QTL Mapping and Marker Mean:
F₂ Population

Example:

Say u₁=2, u₂=1, u₃=0. The marker and QTL are in coupling phase linkage for an F₂ population and r = 0.2. Find the mean for each marker genotype and the overall mean.

u_AA = (1-r)²u₁ +2r(1-r)u₂ + r²u₃
      = (0.8)²(2) + 2(0.2)(0.8)(1) + (0.2)²(0)
      = 1.28 + 0.32 + 0
      = 1.6

u_Aa = r(1-r)u₁ + [(1-r)² + r²]u₂ + r(1-r)u₃
      = 0.2(0.8)(2) + [(0.8)² + (0.2)²](1) + (0.2)(0.8)(0)
      = 0.32 + 0.68 + 0
      = 1

uaa = r²u₁ + 2r(1-r)u₂ + (1-r)²u₃
      = (0.2)²(2) + 2(0.2)(0.8)(1) + (0.8)²(0)
      = 0.08 + 0.32
      = 0.40

u = 1/4u_AA + 1/2u_Aa + 1/4u_aa
   = 1/4(1.6) + 1/2(1) + 1/4(0.4)
   = 0.4 + 0.5 + 0.1
   = 1

Now let's say that r=0.4, u₁=4, u₂=1, u₃=-2. Find the u_AA, u_Aa, u_aa and the overall mean(u).

u_AA = (1-r)²u₁ + 2r(2-r)u₂ + r²u₃
      = (0.6)²(4) + 2(0.4)(0.6)(1) + (0.4)²(-2)
      = 0.36(4) + 0.48(1) + 0.16(-2)
      = 1.44 + 0.48 - 0.32
      = 1.6

u_Aa = r(1-r)u₁ + [(1-r)² + r²]u₂ + r(1-r)u₃
      = 0.4(0.6)(4) + [(0.6)² + (0.4)²](1) + 0.4(0.6)(-2)
      = 0.96 + 0.52 - 0.48
      = 1

u_aa = r²u₁ + 2r(1-r)u₂ + (1-r)²u₃
      = (0.4)²(4) + 2(0.4)(0.6)(1) + (0.6)²(-2)
      = 0.64 + 0.48 - 0.72
      = 0.4

u = 1/4u_AA + 1/2u_Aa + 1/4u_aa
u = 1/4(1.6) + 1/2(1) + 1/4(0.4)
u = 1

The results show that when r=0.2 and u₁=2, u₂=1, u₃=0 the means of each marker class are u_AA=1.6, u_Aa=1, u_aa=0.4. The results also show that when r=0.4 and u₁=4, u₂=1, u₃= -2 the means of each marker class are u_AA=1.6, u_Aa=1, u_aa=0.4.

The marker class means did not change as r was varied, but the means of each QTL genotype did change.

Conclusion:

For the case of a single marker, linkage values between the marker (only true for the case of single marker) and QTL are confounded with QTL value. The only information we have is the mean of each marker class. In out example we know that the means of the AA, Aa, and aa marker classes are 1.6, 1.0, and 0.4, respectively. If r = 0.2 then the associated effects of the QQ, Qq, and qq QTL genotypes are 2, 0, and 1, respectively. If r = 0.4, then the marker class means are the same, but the effects of the QQ, Qq, and qq QTL genotypes are 4, 1, and -2, respectively.