Cyiogenet. Ceil Genet. IS: 320-326 (1977)
A new approach to quantitative analysis of somatic association J.-R. L acadena,1 E. F errer,1 and I. Y añez2 'Departamento de Genética, Facultad de Biología, and "Departamento de Estadística, Facultad de Matemáticas, Universidad Complutense, Madrid
Abstract A formula is proposed for estimating statistically the degree of somatic chromo some association during mctaphasc. Standard chi-square tests can then be used to determine the presence or absence of somatic association between homologous chromosomes in selected species. The method requires that the chromosomes be morphologically distinguishable and that their circular arrangement on the meta phase plate has not been distorted by the squash used during their preparation.
Request reprints from: Prof. D r. J uan-R amón L acadena. Departamento de Genética, Facultad de Biología, Universidad Complutense de Madrid, Madrid 3 (Spain).
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
Homologous chromosome association at somatic metaphase is evident in some organisms, e.g.. Díptera (Stevens, 1908; M etz. 1916). On the other hand, distortion can either alter the relative positions of some chro mosomes in their circular arrangement on the metaphase plate or even destroy the arrangement scattering the chromosomes over the plate. In order to ascertain the degree of somatic association of homologous chromosomes, it can be interesting to analyze statistically the location of homologous chromosome pairs when the distortion has not changed the circular arrangement of metaphase plates but has altered the relative position of the chromosomes. In this paper a formula is developed to calculate the frequencies of a given circular arrangement in which 0, 1 , 2 , . . . , « pairs of homologous chromosomes are associated, assuming the random location of chromosomes on the equatorial ring. Let Nj" be the number of different circular arrangements of the n pairs of homologous chromosomes of the complement of which any j pairs are associated; i.e., both homologs of each pair lie together on the metaphase
Lacadena et al.
Quantitative analysis of somatic association
E f f e c t p r o d u c e d b y in t r o d u c in g a n e w
F re q u e n c y
321
C ritic a l
p a ir o f h o m o lo g o u s c h ro m o s o m e s
c o n fig u ra tio n
T h e n ew p a ir b re a k s tw o a s s o c ia tio n s
\ l /
A'% l2 K /+ 2 ) f /+ l >1 -
T h e n e w p a ir b re a k s o n e a s s o c ia tio n
^
¡
W
1 2 0 + D (2 « -/-3 )i '
T h e n e w p a ir n e ith e r b r e a k s n o r
jV
c re a te s a n y a s s o c ia tio n
•
T h e n e w p a ir b re a k s o n e a s s o c ia tio n b u t . in t u r n , c r e a t e s a n e w o n e
A’" ' 1 ( 2 / )
T h e n ew p a ir c re a te s o n e a s s o c ia tio n
a
'
'
, ; ; | i 2 ( 2 'i - / - i ) i
\ ^
1 \ « \
/ 11 \ \
/ /
N" 1
/y
\*
" 1 ((2 /» -j-2 ) (2 /1 -/-3 )] '
T o ta l
\
t\ 1
1
N
Fig. I. The new pair of homologous chromosomes introduced into the circular arrangement of 2/1-2 chromosomes are indicated by arrows. Continuous lines represent the homologous chromosome pairs whose associations are broken by the introduction of the new pair of chromosomes. Dashed lines indicate homologous chromosomes (associated or not) whose relative positions are not changed by the introduction of the new pair. For each situation, only one possible configuration is shown. Table I. Absolute frequencies (N¡") of the different circular arrangements according to the number of associations present.
n
0
1
2
i
0 2 32 1488 112512
1 0 48 1920 138240
_
2 3 4 5
4 24 1152 78720
3
4
5
_
_
—
-
-
-
-
-
16 384 26880
96 5760
-
768
Number of circular arrangements (2/1-1)! 1
6 120 5040 362880 Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
Number of homologous chromosome pairs associated (j) -------------------------------------------------------------------------
L acadena et al.
