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Experimental design and statistical analysis

This document discusses genetic linkage analysis and map construction. It begins by presenting Mendel's experiments with plant hybrids and rediscovery of his work in 1900. It then discusses estimation of recombination frequency using different genetic populations like backcross, doubled haploid, recombinant inbred lines and describes maximum likelihood estimation. The document also discusses significance tests for linkage and examples of linkage analysis in different populations like F2, BC1F2. It describes expected genotypic frequencies and effect of distortion. Finally, it mentions three-point analysis and construction of linkage maps.

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Lecture 12 Genetic Linkage Analysis and Map Construction 1
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Experiments with Plant Hybrids (1866) Seed shape: 5474 round vs 1850 wrinkled Cotyledon color: 6022 yellow vs 2001 green Seed coat color: 705 grey-brown vs 224 white Pod shape: 882 inflated vs 299 constricted Unripe pod color: 428 green vs 152 yellow Flower position: 651 axial vs 207 terminal Stem length: 787 long (20-50cm) vs 277 short (185-230cm) Rediscovered in 1900
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Ear length of maize (East 1911) P1: 7cm; P2: 17cm One locus a=(17-7)/2=5; F2: 1/4 aa (7) + 2/4 Aa (12) + 1/4 AA (17) Two locus a=(17-7)/4=2.5 F2: 1/16 (7) + 4/16 (9.5) + 6/16 (12) + 4/16 (14.5) +1/16 (17) 5
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1 2 VF 2 2 ka
P1 ka P2 ka 2 (P1 P2 ) k 1 8[VF2 2 (VP1 VP2 )]
1 2 1 2 VA 2 a 2 ka 2 (P1 P )2 k 8V A
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Mendel and Fisher Annuals of Science 1:115- close to the values that Mendel expected under his theory that there must have been some manipulation, or omission, of data Dominant trait: 1/3 AA + 2/3 Aa Family size: 10 Non-segregating (AA) : Segregating (Aa) = 1:2 (Mendel) Fisher: Pro {Aa family classified as AA} = 0.75^10=0.0563 Pro {Non-segregating (AA)} =2/3*(1-0.0563)=0.6291 Non-segregating (AA) : Segregating (Aa) = 0.3709 : 0.6291 = 1 : 1.6961 13
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Genetic populations and pair- wise linkage analysis 15
Populations handled in QTL IciMapping Parent P1 Parent P2 Legends Hybridization F1 Selfing 1. P1BC1F1 7. F2 2. P2BC1F1 Repeated selfing 9. P1BC2F1 13. P1BC1F2 8. F3 14. P2BC1F2 10. P2BC2F1 Doubled haploids 15. P1BC2F2 16. P2BC2F2 11. P1BC2RIL 5. P1BC1RIL 4. F1RIL 6. P2BC1RIL 12. P2BC2RIL BC3F1, BC4F1 etc. P1BC2F1 P1BC1F1 F1 P2BC1F1 P2BC2F1 Marker-assisted selection 19. P1BC2DH 17. P1BC1DH 3. F1DH 18. P2BC1DH 20. P2BC2DH CSS lines or Introgression lines P1 × CP P2 × CP P3 × CP Pn × CP CP=common parent RIL family 1 RIL family 2 RIL family 3 RIL family i RIL family n One NAM population
Example: 10 RILs in a rice population (Linkage map of Chr. 5) Grain Marker C263 R830 R3166 XNpb387 R569 R1553 C128 C1402 XNpb81 C246 R2953 C1447 width (mm) Position 0.0 3.5 8.5 19.5 32.0 66.6 74.1 78.6 81.8 91.9 92.7 96.8 (cM) RIL1 0 0 0 0 0 0 0 0 0 0 0 0 2.33 RIL2 2 2 2 2 2 0 0 0 0 2 2 2 1.99 RIL3 0 2 2 2 2 2 2 2 2 2 2 2 2.24 RIL4 0 0 0 0 0 0 2 2 2 2 2 2 1.94 RIL5 0 0 0 0 0 2 2 0 0 0 0 0 2.76 RIL6 0 0 0 2 2 2 2 2 2 2 2 2 2.32 RIL7 0 0 0 0 0 0 0 0 0 0 0 0 2.32 RIL8 2 2 0 2 2 0 0 0 0 2 2 2 2.08 RIL9 0 0 0 0 2 2 0 0 0 0 0 0 2.24 17 RIL10 0 0 0 0 2 2 0 0 0 0 0 0 2.45
Genetic markers in linkage analysis Morphological traits hybridization experiments Cytogenetic and bio-chemistry markers (e.g. isozyme) DNA molecular markers RFLP, SSR, SNP etc.
The four gametes (haplotypes) of an F1 P1: AABB P2: aabb A B a b A B a b F1: AaBb A B a b Meiosis A B A b a B a b (1-r)/2 r/2 r/2 (1-r)/2 19 Parental type Recombinant Recombinant Parental type type type
Expected genotypic frequency in backcross and DH populations P1: AABB; P2: aabb 20
MLE of recombination frequency Likelihood function n1 n2 n3 n4 n! 1 1 1 1 L (1 r ) r r (1 r ) C (1 r ) n1 n4 ( r ) n2 n3 n1!n2 !n3! n4 ! 2 2 2 2 Logarithm of likelihood ln L ln C (n1 n4 ) ln(1 r ) (n2 n3 ) ln r n2 n3 n2 n3 r MLE of r n1 n2 n3 n4 n Fisher information d 2 ln L n1 n4 n2 n3 n I E( 2 ) E d r (1 r ) 2 r2 r (1 r ) Variance of estimated r Vr 1 r (1 r ) I n
Significance test of linkage Null hypothesis H0: r = 0.5 (no genetic linkage, or locus A-a and B-b are independent) Alternative hypothesis HA Likelihood ratio test (LRT) or LOD score L(r 0.5) 2 LRT 2 ln[ ]~ (df 1) L(r ) L(r ) LOD L(r 0.5)
An example P1BC1 population Genotypes of two inbred parents P1 and P2 are AABB and aabb Observed samples of the four genotypes in P1BC1 AABB 162 AABb 40 AaBB 41 AaBb 158 40 41 81 r 20.20% 162 40 41 158 401 r (1 r ) 4 Vr 4.02 10 23 n
Test of linkage Null hypothesis H0: r = 0.5 Alternative hypothesis HA L( r ) (1 r ) n1 n4 r n2 n3 6.3 10153 L( r 0.5) ( 1 ) n1 n2 n3 n4 4 Likelihood ratio test (LRT) (P<0.0001) and LOD score L( r ) LRT 2 * ln[ ] 708.27 L( r 0.5) L(r ) LOD log[ ] 153.80 24 L (r 0.5)
Genotypic frequencies in RIL populations, compared with DH DH Theoretical RIL Theoretical population frequency population frequency AABB f1=(1-r)/2 AABB f1=(1-R)/2 AAbb f2=r/2 AAbb f2=R/2 aaBB f3=r/2 aaBB f3=R/2 aabb f4=(1-r)/2 aabb f4=(1-R)/2 25 R=2r/(1+2r)
Parent type or RIL Marker 1 Marker 2 recombinant C263 XNpb387 n1=6 RIL1 0 or A 0 or A P1 type n2=2 n3=0 RIL2 2 or B 2 or B P2 type n4=2 RIL3 0 or A 2 or B Recombinant RIL4 0 or A 0 or A P1 type R=2/10=0.2 RIL5 0 or A 0 or A P1 type r=0.125 RIL6 0 or A 2 or B Recombinant RIL7 0 or A 0 or A P1 type LRT=17.72 (P=2.56 10-5) RIL8 2 or B 2 or B P2 type LOD=3.85 RIL9 0 or A 0 or A P1 type RIL10 0 or A 0 or A P1 type
Expected genotypic frequencies in F2 populations
MLE of r in F2: dominant markers 2 Logarithm of the likelihood ratio k (1 r ) ln L C n1 ln(3 2r r 2 ) (n3 n7 ) ln(2r r2) n9 ln(1 2r r2) C n1 ln(2 k ) (n3 n7 ) ln(1 k ) n9 ln k MLE of r 2 ( 2n 3n1 n9 ) ( 2n 3n1 n9 ) 2 n n9 k (1 r ) 2n Variance of the estimated r (1 k )(2 k ) (2r r 2 )(3 2r r 2 ) Vr 2n(1 2k ) 2n(3 4r 2r 2 )
