J. Embryo/, exp. Morph. Vol. 43, pp. 195-219, 1978
Printed in Great Britain © Company of Biologists Limited 1978
195
Variability of chimaeras and mosaics
By D. S. F A L C O N E R 1 AND P. J. A V E R Y 1
From the A.R.C. Unit of Animal Genetics and the
Department of Genetics, University of Edinburgh, Scotland
SUMMARY
Aggregation chimaeras and X-inactivation mosaics in mice are alike in general appearance,
but chimaeras are very much more variable in the proportions of the cell types (p) seen for
example in coat pigmentation. The distribution of p in chimaeras is not binomial, but is
uniform, or flat, between the two extremes. The greater variability of chimaeras arises from
two sampling events that occur when cellular heterogeneity is already present in chimaeras
but before it arises in mosaics. These are the differentiation of the inner cell mass from the
trophectoderm and of the primary ectoderm from the primary endoderm. The second of
these generates the flat distribution of chimaeras as a consequence of the two cell types being
unmixed at that time. The two sampling events generate single-colour individuals in roughly
the proportions observed. Consideration of the second sampling event provides evidence that
the primordial germ cells must originate in the primary ectoderm and not in the yolk-sac.
Estimation of numbers of progenitor cells on the supposition of binomial sampling is not
valid unless the clone-size is known or the cells in the sample are not contiguous. Data on coat
pigmentation are consistent with the assumptions that X-inactivation is random in about
21 cells, that the sampling of melanoblasts is binomial (because they are not contiguous), and
that the melanocytes of the head and body are descended from about 34 progenitor cells.
INTRODUCTION
This paper is concerned with differences between 'chimaeras' and 'mosaics'
in mice. The chimaeras to be considered are those produced by aggregation of
cleavage-stage embryos, the strains aggregated differing in some recognizable
genetic marker. The mosaics to be considered are those resulting from Xchromosome inactivation in females heterozygous for a sex-linked gene with
distinctive phenotypes. Both are therefore heterogeneous for their cell types
and both show variegation in the adult tissues, most readily seen in the coat
colour, when the marker gene affects pigmentation. On the whole chimaeras
and mosaics look very much alike. Many authors, however, have noted that
chimaeras tend to be more variable than mosaics (see McLaren, 1976 a), and
our main purpose is to show how this difference of variability arises from the
different origins of the cellular heterogeneity.
Most of the work has been done with albino versus coloured as the genetic
marker distinguishing the two cell types, and the animals have been classified by
1
Authors' address: Institute of Animal Genetics, West Mains Road, Edinburgh EH9 3JN,
Scotland, U.K.
196
D. S. FALCONER AND P. J. AVERY
the proportion of albino in the coat pigmentation. For the sake of generality
and simplicity one cell type will be called 'white' and the other 'black'. The
proportion of 'white' (e.g. albino) cells in any organ or tissue will be symbolized
by p. We are concerned, then, with the distribution and the variance of p among
individuals. Individuals with 0 % or 100 % white will be referred to as 'pure'
classes. Attempts have been made to deduce the number of ' progenitor cells ' from
the frequency of pure classes, on the supposition that the variation arises from
binomial sampling. The second part of the paper will deal more specifically
with this problem. But since the whole problem of the variability concerns the
sampling of progenitor cells we must be clear at the outset what the idea of
progenitor cells implies.
Progenitor cells
The following account is based on the explanation given by McLaren
(1976«, p. 112). After the cellular heterogeneity has been created in the embryo
the cells proliferate and the two types mingle with each other. The mingling
may be complete, leading to a random mixture; or the growth may be partially
clonal, leading to a non-random arrangement in patches. Then some cells
become 'allocated' (McLaren's term) to the formation of the organ or tissue
that is studied in the adult. Allocation means, strictly speaking, that the descendants of these cells make up the whole of the organ, and contribute to no
other organ or tissue. The progenitor cells are those cells that have become
thus allocated. The cells that make up the embryonic tissue before allocation
are not progenitor cells ; only after allocation are the allocated cells defined as
progenitors. Allocation is an event that is inferred from the observed variation
of cell proportions in the adult organ and, as McLaren points out, it need not
coincide with any event of cellular differentiation that might be detected by
other means. The concept of progenitor cells is therefore a statistical rather than
an embryological one. The statistical requirement is that the cell proportions
(p) among the progenitor cells should not be changed in their descendants by
subsequent sampling events during the development of the organ. The variance
of p in the adult organ is then a measure of the variance due to the sampling
of the progenitor cells from the embryonic tissue of which they form a part.
An ' organ ' in this context is any assemblage of cells among which the proportions of the two cell types are observed. Thus when the proportion of one cell
type is assessed from the proportion of albino in the coat, the 'organ' consists
only of the melanocytes that contribute to coat pigmentation. The strict definition
of allocation given above is unrealistic and unnecessarily restrictive. The
developing organ may gain cells by immigraton of cells not descended from the
progenitors, or it may lose cells by emigration to other tissues. Neither of these
events will affect the issue provided the cell proportions in the developing organ
are not changed by them.
Variability of chimaeras and mosaics
197
fU
%
rTTTh,,, m60 n m
11111
20
Per cent of marker cells in coat pigmentation
Fig. 1. Distributions of per cent marker cells in coats of mosaics and chimaeras. The
parameters of the distributions are given in Table 1. (a) Mosaics marked by brindled ;
359 mice (Falconer & Isaacson, 1972). (b) Binomial with same mean as (a) and
N = 11. (c) Mosaics marked by albino with Cattanach's translocation; 275 mice.
(Data supplied by Dr Cattanach.) (d) Binomial with same mean as (c) and N = 17.
(e)-(g) Chimaeras marked by albino (Roberts et al, 1976, with single-colour mice
added), (e) Large <-> Small; 18 mice, (ƒ) Control <-> Large; 24 mice, (g) Control <->
Control; 30 mice, (h) Chimaeras marked by multiple récessives; 22 mice (Grüneberg
& McLaren, 1972). (/) Chimaeras marked by albino; 37 mice. (Data supplied by
Dr M. F. Lyon.)
DISTRIBUTION AND VARIANCE
Distributions of mosaics have been recorded by Nesbitt (1971), and of
chimaeras by Mullen & Whitten (1971). In Fig. 1 we show some additional
distributions to illustrate the differences between mosaics and chimaeras. One
group of mosaics (a) was marked by the sex-linked gene brindled which affects
pigmentation; the data are from an experiment described by Falconer &
Isaccson (1972). The other group of mosaics (c) was marked by albino with
Cattanach's translocation; the data were kindly supplied by Dr Cattanach.