322
Quantitative analysis of somatic association
Table II. Probabilities of occurrence of different circular arrangements
1
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Poisson
Number o f homologous chromosome pairs associated ( j ) 0 1 2 3 4 5 6 7 8
o
0.333 0.267 0.295 0.310 0.320 0.327 0.332 0.336 0.340 0.342 0.344 0.346 0.348 0.349 0.350 0.351 0.352 0.353 0.354 0.355 0.355 0.356 0.356 0.357
i
00.000 0.400 0.381 0.381 0.380 0.379 0.378 0.377 0.376 0.375 0.375 0.374 0.374 0.374 0.373 0.373 0.373 0.372 0.372 0.372 0.372 0.372 0.372 0.371
0.667 0.200 0.229 0.217 0.211 0.206 0.203 0.201 0.199 0.198 0.196 0.195 0.194 0.194 0.193 0.192 0.192 0.192 0.191 0.191 0.190 0.190 0.190 0.190
0.133 0.076 0.074 0.071 0.069 0.068 0.067 0.067 0.066 0.066 0.065 0.065 0.065 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.063 0.063
0.019 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016
0.002 0.002 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003
2x10 ‘ 3 x 10 4 3 x 10 1 3x10 • 4 x 10* 4 x 10 * 4 x 10 * 4 x 10 4 4 x 10 • 4 x 10 ‘ 4 x 10 * 4 x 10 4 4 x 10 4 4 x 10 4 4 x 10 4 4 x 10 4 5 x 10 4 5 x 10 4 5 x 10 4 5 x 10 4
1 x 10 1 2x10s 3 x 10 1 4 x 10— 4x10s 4x10s 5x10s 5 x 10 ‘ 5x10 ‘ 5 x 10 a 5 x10 s 5x10s 5 x 10 4 6x10 1 6x10s 6 x 10 1 6 x 10 s 6 x10 s 6x10s
1 x 10 * 2x10 « 3 x !0 « 3x10« 4x10« 4 x l0 -‘ 4 x 10 « 5 x 10 " 5 x 10 " 5 x 10 " 6 x 10 -" 6 x 10 « 6x10* 6x10* 6 x 10 « 6 x 10 « 6x10* 7x10 «
0.368 0.368 0.184 0.061 0.015 0.003 5 x 1 0 4 7 x 1 0 1 9 x 1 0 *
9
10
II
6x10 : 1 x IO 2x10 T 2 x 10'* 3x103x10 7 4 x 10 7 4 • 10 7 4x105 • 10 7 5 • 10 7 5x106 x I O'7 6 x, 10 7 6 x 10-7 6x10 7 6 ;-10 7
3 x 10 # 7 x 10 * 1 x 10 « 2 x 10 * 2 x 10 « 2 x 10 * 3 x 10 * 3 x 10 ’ 3 x 10 * 4 x 10 * 4 x 10 * 4 x 10 * 5x10 = 5 • ID ■ 5 x 10 5 x 10 '
1 x 10 “» 6x 4 x 10 7 x 1 0 '° 2 / 1 X 10 ’ 3 x 1 x 10 ' 5 x 2x10 ’ 7 x 2x10 * 9 x 2 x 10 * 1 x 2 x 10 * 1 x 3 x 10 • 2 x 3 x I0 W 2 x 3 x 10 * 2 x 3x10 » 2 x 4 x 10 * 2 x 4x10 " 2x
1x10*
1 x 10 ’ 9 x 10 » 7 x 1 0 "' 5 x 1 0 “
12
10 11 10“ 10“ 10 " 10“ 10 " 10 »• 10 ,u 10 •“ 10 ■"» 10 10 10“ 10 10“
13
14
3 x 10 ** 8 x 10 11 2 x 10 11 3 x 10 " 4 x 10 ** 5 x 10 14 6 x 10 14 7 x 10 « 8 x 10 14 1 x 10“ 1 x 10 “ I x 10 “ 1 X 10“
9 x 10 “ 3 x 10 14 7 x 10“ 1 x 10“ 2x10“ 2 * 10“ 3 x 10 14 4 x 10“ 4 x 10“ 5 x 10 “ 6 x 1 0 14 7 x 10 4*
ring. The value of Nj" can then be deduced by recurrence from previously known values of A'/' - 1 for n —1 pairs of chromosomes, i taking the values of ¡+2, / T l , /, or / —1. These different values of / depend on the relative positions taken on the circular arrangement by both members of a new pair of homologous chromosomes when a hypothetical cell passes from 2n—2 to 2/j chromosomes. Thus, the following five different situations are possible: 1. The new pair of homologous chromosomes that is introduced breaks the previously existing associations of two pairs of homologous chromo somes. In this case, /= /+ 2 . 2. The new pair only breaks one previous association (i= /+ l). 3. The new pair neither breaks nor creates any association (/=/). 4. The new pair breaks one existing association but, in turn, creates a new one. In this case, i=j too. 5. The introduction of a new pair of homologous chromosomes creates one additional association ( /= /—1). Taking into account the five situations described and the different possibilities of location on the metaphase ring for the new chromosome pair corresponding to each case, N/' is given by:
4 x 1 0 '"
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
n
L acadena et al.