MLE of r in F2: co-dominant markers (Newton-Raphson algorithm) Log-likelihood function ln L ln C (2n1 2n9 n2 n4 n6 n8 ) ln(1 r ) ( n2 n4 n6 n8 2n3 2n7 ) ln r n5 ln(1 2r 2r 2 ) The first-order derivative of LogL f'(r) ) d dr L 2n 2n n 1n n n n n n rn 2n 2n 1n (24rr 22r) ln r 1 9 2 4 6 8 2 4 6 8 3 7 5 2 The second-order derivative of LogL 2 2 d ln L 2 n 2n n n n n n n n n 2n 2n n ( 4r 4r ) f''(r) d r ) 2 ( r 1) 1 9 r 2 2 4 (1 2r 2r ) 6 8 2 4 6 2 8 3 7 5 2 2 The iteration algorithm: ri+1 = ri - f'(ri)/f''(ri)
MLE of r in F2: co-dominant markers (EM algorithm) EM for expectation and maximization E-step: for an initial r0, calculate the probability of crossover in each marker type M-step: Update r, and repeat from the E-step 1 r' n nk Pk ( R | G) k
Expected probability of crossover r= [n1 0+ n2 0.5+ n3 1 n8 0.5+ n9 0]/n
Estimated r after 3 EM iterations (r0=0.5)
Estimated r after 3 EM iterations (r0=0.25)
Estimated r after 3 EM iterations (r0=0.0)
Co-dominant markers in other populations R=2r/(1+2r)
More populations (e.g. BC1F2, F3 etc): Generation transition matrix of
Distortion has little effect on linkage analysis! DH pop Theo. Freq. Distortion Freq. in distortion AABB f1=(1-r)/2 (1-r)/2 (1-r)/(1+s) AAbb f2=r/2 r/2 r/(1+s) aaBB f3=r/2 s r/2 r s/(1+s) aabb f4=(1-r)/2 s (1-r)/2 (1-r) s/(1+s) Sum 1 (1+s)/2 1 r r /(1 s) r s /(1 s) r (1 s) /(1 s) r
Three-point analysis and linkage map construction 38
Linkage analysis of three markers r13 r12 r23 21 r12 r23 When 0 interference), (no (1 r13 ) (1 r12 )(1 r23 ) r12 r23 r13 r12 (1 r23 ) (1 r12 ) r23 r12 r23 2r12 r23 When 1 (complete interference), r13 r12 r23 The order of the three loci can be determined after linkage analysis (3!/2=3 potential orders) 39 1 2 3, or 1 3 2, or 2 1 3
Mapping distance and recombination frequency Mapping distance m13 m12 m23 Unit of mapping distance M (Morgan) or cM (centi-Morgan), 1M=100cM The function of mapping distance on recombination frequency (Mapping function): m f (r ) 40
Common mapping functions Morgan function (complete interference) In M: m =r (M) In cM: m =r 100 (cM) Haldane function (no interference) 1 2m In M: m f (r ) 2 ln(1 2r ) r 1 2 (1 e ) m / 50 In cM: m f (r ) 50 ln(1 2r ) r 1 2 (1 e ) Kosambi function (interference depends on length of interval) 4m In M: m 1 1 2r ln r 1 e 1 4m 4 1 2r 2 e 1 m / 25 1 2r 1e 1 m 25 ln r 41 In cM: 1 2r 2 em / 25 1
Comparison of the three functions Mapping distance (cM) (M) 42 Recombination frequency
Three steps in linkage map construction Step 1: Grouping. Grouping can be based on (i) a threshold of LOD score (ii) a threshold of marker distance (cM) (iii) anchor information Step 2: Ordering. Three ordering algorithms are (i) SER: SERiation (Buetow and Chakravarti, 1987. Am J Hum Genet 41:180 188) (ii) RECORD: REcombination Counting and ORDering (Van Os et al., 2005. Theor Appl Genet 112: 30 40) (iii) nnTwoOpt: nearest neighbor was used for tour construction, and two-opt was used for tour improvement, similar to Travelling Salesman Problem (TSP) (Lin and Kernighan, 1973. Oper. Res. 21: 498 516.
Three steps in linkage map construction Due to the large number of markers (n), it is impossible to compare all possible orders (say n=50, possible orders are n!/2=1.52x1064). Orders from the above algorithms are regional optimizations. Step 3: Rippling. Five rippling criteria are (i) SARF (Sum of Adjacent Recombination Frequencies) (ii) SAD (Sum of Adjacent Distances) (iii) SALOD (Sum of Adjacent LOD scores) (iv) COUNT (number of recombination events)
The MAP functionality in QTL IciMapping 45
Interface of the MAP functionality
A. Map of one chromosome B. Map of all chromosomes Map outputs: Linkage map for each chromosome (A) or all chromosomes (B)
An example map of seven chromosomes or groups 48
Linkage map and physical map Species Size of haploid Size of linkage kb/cM genome (kb) map (cM) Yeast 2.2 104 3700 6 Neurospora 4.2 104 500 80 Arabidopsis 7.0 104 500 140 Drosophila 2.0 105 290 700 Tomato 7.2 105 1400 510 Human 3.0 106 2710 1110 Wheat 1.6 107 2575 6214 Rice 4.4 105 1575 279 49 Corn 3.0 106 1400 2140
What is QTL Mapping? The procedure to map individual genetic factors with small effects on the quantitative traits, to specific chromosomal segments in the genome The key questions in QTL mapping studies are: How many QTL are there? Where are they in the marker map? How large an influence does each of them have on the trait of interest?
Grain Marker C263 R830 R3166 XNpb387 R569 R1553 C128 C1402 XNpb81 C246 R2953 C1447 width (mm) Position 0.0 3.5 8.5 19.5 32.0 66.6 74.1 78.6 81.8 91.9 92.7 96.8 (cM) RIL1 0 0 0 0 0 0 0 0 0 0 0 0 2.33 RIL2 2 2 2 2 2 0 0 0 0 2 2 2 1.99 RIL3 0 2 2 2 2 2 2 2 2 2 2 2 2.24 RIL4 0 0 0 0 0 0 2 2 2 2 2 2 1.94 RIL5 0 0 0 0 0 2 2 0 0 0 0 0 2.76 RIL6 0 0 0 2 2 2 2 2 2 2 2 2 2.32 RIL7 0 0 0 0 0 0 0 0 0 0 0 0 2.32 RIL8 2 2 0 2 2 0 0 0 0 2 2 2 2.08 RIL9 0 0 0 0 2 2 0 0 0 0 0 0 2.24 RIL10 0 0 0 0 2 2 0 0 0 0 0 0 2.45
Bi-parental mapping populations (linkage mapping) Temporary population: F2 and BC Permanent population: RIL, DH, CSSL Secondary population Association mapping Natural populations: human and animals
Single marker analysis (Sax 1923; Soller et al. 1976) The single marker analysis identifies QTLs based on the difference between the mean phenotypes for different marker groups, but cannot separate the estimates of recombination fraction and QTL effect. Interval mapping (IM) (Lander and Botstein 1989) IM is based on maximum likelihood parameter estimation and provides a likelihood ratio test for QTL position and effect. The major disadvantage of IM is that the estimates of locations and effects of QTLs may be biased when QTLs are linked. Regression interval mapping (RIM) (Haley and Knott 1992; Martinez and Curnow 1992 ) RIM was proposed to approximate maximum likelihood interval mapping to save computation time at one or multiple genomic positions.
