Polyploid incidence and evolution.

Polyploid incidence and evolution.
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Annu. Rev. Genet. 2000. 34:401–37
c 2000 by Annual Reviews. All rights reserved
Copyright POLYPLOID INCIDENCE AND EVOLUTION
Sarah P Otto1 and Jeannette Whitton2
1Department
of Zoology and 2Department of Botany,
University of British Columbia, Vancouver BC V6T 1Z4 Canada;
e-mail: [email protected]; [email protected]
Key Words triploid, tetraploid, animals, plants, gene duplication, rate of evolution
■ Abstract Changes in ploidy occurred early in the diversification of some animal
and plant lineages and represent an ongoing phenomenon in others. While the prevalence of polyploid lineages indicates that this phenomenon is a common and successful
evolutionary transition, whether polyploidization itself has a significant effect on patterns and rates of diversification remains an open question. Here we review evidence for
the creative role of polyploidy in evolution. We present new estimates for the incidence
of polyploidy in ferns and flowering plants based on a simple model describing transitions between odd and even base chromosome numbers. These new estimates indicate
that ploidy changes may represent from 2 to 4% of speciation events in flowering plants
and 7% in ferns. Speciation via polyploidy is likely to be one of the more predominant
modes of sympatric speciation in plants, owing to its potentially broad-scale effects on
gene regulation and developmental processes, effects that can produce immediate shifts
in morphology, breeding system, and ecological tolerances. Theoretical models support the potential for increased adaptability in polyploid lineages. The evidence suggests that polyploidization can produce shifts in genetic systems and phenotypes that
have the potential to result in increased evolutionary diversification, yet conclusive
evidence that polyploidy has changed rates and patterns of diversification remains
elusive.
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
INCIDENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incidence in Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incidence in Animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phenotypic Effects of Polyploidy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Genetic Consequences of Polyploidization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maintenance of Duplicate Genes and Divergence in Gene Function . . . . . . . . . . .
DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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INTRODUCTION
...polyploidy has contributed little to progressive evolution.
Stebbins (116, p. 132)
...polyploidy, far from playing a secondary role in evolution, has provided
the additional, uncommitted gene loci necessary for major steps in the
evolution of animals.
Schultz (104)
Polyploidy is widespread in plants, with an estimated frequency between 30 and
80% (79), and occurs sporadically among animals (69). The evolutionary significance of polyploidy remains a mystery, however. As illustrated by the above quotes,
two diametrically opposing views exist, one assigning polyploidy a marginal role
in evolution and the other granting it a primary creative role. On the one hand, polyploids may represent a relatively frequent class of mutation, one that occasionally
establishes within populations when its phenotypic effects are relatively mild. Simply put, polyploidy may be widespread because it arises repeatedly, without playing
a significant role in evolution. Conversely, polyploidy may be common because
polyploid species evolve faster or in more novel directions than related diploid
species. Under this view, polyploidy promotes adaptive evolutionary change.
Understanding the evolutionary role of polyploidy provides insight into one
of the most fundamental questions in evolutionary biology: To what extent is
the pace of phenotypic evolution set by the structure and content of the genome?
The tempo of evolution may be determined largely by the rate of environmental
change, spurred by changes in the abiotic or biotic context in which a species
lives, but slowed during periods of stasis (see Reference 67 for an experimental
demonstration of the importance of the environment on the rate of evolution in
Escherichia coli). Alternatively, the tempo may be set internally and may depend
strongly on genomic structure and content. The rate of evolutionary change in a
trait not only depends on the strength of selection but also on the form and extent
of genetic variability present within a population (39). Thus, evolutionary change
could be constrained by the number and type of genes present. If this constraint
is substantial, then doubling the number of genes within a genome could have
profound long-term effects on genetic variability and on evolution. If, on the other
hand, genetic constraints on evolutionary change are minor, polyploidization may
make little difference to the tempo of evolution.
In this review, we address the fundamental question of whether polyploidy
has played a secondary or an essentially creative role in evolution. We begin by
reexamining the incidence of polyploidy in plants and animals. We then review
the common phenotypic correlates of polyploids and their immediate selective
effects. Third, we address the population genetic consequences of polyploidy on
the mutation load and the rate of adaptation. Finally, we review recent genomic
studies that shed light on past polyploidization events and their influence on evolution. Although our conclusions are by necessity tentative, evidence suggests that
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the tempo and mode of evolution may have been extensively influenced by the
occurrence of polyploidy.
Throughout this paper, we adopt the terminology and conventions employed by
Ramsey & Schemske (99). We use x to denote the base chromosome number of
a lineage and 2n to refer to the chromosome number in somatic tissue, regardless
of whether an individual is diploid (2n = 2x), triploid (2n = 3x), etc. Thus, n
represents the gametic or haploid chromosome number expected after meiosis
in a sexually reproducing organism. We use the term allopolyploidy to refer to
polyploids of hybrid origin, consistent with its first use by Kihara & Ono (59).
By contrast, autopolyploids are formed by the coming together of entire genomes
from within a species. It is often assumed that polyploids that form bivalents during
meiosis (i.e. exhibit disomic segregation) are allopolyploids, whereas those that
form multivalents during meiosis are autopolyploids. These rules are often broken,
however (99). In particular, polyploids with tetrasomic segregation (pairing of
four homologous chromosomes during meiosis) tend to rediploidize over time as
mutations accumulate and chromosomes diverge. Furthermore, autopolyploids in
groups with small chromosomes or low chiasma frequencies may exhibit disomic
inheritance immediately after their formation (115). Therefore, it is important to
keep the mode of origin of polyploidy and the mode of chromosomal segregation
distinct.
Background
Our review aims to uncover the long-term role of polyploidy in evolution. Recent
reviews have focused solely on plants and emphasized shorter-term questions. In
particular, Ramsey & Schemske (99) examined the rate of formation of polyploid
plants, and Soltis & Soltis (110, 111) reviewed data on the tendency of polyploid
taxa to descend from multiple polyploidization events. Here, we briefly review
their findings concerning how often and in what manner polyploid populations
arise.
Polyploidy represents a special class of mutation and can occur via several
routes: genomic doubling, gametic nonreduction, and polyspermy. Genomic doubling occurs in both animals and plants and involves a failure of cell division
following mitotic doubling. Similarly, gametic nonreduction involves a failure of
cell division during meiosis. Unreduced eggs are a common route to polyploidy
in both animals and plants, as is unreduced pollen in plants. Unreduced sperm,
however, appear to play a minor role in polyploidization in animals, perhaps
because diploid sperm are rarely successful in competition with reduced sperm
(see Reference 3 for an exception). Polyspermy is also known in plants and animals and is, for example, the most common mechanism leading to human triploids
(119), which comprise ∼1–3% of conceptions (80). Ramsey & Schemske (99)
estimate that autotetraploid plants are formed at a rate on the order of 10−5 per
individual per generation. Allotetraploids are formed at similar rates only if hybrid
matings represent at least 2% of matings in outcrossing groups (99). Recent data
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(reviewed in 118) confirm that autopolyploidy may be a more common
phenomenon in plants than had once been thought (50, 114).
One of the biggest stumbling blocks to the successful establishment of polyploidy in sexual species is the requirement for a genetically compatible mate.
Perhaps because of this obstacle, successful polyploid establishment appears to
be facilitated by selfing, asexuality, and perenniality (12, 99, 113). For example,
polyploidy is more often found among perennial plants (113) and insects (71),
probably because having a long life span increases the chances that rare events
will occur (e.g. polyploidization following hybridization), and allows for mating between polyploids and their offspring. In predominantly outcrossing taxa,
an obstacle facing newly formed tetraploids is that they often mate with diploid
relatives, producing triploids. Triploidy has generally been thought to be an evolutionary dead-end, because triploids have very low fertility and tend to produce
aneuploid gametes, owing to problems of chromosomal pairing and segregation
during meiosis. Nevertheless, triploids do produce euploid (haploid, diploid, or
triploid) gametes at a low rate (99, 104). These euploid gametes can then lead
to the production of triploid and tetraploid offspring. Consequently, triploids
may actually facilitate the transition from diploidy to tetraploidy by allowing the
propagation of polyploid lineages, albeit at a low rate (54a, 99).
A potentially important observation is that the rate of polyploid formation in
plants and animals varies greatly with environmental circumstances [especially
temperature (15, 35, 99)], parental genotype (13, 99), and parental origin [being
substantially higher in hybrids than nonhybrids (54a, 62, 99)]. This rate variation
may be critical to the successful establishment of polyploids in that, occasionally,
circumstances may conspire to produce several polyploids within the same generation. A sudden freeze during egg development, the drift to high frequency of a
strain tending to produce unreduced gametes, or an unusually high rate of hybrid
formation (e.g. in hybrid zones) may generate multiple polyploids that, if there
is any tendency towards reproductive isolation from the diploid progenitor, may
successfully establish a new polyploid lineage.
The formation of a polyploid lineage from multiple individuals has important
ramifications. Genetic variation would be drastically reduced within polyploids
formed from a single ancestor, reducing the ability of a polyploid lineage to adapt
to its environment and to keep apace of its genetically diverse diploid progenitor.