Quantitative analysis of somatic association
323
showing j associations and mean number of associated pairs per cell (xj).
15
16
17
18
19
20
21
22
23
24
25
I
I x 10 17 4 x 10 17 9 x 10 17 2 x 10“ 3 x 10“ 4 x 10“ 6 x 10“ 7 x 10 16 9 x 10 >•
3 x 10 '*
1 x 10“
3 x 10 '■ 1 x 10“ 3x10“ 6 x 10“ ! x 10 17 2 x 10 17 2 - 10 17 3 x 10 17 4 x 10“
2 x 10 11
1 x 10“
1 x 10“
9 x 10 “ 4 x 10“ 1 x 10“ 2 x 10 *• 4 x 10“ 5 x 10“ 8 x 10“ 1 x 10“
2x 1x 3x 6x 1x 2x
5 x 10“
10“ 10 11 10“ 10“ 10 70 10 70
6x 3x 9x 2 • 3x
3 x 10 3U
10 71 10 43 10“ 10“ 10“ 6 x 10“
2 x 8 x 2x 5 x 1x
3 x 10“
1 x 10“
7 x 10“
10 43 10“ 10“ 10“ 10“
4x : 6> 1x
10 47 io * 10“ 10“
3 x 1 0 “«
8 x 10“ 4 x 10“ 1 X 10 37
2 x 10“ 1x 10*“
3 x IO-33
1.0246 1.0224 1.0221 1.0219 1.0206
1 X 10“
5 x 10“
2 x 10“
1
N f = N } . ,» '[(/ + 2 )(/+ l)j + N ; t ,«-«[2 (H -l)(2 /i-/-3 )]
+ N j" - ' [ ( 2 n - j - 2 ) ( 2 n - j - 3 ) ] + N /'- ' (2j) + Nj _i " - 1[2(2/i—/ —1 )]. Each term in the equation corresponds to one of the five situations mentioned above, respectively, and they can be easily derived from the critical configurations shown in fig. 1. Obviously, this recurrent equation is only valid from n = 3. In table I the absolute frequencies up to n = 5 are shown. Probability values [NjM /(2n—1)!] for each type of circular arrange ments are given in table II for values of n up to 25. The last column of table II indicates the mean number of associated pairs expected per cell (Xj). For large values of n, the frequency distribution fits a Poisson distribution with a mean of I (/. = !). Of course, the fit is better as n increases. To facilitate the comparison, the last row of table II indicates the corresponding Poisson values. Standard chi-square tests can be used to compare the random expected values with the observed (xj metaphase cells with j pairs of homologous chromosomes associated, 0 < / < n). The presence or absence of somatic association between homologous chromosomes can then be tested.
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
3 x I0-1“ 1 x 10 •* 3 x 10 11 5 x 1 0 14 7x 10“ 1 x 10 11 1 x 10-'* 2 x 10 '* 2 x 10 " 2 x 10 "
1.3340 1.1990 1.1436 1.1115 1.0959 1.0765 1.0664 1.0593 1.0527 1.0478 1.0429 1.0389 1.0362 1.0358 1.0320 1.0291 1.0289 1.0274 1.0252
L acadena et al.
Quantitative analysis of somatic association
Since the biological meaning of finding more than the expected number of associations would be very different from the meaning of fewer than the expected number, the most sensitive test seems to be a one-tailed chi-square test with 1 degree of freedom testing whether the total numbers of associated and unassociated pairs observed differ significantly from the expected ones. These expected values are, respectively, Nxj and N(n—Xj), N being the number of cells observed and xj the mean number of associated pairs per cell (see table II). We have applied the proposed approaches to the experimental data obtained by us in root tips of Crepis rubra (2/t=10), in which the five chromosomes of the complement are distinguishable. The results obtained (table III) show that the arrangement of chromosomes on the equatorial plate deviates from the expected values calculated under the assumption that the chromosome distribution is random (P
A new approach to quantitative analysis of somatic association J.-R. L acadena,1 E. F errer,1 and I. Y añez2 'Departamento de Genética, Facultad de Biología, and "Departamento de Estadística, Facultad de Matemáticas, Universidad Complutense, Madrid
Abstract A formula is proposed for estimating statistically the degree of somatic chromo some association during mctaphasc. Standard chi-square tests can then be used to determine the presence or absence of somatic association between homologous chromosomes in selected species. The method requires that the chromosomes be morphologically distinguishable and that their circular arrangement on the meta phase plate has not been distorted by the squash used during their preparation.
