Download The Genetic Structure and Evolutionary Fate of Parthenogenetic

A M . ZOOLOCIST, 11:381-398 (1971).
The Genetic Structure and Evolutionary Fate of Parthenogenetic
Amphibian Populations as Determined by Markovian Analysis
JAMES H. ASHER, JR. AND GEORCE W. NACE
The Department of Zoology
and
The Center for Human Growth and Development,
The University of Michigan, Ann Arbor, Michigan 48104
SYNOPSIS. One-locus, two-allele models are presented which describe the genetic consequences of naturally occurring and experimentally induced parthenogesis in triploid
and diploid amphibians. The models may in general be used to investigate genetic
change resulting from apomictic (ameiotic) and automictic (meiotic) parthenogenetic
reproduction.
These models quantify the influence of mutation, segregation, and selection upon
genetic variability in parthenogenetic populations. They also allow an estimate of the
relative importance of stochastic forces in altering this variability. They thus provide a
basis for understanding evolution in these populations.
Some of the conclusions derived from this study contradict previous predictions
regarding genetic variability in parthenogenetic populations. First, if mutation is the
sole source of genetic change (i.e., strict apomixis), parthenogenetic populations should
not become completely heterozygous. Second, small amounts of segregation occurring
in apomictic populations have enormous effects upon the genetic variability of these
populations, i.e., they should lose much of their heterozygosity.
In addition to these conclusions, the results of this study suggest that studies of
protein variability in parthenogenetic species should contribute toward answering the
question: How much of the genetic variability observed in nature is evolutionarily
relevant?
Many reproductive mechanisms are em- "populations," and (3) considering the
ployed by animals in their adaptation to evolutionary implications o£ these models,
Polyploidy and parthenogenesis are relaconstant and changing environments. Parthenogenesis as one of these mechanisms is tively rare in the amphibia. Naturally ocexamined here by: (1) reviewing natural curring polyploidy has been reported in
and experimental parthenogenesis in am- three genera of anurans, Ceratophrys,
phibians, (2) presenting mathematical Odontophrynus, and Hyla (Be^ak, Bec,ak,
models which describe changes in the gen- and Rabello, 1966, 1967; Saez and Brumetic structures of some parthenogenetic Zorrilla, 1966; Bogart, 1967; Wasserman,
1970) and in three genera of urodeles,
We thank Henry Wilbur for his helpful sugges- Ambystoma, Eurycea, and Notophthalmus
tions and discussions made during the preparation
( F a n k h a u s e
v
of this manuscript. We also wish to extend our
lyb4
1 9 3 8
'
1938&
1 9 3 9
,nAn
U z z d l
,
appreciation to George Estabrook for suggesting
the use of Markovian Theory in modeling the
parthenogenetic systems presented here and for his
patience in explaining this and other mathematical
concepts. Finally, we wish to thank Dr. Dan L.
Hartl for his very valuable suggestions made during and after the symposium presentation of this
5
Book,
1940;
Fankhauser
and
Humphrey, 1942). Parthenogenesis, in the
form of natural gynogenetic reproduction,
j
,
,
o ( x u r s
{
f
u r o d d e s
;
°
°
(Ambystoma)
and involves a complex
composed of four species (Uzzell, 1963,
paper.
These investigations were supported by a Horrace H. Rackham Postdoctoral Fellowship to one of
us (Asher) and by a grant from the National
UzzeU a n d Goldblatt 196?.Wilbur>
„,
. . , ' . ' .
T h e fro u r s e c i e s
Science Foundation, GB-8187.
plex are: A. jeffersonanium (2n) , A. later-
1964, 1969; Macgregor and Uzzell, 1964;
1971).
'
P
involved in this COm-
SSI
382
JAMES H. ASHER, JR. AND GEORGE W. NACE
Ancestral
Form
GLACIATION
WISCONSIN
A. laterale (2n)
A.
jeffersonianum
(2 n)
X
2n Hybrid (LJ)
A. laterale
X
UNREDUCED EGG
J
3n Hybrid (2LJ)
A. tremblayi (3n)
X
A. jeffersonianum
I
3n Hybrid (L2J)
A. platineum (3n)
FIG. 1. Diagrammatic representation o£ the origin
of triploid parthenogenetic Ambystoma estimated
at 10,000 years or 5000 generations ago (proposed
by Uzzell, 1963, 1964; and Uzzell and Goldblatt,
1967).
ale (2n) , A. platineum (3n), and A. tremblayi (3n). The parthenogenetic triploids
probably arose immediately following the
Wisconsin glaciation (10,000 years ago) as
a result of hybridization between the two
diploid bisexual species (U/zell, 1963,
1964). If we assume the length of each
generation in the triploids is two years
(Wilbur, personal communication), the
Lriploid populations have had at most 5,000 generations in which to diverge. Figure 1 presents a summary of this proposed
evolutionary relationship. The genomic
compositions of the triploids were deduced
from a consideration of morphological
and serum protein phenotypes (Uzzell,
GENETICS OF PARTHENOGENETIC POPULATIONS
1964; Uzzell and Goldblatt, 1967). Based
on the differences in electrophoretic mobilities and relative concentrations of serum
proteins of the "modern" forms, it was hypothesized that A. jeffersonianum contributed two chromosomal sets and A.
laterale one set to A. platineum (L2J)
while A. jeffersonianum contributed one
set and A. laterale two sets to A. tremblayi
(2LJ).
