The calculator is based on estimable meiotic parameters that influence chromosome segregation during meiosis of autotetraploids. For instance, parameter β determines, whether or not there is preferential pairing (disomic inheritance) between chromosomes. β is restricted to have values of 0 or 1. In case of tetrasomic inheritance or no preferential pairing β = 1. With four alleles per locus (i.e. ABCD) six gamete combinations are possible (AB, AC, AD, BC, BD, CD) under random pairing. In case of preferential pairing of homologues, β = 0. Furthermore, preferential pairing between genomes results in only four gamete combinations. To illustrate this further we named each genome jklm and alleles at a locus correspond to each genome in that order. For example, a genotype ABCD with preferential pairing between jk/lm has gametes with haplotypes AC, AD, BC, BD. In case of preferential pairing, only one out of the three possible pairing scenarios jk/lm, jl/km, or jm/kl can be true. This has been implemented in our calculations by defining a Δ value to test all three possible scenarios sequentially. Technically, if the Δ value is equal to 1 for one scenario, it is 0 for the other. Usually, the values for β and Δ do not differ for S and Z.

S and Z are independent loci. Therefore, normal chromosome segregation of a genotype with four different alleles for S or Z and with no preferential pairing between chromosomes jk, jl, jm, kl, km, and lm will produce six possible pollen haplotype combinations for each locus. Thus, the probability of a haplotype is 1/6. It is possible that recombination during multivalent formation in meiosis followed by double reduction can cause sister chromatids to segregate together during meiosis. Therefore, there is a chance of two identical alleles to end up in the same gamete by double reduction 12. If we consider a single chromatid, only 1 out of the seven remaining is its corresponding sister chromatid. Therefore, the probability (α) of two sister chromatids to end up in the same gamete ranges from 0 to 1/7. The value of α is species-specific, and depends on the distance between the locus and the centromere 1216. For the jklm genotype, there are four possible gamete combinations that have two identical by descent alleles (IBD) i.e., jj, kk, ll, mm. Therefore, the frequency of each of these pollen genotypes is α/4. The closer S and Z are linked with the centromere, the smaller is the α value 16.

The input form of the program has a cell to enter α values to calculate gamete frequencies that can be different for each locus. For example, Phalaris coerulescens is a diploid species for which S and Z have been mapped. The S locus is located in a subcentromeric region of chromosome 1, and the Z locus is located on the long arm at the end of chromosome 2 17. In P. coerulescens, the α value for S will, therefore, be close to 0, whereas the one for Z should be larger. If the α value is unknown a default value such as 1/14 (average between the extremes of 0 and 1/7) could be used.

The input window (Figure 2) has boxes for meiotic parameters for both loci (S and Z) as described above and for a given pair of genotypes. The default setting is disomic inheritance and preferential pairing between jl/km for both loci, which can be used as a null hypothesis. The parameters settings have some restrictions. For example, if β = 1, all Δ values are set to 0. If an α value is given, then β = 1. It is possible to enter the genotype for a pair of individuals. To our knowledge there is no information about allelic diversity for S and Z loci since the determinants have not been identified yet. However, we developed the software assuming that the maximum difference between two genotypes is eight different alleles at each locus. Therefore, the program accepts values for S defined by letters from A-H and numbers from 1-8 for Z. Settings will be summarized in words once input parameters are set. All the input settings will be summarized in words once input parameters are set.

The initial output (Figure 3) obtained is a summary table of the input values. After the analysis is performed, the pollen haplotypes for each genotype and each locus are displayed. Pollen haplotypes are calculated based on the formulas in Table 1. Stift (2008) published formulas to calculate pollen haplotypes, which differ from ours in nomenclature. The second output contains tables for pollen genotypes and frequencies as well as pollen compatibility in reciprocal crosses between genotypes 1 and 2 (Figure 4a). For example for a genotype ABCD1234, the matrix for pollen compatibility has the 10X10 possible pollen haplotypes. Pollen haplotypes are arranged in column 1. In the column header of columns 2-5 and 6-9, the four female alleles for both, S and Z are displayed. Under each female allele, the numbers of identical alleles in the pollen are counted. Since pollen genotypes have only two alleles for each locus, the number of alleles it can possibly share with a female genotype at the S or Z locus, respectively, has to be 0, 1, or 2 alleles (see Additional file 1 for more description on input parameters and output).

Genomes are named generically jklm. Therefore, alleles for S (i.e. ABCD) correspond to genomes jklm respectively; and alleles for Z (i.e. 1234) also correspond to genome jklm respectively. α is the probability of double reduction and ranges from 0- 0.14. β is a parameter for chromosome pairing during meiosis and can have values of 0 (disomic) or 1 (tetrasomic). Δ is a parameter that defines which chromosomes show preferential pairing.

