Efficiency of mapping epistatic quantitative trait loci

Abstract

Most theoretical studies on epistatic QTL mapping have shown that this procedure is powerful, efficient to control the false positive rate (FPR), and precise to localize QTLs. The objective of this simulation-based study was to show that mapping epistatic QTLs is not an almost-perfect process. We simulated 50 samples of 400 F₂ plants/recombinant inbred lines, genotyped for 975 SNPs distributed in 10 chromosomes of 100 cM. The plants were phenotyped for grain yield, assuming 10 epistatic QTLs and 90 minor genes. Adopting basic procedures of r/qtl package, we maximized the power of detection for QTLs (56–74%, on average) but associated with a very high FPR (65%) and a low detection power for the epistatic pairs (7%). Increasing the average detection power for epistatic pairs (14%) highly increased the related FPR. Adopting a procedure to find the best balance between power and FPR, there was a significant decrease in the power of QTL detection (17–31%, on average), associated with a low average detection power for epistatic pairs (8%) and an average FPR of 31% for QTLs and 16% for epistatic pairs. The main reasons for these negative results are a simplified specification of the coefficients of epistatic effects, as theoretically proved, and the effects of minor genes since 2/3 of the FPR for QTLs were due to them. We hope that this study, including the partial derivation of the coefficients of epistatic effects, motivates investigations on how to increase the power of detection for epistatic pairs, effectively controlling the FPR.

Appendix: The coefficients of the epistatic effects in F2 generation

Assume that there is a QTL in two intervals with the independent assortment (due to independence or free recombination; Q1/q1 and Q2/q2). Assume also association for the two QTLs and the four flanking SNPs (A/a, B/b, C/c, and D/d) in the F₁ (AQ1BCQ2D/aq1bcq2d). Assuming no interference, the F₁ gamete probabilities in relation to the first QTL and their flanking markers are well known, given by:

$$P(AQ1B)={P}_{111}=P(aq1b)={P}_{000}=(1-{r}_{aq1})(1-{r}_{q1b})/2$$

$$P(Aq1b)={P}_{100}=P(aQ1B)={P}_{011}={r}_{aq1}(1-{r}_{q1b})/2$$

$$P(AQ1b)={P}_{110}=P(aq1B)={P}_{001}=(1-{r}_{aq1}){r}_{q1b}/2$$

$$P(Aq1B)={P}_{101}=P(aQ1b)={P}_{010}={r}_{aq1}{r}_{q1b}/2$$

Defining by R the F₁ gamete probabilities in relation to the second QTL and their flanking markers, we have, for example,

$$P(Q1Q1Q2Q2|AABBCCDD)=\frac{P(AAQ1Q1BB)P(CCQ2Q2DD)}{P(AABB)P(CCDD)}=\frac{{P}_{111}^{2}{R}_{111}^{2}}{{P}_{1.1}^{2}{R}_{1.1}^{2}}$$

The expectation of the genotypic values for the F₂ individuals with SNP genotype AABBCCDD is

$$\begin{array}{ll}E(G|AABBCCDD)\\ =P(Q1Q1Q2Q2|AABBCCDD)\left[\right.{m}_{1}+{m}_{2}+{a}_{1}+{a}_{2}\,+\,4({\alpha }_{Q1}{\alpha }_{Q2})\\ \quad+\,2({\alpha }_{Q1}{\delta }_{Q2Q2}) +2({\delta }_{Q1Q1}{\alpha }_{Q2})+({\delta }_{Q1Q1}{\delta }_{Q2Q2})\left]\right.+ \ldots \\ \quad+\,P(q1q1q2q2|AABBCCDD)\left[\right.{m}_{1}+{m}_{2}-{a}_{1}-{a}_{2}\,+\,4({\alpha }_{q1}{\alpha }_{q2})\\ \quad +\,2({\alpha }_{q1}{\delta }_{q2q2})+ 2({\delta }_{q1q1}{\alpha }_{q2})+({\delta }_{q1q1}{\delta }_{q2q2})\left]\right.\\ ={m}_{1}+{m}_{2}+{\alpha }_{1}{a}_{1}+{\alpha }_{2}{a}_{2}+{\delta }_{1}{d}_{1}+{\delta }_{2}{d}_{2}+{I}_{2222}\end{array}$$

