Supporting Information S2. The expected genetic values of Z1i , Z2i and Z3i under the F2
and the F∞ metric models in the F2-based TTC design
We assume that a quantitative trait is controlled by two QTL, A and B, each with two alleles ( A , a and B , b ).
The recombination fraction between A and B was r . The genotypes of the two inbred lines (P1 and P2) and their
F1 are AABB , aabb and AaBb , respectively. The F1 produces four types of gametes AB , Ab , aB and ab
with frequencies
1
2
(1  r ) ,
1
2
r,
1
2
r and
1
2
(1  r ) . The gametes from F1 are combined to produce F2 population
with 10 type genotypes (Table S1). Note that the two double heterozygotes AB / ab and Ab / aB , with different
gametes constitutions under linkage disequilibrium, should be distinguished. Individuals from the F2 are cross to
P1 (with gamete AB ), P2 (with gamete ab ) and F1 to produce L1, L2 and L3 families. The genetic constitutions of
the three families are given in Table S1.
The genotypic values of the nine genotypes in L1i, L2i and L3i families under the F2 metric are defined as Kao and
Zeng [28],
1 1
GAABB  
G
 1 1
 AABb  
GAAbb  1 1

 
GAaBB  1 0
G
  1 0
 AaBb  
GAabb  1 0
G
 
 aaBB  1 1
GaaBb  

 1 1
Gaabb  1 1

 21
1  21

0




1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1 
1 
0
1 
1 
0
1 
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2

  
0
0   a 
 1 
1
1
1 
2
4   d1 


1
0
0
 14   a2 
2

1 
d 
0
0
0
4  2 

1
1
0
0  2  4   ia1a2 
 i 
1
1
1
 21
 a1d2 
2
4
1
1   id a 
0 2
0  4 1 2 
id1d2 
1
1
1 
1
2
2
4
1  21
1
2
1
2
 21
1
4
1
4
where  is model mean (mean genotypic values of F2 population when r  0.5 ); a1 is additive effect of QTL
A, i.e. one-half of the difference in genotypic value between the two homozygote means of AA ( AABB ,
AABb and AAbb ) and aa ( aaBB , aaBb and aabb ); d1 is dominance effect of QTL A, i.e. the departure in
genotypic value of the heterozygote mean of Aa ( AaBB , AaBb , and Aabb ) from the midpoint between the
two homozygote means of AA ( AABB , AABb and AAbb ) and aa ( aaBB , aaBb and aabb ); a2 and d2
are additive and dominance effects of QTL B, with similar definitions to QTL A; ia a , ia d , id a
1 2
1 2
1 2
and id d
1 2
are
additive × additive, additive × dominance, dominance × additive and dominance × dominance epistatic effects.
While under the F∞ metric, genotypic values of the nine genotypes are defined as Kao and Zeng [28],
1
1
GAABB   1 1 0
G
 1 1 0 0
 AABb  
GAAbb   1 1 0 1

 
GAaBB   1 0 1 1
G
  1 0 1 0
 AaBb  
GAabb   1 0 1 1
G
  1 1 0
1
 aaBB  
GaaBb   1 1 0 0

 
Gaabb   1 1 0 1
0
1
1
0
0
1
0 1
0
0
0
0
1
0
0
0
0
0
0 1
0
1
0 1
0
1
0
0 0   
0 0   a1 