Cytological studies of oocytes from the
triploids (Macgregor and Uzzell, 1964) revealed a 6n ploidy prior to the completion
of meiotic events. This suggested the occurrence of an endomitotic duplication of
the somatic 3n ploidy. Activation of such
eggs with sperm from the diploid species—
A. jeffersonianum males in the case of
A. platineum, and A. laterale males in
the case of A. tremblayi—is accompanied by a normal meiosis. Thus, Ambystoma triploids are technically automictic
parthenogens.1 The cytological analysis further suggested that only sister chromosomes produced by the pre-meiotic endomitosis pair during the prophase of meiosis I. Hence, the regular meiosis should result in the production of progeny that are
genetically identical to the mother and to
each other. Figure 2 summarizes these cytological observations. Two events can alter
the genetic consequences of this mode of
reproduction: (1) segregation resulting
from the infrequent formation of quadrivalents which does occur (See Fig.
2), and (2) mutation. The first objective
of this paper is to present and evaluate
several models which predict the genetic
consequences of these two "rare" events.
These models are also applicable to any
apoomictic (ameiotic) parthenogenetic reproduction.
Experimentally induced parthenogenesis
is well documented in both the anurans
383
and urodeles (Guyer, 1907; Loeb, 1913;
Parmenter, 1933; Kawamura, 1939; Tyler,
1941, 1955; Rostand, 1950; Beatty, 1967;
Kawamura and Nishioka, 1967). These
reproductions have been used to investigate a variety of problems including those
of fertilization, cleavage, sex differentiation, and genetic mapping (Fankhauser,
1938a; Kawamura, 1939; Tyler, 1941;
Lindsley, Fankhauser, and Humphrey,
1956; Kawamura and Nishioka, 1967;
Nace, Richards, and Asher, 1970; Volpe,
1970). To employ experimental parthenogenesis in the mapping of genekinetochore distances and in the development of "inbred" strains, a mathematical
model was developed (Nace et ah, 1970;
Asher, 1970) which described the genetic
consequences of parthenogenetic reproduction by inhibition of meiosis I (or central
fusion) or meiosis II (or terminal fusion).
A second objective of this paper is to
present an extension of this earlier model
and to consider the genetic consequences of
both experimentally induced and naturally occurring automictic parthenogenetic
reproduction.
Genetic studies of parthenogenetic species (both apomictic and automictic
forms) are important from the standpoint
of understanding evolution and development in these organisms. They may, however, provide information with far broader
consequences. A third objective of this
paper, then, is to suggest genetic studies
which may provide solutions to general
problems in evolution and development.
THEORETICAL CONSIDERATIONS AND RESULTS
Natural populations
For the purposes of this discussion, we
consider only one-locus, two-allele models.
As a consequence of this limitation,
triploid Ambystoma may exist in only one
of
four genotypic states: (I) AAA, (II)
i Throughout the text, we use two terms as they
are defined and used by White (1948) and A'A'A', (III) AA'A', and (IV) AAA'. The
Suomalainen (1950). By apomictic parthenogenetic Roman numerals are used throughout the
reproduction, we mean parthenogenesis in the ab- text as synonyms for these four states. We
sence of meiosis. By automictic reproduction, we
symbolize the frequency of mutation from
mean parthenogenesis employing some form of
A to A' by p and the reverse frequency by
meiosis.
384
JAMES H. ASHER, JR. AND GEORGE W. NACE
3N
OOGONIUM
PRE-MEIOTIC MITOShS
WITHOUT
CYTOKINESIS
6N
CHROMATID
REPLICATION
OOCYTE
FOR
QUADRIVALENT FORMATION
3/8 OF ZYGOIDS SURVIVE
5/8
DIE
1/3/
\2/3
METAPHASE
MEIOSIS I
METAPHASE
AAA (P/24)
A (4P/24)
(P/24)
MEIOSIS E
Y= 2/3
AAA (27P/56O)
(86P/56O)
(27P/56O)
ZYGOID
AAA1 (I-P)
FIG. 2. Meiosis and quadrivalent formation obser\ed in Ambystoma triploids (based on the cytological analyses of Macgregor and Uzzell, 1964).
The genetic consequences of quadrivalent formation
are given assuming that: P is the probability that
a quadrivalent forms; Y is the probability that
(3P/24)
recombination occurs where Y = 0 implies absolute
linkage and Y z= 2/3 implies independent assortment; quadrivalent formation is independent of
chromosomal origin; and /v^oids with deviations
in the 3n kinetochore number are lethal.
GENETICS OF PARTHENOGENETIC POPULATIONS
GENERATION
GENOTYPIC
385
STATES
N
-3-J
N+l
AAA
AAA
FIG. 3. Diagrammatic representation of mutational
changes in triploid apomictic populations. The frequency of mutation from A to A' is given by /i.
The frequency of the reverse mutation is given by
v. Only first order terms are considered in this
model.
v. We also make three assumptions regarding the ancestral triploid Ambystoma
populations: (1) ancestral A. tremblayi
were AAA', (2) ancestral A. platineum
were AA'A', and (3) ancestral forms arose
10,000 years ago (Uzzell, 1964; Uzzell and
Goldblatt, 1967). Finally, we assume that
generations of reproduction do not overlap
and that population size remains constant.
Mutation: Figure 3 is a schematic representation of genetic change in the
triploid populations caused by mutation
alone. This model assumes that P (the
probability of quadrivalent formation given in Fig. 2) is zero, and considers only
first-order mutation effects. Hence, the values fi, v, 2/i, 2v, etc. of Figure 3 represent
the probabilities that an individual in any
genotypic state produces individuals in
other states in the next generation. These
values represent transition probabilities
(probabilities of making the transition
from one state to another) and can be
arranged to form a transition matrix T
(below).
