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CYTOLOGICAL, STATISTICAL AND TRANSMISSION ELECTRON
MICROSCOPY STUDIES OF SECONDARY ASSOCIATION
IN THE GENUS PHASEOLUS

BY

John Henry Blackson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Botany and Plant Pathology

1981

ABSTRACT
CYTOLOGICAL, STATISTICAL AND TRANSMISSION ELECTRON

MICROSCOPY STUDIES OF SECONDARY ASSOCIATION
IN THE GENUS PHASEOLUS

BY

John Henry Blackson

Four cultivars of Phaseolus coccineus L., five
cultivars of P. vulgaris L. and one collection of
P. vulgaris var. aborigineus were examined cytologically
for the presence of secondary association of bivalents
(bivalent pairing) during metaphase I of meiosis in pollen
mother cells. Statistical methods were presented for
evaluating the deviation of the observed degree of
association to that expected at random. The degree of
secondary pairing was found to be highly significant.
Transmission electron microscope techniques were modified
to enable the viewing of squashed but intact pollen
mother cells which showed that a physical connection

could occur between bivalents secondarily associated.

ACKNOWLEDGMENTS

I would like to thank Dr. William Tai for being
my major professor and for the use of his research
facilities. I would also like to thank Drs. Susan Kephart
and Wayne Adams, members of my committee, for their
guidance and time and Dr. Joanne Whallon for her support
and friendship.

I would specially like to express my sincere
gratitude to my wife, Marcia, whose support, love and
encouragement has been unfailing and for the fine job
she has done in typing this manuscript.

A grant from Sigma Xi helped to finance the
electron microscopy portion of this study. This support

is very gratefully acknowledged.

ii

LIST OF TABLES

LIST OF ILLUSTRATIONS

INTRODUCTION

MATERIALS AND METHODS

RESULTS . .

DISCUSSION .

BIBLIOGRAPHY

TABLE

OF CONTENTS

iii

Page

iv

10

15

26

37

Table

LIST OF TABLES

Frequency of Bivalent Configurations

Mean Number of Associations Involving
k Bivalents . . . . . . . . . . . .

Pollen Viability as Determined by Counting

Iz-KI and Unstained Pollen Grains .

Expected Number of Clumps of Size k (C )
and Mean Number of Isolated Clumps of

Sizes

3 and 4 O O O O O O O O I O O O O 0

Range of the Expected Number of Clumps of
Size k (Ck) as Calculated using Equation 5

Range of the Expected Number of Clumps of

(C) as Calculated using Equations

11

Size k (Ck) using Equation 8 . . . . . . .

The Kolmogorov-Smirnov Values (D) and Their
Significance, Calculated using the Expected

X

k

and Observed Xk for each Accession

iv

Page
19
20

25

28

29

31

33

LIST OF ILLUSTRATIONS

Figure Page

1 The hypothesized scheme put forth by
Darlington and Moffett (1930) to
explain the maximum association of
bivalents seen in Malus species 2n = 34 . . . 3

2 Meiosis in Phaseolus . . . . . . . . . . . . 18

3 Comparison of a light micrograph of a
Phaseolus vulgaris (accession number 150943),
PMC at Diakinesis with electron micrographs
of the same cell . . . . . . . . . . . . . . 22

4 Electron micrographs of bivalents in
aSSOCiatj-on O O O O O O O O O O O O O O O O O 24
Graph
1 Comparison of the number of isolated

clumps expected and the number found
by experiment . . . . . . . . . . . . . . . . 28

INTRODUCTION

For this study secondary association will be
differentiated from meiotic synapsis using the definition
of Rieger, et al. (1976).

Secondary association or pairing of bivalents:
The nonrandom distribution of bivalents at

first metaphase of meiosis in some polyploid
species. The bivalents occur in pairs or groups
if they are related by genetic and evolutionary
factors.

Meiotic synapsis or pairing: The intimate
association of homologues easily recognized

during first prophase of meiosis and leading to
bivalent or multivalent formation.... When the
pairing process is viewed under the electron
microscope the visibly single zygotene chromosomes
form a tripartite ribbon called a synaptonemal
complex which is absent in achiasmate meiosis.

 

The phenomenon of secondary association of bivalents
or bivalent pairing during meiosis was first noted by
Kuwada (1910) in Oryza sativa. However, it was not
until the early 1930's that the phenomenon was adequately
defined and characterized.