Composite interval mapping (CIM) (Zeng 1994) CIM combines IM with multiple marker regression analysis, which controls the effects of QTLs on other intervals or chromosomes onto the QTL that is being tested, and thus increases the precision of QTL detection. Multiple interval mapping (MIM) (Kao et al. 1999) MIM is a state-of-the-art gene mapping procedure. But implementation of the multiple-QTL model is difficult, since the number of QTL defines the dimension of the model which is also an unknown parameter of interest. Bayesian model (Sillanpää and Corander 2002) In any Bayesian model, a prior distribution has to be considered. Based on the prior, Bayesian statistics derives the posterior, and then conduct inference based on the posterior distribution. However, Bayesian models have not been widely used in practice, partially due to the complexity of computation and the lack of user-friendly software.
mm Mm MM mm Mm MM A. B. QTL QTL
Backcrosses (P1BC1 and P2BC1) of P1: MMQQ and P2: mmqq BC1 BC2 Genotypic Genotypic Genotype Frequency Genotype Frequency value value 1 1 MMQQ 2 (1 r ) m+a MmQq 2 (1 r ) m+d 1 1 MMQq 2 r m+d Mmqq 2 r m-a 1 1 MmQQ 2 r m+a mmQq 2 r m+d 1 1 MmQq 2 (1 r ) m+d mmqq 2 (1 r ) m-a
Two marker types: MM (1 r ) MMQQ r MMQq (1 r )(m a) r (m d ) m (1 r )a rd Mm r MmQQ (1 r ) MmQq r (m a) (1 r )(m d ) m ra (1 r )d Difference in phenotype between the two types MM Mm (1 2r )(a d )
Linear model (j=1 2 n) yi b0 b* x* e j j b* represent QTL effect x * is the indicator j variable (0 or 1) for QTL genotype Likelihood profile Support interval: One-LOD interval
P1: Mi Q Mi +1 P2: mi q mi +1 Mi Q Mi +1 mi q mi +1 F1: Mi Q Mi +1 P1: Mi Q Mi +1 mi q mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q mi +1 mi q Mi +1 mi q mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi q mi +1 mi Q Mi +1 1 4
Assumption: No more than one QTL per chromosome or linkage group Large confidence interval Biased effect estimation Composite interval mapping (CIM) (Zeng 1994)
In the algorithm of CIM, both QTL effect at the current testing position and regression coefficients of the marker variables used to control genetic background were estimated simultaneously in an expectation and maximization (EM) algorithm. Thus, this algorithm could not completely ensure that the effect of QTL at current testing interval was not absorbed by the background marker variables and therefore may result in biased estimation of the QTL effect.
Theoretical basis of ICIM m G ajg j aa jk g j g k j 1 j k E ( g j | X) j xj j xj 1 E( g j gk | X) j k x j xk j k x j xk 1 j k x j 1xk j k x j 1xk 1 m 1 yi b0 b j xij b jk xij xik ei j 1 j k
One-dimensional scanning (interval mapping) yi yi b j xij j k ,k 1 Two-dimensional scanning (interval mapping) yi yi br xir brs xir xis r j , j 1,k ,k 1 r j, j 1 s k ,k 1
40 2 1.5 30 LOD score 1 0.5 Effect 20 0 10 -0.5 11111111111222222222233333333334444444444 -1 0 -1.5 11111111111222222222233333333334444444444 -2 Scanning posoition along the genome Scanning posoition along the genome 80 3 2 60 LOD score 1 Effect 40 0 -1 11111111111222222222233333333334444444444 20 -2 0 -3 11111111111222222222233333333334444444444 -4 Scanning posoition along the genome Scanning posoition along the genome 70 1.5 60 1 LOD score 50 40 0.5 Effect 30 0 20 -0.5 11111111111222222222233333333334444444444 10 0 -1 11111111111222222222233333333334444444444 -1.5 Scanning posoition along the genome Scanning posoition along the genome
Detecting epistasis where the interacting significant additive effects
One-locus model in F2 One-locus model: G aw dv where is mean of the two homozygous genotypes QQ and qq, a is the additive effect, d is the dominance effect . w and v are the indicators for genotypes at the QTL, valued at 1 and 0 for QQ, 0 and 1 for Qq, and -1 and 0 for qq, respectively
The expected genotypic value of an individual with known marker types E (G | x1 , x2 , y1 , y2 ) a E ( w | x1 , x2 , y1 , y2 ) d E (v | x1 , x2 , y1 , y2 )
Probability of the three QTL genotypes under given marker types Left Right QQ (w=1, v=0) Qq (w=0, v=1) qq (w=-1, v=0) marker marker (m+a) (m+d) (m-a) 2 2 1 1 2 2 AA BB 1 4 (1 r1 ) (1 r2 ) 2 1 r (1 r1 )r2 (1 r2 ) r r 4 1 2 2 AA Bb 1 2 (1 r1 ) 2 r2 (1 r2 ) 1 r (1 r1 )(1 r2 ) 2 1 2 1 r (1 r1 )r2 2 1 2 1 r r (1 r2 ) 2 1 2 1 2 2 (1 r1 ) r2 1 1 2 AA bb 4 r (1 r1 )r2 (1 r2 ) 2 1 r (1 r2 ) 2 4 1
Estimation of marker class mean Indicator Marker for marker E (w | x1 , x2 , y1 , y2 ) E (v | x1 , x2 , y1 , y2 ) Genetic mean n Frequency class of the class x1 x2 y1 y2 AABB n1 1 4 (1 r ) 2 1 1 0 0 f1 g1 f1a g1d 1 f2a g2d AABb n2 2 r (1 r ) 1 0 0 1 f2 g2 1 2 AAbb n3 4 r 1 -1 0 0 f3 g3 f 3a g3d 1 2r1r2 /(1 r ) f1 2r1 (1 r1 )r2 (1 r2 ) /(1 r ) 2 g1 [(1 2r1 )r2 (1 r2 )] /( r r ) 2 f2 r1 (1 r1 )(1 2r2 2r22 ) /( r r 2 ) g2 (r2 r1 ) / r f3 2r1 (1 r1 )r2 (1 r2 ) / r 2 g3
Relationship between marker class mean and marker effect (including marker interactions) f1a g1d 1 1 1 0 0 1 0 0 0 (d ) d f 2a g 2d 1 1 0 0 1 0 1 0 0 (a ) A1 f 3a g 3d 1 1 1 0 0 1 0 0 0 (a ) A2 f 4a g 4d 1 0 1 1 0 0 0 1 0 (d ) D1 g5d 1 0 0 1 1 0 0 0 1 (d ) D2 f 4a g 4d 1 0 1 1 0 0 0 1 0 (d ) AA12 f 3a g 3d 1 1 1 0 0 1 0 0 0 AD12 f 2a g 2d 1 1 0 0 1 0 1 0 0 DA12 f1a g1d 1 1 1 0 0 1 0 0 0 (d ) DD12
Relationship between marker effects and QTL effects 1 (d ) d 2 ( g1 g3 )d (a) A1 f2a 1 (a) A2 2 ( f1 f 3 )a 1 1 (d ) D1 ( g 2 1 2 g3 g 4 )d 1 1 (d ) D2 ( g g2 2 1 2 g 3 )d (d ) AA12 1 2 ( g1 g 3 )d AD12 0 DA12 0 (d ) DD12 ( 1 g1 g 2 2 1 2 g 3 g 4 g 5 )d
The linear model of genotypic values on markers in F2 E(w | x1 , x2 , y1 , y2 ) x 1 1 2 2 x E (v | x1 , x2 , y1 , y2 ) 1 1 y 2 y2 xx 12 1 2 yy 12 1 2
The linear model of genotypic values on markers in F2 E (G | x1 , x2 , y1 , y2 ) (a) A1 x1 (d ) D1 y1 (a) A2 x2 (d ) D2 y2 (d ) AA12 x1 x2 (d ) DD12 y1 y2
Properties of the linear model in F2 The additive and dominance effects of the flanked QTL are completely absorbed by the six variables in the model above. Interactions between marker variables may be declared as interaction between QTL by mistake when using ANOVA. But from our analysis, interactions between marker variables can be caused simply by dominance effects of QTL .
