the curves by conventional methods which aie used if b is zero or measured. The values of the ordinate in counts min- mg-1 (see Fig. 4) were taken to refer to points x at 05, 1-5, 2-5 mm etc. measured from the end of the tendon, and there were eight to thirteen slices in each diaphragm. A computer programme BMD07R (Dixon, 1974) was used to find the least squares fit of the values (x,y) to the curve of eqn. (2). That is, the values of b, 1, #t and ar were found that minimized 82 which is given by:
82
1
=
m
E[i - b- I exp {_(X.,-,)2/20r21]2,
(3
where m is the number of sections for the particular diaphragm, and (xl, Yi) .... (Xm, ym) are the measurements on the sections of the muscle. For each diaphragm the least-squares estimates of u, a, I and b were found. The theoretical curve given by eqn. (2) can be broken up into the sum of two curves y,(x), and y2(x) where y1(x) = b y2(X) = I exp {-(x-,/)2/2o-2}. With these definitions eqn. (2) can be rewritten as Y
=
Y1(X)+y2(x).
The curve represented by yl(x) is called the base curve, and the curve represented by y2(x) is called the normal curve. An is an estimate of the total number of counts contributed by the normal curve. It is the total area under the curve y2(x) in the region corresponding to the sections measured. Precisely A=
Z
1 exp -(z-,u)2/2a'2}dz,
where the limits of integration for the ith slice are (xi- 1) and (xi + i). For most diaphragms A. was calculated using the formula
An= (2ir)i1.(
(4)
R. CASE AND OTHERS 292 In those cases where measmements from a section were missing or there was a significant portion of the normal curve in the region beyond the sections the value using (5) was multiplied by the factor p when p is given by m j'z2 p = z § (2ir)- exp(-z2/2) dz (6)
i=1 Jz where the limits of integration for the ith slice are (xi+ -j-)/cr and (xi-i -#s)/or. An given by eqn. (4) is equivalent to p(2ir)ii". A b is an estimate of the total number of counts contributed by the base curve. It is the total area under the curve y,(x) in the regions corresponding to the sections measured. It is given by Ab= mb. (7) At is an estimate of the total number of counts that should be observed for all measured sections of the diaphragm. It is the area under the curve given by eqn. (2) in the regions corresponding to the sections measured. It is given by (8) At = An +Ab. Then ac is the ratio of the number of counts contributed by the normal curve to the total number of counts. It is given by a = AnIAt. (9) and The results are summarized in Table 1 were used to calculate the apparent diffusion coefficient for longitudinal movement. It can be seen from Fig. 4C that at 10 days the fitted curve is still diminishing at the ends, so that some portion of the curve extends beyond the limits of the measured values. The effect was small at 10 days and no correction has been applied. If the experiment had been appreciably prolonged it would have been necessary to consider the effects of reflexion (Jacobs, 1935; Thomasian, 1969). Plot of variance against time In Fig. 5 the estimate of the variance does not have a normal distribution and it would be inappropriate to apply conventional regression analysis to the results as they stand. The estimates of 0-, the standard deviation, have a distribution which approximates to the normal. The values of o are in groups which are listed in Table 1, and the individual results were standardized by dividing each by the group median. For the resultant ratio the median was 1-0, the mean was 1-0065 and the standard deviation was 0-164. Pearson's skewness coefficient, which is defined as (mean-mode)/(standard deviation) may be taken as 3 (mean-median)/(standard deviation), and this comes to 0 118 (Yule & Kendall, 1950).
LABELLED DECAMETHONIUM 293 The regression in Fig. 5 may be extrapolated to the abscissa at a time d which is approximately - 3 days. Equation (1) may be transformed to
2D'(tI+d) or = (2D')1(t+d)i.