Request reprints from: Prof. D r. J uan-R amón L acadena. Departamento de Genética, Facultad de Biología, Universidad Complutense de Madrid, Madrid 3 (Spain).
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
Homologous chromosome association at somatic metaphase is evident in some organisms, e.g.. Díptera (Stevens, 1908; M etz. 1916). On the other hand, distortion can either alter the relative positions of some chro mosomes in their circular arrangement on the metaphase plate or even destroy the arrangement scattering the chromosomes over the plate. In order to ascertain the degree of somatic association of homologous chromosomes, it can be interesting to analyze statistically the location of homologous chromosome pairs when the distortion has not changed the circular arrangement of metaphase plates but has altered the relative position of the chromosomes. In this paper a formula is developed to calculate the frequencies of a given circular arrangement in which 0, 1 , 2 , . . . , « pairs of homologous chromosomes are associated, assuming the random location of chromosomes on the equatorial ring. Let Nj" be the number of different circular arrangements of the n pairs of homologous chromosomes of the complement of which any j pairs are associated; i.e., both homologs of each pair lie together on the metaphase
Lacadena et al.
Quantitative analysis of somatic association
E f f e c t p r o d u c e d b y in t r o d u c in g a n e w
F re q u e n c y
321
C ritic a l
p a ir o f h o m o lo g o u s c h ro m o s o m e s
c o n fig u ra tio n
T h e n ew p a ir b re a k s tw o a s s o c ia tio n s
\ l /
A'% l2 K /+ 2 ) f /+ l >1 -
T h e n e w p a ir b re a k s o n e a s s o c ia tio n
^
¡
W
1 2 0 + D (2 « -/-3 )i '
T h e n e w p a ir n e ith e r b r e a k s n o r
jV
c re a te s a n y a s s o c ia tio n
•
T h e n e w p a ir b re a k s o n e a s s o c ia tio n b u t . in t u r n , c r e a t e s a n e w o n e
A’" ' 1 ( 2 / )
T h e n ew p a ir c re a te s o n e a s s o c ia tio n
a
'
'
, ; ; | i 2 ( 2 'i - / - i ) i
\ ^
1 \ « \
/ 11 \ \
/ /
N" 1
/y
\*
" 1 ((2 /» -j-2 ) (2 /1 -/-3 )] '
T o ta l
\
t\ 1
1
N
Fig. I. The new pair of homologous chromosomes introduced into the circular arrangement of 2/1-2 chromosomes are indicated by arrows. Continuous lines represent the homologous chromosome pairs whose associations are broken by the introduction of the new pair of chromosomes. Dashed lines indicate homologous chromosomes (associated or not) whose relative positions are not changed by the introduction of the new pair. For each situation, only one possible configuration is shown. Table I. Absolute frequencies (N¡") of the different circular arrangements according to the number of associations present.
n
0
1
2
i
0 2 32 1488 112512
1 0 48 1920 138240
_
2 3 4 5
4 24 1152 78720
3
4
5
_
_
—
-
-
-
-
-
16 384 26880
96 5760
-
768
Number of circular arrangements (2/1-1)! 1
6 120 5040 362880 Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
Number of homologous chromosome pairs associated (j) -------------------------------------------------------------------------
L acadena et al.