where

$$\begin{array}{ll}{\alpha }_{1}=\frac{{P}_{111}^{2}-{P}_{101}^{2}}{{P}_{1.1}^{2}}=\frac{{(1-{r}_{aq1})}^{2}{(1-{r}_{q1b})}^{2}-{r}_{aq1}^{2}{r}_{q1b}^{2}}{{(1-{r}_{aq1}-{r}_{q1b}+2{r}_{aq1}{r}_{q1b})}^{2}}\\ \quad\,\,=\frac{{(1-{r}_{aq1})}^{2}{(1-{r}_{q1b})}^{2}-{r}_{aq1}^{2}{r}_{q1b}^{2}}{{(1-{r}_{ab})}^{2}}\end{array}$$

$${\delta }_{1}=\frac{2{P}_{111}{P}_{101}}{{P}_{1.1}^{2}}=\frac{2{r}_{aq1}(1-{r}_{aq1}){r}_{q1b}(1-{r}_{q1b})}{{(1-{r}_{ab})}^{2}}$$

$${\alpha }_{2}=\frac{{R}_{111}^{2}-{R}_{101}^{2}}{{R}_{1.1}^{2}}=\frac{{(1-{r}_{cq2})}^{2}{(1-{r}_{q2d})}^{2}-{r}_{cq2}^{2}{r}_{q2d}^{2}}{{(1-{r}_{cd})}^{2}}$$

$${\delta }_{2}=\frac{2{R}_{111}{R}_{101}}{{R}_{1.1}^{2}}=\frac{2{r}_{cq2}(1-{r}_{cq2}){r}_{q2d}(1-{r}_{q2d})}{{(1-{r}_{cd})}^{2}}$$

