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Statistical significance tests for autoradiographic data.

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THE ANATOMICAL RECORD 211:126-132 (1985)
Statistical Significance Tests for
Autoradiographic Data
BETTY J. SKIPPER AND LINDA J. McGUFFEE
Department of Family, Community, and Emergency Medicine (B.J.S.) and Department of
Pharmacology (L. J. M.), Uniuersity of New Mexico Albuquerque, N M 87131
ABSTRACT
The purpose of this paper is to develop statistical methods that take
radiation spread into account in analyzing data from different autoradiographic
experiments. The method uses the probability circle analysis of Salpeter and McHenry (1973) to obtain the probable source of each radioactive emission and the
circle and point counting method of Williams (1969) to estimate the relative area
occupied by each cellular site. Two levels of analysis are presented. The first level of
analysis is concerned with estimating relative activities and standard errors for
cellular items that are larger than the probability circle. The second level of analysis
involves estimating relative activities and standard errors for cellular sites that are
smaller than the probability circle and are therefore observed in circles containing
another item such as cytoplasmic matrix. Two different tests of hypotheses are
discussed. The first null hypothesis is that the radioactivity is randomly distributed
among the cellular sites. The second null hypothesis is that there is no difference
between two different treatments in the relative activities for a given site.
Autoradiography a t the level of the electron microscope can be used to determine the relative distribution
of a substance among the various components of a cell.
The methodology for carrying out this type of experiment is well documented (Kent and Williams, 1974;
Salpeter and McHenry, 1973; Salpeter et al., 1978; Williams, 1977). Briefly, the method consists of exposing a
tissue to a radioactive beta-emitting isotope, fixing the
tissue, and slicing it into thin sections. These sections
are then overlayed with a photographic emulsion containing silver halide crystals in a gelatin matrix. The
isotope will emit beta radiation in all directions. Some
of the beta particles will enter the emulsion layer and
hit silver halide crystals. When a beta particle hits a
sufficient cross-sectional area of a crystal, a latent image, a speck of reduced silver, appears somewhere in the
crystal. When this crystal is exposed to photographic
developer, it is subject to faster reduction than are crystals that are not hit by beta particles. After development, the silver grain is visible at the light microscopic
level as a black dot or at the electron microscopic level
as a darkened area of filamentous silver. An illustrative
electron micrograph is shown in Figure 4 of the paper
by McGuffee et al. (1985).
Because the radioactive particles are emitted in all
directions, many particles will not travel in a path that
is perpendicular to the thin section and may not appear
over the site of origin. Depending on the energy of the
beta particle, most of the grains originating from small
sources will fall outside of their sources owing to this
radiation spread. In smooth muscle, mitochondria and
SR can be considered small sources. The “Probability
Circle” analysis as described by Salpeter and McHenry
(1973) takes the radiation spread into consideration. The
method involves experimentally determining the half
0 1985 ALAN R. LISS. INC.
distance (HD) for the isotope, i.e., the distance from a
line source within which 50%of the grains lie. When a
circle with a radius equal to 1.7 HD is drawn around a
grain, there is a 50%probability that the circle contains
the point source. Salpeter and McHenry (1973) have
pointed out the fact that one cannot use a circle that is
too large because as the circle size increases toward
loo%, the selective association of grains with structures
is lost.
In the initial step of the analysis, the center of the
developed grain, i.e., the midpoint between the extremities, is chosen for each grain (Salpeter and McHenry,
1973) and the circle of radius equal to 1.7 HD is placed
over the grain. All structures that lie within this circle,
e.g., mitochondria, SR, plasma membrane, or nucleus in
smooth muscle, are considered to be possible sources of
the radioactive decay. If no organelle occurs within the
circle, the cytoplasmic matrix is scored as the probable
source.
Because different sites occupy different proportions of
the area of the cell, the information that can be obtained
from the grain count distribution is limited unless one
takes into account the relative sizes of the various components of the cell. The stereological methodology used
to determine the area distribution for this study is the
point and circle analysis of Williams (1969). For this
analysis, one uses a plastic overlay on which a n array of
circles has been drawn. The radius of each circle is the
same as the size of the circle used for obtaining the
grain counts and the center of the circle is denoted by a
point. The overlay is placed over each autoradiogram.