Lawrence (1929, 1931) described the occurrence of
secondary association in several species in the genus
Dahlia. He noted that secondary association was not
widely accepted, many cytogeneticists believing the
phenomenon to be nothing more than an artifact of

fixation (Lawrence 1931). Lawrence described the

phenomenon as a result of the juxtaposition of homologous

chromosomes following primary association and used three

criteria to support his hypothesis of nonrandomness:

1. Secondary association was found to be constant in
the best of fixations.

2. The average number of chromosomes per association and
the average frequency of each kind of association was
characteristic for a given species.

3. At metaphase,associated bivalents are similar in size
and configuration, structurally similar with respect
to the position and number of chiasmata and the
position of their attachments.

Based on these postulates of nonrandomness and homology

between associates, Lawrence proposed a basic number of

x = 8 to explain the maximum association observed in

Dahlia merckii of two groups of three bivalents, 2(3),

and five groups of two bivalents 5(2).

A second paper (Darlington and Moffett 1930) which
appeared about the same time used arguments similar to
those put forth by Lawrence (1931) to hypothesize that
the basic number of x = 17 in species of Malus was of
secondary origin and that the true basic number was
therefore x = 7. The authors presented a scheme to

explain the maximum association observed (figure 1)‘.

1In this scheme each letter represents a bivalent,
the same letter used more than once indicates homologous
bivalents.

3

To further substantiate this contention they state that
the basic number of x = 7 occurs in other subfamilies
of the Rosaceae.

AAA

BBB

CCC

DD

EE

FF
GG

x = 7
Figure l. The hypothe-
sized scheme put forth by
Darlington and Moffett (1930)_
to explain the maximum associ-
ation of bivalents seen in
Malus species 2n = 34.

Several authors published papers describing similar
occurrences of secondary association in a number of
unrelated genera. Muntzing (1933) proposed a basic number
of x = 6 rather than x = 12 for Solanum tuberosum based
on the frequent secondary association of chromosomes in
groups of two (38% of the chromosomes).

Catcheside (l934)_employed Lawrence's (1931)
postulates to hypothesize that secondary balanced
polyploidy occurs within a group of Brassica species
(2n = 18). Based on secondary pairing a basic number of
x = 6 was determined for the genus.

Subsequent authors have disputed several contentions

put forth in the papers by Lawrence (1931) and Darlington

and Moffett (1930) which laid the ground work for the

study of the phenomenon of secondary association.
Gustafsson (l934)_noted a physical connection between
the associated bivalents in Taraxacum a point which both
Darlington and Moffett (1930) and Lawrence (1931).
claimed did not occur nor should be seen in secondary
association. Gustafsson attributed the physical contact
made by bivalents in Taraxacum to the presence of a
greater degree of homology, physical contact implying
that the chromosomes were more closely related than in
the case when only a juxtaposition occurred. Gustafsson
(1934) attempted to explain what he thought to be the
mechanics of secondary association by hypothesizing,
"that the secondary associates are due to terminal
affinities and to the fusion of the pellicles of
chromosomes."?

Heilborn (1936) was one of the first to determine
the percentage of associations between distinguishable
bivalents. He looked at the degree of association
between the two smaller chromosomes in Carex pilulifera
and between the three larger chromosomes of C. panicea
and found that a definite association existed. However,
unlike earlier workers,.Heilborn found secondary associ-

ation to be as prevalent during prOphase as metaphase

 

2Electron microscope studies have not supported
the presence of chromosome pellicles.

and suggested a different explanation of the mechanisms
involved in secondary association: "secondary association
of chromosomes results from the action of the forces of
nuclear division upon chromosomes of different size and
mass." However, as soon as homologous bivalents occur
they become associated on account of their identity in
size and mass and in this way secondary association can
still be used to imply chromosome duplication (Heilborn
1935).

Nandi (1936) gave yet a different explanation to
describe the secondary association found in the genus
Oryza. He compares the arrangement of the chromosomes
to the arrangement formed by a corresponding number of
floating magnets. When related bivalents are not
associated it is because they are not lying near enough
to each other.

With yet a different interpretation Thomas and
Revell (1946) ascribe the mechanisms of secondary
association in Cicer arietinum to the fusion of hetero-
chromatin noting that the fusion of heterochromatic
regions was not specific even though a high degree of
association does occur among morphologically similar
bivalents.