Five groups of chimaeras (e-i) are plotted separately because they involved
different strain combinations and had different means. Groups (e), (ƒ) and (g)
are our own, reported by Roberts, Falconer, Bowman & Gauld (1976). The
other groups are compiled from published sources, as noted under the figure,
198
D. S. FALCONER AND P. J. AVERY
Table 1. Parameters of the distributions in Fig. 1
p = mean proportion of marker cells in coat pigmentation, q = 1 — p. a2 = variance of p. cr2/pq gives the variance adjusted for differences of mean, pq/a2 gives the
number of progenitor cells needed to generate the observed variance by binomial
sampling.
Proportion of
Group
Mosaics
Chimaeras
P
(a)
(c)
(e)
(f)
(*)
(h)
(0
0-602
0-511
0-494
0-555
0-347
0-398
0-496
O"2
<r2/pq
00212
00149
00885
00596
0-4765
0-4643
0-6508
0-4487
0-3848
01191
01147
01474
0-1075
00962
pq/a2
11-3
16-8
21
2-2
1-5
2.2
2-6
'0%'
'100%'
0
0
0
0
0-22
0
0-20
017
008
011
014
017
008
003
and from data kindly supplied by Dr Lyon. Group (h) was marked by multiple
recessive genes; all the other chimaeras were marked by albino. The coat
pigmentation of all mosaics and chimaeras was by class-intervals of 5 percentage points, except group (i) where the class interval was 10 percentage points.
The two mosaic groups (a, c) and three of the chimaeras (e, ƒ, g) were all classified by the same standards developed by Dr Cattanach and Mr J. H. Isaacson.
The differences between them therefore cannot be attributed to different
criteria of classification.
It is immediately obvious that the mosaics and chimaeras are very different.
The chimaeras are not only much more variable, but their distributions are
very clearly not binomial. Both groups of mosaics, in contrast, fit closely
to binomial distributions, though they would not be expected to be exactly
binomial for reasons to be stated later. For the sake of illustration, binomial
distributions are shown in Fig. 1(b) and (d). These are the binomials best fitting
the observed mosaic distributions (a) and (c) respectively, having the same
mean and variance. The means and variances of the distributions shown in
Fig. 1, with other parameters for discussion later, are given in Table 1.
The essential feature of the distributions of chimaeras is that, excluding the
pure classes (0 % and 100 %), they are more or less flat, as far as can be judged
from the rather small numbers. That is to say, all values of p are equally probable. We have, therefore, to look for a sampling process that will give a
distribution that is flat, or nearly so, between the two extremes. The frequencies
of the pure classes seem to differ between groups, though not significantly,
ranging from 11 to 37 % for the two classes combined. We hope to show that
these single-colour animals are not necessarily technical failures, but could
result from the sampling processes in chimaeras.
Variability of chimaeras and mosaics
199
SAMPLING EVENTS AND THE ORIGIN OF CELLULAR HETEROGENEITY
The events in the early embryo that we have to consider are the following.
The embryonic stages at which they occur are illustrated diagrammatically in
Fig. 2.
In chimaeras the cellular heterogeneity is, of course, present from the beginning, when the embryos are stuck together. The aggregated embryo is then
composed of two halves with different cell types. It has been shown by Garner &
McLaren (1974) that virtually no mingling of the cell types occurs during
the two cell divisions after aggregation of 8-cell embryos. Thus in the blastocyst of 64 cells the two cell types are still unmixed. This is one of the two main
points in our argument and we shall return to it later.
(1) The first sampling event is the separation of the inner cell mass from
the trophectoderm at 3-3-| days. The inner cell mass gives rise later to the
whole of the embryo. It is composed of cells lying inside, the outside layer
being the trophectoderm. In principle, a circumferential division would not
affect the cell proportions, but the number of cells in the inner cell mass is not
large, and chance could affect the numbers of the two cell types that find
themselves in the inside. Garner & McLaren (1974) counted the cells in chimaeric blastocysts after one of the components had been labelled with [3H]thymidine. In six blastocysts the mean proportion of labelled cells in the inner
cell mass was 0-45 ±0-05. The variance of the cell proportions was 0-0161,
and the range was 0-273 to 0-565. Thus it seems that some variation does arise
in chimaeras from the sampling of cells to form the inner cell mass, but not
enough to generate the flat distribution.
(2) The second sampling is the division of the inner cell mass into 'primary
ectoderm' and 'primary endoderm' at 4-5 days. There are then about 45
cells in the whole inner cell mass of non-chimaeric embryos, and roughly half
of these, i.e. about 22 form the primary ectoderm from which the whole
embryo develops (McLaren, 1976&). Chimaeras at this stage have double the
normal number of cells in the whole embryo, and probably about three times
as many in the inner cell mass, for reasons of geometry explained by Buehr &
McLaren (1974). Thus there are probably roughly 60-70 cells in the primary
ectoderm of chimaeras when it differentiates. This is the sampling that we believe
gives rise to the flat distribution in chimaeras, and we shall return to it later.
(3) The cellular heterogeneity of mosaics arises when X-inactivation takes
place. This occurs sometime between 4 | and 7 days (McLaren, 1976 a, p. 23),
and is therefore probably after the second sampling. If X-inactivation takes
place just before or just after the separation of the primary ectoderm from
primary endoderm, then the number of cells present would be about 22. If it
takes place later the number of cells would be greater. The important difference
between the origins of cellular heterogeneity in mosaics and chimaeras is that
in mosaics it arises randomly among the cells present, so that the two cell
200
D . S. F A L C O N E R a n d P . J. A V E R Y
Chimaera
Mosaic
Inner
cell mass
(I) 3d
Inner
cell mass
Primary
ectoderm
Primary
endoderm
(2) 4\ d
(3) >4l d
X-inactivation
Fig. 2. Left: Diagrammatic representation of the early stages of embryonic development, showing the sampling events. Cells destined to give rise to the whole of the
embryo are in heavy outlines. (1) Differentiation of inner cell mass from trophectoderm, (2) differentiation of primary ectoderm from primary endoderm, (3)
X-inactivation.
Right: Diagrams to show how sampling event (2) generates the flat distribution
and the pure classes. The circles represent the inner cell mass, which is supposed to
be spherical. A-A is the plane of aggregation separating white cells from black
(stippled). B-B is the plane of differentiation separating primary ectoderm (heavy
outline) from primary endoderm, a = proportion of white cells in the inner cell
mass, varying as a result of sampling event (1). b = proportion of inner cell mass
cells that become primary endoderm giving rise to the whole embryo.