Received May 30, 1984; accepted August 8, 1984.
STATISTICAL METHODS FOR AUTORADIOGRAPHY
Cov(G,, GJ
=
-GglgJ
127
Relative Activities and Standard Errors of Individual
Constituents of Compound Items
If the sample size is relatively large, then the Gi’s are
Some cellular sites are smaller than the probability
approximately normally distributed.
circle so their relative activites cannot be estimated
For the proportion of grains in category i, the esti- using the above method. The purpose of this section is
mated mean is g, and the estimated variance is
to obtain estimated relative activities and standard erV(gJ = g, (1 - gJG.
(la) rors for these individual constituents of compound items.
Let aij denote the relative activity for a compound item
The estimated standard error is
containing sites i and j. Assume that site i is large
enough to be observed by itself, but site j is too small to
SE(gJ =
= Jg, (1 - gJG.
(lb)
be observed by itself. For example, site i could be cytoplasmic matrix while site j could be mitochondria. Let
The estimated covariance between proportions g, and gj ai denote the relative activity for the observed item
is
containing site i by itself. Let C- denote the number of
circles containing items i and j. f!et Pi be the number of
(IC)
Cov(gi, gj) = -gi gjIG.
points over site i and let Pj be the number of points over
site j. Then pi = Pi/Cij is the proportion of the area of
Since not all items occupy the same proportion of the the compound item occupied by site i and pj = P;/Cij is
area of the cell, the grain count for each compound item the proportion of the area occupied by site j. Let s; denote
must be adjusted for the relative amount of the area the relative activity of site j, the component that is too
that is occupied by that item. As described previously, small to be observed by itself. The equation given by
the point and circle counting method is used to deter- Williams (1969) and Kent and Williams (1974) for estimine these relative areas. This procedure is carried out mating s; is
using the same set of micrographs and the same set of
compound items, but is done without regard to the positions of the grains.
(5)
Let C be the total number of circles counted. Let Ci be
the number of circles observed to occur over category i.
Since the variance of the difference between two ranLet ci = Ci/C denote the proportion of circles associated
with category i. The equation for the expected number dom variables X and Y is
of grains in category i is given by Kent and Williams
v:v - Y ) yx) t VEf -9 Cnv<X,V!
(1974)and Williams (1977)as expected number of grains
= G x (Ci/C). The relative activity is
the variance of the relative activity for site j can be
obtained as
~
a
-
Gi
C, GIC
This expression may be rewritten as
That is, the relative activity is the proportion of grains
associated with observed item i divided by the proportion of the area associated with that item. Since gi must
be between 0 and 1, the relative activity, ai, will be
between 0 and Uci.
It has been previously suggested (Williams, 1969)that
if one counts a large number of circles, then the standard errors for the circle count will be so small that they
can be ignored. Since the ci’s are constant under this
assumption, the estimated variance of the relative activity can be obtained as
The estimated standard error is
(7a)
(7b)
The covariance is obtained as follows:
128
B.J. SKIPPER AND L.J. MCGUFFEE
Some circles will include a single cellular site, whereas
other circles will fall over portions of more than one
cellular site. One then records the frequency of occurrence of single items, such as cytoplasmic matrix, and
compound items, such as mitochondria and cytoplasmic
matrix or SR and cytoplasmic matrix. To obtain results
for the individual constituents of these compound items,
a n estimate of the portion of the circle that is occupied
by each constituent item must be obtained. This is done
by recording the frequency with which points (i.e., centers of the circles) fall over each of the individual items.
For example, assume that in a series of autoradiograms
100 circles fall over areas that contain portions of mitochondria and cytoplasmic matrix. For those 100 circles,
25 have their centers over mitochondria and 75 have
their centers over matrix. One would then estimate that
the mitochondria occupy 25% of the area in circles containing mitochondria and matrix while the matrix occupies the remaining 75% of the area.
Williams (1977) has used the relative areas obtained
from the point and circle analysis to calculate the expected number of grains for a given item assuming random incorporation of the radioisotope. He then divides
the observed grain count for each item by the expected
grain count to obtain the quantity that he called the
“relative crude specific activity.” In this paper, this
quantity will be called the “relative activity for observed compound items.”