322
Quantitative analysis of somatic association
Table II. Probabilities of occurrence of different circular arrangements
1
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Poisson
Number o f homologous chromosome pairs associated ( j ) 0 1 2 3 4 5 6 7 8
o
0.333 0.267 0.295 0.310 0.320 0.327 0.332 0.336 0.340 0.342 0.344 0.346 0.348 0.349 0.350 0.351 0.352 0.353 0.354 0.355 0.355 0.356 0.356 0.357
i
00.000 0.400 0.381 0.381 0.380 0.379 0.378 0.377 0.376 0.375 0.375 0.374 0.374 0.374 0.373 0.373 0.373 0.372 0.372 0.372 0.372 0.372 0.372 0.371
0.667 0.200 0.229 0.217 0.211 0.206 0.203 0.201 0.199 0.198 0.196 0.195 0.194 0.194 0.193 0.192 0.192 0.192 0.191 0.191 0.190 0.190 0.190 0.190
0.133 0.076 0.074 0.071 0.069 0.068 0.067 0.067 0.066 0.066 0.065 0.065 0.065 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.063 0.063
0.019 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016
0.002 0.002 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003
2x10 ‘ 3 x 10 4 3 x 10 1 3x10 • 4 x 10* 4 x 10 * 4 x 10 * 4 x 10 4 4 x 10 • 4 x 10 ‘ 4 x 10 * 4 x 10 4 4 x 10 4 4 x 10 4 4 x 10 4 4 x 10 4 5 x 10 4 5 x 10 4 5 x 10 4 5 x 10 4
1 x 10 1 2x10s 3 x 10 1 4 x 10— 4x10s 4x10s 5x10s 5 x 10 ‘ 5x10 ‘ 5 x 10 a 5 x10 s 5x10s 5 x 10 4 6x10 1 6x10s 6 x 10 1 6 x 10 s 6 x10 s 6x10s
1 x 10 * 2x10 « 3 x !0 « 3x10« 4x10« 4 x l0 -‘ 4 x 10 « 5 x 10 " 5 x 10 " 5 x 10 " 6 x 10 -" 6 x 10 « 6x10* 6x10* 6 x 10 « 6 x 10 « 6x10* 7x10 «
0.368 0.368 0.184 0.061 0.015 0.003 5 x 1 0 4 7 x 1 0 1 9 x 1 0 *
9
10
II
6x10 : 1 x IO 2x10 T 2 x 10'* 3x103x10 7 4 x 10 7 4 • 10 7 4x105 • 10 7 5 • 10 7 5x106 x I O'7 6 x, 10 7 6 x 10-7 6x10 7 6 ;-10 7
3 x 10 # 7 x 10 * 1 x 10 « 2 x 10 * 2 x 10 « 2 x 10 * 3 x 10 * 3 x 10 ’ 3 x 10 * 4 x 10 * 4 x 10 * 4 x 10 * 5x10 = 5 • ID ■ 5 x 10 5 x 10 '
1 x 10 “» 6x 4 x 10 7 x 1 0 '° 2 / 1 X 10 ’ 3 x 1 x 10 ' 5 x 2x10 ’ 7 x 2x10 * 9 x 2 x 10 * 1 x 2 x 10 * 1 x 3 x 10 • 2 x 3 x I0 W 2 x 3 x 10 * 2 x 3x10 » 2 x 4 x 10 * 2 x 4x10 " 2x
1x10*
1 x 10 ’ 9 x 10 » 7 x 1 0 "' 5 x 1 0 “
12
10 11 10“ 10“ 10 " 10“ 10 " 10 »• 10 ,u 10 •“ 10 ■"» 10 10 10“ 10 10“
13
14
3 x 10 ** 8 x 10 11 2 x 10 11 3 x 10 " 4 x 10 ** 5 x 10 14 6 x 10 14 7 x 10 « 8 x 10 14 1 x 10“ 1 x 10 “ I x 10 “ 1 X 10“
9 x 10 “ 3 x 10 14 7 x 10“ 1 x 10“ 2x10“ 2 * 10“ 3 x 10 14 4 x 10“ 4 x 10“ 5 x 10 “ 6 x 1 0 14 7 x 10 4*
ring. The value of Nj" can then be deduced by recurrence from previously known values of A'/' - 1 for n —1 pairs of chromosomes, i taking the values of ¡+2, / T l , /, or / —1. These different values of / depend on the relative positions taken on the circular arrangement by both members of a new pair of homologous chromosomes when a hypothetical cell passes from 2n—2 to 2/j chromosomes. Thus, the following five different situations are possible: 1. The new pair of homologous chromosomes that is introduced breaks the previously existing associations of two pairs of homologous chromo somes. In this case, /= /+ 2 . 2. The new pair only breaks one previous association (i= /+ l). 3. The new pair neither breaks nor creates any association (/=/). 4. The new pair breaks one existing association but, in turn, creates a new one. In this case, i=j too. 5. The introduction of a new pair of homologous chromosomes creates one additional association ( /= /—1). Taking into account the five situations described and the different possibilities of location on the metaphase ring for the new chromosome pair corresponding to each case, N/' is given by:
4 x 1 0 '"
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
n
L acadena et al.
