PLSC 781: Quantitative Genetics

Flanking Markers to Map QTL's

Reference: Haley, C.S. and S.A. Knott. 1992. A simple regression method for mapping quantitative trait loci in line crosses using flanking markers. Heredity 69:315-324.

The position of a QTL can only be determined when there are markers linked on both sides of the QTL. Previously we showed that the effect of the QTL and the position of the QTL are confounded when there is only one marker linked to the QTL. This means that when there is a marker on only one side of the QTL, the position of the QTL cannot be determined. When there are markers linked to a QTL on both sides of it, these are called flanking markers. We will show that the closer our estimate of the position of the QTL is to the actual QTL position, the larger the F-value for regression becomes. When the F-value due to regression is maximized, this is the same position as the peak LOD score for a program such as Mapmaker that uses the maximum likelihood approach. The least squares solution which is used to estimate the position of the QTL is easier to understand than the maximum likelihood method and the results are identical.

Let r represent the observed recombination fraction between the two markers A and B. 2r is the probability of a crossover event between the two loci. Suppose A1 and B1 are in coupling phase linkage. The probability of an A1B1 gamete is (1-r)/2 and the probability of a A2B2 gamete is (1-r)/2. The probability of a gamete that results from recombination, such as the A1B2 or the A2B1 gametes is r/2 for either case.

Suppose we have a QTL located between the A and B marker loci. There are two alleles at the QTL locus and there are labeled Q1 and Q2. The observed recombination between the A1 and Q1 loci is rA. The observed recombination between the Q1 and B1 loci is rB. Then A1, Q1, and B1 are all linked in coupling phase.

Let (1-2r) = (1-2rA)(1-2rB). In words this equation states that the probability of no recombination between the A and B loci equals the probability of no recombination between the A and Q loci times the probability of no recombination between the Q and B loci. The above formula reduces to r = rA + rB - 2rArB. 2rArB is the probability of a double crossover event between the the A and Q as well as the Q and B loci. An observed recombinant gamete that is the result of a double cross-over, such as A1Q2B2, would occur with probability rArB/2. Likewise, the probability of an A2Q1B2 gamete would be rArB/2. The probability of observing an non-recombinant gamete such as A1Q1B1 would be (1-rA)(1-rB)/2 and this probability would also be identical for the probability of observing an A2Q2B2 non-recombinant gamete. We develop the following table that provides the probability of each type of gamete.

  Gametes           Probability            Type      
  A1Q1B1 (1-rA)(1-rB)/2   parental
  A1Q1B2 (1-rA)rB/2   single c.o.
  A1Q2B1 rArB/2   double c.o.
  A2Q1B1 (1-rB)rA/2   single c.o.
  A1Q2B2 rA(1-rB)/2   single c.o.
  A2Q2B1 (1-rA)rB/2   single c.o.
  A2Q2B2 (1-rA)(1-rB)/2   parental
  A2Q1B2 rArB/2   double c.o.
The probability of A1Q1 is (1-rA)/2 and the probability of A1Q2 is rA/2. The probability of either A1Q1 or A1Q2 is
(1-rA)/2 + rA/2 = ½.

The probability of the A1Q2B1 gamete is the probability of a crossover between A1 and Q2 which is rA times the probability of a crossover between Q2 and B1 which is rB, divided by one-half. The reason we divide by one-half is that when a double crossover occurs the A1Q2B1 and A2Q1B2 gametes will both occur. One-half of the double crossover gametes will be A1Q2B1 and the other half of the gametes will be A2Q1B2. Then the probability of an A1Q2B1 gamete is rArB/2. Likewise, the probability of an A2Q1B2 double crossover gamete is rArB/2.

We can develop a table that shows the probability of each type of marker gamete, without considering the genotype of the QTL which is between the flanking markers. When we evaluate the progeny of a cross we can only determine the genotypes of the markers for each individual or line which is scored. The QTL is a quantitative trait and is affected by the environment as well as other loci. A quantitative trait is by definition a trait which has many loci that determine the genotype. The QTL which is positioned between the two markers is only one of many loci that influence the inheritance of that trait. Because there are many loci that affect the trait of interest, we cannot separate the genetic effect of the QTL which is positioned between markers A and B from the genetic effects of all the other loci. However, after we estimate r, rA, and rB, then we can estimate the effect of the QTL by using all the data from the various genotypes which we have scored for the markers and measured for the quantitative trait of interest.

The expected relative frequency of the A1Q2B1 gamete is the probability of the double cross-over gamete multiplied by one-half = rArB(½). Likewise, the probability of the A2Q1B2 gamete is the probability of the double cross-over gamete multiplied by one-half = rArB(1/2).