Interestingly, recent studies have suggested that polyploid populations more often
have multiple origins than single origins. The plant literature on multiple polyploid
origins is reviewed in (110, 111). In Table 1 (available at www.annualreviews.org
Supplementary Materials), we review the incidence of polyploidy in insects and
vertebrates; nine of these polyploid taxa are thought to have multiple origins compared with one considered to have arisen via a single origin. Other invertebrate
polyploids with multiple origins are discussed in (25, 33). Multiple genetic origins can be explained either by a high rate of polyploidization during the initial
establishment of the lineage (as suggested by the previous paragraph) and/or by
on-going gene flow with related diploids. In either case, genetic diversity within
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the newly formed polyploid population will be substantially affected. Indeed,
studies examining genetic diversity have found that genetic diversity in polyploids
is often similar to or higher than related diploids (19, 63, 110).
INCIDENCE
Incidence in Plants
Many familiar crop species, including wheat, maize, sugar cane, coffee, cotton,
and tobacco, are polyploid, either through intentional hybridization and selective
breeding (e.g. some blueberry cultivars) or as a result of a more ancient polyploidization event [e.g. maize ∼11 MYA (44)]. Polyploidy is also thought to be
the mechanism by which certain species have evolved particularly high chromosome numbers, including the stonecrop, Sedum suaveolens [with 2n = 640 ≈ 80x,
the highest known chromosome number among angiosperms (120)] and the fern
Ophioglossum pycnostichum [with 2n = 1260 ≈ 84x, the highest among plants
(73)].
Detecting polyploidization events can be enormously challenging. Commonly,
polyploidy is inferred when chromosome numbers among closely related species
follow a polyploid series (e.g. 2n = 16, 32, 64). Less often, polyploidy is demonstrated by the observation of multivalents during meiosis. Both of these methods,
however, are better able to detect recent polyploidization events than ancient ones.
Ancient polyploidization events are especially difficult to detect because time
erases the signals of duplication; disomic segregation reestablishes, rearrangements scramble chromosomal synteny, and differentiation (or loss) obscures gene
copies. Soltis & Soltis (110) have coined this problem the “paradox of highly
diploidized polyploid populations.” Over the past few years, extensive genomic
analyses have confirmed polyploidization events in ancestors to maize (2), yeast
(131), and Xenopus (57). Even genomic analyses can be ambiguous, however. For
example, Ohno’s (87) hypothesis that two rounds of genome duplication occurred
early in the evolution of vertebrates has received both supporting (4, 47, 98) and
contradictory (77, 109) evidence from comparative genomic analyses.
To assess the evolutionary significance of polyploidization, it is critical to move
beyond identifying particular cases and towards estimating the overall incidence
of polyploidy. Previous estimates for the extent of polyploidy in angiosperms vary
widely: 20–40% (113), 57% (49), and 70% (48, 79). These estimates measure different quantities, and each suffers from a large degree of extrapolation. Stebbins’
(113) estimate was based on the percentage of polyploid species within a genus
that had “haploid numbers that were multiples or near multiples of the lowest one
found in the genus,” which estimates the fraction of species with polyploidy in
their history, within the time frame of generic evolution. Grant’s (49) estimate was
based on the assumption that the ancestral number of chromosomes in angiosperms
was 7–9 and that any flowering plant with n ≥ 14 chromosomes had undergone
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polyploidization at some point during angiosperm evolution. Goldblatt (48) and
Masterson (79) argued that Grant’s threshold was too conservative and that n ≥ 11
was more appropriate, from which they inferred that 70% of angiosperm species
are polyploid. These methods are crude, however, and can only tell us whether
polyploidy has occurred at some point in the past. Groups that have polyploidized
once and those that have done so repeatedly are treated equally. Using Stebbins’
method, for example, a polyploid index of 80% would be inferred both for a genus
containing one species with n = 12 and four species with n = 24 and also for a
genus containing five species with haploid chromosome numbers n = 12, 24, 48,
96, and 192, respectively; clearly, however, more polyploidization has occurred
within the second group. Grant’s threshold method would also estimate the degree
of polyploidy as 80% in both of these genera, whereas Goldblatt’s and Masterson’s
threshold method would identify 100% of these species as polyploids. In addition, the threshold methods suffer from the fact that chromosome numbers will
have increased above the threshold in some nonpolyploid taxa by chromosomal
fission and will have decreased below the threshold in some polyploid taxa by
chromosomal loss and fusion [e.g. Zea mays, which has n = 10 but is known to
be tetraploid (82)]. Estimating the fraction of species that have undergone genome
duplication at some point in their evolutionary history has its value, but it provides
no clear insight into whether polyploidization has been a rare or frequent event
during evolution.
Here we develop a new method to estimate the incidence of polyploidy, which
utilizes the distribution of haploid chromosome numbers (Figure 1). A remarkable feature of this distribution is the large excess of even over odd haploid numbers. This saw-toothed pattern is difficult to explain by any mechanism other than
frequent polyploidization. This signature of polyploidy arises because the haploid number is always even following genomic doubling, either within a species
(autotetraploidy) or following hybridization between species with the same number of chromosomes (monobasic allotetraploidy). Other processes, including chromosome fission, fusion, gain, or loss, tend instead to move species between the
categories of even and odd chromosome numbers (Figure 2). Indeed, we can use
this saw-toothed signature to provide a simple index of the incidence of polyploidy.
If ρ is the rate of polyploidization per unit time, and χ is the rate by which haploid
chromosome numbers increase or decrease by one (“dysploidy”), then the fraction
of species ( f ) that have an even haploid chromosome number obeys the differential
equation:
df
= −χ f + (ρ + χ)(1 − f ).
1.
dt
Over time, the fraction of species with an even n will approach the steady-state
value, f = (ρ + χ)/(ρ + 2χ). Consequently, the fraction of all karyotypic
changes that involve genomic doubling can be estimated by:
PI =
f − (1 − f )
# evens − # odds
ρ
=
=
,
ρ+χ
f
# evens
2.
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Figure 1 Haploid chromosome numbers in ferns. The distribution of haploid chromosome
numbers in Pteridophyta (Filicopsida) was compiled from Löve et al (73) by J Tzenova.
Multiple chromosome numbers within species were included separately; circa or unreliable
counts were excluded; noninteger haploid numbers (e.g. triploids) were also excluded.
which we call the polyploid index. The polyploid index will be most accurate
when there is a low degree of phylogenetic inertia in chromosome number and
when long evolutionary time frames are considered. Misleading estimates will
be obtained when many species share a particular haploid chromosome number
because they are closely related and few karyotypic changes have occurred. To
guard against this problem, we calculate the polyploid index only for large taxonomic groups that have a broad array of haploid chromosome numbers with
the most common chromosome number (the modal number) represented by at
most 15% of species. Results for ferns, monocots, dicots, crustacean decapods,
cockroaches, and mammals are provided in Table 2. Under the null hypothesis
that polyploidization does not occur, one can determine the significance of the
polyploid index from the binomial distribution with an expectation of 50% even
and 50% odd chromosome numbers (PI = 0). This calculation assumes, however, that the data points are independent, whereas many sibling species share a
common chromosome number. Consequently, the number of independent data
points will be fewer than the number of species. To address this issue, we calculate an index of support, which indicates the fraction of the data that must
be independent for the polyploid index to remain significantly different from
zero. When this fraction is low, there is strong support for a non-zero polyploid
index.
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Figure 2 A model describing transitions among even and odd chromosome numbers. The polyploid index (PI ) is based on the steady-state
expectation of this model.
The ferns are well known for having a high degree of polyploidy (127). The
ancestral or basal chromosome number within ferns is disputed but is thought to
lie within the range of 7 to 13 (127). Using the threshold method to estimate levels
of polyploidy (i.e. the proportion of taxa with n ≥ 14), 99.7% of fern species
listed in Löve et al (73) have polyploidy in their history. Given that most ferns
have high chromosome numbers relative to angiosperms, the relative importance
of polyploidy early in the evolution of ferns (“paleopolyploidy”) versus more recent events (“neopolyploidy”) has been debated (127). If polyploidization has been
rare after the initial radiation of ferns, we would expect a low polyploid index. In
TABLE 2
Polyploid indices (PI ) for various groups of plants and animals
Taxa
Ferns
Monocots
Dicots
# Speciesa
1729
# Even
Modeb
PIc
p [support]d
Reference
1092
14.2% [41]
41.7%
10−28 [3.9%]
73
4988
2963
12% [7]
31.7%
10−40
[2.5%]
50
11742
6441
9.8% [9]
17.7%
10−25 [3.7%]
50
11.3% [8,9]
26.3%
10−28
52
Herbaceous
5287
3043
Woody
[3.5%]
2665
1318
21.5% [13]
−2.2%
0.586
52
Decapod
crustaceans
38
24
10.5% [53]
41.7%
0.143
130
Cockroaches
107
52
14.0% [19]
−5.8%
0.847
27
Mammals
924
442
8.9% [19]
−9.0%
0.188
130
a
Number of species with even and odd haploid chromosome numbers were obtained from tables or figures in the references (see
Figure 1 for information on ferns).
b
c
Mode describes the percentage of species counts with the modal chromosome number (indicated in brackets).
The PI index is based on equation (2).
d
p is the exact binomial probability of getting even:odd counts as far or farther from 1:1 as observed (two-tailed test). The index
of support is the percentage of the data that is required to maintain a significant result ( p < 0.05), which indicates how sensitive
a result is to the assumption that chromosome counts are independent.