The Roman numeral in the column at
the left of the matrix represents the i th
starting genotypic state at generation n,
while a Roman numeral labeling the column represents the j t h transition state at
generation n -f- 1. The ij th entry in the matrix represents the probability of passing
from the ith state to the j t h state in a single
generation. Hence, the probability that an
AAA (I) individual gives rise to an AAA'
(IV) is 3/x (three times the mutation frequency from A to A') while the probability that an A'A'A' (II) individual gives rise
to an AA'A' (III) is 3i/ (three times the
mutation frequency from A' to A). To
obtain the frequencies of each genotypic
state in the n -|- 1 generation, a row vector
FW whose elements are the genotypic fre-
I
II
I "l - $>X
0
II
0
1 - 3v
T = III
0
M
1
IV
V
0
IV
III
0
3v
3M
0
2v
2M
2v
1 - 2M'- v
386
JAMES H. ASHER, JR. AND GEORGE W. NACE
quencies /("> in generation (n) is postmultiplied by the transition matrix T.
Hence,
0)
where
(2)
and
TABLE
1.
Genotypic equilibria for mutation
fi/v
AAA
Genotypes
AAA'
AA'A'
1
2
5
10
.1250
.0370
.0046
.0008
.3750
.2222
.0694
.0225
.3750
.4444
.3472
.2254
alone*
A'A'A'
.1250
.2963
5787
.7513
* Computed from equations 5 through 8, recall
that the Roman numerals I, II, III, and IV found
in these equations refer to genotypes AAA, A'A'A',
AA'A', and AAA' respectively.
3 represents an independent trials process, then the expected distribution of
genotypic frequencies for this model at
generation n, considering a finite population size N, is given by (/i^n'-)-/ii(n) +
/m ( n > +/iv ( n ) ) N - Hence, the probability
that all individuals in a population will
retain
the ancestral genotype is given by
Tn represents the product of the transition (/iv*n')N f° r t n e triploid A. trimblayi. Some
matrix multiplied by itself n times. The numerical values for this expectation are
process described by Figure 3 and equa- given in Table 2.
tions 1 through 4 represents a Markov
Segregation: Two variables can affect
chain. From the theory of Markovian pro- the proportion of segregants produced by
cesses (Kemeny et ah, 1959), we determine triploid Ambystoma: (1) frequency of
the equilibrium state, F^), for the model synapsis of unlike chromosomes, and (2)
represented by Figure 3 to be:
probability of recombination between the
locus in question and its kinetochore.
1
Co)
Accepting the cytological mechanism
(5)
(Fig. 2) attributed to these triploids, the
1 +3 (£
first variable is a function of quadrivalent
formation. Based on data from Macgregor
(6) and Uzzell (1964), an estimate of the frequency of quadrivalent formation in these
1+
triploids is P = 1 quadrivalent/208 bivalent equivalents (approximately .005). To
(7) obtain an estimate of the proportion of
fin™ =
The values fjW, /„<">, fllIW> a n d fIYW,
are the frequencies of each designated genotype at generation (n). The genotypic frequencies occurring n generations following
some ancestral state 2""<0> are given by:
WWW
1 +M-
1
£)+(")+KfT
TABLE 2. Probability* that all members of a small
apomictic population are of genotype AAA'.
(8)
Thus, given the starting genotypic frequencies and the mutation frequencies, /x and v,
the population structure at each generation is described by equation 4, and the
equilibrium conditions are described by
equations 5 through 8. Some numerical
values obtained from these equations are
given in Table 1. If we assume that Figure
Generation
*
n
1000
2000
3000
4000
5000
.9708
.9431
.9168
.8919
.8682
20
.5527
.3097
.1760
.1014
.0593
Population Size (N)
30
50
100
.4109
.1724
.0739
.0323
.0144
.2271
.0534
.0130
.0033
.0009
.0516
.0029
.0002
.0000
.0000
• Probability that all individuals are AAA'
—
(I AAA') •
** Expected frequency of genotjpe AAA' at generation (n) for any population size where: u. = v —
10-*;/<»!.= l:/«2>, =.3750.
387
GENETICS OF PARTHENOGENETIC POPULATIONS
GENERATION
GENOTYPIC
N
AAA
N+ l
AAA
AAA
STATES
AA'A'
A'A'A'
AAA
A'A'A'
FIG. 4. Diagrammatic representation of the segregational changes in triploid apomictic populations.
P is the probability of quadrivalent formation while
the segregation ratio expected from heterozygous
females (AAA') is g± (AAA): g2 (AAA'): gt
(AA'A').
segregants produced by this mode of reproduction (Fig. 2), we must make three additional assumptions: (1) The frequency of
quadrivalent types is independent of the
ancestral chromosome type (i.e., independent of whether the chromosome was derived from A. laterale or A. jeffersonianum); (2) Kinetochore segregation from
quadrivalents at anaphase of meiosis I is
an independent trials process (i.e., the
probability that a cell gets 0, 1, 2, 3, or 4
kinetochores is binomially distributed);
and (3) Zygoids with an aneuploid number of kinetochores do not survive. From
assumption one, we observe that 2/3 of the
quadrivalents should result in segregation
while 1/3 should not. From assumptions
two and three, we observe that 3/8 of the
eggs with quadrivalents should yield viable
zygoids while 5/8 should yield inviable
ones. Thus, the frequency of quadrivalents
leading to viable segregants and nonsegregants should be 6P/24 and 3P/24 respectively.
and (2) independent assortment of the locus and its kinetochore (y = 2/3). If the
ratio of segregants from a heterozygous female (AAA') is given by gt (AAA) : e»
(AAA') : gl (AA'A') where 2gl + g2 = " l ,
then the value of gr (non-parental segregants) for these two extreme cases of linkage are given by:
Once a synaptic configuration occurs
which can lead to segregation (occurring
with a frequency of 6P/24), the proportion of segregants will depend upon the
probability of recombination between the
locus and its kinetochore. For the purposes
of this discussion, we consider two extreme values of recombination: (1) absolute linkage to the kinetochore (y = 0),
gl
= (P/24)/(l - 5P/8)
fory = 0
(9)
gi = (27P/560)/(l - 5P/8)
for^v = 2/3 (10)
Substituting P =: .005 (the value observed
for triploid Ambystoma) into equations 9
and 10, we find that 2.0 X 10- 4 <gx<2.5 X
10~4. Thus, we expect approximately two
non-jsarental segregants among every 5,000
progeny produced by a heterozygous female.
