Single Marker Analysis Using F2 Progeny

Reference: Ben Hui Lui. Statistical Genomics: Linkage, Mapping and QTL Analysis. pgs. 402-405.

In this situation we have one marker linked to a QTL. We do not have a marker on either side of the QTL, which would be the case for flanking markers. Marker genotypes are MM, Mm, mm and the QTL genotypes are QQ, Qq, and qq. The marker and the QTL are in coupling phase linkage.

In the F₂ population the relative frequencies and means of the three marker classes are:

Marker Class	Relative Frequency	Marker Mean
MM	0.25	u1
Mm	0.5	u2
mm	0.25	u3

In this same F₂ population the relative frequencies and means of the three QTL classes are:

QTL Class	Relative Frequency	QTL Mean
QQ	0.25	m + a
Qq	0.5	m + d
qq	0.25	m - a

The observed recombination fraction between the marker and the QTL is ‘r'. The relative frequency of crossovers events between the marker and QTL loci is 2r. This is because for each crossover event only two out of four strands are involved in recombination. The gametic relative frequencies for each marker-QTL combination are:

Type of Gamete	Relative Frequency
QM	(1 - r)/2
qm	(1 - r)/2
Qm	r/2
qM	r/2

A Punnett square shows the genotypes in the F₂ population.

                             (1 - r)/2    (1 - r)/2      r/2           r/2

		QM	qm	Qm	qM
(1 - r)/2	QM	QQMM	QqMm	QQMm	QqMM
(1 - r)/2	qm	QqMm	qqmm	Qqmm	qqMm
r/2	Qm	QQMm	Qqmm	QQmm	QqMm
r/2	qM	QqMM	qqMm	QqMm	qqMM

We can collect like genotypes to derive the relative frequencies of each class.

  Genotypes                             Relative Frequency

QQMM	(1 - r)2/4
QqMM	(r - r2)/2
qqMM	r2/4
QQMm	(r - r2)/2
QqMm	(1 - r)2/2 + r2/2
qqMm	(r - r2)/2
QQmm	(r/2)2
Qqmm	(r - r2)/2
qqmm	(1 - r)2/4

We can rearrange the results from the above table into a 3 X 3 table.

	QQ	Qq	qq
MM	(1 - r)2/4	(r - r2)/2	r2/4
Mm	(r - r2)/2	{(1 - r)2 + r2}/2	(r - r2)/2
mm	r2/4	(r - r2)/2	(1 - r)2/4

The marginal frequency for the MM marker class is the relative frequency of this marker genotype averaged across the three QTL genotypes:

(1-r)2	+	(r-r2)	+	r 2	=(1/4)[1-2r+r2+2r+2r2+r2]=1/4
4		2		4

The marginal frequency for the Mm marker class is:

(r-r2)	+	[(1-r2) + r2]	+	(r-r2)	=(1/2){r-r2+1-2r+r2+r2+r-r2}=1/2
2		2		2

The marginal frequency for the mm marker class is:

r2	+	(r-r2)	+	(1-r2)	=(1/4){r2+2r-2r2+1-2r+r2}=1/4
4		2		4

The MM marker class can be identified with banding patterns and includes QTL genotypes QQ, Qq, and qq. The MM marker class is a mixture of QTL classes due to recombination between the marker and the QTL. We can only identify the means of each marker class, because we cannot identify the QTL genotypes associated with a given marker in an F₂ plant. We can only estimate the means of each marker class.

To determine the means of each marker class we must first find the conditional probability of each QTL class for a given marker class:

P(Qj/Mi) =	P(QjnMi)
	P(Mi)

The conditional probability of the QQ genotype given that the MM marker genotype is present in that F₂ is:

P(QQ/MM)=	P(QQMM)	=	{(1-r)2/4}	=(1-r)2
	P(MM)		4

We are dividing the probability of each marker-QTL genotype by the probability of the marker genotype. We can develop the following table:

	QQ	Qq	qq
MM 1/4	(1 - r)2	2r(1 - r)	r2
Mm 1/2	(r - r2)	{(1 - r)2 + r2}	(r - r2)
mm 1/4	r2	2r(1 - r)	(1 - r)2

Now we can calculate the mean of each marker class. The markers do not contribute to the mean value. Only the QTL loci contribute to the mean value.