Five cases with different values of a or b are shown. In each case the two planes
can have any orientation with respect to each other. The outcome of each case can
be seen by imagining plane A to be rotated with respect to plane B. The pure
classes generated by each case are as follows.
Case
(0
00
(iii)
(iv)
(V)
a
0-5
0-5
0-5
>0-5
<0-5
b
0-5
<0-5
>0-5
0-5
0-5
Pure classes generated
Both, but very rare
Both
Neither
White only
Black only
Variability of chimaeras and mosaics
201
types are randomly mixed immediately after X-inactivation. Consequently
the variation arising is binomial with a sample size of not less than about 22
cells. Nesbitt (1971) made a penetrating analysis of mosaics and concluded
that the variation arising from X-inactivation is binomial with a sample size
of about 21 cells at X-inactivation.
The three sampling events considered so far - two in chimaeras and one in
mosaics - produce samples that give rise to the whole embryo. The variation
that arises from them is variation of the cell-proportions in the animal as a
whole. Subsequent events, or processes, give rise to variation between different
organs of the same individual, and therefore add to the variation between
the same organ in different individuals. The processes that need to be considered are the following.
(4) The allocation of the progenitor cells of the organ. This will be considered
later in the paper.
(5) Differential growth of the two cell types : the relative growth rates may
vary from one organ to another, as found for example by Barnes, Tuffrey,
Drury & Catty. (1974).
(6) Finally, there will be variation due to error of measurement of the cell
proportions in the organ studied.
ORIGIN OF FLAT DISTRIBUTION IN CHIMAERAS
The differentiation of the inner cell mass into primary ectoderm and primary
endoderm is the second sampling event illustrated in Fig. 2. It takes place two,
or at most three cell divisions after Garner & McLaren (1974) found the cell
types to be unmixed. We assume that they are still unmixed when the sampling
occurs. The consequences are then as follows.
For simplicity, let us suppose that the inner cell mass then consists of a
roughly spherical ball of cells, divided in two equal halves by the plane of
aggregation, with 'white' cells in one half and 'black' cells in the other. The
cells that are nearest the blastocoel become the primary endoderm, the remainder becoming the primary ectoderm from which the whole of the embryo
proper develops. According to the evidence summarized by McLaren (1976&),
the division is roughly equal, so that roughly half the inner cell mass becomes
primary ectoderm. The situation is illustrated in Fig. 2(i). There is no reason to
suppose that the plane of aggregation will have any specific orientation with
respect to the plane of differentiation. The angle between the two planes will
presumably, therefore, be random. The proportion, p, of white cells in the
sample (i.e. the primary ectoderm) depends on this angle. So if all angles are
equally probable, all values of p will be equally probable, including the pure
classes (0 % and 100 %). Thus, if the conditions specified hold, this sampling
can in principle generate a flat distribution. Some further elaboration of the
model is, however, necessary.
202
i i
D. S. FALCONER AND P. J. AVERY
rt = 0-5
6 = 0-3
i
i
-
I
i
l
1
1
1
1
1
1
15
o = 0-5
6 = 0-7
10 -
-
5
1
1
1
1
1
1
1
1
1
1
i—i—i—r
1
JO
6 = 0-5
« = 0-6
-
1
15
1
10
i
;>
i
i
1
0-2
1
0-4
1
1
0-6
1
1
0-8
1
.10
Fig. 3. Theoretical distributions of cell proportions, resulting from the first two
sampling events in chimaeras, i.e. of inner cell mass and of primary ectoderm.
a = proportion of white cells in inner cell mass, b = proportion of inner cell mass
cells that form primary ectoderm, p = proportion of white cells in adult, grouped in
class-intervals of 005, i.e. 5 percentage points. The distributions are derived from
the continuous distributions, described in Appendix I. When b > 0-5 the distribution
is restricted within certain limits. These limits fall within the terminal class-intervals
of p, and this is why the terminal classes are shown with narrower columns.
1
Variability of chimaeras and mosaics
203
First, there has already been some variation in cell proportions introduced
by the first sampling (inner cell mass/trophectoderm), so the plane of aggregation will not be equatorial in all embryos (Fig. 2, iv and v). Second, the
primary ectoderm may not comprise exactly half the cells, so the plane of
differentiation may not be through the centre of the sphere (Fig. 2, ii and iii).
Dr McLaren kindly provided data on this point, consisting of cell counts in
10 non-chimaeric embryos at the appropriate stage. The total number of inner
cell mass cells ranged from 31 to 177. The proportion of cells that had differentiated into primary ectoderm ranged from 0-41 to 0-63, and this proportion was
not correlated with the total cell number. The mean proportion was 0-51, with
a standard error of 0-02 and a standard deviation of 0-06. Thus it is clear that,
on average, half the inner cell mass cells differentiate into primary ectoderm,
and that there is some variation in this proportion. It is not known whether
chimaeras, with twice as many cells, do the same but we shall assume that they
do.
We have worked out the geometry of the sampling at the differentiation of
the inner cell mass into primary ectoderm and endoderm. Details of how the
distributions were derived are given in Appendix I, and the conclusions are
illustrated in Fig. 3. Figure 3 shows frequency distributions of the proportion
of white cells, /?, in the embryonic ectoderm and therefore in the whole of the
embryo proper. Values of/? are grouped in five percentage-point classes, so
that the distributions are comparable with the observed distributions in Fig. 1.
There are difficulties in showing the frequencies of pure classes in these histograms of theoretical distributions. Pure classes will be dealt with separately
later; in Fig. 3 they are combined with the end-classes, 5 % and 95 %. There are
two variables that affect the distributions: the proportion, a, of white cells
in the inner cell mass, which as noted earlier has been seen to vary between
about 0-3 and 0-6; and the proportion, b, of inner cell mass cells that form
primary ectoderm, which we saw in the previous paragraph may vary between
about 0-4 and 0-6. The four upper graphs show what happens if a is always
0-5 and b varies between 0-3 and 0-7. Consideration of Fig. 2 will show that
when b is less than 0-5 (case ii) all values of/? can occur; but when b is greater
than 0-5 (case iii), p is restricted within narrower limits. The lower four graphs
show what happens when b is always 0-5 and a varies between 0-3 and 0-7. The
distributions are now asymmetrical, reaching one extreme but not the other.
The reason for this can be seen from Fig. 2, (iv) and (v) : with the proportions
shown in (iv) the primary ectoderm can be all-white but not all-black; with
the proportions in (v) it can be all-black but not all-white, the greatest possible
proportion of white cells in the primary ectoderm being as shown in the drawing.
The graph for a = 0-5 with b = 0-5 is not shown; it is perfectly flat from one
extreme to the other. We have not explored the consequences of both a and b
varying simultaneously.