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contrast, ferns have a high polyploid index (41.7%), meaning that nearly half of
recent changes in haploid chromosome number have occurred via polyploidization. To calibrate this measure per speciation event, we estimated the fraction of
speciation events in ferns associated with a change in chromosome number, by
totaling
P the minimum number of chromosome changes found within each genus,
one), and dividing this by the total
i.e. genera (# unique n values per genus minusP
number of speciation events represented, i.e.
genera (# species per genus minus
one). We ignored genera with a chromosome count from only one species. This
procedure indicated that ≈16% of recent speciation events in ferns involved a
change in chromosome number. Because chromosome numbers are known from
only ≈15% of fern species (73), this estimate is necessarily coarse. Furthermore,
there will generally be more chromosome changes and speciation events in the
history of these taxa than estimated above. Keeping in mind our uncertainty, we
estimate that ∼7% (= 41.7% × 16%) of speciation events involve polyploidization in ferns. We conclude that polyploidy is an active and on-going process in
ferns and one that is involved in a substantial fraction of speciation events.
Polyploidy is also considered to be an important on-going process among angiosperms (48, 49, 79, 113). Consistent with this view, the polyploid index is significantly greater than zero in both monocots and dicots. Interestingly, there is no
evidence that recent chromosomal changes in woody dicots involve polyploidization. Instead, the high polyploid index of dicots is entirely due to a high rate
of polyploidization among herbaceous dicots. Why there is such a low rate of
polyploidization in woody angiosperms is not well understood, but one possibility is that the proper formation of wood fibers by the vascular cambium may
constrain cell size changes and hence constrain changes in ploidy level among
woody species (113). Similar developmental constraints may explain the exceedingly low representation of polyploidy among gymnosperms (31). To estimate
the rate of polyploidization per speciation event in angiosperms, we again calculated the fraction of speciation events associated with a change in chromosome
number. We used data from plant families starting with the letters A, F, L, R,
and X in Federov (36), focusing only on genera whose chromosome numbers
were summarized in table format. To be conservative, we excluded species that
were polymorphic for chromosome number. There were at least 987 chromosomal
shifts in 8884 speciation events, implying that the rate of change of chromosome
number is ∼11% per speciation event. Multiplying this by the polyploid index,
we estimate that ∼2–4% of speciation events in angiosperms involve polyploidization.
Incidence in Animals
Polyploidy is far rarer in animals than in plants, yet there are hundreds of
examples of recent and ancient polyploidization events throughout the animal kingdom. Table 1 (see www.annualreviews.org/Supplementary Materials) lists known
polyploid insects and vertebrates, along with their mode of reproduction [summarized in Table 3; for reviews of polyploidy in other invertebrates, see (12, 21, 33, 65,
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TABLE 3 Incidence of polyploid species of insects and vertebratesa
Taxa
Reproductive mode
2x/3x
3x
4x
Other
Insects
Parthenogens
Sexuals
13
0
34
[1]
16
0
16
[2]
Fish
Parthenogens
Sexuals
?
1
0
0
6
0
2
0
10 [1]
12 [1]
2
12
3
Amphibians
Parthenogens
Sexuals
?
1
0
0
0
0
0
0
18 [2]
0
2
6
1
Reptiles
Parthenogens
Sexuals
3
0
11 [1]
0
0
0
0
1
Mammals
Sexuals
0
0
1
0
a
This table summarizes Web Table 1, ignoring cases where polyploids are rare. Polyploid species are
grouped by ploidy level and mode of reproduction, where ‘?’ denotes that the reproductive mode is
unknown to the authors. Values in brackets refer to additional cases where there is doubt about the
polyploid nature of the species.
101, 122)]. Support for polyploidization varies from karyotypic analyses of closely
related taxa [e.g. in red viscacha rats (43)] to genomic sequence analysis (e.g. the
putative genome duplications early in the evolution of tetrapods (4, 98)].
We calculated the polyploid index for three animal groups (Table 2), and none
exhibited a polyploid index significantly different than zero. Decapod crustaceans
have a wide range of chromosome numbers, n = 27–188 (65), and a high polyploid index (41.7%), but few chromosome counts were available for analysis. The
high value of the index is consistent with karyotypic comparisons, suggesting polyploidy in several species of lobsters, crayfish, spiny lobsters, and hermit crabs [in
Astacidea, Scyllaridea, and Anomura (65)]. More chromosome data are necessary
to improve the power of the analysis and to assess the frequency of polyploidization
in Decapoda. In contrast, the polyploid indices of mammals and cockroaches (Blattaria) were slightly negative (i.e. n was more often odd than even), but these values
were not significantly different from zero despite the inclusion of >100 species.
Given that polyploidy is known to be rare in both groups, these analyses serve as a
negative control, confirming that the polyploid index is approximately zero when
polyploidy has played a relatively minor role in the on-going evolution of a group.
Müller was the first to explain the relative paucity of polyploidy in animals
based on the idea that polyploidization would interfere with sex determination in
animals (83). Dioecy, where male and female reproductive organs are found in
separate individuals, is rare in plants [comprising ∼6% of taxa (100)] but common in animals, where it is often genetically based (20). As noted by Müller,
whenever sex is determined by the ratio of X-chromosomes to autosomes (as
in Drosophila), newly arisen polyploid species will produce offspring with
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unbalanced ratios leading to disturbed sexual development and sterility. Evidence
suggests, however, that sex determination is more often based on the presence or
absence of a Y chromosome (88). With a dominant-Y mechanism for sex determination (or equivalently, a dominant-W mechanism in WZ/ZZ species), polyploidy
is less likely to disrupt sexual development, but other obstacles arise. In a newly
formed population of tetraploids, for example, the sex ratio will be strongly biased
towards the heterogametic sex. With male heterogamety, for example, the Y will
have an initial frequency of 50%, causing males (XYYY, XXYY, XXXY) to be
common and females (XXXX) to be rare. This problem will be short-lived, however, because selection for an even sex ratio is strong and rapid (9, 39). A second
obstacle to the establishment of polyploidy in dioecious taxa is the disruption of
dosage compensation whenever there are degenerate sex chromosomes that have
evolved mechanisms to balance the ratio of X to autosomal gene products (88).
These considerations generate the predictions that polyploidy should be more
common in animal taxa (a) with asexual or hermaphroditic reproduction, (b) with
sex determination based on the presence of a Y chromosome rather than the X to
autosomal ratio, and (c) with non-degenerate sex chromosomes and an absence
of dosage compensation. Indeed, polyploidy is highly associated with parthenogenetic reproduction in animals (Tables 1 and 3), even though parthenogenesis is
rare (12). Furthermore, among those amphibia and reptiles where polyploidy does
occur in sexual groups, heteromorphic sex chromosomes are rare (14), supporting
Orr’s (88) contention that sex chromosome degeneration may prevent successful polyploidization. Nevertheless, organisms may circumvent potential obstacles
to the establishment of polyploidy in unexpected ways. For example, although
there is genetic sex determination in diploid Xenopus (WZ/ZZ system), artificially
formed hybrids often show temperature-dependent sex determination (63). This
flexibility in the sex determination mechanism may allow the continued production
of both sexes following polyploidization. Such flexibility may have contributed to
the successful establishment of dioecious polyploid animals (Table 1), which are
particularly common among amphibia.
Although sex determination and dosage compensation may be important obstacles to the success of polyploidy in animals, additional factors must be at work.
Studies investigating asexual species or species lacking chromosomal sex determination find that polyploidy is “surprisingly uncommon” (21, 33, 130, p. 181).
An additional contributing factor to the rarity of polyploids in animals may be that
interspecific hybridization is less common, reducing the frequency of allopolyploids (33). It is not clear whether and to what extent this generalization is true,
however; certainly hybridization is not uncommon in several animal taxa [e.g. 3130
hybrid species combinations are known among fish (104)]. Indeed, the opposite
problem also contributes to the rarity of polyploid animals: hybridization may
be too common between newly formed polyploids and their diploid progenitors
(130, p. 456). Shifts in flowering time and ecological habitat are frequently found
among plants of differing ploidy level (118), which would increase the mating
isolation between diploids and polyploids. In contrast, animal mobility would
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reduce the chance that newly formed polyploids are spatially isolated. Interestingly, it has been argued that polyploidy may be found more often among anurans
because increased cell volume can affect vocalization, which is a major determinant of reproductive compatibility, thus reducing the extent of polyploid-diploid
hybridization (14).
Finally, developmental constraints severely restrict the appearance of polyploids in at least some animal taxa (114, p. 368; 126). Extensive data from chickens and humans suggest that a generalized disruption of development may explain the near absence of polyploid birds and mammals. ZZW chickens that do
hatch have an intersex phenotype (1), which is consistent with a Z:autosomal sexdetermination mechanism in birds (26) and with Müller’s (83) explanation for
the lack of polyploidy in animals. However, mortality is high even among ZZZ
individuals, which develop as males (26). In a study of 4182 chicken embryos,
haploids (1.4%), triploids (0.8%, 9 ZZZ, 7 ZZW, 15 ZWW), and tetraploids (0.1%,
1 ZZZZ, 1 ZZWW) were found, none of which survived to hatching (13). Disruption of sex determination is unlikely to explain the fact that all of these genotypes
die early in development. Furthermore, dosage compensation is not known to
occur in birds (103), which argues against Orr’s (88) explanation for the lack of
polyploid birds. Indeed, the fact that both ZZZZ and ZZWW tetraploids die as
embryos bolsters the claim that polyploids suffer from a general disruption of
development rather than a particular problem associated with having chromosomal sex determination. Similar observations apply to human polyploids. Approximately 5% of natural abortuses are polyploid (28). Although most triploids and
tetraploids have severe defects and fail to survive to term, livebirths do occasionally occur with infants surviving for up to two years (55, 107). Imprinting has
been implicated as a possible mechanism causing developmental abnormalities
in triploids (32, 80), based on the observation that survival is higher for digynic
triploids (formed from polyploidization in the egg) than for dispermic triploids
(formed from polyploidization in the sperm or, more commonly, fertilization by
two sperm). Consistent with what is known about imprinting in mammals, digynic
triploids develop at an appropriate rate but display stunted growth, while dispermic triploids are larger than expected based on their developmental stage (80).