Multiple QTL model in F2 For multiple QTL, assume there are m QTL located on m intervals defined by m+1 markers on one chromosome, then the genotypic value of an F2 individual is defined as: m G [a j w j d jv j ] j 1
The linear model in F2 under multiple QTL The genotypic value of an F2 individual with known marker types can be re-organized as: m 1 m 1 E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
The linear model for QTL mapping in F2 m 1 m 1 P E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
Property of the linear model for QTL mapping in F2
ICIM (Inclusive Composite Interval Mapping) in F2 Pi Pi [ j xij j yij ] j k ,k 1 [ j , j 1 ijx xi , j 1 j, j 1 yij yi , j 1 ] j k
Hypothesis test of QTL mapping in F2 The two hypotheses used to test the existence of QTL at the scanning position are: vs. H 0 : 1 2 3 H A : at least two of 1 , 1 and 3 are not equal The logarithm likelihood under HA is 9 3 2 LA log[ jk f ( Pi ; k , )] j 1 i Sj k 1 where S j denotes individuals belonging to the j th marker class (j=1, th jk k=1, 2, 3) is the proportion of the k QTL genotype in th the j class, and f ( ; k , 2 ) is the density function of the normal 2 distribution N ( k , ) .
EM algorithm of QTL mapping in F2 Use EM algorithm to get the estimation of 1 , 2 and 3 So the genetic effects in G aw dv were therefore estimated by 1 1 2 ( 1 3 ) a 2 ( 1 3 ) d 2
EM algorithm of QTL mapping in F2 Parameters under H0 were calculated as: n n 1 2 1 2 0 n Pi 0 n ( Pi 0 ) i 1 i 1 From which the maximum likelihood under H0, and the LOD score between HA and H0 can be calculated.
QTL distribution models in simulation
QTL distribution models in simulation
QTL distribution models in simulation F2 populations were simulated by the genetics and breeding simulation tool of QuLine. QTL mapping using ICIM was implemented by the software QTL IciMapping.
Theoretical marker effects in the genetic model used in simulation The expected additive, dominance, additive by additive, and dominance by dominance effects of the two flanking markers associated with each QTL is shown in the following table. It indicated that the dominance of a QTL could complicate the coefficients of the two markers flanking a QTL, and cause the interactions between markers.
The expected marker effects in simulation Interaction QTL (d) d (a) A1 (a) A2 (d ) D1 (d ) D2 (d ) AA12 (d ) DD12 variation (%) QTL1 0.000 0.498 0.498 0.000 0.000 0.000 0.000 0.0 QTL2 0.253 0.000 0.000 0.248 0.248 -0.248 0.243 21.8 QTL3 0.253 0.498 0.498 0.248 0.248 -0.248 0.243 5.7 QTL4 -0.253 0.498 0.498 -0.248 -0.248 0.248 -0.243 5.7 QTL5 0.379 0.498 0.499 0.371 0.371 -0.371 0.364 9.6 QTL6 -0.379 0.498 0.498 -0.371 -0.371 0.371 -0.364 9.6
QTL mapping in simulated F2 populations
QTL LOD PVE True Est. True Est. add. True Est. score (%) Position Position add. effect dom. dom. (cM) (cM) effect effect effect QTL distribution model I QTL1 16.52 6.67 25 28 1 0.88 0 -0.11 QTL2 7.67 3.27 55 53 0 0.03 1 0.85 QTL3 25.11 11.28 25 24 1 0.86 1 1.08 QTL4 35.46 16.43 55 57 1 0.74 -1 -1.58 QTL5 37.12 16.74 25 26 1 1.05 1.5 1.38 QTL6 28.44 13.16 55 55 1 0.84 -1.5 -1.22
180 individuals The cross was made in Chengdu, China, in July 2002 between the indica rice variety and Nipponbare. 137 SSR markers. The whole genome was of 2046.2 cM, and the average marker distance was 17.1 cM. A number of agronomic traits were investigated in the field.
QTL mapping in the actual F2 population
QTL distribution Trait R2 of R2 of Absolute degree of dominance (|d/a|) Total additive additive and dominance (%) <=0.25 (0.25, 0.75] (0.75, 1.25] >1.25 (%) PH 25.84 51.56 2 1 1 5 9 HD 16.12 41.37 1 1 1 3 6 PL 25.58 61.26 5 3 1 8 17 FL 20.86 40.00 0 2 0 3 5 SPK 25.64 27.09 1 1 1 1 4 TKW 20.11 20.11 2 0 2 1 5 DP 19.45 24.87 1 1 0 1 3 GL 30.69 41.96 1 1 0 0 2 GW 26.63 26.63 2 2 0 0 4 RLW 37.63 45.70 1 3 1 1 6 Total 16 15 7 23 61
PVE distribution 20 18 Frequency across traits 16 14 12 10 8 6 4 2 0 Phenotypic variation explained(%)
Trait QTL Chr Distance to Add Dom LOD PVE(%) left marker Plant QPh1-1 1 12 -0.57 -7.98 8.04 12.03 height QPh1-2 1 19.5 -8.59 0.59 15.54 25.57 (Ph) QPh3-1 3 16.9 4.35 -4.86 6.51 13.30 QPh3-2 3 11.4 -4.69 -1.00 5.04 6.84 QPh4 4 13.7 -3.56 -2.09 4.61 5.53 QPh5 5 13 -0.44 -4.48 3.13 3.86 QPh6 6 6.2 -0.79 -5.05 3.17 4.96 QPh7 7 7 0.26 6.48 5.27 7.56 QPh12 12 2.4 -1.66 3.93 3.98 5.44 Heading QHd1 1 22.1 1.74 -0.30 3.65 7.27 date (Hd) QHd3 3 19.9 0.88 -3.70 6.04 21.09 QHd4 4 0.2 -0.77 1.85 3.58 5.24 QHd8 8 5.7 -1.41 -1.46 4.79 8.20 QHd10 10 0.3 -1.78 -0.80 4.85 7.21 QHd11 11 6.2 0.15 -3.03 5.71 11.70
Conclusions m 1 m 1 P E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
Six methods in BIP SMA: single marker analysis (Soller et al., 1976. Theor. Appl. Genet. 47: 35-39) IM-ADD: the conventional simple interval mapping (Lander and Botstein, 1989. Genetics 121: 185-199) ICIM-ADD: inclusive composite interval mapping of additive (and dominant) QTL (Li et al., 2007. Genetics 175: 361-374. Zhang et al., 2008. Genetics 180: 1177- 1190) IM-EPI: interval mapping of digenic epistatic QTL ICIM-EPI: inclusive composite interval mapping of digenic epistatic QTL (Li et al., 2008. Theor. Appl. Genet. 116: 243-260) SGM: selective genotyping mapping (Lebowitz et al., 1987. Theor. Appl. Genet. 73: 556 562)
Interface of the BIP functionality
LOD profile of ICIM additive mapping (ICIM-ADD)
Figures of interacting QTL from ICIM epistatic mapping (ICIM-EPI)

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Experimental design and statistical analysis

  • 1.
    Lecture 12 Genetic LinkageAnalysis and Map Construction 1
  • 2.
  • 3.
    Experiments with PlantHybrids (1866) Seed shape: 5474 round vs 1850 wrinkled Cotyledon color: 6022 yellow vs 2001 green Seed coat color: 705 grey-brown vs 224 white Pod shape: 882 inflated vs 299 constricted Unripe pod color: 428 green vs 152 yellow Flower position: 651 axial vs 207 terminal Stem length: 787 long (20-50cm) vs 277 short (185-230cm) Rediscovered in 1900
  • 4.
  • 5.