They also examined the effect that the squash
technique had on the arrangement of bivalents. By studying

many individual cells, before and after applying pressure

to the coverslip, they showed that the degree of
association in fixed unsquashed material was not altered
by squashing (Thomas and Revell 1946). However, Brown
(1950) arrived at a different conclusion when comparing
squashed and sectioned material in Luzula campestris
2n = 12. He noted that the high degree of secondary
association present in squashed preparations of this
species suggests a secondary polyploid nature,
particularly since a related species, L. purpurea, has
a diploid complement of 2n = 6. Unfortunately, an
examination of sectioned cells of the same species
showed a lower incidence of secondary association and
Brown (1950) concluded that the observed secondary
association in squashed material was only an artifact
of the technique. He did not, however, dismiss all
secondary association as an artifact and believed the
work of Darlington which described sectioned material,
to represent a good example of true secondary pairing.

Jakob (1957) working with Ricinus communis,
2n = 20, postulates that the secondary association found
in this species is due in part to either centromeric
attraction and/or in the exchange of portions of pairing
strands without the formation of chiasmata in the
regions of interchromosomal contact.

Riley (1960) and Kempanna and Riley (1964) using

ditelocentric Triticum aestivum looked at the number of

intervening bivalents which occurred between the

marked bivalents and in their 1964 paper concluded that,
"...secondary pairing genuinely occurs between bivalents
with genetically and structurally similar chromosomes,
moreover there is no association between genetically
unrelated bivalents."

Hu (1962) agreeing with the findings of Kuwada
(1910) and Nandi (1936), concluded that the 2n = 24
species of Oryza showed a maximum association of two
groups of three, 2(3), and three groups of two, 3(2).
Comparing the observed number of bivalents in association
to that expected from a Poisson distribution, he concluded
that the genus Oryza is of a secondary polyploid origin
and "the doubling of genetic material in remote ancestry
might have played an important role in the evolution of
the genus" (Hu 1962).

Kempanna and Setty (1967) working with Eleusine
coracana, 2n = 36, used a Chi? test to compare the
observed number of bivalent groups in each cell to the
expected number as defined by the equal probability of
obtaining one group of 18 bivalents, two groups....18
groups of bivalents. They found that the nine group
class deviated the most from the expected value and thus
proposed a basic number of x = 9 for this species.

Sastrapradja and Rijanti (1972) noted secondary

association in Colorasia gigantea and C. esculenta,

2n = 28, and indicated that the maximum association was
6(1) + 1(2) + 2(3) and that "the bivalents were held
together by one or two chiasmata." The authors were
hesitant to declare these species to be of a secondary
polyploid origin and concluded that the phenomenon
needed further substantiation in this genus.

Gupta and Roy (1973) noted secondary association
in Euryale ferox Salisb. 2n = 58. With the formation of
groups of 2, 3 and 4 bivalents they suggested its gametic
number to have originated secondarily by means of
allopolyploidy.

Mursal (1979) used the evidence of secondary
association in haploids and hybrids of Gossypium to
give additional support to the homoeology between the
A and D genomes of cotton. Eighty-three percent of the
associates were of the AD type and seventeen percent
either of the AA type or DD type. (Here the letters
represent chromosomes belonging to either the A or
D genome of cotton.)

In yet another study, Rejon and Oliver (1980).
combining electrophoretic studies and the presence of
secondary association during meiosis point to a remote
polyploid origin in the genus Asphodelus, 2n = 28 to
7 for

2n = 84, and hypothesize a basic number of x

the genus.

From the evidence available to date, it seems clear
that the presence of secondary association can be used
to denote homology between the chromosomes despite the
lack of agreement on the mechanisms involved. The exact
mechanism of secondary association has not yet been
clarified.

The purpose of the present study was to investigate
the occurrence of secondary association in two species
in the genus Phaseolus and to statistically evaluate its
significance if present. Secondary association in the
genus Phaseolus L. has never been reported. Two authors
Hussein (personal communique) and Machado (1978), working
with the closely allied genus Vigna Savi., 2n = 22,have
noted the occurrence of secondary association. Based
on the occurrence of secondary association and multipolar
meiosis (grouping of chromosomes of a genome) they
proposed a basic number of x = 5 or 6 for the genus.

A diploid number of 2n = 22 has been reported for
all species of Vigna and Phaseolus with occasional
reports of 2n = 20-24 (Frahm-Leliveld 1965, Bauchan &
Tai, in press). Meiosis in the genus Phaseolus L. has
been described as highly regular except for an occasional
inversion and the occurrence of precocious separation at
metaphase I (Honma 1968, Sarbhoy 1977, Machado 1978,

Sinha & Roy 1979).

MATERIALS AND METHODS

Seeds of the following accessions were obtained

from the W-6 Regional Plant Introduction Station of

the U.S. Department of Agriculture, Pullman, WA 99164.

Phaseolus coccineus L.