o- =
(10)
or (11) Hence if d were known then o- could be plotted against (t + d)i and conventional regression analysis could be applied. In practice the mean value of or at each time was plotted, with weighting coefficient n/var where n is the number of muscles and var the variance. Different values of d were tried and it was found that for d equal to 3-0012 days the weighted regression had an origin of zero. With these values the slope which was (2D')4 came to 0-4534, so D' was 0-1028 mm2 day-'. The error variance of the weighted regression was 0-00148; since four mean values of C were used, the statistic 't' was 4-30 and the 95 % confidence internal for the slope (2D')i was from 0-4382 to 0-4685. Hence the limits for D' were 0-096-0-110 mm2 day-', or 1-1-1-3 x 10-8 cm2 sec-L. The values of a, the standard deviations computed from the histograms of muscle in vivo, had a sampling distribution which was found to be approximately normal. Other possible distributions were tried for plotting as and o against time, including the use of the median, the censored mean (Dixon, 1960) and the transformation 1/lo, but in practice these all gave a larger variance from regression when plotted against (t+d)i (or its reciprocal where applicable). It was concluded that there was no advantage in discarding the arithmetic mean, which was used in the plot shown in Fig. 5. REFERENCES
BROOKES, N. & M.ACKAY, D. (1971). Diffusion of labelled substance through isolated rat diaphragm. Br. J. Pharmac. 41, 367-378. COOKSON, J. C. & PATON, W. D. M. (1969). Mechanisms of neuromuscular block. Anaeathe8ia 24, 395-416. CREESE, R. (1968). Sodium fluxes in diaphragm muscle and the effects of insulin and serum proteins. J. Phygiol. 197, 255-278. CREESE, R., FRANKLIN, G. I., HUMPHREY, P. P. A. & MITCHELL, L. D. (1976). Dose response curves with labelled sodium and labelled decamethonium in rat muscle. J. Phygiol. 254, 43-44 P. CREESE, R. & MACLAGAN, J. (1970). Entry of decamethonium in rat muscle studied by autoradiography. J. Phygiol. 210, 363-386. CREESE, R. & MACLAGAN, J. (1976). Labelled decamethonium in cat muscle. Br. J. Pharmacy. 58, 141-148. CHEESE, R. & MITCHELL, L. (1975). Sodium entry in junctional region of rat muscle. J. Physiol. 246, 44-45 P. CREESE, R. & NORTHOVER, J. (1961). Maintenance of isolated diaphragm with normal sodium content. J. Physiol. 155, 343-357. CREESE, R., TAYLOR, D. B. & CASE, R. (1971). Labeled decamethonium in denervated skeletal muscle. J. Pharmacy. exp. Ther. 176, 418-422. CREESE, R., TAYLOR, D. B. & TILTON, B. (1963). The influence of curare on the uptake and release of a neuromuscular blocking agent labeled with radioactive iodine. J. Pharmac. exp. Ther. 139, 8-17.
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DIXON, W. J. (1960). Simplified estimation from censored normal samples. Ann. math. Stati8t. 31, 385- 391. DixON, W. J. (1974) (ed). Biomedical Computer Programs. Los Angeles: University of California Press. DREYER, F. & PEPER, K. (1974). The acetylcholine sensitivity in the vicinity of the neuromuscular junction of the frog. Pflugere. Arch. gee. Phyeiol. 348, 273-286. DREYER, F., Miser, K.-D., PEPER, K. & STERZ, R. (1976). The M. omohyoideus of the mouse as a convenient mammalian muscle preparation. A study of junctional and extra-junctional acetylcholine receptors by noise analysis and cooperativity. Pfluger8 Arch. gee. Phyeiol. 367, 115-122. ENDO, M. (1966). Entry of fluorescent dyes into the sarcotubular system of frog muscle. J. Phy8iol. 185, 224-238. ENGLAND, J. M. (1970). The localization of end-plates in unstained muscle. J. Anat. 106, 311-321. EvANs, R. H. (1974). The entry of labelled calcium into the innervated region of the mouse diaphragm muscle. J. Phy8iol. 240, 512-533. HEBB, C. O., KRNJEVI6, K. & SILVER, A. (1964). Acetylcholine and choline acetyl transferase in the diaphragm of the rat. J. Phy8iol. 171, 504-513. JACOBS, M. H. (1935). Diffusion processes (section 11) Ergebn. Biol. 12, 1-60. KUFFLER, S. & YosmE.m, D. (1975). The distribution of acetylcholine sensitivity at the post-synaptic membrane of vertebrate skeletal twitch muscles: iontophoretic mapping in the micron range. J. Phyeiol. 244, 703-730. MLE-DI, R. (1960). Junctional and extra-junctional acetylcholine receptors in skeletal muscle fibres. J. Phyeiol. 151, 24-30. MILEDI, R. (1962). Induction of receptors. In Enzymes and Drug Action (CIBA Foundation Symposium), ed. MONGAR, J. L. & DE REUCK, A. V. S., pp. 220-235. London: Churchill. TAYLOR, D. B., DIXON, W. J., CREESE, R. & CASE, R. (1967). Diffusion of decamethonium in the rat. Nature, Lond. 215, 989. TAYLOR, D. B. & NEDERGAARD, O. A. (1965). Relation between structure and action of quaternary ammonium neuromuscular blocking agents. Phyeiol. Rev. 45, 523556. TitMSIAN, A. J. (1969). The Structure of Probability Theory with Applications, ch. 13 and ch. 24. New York: McGraw-Hill. YuLE, G. U. & KENDATT, M. G. (1950). An Introduction to the Theory of Statiiticm, 14th edn., ch. 7. London: Griffin.