    Ear length ofmaize (East 1911) P1: 7cm; P2: 17cm One locus a=(17-7)/2=5; F2: 1/4 aa (7) + 2/4 Aa (12) + 1/4 AA (17) Two locus a=(17-7)/4=2.5 F2: 1/16 (7) + 4/16 (9.5) + 6/16 (12) + 4/16 (14.5) +1/16 (17) 5
  • 6.
  • 7.
  • 8.
  • 9.
    1 2 VF 2 2 ka
  • 10.
    P1 ka P2 ka 2 (P1 P2 ) k 1 8[VF2 2 (VP1 VP2 )]
  • 11.
    1 2 1 2 VA 2 a 2 ka 2 (P1 P )2 k 8V A
  • 12.
  • 13.
    Mendel and Fisher Annualsof Science 1:115- close to the values that Mendel expected under his theory that there must have been some manipulation, or omission, of data Dominant trait: 1/3 AA + 2/3 Aa Family size: 10 Non-segregating (AA) : Segregating (Aa) = 1:2 (Mendel) Fisher: Pro {Aa family classified as AA} = 0.75^10=0.0563 Pro {Non-segregating (AA)} =2/3*(1-0.0563)=0.6291 Non-segregating (AA) : Segregating (Aa) = 0.3709 : 0.6291 = 1 : 1.6961 13
  • 14.
  • 15.
    Genetic populations andpair- wise linkage analysis 15
  • 16.
    Populations handled inQTL IciMapping Parent P1 Parent P2 Legends Hybridization F1 Selfing 1. P1BC1F1 7. F2 2. P2BC1F1 Repeated selfing 9. P1BC2F1 13. P1BC1F2 8. F3 14. P2BC1F2 10. P2BC2F1 Doubled haploids 15. P1BC2F2 16. P2BC2F2 11. P1BC2RIL 5. P1BC1RIL 4. F1RIL 6. P2BC1RIL 12. P2BC2RIL BC3F1, BC4F1 etc. P1BC2F1 P1BC1F1 F1 P2BC1F1 P2BC2F1 Marker-assisted selection 19. P1BC2DH 17. P1BC1DH 3. F1DH 18. P2BC1DH 20. P2BC2DH CSS lines or Introgression lines P1 × CP P2 × CP P3 × CP Pn × CP CP=common parent RIL family 1 RIL family 2 RIL family 3 RIL family i RIL family n One NAM population
  • 17.
    Example: 10 RILsin a rice population (Linkage map of Chr. 5) Grain Marker C263 R830 R3166 XNpb387 R569 R1553 C128 C1402 XNpb81 C246 R2953 C1447 width (mm) Position 0.0 3.5 8.5 19.5 32.0 66.6 74.1 78.6 81.8 91.9 92.7 96.8 (cM) RIL1 0 0 0 0 0 0 0 0 0 0 0 0 2.33 RIL2 2 2 2 2 2 0 0 0 0 2 2 2 1.99 RIL3 0 2 2 2 2 2 2 2 2 2 2 2 2.24 RIL4 0 0 0 0 0 0 2 2 2 2 2 2 1.94 RIL5 0 0 0 0 0 2 2 0 0 0 0 0 2.76 RIL6 0 0 0 2 2 2 2 2 2 2 2 2 2.32 RIL7 0 0 0 0 0 0 0 0 0 0 0 0 2.32 RIL8 2 2 0 2 2 0 0 0 0 2 2 2 2.08 RIL9 0 0 0 0 2 2 0 0 0 0 0 0 2.24 17 RIL10 0 0 0 0 2 2 0 0 0 0 0 0 2.45
  • 18.
    Genetic markers inlinkage analysis Morphological traits hybridization experiments Cytogenetic and bio-chemistry markers (e.g. isozyme) DNA molecular markers RFLP, SSR, SNP etc.
  • 19.
    The four gametes(haplotypes) of an F1 P1: AABB P2: aabb A B a b A B a b F1: AaBb A B a b Meiosis A B A b a B a b (1-r)/2 r/2 r/2 (1-r)/2 19 Parental type Recombinant Recombinant Parental type type type
  • 20.
    Expected genotypic frequencyin backcross and DH populations P1: AABB; P2: aabb 20
  • 21.
    MLE of recombinationfrequency Likelihood function n1 n2 n3 n4 n! 1 1 1 1 L (1 r ) r r (1 r ) C (1 r ) n1 n4 ( r ) n2 n3 n1!n2 !n3! n4 ! 2 2 2 2 Logarithm of likelihood ln L ln C (n1 n4 ) ln(1 r ) (n2 n3 ) ln r n2 n3 n2 n3 r MLE of r n1 n2 n3 n4 n Fisher information d 2 ln L n1 n4 n2 n3 n I E( 2 ) E d r (1 r ) 2 r2 r (1 r ) Variance of estimated r Vr 1 r (1 r ) I n
  • 22.
    Significance test oflinkage Null hypothesis H0: r = 0.5 (no genetic linkage, or locus A-a and B-b are independent) Alternative hypothesis HA Likelihood ratio test (LRT) or LOD score L(r 0.5) 2 LRT 2 ln[ ]~ (df 1) L(r ) L(r ) LOD L(r 0.5)
  • 23.
    An example P1BC1population Genotypes of two inbred parents P1 and P2 are AABB and aabb Observed samples of the four genotypes in P1BC1 AABB 162 AABb 40 AaBB 41 AaBb 158 40 41 81 r 20.20% 162 40 41 158 401 r (1 r ) 4 Vr 4.02 10 23 n
  • 24.
    Test of linkage Nullhypothesis H0: r = 0.5 Alternative hypothesis HA L( r ) (1 r ) n1 n4 r n2 n3 6.3 10153 L( r 0.5) ( 1 ) n1 n2 n3 n4 4 Likelihood ratio test (LRT) (P<0.0001) and LOD score L( r ) LRT 2 * ln[ ] 708.27 L( r 0.5) L(r ) LOD log[ ] 153.80 24 L (r 0.5)
  • 25.
    Genotypic frequencies inRIL populations, compared with DH DH Theoretical RIL Theoretical population frequency population frequency AABB f1=(1-r)/2 AABB f1=(1-R)/2 AAbb f2=r/2 AAbb f2=R/2 aaBB f3=r/2 aaBB f3=R/2 aabb f4=(1-r)/2 aabb f4=(1-R)/2 25 R=2r/(1+2r)
  • 26.
    Parent type or RIL Marker 1 Marker 2 recombinant C263 XNpb387 n1=6 RIL1 0 or A 0 or A P1 type n2=2 n3=0 RIL2 2 or B 2 or B P2 type n4=2 RIL3 0 or A 2 or B Recombinant RIL4 0 or A 0 or A P1 type R=2/10=0.2 RIL5 0 or A 0 or A P1 type r=0.125 RIL6 0 or A 2 or B Recombinant RIL7 0 or A 0 or A P1 type LRT=17.72 (P=2.56 10-5) RIL8 2 or B 2 or B P2 type LOD=3.85 RIL9 0 or A 0 or A P1 type RIL10 0 or A 0 or A P1 type
  • 27.
  • 28.
    MLE of rin F2: dominant markers 2 Logarithm of the likelihood ratio k (1 r ) ln L C n1 ln(3 2r r 2 ) (n3 n7 ) ln(2r r2) n9 ln(1 2r r2) C n1 ln(2 k ) (n3 n7 ) ln(1 k ) n9 ln k MLE of r 2 ( 2n 3n1 n9 ) ( 2n 3n1 n9 ) 2 n n9 k (1 r ) 2n Variance of the estimated r (1 k )(2 k ) (2r r 2 )(3 2r r 2 ) Vr 2n(1 2k ) 2n(3 4r 2r 2 )
  • 29.