Quantitative analysis of somatic association
323
showing j associations and mean number of associated pairs per cell (xj).
15
16
17
18
19
20
21
22
23
24
25
I
I x 10 17 4 x 10 17 9 x 10 17 2 x 10“ 3 x 10“ 4 x 10“ 6 x 10“ 7 x 10 16 9 x 10 >•
3 x 10 '*
1 x 10“
3 x 10 '■ 1 x 10“ 3x10“ 6 x 10“ ! x 10 17 2 x 10 17 2 - 10 17 3 x 10 17 4 x 10“
2 x 10 11
1 x 10“
1 x 10“
9 x 10 “ 4 x 10“ 1 x 10“ 2 x 10 *• 4 x 10“ 5 x 10“ 8 x 10“ 1 x 10“
2x 1x 3x 6x 1x 2x
5 x 10“
10“ 10 11 10“ 10“ 10 70 10 70
6x 3x 9x 2 • 3x
3 x 10 3U
10 71 10 43 10“ 10“ 10“ 6 x 10“
2 x 8 x 2x 5 x 1x
3 x 10“
1 x 10“
7 x 10“
10 43 10“ 10“ 10“ 10“
4x : 6> 1x
10 47 io * 10“ 10“
3 x 1 0 “«
8 x 10“ 4 x 10“ 1 X 10 37
2 x 10“ 1x 10*“
3 x IO-33
1.0246 1.0224 1.0221 1.0219 1.0206
1 X 10“
5 x 10“
2 x 10“
1
N f = N } . ,» '[(/ + 2 )(/+ l)j + N ; t ,«-«[2 (H -l)(2 /i-/-3 )]
+ N j" - ' [ ( 2 n - j - 2 ) ( 2 n - j - 3 ) ] + N /'- ' (2j) + Nj _i " - 1[2(2/i—/ —1 )]. Each term in the equation corresponds to one of the five situations mentioned above, respectively, and they can be easily derived from the critical configurations shown in fig. 1. Obviously, this recurrent equation is only valid from n = 3. In table I the absolute frequencies up to n = 5 are shown. Probability values [NjM /(2n—1)!] for each type of circular arrange ments are given in table II for values of n up to 25. The last column of table II indicates the mean number of associated pairs expected per cell (Xj). For large values of n, the frequency distribution fits a Poisson distribution with a mean of I (/. = !). Of course, the fit is better as n increases. To facilitate the comparison, the last row of table II indicates the corresponding Poisson values. Standard chi-square tests can be used to compare the random expected values with the observed (xj metaphase cells with j pairs of homologous chromosomes associated, 0 < / < n). The presence or absence of somatic association between homologous chromosomes can then be tested.
Downloaded by: Lund University Libraries 130.235.66.10 - 12/31/2018 5:04:37 PM
3 x I0-1“ 1 x 10 •* 3 x 10 11 5 x 1 0 14 7x 10“ 1 x 10 11 1 x 10-'* 2 x 10 '* 2 x 10 " 2 x 10 "
1.3340 1.1990 1.1436 1.1115 1.0959 1.0765 1.0664 1.0593 1.0527 1.0478 1.0429 1.0389 1.0362 1.0358 1.0320 1.0291 1.0289 1.0274 1.0252
L acadena et al.
Quantitative analysis of somatic association
Since the biological meaning of finding more than the expected number of associations would be very different from the meaning of fewer than the expected number, the most sensitive test seems to be a one-tailed chi-square test with 1 degree of freedom testing whether the total numbers of associated and unassociated pairs observed differ significantly from the expected ones. These expected values are, respectively, Nxj and N(n—Xj), N being the number of cells observed and xj the mean number of associated pairs per cell (see table II). We have applied the proposed approaches to the experimental data obtained by us in root tips of Crepis rubra (2/t=10), in which the five chromosomes of the complement are distinguishable. The results obtained (table III) show that the arrangement of chromosomes on the equatorial plate deviates from the expected values calculated under the assumption that the chromosome distribution is random (P
