Family-based association mapping in crop species

Abstract

Identification of allelic variants associated with complex traits provides molecular genetic information associated with variability upon which both artificial and natural selections are based. Family-based association mapping (FBAM) takes advantage of linkage disequilibrium among segregating progeny within crosses and among parents to provide greater power than association mapping and greater resolution than linkage mapping. Herein, we discuss the potential adaption of human family-based association tests and quantitative transmission disequilibrium tests for use in crop species. The rapid technological advancement of next generation sequencing will enable sequencing of all parents in a planned crossing design, with subsequent imputation of genotypes for all segregating progeny. These technical advancements are easily adapted to mating designs routinely used by plant breeders. Thus, FBAM has the potential to be widely adopted for discovering alleles, common and rare, underlying complex traits in crop species.

Appendix: Conditional probabilities of QTL genotypes given flanking markers in DHF2 populations

In maize, DH_F2 can be produced through inducing haploids from F₂ plants using inducer lines and doubling them using chemical colchicine (Chase 1951; Bordes et al. 1997; Bernardo 2009). Consider three linked loci A, Q and B, which are fixed with AAQQBB and aaqqbb for two parental lines, respectively. r ₁ is recombination rate between A and Q loci, r ₂ between Q and B loci and r between A and B loci. Due to the high density of markers we assume that probability of double-cross over events per meiosis is zero. Therefore, r = r ₁ + r ₂.

Production of DH_F2 involves three processes. Firstly, F₁ plants (AQB//aqb) produce a total of six gametes: (1) No recombination event. AQB and aqb, with a frequency of (1 − r)/2, respectively. (2) A single recombination event between A and Q. aQB and Aqb, with a frequency of r ₁/2, respectively. (3) A single recombination between Q and B. AQb and aqB, with a frequency of r ₂/2, respectively. Secondly, female and male gametes are randomly mated to produce a total of 21 F₂ phased genotypes: (1) No recombination event. AQB//AQB, aqb//aqb and AQB//aqb. (2) A recombination event in one gamete between A and Q. aQB//AQB, aQB//aqb, Aqb//AQB and Aqb//aqb. (3) A recombination event in one gamete between Q and B. AQb//AQB, AQb//aqb, aqB//AQB and aqB//aqb. (4) Recombination events in both gametes between A and Q. aQB//aQB, Aqb//Aqb and aQB//Aqb. (5) Recombination events in both gametes between Q and B. AQb//AQb, aqB//aqB, AQb//aqB. (6) Recombination events in both gametes, one occurs between A and Q, but the other occurs between Q and B. AQb//aQB, aqB//aQB, AQb//Aqb and aqB//Aqb. The frequency of one phased genotype is two times the product of two F₁ gamete frequencies if the two gametes have different genotypes and the product of two F₁ gamete frequencies if the two gametes have the same genotype. For example, AQB//AQB is (1 − r)²/4 and aQB//AQB is r ₁(1 − r)/2. F₂ plants produce haploids which are equivalent to gametes. For each phased F₂ genotype, the frequencies of gametes are produced as with F₁ described above and then be multiplied by its F₂ genotype frequency to obtain the frequencies of corresponding double haplotype frequencies. For example, an F₂ genotype of aQB//Aqb, which has a frequency of (r ₁)²/2, produces haploids: aQB with a frequency of (1 − r)/2, Aqb with a frequency of (1 − r)/2, AQB with a frequency of r ₁/2, aqb with a frequency of r ₁/2, aQb with a frequency of r ₂/2 and AqB with a frequency of r ₂/2. The frequencies of haploids aQB, Aqb, AQB, aqb, aQb and Aqb are r ²₁ (1 − r)/4, r ²₁ (1 − r)/4, r ³₁ /4, r ³₁ /4, r ³₂ /4, and r ³₂ /4, respectively. A total of eight double haploid genotypes are produced from F₂ plants: AAQQBB, aaqqbb, AAqqBB, aaQQbb, aaQQBB, AAqqbb, aaqqBB and AAQQbb. Their frequencies are as follows:

$$\begin{aligned} & P({\text{AAQQBB}}) = P({\text{aaqqbb}}) = p_{1} (2 - r) + p_{2} (1 - r_{2} /2) + p_{3} (1 - r_{1} /2) + p_{4} r_{1} + p_{5} r_{2} + 0.5p_{6} r \\ & P({\text{AAqqBB}}) = P({\text{aaQQbb}}) = 0.5p_{2} r_{2} + 0.5p_{3} r_{1} + p_{4} r_{2} + p_{5} r_{1} + 0.5p_{6} r \\ & P({\text{aaQQBB}}) = P({\text{AAqqbb}}) = p_{1} r_{1} + p_{2} (1 - r_{2} /2) + 0.5p_{3} r_{1} + p_{4} (2 - r) + p_{6} (1 - r/2) \\ & P({\text{aaqqBB}}) = P({\text{AAQQbb}}) = p_{1} r_{2} + 0.5p_{2} r_{2} + p_{3} (1 - r_{1} /2) + p_{5} (2 - r) + p_{6} (1 - r/2) \\ & \quad {\text{where }}p_{1} = (1 - r)^{2} /4,p_{2} = \, r_{1} (1 - r)/2,p_{3} = r_{2} (1 - r)/2,p_{4} = r_{1}^{2} /4,p_{5} = r_{2}^{2} /4{\text{ and }}p_{6} = r_{1} r_{2} /2. \\ \end{aligned}$$

On the ends of chromosomes a single marker could be used to impute the QTL genotypes. Assume that Q locus is linked with marker locus with recombination rate of r between them. The frequencies of genotypes in DH_F2 are:

$$\begin{aligned} & P({\text{AAQQ}}) = P({\text{aaqq}}) = p_{1} (2 - r) + p_{2} + p_{3} r \\ & P({\text{AAqq}}) = P({\text{aaQQ}}) = p_{1} r + p_{2} + p_{3} (2 - r) \\ & \quad {\text{where }}p_{1} = (1 - r)^{2} /4,p_{2} = r(1 - r)/2{\text{ and }}p_{3} = r^{2} /4 \\ \end{aligned}$$