The genetic change resulting from this
segregation is represented schematically in
Figure 4. As in the case of mutation alone,
the process modeled by Figure 4 is a
Markov chain with a transition matrix Z
given by:
I II III IV
I
0
0
0
0
II
0
0
1
7 =
III
gl
gl
gl
IV
gl
0
The Markov chain described by Z is an
388
JAMES H. ASHER, JR. AND GEORGE W. NACE
absorbing chain, i.e., it is impossible to
leave states I and II. In terms of the
Ambystoma triploids, this means that for
segregation alone, the population should
eventually become completely homozygous
at the ^4-locus. We may ask two questions
regarding this process: (1) What is the
average number of generations to complete
homozygosity, and (2) What is the proportion of each homozygous class observed
at this equilibrium point? Using procedures outlined by Kemeny et al.
(1959), both of these questions may be
answered.
The average number of generations to
complete homozygosity, i.e., fixation, were
obtained for various values of P (probability of quadrivalent formation) and
y (probability of recombination) and
are presented in Table 3. The final frequency of homozygotes depends only upon
the starting (ancestral) genotypic frequencies. For the A. tremblayi population
with the ancestral form of AAA', the final
genotypic proportion is expected to be 2/3
(AAA): 1/3 (A'A'A'). Thus, the time
required to reach this final equilibrium
state depends upon the values of P and y
(Table 3); however, the final state is itself
independent of P and y provided P > 0 .
Mutation and segregation: The two
models presented to this point describe the
independent influences of mutation and
segregation upon the genetic structure of
the triploid Ambystoma populations. To
determine the composite effect of these two
events (mutation and segregation), the
two transition matrices T and Z are multiplied to give a new transition matrix. This
multiplication, TZ, produces a transition
TABLE
3. Generations to fixation for
alone*.
P
y= 0
y = 2/3
.0001
.0005
.0010
.0050
239,984
47,985
23,985
4,785
207,394
41,469
20,728
4,135
P
.0100
.0500
.1000
.5000
segregation
y = o ,i = 2/3
2,385
465
225
33
2,061
402
194
29
matrix for the case of pre-meiotic mutation while the reverse order, ZT, gives a
transition matrix for the case of postmeiotic mutation. For the purposes of this
paper, we will present only the latter, for
which the transition matrix is given by ZT
= M where:
II
0
I
I "l - 3M
0
II
M =
III gi"
IV _ fe
III
0
3»
&
&
1 — 3v
gllx
IV
3M
0
h
where
- 3M - 2v)
& =
v + 4M - 2)
- 2v -
/i + 4» - 2)
- 2n - v
- 2M - 5v)
2v
- 2 v -
2M
5M)
As in the case of mutation alone, this process is a Markov chain with an equilibrium
state, F^i, given by:
feai
-
i
where
+ k)
(gin
3M")
Sample values obtained from equations 11
through 14 are given in Table 4 for this
model which incorporates both post• P — probability of quadrivalent formation; y = meiotic mutation (modeled in Fig. 3) and
probability of recombination between the locus and
segregation (modeled in Fig. 4).
its kinetochore.
GENETICS OF PARTHENOGENETIC POPULATIONS
389
TABLE 4. Genotypic equilibria* for mutation and
segregation.
1/10
1/5
1/2
1
2
5
10
AAA
AAA'
AA'A'
A'A'A'
.8395
.7536
.5913
.4398
.2918
.1455
.0086
.0815
.0873
.0775
.0602
.0409
.0207
.0012
.0456
.0647
.0712
.0602
.0428
.0222
.0014
.0333
.0944
.2600
.4398
.6245
.8116
.9888
""b £i/ni(n'Win,} WH, (17)
(18)
• Equilibria computed from equations 11 through
14 where: p. — 10"5; P = .005; and y = 0.
Mutation, segregation, and selection:
Two additional factors may influence the
genetic structure of any parthenogenetic
population: migration and selection. If we
assume that current triploid populations
arose as a consequence of a single or a few
hybridization events (Fig. 1) and that migration is proportional to the genotypic
frequencies, then migration would have no
influence upon the genetic structure of
this population. Selection, on the other
hand, could play a very important role in
altering the genetic structure. Since selection alters the transition probabilities from
generation to generation, we were unable
to use fully the theories of Markovian processes to answer questions concerning
equilibrium conditions where mutation,
segregation, and selection are acting simultaneously. The transition matrix M previously given may be used, however, to obtain a set of recurrent equations describing
the genetic structure of such a population
where two fitness components are considered: (1) fecundity, and (2) zygoid survival. Multiplying M on the left by a matrix whose principal diagonal elements are
the fitness components with respect to
fecundity and on the right by a matrix
whose principal diagonal elements are the
fitness components with respect to zygoid
survival gives a new matrix S. The recurrent equation which considers post-meiotic
mutation, segregation and selection thus
takes the form:
F<.n+i)
(15)
Expanding this equation produces a set of
simultaneous equations for the frequency
of each genotype:
(19)
The symbol W represents the component
of fitness considered where subscript 1
refers to fecundity while 3 refers to zygoid
survival, and the Roman numerals refer to
the genotypic states as previously defined.