All the distributions in Fig. 3 are very nearly flat between the limiting values
204
D. S. FALCONER AND P. J. AVERY
Table 2. Frequencies (%) of each pure class (0 % or 100 % white)
expected from the sampling illustrated in Fig. 2
a = proportion of white cells in the inner cell mass, b = proportion of inner cell
mass that forms the primary ectoderm, n = initial number of cells in the primary
ectoderm.
The frequencies of pure classes are tabulated under three values of n. Each line of the
table refers to two alternative combinations of a and b, as shown on the right; for
example the last line refers to either a = 0-5 with b = 0-2, or to b = 0-5 with
a = 0-8. The frequencies entered are those of one pure class. In the cases where
a > 0-5 there is only one pure class; otherwise there are two pure classes with equal
frequencies. For details of how these frequencies were calculated, see Appendix I.
Pure! classes (%)
Conditions
r
r
< b = 0-5
a = 0-5 or
b=
b=
b=
b=
0-5
0-4
0-3
0-2
a=
a=
a=
a=
0-5
0-6
0-7
0-8
n = 2C
n = 50
4-8
11-6
171
22-5
20
8-5
13-9
19-4
n-
100
10
7-2
12-4
17-9
of p. Any real distribution is likely to be a mixture of distributions such as
those shown, and with other combinations of a and b. Predictions about the
expected distributions therefore cannot be precise, but the expectation of a
more or less flat distribution seems fully justified, provided the mean values of
a and b are close to 0-5. Certainly the observed distributions in Fig. 1 can
easily be accounted for by the theoretical distributions in Fig. 3.
We come now to the question of what will be the frequencies of the pure
classes. The number of possible values of/?, the cell proportions in the sample,
is limited by the number of cells in the sample. If there are n cells, then there
are only n+\ possible values of p, i.e. 0/n, 1/n.. .n\n. If the distribution is
perfectly flat, then all values of p are equally probable and the frequency of
each pure class will be l/(/i+l). As noted earlier, there are probably about
60 or more cells in the primary ectoderm of chimaeras, so a perfectly flat distribution would have pure classes represented with a maximum frequency of
about 1/61 = 1-6 % for each class. Table 2 gives the frequencies of pure classes
for each of three values of n, with different combinations of a and b. It will be
seen that the value of n, within the range 20-100, does not very greatly affect
the frequency of pure classes. Three points need to be noted about Table 2
before we compare observed with expected frequencies. First, consider what
happens if b varies, with a = 0-5. If b is always less than 0-5 (Fig. 2, ii) we shall
get both pure classes, each with the frequency given in Table 2. But if b varies
round a mean of 0-5 it will be greater than 0-5 as often as it is less than 0-5,
and when b is greater than 0-5 (Fig. 2, iii) we get no pure classes. Thus with
variable b the overall frequency of each pure class will be only half of the value
entered in Table 2. Second, consider what happens if a varies, with b = 0-5. If
Variability of chimaeras and mosaics
205
a is always greater than 0-5 (Fig. 2, iv) there will be only one pure class with the
frequency shown in Table 2. If a is always less than 0-5 (Fig. 2, v), there will be
only the other pure class, with the same frequency. But if a varies round 0-5 there
will be both pure classes, each with half the frequency shown. In summary, if either
b or a vary round a mean of 0-5 the frequencies in Table 2 are those of both pure
classes combined, except in the case of both a and b = 0-5 when the frequency
entered is of one pure class. The third point concerns the generation of asymmetry,
i.e. unequal frequencies of the two pure classes. Asymmetry can be generated only
by a having a mean different from 0-5, such as would result from one cell-type
being preferentially included in the inner cell mass when this differentiates
from the trophectoderm. If the mean of b differs from 0-5 the consequence
is an increase (5 < 0-5) or a decrease (5 > 0-5) of both pure classes.
The observed frequency of pure classes differs among experimental groups.
In the five groups in Fig. 1 and Table 1 the combined frequency of both pure
classes ranges from 11 % to 37%, with an unweighted mean of 23 %. The
differences between these five groups are, however, not significant (x2[4] =
8-52, P > 0-05). The three 'balanced' strain combinations of Mullen & Whitten
(1971) are nearly alike, with a mean of 33 %. Sanyal & Zeilmaker (1976) found
36 % of pure classes for coat pigmentation. It is clear from these data that the
observed frequencies of pure classes are generally higher than would be expected
from the variation of a and b as shown in Table 2. The expectations in Table 2,
however, refer to the whole body, whereas the observed frequencies are ' singlecolour' individuals classified by coat pigmentation. Single-colour individuals
have often been found to be chimaeric in other organs. The data are very
scanty, but we have made a rough estimate of the proportion of single-colour
individuals that are chimaeric elsewhere. In our own chimaeras (group ƒ in
Table 1 and others not included) chimaerism of an enzyme marker was looked
for in 8 other organs or tissues (I. K .Gauld, unpublished) ; 2 out of 11 single-colour
individuals were chimaeric elsewhere. In group (h) of Table 1, 2 out of 6 were
chimaeric in hair-follicles. In Sanyl & Zeilmaker's (1976) data 3 out of 24 were
chimaeric in eye-pigmentation or in retinal cells. The total is 7 chimaeric out of
41, or 17 %. This, of course, is a minimal estimate because of the limited search
of other tissues. To get the frequency of ' true ' pure classes we have to reduce the
observed frequency of single-colour individuals by 17 % or more. Taking the
observed frequency as very roughly 30 %, this gives 25 %, which still seems too
high to be accounted for by the variation of a and b. We think, nevertheless,
that it is not necessary to attribute single-colour individuals to technical failure of
the aggregation, for four reasons. First the proportion that are chimaeric elsewhere
may be higher. Second, the expectations in Table 2 are for either a or b varying, but
not both together. If both vary simultaneously the proportion of pure classes
could be much higher. Third, the expectations are based on the assumption that
there is no cell selection. If the two cell types proliferate at different rates the mean
cell proportions observed will not be 0-5 and the frequency of pure classes will be
14
EMB 43
206
D . S. F A L C O N E R A N D P . J. A V E R Y
increased above the expectations based on no selection. Some of the data cited
above on the frequency of pure classes may not meet this requirement. And,
finally, the expectations can only be approximate because the inner cell mass
is not in reality spherical.
SAMPLING FROM CLONES
So far we have considered the sampling of cells that give rise to the whole
embryo and adult. In the case of chimaeras we have seen that there are two
samplings and they give rise to a large amount of variation, ranging from one
extreme to the other. In the case of mosaics there is only one such sampling,
at X-inactivation, and this gives rise to binomial variation dependent on the
number of cells at the time of X-inactivation. In this section we consider the
subsequent sampling of progenitor cells to form particular organs or tissues.