The marker genotypes A1A1B1B1 and A2A2B2B2 have expected frequencies of (1-r)2/4, because the probability of marker gametes A1B1 and A2B2 are (1-r)/2. The marker genotypes A1A1B2B2 and A2A2B1B1 have expected frequencies of r2/4, because the probability of marker gametes A1B2 and A2B1 are r/2. The probability of recombination is r and the probability of a crossover event is 2r because only two of four strands are involved. The marker genotypes A1A1B1B2, A1A2B1B1, and A1A2B2B2, and A2A2B1B2 occur with probabilities r(1-r)/4, because the probability of a crossover gamete is r/2 and the probability of a non-crossover gamete is (1-r)/2. The marker genotype A1A2B1B2 can occur in two ways. One way is due to two non-crossover gametes uniting, and the second way is due to two crossover gametes uniting. The probability of two non-crossover gametes uniting is [1-r/2]2 = (1-r)2/4. The probability of two crossover gametes uniting is [r/2]2 = r2/4.

The probability of these two mutually exclusive events is the sum of the separate probabilities

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


Marker     Genotypic value of QTL genotypes
 genotype    Probability   Q1Q1 Q1Q2 Q2Q2
  A1A1B1B1   (1-r)2/4      a    d   -a
  A1A1B1B2   (1-r)r/2      a    d   -a
  A1A1B2B2   r2/2      a    d   -a
  A1A2B1B1   (1-r)r/2      a    d   -a
  A1A2B1B2   (1-r)2/2 + r2/2      a    d   -a
  A1A2B2B2   (1-r)r/2      a    d   -a
  A2A2B1B1   r2/4      a    d   -a
  A2A2B1B2   r(1-r)/2      a    d   -a
  A2A2B2B2   (1-r)2/4      a    d   -a

We can also develop a table for the probability of each QTL-marker genotype.

    Marker          QTL                  Probability      
  A1A1B1B1       Q1Q1              (1-rA)2(1-rB)2/4
  A1A1B1B1       Q1Q2            2(1-rA)(1-rB)rArB/4
  A1A1B1B1       Q2Q2                   rA2rB2/4

  A1A1B1B2       Q1Q1              (1-rA)2(1-rB)rB/2
  A1A1B1B2       Q1Q2       2[(1-rA)(1-rB)2rA/4 + rArB2(1-rA)/4]
  A1A1B1B2       Q2Q2                 rA2rB(1-rB)/2

  A1A1B2B2       Q1Q1                (1-rA)2rB2/4
  A1A1B2B2       Q1Q2            2(1-rA)rB(1-rB)rA/4
  A1A1B2B2       Q2Q2                rA2(1-rB)2/4

  A1A2B1B1       Q1Q1              (1-rA)(1-rB)2rA/2
  A1A2B1B1       Q1Q2        2[rA2rB(1-rB)/4 + (1-rA)2rB(1-rB)/4]
  A1A2B1B1       Q2Q2                 rArB2(1-rA)/2

  A1A2B1B2       Q1Q1    2[(1-rA)(1-rB)rArB/4 +(1-rA)rB(1-rB)rA/4]
  A1A2B1B2       Q1Q2     2[(1-rA)2(1-rB)2/4 + rA2rB2/4 + (1-rA)2
                                           rB2/4 + (1-rB)2rA2/4]
  A1A2B1B2       Q2Q2    2[rArB(1-rA)(1-rB)/4 + (1-rA)rBrA(1-rB)/4]

  A1A2B2B2       Q1Q1                (1-rA)rB2 rA/2
  A1A2B2B2       Q1Q2      2[(1-rA)2rB(1-rB)/4 + rA2(1-rB)rB/4]
  A1A2B2B2       Q2Q2              rA(1-rB)2(1-rA)/2

  A2A2B1B1       Q1Q1                 (1-rB)2rA2/4
  A2A2B1B1       Q1Q2             2rArB(1-rA)(1-rB)/4
  A2A2B1B1       Q2Q2                 (1-rA)2rB2/4

  A2A2B1B2       Q1Q1                 (1-rB)rA2rB/2
  A2A2B1B2       Q1Q2      2[(1-rB)2rA(1-rA)/4 + (1-rA)rB2rA/4]
  A2A2B1B2       Q2Q2              (1-rA)2rB(1-rB)/2

  A2A2B2B2       Q1Q1                  rA2rB2/4
  A2A2B2B2       Q1Q2            2rArB(1-rA)(1-rB)/4
  A2A2B2B2       Q2Q2              (1-rA)2(1-rB)2/4

In the above table there is a coefficient of 2 for heterozygous QTL's. The probability of a given QTL genotype is conditional on having the specified marker genotype. Because of double crossovers, the probability of A1A1B1B1 is different than the probability of A1A1Q1Q1B1B1.