J. Phy8iol. (1977), 272, pp. 283-294 With 5 text-figurem Printed in Great Britain
MOVEMENT OF LABELLED DECAMETHONIUM IN MUSCLE FIBRES OF THE RAT
BY R. CASE, R. CREESE,* W. J. DIXON, F. J. MASSEY AND D. B. TAYLOR From the Departments of Pharmacology and of Public Health and Biomathematics, U.C.L.A. School of Medicine, Los Angeles, California 90024, U.S.A.
(Received 2 December 1976) SUMMARY
1. Tritium-labelled decamethonium accumulated in diaphragm muscles of the rat in vitro with a peak at the end-plate region and the distributions were fitted by Gaussian curves. 2. Prolonged wash in physiological saline (10 hr) produced some loss in radioactivity but no detectable spread of the labelled compound along the fibres, which indicated that the decamethonium was not in a mobile form. 3. Rats injected with labelled decamethonium showed radioactivity in the diaphragm muscles after 21 days. 4. A slow spread of the labelled compound along the fibres was detected, and from the widening of the Gaussian curves the apparent diffusion coefficient was 1 2 x 10-8 cm2 sec-', which is less than 1/500 of that in free solution. INTRODUCTION
Decamethonium and similar compounds are known to enter skeletal muscle at the end-plate region (Taylor & Nedergaard, 1965; Creese & Maclagan, 1970). These substances are thought to pass through channels opened by the depolarizing drugs (e.g. Cookson & Paton, 1969), and the entry can be used to indicate the change in permeability of the end-plate (Creese, Franklin, Humphrey & Mitchell, 1976). An attempt was made to measure the internal diffusion along the muscle fibres, and the mobility was found to be much lower than expected. A brief account has been given previously (Taylor, Dixon, Creese & Case, 1967). * Present address: Department of Physiology, St Mary's Hospital Medical School, London, W2 1PG.
284
R. CASE AND OTHERS METHODS
Albino rats of approximately 100 g were injected i.v. with [3H-methyl]decamethonium chloride (1.64 mg/kg) in a volume of 0 01 ml. (Creese, Taylor & Case, 1971). After an interval which varied between 2 hr and several days the animals were stunned and the left hemidiaphragm was removed, frozen, sliced, weighed, dissolved and counted by liquid scintillation methods (Creese et al. 1971). In other rats the hemidiaphragm was removed and suspended in physiological saline (Creese & Northover, 1961) at 38 'C. For measurements of outwardmovement of labelled drug (as in Fig. 2 below) the left and right hemidiaphragm was used alternately. The muscles were exposed to a solution containing labelled decamethonium and then passed through a succession of vials containing 10 ml. physiological saline. At the end of the run the vials were freeze-dried and scintillator was added for counting. The radioactivity which remained in the muscles and tendon was found by dissolving the tissue and counting. An estimate of the thickness was obtained from the other hemidiaphragm by means of a cork-borer, which was used to punch out a segment of the muscle (Creese, 1968). In experiments designed to detect the longitudal movement of labelled decamethonium in vitro (as in Fig. 3 below), each rat was used to provide two strips from the left diaphragm, an anterior (or medial) strip and a posterior (or lateral) strip. The diaphragms were mounted on holders and a crocodile clip was attached to the thread (see Creese & Northover, 1961). The muscles were exposed to labelled decamethonium for 2 hr and washed either for 30 min or for 10 hr in inactive saline. The solution was changed each min for the first five min, then every 5 min for 30 min and subsequently every 15 min. When the muscle was removed marks were made on the thread and holder so that the alignment could be maintained while the muscle was being frozen on solid carbon dioxide. The diaphragms were sliced and counted as before. RESULTS