This gives rise to variation between different organs of the same individual
and adds to the variation between the same organ in different individuals. As
Nesbitt (1971) has pointed out, if an analysis of variance is made between and
within individuals, the component between individuals, i.e. the covariance,
estimates the variation due to the first sampling, and the component within
individuals estimates the variation due to the later samplings of the progenitor
cells.
The sampling of progenitor cells has in the past been assumed to be binomial,
so that the number of progenitor cells, N, can be deduced from the variance
or2 = pq\N, or from the frequencies of pure classes, which are pN and qN
respectively, where p and q are the mean proportions of the two types of cell.
The supposition of binomial variation, however, is not valid, if the progenitor
cells are contiguous cells, sampled from a tissue in which there has been some
coherent clonal growth so that the two cell types are not randomly dispersed
in the tissue. The variation then depends not only on the sample size but also
on the clone size in the tissue from which it is taken. It turns out that the
binomial supposition can lead to estimates of cell number that are grossly
wrong. Other reasons for doubting the validity of the binomial supposition are
discussed by McLaren (1976a).
If, however, the progenitor cells are sampled singly, and not as a contiguous
group, then the variation arising will be binomial, whatever the clone size in
the tissue may be. It is possible that the progenitor cells of melanocytes become
allocated as single cells in the neural crest. If this is so, then the variation of
skin pigmentation arising from the sampling of progenitor cells would be
binomial, and estimates of their number would not be subject to the type of
error discussed below. Melanocyte progenitor cells are the subject of the next
section.
A 'clone', in the sense we use it, means a group of contiguous cells that are
descended from a single cell one or more cell divisions previously. Adjacent
Variability of chimaeras and mosaics
207
Fig. 4. Variance arising from sampling n cells from a one-dimensional array in
which the clone size is k. The curve for k = 1 represents binomial sampling variance.
clones of the same cell type form a single 'patch'. There will, of course, be
patches even if the arrangement of the cells is completely random (see West,
1975, 1976). Sampling of contiguous cells from a random arrangement will
yield a binomial distribution. We are concerned with the embryonic tissue in
which the progenitor cells become allocated. In a rapidly growing tissue, cell
mingling would have to be very rapid if it were to break up the clones and maintain a random arrangement. So in embryonic tissue one must expect clones
to be present and patches to be larger than the random size.
We have worked out the consequences of sampling in a one-dimensional
array, i.e. a line, of cells. We assume there is an initial state of random arrangement, corresponding to a clone size of k = 1. Then there is cell division with
no mixing, so that daughter cells lie side by side, and successive divisions give
clone sizes of k = 2, 4, 8, etc. We have assumed for simplicity that all cells
divide at the same rate, so that all clones are the same size. We assume, further,
that after the sampling of progenitor cells the two cell types proliferate at the
same rate, so that the cell proportions observed in the adult are the same as
they were in the sample. Details of the derivation of the variance are given
in Appendix II. The results are shown in Fig. 4. This shows the variance of
cell proportions, of„ plotted against the sample size, n, when the clone size,
it, is 1, 2, 4 or 8. A clone size of k = 1 represents a random arrangement of
14-2
208
D. S. FALCONER AND P. J. AVERY
Table 3. Estimates of the number of progenitor cells, N, that would be obtained
from the variance, a-2, and the mean proportion, p, by assuming binomial sampling :
N = p(l — p)/o"2. k = clone size; n = true sample size
The table refers to sampling from a one-dimensional array of cells, and is not
restricted top = 0-5.
k
n
1
2
5
10
15
20
30
40
2
4
8
10
1-3
2-8
5-3
7-8
10
11
1-7
2-9
41
10
11
1-3
1-7
2-3
10-3
5-3
2-9
15-3
20-3
7-8
10-3
4-1
5-2
cells, so the variance shown for k = 1 is the binomial variance. It can be seen
from the figure that for any sample size, n, the variance increases as the clone
size, k, increases. This means that if we estimate n from the binomial variance
(k = 1) we will always get an underestimate if A: is greater than 1. For example,
suppose the observed variance were 0-05, and the clone size k = 4. Reading,
correctly, against the curve of k = 4 shows that the sample size was n = 19.
But reading erroneously against the curve for k = 1 would give a 'binomial'
estimate of n = 5. The conclusion is that the number of progenitor cells cannot be estimated from the variance unless the clone size in the tissue is known.
Table 3 gives some examples from which the range or error can be appreciated.
It shows the estimates (N) that would be obtained by supposing the sampling
to be binomial, with various true values of n and clone-sizes of 2, 4 and 8.
When the true n is large relative to k, the binomially estimated iV approximates to
n\k, which is the mean number of clones in the sample of progenitor cells.
The table shows that the approximation is quite close if n is greater than about
2k. Thus the binomial-N is much better regarded as an estimate of n\k than
of«.
The foregoing discussion refers to the rather unrealistic situation of a
tissue composed of a single line of cells. For a two-dimensional, or for a solid
tissue, the details may be different, but the same general conclusion must hold :
that the sample size will be underestimated by assuming binomial sampling. If
the sample size were larger than the clone size, the binomial estimate would
approximate to the number of clones included in the progenitor cells, as we have
seen it does in a one-dimensional tissue.
Variability of chimaeras and mosaics
209
SEQUENTIAL SAMPLING
One source of variation remains to be examined, and that is the sampling
of progenitor cells of melanoctyes. In chimaeras, this sampling will give rise to
some individuals that are chimaeric in some tissues, but not in the coat melanocytes, i.e. 'single-colour chimaeras'. We want to find out what proportion of
these would be expected. The number of melanocyte progenitor cells has been estimated in different ways from mosaics and chimaeras. We consider mosaics first.
The variation of coat pigmentation in mosaics results from two known
sampling events, X-inactivation and the allocation of progenitor cells of the
coat melanocytes. There may be other sampling events between these two, but
we shall suppose for the moment that there are not. Errors of measurement
add a third source of variation of the observed cell proportions. The first sampling is binomial, and so is the second if the progenitor cells are single and
not contiguous at allocation, which seems fairly likely. The consequences of
two or more sequential binomial samplings have been described by Nesbitt
(1971), and we have little to add, except in relation to the variation arising from
measurement error. Nesbitt showed that if nx and n2 are the sequential sample
sizes, then
1 1 1 1
N
nx
n2
nxn2
where N is the 'effective' sample size. The variance resulting from the two
samplings is given by cr2 = pq/N, p and q being the mean cell proportions.