Tetraploid newborns, both XXXX [female (107)] and XXYY (male (72)], have
been reported and are thought to result from a failure of cell division in the zygote.
Although polyploidy can disrupt normal sexual development and cause abnormal
genital formation in humans (102), the fact that polyploid survival is much lower
than that of trisomics involving the sex chromosomes (XYY, XXY, and XXX) and
is associated with much more severe abnormalities and early mortality regardless
of genotype suggest that a general disruption of development and not a disruption of sexual development, per se, explains the absence of polyploidy in adult
humans. One factor contributing to the high rate of natural abortion of polyploid
humans is abnormal placental growth (28, 45), perhaps related to changes in cell
size or to a disruption of the balance between maternally and paternally imprinted
genes.
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Although the above obstacles undoubtedly contribute to the rarity of polyploidy in animals, they are not absolute. Polyploid races or species are found
sporadically in many groups of animals (Table 1). These species result from the
occasional congruence of factors lessening the developmental and ecological constraints on polyploid formation. Nevertheless, these polyploid species may represent nothing more than evolutionary dead-ends. If this were true, however,
we would expect very few animal groups to be anciently polyploid. Although
it is difficult to confirm ancient polyploid events, several examples are known
(Table 1). Among fish, larger taxonomic groups of polyploids include the Actynopterygii (the highly speciose ray-finned fishes), Catostomidae, Salmonidae,
the subfamily Schizothoracinae (Cyprinidae), the monophyletic group Corydoras–
Aspidoras–Brochis (Callichthyidae), the subgenus Barbus (Cyprinidae), and the
subgenus Labeobarbus (Cyprinidae). Among amphibia, polyploid groups include
the Sirenidae and Xenopus (Pipidae). In addition, vertebrate genomes contain
extensive collinearity in genetic content on different chromosomes (47, 98), supporting Ohno’s (87) contention that vertebrates underwent two genomic doubling
events early in their evolution. These ancient polyploids indicate that polyploid
groups can flourish over evolutionary time, disproving the claim (e.g. 116) that the
rate of evolution will be so slowed when multiple homologous chromosomes are
present.
Phenotypic Effects of Polyploidy
The role of polyploidization as a creative force in producing evolutionary novelty
will be mediated through its effects on the phenotype. Therefore the fundamental question that must be addressed is whether polyploidization, in and of itself,
tends to produce phenotypic shifts that have adaptive potential. Levin (68) considered the phenotypic effects of polyploidy in flowering plants and their role in
producing evolutionary novelty. He concluded that there was ample evidence from
flowering plants that “chromosome doubling may propel a population into a new
adaptive sphere” and “bring about abrupt, transgressive, and conspicuous changes
in the adaptive gestalt of populations within microevolutionary time.” Here we
briefly summarize and update his presentation, reaching a similar, if still tentative,
conclusion.
Isolating the direct effects of genome doubling using naturally occurring
polyploids is complicated by the difficulty of identifying populations that differ only in ploidy. The effects of traits like hybridity and asexual or vegetative
reproduction that frequently occur in polyploid lineages can be confounded with
the effects of polyploidy. For example, in allopolyploids, the direct effects of polyploidy will be difficult to separate from the effects of hybridity. While comparisons
with diploid hybrids could help tease apart these effects, in practice, suitable populations may not exist. In addition, as noted by Levin (68) among others, it is
difficult to isolate the immediate effects of polyploidization from changes that
have accumulated subsequent to changes in ploidy.
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Effects on Cell Size and Growth Rates Among the most common and universal effects of polyploidization is increased cell volume (24), although the extent
to which this occurs varies (114) and may depend on environmental conditions
(B Mable, personal communication). Indeed, stomatal cell size in plants is frequently used as an indirect measure of ploidy level (e.g. 79). Likewise, the diameter
of pollen grains, comprising 2–3 nuclei in flowering plants, is also used to infer
ploidy. In addition, there is evidence from some species of amphibia (11) and plants
(23) that over time, the cell volume of polyploids tends to decrease towards the
levels of related diploids. Changes in cell volume result in changes in surface to volume ratios of cells, which may alter the rate of metabolic processes, especially those
that involve membranes (24, 68, 129). These sorts of metabolic changes are likely
at the root of changes in growth rates that are frequently detected in polyploid lineages. Developmental rates have been shown to be altered in polyploid plants, amphibia, and insects, with polyploids generally having slower developmental rates
than related diploids (e.g. 35, 73a, 125). On the other hand, polyploid plants frequently have larger seeds than related diploids, and this increase can lead instead to
higher rates of early development (123, 125). Such changes in developmental rates
may affect the likelihood of establishment of seedlings in resource-limited environments and may result in niche differentiation as a byproduct of polyploidization.
Effects on Size and Shape Given the changes in cell shape and metabolic and
developmental processes described above, effects on overall size and shape of
organs and whole organisms would be predicted. Although these are frequently
found, many polyploids are described as falling within the range of morphological variation of their diploid progenitors, with only subtle phenotypic effects
(e.g. 16, 22, 85). In plants, polyploidy is sometimes, but not always, associated
with larger overall size (114). In vertebrates, polyploidy tends to have little to
no effect on body size (35, 76, 104), but it tends to increase size in invertebrates
(12, pp. 266, 300–301).
Polyploidization often has some effect on organ structure and function, but
such effects tend to be highly idiosyncratic (114). Bretagnolle & Lumaret (17) attempted to isolate the direct effects of polyploidization in triploids and tetraploids
experimentally produced from the diploid plant, Dactylis glomerata subsp. lusitanica. Although diploids and polyploids did not differ in overall biomass, polyploids were found to have broader, thicker leaves, fewer stems per plant and fewer
inflorescences, increased seed weight, but fewer seeds overall. In hylid frogs,
vocalization differs between the diploid Hyla chrysoscelis and the related tetraploid
H. versicolor (81) but is very similar in the diploid and tetraploid leaf-frogs,
Phyllomedusa distincta (53).
Effects on Reproductive Systems Among the most common shifts in reproductive system that correlate with the occurrence of polyploidy is asexual reproduction. While most apomictic plants are polyploid, most polyploid plants are not
apomictic. In animals, however, approximately two thirds of polyploids reproduce
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parthenogenetically (Table 1, Table 3). Interestingly, this fraction is not consistent
across taxa. The majority of polyploid insects and reptiles reproduce parthenogenetically, but the majority of polyploid fish and amphibia reproduce sexually.
Although the association between parthenogenesis and polyploidy is common to
both plants and animals, the reasons for the association may be quite different
(117). In animals, parthenogenesis tends to arise in diploids; thereafter, matings
between parthenogenetic diploids and related sexuals frequently generate polyploids through the union of unreduced eggs and sperm. In contrast, in flowering
plants, polyploidization often appears to predate apomixis (117). For example,
recent studies indicate that the genes for apomixis can only be transmitted in
unreduced gametes (86, 96).
In addition to shifts to asexual reproduction, other changes in breeding systems
have been noted in plants. For example, genetic self-incompatibility systems may
break down in polyploids, resulting in higher selfing rates in polyploids than in
their diploid progenitors (114, pp. 306–307; 128a). Further, polyploidization may
alter floral traits, including the relative sizes and spatial relations of floral organs
(18). These changes may shift biotic interactions with pollinators, which may lead
to further selection for divergence in reproductive traits.
Geographical Distribution and Ecological Correlates Polyploids are frequently
said to have broader ecological tolerances than their diploid progenitors (68, 71,
106, 121). Among the explanations for this observation is the idea that increased
heterozygosity can provide metabolic flexibility to cope with a broader array
of conditions. For example, multimeric enzymes assembled from two or more
subunits will have more different forms of the enzyme available in polyploid
heterozygotes, each of which may be most effective under slightly different conditions. Another possibility is that the polyploid species that successfully establish are a biased subset of all polyploids that arise. A particularly strong bias
may be that polyploids that are able to persist are more likely to inhabit different niches than their diploid progenitors. Niche separation reduces competition
and reduces the chance of cross-ploidy matings, which tend to produce low fitness
progeny (e.g. triploids). Consequently, niche partitioning may be highly selectively
favored in newly established polyploids (78, 122).
In what may be a related phenomenon, polyploid plants and animals frequently
occur at higher altitudes and more polar latitudes than their diploid progenitors
(6, 7, 10, 12, 34, 56, 70). While numerous taxa display these trends, no satisfactory
general explanation has been proposed, and exceptions have been noted (114).
Because cold treatments can increase the frequency of unreduced gamete formation
(15, 35, 99), and these are involved in most polyploidization events, it is tantalizing
to hypothesize that this relationship may reflect different rates of origination of
polyploids in colder climates. However, we do not know the historical distributions
of most taxa, and therefore, in most cases, we do not know if polyploids originated
in these colder climes, or whether they expanded their ranges into these regions
owing to their broader tolerances. It has also been proposed that larger cell size
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may be at the root of the selective advantage that some polyploids appear to display
in these areas (34).