Because of the non-linearity of these
four simultaneous equations, iterative
procedures were used to solve for the
equilibrium values for various values of
the fitnesses considered. Under all conditions investigated, the equilibrium values
appeared to be stable and non-oscillatory.
Sample iterative solutions which describe
triploid populations where segregation, selection, and post-meiotic mutation occur
are presented in Table 5. Analytical solutions for these equilibrium states are at
present being developed using the eigenvalues of S as the proportionality constants
of F(eq)
SccFti).
Summarizing the results of this section,
we observed that in apomictic populations
mutation, segregation, and selection all
play roles in maintaining heterozygosity.
In this process, very small amounts of
segregation appear to dominate mutational forces. In the presence of both mutation
and segregation, intense selective forces
are needed to maintain the ancestral genotypic state.
Experimental Populations
Experimentally
induced
parthenogenesis, or reproduction involving the
manipulation of normal meiotic mechanisms, in amphibians has been used for genetic investigations in two genera: Rana
and Ambystoma. These studies have been
390
JAMES H. ASHER, J R . AND GEORGE W. NACE
TABLE 5. Genotypic
frequencies
for mutation*,
segregation**,
and
selection*
Genotype
Fitness
5000
Equl
Fitness
5000
Equl
Fitness
5000
Equl
AAA
1.000
1.000
1.000
1.000
.456
.210
.164
.170
.440
.060
.060
.440
1.000
.999
.999
.999
.952
.029
.010
.008
.975
.021
.003
.001
.999
1.000
.999
.999
.204
.620
.134
.042
.209
.612
.133
.045
.999
.999
.999
1.000
.109
.052
.054
.786
.001
.003
.021
.975
.000
1.000
.999
.000
.000
.822
.178
.000
.000
.821
.179
.000
.000
1.000
.970
.000
.000
.993
.007
.000
.000
.993
.007
.000
AAA'
AA'A'
A'A'A'
AAA
AAA'
AA'A'
A'A'A'
*n=v=10r>.
* * P = . O O 5 ; y = 0.
*** Differences in fitness are attributed to zygoid survival alone and the frequencies are given
at generations 5000 (Fig. 1) and at equilibrium.
concerned with mapping gene-kinetochore
distances for previously identified loci,
uncovering new mutants at other loci, and
producing homozygous strains (Lindsley et
STAGE
ADULTS
al., 1956; Nace et al., 1970; Volpe, 1970).
The development
off theory needed
l
d d to
predict the genetic consequences of these
meiotic manipulations was presented by
STATES
GENOTYPIC
AA
AA'
A'A'
WAA.
"A'A1,
I '
EGGS
A A'
AA
GENERATION
A'A'
MEIOSIS
N
N
I
K2
{
ZYGOIDS
AA
A ' A>
AA'
\
2M
AA
N+
I
ZYGOIDS
AA
AA'
A'A'
N+ 1
ADULTS
AA
AA'
A'A1
N +1
FIG. 5. Diagrammatic representation of genotypic
change in a diploid automictic parthenogenetic
population uhere segregation, post-mciotic muta-
tion, and selection occurs. Fitness (W) in this
model has two components: (1) fecundity, stage 1,
and (2) z>goid survival, stage 3.
391
GENETICS OF PARTHENOGENETIC POPULATIONS
Nace et al. (1970) and Asher (1970).
When meiosis I is inhibited, heterozygous
(AA') females are expected to produce
three genotypes in the ratio y/i (AA):
1 _ y/z (AA'): y/4 (A'A') where y is the
probability of recombination between the
locus and its kinetochore. In the case of
inhibition of meiosis II, segregation ratios
are (1 — y)/2 (AA): y (AA'): (1 — y)/2
(A'A'). Map distances may be computed
from these ratios with the aid of a mapping function (Barratt et al., 1954; Nace et
al., 1970) which describes the relationship
between y (the probability of recombination) and x, the corrected map distance
between the locus and its kinetochore.
Both mechanisms of reproduction (inhibition of meiosis I or II) should produce
homozygosity at a very rapid rate in the
absence of selection (Nace et al., 1970;
Asher, 1970).
Since these mechanisms of reproduction
have naturally occurring counterparts,
they warrant further discussion. Inhibition
of meiosis I has been shown to be genetically equivalent to central fusion (observed in Drosophila mangabeirai, Murdy
and Carson, 1959) while inhibition of
meiosis II is equivalent to terminal fusion
(also observed in D. mangabeirai, Murdy
and Carson, 1959), or fusion of the second
polar body with the egg nucleus (suggested
for Lacerta saxicola, Darevsky, 1966).
The mathematical models presented by
Nace et al. (1970) and Asher (1970) predict that in these natural automictic parthenogenetic populations as well as in the
experimental populations, in the absence
of selection for heterozygosity, the populations should become genetically uniform.
However, if selection does occur, heterozygosity can be maintained indefinitely. The
models presented by Asher (1970) considered frequent segregation (a consequence of automictic reproduction) and
selection acting only upon zygoid survival.
In this paper, we wish to extend this model in two ways: (1) to consider both fecundity and zygoid survival as components of
fitness, and (2) to include mutation as
another source of genetic change. Figure 5
diagrams the genetic change between generations of automictic parthenogenetic organisms reproducing by either or both inhibition of meiosis I and meiosis II. In this
model, many of whose components are
more completely described in Nace et al.
(1970) and Asher (1970), only a single
locus with two allelic states in diploid populations is considered. The mutation frequencies fx, and v are as previously defined.