Nesbitt, working with Cattanach's translocation, estimated nx = 21 from the
covariance of/? in several different organs, and then estimated n2 = 22 for
coat pigmentation from the observed variance. The observed variance, however,
included variance due to error of measurement which was not separately
estimated for coat pigmentation. If allowance is made for the error variance,
the estimate of n2 is considerably increased. Falconer & Isaccson (1972) classified a number of brindled mosaics each twice and found the correlation between
the two scores, i.e. the repeatability, to be 0-92 in the strain whose distribution
is shown here in Fig. 1 (a). This means that 8 % of the variance of single scores
was attributable to measurement error. The variance of p in those mice was
0-0141, so the estimated error variance was <T\ = 0-0011. The classification,
however, was made by judgement, not measurement, so the real error variation
was probably greater than that arising from inconsistency of classification,
but how much greater can only be guessed at. We shall show that a reasonable
guess leads to an estimate of n2 that is consistent with the estimate derived from
the striping pattern of chimaeras.
It is easier to think of the standard deviation of percentage scores attributable
to error, rather than the variance of proportions. The estimate of cr\ given
above corresponds to a standard deviation of about three percentage points.
Remembering that the classification was made in five percentage-point classes,
210
D. S. FALCONER AND P. J. AVERY
one might guess that the total error variance might be equivalent to a standard
deviation of five percentage points, which means a variance of proportions
of <J\ = 0-0025. Subtracting <r\ from the observed variance gives the 'true'
variance from which N is to be derived. Taking the observed variance of the
brindled mosaics from Table 1 gives an estimate of N = 13, which is only a
little higher than Nesbitt's estimate of 11. The effect on the estimate of n2,
however, is greater. If we take Nesbitt's estimate of nx = 21, we get n 2 = 34.
The effects of assuming different levels of error variance are as follows.
(Te
N
n2
0
11
23
0-03
12
28
0-05
13
34
0-07
15
52
Turning now to chimaeras, the number of melanocyte progenitors has been
estimated from the pattern of striping. From the 'standard pattern', Mintz
(1967) concluded that there are 34 primordial cells, 6 for the head, 12 for the
body and 16 for the tail. Wolpert & Gingell (1970) increased the number to
64, on the ground that the stripes are patches and not clones, and West (1975),
on the same grounds but by a different method, arrived at a number of 68.
For comparison with the estimate from mosaics, we are concerned only with
the head and body, since tails were not taken into consideration in the classification. Taking Wolpert & Gingell's total number and dividing it in Mintz'
proportions to head and body gives an estimate from the striping pattern of
«2 = 34. This is the same as the estimate based on an error standard deviation
of five percentage points. Making a reasonable guess at the error variance
thus gives an estimate of the number of melanocyte progenitor cells in the head
and body that is consistent with the independent estimate from the striping
pattern. The estimate of n% derived from the variance is, of course, dependent
on the value taken for nx. Confidence limits of nx can be calculated from Nesbitt's
data. In her Table 3 she gives 15 estimates of nx based on covariance of various
tissues in pairs. The mean of these estimates is 20-13 with a standard error of
1-18. Taking 95 % confidence limits as ± 2 S.E. gives upper and lower confidence
limits for nx of 22-5 and 17-8. Using these with N = 13 (corresponding to <re =
0-05) gives lower and upper limits for n2 of 31 and 48. The real confidence
limits must be wider than these because we have disregarded the error variance
in estimating N from the observed variance. The estimates of n2 are therefore
not to be regarded as being very precise.
The conclusions to be drawn from the foregoing calculations are that a
consistent picture results from the assumptions that (i) there are about 21
cells in the embryo proper at the time of X-inactivation, (ii) the melanocytes in
the coat of the head and body are derived from about 34 progenitor cells, and
(iii) the sampling resulting from X-inactivation and from the allocation of
progenitor cells of melanocytes are both binomial.
We come now to the main question about chimaeras : what proportion of
Variability of chimaeras and mosaics
211
single-colour individuals can be attributed to the sampling of melanocyte
progenitor cells ? These are individuals that are chimaeric elsewhere in the body
but not in the coat pigmentation. In considering single-colour individuals we
can no longer exclude the tail, because the pigmentation on the tail is taken
into account in distinguishing single-colour individuals from overt chimaeras.
For this purpose, therefore, we shall adopt Wolpert & Gingell's total of 64
melanocyte progenitors.
The proportion of pure classes among all individuals will be denoted by
Px and the proportion of chimaeras by Q = 1 — P. Let:
n = the number of melanocyte progenitor cells, taken
to be 64.
Qx = the proportion of individuals that are chimaeric before
melanocyte sampling.
Q2 = the proportion that are chimaeric in the coat after
melanocyte sampling, i.e. overt chimaeras.
Then the difference, Qx- Q2, represents the proportion of individuals that have
become single-colour by the melanocyte sampling but are chimaeric elsewhere.
Let AP be this increment to the pure classes resulting from melanocyte sampling. It is shown in Appendix III that AP = 2QJ(n +1). (We have assumed that
the number of cells in the tissue from which the n melanocytes are sampled
is large relative to n, and that among those that are chimaeric in this tissue all
values of p are equally probable, i.e. the distribution is flat.)
We want to know AP as a proportion of the overt chimaeras, Q2, and of the
single-colour animals, P 2 . As stated above,
Ô i - Ô 2 = AP = 2ß 1 /(» + l).
Algebraic rearrangement leads to
a-ß.(£r).
and substitution gives, after further rearrangement,
A P / ß a = 2/(/f-l)
= 0-03 when n = 64.
Thus the proportion (or number) of individuals expected to be single-colour
chimaeras is 3 % of the proportion (or number) of overt chimaeras. Rearranging the last formula above and dividing by P 2 leads to
AP/P 2 = 003ß 2 /P 2 .
This gives the proportion of single-colour individuals that are expected to be
chimaeric elsewhere. The expectations according to the value taken for the
frequency of single-colour individuals (P2) a r e a s follows :
P2(%):
AP/P 2 (%):
10
29
15
18
20
13
30
7
40
5
212
D . S. F A L C O N E R A N D P . J. A V E R Y
We saw earlier that P 2 ranges roughly from 10 to 40 %, and that AP/P^,
estimated from very limited data, is 17 % or more. The mean value of P2 in
the groups from which the estimate of AP/P2 came was about 30 %, for which
the expectation of Ai>/i>2 would be 7 %. The discrepancy may not be very
serious considering the paucity of data; the estimate of Ai>/iJ2 = 17 % has
a standard error of 6 %, and the estimate of P2 = 30 % has a standard error
of 4 %. An adjustment of one standard error in each would bring them into
agreement. On the other hand, the discrepancy could be accounted for by an
additional sampling event between X-inactivation and the allocation of melanocyte progenitors. This would result in some individuals having only one cell
type in the neural crest tissue before melanocyte sampling, though still being
chimaeric elsewhere. These individuals would be found as single-colour chimaeras, but would not have resulted from the melanocyte sampling.