175855

311977

3580911

361520

P. vulgaris L.

150943

226898

281996

311798

Collected in a market, Yozgat, Yozgat,
Turkey, Sept. 20, 1948
Collector J. R. Harlan

Ayocote' Mexico, 1965
Collector H. S. Gentry No. 21318

Probistipski Buciste, Yugoslavia,
Received Feb. 15, 1970
Collector Lazar Aladzajkov

Sarhan, Gopalpur, India
Collector Himachal Pradesh

Tlalnepantla, Mexico, Received May 1, 1945,
Presented by Dario L. Arrieta

'Pan de Libano', Spain, Received July 11, 1955
Presented by 0. H. Pearson

'Pajaritos' Chile, Received July 19, 1962
Presented by the Rockefeller Foundation
Santiago

'Frijol negro' from market in Jutiapa
Guatemala, 1965 H. S. Gentry, No. 21363

 

1This accession was listed under P. multiflorus in
the Bean Inventory (Phaseolus species) Catalog of Seed
Available (Western Regional Plant Introduction Station;
1978; Pullman, Washington) but is now considered a
synonym of P. coccineus (Bailey and Bailey 1976).

10-

11

P. vulgaris L. (continued)
314729 ' From a market in Alma Ata. U.S.S.R., in

the vicinity of Samarkand, Uzbeckistan,
May 9, 1966. Quentin Jones and Wesley
Keller, No. 335.

P. vulgaris var. aborigineus (Burk.) Baudet.

2669102 Collected from the wild, Argentina.
Received June 23, 1960
Presented by Instituto de Botanica
Darwinion San Isidro.3
All plants were grown under greenhouse conditions.

Plants were fertilized with "Ra~pid°gro"“ approxi-

mately every two weeks, but no other chemicals were

applied. Plants of P. vulgaris var. aborigineus

required a short day length to promote flowering

and were grown under 8 hours light and 16 hours of

dark. All other taxa were grown under 16 hour days.

Buds were collected between the hours of 9 and 10 a.m.

and were fixed in six parts ethanol : three parts glacial

acetic acid : one part chloroform for 30 hours at which

time they were transferred to 70% ethanol and refrigerated

until they were used.

Approximately fifty well spread metaphase I pollen

mother cells were observed, photographed and scored for

 

2This accession was listed under P. aborigineus in
the Bean Inventory but according to a recent revision of
the genus is now considered a variety of P. vulgaris
Marechal 1978).

3Collection data was obtained from Plant Inventory,
United States Department of Agriculture, Washington D.C.

“Ra-pid-gro Corporation, Dansville, N.Y.

12

the degree of secondary association present for each
of the accessions.

Staining of chromosomes in squashed pollen mother
cells was found optimum when proprionic carmine was
utilized as Opposed to acetocarmine which gave unsatis-
factory results. All usable slides were made semi-
permenant using Dental Sticky wax. In this study, cells
in metaphase I were of primary importance, however, other
stages were observed for irregularities. All squashed
but intact metaphase I cells with 11 countable bivalents
were photographed using a Zeiss Photoscope II light
microscope with a Planapo 63x N.A. 1.4 objective. The
recording medium was Kodak Panatomic x film with 10
minute development in undiluted Kodak Microdol x at
70° F.

Drawings were made during observation along
with photomicrographs of each cell and each cell was
scored for the degree of secondary association present.
Two bivalents were only scored as associated if they
were in physical contact or close together and suffi—
ciently isolated from all other bivalents as to appear
grouped.

For a representative number of slides, stage
coordinates were recorded for photographed cells to aid
in their relocation after treatment for transmission

electron microscope studies described below.

13

Several slides were then processed using a modifi-
cation of a technique developed by Ris (1978). The
process enables one to transfer cells from glass slides to
EM grids. With wax removed, the slides were placed on solid
C02 and coverslips were removed. The slides were then
immediately immersed in distilled water (DHZO). After
removal from the DHZO the slides were covered with aqueous
uranyl acetate for three hours at which time they were
rinsed with 50% ethanol and run through a dehydration
series: 70%, 80%, 90%, 100%, 100% ethanol. Ten minute
intervals were used. Excess ethanol was then blotted
from areas of the slide not containing cells and the slides
were covered with a 2% parlodion solution using iso-
amylacetate as a solvent. The slides were allowed to
dry in a horizontal position in a dust free environment
over night. The slides could then be returned to the
photosc0pe and cells relocated using the coordinates
recorded. Once a cell of interest was located a circle,
approximately the size of an EM grid,was drawn around the
cell using a diamond marking tool. Next a drop of
water was placed over the embedded cell which was carefully
lifted off of the slide and floated on the water surface.
The cell was then picked up on a 100 mesh 0.5% formvar-
coated grid. After blotting off excess water the grid
was immersed in iso-amylacetate and left over night to

remove the parlodion. The grid was then critical point

14

dried, carbon-coated and viewed in a Philips 201 TEM at
80 or 100 KEV. Micrographs were made using Kodak Electron
Microscope film 4489 and develOped in D-19 for four
minutes at 700 F.