Diffusion of decamethonium in extracellular spaces Fig. 1 shows results from a diaphragm muscle which was loaded by immersion in a solution containing labelled decamethonium (20 min at 60 AM) and then passed through a succession of vials of inactive solution every 1-5 min. The radioactivity in the vials was counted as described above and at the end of the wash the muscle was dissolved and also counted. The commutative counts have been plotted and show a double exponential curve with a rapid phase having a half-time of 1-3 min and a slower component with half-time of 29-4 min. Results from six diaphragms were averaged. The half-time for the fast fraction, obtained by methods used by Creese, Taylor & Tilton (1963), gave a mean value of 1-1 min (range 0-9-1-5). The thickness was 0-472 mm (mean of 6, range 0-42-0-70). The apparent diffusion coefficient from these values (Creese, 1968) was 2-4 x 10-6 cm? sec- in extracellular fluid, which can be compared with the value of 7-7 x 10-6 cm2 sec1 obtained in solution at 36 'C by conductance measurements (Brookes & Mackay, 1971). The apparent coefficient in the extracellular spaces is approximately 1/3
285 LABELLED DECAMETHONIUM of that in free solution. The extracellular space, estimated from the fast fraction, was 0-36 ml. g'1 (mean of 6, range 0.29-0-45), and with a specific gravity of 1-07 this gives the volume fraction of 0 34, which is similar to the value of 0 33 obtained with the use of labelled sodium (Creese, 1968). Fig. 2 shows a longer experiment. The diaphragm was immersed for 2 hr in labelled decamethonium (60 pUM), and washed for a further 101 hr, 40,000 T-
20,000 1- - \0 50,000.
10,000
0
"0
40,000
5,000 0
30,000- - o
0 0
C
en *-)
Co
cJ
2,000
l
I
'O
20,000]
0 1 2 3 4 5 6
,do
10,000I
0
10
I
20
l
30
Min Fig. 1. Outward movement of labelled decamethonium in rat diaphragm (uptake 20 min in 60 FM). Ordinate shows counts remaining. After a short time the curve is described by the sum of two exponential, with halftimes of 1-3 min and 29 min. Inset shows the fast fraction.
the saline being changed as for Fig. 1 for the first 30 min and thence at intervals of 15 min. The outward movement consists of a rapid fraction as above, plus a slower fraction with a half-time of 19 min, and a much slower fraction with a half-time of 8-9 hr. Attempted measurement of longitudinal diffusion in vitro Fig. 3 shows the distribution of labelled decamethonium in experiments similar to that shown in Fig. 2. The design was symmetrical, and the two strips of muscle were obtained from the left diaphragm of each rat. The muscles were exposed to labelled decamethonium (60 /M) for 2 hr and then washed either for 30 min or for 101 hr, as described above. The muscles were frozen on solid carbon dioxide, sliced and counted.
R. CASE AND OTHERS
286 1-2
100,000
0
T 10 o 0) E o 0 Cu
80,000 0
E 07
o 0
0
°Q
-
-c
E
o50,000 O°
0.
0)~~~~~~~~~~~~~~~~~~0 ~ ~ ~ ~ 0) ~ ~~
~
~
~
~
~
0
~~0
aQ'^O"
- 30,000
"
03 _
0-3~~~~~~~~~~~~~~~~~~~~~~-
0
.>
-
6 8 10 Hr Fig. 2. Long washout following uptake of 2 hr at 60 saM. Abscissa shows labelled decamethonium which remains, expressed as a clearance, being ,ul. mg-" or (counts min-1mg-')/(counts min-1,al.-1 of radio-active solution). From 2-5 to 10-5 hr there is a very slow fraction with half-time of 8-9 hr. 4
2
A
B
8000
6
6000
E
.~4000-
E C
0m
2000 0
0
2
4
6
8
2 10 12 0 mm from tendon
4
6
8
10
Fig. 3. Distribution of labelled decamethonium in diaphragm muscles following exposure for 2 hr in 60 /SM with washout (A) 30 min and (B) 10-5 hr. Ordinate gives counts min-'mg-", abscissa is distance in mm from tendon. After prolonged wash the radioactivity in (B) is smaller. Gauss curves fitted to the histograms show little difference in the S.D. which is 0-99 mm for A and 1-10 mm for B.