Genetic Consequences of Polyploidization
...the large amount of gene duplication dilutes the effects of new
mutations and gene combinations to such an extent that polyploids have
great difficulty evolving truly new adaptive gene complexes.
Stebbins (116, p. 133).
It is often argued that the evolutionary success of polyploids is hampered by the
inefficiency of selection when multiple alleles are present at each gene. Indeed,
the spread of a favorable allele from a given frequency is slower at higher ploidy
levels (132, p. 50), because the selective effects of an allele are partially off-set
by the presence of alternate alleles. Populations of varying ploidy level are not,
however, expected to start from the same allele frequency, and this difference must
be taken into account in determining the evolutionary effects of polyploidy. In
particular, because polyploid individuals carry more alleles, each individual has a
greater chance of carrying a new beneficial mutation (93). Similarly, alleles that
have recently become beneficial will tend to be at higher frequency in polyploid
populations if they were previously deleterious, because deleterious alleles persist for longer and reach higher frequencies when masked. Surprisingly, there
has been relatively little work on the basic population genetic consequences of
polyploidy. Here we explore the mutation load and rate of adaptation of polyploid species. Although these evolutionary factors may affect the persistence of
a newly formed polyploid lineage, the ecological effects of polyploidization are
more likely to determine establishment in the short-term. Nevertheless, the genetic load and rate of adaptation may have longer-term effects on the persistence
and diversification of polyploids once formed. As we shall see, polyploid populations can have a transiently lower deleterious mutation load than populations of
lower ploidy level, and they can, under certain circumstances, adapt at a faster
rate.
Notation In the following sections, we consider the dynamics at one locus in
a population of size N and of ploidy level c. Matings between individuals of
different ploidy levels are not considered, nor are ecological interactions. The
locus is subject to mutations at rate µ (if deleterious) or ν (if beneficial) per
generation. Let allele A be wild type with a relative fitness of 1 in haploids, and
let allele a be mutant with fitness 1 ± s, depending on whether beneficial (+)
or deleterious (−) mutations are considered. We will assume that the fitnesses
of A, AA, and AAAA individuals are equal, as are the fitnesses of a, aa, and aaaa
individuals. This symmetry ensures that there are no intrinsic fitness advantages of
one ploidy level over another and allows us to explore the effects of masking alone.
In diploids, heterozygotes have fitness 1 ± hs, where h measures the dominance
of the mutant allele. In tetraploids, there are three heterozygous types, AAAa,
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AAaa, and Aaaa, whose fitnesses are defined as 1 ± h1s, 1 ± h2s, and 1 ± h3s,
respectively. To simplify notation, we shall write σc for the fitness effects of a
single mutation in an organism of ploidy level c. That is, σ 1 = s, σ2 = hs, and
σ 4 = h1s, which are all assumed to be non-zero (completely recessive alleles are
not treated).
When considering polyploids with multisomic inheritance, we assume, for
simplicity, that the gene of interest is located near the centromere. Hence, we ignore
the phenomenon of double reduction, where recombination between the centromere and a gene allows an A1A2A3A4 tetraploid to produce A1A1 gametes, for
example (30). Double reduction tends to make selection in tetrasomic tetraploids
more similar to that in diploids, because the double-diploid tetraploids (AAAA,
AAaa, aaaa) become more common relative to the AAAa and Aaaa genotypes.
We also assume that selection is sufficiently weak that the population will be in
nearly Hardy-Weinberg proportions (132).
Masking Deleterious Mutations Increasing ploidy can provide a selective advantage by masking the deleterious fitness effects of mutations (54, p. 110). Previous models have shown that the selective advantage of diploid life cycles over
haploid life cycles is greatest when deleterious mutations are recessive or nearly
so (64, 91, 95). Similarly, we might expect polyploids to have a fitness advantage
over lower ploidy levels through masking. The situation is complicated, however,
by the fact that individuals of higher ploidy have a higher chance of bearing a
mutation and the fact that deleterious mutations persist for longer in populations
of higher ploidy levels [as shown by (91) in a comparison of diploid and haploid
life cycles]. Assuming a large population size, low deleterious mutation rate (µ),
and tetrasomic inheritance in tetraploids, the equilibrium frequency of the mutant
allele (q) becomes µ/σ c in both asexual and randomly mating sexual populations.
This frequency increases with ploidy level for deleterious mutations that are partially recessive and masked (i.e. h < 1/2 and h1 < 1/2 h), which is consistent with
data from diploids (108). At equilibrium, the amount by which mean fitness is
reduced by the presence of deleterious mutations (the mutation load) is the frequency of heterozygous individuals (≈cµ/σc ) times their average selective cost
(σ c). Therefore, the mutation load is approximately cµ and rises linearly with
ploidy level, c.
For a newly produced tetraploid population, however, these equilibrium calculations are inappropriate. If there were multiple origins of the tetraploid population,
then the initial mutant allele frequency in the tetraploid population is expected to
equal µ/σ2, i.e. the diploid mutant allele frequency at equilibrium. Newly formed
tetraploid populations would then have a mutation load equal to 4µσ4/σ2 =
4µh1/h. This load will be lower than the equilibrium load of the diploid progenitor population whenever h1 < h/2, i.e. whenever AAAa tetraploids are nearer
in fitness to AA diploids than Aa diploids (the equivalent of being partially
recessive). Essentially, there is an immediate advantage to polyploidization whenever individuals of higher ploidy levels are better able to mask single-copy
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Figure 3 The time required for the mutation load in a newly formed tetraploid population
to rise above the load of a diploid population. Assuming that mutations are rare and that
selection is relatively weak, this time equals Loge[2(1 − f )]/( fhs), where f = h1/h. For
tetraploid populations to have a reduced genetic load initially, f must be less than 1/2 ,
consistent with deleterious alleles being partially recessive. Here, h s is set to 0.005, 0.01,
and 0.02 (the scale on the y-axis is inversely proportional to hs).
deleterious mutations than individuals of lower ploidy levels. This advantage is
transient, however, because deleterious alleles increase in frequency over time until
AAAa genotypess are relatively common. In a newly formed tetraploid population,
mean fitness decreases exponentially towards its equilibrium level at rate σ4. The
number of generations that a tetraploid population is expected to have a lower
genetic load than its diploid ancestor is shown in Figure 3, which shows that the
transient advantage of a lower load may persist for many generations, especially
when masking is strong. The picture is complicated if the tetraploid population
starts from a single individual, but the qualitative outcome is similar when averaged over the genome. As long as a single copy of a deleterious mutation is more
masked in a tetraploid than in a diploid, a newly formed tetraploid population will
have, on average, a lower genetic load initially. This advantage dissipates over
time, with tetraploid populations ultimately having a higher genetic load.
Before leaving the topic of deleterious mutations, it is worth discussing another possible advantage to sexual tetraploids: deleterious mutations are masked
in the gametophyte or gamete phase. When selection acts on the gametophyte (as in
many plants), a new tetraploid population may gain an immediate advantage due to
masking deleterious mutations in the diploid phase. Again, this advantage is transitory, because such deleterious alleles will eventually rise to a higher frequency in
tetraploid populations as a side-consequence of their being masked. Nevertheless,
the initial success of a sexual tetraploid population may be enhanced by the additional protection derived from having two copies of every gene in the gametophyte.
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Altered Rates of Adaptation In polyploids, any benefit imparted by a new advantageous mutation is diluted by the effects of alternative alleles at the same locus.
As noted by Paquin & Adams (93), however, organisms with a higher ploidy level
have a higher chance of bearing new beneficial alleles. In the final balance, whether
polyploids evolve more or less rapidly depends on several key factors, including
their mode of reproduction, population size, and dominance coefficients.
Asexuals Because polyploidy often occurs in asexual lineages especially in
animals, we begin by discussing the rate of evolution in asexual populations.
The spread of beneficial mutations is hampered in asexual populations by the fact
that beneficial mutations occurring in separate individuals cannot be recombined
into the same genome (39). Because only one asexual lineage will be the ultimate
ancestor of the descendant population, the rate of evolution of an asexual population is limited to those beneficial mutations that appear in this lineage. In the
Appendix, we estimate the effect of ploidy on the rate of increase of fitness owing
to the accumulation of beneficial mutations, which arise within a population at rate
cNν. For an additively acting allele (with σc = σ 1/c), the rate of fitness increase
is always highest for haploids (c = 1). More generally, we note that the rate of
fitness increase lies within the ranges:
h2
.
Rate of increase of fitness in diploids
Rate of increase of fitness in haploids
.
2h 2
3a.
and
h 21
h2
.
Rate of increase of fitness in tetraploids
Rate of increase of fitness in diploids
.
2
h 21
.
h2
3b.
The rates approach the lower limits of these ranges in large populations (Nν 1)
and approach the upper limits in small populations where few beneficial mutations
appear each generation (Nν 1).