Pollen stainability was determined for each
accession utilizing standard iodine techniques on fresh

or fixed material.

RESULTS

Meiosis in PMC's of all species observed was normal
with 2n = 22 except for the occurrence of precocious
separation,a rare anaphase I bridge (figure 2E) and
frequent secondary association. At diakinesis.two
pairs of chromosomes were seen attached to the nucleolus
and the loose association of bivalents was apparent
(figure 2A). At metaphase I,some degree of secondary
association was evident in 95.74% of all cells observed
(figures 2B & C). Equal separation at anaphase I without
cytokinesis was the normal case. Metaphase II again
exhibited strong secondary association (figure 2D).
Secondary association was also apparent during early
telophase II (figure 2F). The frequency of bivalent
association for P. coccineus, P. vulgaris and P. vulgaris
var. aborigineus are recorded in table 1 and the mean
number of configurations of k bivalents per association,
where k = 1, 2-7, for each accession are recorded in
table 2. Transmission electron microsc0py studies of
PMC's showed that a physical connection could exist
between bivalents in association (figure 3 &‘4). The
technique enabled direct correlation of light micrographs

with electron micrographs, beneficial for the study of

15

16

cytogenetic phenomena. Pollen fertility was high for

all accessions (table 3).

Fig.

2.

A.

Meiosis in Phaseolus

Diakinesis showing two bivalents attached to the
nucleolus in Phaseolus vulgaris var. aborigeneus
(accession number 266910)

Metaphase I showing the maximum secondary
association of 4(2) 1(3) in Phaseolus coccineus
(accession number 358091)

Metaphase I showing the maximum association of
4(2) 1(3) in Phaseolus coccineus (accession number
361520)

Anaphase I with secondary association in Phaseolus
vulgaris var. aborigeneus (accession number 266910)

Late anaphase I with chromatin bridge in Phaseolus
coccineus (accession number 175855)

Anaphase II showing secondary association in
Phaseolus vulgaris (accession number 150943)

17

18

 

l9

 

 

oo. m.nm o m.~ o.c 0.5 w.o m.- 0.5 ¢.q_ o.m 0.0— o.m unmouom
mm_ on o n o o_ m m. o. m. s q. q mHHoo «0 s
mamcwmwhonm
.um> mwummwsb .m
00. m.m~ m.o w.o m.m o.m 0.0 n.m_ ~.m m.¢_ m.~ m.m m.m usoouwm
«cm no N N n. «N 0. mm om mm o q_ «— mHHou mo a
.A mwumowzb .m
00. o.m~ ~.m N.m m.o n.o m... m.m_ o.w o... o «.0 m.m ucooumm
men mm __ __ mm mm _q me mm mm _ mm ~_ magma mo e
.4 mzmcwoooo .m
Hmuoe Hwfio Ame. Amy. Amv_ Ame. mmwomam
ANVS ANVn Amen Amos Amvm Amen “NV. “mom Anv_ A~v_ A_v__

 

 

monH<MDuHmzou Hzm4<>Hm

p mqm<e

so sozmscmmm

2(J

 

 

000 000.0 000.0 000.0 000.0 000.0 000.0 000.0 :0
0000.0 0000.0 0000.0 000.0 000.0 000.0 000.0 w 000000
maoswowuoam
.uu> unknowns .m
00 000.0 000.0 000.0 000.0 000.0 um
0000.0 000.0 000.0 000.0 000.0 m 000000
00 000.0 000.0 000.0 000.0 000.0 000.0 mm
0000.0 0000.0 0000.0 000.0 00.0 000.0 m 000000
00 000.0 000.0 000.0 000.0 M0
0000.0 0000.0 000.0 000.0 m 000000
00 000.0 000.0 000.0 000.0 000.0 0.0
0000.0 000.0 000.0 000.0 000.0 m 000000
00 000.0 000.0 000.0 000.0 000.0 000.0 000.0 W0
0000.0 0000.0 0000.0 0000.0 000.0 000.0 000.0 w 000000
.4 mwumowas .m
00 000.0 000.0 000.0 000.0 000.0 000.0 W0
0000.0 0000.0 000.0 000.0 000.0 000.0 m 000000
000 000.0 000.0 000.0 000.0 000.0 000.0 W0
0000.0 0000.0 000.0 000.0 000.0 000.0 m 000000
000 000.0 000.0 000.0 000.0 000.0 W0
0000.0 000.0 000.0 000.0 000.0 w 000000
00 000.0 000.0 000.0 000.0 000.0 000.0 #0
0000.0 0000.0 000.0 000.0 000.0 000.0 m 000000
.4 moonwuooo .m
000000 0 0 0 0 0 0 0 000502
ozm uo ceaumusmwucoo nod nuco0n>0m mo umbssz sodomooo<
uonEsz 0uuoe x a uoaoodm