DISCUSSION
Examination of the events in the early embryo by which cells are sampled
has provided a coherent explanation of many features of chimaeras and mosaics,
and particularly of the much greater variation of cell proportions found in
chimaeras. There are three eaily sampling events - the differentiation of inner
cell mass from trophectoderm, the differentiation of primary ectoderm from
primary endoderm, and X-chromosome inactivation. The first two events
cause variation in chimaeras, but not in mosaics because the cellular heterogeneity is not then present in mosaics. The third event causes variation
in mosaics, but not in chimaeras marked by an autosomal gene. Most of
the variation in chimaeras comes from the second sampling and the reason
why this is so much more than arises from X-inactivation in mosaics is that
the two cell types are still largely unmixed when the primary ectoderm
differentiates.
All three of these sampling events occur before organogenesis has started.
Consequently the variation they produce is variation of cell proportions
throughout the whole body. If no variation arose subsequently in separate
organs, all organs would therefore be completely correlated in respect of
cell proportions, in both chimaeras and mosaics. Subsequent variation affecting
organs separately arises from the sampling of progenitor cells, and from other
causes such as differential proliferation. It is tempting to think that if two organs
are correlated in respect of cell proportions they must share a common cell
lineage. A great deal of caution, however, is needed in drawing such a conclusion, for two reasons. First, a correlation between two organs by itself tells us
no more than that they are both derived from the primary ectoderm. Second, if
two organs are more highly correlated than either is with a third, there would
be some grounds for concluding that the first two have some cell lineage in
common that they do not share with the third. But there could be another
Variability of chimaeras and mosaics
213
explanation: the third organ could have fewer progenitor cells than the first
two, and so have a greater variance. This alternative explanation, however,
would not apply if it were the covariances, rather than the correlations, that
differed in the way described.
The consequences of the sampling of cells to form the primary ectoderm has
a bearing on the origin of germ cells. The primordial germ cells are first seen
in the yolk sac, which is derived from the primary endoderm, but there is
doubt about whether this is their site of origin. Evidence from injection chimaeras suggests that they do not originate in the yolk-sac, and Gardner &
Rossant (1976) come to the following conclusion: 'Hence, it would appear that
primordial germ cells originate from the embryonic ectoderm rather than from
the extra-embryonic endoderm, and that they secondarily migrate into the
latter.' Our evidence supports this conclusion for the following reason. If
our description of the sampling that results from the differentiation of primary
ectoderm from primary endoderm is correct, then the cell proportions in the
primary endoderm must be the complement of the cell proportions in the primary ectoderm, at least roughly. For example, if the inner cell mass contains equal
numbers of two cell types and half its cells become primary ectoderm ; then if the
sampling results in the primary ectoderm having 60 % of white cells the primary
endoderm must have 40 % of white cells. Consequently any extra-embryonic
tissue derived from the primary endoderm should be negatively correlated
in respect of cell proportions with tissues in the embryo proper. If primordial
germ cells originate in the yolk-sac, then the gametic output of chimaeras
should be negatively correlated with the somatic cell proportions. In fact the
correlation is positive. This is clearly seen in the data of Ford et al. (1975).
Ten of the overt chimaeras made in this department provided data (I. K.
Gauld, unpublished). The correlation between their gametic output and
coat pigmentation was +0-70 (P < 0-05). Thus, if our description of the
sampling event is right, the primordial germ cells must originate from the primary
ectoderm, and not from the yolk-sac.
APPENDIX I
The inner cell mass is assumed to be spherical with the proportion of white
cells being a and the proportion sampled to form the primary ectoderm being b.
Let us also assume that there are many cells in the inner cell mass so that we
can assume that/?, the proportion of white cells in the primary ectoderm, can
take all real values from 0 to L A correction due to discreteness will be made
later.
Let the angles a and ß, measured in radians, be defined as in Fig. 5 and let
the radius of the sphere be 1. First let us consider a = 0-5, b < 0-5 as is illustrated
in Fig. 5. Then by using cylindrical polar coordinates (the Z-axis being perpendicular to the plane of differentiation), it can be proved that V, the volume
214
D . S. F A L C O N E R A N D P . J. A V E R Y
Z-axis
Plane of
differential
Black cells
White cells
Plane
aggreg;
Fig. 5
cut off by the plane of aggregation, the plane of differentiation and the surface
of the sphere, is given by
V = J ^ f (1 -ZW-*sin2^) dZ (0 < ß ^ a),
(Al)
= ^-\C™fa(\-Z*){f-\sm2*)dZ
where
(0 > ß > - a),
cos f = (Ztan/?)/V(l -Z 2 ).
Thus for any particular value of ß
p = 3VI(4nb).
(A2)
When b = 0-5, (Al) and (A2) simplify to give
(A3)
By putting ß = 0 in the two expressions for V and equating them, we find that
b = 7 [2-3 cosa + cos3a].
(A4)
Thus given any value for b, cos a, and thus a, can be found by using the
Newton-Raphson iterative method.
Variability of chimaeras and mosaics
215
The angle ß can have any value between - nß and + -nß with equal likelihood. Thus/7 is a uniform random variable on (-n/l, nß) and its probability
density function, ƒ(/?), is given by
Aß)-l
H </»<!)•
(AS)
If a ^ ß ^ 7T/2, then/? = 0, while if - a ^ /? ^ -nß,p
= 1. For all other
values of /?, (A2) defines a unique value of p lying in (0, 1). From (A5), we can
see that
Prob (p = 0) = Prob (p = 1) = ^ ( ^ - « ) -
(A6)
To find Prob (p' ^ p ^ //' ), the values /?' and /#" need to be found where
ß' and ß" are those values of ß which on substitution in equation (A 2) give
p = p' and p" respectively. They can be found by use of the Newton-Raphson
interative technique applied to equations (Al) and (A2).
Then
Prob tp' <p^
p") = Prob {ß' > ß > ß") = \{ß'-ß").
(A7)
This method was used to produce Fig. 3, p' and p" being the limits of the intervals. The probabilities of the end intervals (e.g. (0, 0-05)) must be increased by
the addition of the discrete probabilities given by (A 6).