Because it is reasonable to assume that h1 < h < 1, the lower limits of Equations 3 indicate that haploids will adapt fastest, followed by diploids, and finally
tetraploids in large asexual populations with high rates of beneficial mutations
(Nν 1). Under these circumstances, the rate of adaptation is not limited by the
appearance of beneficial mutations but by their rate of spread to high frequency,
which is always faster in haploids (132, p. 50). For smaller populations, however, the rate at which beneficial mutations appear and avoid loss, while rare, is
low enough to limit the rate of adaptation. Because more beneficial mutations
appear in polyploid populations, the rate of adaptive evolution can increase with
ploidy level. The upper limits of Equation 3 predict that, in small populations
(Nν 1), diploids will adapt faster than haploids when h > 0.71 and that tetraploids
will adapt even faster if h1 > 0.71 h. Unfortunately, we have very few data on
the dominance coefficient of newly arisen beneficial alleles in diploids, let alone
in tetraploids. Nevertheless, we might expect the fitness benefits of an allele to
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Figure 4 Relationship between allele dosage and fitness effect. The fraction of alleles
within an individual that are of the new beneficial type a is drawn on the x-axis. Wild-type
individuals have an x-value of zero, individuals carrying only a alleles have an x-value
of one, and intermediate genotypes have intermediate values (e.g. 1/2 for Aa or AAaa
individuals and 1/4 for AAAa individuals). The y-axis is the selection coefficient acting on
individuals of the given allele dosage. Hypothetical curves are drawn based on an additive
model (straight line, denoted by a star), on a hyperbolic model (curves denoted by d1–d3,
which exhibit dominance), and on an exponential model (curves denoted by r1–r3, which
exhibit recessivity). The hyperbolic model is expected based on the relationship between
flux and enzyme activity in metabolic control theory (58).
scale with the dose of that allele according to functions of the shapes illustrated in
Figure 4 [the hyperbolic curves d1–d3 are based on the effects of varying the activity
of an enzyme in a metabolic pathway; akin to Figure 4 in (58)]. Interestingly, for
the types of curves shown, diploids never have the highest rate of fitness increase
according to Equation A2 (see appendix). When dominance is high enough (e.g.
curves d2 and d3), fitness is expected to increase fastest in tetraploid populations,
otherwise it will increase fastest in haploid populations. In summary, asexual populations of higher ploidy level can adapt faster than those of lower ploidy levels
if the population is relatively small and if beneficial alleles tend to be partially
dominant.
Sexuals In sexual populations with ample genetic mixing, beneficial mutations
that occur in separate individuals can be brought together by recombination (39).
Indeed, the fixation probability of a mutation is only slightly affected by the dynamics at other loci unless linkage is tight (8). Consequently, the fixation probability
of each beneficial mutation is very nearly 2σc (assuming directional selection
on an allele that is not fully recessive). Therefore, mutations that successfully
become established appear within a population at a rate of 2σc cNν per generation.
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Multiplying this rate by the resulting increase in fitness once the mutant allele
is fixed, we estimate that the rate of increase in fitness is 1W1n = (2sνN ) s
in haploids, 1W2n = (4hsvN ) s in diploids, and 1W4n-tetrasomic = (8h1svN ) s in
tetraploids with tetrasomic inheritance. Therefore, sexual tetrasomic tetraploids
will adapt faster than diploids whenever h1 > h/2. For tetraploids with disomic
segregation, however, new beneficial alleles can spread to fixation only at one
of the two gene copies (unless gene conversion occurs). If the fitness of such
AAaa individuals equals 1 + h2s, then the rate of increase in fitness will equal
1W4n-disomic = (8h1sνN ) h2s. Now, h1h2 must be greater than h/2 for tetraploids
with disomic inheritance to evolve faster than diploids. Figure 5 summarizes these
results. It is necessary, however, to determine the biologically relevant part of
this parameter space. Once again, we use the relationships between allele dosage
and fitness illustrated in Figure 4, which suggest that parameters near the dotted
curve may be most common. Along this curve, either haploids have the highest average rate of fitness increase (for beneficial alleles that are partially recessive) or tetraploids do (for beneficial alleles that are partially dominant). There
is no justification for the widespread notion that evolution proceeds fastest in
diploids.
Figure 5 Dominant expression of new beneficial alleles favors higher ploidy levels. In
sexual populations, the rate of adaptation will be highest for the ploidy level indicated on
the graph, all else being equal. The x-axis and y-axis correspond to the dominance level of
a single-copy beneficial allele in diploids and tetraploids, respectively (i.e. h and h1). For
tetraploids with disomic segregation, the fitness of Aa and AAaa individuals are set equal
(h = h2), in which case tetraploids evolve faster than diploids in the region above the
dashed line. A star is placed at the point where an additively acting allele would lie. The
dotted curve corresponds to the dominance relationships expected based on curves of the
form illustrated in Figure 4.
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The above discussion is based, however, on the assumption that newly arisen
mutations are the basis of adaptive change. It may be that beneficial alleles are
previously deleterious alleles whose selection coefficients have become positive
in response to environmental changes. Because deleterious mutations tend to
be more masked and more frequent in populations of higher ploidy level, polyploids may be better poised to respond rapidly to changes in the environment.
As we shall show, whether polyploids adapt faster under these circumstances
depends critically on the population size. We consider a population at mutationselection balance, with deleterious alleles at frequency qcd = µ/σcd, where the
subscript d emphasizes that σ cd refers to past selection against the allele while
it was deleterious. If the number of copies (cNqcd) of the previously deleterious allele is small, such that the probability that at least one of them fixes is
much less than one (i.e. for populations of small to moderate size; Nµ 1),
then the increase in fitness based on the fixation of beneficial alleles that were
previously held at a mutation-selection balance is approximately 2σ c (cNqd) s
for haploids, diploids, and polyploids with multisomic segregation. As long as
selection for the allele when beneficial is not much weaker than it was when deleterious (σ c/σ cd > 1/c), increasing ploidy will result in a faster rate of adaptation.
The advantage of polyploidy will be most pronounced when the allele is absent
or in few copies in diploid populations but tends to be found in multiple copies in
polyploid populations (i.e. 2Nq2d < 1 for diploids but 4Nq4d 1 for tetraploids),
which can occur if deleterious alleles are well masked and if the population size is
moderately small. In this case, the newly beneficial allele can rise immediately in
frequency in the polyploid population, whereas haploid and diploid populations
must wait for the next appearance of the allele by mutation.
If, on the other hand, the population is so large that enough copies of the previously deleterious allele are present to guarantee that at least one will fix, then
the rate of adaptation is no longer set by the probability of carrying a beneficial
allele but rather the rate of spread of these alleles. In this case, haploid populations
nearly always adapt faster than diploid populations, which adapt faster than polyploid populations, because of the decreasing efficiency of selection with ploidy
level (132).
Overall The blanket statement that polyploids adapt more slowly than diploids is
false. In small to moderately sized populations, the rate of fitness increase depends
less on the efficiency of selection and more on how often mutations appear and
establish within populations. Consequently, the rate of adaptation can be faster
for higher ploidy levels as long as beneficial alleles are partially dominant. This
basic conclusion holds for sexual and asexual polyploids, whether adaptive change
results from novel mutations or previously deleterious alleles.
The above calculations are predicated on several simplifying assumptions,
which were chosen to focus our attention on the costs and benefits of changing ploidy level alone. In particular, we assumed that (a) the populations being
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compared were of equal size, (b) the selective advantage or disadvantage of an
allele was the same regardless of ploidy level if it was the only allele present
(i.e. s was treated as constant), and (c) the deleterious (µ) and beneficial (ν) mutation rates were constant per allele. As with any two populations, however, these
parameters are likely to differ between a polyploid population and a related diploid
population. For example, polyploid populations sometimes inhabit marginal habitats with relatively low population densities (suggesting a lower N ), although polyploids occasionally have much wider ranges (and a higher N ) than related diploid
species (122). Furthermore, polyploids are generally not morphologically or physiologically identical to diploids, so the selective advantages and disadvantages of
mutant alleles in polyploids need not be related to these coefficients in diploids.
For example, immediate changes in gene expression occur upon polyploidization
in yeast (42). Some of these changes appear to regulate cell cycle progression (and
hence cell size), cell shape, cell clumping, and fungal invasiveness. Metaphorically, a newly produced polyploid population may occupy a different position on
the adaptive landscape (132), and the spectrum of fitness effects of alleles may be
greatly altered.
An intriguing possibility is that the rate of beneficial mutations may differ
among ploidy levels. In particular, if survival and reproduction depend on the
proper expression of a particular allele, any mutation that changes its function or
pattern of expression will be deleterious in haploids, but some of these mutant alleles may be beneficial in diploids or tetraploids as long as the essential allele is still
present and active. Similarly, if the proper expression of an allele within gametes
is essential, mutations to this allele may be strongly deleterious in diploids but potentially advantageous in tetraploids. Such phenomena would lead the beneficial
mutation rate (ν) to increase with ploidy level. Of course, the ultimate outcome
of such selection also depends on the ploidy level and mode of segregation. In
diploids and tetrasomic tetraploids, a balanced polymorphism will result, where
neither the newly functioning allele nor the old essential allele is stably inherited.
In disomic tetraploids, on the other hand, the fixation of both alleles on different pairs of chromosomes is possible, allowing their stable inheritance. Indeed,
changes in gene function or in the timing and tissue specificity of gene expression
have been found between duplicate genes in yeast (131), catostomid fish (38), and
Xenopus (57). The possibility exists that genomic duplication released constraints
on selection and allowed evolution to proceed in a novel direction, which was,
perhaps, prohibited in the absence of gene copies (131).