287 LABELLED DECAMETHONIUM The wash out period was 30 min for Fig. 3A and 10 hr for Fig. 3B. The radioactivity in Fig. 3 B is diminished, but Gauss curves fitted to the histograms gave a standard deviation which was close to 1 mm in each case. In the series the S.D. for 30 min wash was 1-06 mm (mean of 8, range 0.75-1.42), while for muscles washed for 101 hr the S.D. was 1-10 (mean of 8, range 0.87-1.27). There is no significant difference in these results. 300 r-
200 H E
cI E C
0
100 H
U
0 0
2
4
6
1H A
8 10 mm
0 2 from
4
6
8 10
0
2
tendon
6 8 10 12
4
C B
Fig. 4. Distribution of labelled decamethonium in diaphragm muscles injected with labelled decamethonium and removed after intervals of (A) 2 hr, (B) 6 days and (C) 10 days. There is a reduction in radioactivity with time, and also a widening of the Gaussian curves fitted to the histograms. The
S.D.
is indicated beneath each
muscle, and
was
0-81
mm
for
(A),
and 1-35 mm for (B) and 1-82 mm for (C).
Longitudinal diffusion of decamethonium in vivo It was clear from Fig. 3 and similar experiments that the movements of decamethonium in the fibre was very slow. Fig. 4 shows the results of experiments in which labelled decamethonium was injected into rats in vivo (1.64 mg/kg). In Fig. 4A the left diaphragm was removed after 2 hr, while Fig. 4B and Fig. 4C show the distribution in diaphragm removed after 6 days and 10 days respectively. Gaussian curves have been fitted to the histograms by a method which is described below (see Appendix). In Fig. 4 there is a fall in radioactivity with time, and also a progressive widening of the Gauss curves, shown by the computed values of the
R. CASE AND OTHERS 288 standard deviation which are shown for each of the curves. The results for the whole series are listed in Table 1. If the widening of the Gaussian curves is attributed to a process with diffusion-like kinetics then the increase in variance should be linear with time and
0-12-0o2= 2D't,
(1)
TAiRZ 1. Labelled decamethonium in vivo
No. of muscles 12 6 6 12
Time 2hr 2 days 6 days 12 days
Standard deviation (mm) 0802 + 0140 0987 + 0O197 1*352 + 0*237 1-643 + 0253
Total counts (min- mg-L) 651 655 446 619
Ratio 058 072 0-69 071
Mean values of o, the computed standard deviation from histograms similar to those of Fig. 4, are listed together with the S.D. of the group. Mean values of the total counts min-' mg-' are also shown. The last column gives the fraction of the total count which is alloted to the Gauss curve, obtained as described in the text.
3H .1 .1
.1
11
.1
2 CN
E E 0)
., .1
>
.1
1
I
.1
0
I
-
O
2
I
6
4
8
J 10
Days
Fig. 5. Plot of variance against time in days. The points show (mean value of the standard deviation)', obtained from histograms similar to those of Fig. 4. The limits show the 95 % confidence values. The interrupted line is the regression, obtained as described in the text.
LABELLED DECAMETHONIUM ass where ro- is the S.D. at zero time and o the S.D. after an interval t; D' is an apparent diffusion coefficient (e.g. Jacobs, 1935). In Fig. 5 the variance in mmx is plotted against time in days. The points give values of (mean S.D.)., and the limits give the estimated 95 % confidence values. There is a progressive increase of variance with time, and the interrupted line gives the regression where the slope is 0-2055 mm2 day-1, and hence D' is 1-2 x 1o-8 cm sec-1 with 95% limits 1-1-13 x 10-8 cm2 sec-1 (see Appendix). The deviation ao at zero time is 0 79 mm. The histograms shown in Fig. 4 are representative of muscles removed at various times after injection. A fall in the total radioactivity occurred during the first few days, but there was little change between the sixth and the tenth days apart from the widening of the Gaussian curves. It was found in other experiments that it was possible to detect radioactivity in diaphragms removed 21 days after injection. The total radioactivity in the individual muscles was variable, and this was partly attributed to variations in the dose which was injected. The fraction of total radioactivity attributable to the Gaussian curve is also shown in Table 1. This was 0-58 after 2 hr and later became nearly constant at approximately 0-71. It can be seen in Fig. 4A that the values at the end of the muscles at 2 hr are much larger than values in Fig. 4 B and Fig. 4C, and the change in the ratio could be attributed to loss of non-specific component with a comparatively rapid turnover. After 2 days the ratio is nearly constant, and no difference is then detectable in the rate of exchange of the
different components. DISCUSSION