    MLE of rin F2: co-dominant markers (Newton-Raphson algorithm) Log-likelihood function ln L ln C (2n1 2n9 n2 n4 n6 n8 ) ln(1 r ) ( n2 n4 n6 n8 2n3 2n7 ) ln r n5 ln(1 2r 2r 2 ) The first-order derivative of LogL f'(r) ) d dr L 2n 2n n 1n n n n n n rn 2n 2n 1n (24rr 22r) ln r 1 9 2 4 6 8 2 4 6 8 3 7 5 2 The second-order derivative of LogL 2 2 d ln L 2 n 2n n n n n n n n n 2n 2n n ( 4r 4r ) f''(r) d r ) 2 ( r 1) 1 9 r 2 2 4 (1 2r 2r ) 6 8 2 4 6 2 8 3 7 5 2 2 The iteration algorithm: ri+1 = ri - f'(ri)/f''(ri)
  • 30.
    MLE of rin F2: co-dominant markers (EM algorithm) EM for expectation and maximization E-step: for an initial r0, calculate the probability of crossover in each marker type M-step: Update r, and repeat from the E-step 1 r' n nk Pk ( R | G) k
  • 31.
    Expected probability ofcrossover r= [n1 0+ n2 0.5+ n3 1 n8 0.5+ n9 0]/n
  • 32.
    Estimated r after3 EM iterations (r0=0.5)
  • 33.
    Estimated r after3 EM iterations (r0=0.25)
  • 34.
    Estimated r after3 EM iterations (r0=0.0)
  • 35.
    Co-dominant markers inother populations R=2r/(1+2r)
  • 36.
    More populations (e.g.BC1F2, F3 etc): Generation transition matrix of
  • 37.
    Distortion has littleeffect on linkage analysis! DH pop Theo. Freq. Distortion Freq. in distortion AABB f1=(1-r)/2 (1-r)/2 (1-r)/(1+s) AAbb f2=r/2 r/2 r/(1+s) aaBB f3=r/2 s r/2 r s/(1+s) aabb f4=(1-r)/2 s (1-r)/2 (1-r) s/(1+s) Sum 1 (1+s)/2 1 r r /(1 s) r s /(1 s) r (1 s) /(1 s) r
  • 38.
    Three-point analysis andlinkage map construction 38
  • 39.
    Linkage analysis ofthree markers r13 r12 r23 21 r12 r23 When 0 interference), (no (1 r13 ) (1 r12 )(1 r23 ) r12 r23 r13 r12 (1 r23 ) (1 r12 ) r23 r12 r23 2r12 r23 When 1 (complete interference), r13 r12 r23 The order of the three loci can be determined after linkage analysis (3!/2=3 potential orders) 39 1 2 3, or 1 3 2, or 2 1 3
  • 40.
    Mapping distance and recombination frequency Mapping distance m13 m12 m23 Unit of mapping distance M (Morgan) or cM (centi-Morgan), 1M=100cM The function of mapping distance on recombination frequency (Mapping function): m f (r ) 40
  • 41.
    Common mapping functions Morganfunction (complete interference) In M: m =r (M) In cM: m =r 100 (cM) Haldane function (no interference) 1 2m In M: m f (r ) 2 ln(1 2r ) r 1 2 (1 e ) m / 50 In cM: m f (r ) 50 ln(1 2r ) r 1 2 (1 e ) Kosambi function (interference depends on length of interval) 4m In M: m 1 1 2r ln r 1 e 1 4m 4 1 2r 2 e 1 m / 25 1 2r 1e 1 m 25 ln r 41 In cM: 1 2r 2 em / 25 1
  • 42.
    Comparison of thethree functions Mapping distance (cM) (M) 42 Recombination frequency
  • 43.
    Three steps inlinkage map construction Step 1: Grouping. Grouping can be based on (i) a threshold of LOD score (ii) a threshold of marker distance (cM) (iii) anchor information Step 2: Ordering. Three ordering algorithms are (i) SER: SERiation (Buetow and Chakravarti, 1987. Am J Hum Genet 41:180 188) (ii) RECORD: REcombination Counting and ORDering (Van Os et al., 2005. Theor Appl Genet 112: 30 40) (iii) nnTwoOpt: nearest neighbor was used for tour construction, and two-opt was used for tour improvement, similar to Travelling Salesman Problem (TSP) (Lin and Kernighan, 1973. Oper. Res. 21: 498 516.
  • 44.
    Three steps inlinkage map construction Due to the large number of markers (n), it is impossible to compare all possible orders (say n=50, possible orders are n!/2=1.52x1064). Orders from the above algorithms are regional optimizations. Step 3: Rippling. Five rippling criteria are (i) SARF (Sum of Adjacent Recombination Frequencies) (ii) SAD (Sum of Adjacent Distances) (iii) SALOD (Sum of Adjacent LOD scores) (iv) COUNT (number of recombination events)
  • 45.
    The MAP functionalityin QTL IciMapping 45
  • 46.
    Interface of theMAP functionality
  • 47.
    A. Map ofone chromosome B. Map of all chromosomes Map outputs: Linkage map for each chromosome (A) or all chromosomes (B)
  • 48.
    An example mapof seven chromosomes or groups 48
  • 49.
    Linkage map andphysical map Species Size of haploid Size of linkage kb/cM genome (kb) map (cM) Yeast 2.2 104 3700 6 Neurospora 4.2 104 500 80 Arabidopsis 7.0 104 500 140 Drosophila 2.0 105 290 700 Tomato 7.2 105 1400 510 Human 3.0 106 2710 1110 Wheat 1.6 107 2575 6214 Rice 4.4 105 1575 279 49 Corn 3.0 106 1400 2140
  • 54.
    What is QTLMapping? The procedure to map individual genetic factors with small effects on the quantitative traits, to specific chromosomal segments in the genome The key questions in QTL mapping studies are: How many QTL are there? Where are they in the marker map? How large an influence does each of them have on the trait of interest?
  • 56.
    Grain Marker C263 R830 R3166 XNpb387 R569 R1553 C128 C1402 XNpb81 C246 R2953 C1447 width (mm) Position 0.0 3.5 8.5 19.5 32.0 66.6 74.1 78.6 81.8 91.9 92.7 96.8 (cM) RIL1 0 0 0 0 0 0 0 0 0 0 0 0 2.33 RIL2 2 2 2 2 2 0 0 0 0 2 2 2 1.99 RIL3 0 2 2 2 2 2 2 2 2 2 2 2 2.24 RIL4 0 0 0 0 0 0 2 2 2 2 2 2 1.94 RIL5 0 0 0 0 0 2 2 0 0 0 0 0 2.76 RIL6 0 0 0 2 2 2 2 2 2 2 2 2 2.32 RIL7 0 0 0 0 0 0 0 0 0 0 0 0 2.32 RIL8 2 2 0 2 2 0 0 0 0 2 2 2 2.08 RIL9 0 0 0 0 2 2 0 0 0 0 0 0 2.24 RIL10 0 0 0 0 2 2 0 0 0 0 0 0 2.45
  • 57.
    Bi-parental mapping populations(linkage mapping) Temporary population: F2 and BC Permanent population: RIL, DH, CSSL Secondary population Association mapping Natural populations: human and animals
  • 58.
    Single marker analysis(Sax 1923; Soller et al. 1976) The single marker analysis identifies QTLs based on the difference between the mean phenotypes for different marker groups, but cannot separate the estimates of recombination fraction and QTL effect. Interval mapping (IM) (Lander and Botstein 1989) IM is based on maximum likelihood parameter estimation and provides a likelihood ratio test for QTL position and effect. The major disadvantage of IM is that the estimates of locations and effects of QTLs may be biased when QTLs are linked. Regression interval mapping (RIM) (Haley and Knott 1992; Martinez and Curnow 1992 ) RIM was proposed to approximate maximum likelihood interval mapping to save computation time at one or multiple genomic positions.
  • 59.