For each possible pollen-stigma combination the number of shared alleles at the S and Z locus are given and summarized independently for S and Z. ΣS and ΣZ reflect the sum of shared alleles at S and Z, respectively (columns 10 and 11). In autotetraploids, for a given allele a locus can be monoallelic (AAAA), biallelic (AABB) or triallelic (AABC) 12. Therefore, the program is designed to avoid double counting of alleles.

Pollen compatibility (C) is calculated for each pollen- stigma combination for the two models described above. Mathematically, the Model I algorithm states that if ΣS or ΣZ < 2, the pollen is compatible, therefore C = 1; otherwise C = 0. For Model II, the algorithm states that if ΣS or ΣZ = 0 then pollen is compatible and C = 1; otherwise C = 0. Thereafter, each C value is multiplied with the respective pollen haplotype frequency. The percentage of compatible pollen for a particular pollen-stigma genotype combination is the sum of the products between C values and pollen frequencies (Figure 4b).

Our program allows the user to continue the analysis with similar meiotic parameters. Thus, the final output (Figure 5) is a summary table containing meiotic parameters used, genotype of individuals and the compatibility percentages under both models.

A male with genotype ABCD1234 will produce gametes with different haplotypes depending on the mode of pairing during meiosis. Tables 2 & 3 show possible pollen haplotypes, where shaded pollen haplotypes were produced under the assigned pairing and meiotic parameters. If such a male is crossed onto a female with different alleles at S and Z (i.e., EFGH5678), it is possible to determine the meiotic behavior based on the progeny. Based on output information, it is also possible to determine the number of progeny needed, to screen to find a certain haplotype.

Possible pollen haplotypes produced under the designed preferential pairing are in bold (Δ1-3).

Pure tetrasomic inheritance produces all 36 possible genotypes are shown in regular format. Haplotypes in bold are the ones formed by chromatid segregation and haplotypes in italic are formed by double reduction. The probabilities of each S or Z allele combination are given; the gamete frequency can be calculated as joined probability of the frequency at each locus.

Assuming that markers are available to distinguish alleles for S and Z, the stepwise procedure to determine meiotic behavior for S and Z loci would be as follows: first, distinguish between disomic and tetrasomic inheritance. If disomic inheritance occurs, under any type of preferential pairing only 16 pollen haplotypes are possible. In contrast, if tetrasomic inheritance occurs, 36 pollen haplotypes are possible. To distinguish between both, it is necessary to evaluate the presence or absence of any of the 20 (36-16) haplotypes, that are indicative of tetrasomic inheritance.

If disomic inheritance is true, it can be determined, which genomes are pairing during meiosis. One haplotype that distinguishes between jk/lm, jl/km or jm/kl needs to be identified. In the example presented in Table 2, AB12 is a haplotype that can differentiate between jk/lm and jl/km. The probability of occurrence of these haplotypes in the progeny is 0.0625, to find at least one progeny we use the formula described by Sedcole (method III) 18: carrying a specific pollen haplotype,

n = 2 r − 0.5 + z 2 1 − q + z z 2 1 − q 2 + 4 1 − q r − 0.5 1 2 2 q

Where n = total number of plants to be screened

r- required number of plants with the desired genotype

q = frequency of plants with the desired genotype

p = probability of recovering the required number of plants with the desired genotype

z- probability based on a z distribution

Therefore, we need to screen at least 56 plants in this example to be 95% confident to find one offspring carrying the AB12 haplotype. If Δ2 is true, it is necessary to select a haplotype to distinguish it from Δ3 (i.e., AD12). Using the same calculation as above, 56 individuals need to be screened to find at least one genotype carrying the AD12 haplotype with a 95% probability.

On the other hand, if tetrasomic inheritance is true for our species, it is necessary to determine, whether chromosome segregation or double reduction occurs. The probability of finding one haplotype produced by chromatid segregation is 0.2688 (the sum of the frequencies of haplotypes of light shaded boxes in Table 3). Using the same formula as above, to find at least one of these haplotypes in the progeny with a 95% confidence, a total of 11 individuals need to be analyzed. Furthermore, if we want to assess the occurrence of double reduction, which occurs with a frequency of 0.0256 (the sum of the frequencies of no shaded lines in Table 3), a total of 139 individuals need to be screened.