Altered Linkage Arrangements The evolutionary consequences of polyploidy
discussed above result from a change in allelic copy number, which alters both the
chance of mutation and the strength of selection on any particular allele. Polyploidization has additional genetic effects that may influence the success of polyploid lineages. With an increasing number of chromosomes and disomic segregation, the average rate of recombination between loci rises. Thus, Grant (51,
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Figure 6 Effect of a change in chromosome number on the rate of recombination between
an average pair of loci. This graph assumes that there are n chromosomes (with disomic
inheritance) of approximately equal length, M, measured in Morgans. The average rate of
recombination is 1/2 times the chance that the loci are on different chromosomes plus the
average rate of recombination between loci on the same chromosome, calculated using
Haldane’s mapping function and multiplied by the chance that the two loci are on the same
chromosome.
p. 394) argued that larger chromosome numbers would be “favored by selection
for open recombination systems.” This effect is likely to be weak, however, in
any organism with more than a handful of chromosomes (Figure 6). With more
than five chromosomes (≥0.1 Morgan in length), doubling the number of chromosomes will at most increase the average rate of recombination between randomly
chosen pairs of loci by 10%. While the effect on the average level of recombination is modest, selection for increased recombination between certain pairs of loci
may favor polyploids. That is, linked loci in a diploid may become unlinked in a
polyploid if silencing occurs at opposite genes on the duplicate chromosomes. If
the two loci are routinely under directional selection for novel function, the rate
of adaptation may be increased by this physical separation (8, 90). Recombination is not always advantageous (37), however, and the increase in recombination may reduce polyploid fitness, especially if co-adapted gene complexes are
dispersed.
Gene expression and regulation may also be affected by changes in the
genomic context in which genes occur following polyploidization. Evidence is
mounting that genomic repatterning in polyploids is often extensive and can be
rapid (66, 111). As a striking example, Song et al (112) created polyploid Brassica
hybrids and observed extensive genomic rearrangements within five generations.
One possible explanation for such rapid changes is that transposable elements that
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are silent in one parental line become active in polyploids of hybrid origin (129a).
Such transposable elements may contribute to physical changes in the karyotype
(e.g. translocations, fusions, fissions) and may hasten gene silencing of duplicate
gene copies. Such massive and rapid changes in genome structure must often be
deleterious, but occasional advantageous variants might arise. Certainly, genomic
rearrangements would augment the genetic variation observed in a new polyploid
population. Although genomic repatterning may often contribute to the rapid
extinction of polyploid lineages, the rare successful lineages may be especially
likely to diverge enough in ecological requirements to coexist with or displace
their diploid progenitors.
Maintenance of Duplicate Genes and Divergence
in Gene Function
...duplications enable genes to make evolutionary experiments which have
previously been forbidden, liberating them from incessant natural selection
whose overwhelming activity is eliminating variants.
Kimura (60, p. 16)
The ultimate contribution of polyploid events to evolution would be minor if the
fate of most duplicate genes is to be silenced. Early work on duplicate gene
evolution predicted just such an outcome (e.g. 40, 54; see 84 for additional references). According to this classical view, partially recessive deleterious mutations
would rise in frequency with little opposition from selection as long as a second
functional gene was present. Once a deleterious allele fixed in one gene, however, selection would act more strongly to preserve function of the duplicate gene,
making the next deleterious mutation more likely to fix in the first gene as well.
Over time, mutational decay would eventually silence one member of most gene
pairs. Only in those very rare instances where a beneficial mutation that causes
divergence in function between duplicate genes arises before the decay process has
proceeded too far would both gene copies be preserved (e.g. 54, 87, 128). Walsh
(128), for example, estimated that about 99% of duplicate genes would devolve
into pseudogenes by this process.
Data from a variety of ancient polyploids suggest, however, that a much larger
fraction of duplicate gene copies is retained over much longer periods of evolutionary time than predicted. The fraction of genes retained in duplicate has been
estimated as ∼8% in yeast over ∼100 MY (105), ∼72% in maize over ∼11 MY
(2, 44), ∼77% in Xenopus over ∼30 MY (57), ∼70% in salmonids over 25–100
MY (5), ∼47% in catastomids over ∼50 MY (38), and ∼33% in vertebrates over
∼ 500 MY (84). Such high levels of maintenance have led evolutionary biologists
to reassess the selective forces acting on duplicate genes, with implications for the
role of polyploidy in evolution.
The maintenance of duplicate genes is rendered much more likely than in the
classical model if (a) deleterious mutations affect fitness even when present in a
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single copy, (b) there is selection for increased gene expression, and/or (c) gene
copies rapidly diverge in function. The first explanation is particularly likely for
genes encoding multidomain proteins, which interact with several other proteins,
because point mutations tend to disrupt protein-protein interactions and have a
partially dominant effect on fitness (46). Consistent with this explanation, many
of the genes maintained in duplicate in Xenopus are involved in protein-protein
interactions (57). Support for the second explanation comes from yeast, where
highly expressed genes (measured by mRNA levels), including heat shock, glucose metabolic, and cytosolic ribosomal genes, were more likely to be retained
in duplicate than genes with low levels of expression (105). Nevertheless, many
duplicate genes with low expression levels in yeast have also been retained, and
Seoighe & Wolfe (105) suggest that, in these cases, divergence in function has
played a role in maintaining duplicate genes. This suggestion is confirmed by
the observation that ∼10% of the genes retained in duplicate belong to slightly
different functional categories in the Yeast Proteome Database (105). This third
explanation, divergence in gene function, is especially likely when degenerative
mutations can occur in different functional or regulatory domains of duplicate
genes (41, 74). In this model, deleterious mutations accumulate in each gene copy
disrupting some but not all functions of the gene, until both genes become essential for some of the subfunctions of the original single-copy gene. This phenomenon, termed sub-functionalization (41), ensures that both gene copies remain under selection and allows their preservation over longer periods of evolutionary time (41, 74). Not only does this increase the probability that adaptive
mutations occur before gene silencing, but it also eases pleiotropic constraints.
Selection on a single gene with multiple functions is constrained to favor the
fittest sequence averaged over all of the roles of the gene; following duplication and sub-functionalization, mutations that are beneficial to only a subset of
the roles of a gene may be favored and spread in the gene copy that eventually takes over these roles (74). The fact that duplicate genes often differ in the
timing of expression and/or the pattern of expression in different tissues is consistent with the sub-functionalization model (38, 41, 57). More direct support comes
from examples where the shared expression pattern of duplicate genes coincides
with the total expression pattern of single-copy orthologues in related organisms
(41).
Under each of these scenarios, duplicate genes maintained over time are not
expected to be released from selection at the amino acid level. This prediction contrasts with that of the classical model, which assumes that selection in both copies
is initially relaxed. Data examining the level of selective constraints in Xenopus
suggest that both duplicate genes are generally subject to purifying selection (57),
with synonymous substitutions occurring more frequently than nonsynonymous
changes for all 17 pairs of genes examined. Consequently, duplicate genes in
polyploid lineages are often preserved in function by purifying selection for long
enough to allow for beneficial mutations to arise in each and, hence, for their
diversification over time.
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DISCUSSION
For greater changes the chance of improvement diminishes progressively,
becoming zero, or at least negligible, for changes of a sufficiently
pronounced character.
Fisher (39)
Based on a compelling geometrical argument, Fisher (39) noted that large mutations would almost surely reduce fitness, either aiming the phenotype away from a
fitness peak or causing it to overshoot it. On the basis of Fisher’s argument, modern biologists give little credence to the role of “hopeful monsters” in evolution.
Consequently, it is puzzling that one of the most drastic possible types of mutation, polyploidization, has occurred frequently throughout evolution. Polyploidy
is widespread in plants and makes several appearances in animals (69; Tables 1
and 3), which suggests that polyploids are either not monsters or happen to be
particularly hopeful monsters.
Although polyploidization would appear to be the most extreme possible
genetic change, the phenotypic changes that it causes are often surprisingly subtle
and follow few general rules. The fact that polyploids often develop functionally
with few inherent effects on phenotype calls into question the assumption that a
change in ploidy is a drastic mutation. Because growth necessarily involves the
cyclic doubling and halving of the genome during mitosis (and an even further
halving during meiosis), not to mention the widespread phenomenon of endopolyploidy, all organisms must have evolved tolerance to changes in ploidy level. In
other words, organisms may be preadapted to withstand changes in ploidy levels,
offering a possible resolution to the puzzle of how such drastic mutations persist.
Nevertheless, the phenotypic changes that polyploidization does produce can be
enormously important for the evolutionary success of newly formed polyploid
lineages. Changes in features such as metabolism, developmental rates, gene regulation, and physiological tolerances can alter biotic interactions, ecological tolerances, and facets of reproductive isolation such as mating behavior and breeding
system. Such traits play critical roles in the establishment and diversification of
newly formed polyploid lineages.
The frequency of polyploidy varies but is particularly high in ferns and
angiosperms. Prior estimates for the level of polyploidy in these groups assessed
not the rate of polyploidization but the extent to which species have a polyploid
history, whether recent or ancient. Such estimates provide a poor assessment of
the on-going role of polyploidy in evolution. We have developed a new method to
estimate the frequency of polyploidy that relies on the fact that polyploidization
leads to an excess of even over odd haploid chromosome numbers (Table 2). We
estimate that ∼2–4% of speciation events in angiosperms and ∼7% in ferns are
associated with polyploidy. Indeed, polyploidization may be the single most common mechanism of sympatric speciation in plants. One reason that polyploidy may
be a particularly common speciation mechanism is that it can be accompanied by
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phenotypic changes that allow niche partitioning between newly formed species
and their ancestors. Although polyploidy is associated with many fewer speciation
events in animals, there are, nevertheless, hundreds of cases of polyploid animals.