    Composite interval mapping(CIM) (Zeng 1994) CIM combines IM with multiple marker regression analysis, which controls the effects of QTLs on other intervals or chromosomes onto the QTL that is being tested, and thus increases the precision of QTL detection. Multiple interval mapping (MIM) (Kao et al. 1999) MIM is a state-of-the-art gene mapping procedure. But implementation of the multiple-QTL model is difficult, since the number of QTL defines the dimension of the model which is also an unknown parameter of interest. Bayesian model (Sillanpää and Corander 2002) In any Bayesian model, a prior distribution has to be considered. Based on the prior, Bayesian statistics derives the posterior, and then conduct inference based on the posterior distribution. However, Bayesian models have not been widely used in practice, partially due to the complexity of computation and the lack of user-friendly software.
  • 60.
    mm Mm MM mm Mm MM A. B. QTL QTL
  • 61.
    Backcrosses (P1BC1 andP2BC1) of P1: MMQQ and P2: mmqq BC1 BC2 Genotypic Genotypic Genotype Frequency Genotype Frequency value value 1 1 MMQQ 2 (1 r ) m+a MmQq 2 (1 r ) m+d 1 1 MMQq 2 r m+d Mmqq 2 r m-a 1 1 MmQQ 2 r m+a mmQq 2 r m+d 1 1 MmQq 2 (1 r ) m+d mmqq 2 (1 r ) m-a
  • 62.
    Two marker types: MM (1 r ) MMQQ r MMQq (1 r )(m a) r (m d ) m (1 r )a rd Mm r MmQQ (1 r ) MmQq r (m a) (1 r )(m d ) m ra (1 r )d Difference in phenotype between the two types MM Mm (1 2r )(a d )
  • 63.
    Linear model (j=12 n) yi b0 b* x* e j j b* represent QTL effect x * is the indicator j variable (0 or 1) for QTL genotype Likelihood profile Support interval: One-LOD interval
  • 64.
    P1: Mi Q Mi +1 P2: mi q mi +1 Mi Q Mi +1 mi q mi +1 F1: Mi Q Mi +1 P1: Mi Q Mi +1 mi q mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q mi +1 mi q Mi +1 mi q mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi q mi +1 mi Q Mi +1 1 4
  • 66.
    Assumption: No morethan one QTL per chromosome or linkage group Large confidence interval Biased effect estimation Composite interval mapping (CIM) (Zeng 1994)
  • 67.
    In the algorithmof CIM, both QTL effect at the current testing position and regression coefficients of the marker variables used to control genetic background were estimated simultaneously in an expectation and maximization (EM) algorithm. Thus, this algorithm could not completely ensure that the effect of QTL at current testing interval was not absorbed by the background marker variables and therefore may result in biased estimation of the QTL effect.
  • 68.
    Theoretical basis ofICIM m G ajg j aa jk g j g k j 1 j k E ( g j | X) j xj j xj 1 E( g j gk | X) j k x j xk j k x j xk 1 j k x j 1xk j k x j 1xk 1 m 1 yi b0 b j xij b jk xij xik ei j 1 j k
  • 69.
    One-dimensional scanning (intervalmapping) yi yi b j xij j k ,k 1 Two-dimensional scanning (interval mapping) yi yi br xir brs xir xis r j , j 1,k ,k 1 r j, j 1 s k ,k 1
  • 70.
    40 2 1.5 30 LOD score 1 0.5 Effect 20 0 10 -0.5 11111111111222222222233333333334444444444 -1 0 -1.5 11111111111222222222233333333334444444444 -2 Scanning posoition along the genome Scanning posoition along the genome 80 3 2 60 LOD score 1 Effect 40 0 -1 11111111111222222222233333333334444444444 20 -2 0 -3 11111111111222222222233333333334444444444 -4 Scanning posoition along the genome Scanning posoition along the genome 70 1.5 60 1 LOD score 50 40 0.5 Effect 30 0 20 -0.5 11111111111222222222233333333334444444444 10 0 -1 11111111111222222222233333333334444444444 -1.5 Scanning posoition along the genome Scanning posoition along the genome
  • 71.
  • 74.
    One-locus model inF2 One-locus model: G aw dv where is mean of the two homozygous genotypes QQ and qq, a is the additive effect, d is the dominance effect . w and v are the indicators for genotypes at the QTL, valued at 1 and 0 for QQ, 0 and 1 for Qq, and -1 and 0 for qq, respectively
  • 75.
    The expected genotypicvalue of an individual with known marker types E (G | x1 , x2 , y1 , y2 ) a E ( w | x1 , x2 , y1 , y2 ) d E (v | x1 , x2 , y1 , y2 )
  • 76.
    Probability of thethree QTL genotypes under given marker types Left Right QQ (w=1, v=0) Qq (w=0, v=1) qq (w=-1, v=0) marker marker (m+a) (m+d) (m-a) 2 2 1 1 2 2 AA BB 1 4 (1 r1 ) (1 r2 ) 2 1 r (1 r1 )r2 (1 r2 ) r r 4 1 2 2 AA Bb 1 2 (1 r1 ) 2 r2 (1 r2 ) 1 r (1 r1 )(1 r2 ) 2 1 2 1 r (1 r1 )r2 2 1 2 1 r r (1 r2 ) 2 1 2 1 2 2 (1 r1 ) r2 1 1 2 AA bb 4 r (1 r1 )r2 (1 r2 ) 2 1 r (1 r2 ) 2 4 1
  • 77.
    Estimation of markerclass mean Indicator Marker for marker E (w | x1 , x2 , y1 , y2 ) E (v | x1 , x2 , y1 , y2 ) Genetic mean n Frequency class of the class x1 x2 y1 y2 AABB n1 1 4 (1 r ) 2 1 1 0 0 f1 g1 f1a g1d 1 f2a g2d AABb n2 2 r (1 r ) 1 0 0 1 f2 g2 1 2 AAbb n3 4 r 1 -1 0 0 f3 g3 f 3a g3d 1 2r1r2 /(1 r ) f1 2r1 (1 r1 )r2 (1 r2 ) /(1 r ) 2 g1 [(1 2r1 )r2 (1 r2 )] /( r r ) 2 f2 r1 (1 r1 )(1 2r2 2r22 ) /( r r 2 ) g2 (r2 r1 ) / r f3 2r1 (1 r1 )r2 (1 r2 ) / r 2 g3
  • 78.
    Relationship between marker class mean and marker effect (including marker interactions) f1a g1d 1 1 1 0 0 1 0 0 0 (d ) d f 2a g 2d 1 1 0 0 1 0 1 0 0 (a ) A1 f 3a g 3d 1 1 1 0 0 1 0 0 0 (a ) A2 f 4a g 4d 1 0 1 1 0 0 0 1 0 (d ) D1 g5d 1 0 0 1 1 0 0 0 1 (d ) D2 f 4a g 4d 1 0 1 1 0 0 0 1 0 (d ) AA12 f 3a g 3d 1 1 1 0 0 1 0 0 0 AD12 f 2a g 2d 1 1 0 0 1 0 1 0 0 DA12 f1a g1d 1 1 1 0 0 1 0 0 0 (d ) DD12
  • 79.
    Relationship between marker effects and QTL effects 1 (d ) d 2 ( g1 g3 )d (a) A1 f2a 1 (a) A2 2 ( f1 f 3 )a 1 1 (d ) D1 ( g 2 1 2 g3 g 4 )d 1 1 (d ) D2 ( g g2 2 1 2 g 3 )d (d ) AA12 1 2 ( g1 g 3 )d AD12 0 DA12 0 (d ) DD12 ( 1 g1 g 2 2 1 2 g 3 g 4 g 5 )d
  • 80.
    The linear modelof genotypic values on markers in F2 E(w | x1 , x2 , y1 , y2 ) x 1 1 2 2 x E (v | x1 , x2 , y1 , y2 ) 1 1 y 2 y2 xx 12 1 2 yy 12 1 2
  • 81.
    The linear modelof genotypic values on markers in F2 E (G | x1 , x2 , y1 , y2 ) (a) A1 x1 (d ) D1 y1 (a) A2 x2 (d ) D2 y2 (d ) AA12 x1 x2 (d ) DD12 y1 y2
  • 82.