If selection, segregation, and first-order,
post-meiotic mutation are considered, the
following non-normalized transition matrix
is derived from the process modeled by
Figure 5:
AA'
AA
B = A'A'\ 0
AA'ly*
where
7 i = (1 72 =
2ix)WAAlWAA,
2nWAAlWAA,z
7 3 = (1
-2v)WA.A,,WA,A.3
74 = 2VWA.A.1WAA,3
7s = (1 2ti)K1WAA,lWAA,
+ vKtWAA-JVAA,
76 = (1 +
2»)K1WAA'JVA>A>,
IIKJVAA'JVA'A^
77 = 20* + ti
(
VIWAA'I
K2 = Ep + EtQ. -y/2)
Ei = the probability of inhibition of meiosis
II or terminal fusion
£2 = the probability of inhibition of meiosis I
or central fusion
E2 = 1 - Ei
y = the probability of recombination between the gene and its kinetochore
392
JAMES H. ASHER, JR. AND GEORGE W. NACE
and the number subscripts on the selection
terms indicate the stage in the life cycle at
which selection occurs (Fig. 5).
An understanding of this complete model can be attained most directly by simplifications that consider first, mutation and
segregation alone, and then segregation
and selection. Thus, the joint influence of
segregation and post-meiotic mutation
upon genetic change in these diploid automictic populations, is determined by setting the fitness components all equal to
one thus reducing matrix B to:
AA
AA'
A'A'
0
AAr 1 - 2
= A'A'l
0
AA'\_
: j.
2M
2,
X
•]
where
Xi =
(1 -
2M)A"I +
vKt
X2 = (1 - 2v)K~i + M/G
X3 ^ 1 — Xi — X2
Again, t h e process is M a r k o v i a n a n d F(ei">
is given by:
JAA
—
\*V)
n
(21)
2v
1
(22)
AI
271
+
2v
+ l
Assuming that //. — v, equation 22 reduces
to:
1
(23)
+2
The maximum value thai equilibrium
heterozygosity may attain for equation 22
occurs when K1 = 0. This result obtains
when reproduction is by central fusion (£2
:= 1) and when the gene is absolutely
linked to the kinetochore (y = 0). Under
these conditions, / / i./ ( c q ) " 1/2. Rephrased,
this states that in the case of reproduction
by inhibition of meiosis I, the frequency of
heterozygotes at equilibrium for genes absolutely linked to the kinetochore will be
1/2 provided all genotypes are equally fit
and the frequencies of mutation from one
allelic state to another are equal.
If we assume that the smallest value that
y actually attains is y = .001, which is the
probability of recombination between adjacent nucleotides (Guest and Yanofsky,
1966), we may obtain another estimate of
heterozygosity for this mode of reproduction. With E2 = 1 and y = .001, Kx =
.00025. If we further assume that ^ = v =
IO-5, then /4/(«i> = .037.
The two values calculated above for
fAA(e(l) (-500 and .037) represent a wide
range, but they do not indicate the extremes of the heterozygosity maintained by
a balance of mutation and segregation in
populations reproducing by central fusion
or by inhibition of meiosis I. This is appreciated by recalling that the value of
/^/< e i) depends upon the value of K±. For
reproduction by central fusion (E2 = 1),
O^K^l/6
where Kx = y/4 (y is the
probability of recombination between the
gene and kinetochore). Substituting these
extreme values into equation 23, 6/A <
/^/(eq) ^ 1/2. The frequency of equilibrium heterozygotes decreases rapidly,
however, as y increases from zero. This
rate of decrease can be observed by substituting x/t = .Ki into equation 23. Hence,
f.i/^il = l / ( x + 2). For a gene located 2
map units from the kinetochore, y ss .04
(Nace et al, 1970) and K = .01. If /x =
10- 5 , x = 1000 and fAA' =3 10- 3 . Thus
populations reproducing by means of central fusion should be homozygous for all
loci having neutral allelic variation except
for those loci which are very closely
linked to the kinetochore. These results
thus verify and quantify the prediction
made by Carson (1967) that heterozygosity
can be retained in automictic parthenogenetic populations reproducing by central
tusion or inhibition of meiosis I.
The limits of heterozygosity maintained
in populations reproducing by terminal fu-
393
GENETICS OF PARTHENOGENETIC POPULATIONS
sion (£i = 1) or inhibition of meiosis II
can be obtained in a similar manner. For
this mode of reproduction,
l/Q^K^l/2
where K1 =: (1 - y)/2. Substituting these
extreme values into equation 23 gives
2/A</AA'(eq)<6/x. Hence, these automictic
populations are expected to be homozygous for all neutral alleles regardless of
linkage relationships.
These results indicate that neutral genetic variability, i.e., heterozygosity, though
not retained in automictic parthenogenetic
populations reproducing by terminal fusion, is retained by central fusion; however, since equilibrium heterozygosity decreases rapidly as y increases, this variability is restricted to an area very closely
linked to the kinetochore.
Turning now to the joint influence of
segregation and selection, we obtain from
the transition matrix B a set of recurrent
equations which take these two factors into
consideration by setting ^ = v = 0.