Decamethonium has been found to induce the entry of labelled sodium in the end-plate region (Creese & Mitchell, 1975), and labelled decamethonium also enters the fibres (Creese & Maclagan, 1970). The compound appears to become internally bound, for 10 hr of wash in the present study produced no change in the distribution apart from a slow loss of radioactivity to the outside solution. This is consistent with the autoradiographic findings (Creese & Maclagan, 1970) in which no change was detected in the distribution of radioactivity in single fibres over a period of 2 hr. Labelled calcium also enters at the junctional region and becomes attached to internal structures so that it is not easily removed by subsequent washing (Evans, 1974). The initial distribution of decamethonium in whole muscles seen in Fig. 4A has a S.D. of 0-8 mm, and this may be due chiefly to the scatter of end-plates in the various layers of the diaphragm. The normal curves shown in Fig. 4 slowly widen with time, and this indicates a very sluggish longitudinal movement with an apparent diffusion coefficient which is less than 1/500 of that in free solution. In 10 days the standard deviation more than doubles, and this is not likely to be produced solely by growth of the muscle. Simple diffusion through narrow
R. CASE AND OTHERS 290 tubules or endothelial reticulum is also unlikely to explain this slow rate, and Endo (1966) found that a dye could be cleared from the tubule spaces within a couple of minutes. The slow spread in Fig. 4 could be attributed to random movements along a series of adsorption sites, which would produce a gradual increase in the scatter. In Fig. 5 the slope gives variance/ time and is proportional to an apparent diffusion coefficient with dimensions L2T-1. In rat peroneus muscle the distribution of labelled decamethonium three minutes after injection was found by autoradiography to be maximal at the end-plate region, but the radioactivity extended for several hundred micra along the fibres (Creese & Maclagan, 1970). Similar results have been obtained in cat muscle (Creese & Maclagan, 1976). The simplest explanation is in terms of extra-junctional receptors, for the distribution found by autoradiography would not be produced if the drug entered only at the end-plate with diffusion coefficient as small as 1-2 x 10-8 cm2 sec-1. The existence of extra-junctional receptors is needed to explain the effects produced by iontophoretic application of acetylcholine in rat diaphragm (Miledi 1960, 1962) and also in mouse omohyoideus muscle (Dreyer, Muller, Peper & Sterz, 1976). These results differ from those in some frog muscles (Dreyer & Peper, 1974; Kuffler & Yashikami, 1975). From Fig. 5 the initial distribution of radioactivity has a S.D. of 0*79 mm, and a similar value is seen in Fig. 4A. This must be mainly due to the scatter of individual end-plates in the muscle. The irregular zig-zag appearance of end-plates in the diaphragm can be seen in the photographs published by Hebb, Krnjevi6 & Silver (1964) and England (1970), and the distribution may be further distorted when the diaphragm is mounted in holders and frozen. Figs. 4 and 5 show a steady spread of radioactivity, and from the increase of variance with time the internal diffusivity has been estimated by means of eqn. (1). During the process of diffusion, the distribution in the whole muscle will be the algebraic sum produced by individual fibres whose end-plates have themselves a considerable scatter. The sum of a number of variates when the origins are not aligned has been considered in connexioan with the central limit theorem of statistics, and Thomasian (1969) points out that if one variate is drastically greater than all others then 'the distribution of the sum of errors might well be about the same as the distribution of the one large error'. The estimation of the diffusion coefficient is thus not feasible by this method for short runs, and in practice the experiment was allowed to proceed until the width of the Gaussian curve as shown by the standard deviation had more than doubled. This research was supported by U.S. Public Health Service Grant NB-00738.
LABELLED DECAMETHONIUM
291
APPENDIX
Fitting of Gau8sian curves to distribution of labelled decamethonium The histograms shown in Figs. 3 and 4 were fitted to normal curves of the form y = b+l exp{-(xX _)2/2o4}, (2) where y is counts min-' mg-', x is distance in mm from the tendon, and b, 1, Ad and o- are constants. This is a Gauss curve plus the term b which is the value of the non-junctional uptake at the ends of the muscle. This term is in general unknown, and because of this additional constant it was not possible to fit the curves by conventional methods which aie used if b is zero or measured. The values of the ordinate in counts min- mg-1 (see Fig. 4) were taken to refer to points x at 05, 1-5, 2-5 mm etc. measured from the end of the tendon, and there were eight to thirteen slices in each diaphragm. A computer programme BMD07R (Dixon, 1974) was used to find the least squares fit of the values (x,y) to the curve of eqn. (2). That is, the values of b, 1, #t and ar were found that minimized 82 which is given by:
82
1
=
m
E[i - b- I exp {_(X.,-,)2/20r21]2,
(3
where m is the number of sections for the particular diaphragm, and (xl, Yi) .... (Xm, ym) are the measurements on the sections of the muscle. For each diaphragm the least-squares estimates of u, a, I and b were found. The theoretical curve given by eqn. (2) can be broken up into the sum of two curves y,(x), and y2(x) where y1(x) = b y2(X) = I exp {-(x-,/)2/2o-2}. With these definitions eqn. (2) can be rewritten as Y
=
Y1(X)+y2(x).
