Mapping Ol-4, a gene conferring resistance to Oidium neolycopersici and originating from Lycopersicon peruvianum LA2172, requires multi-allelic, single-locus markers

Abstract

Lycopersicon peruvianum LA2172 is completely resistant to Oidium neolycopersici, the causal agent of tomato powdery mildew. Despite the large genetic distance between the cultivated tomato and L. peruvianum, fertile F₁ hybrids of L. esculentum cv. Moneymaker × L. peruvianum LA2172 were produced, and a pseudo-F₂ population was generated by mating F₁ half-sibs. The disease tests on the pseudo-F₂ population and two BC₁ families showed that the resistance in LA2172 is governed by one dominant gene, designated as Ol-4. In the pseudo-F₂ population, distorted segregation was observed, and multi-allelic, single-locus markers were used to display different marker-allele configurations per locus. Parameters for both distortion and linkage between genetic loci were determined by maximum likelihood estimation, and the necessity of using multi-allelic, single-locus markers was illustrated. Finally, a genetic linkage map of chromosome 6 around the Ol-4 locus was constructed by using the pseudo-F₂ population.

Appendices

Appendix

Estimation of recombination frequency between Ol-4 and (multi-allelic) markers from the F₂ population

Altogether, we used four types of markers (all possible marker-allele configurations in the F₂ population), for each of which we discuss the linkage analysis below.

Appendix 1

Markers that distinguish between the alternative L. peruvianum alleles (p, p*), but only one of these alleles is distinct from the L. esculentum allele (p*≠p=e)

Marker By-4/HypCH4IV is of this type (Table 4; Fig. 2). The configuration of the cross between the two F₁ genotypes (F_1a and F_1b, see Fig. 1) looks as follows:

Since neither of the parents is heterozygous at both loci, markers of this type are uninformative for linkage. In fact, the loci behave as if they segregate independently (see below). Not being aware of this non-informative nature, the joint segregation can be misinterpreted as ‘unlinked loci’.

Later on, we will have to introduce a parameter for preferential transmission of marker alleles. Let this parameter be β for allele transmission rate in the male parent (β=0.5 corresponds to a 1:1 Mendelian ratio). This leads to the following table of gamete combinations and their frequencies:

♀	♂
	β (ol-4 p*)	1−β (ol-4 e)
(1/2)  Ol-4 e	1/2β	1/2(1−β)
(1/2)  ol-4 e	1/2β	1/2(1−β)

The following two-way table shows genotypes for the progeny (numbers in parentheses are the observed numbers for the marker By-4/HypCH4IV from Table 4).

	Ol-4/ol-4	ol-4/ol-4	Sum
e/p*	1/2β (78)	1/2β (75)	β (153)
e/e	1/2(1−β) (15)	1/2(1−β) (18)	(1−β) (33)
Sum	1/2 (93)	1/2 (93)	1 (186)

Despite the distorted segregation, we indeed observe independent segregation between this marker and the Ol-4 locus. The distortion parameter β is estimated as $\hat{\beta } = 153/186 = 0.8226.$

Appendix 2

Markers that distinguish between the two alternative L. peruvianum alleles (p, p*), but only one of these is distinct from the L. esculentum allele (p≠p*=e)

This is the ‘mirror’ situation of (p*≠p=e), discussed above. Marker By-4/ApoI is of this type. The configuration of the cross looks like:

This represents the classical test cross configuration, from which recombination frequency (r), is readily estimated by counting the recombinant genotypes among the offspring. However, for the sake of completeness we introduce a parameter, α, for preferential transmission of marker alleles by the female parent (α=0.5 corresponds to a 1:1 ratio). This leads to the two-way table of genotype frequencies below (observed numbers are in parentheses, which are taken from Table 4 for marker By-4/ApoI).

	Ol-4/ol-4	ol-4/ol-4	Sum
e/p	α(1−r) (85)	αr (4)	α (89)
e/e	(1−α)r (8)	(1−α)(1−r) (89)	(1−α) (97)
Sum	(93)	(93)	1 (186)

Notice that the boldfaced genotypes together occur with frequency r. So we estimate

$$ \begin{array}{*{20}l} {{\hat{r} = } \hfill} & {{\frac{{12}} {{186}} = 0.0645,\;{\text{and}}} \hfill} \\ {{\hat{\alpha } = } \hfill} & {{\frac{{89}} {{186}} = 0.4785.} \hfill} \\ \end{array} $$

We note that α is close to 0.5, indicating that no significant preferential transmission of alleles by the female parent occurs at this marker locus.

Appendix 3

Markers that distinguish between the two L. peruvianum alleles, as well as the L. esculentum allele (p≠p*≠e)

This represents the most informative class of markers that allow straightforward estimation of the two distortion parameters (α and β) as well as r. Marker By-4 and Aps1/TaqI are of this category. The configuration of the cross reads like:

Using the same parameter notation as above, we have the following table of gamete combinations

♀	♂
	β (ol-4 p*)	1−β (ol-4 e)
α(1−r)   (Ol-4 p)	α β(1−r)	α(1−β)(1−r)
(1−α)r  (Ol-4 e)	(1−α)βr	(1−α)(1−β)r
αr   (ol-4 p)	αβr	α(1−β)r
(1−α)(1−r)  (ol-4 e)	(1−α)β(1−r)	(1−α)(1−β)(1−r)

Thus, the two-way table of genotypes reads:

	Ol-4/ol-4	ol-4/ol-4	Sum
p/p*	αβ(1−r)	α β r	αβ
e/p*	(1−α)βr	(1−α)β(1−r)	(1−α)β
e/p	α(1−β)(1−r)	α(1−β)r	α(1−β)
e/e	(1−α)(1−β)r	(1−α)(1−β)(1−r)	(1−α)(1−β)

Notice that the boldfaced genotypes together represent a proportion, r (independent of α, β). So r is estimated by counting these genotypes. Likewise, α and β are estimated by adding the appropriate classes in the right margin of the above table (pp*+ep for α, pp*+ep* for β). Using the numbers given in Table 4 we obtain for

$$\begin{array}{*{20}l} {{{\text{By-}}4:} \hfill} & {{\ifmmode\expandafter\hat\else\expandafter\^\fi{\alpha } = 0.479,\;\ifmmode\expandafter\hat\else\expandafter\^\fi{\beta } = 0.823;\;\ifmmode\expandafter\hat\else\expandafter\^\fi{r} = 0.0645,} \hfill} \\ {{{\text{and}}\;{\text{for}}} \hfill} & {{} \hfill} \\ {{Aps1/Taq{\text{I:}}} \hfill} & {{\ifmmode\expandafter\hat\else\expandafter\^\fi{\alpha } = 0.479,\;\ifmmode\expandafter\hat\else\expandafter\^\fi{\beta } = 0.828;\;\ifmmode\expandafter\hat\else\expandafter\^\fi{r} = 0.0538.} \hfill} \\ \end{array} $$

We observe that the estimates for α and β are the same as the ones obtained for markers By-4/HypCH4IV and By-4/ApoI, which is not surprising, since they represented the same locus (By-4).

Appendix 4

Markers that distinguish between the L. peruvianum and L. esculentum alleles, but not between the two L. peruvianum alleles (p*=p≠e)

Marker Aps1/Sau96I is of this type. The configuration of the cross reads like:

From this we see that, considering the marker locus only, any segregation distortion at the marker cannot be ascribed to either parent or to both parents. In other words, estimation of the distortion parameters must go along with estimation of r, and vice versa. In this sense, this marker category represents the most ‘difficult’ one: it requires simultaneous estimation of α, β and r.

Proceeding as before, using α and β for female and male transmission frequencies at the marker and r for recombination frequency, we obtain the following two-way table of genotypes (numbers in parentheses are observed numbers for marker Aps1/Sau96I in Table 4).

	Ol-4/ol-4	ol-4/ol-4	Sum
p/p	αβ(1−r) (71)	αβr (3)	αβ (74)
e/p	α(1−β)(1−r)+(1−α)βr (22)	α(1−β)r+(1−α)β(1−r) (73)	α(1−β)+(1−α)β (95)
e/e	(1−α)(1−β)r (0)	(1−α)(1−β)(1−r) (17)	(1−α)(1−β) (17)
Sum	(93)	(93)	1 (186)

For the simultaneous estimation of α, β, and r, we proceed as follows. As can be seen from the table above, α and β can be estimated from the observed frequencies at the marker locus (probabilities in the right margin). We also see that for the marker genotype frequencies, α and β are interchangeable, which means that, in case α and β are not equal, we cannot ascribe the estimate to either parent. However, we observe a 1:1 ratio at the Ol-4 locus, which means that for a marker closely linked to Ol-4, the value of α must be close to 0.5 (α is the female transmission rate: a clear deviation from 0.5 would ‘drag’ along the alleles at Ol-4). Estimates of α and β are obtained by solving the appropriate likelihood equations, using probabilities and observed frequencies at the marker locus. The resulting quadratic equation yields two equivalent solutions, i.e.

$$\begin{array}{*{20}l} {{{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{\alpha },\ifmmode\expandafter\hat\else\expandafter\^\fi{\beta }} \right)} = } \hfill} & {{{\left( {0.483,\;0.823} \right)}\;{\text{and}}} \hfill} \\ {{{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{\alpha },\ifmmode\expandafter\hat\else\expandafter\^\fi{\beta }} \right)} = } \hfill} & {{{\left( {0.823,\;0.483} \right)}.} \hfill} \\ \end{array} $$

We accept the first one, since α should be close to 0.5; it also is in close agreement with the estimates obtained for the other markers. Next, we substitute (α, β)=(0.483,0.823) in the expressions for the probabilities in the body of the table and (numerically) solve the resulting likelihood equation for r. This yields $\hat{r} = 0.0394.$ However, should we have used the ‘mirror’ estimates of$(\hat{\alpha },\hat{\beta }),$ i.e.$(\hat{\alpha },\hat{\beta }) = (0.823,0.483),$ the incorrect estimate of r would have been$\hat{r} = 0.287.$ This, again, shows the necessity of carefully interpreting the joint segregation data in order to avoid wrong conclusions.

Using the obtained estimates, we have calculated the corresponding LOD values for linkage. The table below summarizes the results.

Marker	${\hat \alpha }$	${\hat \beta }$	${\hat r}$	LOD
Aps1/TaqI	0.479	0.828	0.054	39.1
Aps1/Sau96I	0.483	0.823	0.039	27.7
By-4	0.479	0.823	0.065	36.7
By-4/ApoI	0.479	–	0.065	36.7
By-4/HypCH4IV	–	0.823	–	–