After the relative activities have been obtained for
each compound item, one needs to be able to estimate
relative activities for the cellular sites that are smaller
than the probability circle and therefore are only observed in association with another site, such as cytoplasmic matrix. Williams (1977) and Kent and Williams
(1974) have published a n equation for estimating these
relative activities, but no estimates of standard errors
are included in their papers.
In the present paper, we will present equations for the
standard errors of the relative activities of compound
items, as well as standard errors for the relative activities of the smaller individual constituents of compound
items. These standard errors then will be used to test
whether the radioactivity associated with individual
sites is significantly different from what one would expect from a random distribution of radioactivity
throughout the cell. A test of the null hypothesis that
two different treatments result in the same relative
activity for a given cellular site will also be presented.
STATISTICAL METHODS
Theorems Used in Deriving Standard Errors
Three theorems from mathematical statistics (Mendenhall and Scheaffer, 1973)will be used in deriving the
standard errors.
Theorem 3: Let Cov (X,Y) be the covariance between
random variables and let the letter a denote a constant. Then
Cov(uX,Y) = a Cov(X,Y).
Assumptions and Definitions
Assuming that radioactive decay is a random process,
one would expect the number of grains associated with
each cellular item to follow a Poisson distribution (Williams, 1977). The total number of grains would also
follow a Poisson distribution since it is the sum of independently distributed Poisson variables. However, for
analyzing the results of a particular experiment, we
need the distribution of the number of grains associated
with a given cellular item conditional on the total number of grains that are observed. It can be shown that
this conditional distribution is the multinomial distribution (Meyer, 1975).
Let G be the total number of grains counted for the
probability circle analysis. These grains are classified
into N categories based on the cellular sites which are
included in the circles around the grains. Some of the N
categories will include a single site such as cytoplasmic
matrix, while others will include two sites such as mitochondria and cytoplasmic matrix or three sites such
as plasma membrane, cytoplasmic matrix, and extracellular space. The items that include two or more sites
will be called “observed compound items” in contrast to
the individual cellular sites which may be smaller constituents of these items and therefore cannot be directly
counted as single items occupying the entire area of the
probability circle. The statistical methods for analyzing
data from compound items will be developed first and
then this analysis will be expanded to give relative
activity estimates for the smaller individual constituents of these compound items. Single subscripts will be
used for the analysis of grain counts for the observed
compound items. This notation will be expanded to include double or triple subscripts when we are estimating
the relative activities for the individual constituents of
the observed compound items.
Relative Activities and Standard Errors for Observed
Compound items
Let Gi be the number of grains in the ith category.
n
c
1=1
G, = G.
Let gi = Gi/G denote the proportion of grains that are
category i. As discussed above, the number of grains in
category i will follow a multinomial distribution. For
the multinomial distribution (Mendenhall and ScheafTheorem 1: Let V(X) be the variance of a random vari- fer, 1973), the estimated mean is Gi and the estimated
able, X, and let the letter a denote a con- variance is V(Gi) = G(gi)(l-gJ. The estimated standard
stant. Then V(aX) = a2V(X).
error is
Theorem 2: Let V(X) and V(Y) denote the variances of
random variables X and Y, respectively. Let
Cov (X,Y) denote the covariance of X and
Y. Then the variance of the difference be- The estimated covariance between Gi, the number of
tween X and Y can be obtained as
grains in category i, and Gj, the number of grains in
category j, is
V(X-Y) = V(X) + V(Y) - 2 Cov(X,Y).
STATISTICAL METHODS FOR AUTORADIOGRAPHY
129
Testing for Randomness of Radioactivity
The derived relative activities, a;, for individual constituents of compound items are obtained by linear
transformations of the observed grain counts, Gi. Since
the Gi's are approximately normally distributed, the
derived relative activities also will be approximately
normally distributed.
If the radioactivity is randomly distributed throughout the cell, the proportion of the total grains associated
with each site will be equal to the proportion of total
area occupied by that site. That is, the relative activity
will be equal to 1.One can test the null hypothesis that
the relative activity is equal to one by calculating the
It should be noted that the denominator of the standard error (equation 10)includes pJ and &. The standard
error becomes smaller as one increases the number of
grains that are counted. Since it is a square root relationship, one would need to quadruple the number of
grains counted if one wanted to halve the standard error. The pj's will be small for small sites, such as mitochondria and SR. Division by these small quantities
means that standard errors for these sites will tend to
be large. The only way to counteract this effect is to
count a large number of grains and thus reduce the
standard error by increasing the value of G in the
denominator.