 

 

mfizmd<>Hm x 02H>AO>ZH mZOHB<HUOmm< ho mmmzaz 24m:

N mqm<fi

Fig. 3. Comparison of a light micrograph of a Phaseolus
vulgaris (accession number 150943) PMC at Diakinesis
with electron micrographs of the same cell. Electron

micrographs taken at 100KEV

21

22

 

 

 

 

 

 

Figure 3

Fig.

4.

A.

Electron micrographs of bivalents in association.

Secondary association in Phaseolus vulgaris var.
aborigeneus (accession number 266910) 100KEV

Secondary association in P. vulgaris var.
aborigeneus (accession number 266910) 100KEV

Secondary association in P. vulgaris (accession
number 150943) 80KEV Taken 30,000X

Secondary association in P. vulgaris var.
aborigeneus (accession number 266910) 100KEV

23

24

 

 

 

Figure 4

25

TABLE 3

POLLEN VIABILITY AS DETERMINED BY COUNTING Iz-KI AND
UNSTAINED POLLEN GRAINS

 

 

Species & Total Number of
Accession % Viable Pollen Grains
Number Pollen Counted

 

P. coccineus L.

175855 95 217
311977 99 203
358091 98 297
361520 94 409
P. vulgaris L.
150943 81 235
226898 93 413
281996 85 799
311798 74 416
314729 84 242

P. vulgaris var.
aborigineus

266910 93 395

 

DISCUSSION

In order to analyze statistically the significance
of secondary association a method of estimating the
expected number of groups of bivalents given a random
distribution of chromosomes was desired. It was not
possible to utilize methods developed by other workers
studying chromosome arrangement, because of the difficulty
of distinguishing one bivalent from another, and the
small size and uniform appearance of Phaseolus chromosomes
(Barton, et a1. 1967, Irwin 1967, Riley 1960, Kempanna
& Riley 1964, Ferrer and Lacadena 1977, Lacadena &

Ferrer 1978). Considering the limitations it was found
necessary to look to other disciplines to arrive at a
useful solution for purposes of the present study.

Specifically this investigation attempted to find a
method of determining the random expectation of obtaining
n clumps of 5 small particles given N points randomly
distributed over an area A.

Roach (1968) develOped a method of determining the
number of clumps of circular laminae with radius r,
where the centers of the circles are randomly distributed

over a plane surface. In order for a clump to occur the

26

27

center points of two laminae had to fall within 2r of
each other.

The probability that a circle is isolated is equal
to the probability that no laminae center falls within

a distance 2r from a point on the plane. Thus

 

 

_ 2
P1 = e( 4nr N) l
and the probability of an isolated clump of k circles is
Pk = P1(l - P1)k-l 2
The mean number of isolated clumps, Ck’ of size k in
the area A is
NPk NP1(1 - P1)k-l
C = -—— = 3
k k k
and the mean number of isolated clumps of all sizes,
Q, in unit area A is
NPl 1n P1
c = e~—— 4
Pl-l

Roach found that the expected values of the number of
clumps in an area of 20 cm? for densities of 100-2200
formed by circles of r = .25 were remarkably close to

the actual value found by experiment (graph 1). For

the purposes of the present study A = 88 cmfi,a conservative
estimate of the area of the circle surrounding the

space occupied by the chromosomes of a PMC at metaphase I
as determined from photographs at 1350K magnification.

In a similar manner,the radius of the chromosomes is

 

28

set at 0.15 cm and N = 11. The expected values of the

mean number of associations of size 5 are given in

table 4 .