    Properties of thelinear model in F2 The additive and dominance effects of the flanked QTL are completely absorbed by the six variables in the model above. Interactions between marker variables may be declared as interaction between QTL by mistake when using ANOVA. But from our analysis, interactions between marker variables can be caused simply by dominance effects of QTL .
  • 83.
    Multiple QTL modelin F2 For multiple QTL, assume there are m QTL located on m intervals defined by m+1 markers on one chromosome, then the genotypic value of an F2 individual is defined as: m G [a j w j d jv j ] j 1
  • 84.
    The linear modelin F2 under multiple QTL The genotypic value of an F2 individual with known marker types can be re-organized as: m 1 m 1 E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
  • 85.
    The linear modelfor QTL mapping in F2 m 1 m 1 P E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
  • 86.
    Property of thelinear model for QTL mapping in F2
  • 87.
    ICIM (Inclusive Composite Interval Mapping) in F2 Pi Pi [ j xij j yij ] j k ,k 1 [ j , j 1 ijx xi , j 1 j, j 1 yij yi , j 1 ] j k
  • 88.
    Hypothesis test ofQTL mapping in F2 The two hypotheses used to test the existence of QTL at the scanning position are: vs. H 0 : 1 2 3 H A : at least two of 1 , 1 and 3 are not equal The logarithm likelihood under HA is 9 3 2 LA log[ jk f ( Pi ; k , )] j 1 i Sj k 1 where S j denotes individuals belonging to the j th marker class (j=1, th jk k=1, 2, 3) is the proportion of the k QTL genotype in th the j class, and f ( ; k , 2 ) is the density function of the normal 2 distribution N ( k , ) .
  • 89.
    EM algorithm ofQTL mapping in F2 Use EM algorithm to get the estimation of 1 , 2 and 3 So the genetic effects in G aw dv were therefore estimated by 1 1 2 ( 1 3 ) a 2 ( 1 3 ) d 2
  • 90.
    EM algorithm ofQTL mapping in F2 Parameters under H0 were calculated as: n n 1 2 1 2 0 n Pi 0 n ( Pi 0 ) i 1 i 1 From which the maximum likelihood under H0, and the LOD score between HA and H0 can be calculated.
  • 91.
  • 92.
  • 93.
    QTL distribution modelsin simulation F2 populations were simulated by the genetics and breeding simulation tool of QuLine. QTL mapping using ICIM was implemented by the software QTL IciMapping.
  • 94.
    Theoretical marker effectsin the genetic model used in simulation The expected additive, dominance, additive by additive, and dominance by dominance effects of the two flanking markers associated with each QTL is shown in the following table. It indicated that the dominance of a QTL could complicate the coefficients of the two markers flanking a QTL, and cause the interactions between markers.
  • 95.
    The expected markereffects in simulation Interaction QTL (d) d (a) A1 (a) A2 (d ) D1 (d ) D2 (d ) AA12 (d ) DD12 variation (%) QTL1 0.000 0.498 0.498 0.000 0.000 0.000 0.000 0.0 QTL2 0.253 0.000 0.000 0.248 0.248 -0.248 0.243 21.8 QTL3 0.253 0.498 0.498 0.248 0.248 -0.248 0.243 5.7 QTL4 -0.253 0.498 0.498 -0.248 -0.248 0.248 -0.243 5.7 QTL5 0.379 0.498 0.499 0.371 0.371 -0.371 0.364 9.6 QTL6 -0.379 0.498 0.498 -0.371 -0.371 0.371 -0.364 9.6
  • 96.
    QTL mapping insimulated F2 populations
  • 97.
    QTL LOD PVE True Est. True Est. add. True Est. score (%) Position Position add. effect dom. dom. (cM) (cM) effect effect effect QTL distribution model I QTL1 16.52 6.67 25 28 1 0.88 0 -0.11 QTL2 7.67 3.27 55 53 0 0.03 1 0.85 QTL3 25.11 11.28 25 24 1 0.86 1 1.08 QTL4 35.46 16.43 55 57 1 0.74 -1 -1.58 QTL5 37.12 16.74 25 26 1 1.05 1.5 1.38 QTL6 28.44 13.16 55 55 1 0.84 -1.5 -1.22
  • 101.
    180 individuals The crosswas made in Chengdu, China, in July 2002 between the indica rice variety and Nipponbare. 137 SSR markers. The whole genome was of 2046.2 cM, and the average marker distance was 17.1 cM. A number of agronomic traits were investigated in the field.
  • 102.
    QTL mapping inthe actual F2 population
  • 103.
    QTL distribution Trait R2 of R2 of Absolute degree of dominance (|d/a|) Total additive additive and dominance (%) <=0.25 (0.25, 0.75] (0.75, 1.25] >1.25 (%) PH 25.84 51.56 2 1 1 5 9 HD 16.12 41.37 1 1 1 3 6 PL 25.58 61.26 5 3 1 8 17 FL 20.86 40.00 0 2 0 3 5 SPK 25.64 27.09 1 1 1 1 4 TKW 20.11 20.11 2 0 2 1 5 DP 19.45 24.87 1 1 0 1 3 GL 30.69 41.96 1 1 0 0 2 GW 26.63 26.63 2 2 0 0 4 RLW 37.63 45.70 1 3 1 1 6 Total 16 15 7 23 61
  • 104.
    PVE distribution 20 18 Frequency across traits 16 14 12 10 8 6 4 2 0 Phenotypic variation explained(%)
  • 105.
    Trait QTL Chr Distance to Add Dom LOD PVE(%) left marker Plant QPh1-1 1 12 -0.57 -7.98 8.04 12.03 height QPh1-2 1 19.5 -8.59 0.59 15.54 25.57 (Ph) QPh3-1 3 16.9 4.35 -4.86 6.51 13.30 QPh3-2 3 11.4 -4.69 -1.00 5.04 6.84 QPh4 4 13.7 -3.56 -2.09 4.61 5.53 QPh5 5 13 -0.44 -4.48 3.13 3.86 QPh6 6 6.2 -0.79 -5.05 3.17 4.96 QPh7 7 7 0.26 6.48 5.27 7.56 QPh12 12 2.4 -1.66 3.93 3.98 5.44 Heading QHd1 1 22.1 1.74 -0.30 3.65 7.27 date (Hd) QHd3 3 19.9 0.88 -3.70 6.04 21.09 QHd4 4 0.2 -0.77 1.85 3.58 5.24 QHd8 8 5.7 -1.41 -1.46 4.79 8.20 QHd10 10 0.3 -1.78 -0.80 4.85 7.21 QHd11 11 6.2 0.15 -3.03 5.71 11.70
  • 106.
    Conclusions m 1 m 1 P E (G ) j xj j yj j 1 j 1 m m j, j 1 xjxj 1 j, j 1 yj yj 1 j 1 j 1
  • 108.
    Six methods inBIP SMA: single marker analysis (Soller et al., 1976. Theor. Appl. Genet. 47: 35-39) IM-ADD: the conventional simple interval mapping (Lander and Botstein, 1989. Genetics 121: 185-199) ICIM-ADD: inclusive composite interval mapping of additive (and dominant) QTL (Li et al., 2007. Genetics 175: 361-374. Zhang et al., 2008. Genetics 180: 1177- 1190) IM-EPI: interval mapping of digenic epistatic QTL ICIM-EPI: inclusive composite interval mapping of digenic epistatic QTL (Li et al., 2008. Theor. Appl. Genet. 116: 243-260) SGM: selective genotyping mapping (Lebowitz et al., 1987. Theor. Appl. Genet. 73: 556 562)
  • 109.
    Interface of theBIP functionality
  • 110.
    LOD profile ofICIM additive mapping (ICIM-ADD)
  • 111.
    Figures of interactingQTL from ICIM epistatic mapping (ICIM-EPI)