Example: We can determine the additive and dominance effects for the QTL, conditional on the specified marker genotype. The mean of homozygous QTL genotypes for the A1A1B1B1 marker is a[(1-rA)2(1-rB)2/4] - (rA2rB2)]/4. The mean of the heterozygous QTL genotype associated with the A1A1B1B1 marker genotype is d[2(1 - rA)(1 - rB)rArB/4] . The additive effect of the QTL conditional on the marker being A1A1B1B1 is

a[(1-rA)2(1-rB)2]/4-a(rA2rB2)/4
(1-r)2/4

because (1 - r)2/4 is the probability of the A1A1B1B1 marker and a conditional probability is defined as

Prob.(A1A1Q1Q1B1B1 or A1A1Q2Q2B1B1)
Prob. (A1A1B1B1)

for the homozygous genotypes A1A1Q1Q1B1B1 or A1A1Q2Q2B1B1 given the marker genotype is A1A1B1B1. The additive effect of the QTL, conditional on the A1A1B1B1 marker gentotype is

a[(1-rA)2(1-rB)2 - (rA2rB2)]
(1-r)2

because the divisors of 4 cancel out. The dominance effect of the QTL, conditional on the A1A1B1B1 marker genotype is

d[2rArB(1-rA)(1-rB)]
(1-r)2

Now the recombination value between the QTL and flanking markers can be estimated. We have nine equations, one for each marker class. The reason we have nine equations is because we know the phenotypic value for each marker class. We estimate the ‘a' and ‘d' genetic effects for a known ‘r' with rA and rB choosen for each 1 cM interval between markers. For example, let r=0.1648 and set rA = rB = 0.0906 for the first evaluation. Then for the A1A1B1B1 marker class the coefficient for ‘a' is

[(1-rA)2(1-rB)2-(rA2rB2)]
(1-r)2


= [(1-0.0906)2(1-0.0906)2-(0.0906)2(0.0906)2]
(1-0.1648)2

= 0.9804

The coefficient of 'd' is d[2rArB(1-rA)(1-rB)]
(1-r)2

= [2(0.0906)(0.9094)(0.0906)(0.9094)] =0.0194632
(0.8352)2

The coefficients for ‘a' and ‘d' are the relative frequencies used to determine the mean of each marker class. Then the following equations can be set up to solve for the ‘a' and ‘d' effects. The Freq test statistic for each set of rA and rB values:

Marker genotype Marker value Coefficients
  A1A1B1B1 Y1111   0.9804 0.0195
  A1A1B1B2 Y1112   0.4902 0.5
  A1A1B2B2 Y1122   0.0 0.5
  A1A2B1B1 Y1211   0.4902 0.5
  A1A2B1B2 Y1212   0.0 0.9625
  A1A2B2B2 Y1222 - 0.4902 0.5
  A2A2B1B1 Y2211   0.0 0.5
  A2A2B1B2 Y2212 - 0.4902 0.5
  A2A2B2B2 Y2222 - 0.9803 0.0195

Now we can calculate the Yijkl values for each genotype.
For example,
  Y1111 = 0.9803(a) + 0.0195(d)
  Y1112 = 0.4902(a) + 0.5(d)
  Y1122 = 0.5(d)
  Y1211 = 0.4902(a) + 0.5(d)
  Y1212 = 0.9625(d)
  Y1222 = -0.4902(a) + 0.5(d)
  Y2211 = 0.5(d)
  Y2212 = -0.4902(a) + 0.5(d)
  Y2222 = -0.9803(a) + 0.0195(d)

The test statistic for determining plotting the position of the QTL is

(p)MS(regression) =pFreg
Ms(residual)

To solve for unknowns ‘a' and ‘d' we use least squares equations B = (X'X)-1X'Y and

B =  m Y =   Y1111 X =  1   0.9803 0.0195
  a   Y1112   1   0.4902 0.5
  d   Y1122   1   0 0.5
    Y1211   1   0.4902 0.5
    Y1212   1   0 0.9625
    Y1222   1 - 0.4902 0.5
    Y2211   1   0 0.5
    Y2212   1 - 0.4902 0.5
    Y2222   1 - 0.9803 0.0195
 
Now B'X'Y = Sum of Squares regression with p degrees of freedom and p = 3. Y'Y - B'X'Y = Sum of Squares residual with   n - p degrees of freedom. We derive the following ANOVA table:

Source    Sums of Squares     df     Mean Squares   Description 
 model B'X'Y p B'X'Y/p regression
residual    Y'Y-B'X'Y    n - p  [Y'Y-B'X'Y]/n-p   residual  
Total Y'Y n    

The coefficients for ‘a' and ‘d' vary, depending on the distance between marker A and the QTL (rA), the distance between marker B and the QTL (rB), and the distance between markers A and B (r).

In the last numerical example, we set r = 0.1648, rA = rB = 0.0906. The recombination between the markers does not change. We must use the estimate of r, based on the data. Then we vary rA and rB to determine the position of the QTL. As we vary rA, then rB must also vary. We use the equation
(1 - 2r) = (1 - 2rA)(1 - 2rB) with r fixed, as we try a different value of rA we then use the equation to solve for rB. For example, r = 0.1648, and we set rA = 0.12 and solve for rB.
(1 - 0.3296) = (1 - 0.24)(1- 2rB). Solving produces
rB = 0.05895. New values of rA = 0.12 and rB = 0.05895 result in a new F-value. We plot pFreg as we vary rA and rB. The best estimate of rA and rB will maximize the F-value and when the F-value is maximized this is the position of the QTL between markers A and B.