Many species, particularly flowering plants, have usually experienced a state of polyploidy in their evolutionary history. Meiotic recombination creates novel configurations of genetic variants maintained in the genome of a species for facilitating natural and artificial selection. Thus, to understand how a diploid species differs in frequency of meiotic recombination from its polyploid ancestor has a significant impact in both evolutionary biology and plant and animal genetic breeding. It has been well established that the evolution of polyploid genomes is an extremely dynamic process compared to that of diploids, characterized by extensive genetic and epigenetic changes occurring in the nuclear genome following polyploidization 123. Little is known about the mechanism underpinning the genetic changes. To address this fundamental question, Pecinka and colleagues described in a recently published paper by BMC Biology a direct comparison in frequency of meiotic recombination of the diploid and tetraploid genomes of Arabidopsis 4. One of the most striking methodological challenges to the study is to properly evaluate the recombination parameter in populations of the species at different levels of polyploidy, particularly in autopolyploids. In fact, linkage analysis with autotetraploids has been a historical problem that can be traced back to the pioneering works of the prominent mathematical geneticists 567.
There are at least two major challenges to the tetrasomic linkage analysis. Firstly, double reduction, the most distinct characteristic of polysomic inheritance, allows sister chromatids to enter into the same gamete during meiosis and thus cause systematic distortion in allele segregation 8. Secondly, multiplex allele segregation makes it almost impossible to infer the underlying genotype directly from PRCbased phenotype data even for codominant markers such as a single nucleotide polymorphism or simple sequence repeats 9. To avoid these difficulties, these authors firstly developed a seedbased assay by creating transformants with green and red fluorescent markers expressed under a seedspecific promoter in Arabidopsis thaliana 1011. In parallel, they created diploids, allotetraploids and autotetraploids which carry these fluorescent markers. After carrying out a series of backcross and selection breeding, they were able to create the diploid and tetraploid lines which bear only a single copy of the marker alleles linked on the Arabidopsis chromosome III. These lines were used to create the segregation populations from which marker phenotype data were collected and used for estimation of recombination frequency between the markers 4.
The method these authors implemented for modeling and analyzing the marker data from diploid and tetraploid (allo and autotetraploid) populations needs to be formulated on the basis of the disomic and tetrasomic inheritance models. This paper presents statistically appropriate and mathematically rigorous methods for modeling and reanalyzing the datasets.
Notation, model and analysis
We consider segregation of alleles at the two fluorescent marker loci on the Arabidopsis chromosome III in a F_{2 }family from crossing two parental lines at the marker loci. Following notations of Pecinka et al. 4, parental genotypes at the markers can be denoted by GR/GR and BC/BC for diploid, GR/BC/DE/DE and BC/BC/DE/DE for allotetraploid, and GR/BC/BC/BC and BC/BC/BC/BC for autotetraploid. The parental lines were crossed to generate offspring populations of diploids, allotetraploids and autotetraploids accordingly. Regardless of polyploidy, the offspring populations from mating these parents can be grouped into four phenotypes: yellow (carrying both red and green marker alleles), green (green allele only), red (red allele only) and grey (none of the marker alleles). The number of individuals for each of the four phenotype classes is denoted by n_{1},n_{2},n_{3 }and n_{4 }respectively. Let r be recombination frequency between the two markers and α be the coefficient of double reduction at the green marker, which is nearer to the centromere than the red marker locus. The probability of observing each of the four phenotypes in the diploids and allotetraploids depends on only one parameter, r (f_{i}(r), i = 1,...,4), but characterization of the phenotypic distribution in the autotetraploids needs the two parameters, r and α (f_{i}(α,r), i = 1,...,4).
The logarithm of the model parameter(s) given the observations n_{1},n_{2},n_{3 }and n_{4 }can be written as:
L
(
r

n
1
,
n
2
,
n
3
,
n
4
)
∝
∑
i
=
1
4
n
i
log
[
f
i
(
r
)
]
for the diploid and allotetraploid populations or:
L
(
α
,
r