Because the actual proportion, p, can only be 0, 1/«, 2/n,..., njn where n
is the number of cells in the primary ectoderm, we must take account of this
in the calculation of the proportion of pure classes. When a = 0-5 and b = 0-5,
(A7) and (A3) give us
Prob(/>' ^p
^p")
=p"-p'.
In this case, as is argued in the text, Prob (p = 0) should be l/(n+1). Thus we
approximate the probability of a pure class as
3(H +Prob ( o<p s-Tï)'The results for a = 0-5, b ^ 0-5 can be used directly for the other cases
considered, i.e. b = 0-5, a 4= 0-5, and a = 0-5, b > 0-5. In these cases, not all
values of p are possible. The allowable range is (jplt p2) where
= 1-1/(26), p2 = ll(2b) if a = 0-5 and b > 0-5,
p1 = 0,
pt = 2a
if a < 0-5 and b = 0-5,
p1 = 2a-1,
p2 = 1
if A > 0-5 and b = 0-5.
Pl
For each circumstance, the required probabilities (cf. (A 7)) can be found from
216
D. S. FALCONER AND P. J. AVERY
the solution of a case with parameters a' and b' where a' = 0-5 and b' ^ 0-5.
Let this second case give solution p, then :
If a = 0-5, b > 0-5,
l-bp"
bp'
Prob O ' < p < p") = Prob
where b' =• l—b.
^
b "' "
\-b
lia < 0-5, b = 0-5,
Prob (p' ^ p ^ p") = Prob (p'ßa < p ^ p"ßa), where 6' = a.
If a > 0-5, b = 0-5,
I-/?"
„
1-p' where £' = 1 - a.
Prob (p' ^ p ^ p") = Prob
<P
m
[2(\-a)
^2(1-4
As with the simplest case, corrections can be made for discreteness when
looking at pure classes.
APPENDIX II
Let us assume that, originally, cells were randomly distributed along a
conceptually infinite line with the proportion of white cells being p0. Let each
original cell be replaced by k identical clonal descendents. n consecutive cells
are now sampled.
Firstly, let n ^ k. Then the probability of choosing all n from one clone =
( k - « +1)1 k as the first sampled cell can be the 1st, 2nd,.. .Ä:th member of the
clone with equal probability. Otherwise i cells are chosen from one clone and
(n — i) from a neighbouring clone (where i is an integer in the range (1, n— 1)
with probability \\k for each value of /). Let X be the number of white cells in
the sample of size w. Then
Var(/7) = V a r ( l » = i
-7Î+1
"T
1
H _ 1
* n 'T" T 2 J \ * % + y n-i)
(A8)
where Vt = the variance of the number of white cells in a sample of size i
which is all white with probability p0 and all black with probability (1 -p0).
The variances can be added as consecutive clones are independent of each other.
It can easily be shown that Vi = i2p0(l -p0)Thus (A 8) gives us that
Var p = -s2
n
k
= Po(l-Po)
= A)(l -Po)
n2Po(l -Po)+T, 2 A>(1 -Po) iz
n— \
k
n—\
k
2 n~1
kn2 ,-=1
2 n(n — \)(2n 1)
6
kn2
(by use of summation formulae)
n2-\'
= PoO- -Po) 1- 3nk
(A9)
217
Variability of chimaeras and mosaics
If
n>k,
Var O) = p0(l -p0)
k
n
lk2-l
(A 10)
3 H2
This can be proved by induction as follows. Let Var (X\n) be the variance of
the number of white cells from a sample of size n. Let us sample n cells and
assume the variance is given by (A 10). Let us look at the next cell. It can be the
1st, 2nd, 3rd,.. .&th member of a clone with equal probability. If it were the
2nd member, say, then to get the Var (X\n +1) from Var (X\n,), we would need
to add V2 and subtract V1. Thus averaging over all possibilities,
Var (X\n + l) =Var (X\n)+± [{V2-V,) + (V3- V2) +
(V,-V3).
+
(Vk+V1-Vk)}
= Var(X\n) + ^
= Vâi(X\n) +
kp0(l-pQ).
Thus
Var (p\n + l) =
1
[n2 Var (p\n) + kp0 (1 -p0)],
(n+1) 2
which by using (A 10) gives
Var 0 | n + 1 ) =
p0(l-p0)
n+1
1 fc2-l
3(«+l) 2 J
Thus if (A 10) is true for n, it is true for w+1. (A 10) is true for n = k by
comparison with (A9), thus by induction (A 10) is true for all n > k.
APPENDIX III
Let us assume that the n melanocytes are sampled from a large number
(a conceptually infinite number) in which the proportion, p, of white cells takes
any value between 0 and 1 with equal likelihood but is not zero or 1, i.e. using
the flat distribution discussed previously. Thus p is a uniform random variable
on (0, 1) and has a probability density function, ƒ(/?), given by :f(p) = 1 (0 < p < 1).
For any given value ofp, the number of white cells in the n sampled is binomially distributed, probability/?, sample size n. Thus the probability of choosing
r white cells, Prob (r\p) say, is given by
Prob (r\p) =
(n-r) î r\
pr(\ -Py
218
D. S. FALCONER AND P. J. AYERY
Thus to take into account all possible values of/?, we must integrate Prob (r\p)
to get the overall probability of r cells, therefore
Prob (r) = ƒ * {n";)UX
= /
/>"(! -P)n-'dp,
'r7—.B(r+l,
(n-r)\r\
n-r+l),
J
where B(m, n) is a beta-function. For integral m and n,
B{m, n) = ( ( m - l ) ! ( « - l ) ! ) / ( m + n - l ) ! .
Thus
n\
r\(n-r)\
1 ~
„ u . v
Prob (A*)J = .
r- —-, -r—TTT
=
r for nall r.
(n-r)\ t r\ (n+l)\
n+l
Thus, in particular, the probability of getting all white is l / ( n + l ) as is also
the probability of getting all black.
We are very grateful to Dr B. M. Cattanach, Dr Mary F. Lyon, Dr Anne McLaren and
Mr I. K. Gauld for providing us with unpublished data, and to Mr E. D. Roberts for drawing
the figures.
P. J. Avery is grateful to the Science Research Council for financial support.
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proliferation in tetraparental mouse chimaeras derived by fusion of early embryos.
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BUEHR, M. & MCLAREN, A. (1974). Size regulation in chimaeric mouse embryos. J. Embryol.
exp. Morph. 31, 229-234.
FALCONER, D. S. & ISAACSON, J. H. (1972). Sex-linked variegation modified by selection in
brindled mice. Genet. Res. 20, 291-316.
BARNES,
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{Received 12 May 1977, revised 15 July 1977)