0 0   d1 


1 0   a2 

0 1  d2 


1 0   ia1a2 
0 0   ia1d2 

0 0   id1a2 


0 0  id1d2 
where  is model mean (mean genotypic values of four homozygotes, AABB , AAbb , aaBB and aabb ); a1
is additive effect of QTL A, i.e. average substitution effect of a (in aaBB and aabb ) by A ; d1 is dominance
effect of QTL A, i.e. the departure in genotypic value of the heterozygote mean of Aa ( AaBB and Aabb ) from
the midpoint between the two homozygote means of AA ( AABB and AAbb ) and aa ( aaBB and aabb );
a2 and d2 may be defined similarly; and ia1 a2 , ia1 d2 , id1 a2 and id1 d2 are additive × additive, additive ×
dominance, dominance × additive and dominance × dominance epistatic effects. Using the definitions above, the
expected genetic values of L1i, L2i and L3i families can be calculated and are listed in Tables S2, S3 and S4. With
these
expected
values, the
expected
values
of
the
F2-based
Z1i  L1i  L2 i ,
Z2 i  L1i  L2 i
and
Z3 i  L1i  L2 i  2L3 i values can be further calculated and are presented in Table S5 for the F2 metric model and
Table S6 for the F∞ metric model.
According to Table S5, genetic variance between families on Z1i , Z2i and Z3i under the F2-metric model are:
1 2
1
1
( a1 2a2 )
r( 2 5r 2 4r1a 22a ) i( 1d 2 2d i  ) 1a2 2di(  21d i2a
)
2
4
8
1
1
1
1
( 1  r2 a1) a2 
(1
 r 2 a11a)d2(i  2da1 a2 i  ) 1 da1 (a2 i  2 aa1 d2 i  )
r( 22 5r a14ra2 di)1 d2i
 (1
2
2
2
4
1 2
1
1
1
VG ( Z2i )
( d1 2d2 )
(
1 r1a 2 2ai) 
r ( 2 r5 2 r41a 2 2d )i (1d2 2a i  )  (1 1 r2 2
d d
)
 1 ( r1 2d1a) 2a( d i )
2
2
4
2
1
VG (Z3 i )  r 2 (ia12d 2 id a21) 2 r (23  5r  4r )id2d 12 r2 (1 22r )ia d id a 1 2 1 2
2
VG ( Zi1 )
r2 i )d1 a2i
a1 d2
r
2
(2r1a
2d
5 1dr
2a
4i
)i
Likewise, genetic variance of among F2 individuals and between F2:3 families under the F2-metric model are:
1
1
1
1
1
2
VG (F 2)  ( a2 
( d 2 1d ) 2 2 ia1a 22 (ia d 1 22 id a ) 12 2 [1  (1  2r ) ]id d 4 1 2 2
1 a )
2
2
4
4
8
16
1
1
1
(1  2r )a1 a2  (1  2r )2 d1d2  (1  2r )( a1id1a 2 a2 ia d 1) 2 (1  2r )2 ( a1ia d 1 a22 id a ) 1 2
2
2
2
1
1
 (1  2r )( d1  d2 )ia1a 2 r(1  r )(1  2r )ia a 1id 2d 1 2 (1  2r )ia1d2 id1a2
2
4
1 2
1 2
1
1
2
2
2
VG (F2:3 )  ( a1  a2 )  ( d1  d2 )  (1  20r  44r  32r 3 )ia21a 2 [ r 2  (1  r )2 ]( ia2 d 1 2 id2 a ) 1 2
2
16
16
8
1
 [r 2  (1  r )2 ]{1  (1  2 r )4  2[ r 2  (1  r )2 ]3 } id21d2
32
1
1
1
(1  2r )a1 a2  (1  2 r )2 d1 d2  [r 2  (1  r )2 ]( a1ia1d2  a2 id1a2 )  (1  2r )( a1id1a2  a2 ia1d2 )
8
2
2
1
1
1
1
 (1  4r 2 )( d1  d2 )ia1a2  r(1  r )[r 2  (1  r )2 ]( d1  d2 )id1d2  (1  2r )i a1d2 id1a2  r(1  2r )(2  5r  8r 2  7 r 3  2r 4 )i a1a2 i d1d2
8
4
4
2
According to Table S6, genetic variance between families on Z1i , Z2i and Z3i under the F∞-metric model are:
VG ( Zi1 )
1 2
1
( a1 2a2 )
2
4
r ( 2
5r
2
4r1a
2
2a
)i( 1d 2 2d i
1
)
2
r 2d (i  21d 2ai  ) ( 11 2r 2 a )a1
2
1a
2
r2(a
1a
1
a )(i1d
2
2d
2a

i
)
r (2 2 r5 1a
2a
r41d 2di ) i
1a
2d
r1d i2a
i
VG ( Z2i )
1 2
1
( d1 2d2 )
2
2
(1
 r1a 2)2ai(  1d2 i2d
( 1  r2 d1) d2  ( 1
r
VG (Z3 i ) 
)d (  2da1
1
a2
1
)
4
)i ( 
d1 d2 i
r ( 2 r5 2 r 41a2 2d i) (21d 2ai
1
 ) ( a1r1 a2 i d1) di2 
2
1 2 2
r (ia1d 2 id a21) 2 r (23  5r  4r )id2d 12 r2 (1 22r )ia d id a 1
2
2
1
)
r (2 2 
r 5 ar1 d42 i d1 )a2 i
2
Likewise, genetic variance of among F2 individuals and between F2:3 families under the F∞-metric model are:
1
1
1
1
2
VG (F 2)  ( a2 
( d 2 1d ) 2 2 ia1a 22 r(1  r )(ia d 1 22 id a ) 12[2  r (1  r2) ]id d 2 1 22
1 a )
2
2
4
4
4
1 2
2
(1  2r )a1 a2  [   r  (1  r ) ]d1d2  2 r(1  r )( a1ia1d 2 a2 id a 1) 2
2
1
1
1
 (1  2r )( d1  d2 )ia1a 2 [r 2  (1  r )2 ]( d1  d2 )id d 1 2 (1  2 r )[ r 2  (1  r )2 ]ia1a2 id1d2
2
2
2
1 2
1
1
1
1
VG ( F2 : 
a( 1 a2 
d2( 1 d2 
(1 r2 0 2r 4 4 a1 3ra2 32i2 ) r a1 d2r(2 1 i d1 a2)2 ( i 
3)
2)
2)
2
16
16
4
16
1
1
1
( 1  r2 a1) a2 
(1
 2r 2 1d) 2d  r ( 1 ar1 1d2 )a ( i d12 a2 a i  ) 2  ( 1 r14 d)2a1 (a2 d 2 
i)
8
8
8
1
 (1  2r )[r 2  (1  r )2 ][ 1  2 r(1  2 r )(2  r )]ia1a2 id1d2
8
3
) 
r(12 r 22 r 2
] r( d[ 1  d2()i1rd1d2
22
)
2 3
)
d1[ dr
2
2
( i1
) ]