Because of sampling variation, there may be occasions
where sJwill be negative. Such results should be interpreted to mean that the particular experiment does not
show any radiation coming from source j.
An observed compound item may contain three sites,
such as plasma membrane, cytoplasmic matrix, and extracellular space. For such items, assume that sites i
and j are large items, such as cytoplasmic matrix and
extracellular space. Assume that site k is a small site,
such as plasma membrane. The estimated relative activity for site k is
aijk - (Pi a,
sk =
+ PJ aj)
Since this quantity will be approximately normally distributed with a mean of 0 and a standard deviation of 1,
one can use standard tables of the normal distribution
to determine the statistical significance level (P value)
for Z.
Comparison of Results for Two Different Experiments
The purpose of some autoradiographic studies is to
determine if two different treatments result in the same
relative distribution of a given substance within the cell.
Therefore, one will be testing the null hypothesis that
the relative activity for site j in experiment 1is equal to
the relative activity for site j in experiment 2.
Let syl) be the relative activity for site j for treatment
1and jet s;(2) be the relative activity site j for treatment
2. As mentioned previously, these relative activities will
be approximately normally distributed. The difference
between the two relative activities, (sj(l1-sj(2)), will also
be approximately normally distributed with variance
V(s.ci)) + V(S~(~)).
Under the assumption of the null hypotLesis that there is no difference between the two
treatments, the quantity
(11)
Pk
where pi, p;, and Pk are the relative areas of sites i, j,
and k within circles containing these sites. The estimated variance is given by
will be approximately normally distributed with mean
0 and standard deviation 1. Tables of the standard normal distribution can be used to determine the statistical
significance level for Z.
To verify the assumption of normality, we carried out
simulation studies based on data from the experiment
discussed in the following applications section. Figure
la-c shows the histograms of the results of 2,000 simulations comparing relative activities of plasma membrane, mitochondria, and SR for two different experiments. These distributions are approximately normal, as expected.
RESULTS OF SPECIFIC APPLICATION
The numbers of grains observed in association with
each compound item in an experiment using 45Ca in
control and K + -contracted smooth muscle from rabbit
vas deferens are shown in column 1 of Table 1. Details
of the experimental methodology are described in McGuffee et al. (1985).The distribution of the circles that
were counted is shown in column 2 of Table 1 and the
distribution of points is shown in columns 3-8.
130
B.J. SKIPPER AND L.J. MCGUFFEE
500
t
YW
-
-
--
-
;
300-
e
For illustrative purposes, the calculations for the relative activity in the mitochondridcytoplasmic matrix
item are shown below. Extra digits are carried throughout the calculations to minimize rounding error.
(0)
7
200-
g3 =
100I
,
a
LL
56
1.180
=
0.0475.
From equation lb, the standard error is
,
4
W
~
These results are shown in column 1 of Table 2. The
relative proportion of circles in association with this
item is
=tdd€uL- 5 - 4 - 3 Q - I
0
I
2
3
175
c3=--4,578 - 0.0382.
From equation 2, the relative activity is obtained as
4
5611,180
- 1.2415.
17514.578
a3=-(C)
Using equation 4,the standard error is
These results are shown in column 3 of Table 2.
DIFFERENCE BETWEEN RELATIVE ACTIVITIES
FOR CONTRACTU) AND CONTROL TISSUES
Fig. 1. Distribution of the results of 2,000 simulationsof the difference
between the relative activities of contracted and control tissues. a) Plasma
membrane; b) mitochondria; c) SR.
TABLE 1. Raw data for 45Ca in control and K-contracted tissue rabbit vas deferens
Control Tissue
Cyto. matrix
P.M./matrix/Ec.S.
Mito./matrix
S.R./matrix
Nuchatrix
Ec.S.
Total
K-contracted tissue
Cyto. matrix
P.M./matrix/Ec.S.
MitoJmatrix
S.R./matrix
Nuc./matrix
Ec.S.