and I₂₂₂₂ is the average epistatic value for the SNP genotype, given by

$$\begin{array}{ll}{I}_{2222}=(\frac{1}{{P}_{1.1}{R}_{1.1}})4\left[\right.{P}_{111}{R}_{111}({\alpha }_{Q1}{\alpha }_{Q2})+ {P}_{111}{R}_{101}({\alpha }_{Q1}{\alpha }_{q2})+{P}_{101}{R}_{111}({\alpha }_{q1}{\alpha }_{Q2})\\ \qquad\qquad\qquad\,\,+\,{P}_{101}{R}_{101}({\alpha }_{q1}{\alpha }_{q2})\left]\right.\\ \qquad\qquad\qquad\,\,+\,(\frac{1}{{P}_{1.1}{R}_{1.1}^{2}})2\left\{\right.{P}_{111}\left[\right.{R}_{111}^{2}({\alpha }_{Q1}{\delta }_{Q2Q2})+2{R}_{111}{R}_{101}({\alpha }_{Q1}{\delta }_{Q2q2})\\ \qquad\qquad\qquad\,\,+\,{R}_{101}^{2}({\alpha }_{Q1}{\delta }_{q2q2})\left]\right.\\ \qquad\qquad\qquad\,\,+\,{P}_{101}[{R}_{111}^{2}({\alpha }_{q1}{\delta }_{Q2Q2})+2{R}_{111}{R}_{101}({\alpha }_{q1}{\delta }_{Q2q2}) +{R}_{101}^{2}({\alpha }_{q1}{\delta }_{q2q2})]\left\}\right.\\ \qquad\qquad\qquad\,\,+\,(\frac{1}{{P}_{1.1}^{2}{R}_{1.1}})2\left\{\right.{R}_{111}\left[\right.{P}_{111}^{2}({\delta }_{Q1Q1}{\alpha }_{Q2})+2{P}_{111}{P}_{101}({\delta }_{Q1q1}{\alpha }_{Q2})\\ \qquad\qquad\qquad\,\,+{P}_{101}^{2}({\delta }_{q1q1}{\alpha }_{Q2})\left]\right.\\ \qquad\qquad\qquad\,\,+{R}_{101}\left[\right.{P}_{111}^{2}({\delta }_{Q1Q1}{\alpha }_{q2})+2{P}_{111}{P}_{101}({\delta }_{Q1q1}{\alpha }_{q2})\\ \qquad\qquad\qquad\,\,+{P}_{101}^{2}({\delta }_{q1q1}{\alpha }_{q2})\left]\right.\left\}\right.\\ \qquad\qquad\qquad\,\,+\,(\frac{1}{{P}_{1.1}^{2}{R}_{1.1}^{2}})\left\{\right.{P}_{111}^{2}\left[\right.{R}_{111}^{2}({\delta }_{Q1Q1}{\delta }_{Q2Q2})\\ \qquad\qquad\qquad\,\,+\,2{R}_{111}{R}_{101}({\delta }_{Q1Q1}{\delta }_{Q2q2})+{R}_{101}^{2}({\delta }_{Q1Q1}{\delta }_{q2q2})\left]\right.\\ \qquad\qquad\qquad\,\,+\,2{P}_{111}{P}_{101}[{R}_{111}^{2}({\delta }_{Q1q1}{\delta }_{Q2Q2}) +2{R}_{111}{R}_{101}({\delta }_{Q1q1}{\delta }_{Q2q2})+{R}_{101}^{2}({\delta }_{Q1q1}{\delta }_{q2q2})]\\ \qquad\qquad\qquad\,\,+\,{P}_{101}^{2}\left[\right.{R}_{111}^{2}({\delta }_{q1q1}{\delta }_{Q2Q2})\\ \qquad\qquad\qquad\,\,+\,2{R}_{111}{R}_{101}({\delta }_{q1q1}{\delta }_{Q2q2})+{R}_{101}^{2}({\delta }_{q1q1}{\delta }_{q2q2})\left]\right.\left\}\right.\end{array}$$

Note that the coefficients of the a and d deviations are those presented by Haley and Knott (1992). Kempthorne’s assumptions are (since p₁ = p₂ = 1/2):

(i) restrictions for the AA effects: 1) (α_Q1 α_Q2)+(α_Q1 α_q2) = 0; 2) (α_q1 α_Q2)+(α_q1 α_q2) = 0; 3) (α_Q1 α_Q2)+(α_q1 α_Q2) = 0; and 4) (α_Q1 α_q2)+(α_q1 α_q2) = 0.

(ii) restrictions for the AD effects: 1) (α_Q1 δ_Q2Q2)+(α_Q1 δ_Q2q2) = 0; 2) (α_Q1 δ_Q2q2)+(α_Q1 δ_q2q2) = 0; 3) (α_q1 δ_Q2Q2)+(α_q1 δ_Q2q2) = 0; 4) (α_q1 δ_Q2q2)+(α_q1 δ_q2q2) = 0; 5) (α_Q1 δ_Q2Q2)+(α_q1 δ_Q2Q2) = 0; 6) (α_Q1 δ_Q2q2)+(α_q1 δ_Q2q2) = 0; and 7) (α_Q1 δ_q2q2)+(α_q1 δ_q2q2) = 0 (six out of the seven are independent).

(iii) restrictions for the DA effects: 1) (δ_Q1Q1 α_Q2)+(δ_Q1q1 α_Q2) = 0; 2) (δ_Q1q1 α_Q2)+(δ_q1q1 α_Q2) = 0; 3) (δ_Q1Q1 α_q2)+(δ_Q1q1 α_q2) = 0; 4) (δ_Q1q1 α_q2)+(δ_q1q1 α_q2) = 0; 5) (δ_Q1Q1 α_Q2)+(δ_Q1Q1 α_q2) = 0; 6) (δ_Q1q1 α_Q2)+(δ_Q1q1 α_q2) = 0; and 7) (δ_q1q1 α_Q2)+(δ_q1q1 α_q2) = 0 (six out of the seven are independent).