Our program can be used to determine the most likely model underlying pollen compatibility. For example, in a cross between ABCD1234 and ABCE1235, these genotypes have one different allele at each locus. Under Model I, reciprocal crosses between these genotypes reaches 75% pollen compatibility. In contrast, under Model II, this cross is completely incompatible. Multivalent formation, chromatid segregation and double reduction during meiosis have an effect on pollen compatibility. A similar cross, in an autotetraploid species (β = 1) with chromatid segregation and double reduction (α = 1/7), will have 71.6% pollen compatibility under Model I or 6.9% under Model II (Figure 6). In general, Model II is more restrictive than Model I. Model I ranges from 0% to 50% compatibility, when one different allele is present at either S or Z. In contrast, Model II has values lower than 50%, if there are fewer than three different alleles at any locus (Figure 7).

If meiotic parameters are known or can be calculated, but no markers are available for genotyping S and Z, it is possible to infer the number of alleles segregating in a population of a pair cross. In a plant species with tetrasomic inheritance (β = 1) and chromatid segregation/double reduction (α = 1/7), the number of segregating alleles between the pollen donor and female plays an important role on the pollen compatibility in reciprocal crosses. For instance, if the number of total segregating alleles is four or more for at least one locus (i.e., ABCD1234 X AAAB1115), pollen compatibility is higher than 90% under Model I but only 63% for Model II (Table 4). However, the distribution of these alleles in the two genotypes is important for the pollen compatibility in reciprocal crosses. If ABCD1234 is used as the pollen donor it has two and three alleles for S and Z, respectively, which differ from the female AAAB1115. In contrast, when AAAB1115 is used as the pollen donor, it has zero and one allele for S and Z, respectively, that differ from the female ABCD1234, resulting in a pollen compatibility of 47% under Model I, or 4% under model II (Table 3). In addition, if the number of segregating alleles between genotypes drops to two, pollen compatibility is significantly reduced. For instance, a cross between AAAB1115 X AABB1111 will result in no pollen compatibility under any Model. However, if AABB1115 is used as male the single different allele at Z will cause an increase of pollen compatibility to 47% under Model I, or 4% under Model II (Table 4).

Alleles that are IBD are in bold and underlined, First column (total number of segregating alleles) is the number of different alleles between S or Z across genotypes. The columns that follow count the number of different alleles between males and females.

The information provided by PollenCALC can be used to design experiments in order to test hypotheses related to the pairing and segregation of chromosomes during meiosis in tetraploid species with a gametophytic S-Z SI system. The program can be used if linked or functional markers for S and Z are available or not.

The genetic mechanism of SI is still unknown for some self-incompatible tetraploid grasses. Also, there are no markers available for S and Z which makes it difficult to understand and predict pollen compatibility. As described above, there are two possible models for pollen compatibility. In vitro or in vivo pollinations can be performed to determine, which of the two models most likely applies. As demonstrated, only few different alleles are required for Model I. Furthermore, under Model I, if only one different allele is found at either S or Z, pollen compatibility reaches 50%, whereas for Model II values below 50% are common when the number of different alleles is below three for any locus. “Both in vitro and in vivo pollination test provide bias estimates of pollen compatibility. For instance, current in vitro pollination tests rely on imaging and counting to provide estimates of pollen compatibility. In vivo pollination analyzed as seed set can be affected by pollen death due to environmental condition and pre or post zygotic abortion. Nevertheless, PollenCALC can be used to design experiments in order to identify genotypes that are most informative to distinguish the two models i.e. genotypes with extreme pollen compatibilities. Eventually the use of S and Z linked markers are required for confirmation.”

There are other applications for this program when markers are available. For example, genome-wide markers are used in several studies to determine preferential pairing in polyploids. In species with SI, segregation of markers linked or in linkage disequilibrium with S and Z are distorted, which can result in biased interpretation of marker segregation data 1920. PollenCALC can help to identify genotypes that can distinguish between types of pairing. Knowledge about pairing and inheritance in a given species makes it possible to correct mapping distances for segregation distortion due to SI in mapping experiments.

Also, PollenCALC calculates pollen haplotype frequencies of individuals in pair crosses and can be used to calculate pollen compatibility between those genotypes in reciprocal crosses. This information can also be used to maximize seed yield in hybrid breeding programs 21. If markers for S and Z are available, they can be used in combination with our software in breeding programs. For instance in a hybrid seed production programs, PollenCALC can be used to select genotypes with specific S-Z genotypes within one heterotic group that contrast with the S-Z genotype from the individuals in the other heterotic group. If alleles are different between groups, seed yield will be maximized. Also, it is possible to determine potential progeny genotype and frequencies by combining haplotypes and frequencies. In the longer run, this program can be used to simulate allele and genotypes frequencies for S, Z, and loci genetically linked to S and Z over multiple generations. Thus, PollenCALC is a useful tool for plant geneticists and plant breeders working with tetraploid species.