u_MM = ( 1 - r)²u₁ + 2r( 1 - r)u₂ + r²u₃

Now u₁ = u + a; u₂ = u + d; u₃ = u - a;

u_MM = (1 - r)²(u + a) + 2r(1 - r)(u + d) + r² ( u - a)
       = [(1 - r)² + r² + 2r(1 - r)]u + (1 - r)²a - r²a + 2r(1 - r)d
       = (1 - 2r + r² + r² + 2r - 2r²)u + (1 - r)²a - r²a + 2r(1 - r)d
       = u + [1 - 2r + r² - r²]a + 2r(1 - r)d
       = u + (1 - 2r)a + 2r(1 - r)d

We can show that
U_Mm = r(1 - r)(u + a) + [(1 - r)² + r²](u + d) + r(1 - r)( u - a)
       = u + [(1 - r)² + r^2]d

We can also show that
U_mm = r²(u + a) + 2r(1 - r)(u + d) + (1 - r)²(u - a)
       = u - (1 - 2r)a + 2r(1 - r)d

We consolidate these results into a table.

  Marker       n_i           p_i                Marker mean value

MM	n1	0.25	UMM = u + (1 - 2r)a + 2r(1 - r)d
Mm	n2	0.50	UMm = u + [(1 - r)2 + r2]d
mm	n3	0.25	Umm = u - (1 - 2r)a + 2r(1 - r)d

We have four unknowns to estimate that include: u, a , d, and r. We only have the three equations above. With three equations we cannot estimate four unknowns.

The genetic effects which include u, a, and d are confounded with the estimate of the recombination fraction (r). We cannot determine the recombination between the marker locus and the QTL locus. A one-way analysis of variance will only provide evidence that the marker locus is linked to a QTL. The R² is the amount of variation explained by the marker, not the QTL.

The markers are the independent variable and the mean of each marker class is the dependent variable. The marker means are a mixture of QTL genotypes. For example, the mean of the MM marker class is U_MM = (1 - r)²u₁ + 2r(1 - r)u₂ + r²u₃. This equation indicates that there is a proportion (1 - r)² of the QQ genotype, 2r(1 - r) of the Qq genotype, and a proportion r² of the qq genotype associated with the MM marker genotype.

R2 =	Sum of Squares due to regression
	Sum of Squares Total

The R² does not provide information on the amount of variation explained by the QTL locus.

We can also show that the additive and dominance effects cannot be determined when we use data from a single marker-QTL association. The linear contrasts for the additive and dominance effects are so S c_i = 0. Where c_i is the contrast coefficient.

    Contrast                              Marker genotype
                          MM                Mm                 mm

additive	1	0	-1
dominance	1	-2	1

The additive contrast is the difference in the means of the two homozygous genotypes.
Additive contrast = Sc_iY_i = (1)u_AA + (-1)u_aa

Where u_AA = u₁(1 - r)² + 2u₂ r(1 - r) + u₃ r² and
u_aa = u₁ r² + 2r(1 - r)u₂ + (1 - r)² u₃.

The contrast is then u_AA - u_aa = u₁(1 - 2r) + (2r -1)u₃ = (u +a)
(1 - 2r) + (2r -1)(u - a) = 2( 1 - 2r)a.

This shows that the additive effect cannot be separated from the recombination (r) value. Now we will derive the dominance contrast = Sc_iY_i

=(1)[u + (1 - 2r)a + 2r(1 - r)d]-(2){u + [(1 - r)² + r²]d}+(1)
          [u - (1 - 2r)a + 2r(1 - r)d]

= (-2)(1 - 2r)²d

This shows that the dominance effect cannot be separated from the recombination (r) value. For both the additive and dominance contrasts the genetic effects are confounded with the recombination value. When we have a marker located on only one side of the QTL, we cannot determine the additive or dominance effects or the recombination fraction between the marker and the QTL. Because of recombination between the marker and the QTL, a marker may be associated with either the + or - alleles of the QTL locus. Each marker mean includes the effects of homozygous and heterozygous QTL genotypes. Each marker class is a mixture of QTL genotypes. Because we cannot determine the recombination between the marker and QTL, the R² for regression of marker genotype on the mean of each marker class is only a measure of the amount of variation explained by the marker. The amount of variation explained by the QTL cannot be determined for the case of a single marker-QTL association.

"The hypothesis test based on the contrasts for the marker genotypes corresponds to the marker and the QTL being independent or no genetic effects can be detected for the putative QTL, which is r = 0.5 or a = 0 and d=0." From Ben Hui Lui (pg. 405).