These recurrent equations are given by:
WAMWAM
= the fitness of the AA genotype
= the fitness of the AA' genotype
^ A J A A ,
= the fitness of the A'A' genotype
In order for this equation to give a nonzero equilibrium, the following inequalities must be true:
JVAA', >
WA'A-xWA'A't
Since 0^K2^l,
non-zero equilibria can
exist only if the fitness of the heterozygote
is greater than the fitness of either homozygote. To determine the conditions for the
maximum value of the frequency of
heterozygosity with respect to genotypic
fitness, the following partial differential
equation was solved:
(28)
= 0
+ /AA'
M
WAA'.K,} WAA-JK
(24)
WA.A.JK
(25)
WAA^/K
(26)
where
where WK represents the fitness of genotype X. Substituting equation 27 into equation 28, we obtain:
_r
dWx'XIK.WAA^
-
WAAlWAA
"1 __ Q
w.AA'J^A'A's
dWxlKiWAA'iWAA', WA.A^WA-A',}
K=
This set of equations can be solved for the
equilibrium conditions where heterozygosity (/j/ (e(l) ) is maintained. The derivation
is parallel to that presented by Hayman
(1953), Workman and Jain (1966) and
Asher (1970) and yields the equation:
Solving equation 29 with respect to the
various fitness components where WAAl =
and WAM = k,WA.A.a
(kt and
k2 being arbitrary constants), we find that
/A1,(eq) c a n have a maximum only when
the fitness of both homozygotes is equiva(27)
(eq) — .
-
where
]
WAA3
WAAl
AAlW
+
+1
394
JAMES H. ASHER, JR. AND GEORGE W.
NACE
TABLE 6. Summary of equilibrium conditions for diploid parthenogenetic reproduction by either
or both central and terminal fusion.
Genotype
Fitness
Component
Fecundity
W
Zygoid Survival
W
IV
n
AA\
W
AAa
^AAwAAi
Fitness
/AA,(eq> > o ->
AA\
W
AA'
1A'3 > WAA
If WAA — kJF A ,, and W , , =
•A\
W
A'A'
W
AA>
WAA and
w
w
A'A\
A'A'a
W,A'S
A A3
1
fAA,(°'l) to be a maximum
T h e n for
V I
A'A'
AA'
AA
AAr
A
(b) k 1 = l / k 2
lent or when the ratio of fitness components of the homozygotes at one stage is
equal to the reciprocal of the fitness components for the homozygotes at another
stage (i.e., kx = 1/^s) • Table 6 presents a
summary of these statements. Numerical
examples for au tomictic parthenogenetic
reproduction are presented in Nace et al.
(1970) and Asher (1970) and are not repeated here.
Evaluation of the complete model indicated by matrix B which considers segregation, selection, and post-meiotic mutation
will not be presented at this time. However, the comparisons of mutation with
segregation, and segregation with selection
made in this presentation suggest that
segregation and selection have the greatest
influence upon the genetic structure of automictic parthenogenetic populations with
respect to maintenance of heterozygosity.
Mutation can affect this structure; however, this influence is restricted to loci closely linked to the kinetochore.
four factors can influence this structure:
(1) mutation, (2) segregation, (3) selection, and (4) migration. Among these factors, mutation, segregation, and selection
should play major roles. In addition, these
factors may have a directional, or deterministic, component as well as a nondirectional, or stochastic, component
which must be distinguished when evaluating models. Hence, we discuss briefly the
significance of the stochastic component
when considering the triploid Ambystoma
population.
If, for parthenogenetic species that have
had a single origin, we assume that migration into and out of the population is
proportional to the frequencies of each
genotype, migration should play no role in
altering their genotypic structure. If, on
the other hand, migration is not proportional to genotypic frequency, then, migration becomes a component of selection.
Thus, the influence of migration is either
considered in the model, i.e., as selection,
or it is of no consequence.
DISCUSSION
One intent of this paper is to present
mathematical models which describe the
genetic structure of apomictic and automictir parthenogenetic populations, \t least
Mutation
The influence of mutation upon genetic
structure is markedl) different ior apomicfic and. automictic populations, Dof.crmin-
GENETICS OF PARTHENOGENETIC POPULATIONS
isLic predictions state that in the absence of
segregation and selection mutation should
lead to polymorphisms of all loci in a
triploid apomictic population (See equations 10 through 13, and. Table 1). Stochastic predictions, while confirming this
conclusion, add the prediction that this
polymorphic state will not necessarily be
the ancestral state (Table 2) even in cases
of small population size (N ^ 15 to 250)
as observed for the triploid Ambystoma
(Wilbur, 1971). This deviation from the
ancestral state is equally true for so-called
"neutral" variability. Evidence presented
by Uzzell and Goldblatt (1967) suggest to
the contrary that serum protein genotypes
(AAA' and A A'A') postulated for the two
ancestral triploids have been maintained
in the "modern" triploids. The absence of
this divergence from the ancestral genotype with respect to a serum protein and
various enzymes (Uzzell, personal communication) suggests that, while mutations
probably do occur, other factors with homeostatic influence act upon and dominate
the composition of the genome of the
triploids. Hence, the protein phenotypes of
these triploids are not neutral.
Within automictic populations, heterozygosity is maintained by mutation at
loci closely linked to the kinetochore in
populations reproducing by central fusion
or inhibition of meiosis I. The frequency
of equilibrium heterozygosity decreases
rapidly, however, as y increases from zero.
These results support and quantify the
predictions made by Carson (1967). Considerations of equilibrium heterozygosity in
populations reproducing by terminal fusion or inhibition of meiosis II, on the
other hand, indicate that these automictic
populations should be completely homozygous for all loci regardless of linkage relationships.
These observations lead to the general
conclusion that mutation plays a relatively
small role in maintaining heterozygosity
in automictic species and, therefore, that
selection is absolutely essential to the retention of this variability. In the absence
of selection, very few loci should exist in a
polymorphic state.