The curve represented by yl(x) is called the base curve, and the curve represented by y2(x) is called the normal curve. An is an estimate of the total number of counts contributed by the normal curve. It is the total area under the curve y2(x) in the region corresponding to the sections measured. Precisely A=
Z
1 exp -(z-,u)2/2a'2}dz,
where the limits of integration for the ith slice are (xi- 1) and (xi + i). For most diaphragms A. was calculated using the formula
An= (2ir)i1.(
(4)
R. CASE AND OTHERS 292 In those cases where measmements from a section were missing or there was a significant portion of the normal curve in the region beyond the sections the value using (5) was multiplied by the factor p when p is given by m j'z2 p = z § (2ir)- exp(-z2/2) dz (6)
i=1 Jz where the limits of integration for the ith slice are (xi+ -j-)/cr and (xi-i -#s)/or. An given by eqn. (4) is equivalent to p(2ir)ii". A b is an estimate of the total number of counts contributed by the base curve. It is the total area under the curve y,(x) in the regions corresponding to the sections measured. It is given by Ab= mb. (7) At is an estimate of the total number of counts that should be observed for all measured sections of the diaphragm. It is the area under the curve given by eqn. (2) in the regions corresponding to the sections measured. It is given by (8) At = An +Ab. Then ac is the ratio of the number of counts contributed by the normal curve to the total number of counts. It is given by a = AnIAt. (9) and The results are summarized in Table 1 were used to calculate the apparent diffusion coefficient for longitudinal movement. It can be seen from Fig. 4C that at 10 days the fitted curve is still diminishing at the ends, so that some portion of the curve extends beyond the limits of the measured values. The effect was small at 10 days and no correction has been applied. If the experiment had been appreciably prolonged it would have been necessary to consider the effects of reflexion (Jacobs, 1935; Thomasian, 1969). Plot of variance against time In Fig. 5 the estimate of the variance does not have a normal distribution and it would be inappropriate to apply conventional regression analysis to the results as they stand. The estimates of 0-, the standard deviation, have a distribution which approximates to the normal. The values of o are in groups which are listed in Table 1, and the individual results were standardized by dividing each by the group median. For the resultant ratio the median was 1-0, the mean was 1-0065 and the standard deviation was 0-164. Pearson's skewness coefficient, which is defined as (mean-mode)/(standard deviation) may be taken as 3 (mean-median)/(standard deviation), and this comes to 0 118 (Yule & Kendall, 1950).
LABELLED DECAMETHONIUM 293 The regression in Fig. 5 may be extrapolated to the abscissa at a time d which is approximately - 3 days. Equation (1) may be transformed to
2D'(tI+d) or = (2D')1(t+d)i.