(iv) restrictions for the DD effects: 1) (δ_Q1Q1 δ_Q2Q2)+(δ_Q1Q1 δ_Q2q2) = 0; 2) (δ_Q1Q1 δ_Q2q2)+(δ_Q1Q1 δ_q2q2) = 0; 3) (δ_Q1q1 δ_Q2Q2)+(δ_Q1q1 δ_Q2q2) = 0; 4) (δ_Q1q1 δ_Q2q2)+(δ_Q1q1 δ_q2q2) = 0; 5) (δ_q1q1 δ_Q2Q2)+(δ_q1q1 δ_Q2q2) = 0; 6) (δ_q1q1 δ_Q2q2)+(δ_q1q1 δ_q2q2) = 0; 7) (δ_Q1Q1 δ_Q2Q2)+(δ_Q1q1 δ_Q2q2) = 0; 8) (δ_Q1q1 δ_Q2Q2)+(δ_q1q1 δ_Q2Q2) = 0; 9) (δ_Q1Q1 δ_Q2q2)+(δ_Q1q1 δ_Q2q2)=0; 10) (δ_Q1q1 δ_Q2q2)+(δ_q1q1 δ_Q2q2) = 0; 11) (δ_Q1Q1 δ_q2q2)+(δ_Q1q1 δ_q2q2)=0; and 12) (δ_Q1q1 δ_q2q2)+(δ_q1q1 δ_q2q2) = 0 (nine out of the 12 are independent).

Using the restrictions, I₂₂₂₂ can be expressed as follows:

$$\begin{array}{c}{I}_{2222}=4(\frac{{P}_{111}}{{P}_{1.1}})(\frac{{R}_{111}}{{R}_{1.1}}){\alpha }_{1}{\alpha }_{2}({\alpha }_{Q1}{\alpha }_{Q2})+2(\frac{{P}_{111}}{{P}_{1.1}}){\alpha }_{1}(1-2{\delta }_{2})({\alpha }_{Q1}{\delta }_{Q2Q2})\\ \qquad\qquad\qquad+ \,2(\frac{{R}_{111}}{{R}_{1.1}})(1-2{\delta }_{1}){\alpha }_{2}({\delta }_{Q1Q1}{\alpha }_{Q2})- (1-2{\delta }_{1})(1-2{\delta }_{2})({\delta }_{Q1Q1}{\delta }_{Q2Q2})\end{array}$$

Thus, the coefficients for the additive × additive, additive × dominance, dominance × additive, and dominance × dominance effects are not given by the product of the coefficients of the additive and dominance effects. Assuming incomplete linkage between the SNPs B/b and C/c (0 < r_bc < 1/2), we have

$$\scriptstyle\begin{array}{c}P(Q1Q1Q2Q2|AABBCCDD)\,=\,\frac{P{(AQ1BCQ2D)}^{2}}{P{(ABCD)}^{2}}= \frac{{P}_{111}^{2}{[2(1-{r}_{bc})]}^{2}{R}_{111}^{2}}{{P}_{1.1}^{2}{[2(1-{r}_{bc})]}^{2}{R}_{1.1}^{2}}=\frac{{P}_{111}^{2}{R}_{111}^{2}}{{P}_{1.1}^{2}{R}_{1.1}^{2}}\end{array}$$

since $P(AQ1BCQ2D)=P(A).P(Q1|A).P(B|Q1).P(C|B).P(Q2|C).P(D|Q2)=(1/2).(1-{r}_{aq1}).(1-{r}_{q1b}).(1-{r}_{bc}).(1-{r}_{cq2}).(1-{r}_{q2d})={P}_{111}.(1-{r}_{bc}).2{R}_{111}$. Assuming complete linkage (r_bc = 0), the probability is the same, but there will be 32 and not 64 gametes for the F₁.