Furthermore, evidence is accumulating that points to the role of polyploidy early
in the evolution of major groups of animals, including vertebrates, ray-finned fish,
salmonids, and catostomids.
That polyploidy has occurred repeatedly throughout evolution does not,
however, prove that it has played a major creative role in the evolutionary process. Stebbins (114, 116, 117) argued against the creative role of polyploidy on
the basis of several lines of evidence. First, he argued that polyploidy would retard the rate of adaptive evolution because the effects of beneficial alleles would be
“diluted” by the expression of duplicate alleles (116, p. 133). Second, he noted that
polyploidy is often associated with relatively unspecialized development (117).
Third, Stebbins (115, pp. 199–201) noted that most major groups of plants have
a range of haploid chromosome numbers with similar lower limits, suggesting
that morphological divergence among angiosperm families predated the polyploid
events that led to massive increases in chromosome number within the groups.
Finally, he argued that animals have been enormously successful in terms of sheer
taxonomic diversity, the evolution of specialization, and the evolution of complex
morphological features without the aid of frequent polyploidization events (114).
Stebbins concludes that “polyploidy, although it multiplies greatly the number of
species and sometimes genera present on the earth, retards rather than promotes
progressive evolution” (114, p. 369).
While it is important to remain skeptical, there are strong counter-points to
each of the above lines of evidence. First, as we have seen, polyploids can adapt
faster than diploids, especially if beneficial mutations appear within relatively small
populations and have a partially dominant effect on fitness. Second, one can easily
turn the argument that polyploids are rarely specialists on its head; the evolutionary
significance of polyploidy may indeed lie in the tendency of polyploids to have a
broader ecological tolerance (e.g. 114, 117) and more generalist life history. Third,
it is extraordinarily difficult to infer ancient polyploidization events, and it may
very well be that a common ancestor of all modern plants was itself polyploid. And,
finally, the evolutionary “success” of animals may have been intricately connected
with early polyploid events, even if the on-going contribution of polyploidization
is small.
While there are reasons to doubt Stebbins’ sweeping claims against the
evolutionary significance of polyploidy, there is very little hard evidence about
the influence of polyploidy on the tempo or mode of evolution. The fundamental question is: To what extent does the doubling of gene number increase the
rate of morphological evolution or diversification? Certainly, there has been much
speculation that polyploidy promotes adaptive evolution:
Because teleosts are the most species-rich group of vertebrates and exhibit
tremendous morphological diversity, it is tempting to speculate that the
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duplication event [in ray-finned fish] may have provided gene copies that
helped spur the teleost radiation.
Amores et al (4)
It is interesting to speculate that emergence of the complex head structures
of the vertebrates has a basis in the amplification of the homeobox clusters.
Pendleton et al (94)
The proposed genome duplication [in yeast] may have been instrumental in
its evolutionary adaptation to anaerobic growth; for example, the duplicate
genes include several pairs that are regulated differently under aerobic and
anaerobic conditions...as well as several genes encoding sugar transporters.
Wolfe & Shields (131)
Unfortunately, convincing data on this subject remain to be gathered. The definitive
test would examine whether increased rates of speciation or morphological evolution follow polyploid events, assessed in a phylogenetic context involving multiple
independent transitions to polyploidy. It is not enough to prove, for example, that
novel traits appearing over evolutionary time develop from modifications in duplicated genes, as has been suggested (124), because these novel traits may well
have evolved in the absence of gene duplication. Indeed, after polyploidization,
we would expect evolutionary change to involve gene duplicates, not necessarily
because having gene copies was essential for the evolutionary transition, but because selection acts upon the genetic material at hand, which includes the gene
copies. Consequently, polyploidization may not have had an impact on the mode
or direction of evolution, even if we now observe that gene duplicates function in
different ways.
To explore the influence of polyploidy on taxonomic diversity, we determined
the relationship between the number of species in each of 200 dicot genera whose
level of polyploidy was assessed by Stebbins (113). As shown in Figure 7, there
is a small positive relationship between level of polyploidy and species richness
(0.01 < p < 0.05). Similar results were reported by Petit & Thompson (97), who
examined taxonomic richness in 50 angiosperm genera in the flora of the Pyrenees.
In both cases, taxonomic diversity displays a nonlinear relationship with the degree
of polyploidy, with genera having a high fraction of polyploids being less speciose.
Unfortunately, this pattern is consistent with the explanation that older genera have
a greater mixture of ploidy levels and a higher species richness. To get around this
problem requires a phylogenetically correct analysis, where, for example, sister
genera of equal age are compared in terms of polyploidy and species richness.
This analysis is made difficult by several factors: data on chromosome numbers are
lacking for many genera, ploidy level is often difficult to assess, and sister taxa often
have similar levels of polyploidy. As a final complicating factor, species richness
may be higher in clades with a higher level of polyploidy simply because polyploidy
is a speciation mechanism. The ideal study would therefore analyze cases where
the only polyploid event occurred on the basal branch to one of the sister groups.
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Figure 7 Relationship between ploidy level and species richness in dicots. Genera were
classified by Stebbins (113) as 0–25% (101 genera), 25–50% (54 genera), 50–75% (27 genera), and 75–100% (14 genera) polyploid. The number of species per genus was obtained
from Mabberley (75). Species numbers were ln-transformed to satisfy normality assumptions, the means and their 95% confidence limits were obtained and are illustated (on the
original scale of species numbers), and an ANOVA was performed. The ANOVA indicated
a significant interaction between ploidy level and taxonomic diversity (p = 0.017). A
linear regression based on ranks gave a significantly positive relationship, with the average
rank in diversity increasing by 4.3 per ploidy category (p = 0.017). These results were
only marginally significant (both p = 0.11) if small genera (<10 species) were removed
from the analysis.
In the absence of such an ideal analysis, it is difficult to draw firm conclusions
about the effects of polyploidy on speciation and extinction rates. Figure 7 and
the analysis by Petit & Thompson (97) are consistent with, but do not prove, that
polyploidization has played a positive role in generating evolutionary diversity.
There are theoretical reasons to expect polyploidy to boost rates of adaptive
evolution. Polyploids have a greater chance of bearing new beneficial alleles and
are better poised to evolve novel functions in duplicated gene families. Yet, the role
of polyploidy in evolution remains enigmatic. Phylogenetic analyses promise to
clarify the relationship between polyploidy and the tempo of evolutionary change.
Genomic analyses promise to reveal more ancient polyploidy events and to determine the specific genetic changes that have followed. These efforts will undoubtedly shed light on the role that polyploidy has played in the major evolutionary
transitions of eukaryotes.
ACKNOWLEDGMENTS
We are enormously grateful to Jordana Tzenova, Maren Friesen, and Kim Ryall,
who as student research assistants compiled the chromosome data. We thank Ryan
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Garrett, Barbara Mable, Dolph Schluter, and Michael Whitlock for insightful discussions about the evolutionary role of polyploidy. Finally, we thank Wyatt Anderson, Alex Grant, Jana Heilbuth, Kent Holsinger, and Barbara Mable for providing
helpful comments on the manuscript. Support was provided by the Natural Sciences and Engineering Research Council, Canada (to SPO and JW) and the Peter
Wall Institute for Advanced Studies (to SPO).
APPENDIX
We adapt the methods used by Crow & Kimura (29) and Orr & Otto (89) to
estimate the rate at which fitness increases in a population of ploidy level, c. This
method estimates the number of generations that must pass, on average, before an
individual carrying a beneficial mutation will carry a second beneficial mutation.
In a purely apomictic population with c > 1, a beneficial mutation will spread
to fixation on only one of the homologous chromosomes, generating permanent
heterozygosity (61). Therefore, we consider only the selective effects of a mutant
allele when present in a single-copy, σ c.
With a constant number of individuals, the chance that a new beneficial mutation survives loss while rare is approximately 2σ c (54). Once a beneficial allele
becomes established and begins to spread, additional beneficial mutations arising
in individuals that do not carry the first mutation are unlikely to fix (if the selection coefficients are similar). When a second beneficial mutation with selective
advantage σc0 arises in the subset of the population carrying the first mutation, it
will actually have a higher chance of fixing than 2σc0 , because it occurs within a
group of individuals that is growing (because they carry the first beneficial allele).
This important effect has been ignored in previous calculations of the rate of adaptive evolution in asexuals (29, 89). As the frequency ( pt) of the first beneficial
mutation rises logistically over time, pt = p0/(p0 + (1 − p0) e−σc t), the probability
of fixation (Pt) of the second beneficial mutation is:
Pt ≈ 2σc0
σc0 + σc
σc0 + pt σc
A1.
(92). Consequently, the rate at which beneficial alleles appear that survive loss
while rare is Pt times the number of individuals carrying the first allele: pt (cNν).
The total number of generations (g) that must pass, on average, between
R g the appearance of two successful beneficial mutations can be found by setting 0 Pt pt cN ν dt
to one and solving for g. The expected rate of fixation of adaptive mutations is then
the inverse of g. Finally, this rate of fixation must be multiplied by σc to get the
overall rate at which fitness increases due to the incorporation of new beneficial
mutations, giving
1Wasexual =
σc σ 0
¢ σc0 ¤
£ ¡
£ σc c ¤
ln N Exp 2cν N σc − 1 σ 0 +σ
c
A2.
c
Typo: Should be "c N"
Typo: Should be primed
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Equation (A2) is used to derive equations (3) in the text, where it is assumed
that beneficial alleles have a constant selective advantage (σc0 = σc ).
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