n
1
,
n
2
,
n
3
,
n
4
)
∝
∑
i
=
1
4
n
i
log
[
f
i
(
α
,
r
)
]
for the autotetraploid population where f_{i}(r) (i = 1,...,4) can be worked out following the principle of twolocus disomic linkage analysis and listed in Table 1. It needs to be pointed out that the phenotypic distribution is common between the diploid and allotetraploid populations. This is because homoeologous pairing was completely excluded in meiosis of the synthesized allotetraploids of Arabidopsis 1011, thus the allotetraploids show strict disomic inheritance. However, calculation of phenotypic distribution in the autotetraploid segregation population must follow the principle of tetrasomic linkage analysis as is detailed in 11. In the present context, the phenotypic distribution can be worked out as:
<p>Table 1</p>Distribution of seed phenotype and the underlying genotype at the two fluorescence markers in F_{2 }diploid and autotetraploid populations and estimates of the model parameters
Diploids/Allotetraploid
Autotetraploids
Phenotype
Genotype^{a}
Frequency (f_{i}(r))
Obs (n_{i})
Obs (n_{i})
Genotype^{b}
Frequency^{c }
Obs (n_{i})
Yellow
G_R_
3(1r)^{2}/4+r(1r)+r^{2}/2
2,805
1,484
G^{(i)}B^{(4i)}R^{(j)}C^{(4 j)}
f_{1}(α,r)
12,707
Green
G_CC
r(2r)/4
322
275
G^{(i)}B^{(4i)}C^{(4)}
f_{2}(α,r)
1,868
Red
BBR_
r(2r)/4
333
298
B^{(4)}R^{(i)}C^{(4 i)}
f_{3}(α,r)
2,216
Grey
BBCC
(1r)^{2}/4
791
320
B^{(4)}C^{(4)}
f^{4}(α,r)
3,098
Estimates
r
^
±
s
.
d
.
Loglikelihood
r
^
±
s
.
d
.
Loglikelihood
α
^
±
s
.
d
.
r
^
±
s
.
d
.
Loglikelihood
Present study
0.1643 ± 0.0062
4,177.79
0.2770 ± 0.0110
2,553.43
0.0676 ± 0.0121
0.3048 ± 0.0051
20815.3
Pecinka et al.
0.154 ± 0.009
4,179.20
0.241 ± 0.018
2,559.21

0.205 ± 0.011
21010.0
^{a}Stands for any other alleles; ^{b}G^{(i)}B^{(4i)}R^{(j)}C^{(4 i) }is a tetraploid genotype containing i 'G' alleles, 4i 'B' alleles, j 'R' alleles and 4  j 'C' alleles; ^{c}forms of f_{i}(α,r) (i = 1, 2, 3, 4) are presented in context. s.d.: standard deviation.
f
1
(
α
,
r
)
=
g
1
(
α
,
r
)
[
2

g
1
(
α
,
r
)
]
+
2
g
2
(
α
,
r
)
g
3
(
α
,
r
)
f
2
(
α
,
r
)
=
g
2
(
α
,
r
)
[
g
2
(
α
,
r
)
+
2
g
4
(
α
,
r
)
]
f
3
(
α
,
r
)
=
g
3
(
α
,
r
)
[
g
3
(
α
,
r
)
+
2
g
4
(
α
,
r
)
]
f
4
(
α
,
r
)
=
g
4
(
α
,
r
)
2
where g_{i}(α,r) (i = 1, 2, 3, 4) are as given below:
g
1
(
α
,
r
)
=
(
2

α
)
(
1

r
)
2
/
4
+
r
(
1

r
)
/
2
+
(
1

α
)
r
2
/
6
g
2
(
α
,
r
)
=
(
1

α
)
r
(
1

r
)
/
3
+
(
10

α
)
r
2
/
36
g
3
(
α
,
r
)
=
r
(
1

r
)
/
2
+
(
8
+
α
)
r
2
/
36
g
4
(
α
,
r
)
=
(
2
+
α
)
(
1

r
)
2
/
4
+
(
2
+
α
)
r
(
1

r
)
/
3
+
(
2
+
α
)
r
2
/
6
The maximum likelihood estimate (MLE) of the recombination frequency in diploids and allotetraploids can be calculated from solving the normal equations:
∂
L
(
r

n
1
,
n
2
,
n
3
,
n
4
)
/
∂
r
=
0
which is a quadratic equation of r. In the diploid population where n_{1}= 2805, n_{2}= 322, n_{3}= 333 and n_{4}= 791, the quadratic equation has only one real root filling in [0.0, 0.5], which is the MLE
r
^
=
0
.
1643
. Based on the likelihood function, one can calculate the standard deviation of the estimate from the Fisher's information measure for MLE. In the present context, it equals:

1
/
∂
2
L
(
r

n
1
,
n
2
,
n
3
,
n
4
)
/
∂
r
2
r
=
r
^
=
0
.
0062
(
Table 1
)
.
In their original report 4, Pecinka et al. estimated the recombination frequency by equating the probability of the recombinant individuals, in other words, those displaying green and red seeds, to the observed proportion of these individuals. In the present notations, the probability has a form of 2rr^{2 }= 2(n_{2}+n_{3})/n where n = n_{1}+n_{2}+n_{3}+n_{4}. They provided an estimate of
r
^
=
0
.
154
with a standard deviation of 0.009. It is not clear how the standard deviation was calculated. The method for calculating the recombination frequency may not be statistically appropriate in two aspects. Firstly, the calculation did not use the full information of the data. For example, the individuals with yellow seeds were not taken into consideration when counting for recombination events. In fact, there is a proportion of [r(1r)+r^{2}/2]/[3(1r)^{2}/4+r(1r)+r^{2}/2] among this group of individuals which carry recombinant gametes. Secondly, the MLE of r obtained from the present analysis is four times as likely as that provided by the original report, that is:
exp
[
L
(
r
=
0
.
1643
)

L
(
r
=
0
.
154
)
]
=
4
.
096
.
In the allotetraploid population where n_{1}= 1484, n_{2}= 275, n_{3}= 298 and n_{4}= 320, the MLE and corresponding standard deviation are calculated as 0.2770 ± 0.0110. The estimate is in contrast to 0.241 with standard deviation = 0.018 in the original report.
To analyze the likelihood model for the autotetraploid marker data, we firstly noticed that the parameter α involves information of allele segregation at the green marker only. By setting r = 0 in the phenotypic probabilities given above, we can work out:
p
g
(
α
)
=
f
1
(
α
,
0
)
+
f
2
(
α
,
0
)
=
(
12

4
α

α
2
)
/
16
p
0
(
α
)
=
f
3
(
α
,
0
)
+
f
4
(
α
,
0
)
=
(
2
+
α
)
2
/
16
which represent the probability of observing an individual carrying or not carrying the green fluorescent marker respectively. Let n_{g }= n_{1}+n_{2 }be the number of individuals carrying the green marker allele and n_{0 }= n_{3}+n_{4 }be the number of individuals not carrying the allele. The loglikelihood function has the form:
L
(
α

n
g
,
n
0
)
=
n
g
L
o
g
[
p
g
(
α
)
]
+
n
0
L
o
g
[
p
0
(
α
)
]
Solving the following equation:
∂
L
(
α

n
g
,
n
0
)
/
∂
α

n
g
=
14575
,
n
0
=
5314
=
0
results in the MLE of
α
^
=
0
.
0676
with a standard deviation of 0.0121 (Table 1).
Incorporating α^ = 0.0676 into the likelihood function, we found that the equation:
∂
L
(
α
^
,
r
)
/
∂
r

n
1
=
12707
,
n
2
=
1868
,
n
3
=
2216
,
n
4
=
3098
=
0
has only one real root,
r
^
=
0
.
3048
, which is the MLE of the recombination frequency under the tetrasomic model. The estimate has a standard deviation of 0.0051, which was calculated from the second derivative of the likelihood function at the MLE. Based on the estimates of α and r, we can predict the coefficient of double reduction at the red marker from:
β
^
=
[
α
^
(
3

4
r
^
)
2
+
2
r
^
(
3

2
r
^
)
]
/
9
=
0
.
1857
[
12
]
which is below the upper bound of the maximum value of double reduction ¼ 12. The estimates of α and β strongly suggest multivalent pairing among the homologous chromosomes and, in turn, double reduction occurring in the genome region flanked by the fluorescent markers (Table 1).