Total
P.M.
points
Mito.
points
S.R.
points
Nuc.
points
(2)
Cyto.
points
(3)
(4)
(5)
(6)
(7)
2,255
456
142
132
52
0
3,037
0
144
107
1,180
2,255
988
175
163
395
602
4,578
0
0
33
0
0
0
33
0
0
0
31
0
0
31
0
0
0
0
343
0
343
563
136
52
47
49
31
878
3,031
753
263
208
314
263
4,832
3,031
363
227
145
39
0
0
36
0
0
0
36
0
0
0
0
0
0
0
Grains
Circles
(1)
613
277
56
52
75
0
0
0
0
144
0
153
0
0
0
0
0
3,805
153
63
0
0
63
275
0
275
Ec.S.
points
(8)
0
388
0
0
0
602
990
0
237
0
0
0
263
500
Cyto., cytoplasmic matrix; P.M., plasma membrane; EcS., Extra-cellular space; Mito., mitochondria; S.R.,
sarcoplasmic reticulum; Nuc., nucleus.
131
STATISTICAL METHODS FOR AUTORADIOGRAPHY
Since the mitochondria are small relative to the size
of the circle, they will only OCCLU as a constituent of the
compound item including mitochondria and cytoplasmic
matrix. The relative activity and the standard error can
be estimated from equations 5 and 10, respectively.
1.2415 -(142/175)(1.0546)
= 2.0457
33/175
1
(0.0475)(0.9525)
SE ( ~ g =
)
(33/175)J1,180
(0.0382)2
(142/175)2(0.5195)(0.4805)
s3 =
+
(0.4926)2
1
+(2)(142/175)(0.0475)(0.5195)
(0.0382)(0.4926)
%
o.8977
~
These results are shown in column 1of Table 3.
For the mitochondria in the control preparation, one
can determine if the relative activity is significantly
different from 1by calculating
-1
z=-2.05
- 1.17
0.90
From tables for the standard normal distribution, one
finds that for a two-tail test a t the 5% level of significance the critical values are Z = -1.96 and Z = +1.96.
Since 1.17 is less than 1.96, the relative activity for
mitochondria in this preparation is not significantly different from 1.
To determine if there is a significant difference between the relative activities in mitochondria in control
and contracted tissues, one can calculate
Z=
2.05 -1.50
= 0.38.
d(0.90)2 + (1.13)2
This value is shown in column 3 of Table 3. Since 0.38
is less than 1.96, there is no significant difference between the relative activities in mitochondria in the two
preparations.
DISCUSSION
In analyzing the results of autoradiographic experiments, one may want to test the null hypothesis that
the radioactivity associated with a given substance is
randomly distributed throughout the cell or one may
want to compare the amounts of radioactivity associated
with a given cellular site under different experimental
conditions.
To be able to carry out such analyses, one needs to
take into account the effects of radiation spread. One
method for doing this uses the probability circle of Salpeter and McHenry (1973) for determining the most
likely sources of the radioactive emissions. The circle
and point counting method described by Williams (1969)
is used for obtaining the area distribution of the cellular
items. Williams (1977) and Kent and Williams (1974)
have published equations for estimating the relative
activities for observed compound items, as well as sites
that are smaller constituents of the observed compound
items.
The standard errors presented in this paper may be
used in estimating the variability associated with the
estimates of relative activity. As can be seen from equation 4 and equation 10, the estimated standard errors
for the relative activities are inversely proportional to
the square root of G, the total number of grains counted.
Therefore, one can reduce the standard errors by counting more grains. However, if one is already counting a
reasonably large number of grains, one may have to
increase substantially the sample size to obtain the desired results. For example, to halve a n estimated standard error based on counting 1,000 grains, one would
have to count 3,000 more grains.
Some sites are small and can only be counted in conjunction with another item, such as cytoplasmic matrix.
The estimated standard error of the relative activity of
such a site is inversely proportional to pj, the portion of
the area of the circle that is occupied by that site. Since
pj will be small for sites that occupy relatively little
area, the standard errors for these sites will be large
TABLE 2. Relative activities of compound items in control and K-contracted tissues
Relative
activity
(1)
Relative
proportion
of circles
(2)
0.5195 f 0.0145
0.2347 k 0.0123
0.0475 f 0.0062
0.0441 f 0.0060
0.0636 f 0.0071
0.0907 2 0.0084
0.4926
0.2158
0.0382
0.0356
0.0863
0.1315
1.055 0.030
1.088 f 0.057
1.241 k 0.162
1.238 & 0.168
0.737 f 0.082
0.690 i 0.064
0.6412
0.1549
0.0592
0.0535
0.0558
0.0353
0.6273
0.1558
0.0544
0.0430
0.0650
0.0544
1.022 f 0.026
0.994 5 0.078
1.088 k 0.146
1.244 0.176
0.859 k 0.119
0.649 f 0.114
Relative
proportion of
grains f SE
Cellular item
Control Tissue
Cyto. matrix
P.M./matrix/Ec.S.