o- =
(10)
or (11) Hence if d were known then o- could be plotted against (t + d)i and conventional regression analysis could be applied. In practice the mean value of or at each time was plotted, with weighting coefficient n/var where n is the number of muscles and var the variance. Different values of d were tried and it was found that for d equal to 3-0012 days the weighted regression had an origin of zero. With these values the slope which was (2D')4 came to 0-4534, so D' was 0-1028 mm2 day-'. The error variance of the weighted regression was 0-00148; since four mean values of C were used, the statistic 't' was 4-30 and the 95 % confidence internal for the slope (2D')i was from 0-4382 to 0-4685. Hence the limits for D' were 0-096-0-110 mm2 day-', or 1-1-1-3 x 10-8 cm2 sec-L. The values of a, the standard deviations computed from the histograms of muscle in vivo, had a sampling distribution which was found to be approximately normal. Other possible distributions were tried for plotting as and o against time, including the use of the median, the censored mean (Dixon, 1960) and the transformation 1/lo, but in practice these all gave a larger variance from regression when plotted against (t+d)i (or its reciprocal where applicable). It was concluded that there was no advantage in discarding the arithmetic mean, which was used in the plot shown in Fig. 5. REFERENCES
BROOKES, N. & M.ACKAY, D. (1971). Diffusion of labelled substance through isolated rat diaphragm. Br. J. Pharmac. 41, 367-378. COOKSON, J. C. & PATON, W. D. M. (1969). Mechanisms of neuromuscular block. Anaeathe8ia 24, 395-416. CREESE, R. (1968). Sodium fluxes in diaphragm muscle and the effects of insulin and serum proteins. J. Phygiol. 197, 255-278. CREESE, R., FRANKLIN, G. I., HUMPHREY, P. P. A. & MITCHELL, L. D. (1976). Dose response curves with labelled sodium and labelled decamethonium in rat muscle. J. Phygiol. 254, 43-44 P. CREESE, R. & MACLAGAN, J. (1970). Entry of decamethonium in rat muscle studied by autoradiography. J. Phygiol. 210, 363-386. CREESE, R. & MACLAGAN, J. (1976). Labelled decamethonium in cat muscle. Br. J. Pharmacy. 58, 141-148. CHEESE, R. & MITCHELL, L. (1975). Sodium entry in junctional region of rat muscle. J. Physiol. 246, 44-45 P. CREESE, R. & NORTHOVER, J. (1961). Maintenance of isolated diaphragm with normal sodium content. J. Physiol. 155, 343-357. CREESE, R., TAYLOR, D. B. & CASE, R. (1971). Labeled decamethonium in denervated skeletal muscle. J. Pharmacy. exp. Ther. 176, 418-422. CREESE, R., TAYLOR, D. B. & TILTON, B. (1963). The influence of curare on the uptake and release of a neuromuscular blocking agent labeled with radioactive iodine. J. Pharmac. exp. Ther. 139, 8-17.
294
R. CASE AND OTHERS
DIXON, W. J. (1960). Simplified estimation from censored normal samples. Ann. math. Stati8t. 31, 385- 391. DixON, W. J. (1974) (ed). Biomedical Computer Programs. Los Angeles: University of California Press. DREYER, F. & PEPER, K. (1974). The acetylcholine sensitivity in the vicinity of the neuromuscular junction of the frog. Pflugere. Arch. gee. Phyeiol. 348, 273-286. DREYER, F., Miser, K.-D., PEPER, K. & STERZ, R. (1976). The M. omohyoideus of the mouse as a convenient mammalian muscle preparation. A study of junctional and extra-junctional acetylcholine receptors by noise analysis and cooperativity. Pfluger8 Arch. gee. Phyeiol. 367, 115-122. ENDO, M. (1966). Entry of fluorescent dyes into the sarcotubular system of frog muscle. J. Phy8iol. 185, 224-238. ENGLAND, J. M. (1970). The localization of end-plates in unstained muscle. J. Anat. 106, 311-321. EvANs, R. H. (1974). The entry of labelled calcium into the innervated region of the mouse diaphragm muscle. J. Phy8iol. 240, 512-533. HEBB, C. O., KRNJEVI6, K. & SILVER, A. (1964). Acetylcholine and choline acetyl transferase in the diaphragm of the rat. J. Phy8iol. 171, 504-513. JACOBS, M. H. (1935). Diffusion processes (section 11) Ergebn. Biol. 12, 1-60. KUFFLER, S. & YosmE.m, D. (1975). The distribution of acetylcholine sensitivity at the post-synaptic membrane of vertebrate skeletal twitch muscles: iontophoretic mapping in the micron range. J. Phyeiol. 244, 703-730. MLE-DI, R. (1960). Junctional and extra-junctional acetylcholine receptors in skeletal muscle fibres. J. Phyeiol. 151, 24-30. MILEDI, R. (1962). Induction of receptors. In Enzymes and Drug Action (CIBA Foundation Symposium), ed. MONGAR, J. L. & DE REUCK, A. V. S., pp. 220-235. London: Churchill. TAYLOR, D. B., DIXON, W. J., CREESE, R. & CASE, R. (1967). Diffusion of decamethonium in the rat. Nature, Lond. 215, 989. TAYLOR, D. B. & NEDERGAARD, O. A. (1965). Relation between structure and action of quaternary ammonium neuromuscular blocking agents. Phyeiol. Rev. 45, 523556. TitMSIAN, A. J. (1969). The Structure of Probability Theory with Applications, ch. 13 and ch. 24. New York: McGraw-Hill. YuLE, G. U. & KENDATT, M. G. (1950). An Introduction to the Theory of Statiiticm, 14th edn., ch. 7. London: Griffin.