 

5001
n
a
E
3
U
.O' 250-1
5
.0
:E; x Total (C)
2
1’stc.) e
. . 31$(C3) K ,
O 1 1 _
O 1000 2000

 

Number of laminae on lattice

Graph 1. Comparison of the number of
isolated clumps expected and the number
found by experiment.1

TABLE 4
EXPECTED NUMBER OF CLUMPS OF SIZE k (C )

AND MEAN NUMBER OF ISOLATED CLUMPS OF LL
SIZES (C) AS CALCULATED USING EQUATIONS

 

 

 

3 AND 4.
K Ck
1 10.618
2 0.1844
3 0.004268
4 0.000111
5 0.0000031
C 10.807

 

 

1S. A. Roach, The Theory of Random Clumping.

(London:

Methuen & Co. Ltd., 1968). P. 30.

29

Two other authors approached the problem from a
slightly different perspective than did Roach. These are
presented here for sake of comparison to those values
already reported.

Mack (1950, 1948) gives

ck = kHfiHnRZlk'l
where 3 must be defined as the radius of the circle
enclosing the clumped points divided by /A. For our
purposes Ck will be calculated for the two extremes for
which the radius of the circle enclosing the chromosomes
of radius .25 cm can take (table 5). The enclosing
circle radius 3 must be between .15 cm, when the chromo-
somes completely overlap one another and 0.3 cm for the

case when the two chromosomes just come into contact with

each other.

TABLE 5

RANGE OF THE EXPECTED NUMBER OF CLUMPS OF
SIZE k (Ck) AS CALCULATED USING EQUATION 5.

 

 

 

0.000958 5 C3‘5 0.0153

waW‘

 

0.0000027 5,0. 5 0.00175

The values found in table 4 as calculated using the
equation reported in Roach (1968) fall between the

extremes reported in table 5.

30

Mack (1948) also gives an equation for calculating

the probability of obtaining m simultaneous 5 aggregates

m -Ck
Pm(k1N) = (Ck) 9

ml

which could be used for calculating the probability
of any chromosome configuration of m equal sized clumps.

A third paper by Berg (1945) gives yet another set
of equations for the calculation of Ck and Pm' As with
Mack (1948), Berg is concerned with the number of points
likely to be contained within a small area a of a
larger area 5 containing g randomly distributed points.
Berg presents his argument for square areas only; however,
he notes that the result is not dependent on the shape
of the area. For calculating Ck he first gives the
Poisson distribution which takes a similar form to that
presented by Roach with some modifications for dealing
with the area encircling the laminae rather than with the

area of the laminae themselves. Therefore

 

Pk = (Nalke-Na
n!
and Ck can be calculated using
C = P k
k k a

where a = nR2 (k-l) .
—X—

31

Using our data where 3 takes the values discussed in the

previous example one gets values of Ck as found in tablefi.

TABLE 6

RANGE OF THE EXPECTED NUMBER OF CLUMPS OF
SIZE k (Ck) USING EQUATION 8.

 

 

 

k
2 0.0963 5 02 < 0.375

3 0.00169 < c; < 0.0256
4 0.0000333 g c. < 0.00197

 

Again the values found in table 4 are within the
range of those in table 6. The values of table 5 are
comparable to those in table 6, except that the former
overestimates C2 when compared with the latter.

It can be seen that the three papers presented
give results comparable to one another even though they
approach the problem differently. It is therefore
felt that any of the three methods reported could be used
to calculate the mean expected number of clumps containing
E chromosomes. For convenience, equation 3 which gives
single values for Ck will be used to evaluate our data.
The mean number of chromosomes xk involved in an
association of El chromosomes per cell can be calculated
for both the expected and observed values using

Xk = Ck(k) 9

32

To test the null hypothesis of Ho: chromosomes randomly
distributed versus the alternate hypothesis H1: chromo-
somes not randomly distributed, the Kolmogorov-Smirnov
test for goodness of fit will be utilized to compare Xk
observed with Xk expected for each accession (table 7)
(Zar 1974). All observed values of xk for k 3 5 are
lumped in the X5 class.

Considering the results of table 7,it can be seen
that secondary association seems to play a definite role
in the positioning of chromosomes on the metaphase plate.
The maximum association present as determined by observation
appears to be 4(2) + 1(3) from which a basic number of
X = 5 or 6 might be hypothesized using the argument that
secondary association implies homology (Darlington and
Moffett 1930, Lawrence 1931, Gustafsson 1934, Riley 1960,
Kempanna and Riley 1964). A possible diploid chromosome
formula might be:

AAA

BB

CC

DD

EE

huffindings support the work of Machado (1968)

and Hussein (personal communique) on members of the genus
Vigna Savi. which is closely aligned to the genus
Phaseolus.