395
Segregation
Within apomictic species, segregation occurs infrequently because of a lack of
synapsis and genetic exchange. With a
relatively small probability of synapsis,
apomictic populations, in the absence of
mutation and selection, should become homozygous (Table 3). Using values of the
probability of synapsis (.005) suggested by
the data of Macgregor and Uzzell (1964),
triploid Ambystoma should become completely homozygous within 5,000 generations (Table 3). By contrast, complete
homozygosity should be produced within
30 generations for many loci in automictic
populations (Asher, 1970). Thus, any
mode of parthenogenetic reproduction
which allows even small proportions of auto-segregation should, in the absence of
mutation or selection, become completely
homozygous.
The importance of small amounts of auto-segregation in apomictic species was
suggested both by Gustafsson (1942) and
Suomalainen (1961). The data presented
in Table 3 quantify the importance of this
small amount of segregation and contradicts the predictions made by Darlington
(1937), White (1948), and others that this
mode of reproduction must lead to complete heterozygosity.
Mutation, Segregation, and Selection
Values for equilibrium states predicted
by models which incorporate the simultaneous occurrence of mutation and segregation for triploid apomictic populations
are presented in Table 4. The equilibrium
states are markedly different from those
expected with mutation alone. (Compare
the values in Tables 1 and 4, and the
upper left values of Table 5). Since the
data of Uzzell and Goldblatt (1967) which
indicate that triploid Ambystoma maintain their ancestral heterozygosity contradict the predictions made in Table 4 and
the case of equal fitness in Table 5, two
alternative hypotheses are available: (1)
segregation and mutation are not affecting
the genome, or (2) selection is maintaining the constancy of the genome.
396
JAMES H. ASHER, JR. AND GEORGE W. NACE
Table 5 illustrates the influence of selec- obtained, in part, for Ambystoma triploids.
tion upon the genetic structure of triploid Using this information and the models
apomiclic populations where segregation, presented in this paper, we have remutation and selection occur simultane- examined predictions made with respect to
ously. These data suggest that for triploid the role of parthenogenetic reproduction
Ambystoma where both segregation and in the evolution of these species.
mutation are expected to be rare events
It has been stated that species employing
heavy selective pressures are required to the parthenogenetic mode of reproducretain the ancestral genotype through tion represent evolutionary "dead ends,"
time. In this case, it appears that the de- i.e., the probability of parthenogenetic spependence of the triploids, e.g. A. plat- cies becoming extinct is one, while for
ineum, upon the diploids, e.g. A. jefjer- bisexual species this probability is less than
sonianum, for egg activation has allowed one. This evolutionary prediction was
little or no genetic deviation from the based upon two general types of speculatriploid ancestral genome.
tion. First, parthenogenetic populations
must
lack genetic variability: i.e., apomicSince mutation plays such a small role in
maintaining heterozygosity in automictic tic species become completely heterozygous
populations, we have only considered the while automictic species become comjoint effects of selection with segregation in pletely homozygous (Darlington, 1937;
this paper. Equation 27 represents the White, 1948; Suomalainen, 1950; and othequilibrium state for this mode of repro- ers). Second, newly arising benefical mutaduction and represents a generalization of tions cannot be incorporated into the
equation 8 presented by Asher (1970). genomes of asexual species as rapidly as
The conditions for maintaining stable they can in bisexual species (Muller, 1932,
non-zero polymorphisms in these popula- 1964; Crow and Kimura, 1965). The
tions are given in Table 6. In this case, the present paper deals specifically with the
fitness of the heterozygote must be greater first set of speculations. Although the modthan either homozygote.
els are severely limited in their predictive
power because they are restricted to onelocus, two-allele systems, they do provide a
Evolutionary Considerations
series of illuminating predictions with reOne of the goals of population biology is spect to genetic variability in parthenogento develop theories which describe and etic populations.
predict evolutionary change. The miniBased upon these models, it appears that
mum information needed to make such previous speculations made about genetic
predictions is: (1) the mechanism of variability in parthenogenetic populations
reproduction, (2) the number of gener- are not correct. First, apomictic populaations of reproduction, (3) the initial tions should not become completely
composition of the genome, and (4) the heterozygous as a consequence of mutation
fitness of a particular genomic configura- alone but should attain some equilibrium
tion at each generation. The function of state in which there exist both homozypopulation genetics is to develop models gotes and heterozygotes. Selection and small
which use this information to predict the amounts of segregation should drastically
change in the genome from an initial alter this equilibrium state. Second, autostate to some future state, thus permitting mictic populations need not be completely
an understanding of both the past and fu- homozygous provided selection favors hetture evolution of a species and providing eio/.)gOiity. The magnitude of the selecthe data base needed to anticipate the tion which is needed is rigorously determanner in which imposed changes may mined by the mode ot automictic reproinfluence this evolution. It is recognized duction and linkage relationships (Table
that acquisition of this information is a 6, see also Asher, 1970). These two predicmajor undertaking; however, it has been tions thus represent a new starting point
GENETICS OF PARTHENOGENETIC POPULATIONS
for evaluating evolutionary change in parthenogenetic populations.
While analyzing the models developed
to describe apomictic reproduction, we
were led to the conclusion that apomictic
species should not retain their ancestral
genotype. Evidence obtained by Uzzell and
Goldblatt (1967) and Uzzell (personal
communication) indicate that triploid parthenogenetic Ambystoma have retained
this ancestral genotype for several proteins. This contradiction suggests that the
protein phenotype in these parthenogenetic species is not "neutral" and further suggests a technique for answering the question posed by Lewontin and Hubby
(1966) and others: How much of the genetic variability observed in natural populations is evolutionarily relevant?
We are left, at this point, with the conclusion that parthenogenetic populations
need not necessarily represent evolutionary
"dead ends" because of considerations of
the amount of genetic variability that they
may contain. It should be very informative to determine tKs extent of genetic
variability actually existing in these populations.
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