MitoJmatrix
S.R./matrix
Nuclmatrix
Ec.S.
K-contracted tissue
Cyto. matrix
P.M./matrixlEc.S.
MitoJmatrix
S.R. matrix
NucJmatrix
Ec.S.
Abbreviations as in Table 1.
f 0.0162
f 0.0122
f 0.0080
5 0.0076
k 0.0077
f 0.0062
k SE
(3)
132
B.J. SKIPPER AND L.J. MCGUFFEE
TABLE 3. Relative activities of individual cellular sites in control and K-contracted rabbit
vas deferens
K-contracted tissue
relative activity f SE
Cellular site
Control tissue
relative activity k SE
(1)
(2)
2
Cyto. matrix
P.M.
Mito
S.R.
Nuc.
Ec.S.
1.05 0.03
2.27 + 0.50
2.05 0.90
2.02 k 0.92
0.69 f 0.10
0.69 f 0.06
1.02 f 0.03
1.46 + 0.46
1.50 1.13
1.75 f 0.60
0.84 f 0.14
0.65 0.11
0.71
1.19
0.38
0.25
-0.87
0.32
*
*
Abbreviations as in Table 1.
unless one counts a large number of grains to compensate for this effect. As one would expect, it is more
difficult to estimate precisely the relative activities for
small cellular sites than it is to estimate relative activities for large sites.
Two different tests of hypotheses have been presented.
The first one tests the null hypothesis that the relative
activity for a given site is equal to the relative activity
expected from a random distribution. The second test
allows one to compare the relative activities of a given
site under two different experimental conditions. Both
of the tests are relatively easy to calculate once one has
calculated the appropriate relative activities and standard errors. Standard normal tables can be used to determine the statistical significance levels.
The use of these equations for standard errors and
hypothesis testing procedures will permit more rigorous
evaluation of the results of autoradiographic experiments.
ACKNOWLEDGMENTS
This work was partially supported by National Institutes of Health grant 1-R01-GM30003. L.J.M.'s work
was done during the tenure of a n Established Investigatorship from the American Heart Association and with
funds contributed in part by the New Mexico Affiliate.
LITERATURE CITED
Kent, C., and M.A. Williams (1974) The nature of hypothalamo-neurohypophyseal neurosecretion in the rat. J. Cell Biol., 60:554.
McGuffee, L.J., E.S. Wheeler-Clark, B.J. Skipper, and S.A. Little (1984)
The cellular distribution of calcium in freeze-dried rabbit vas deferens using EM autoradiography. Anat. Res. 211t117-124.
Mendenhall, W., and R.L. Scheaffer (1973) Mathematical Statistics
With Applications. D u b u r y Press, North Scituate, MA., pp. 185193.
Meyer, S.L. (1975) Data Analysis for Scientists and Engineers. John
Wiley and Sons, Inc., New York, pp. 335-356.
Salpeter, M.M., and F.A. McHenry (1973) Electron microscope autoradiography. In: Advanced Techniques in Biological Electron Microscopy. J.K. Koehler, ed. Springer-Verlag, Berlin, pp. 113-152.
Salpeter, M.M., F.A. McHenry, and E.E. Salpeter (1978) Resolution in
electron microscope autoradiography. J. Cell Biol., 76127.
Williams, M.A. (1969) The assessment of electron microscopic autoradiographs. In: Advances in Optical and Electron Microscopy, Val.
3. R. Barer and V.E. Cosslet, eds. Academic Press, New York, pp.
219-277.
Williams, M.A. (1977) Quantitative methods in biology. In: Practical
Methods in Electron Microscopy, Val. 6. A.M. Glauert, ed. North
Holland Publishing Co., Amsterdam, pp. 85-169.
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