From the evidence compiled at present,it is

impossible to determine the sequence of events which gave

33

 

 

 

 

 

 

 

mica x mm.a wvvooo.o mmao.o mmwm.o mam.oa x oouoomxm
moo.v mom.o mmm.o mum.o va.H mwm.m mao.m oammmm
msmcwmwuonm
.Hm> wwummwsb .m
mo.v mmm.o wm~.o Nmm.o mm.o vvm.m oam.m mmmvam
moo.v mmm.o ~mm.o mh>.o emm.a m.m oom.v manaam
Ho.v mmv.o oo.o mmmm.o hem.o vmm.v wem.m mmmamm
Hoo.v vmm.o v~.o mwm.o mnv.a vom.v mmn.m mammmm
Hoo.v mmm.o NmH.H mmw>.o mHo.H mom.m «ma.v mvmoma
.A mwhmmwsb .m
Hoo.v Ho>.o nmv.o vmm.o moa.m vmm.v mom.m ommaom
moo.v Hem.o nm~.o mom.o qw>.a cam.m nmm.v Hmommm
Ho.v Hmv.o mwa.o mmm.o mmo.a mm.m wmm.m unmaam
Hoo.v mmm.o Nma.o vmm.o mmm.a vow.v veo.v mmmmna
.A moonwoooo .m
xx pm>uomno HoQEsz
mu v m m H scammooo<
a o x a mmaoomm
.ZOHmmm00¢ mu¢m mom xx am>mmmmo 02¢ xx

Omeummxm mmB OZHmD Dmfiflqouq<u .mUZ<UHmHZOHm MHmmB QZ¢ AOV mmsq¢> >OzmH2ml>OmowOZAOM mmB

h mqmdfi

34

rise to the haploid number of this group. The group
could represent a complex hybridization of segmental
allotetraploids having been derived from the two individ-
uals with different chromosome numbers (Grant 1971) in
which case the probable basic numbers are 5 and 6, or
from the polyploidization of probable diploid hybrids with
a basic number of x = 5, 6 or 7 followed by secondary
aneuploidy.
Stebbins (1971) gives the following chart as a

probable means of deriving a haploid number of n = 11.

10 0 124——134——1+Ll——>-15<—16 LlX

T 1/ \ / \¢

5 4 '6 4 7 *8

2X

 

 

 

From Stebbins 1971, in part.

Indirect support for the hypothesis of polyploidy
in the genus Phaseolus comes from the aneuploidy present
in some members of the Phaseolus-Vigna complex,2n = 20-24
(Frahm-Leliveld 1965, Bauchan and Tai, in press). A
polyploid possessing duplicated genes and chromosomes is
more likely to survive and be maintained when one
chromosome or even a pair of chromosomes is missing than
would a diploid (Elliott 1958).

Additional evidence of polyploidy lies in the
proposed basic number for angiosperms in general and for
members of the Fabaceae in particular. Several authors

report this number to be in the range of x = 6, 7 or 8

35

(Wanscher 1934, Senn 1938, Stebbins 1950, Atchinson 1951,
Turner and Fearing 1959, Bandel 1974, and Sands 1975).
Based on his prOposed basic number of the Fabaceae,

Raven (1975) states that the tribe Phaseoleae Brongn.
might have a basic number of x = 7 with n = 11 possibly
resulting through aneuploidy from n = 14.

Similar cases of polyploidization might have
occurred in two other tribes of the subfamily Faboideae,
Hebysareae D.C. (sensu lat.) where genera with a basic
number of x = 10 or 11 and others with x = 7, 6 or 5
exist and in the tribe Galegeae B & H (sensu lat.) with
genera of both x = 10 or 11 and x = 8, 7 or 6. Turner
and Fearing (1959) suggest that it might be possible that
the n = 10, 11 complex of the Hedysareae might have arisen
from the n = 5, 6 or 7 group.

Considering the absence of n = 5, 6 or 7 members in
the Phaseoleae it is necessary to look to other groups
for a possible connection to a group with this chromosome
number.

The genus Abrus Adans., n = 11, is typically placed
in the tribe Vicieae Adans. but thought by Senn (1938)
more closely aligned to the Phaseoleae. Based on chromosome
number and anatomical features, Senn proposed Abrus as an
intermediate between the Vicieae and Phaseoleae.
Hutchinson (1964) recognizes Abrus as belonging to its own

tribe,Abreae Wight & Arne,intermediate between the

36

Phaseoleae and Vicieae. Based on this relationship the
Phaseoleae might have been derived from the x = 5, 6 or 7
Vicieae or its now extinct ancestor, a hypothesis which

might warrant further investigation.

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