Vol. 32, Special Issue, S 1 S 17 (2011)

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Vol. 32, Special Issue, S 1 S 17 (2011) e-mail:luke154@jcom.home.ne.jp 1. 1.1 7 1900 800 2

S2 1.2 19 1909 Herman Nilsson-Ehle F 2 F 3 1 3 2 3 4 E. M. East 1.3 Wilhelm Ludwig Johannsen 1900 19 574 5,494 1902 5,494 1900 2 1901 1902 gene genotype phenotype

S3 2. 2.1 1910 Ronald Aylmer Fisher The correlation between relatives on the supposition of Mendelian inheritance Biometrika 1918 epistasis population genetics 1960 2 A 3 A 1A 1, A 1A 2, A 2A 2 f 11, 2f 12, f 22 (f 11 +2f 12 + f 22 =1) g 11, g 12, g 22 A 1, A 2 2 A 1 : p = f 11 + f 12 A 2 : q = f 12 + f 22 (1) (p + q =1) ḡ ḡ = f 11g 11 +2f 12g 12 + f 22g 22 (2) ḡ y 11 = g 11 ḡ y 12 = g 12 ḡ

S4 y 22 = g 22 ḡ (3) y 11, y 12, y 22 A 1, A 2 θ 1, θ 2 A 1A 1, A 1A 2, A 2A 2 2θ 1, θ 1 + θ 2,2θ 2 dominance A 1A 1 y 11 2θ 1 2 Q = f 11(y 11 2θ 1) 2 +2f 12(y 12 θ 1 θ 2) 2 + f 22(y 22 2θ 2) 2 (4) Q θ 1, θ 2 θ 1, θ 2 A 1, A 2 average effect y 11, y 12, y 22 θ 1 + θ 1, θ 1 + θ 2, θ 2 + θ 2 A 1 1 Q θ 1, θ 2 (4) θ 1, θ 2 0 1. θ 1, θ 2 g 11 = 100, g 12 = 90, g 22 =60 f 11 =0.25, f 12 =0.25, f 22 =0.25 ḡ = 85, θ 1 = 10, θ 2 = 10. f 11 =0.50, f 12 =0.25, f 22 =0.00 ḡ = 95, θ 1 =2.5, θ 2 = 7.5

Q = 4f 11(y 11 2θ 1) 4f 12(y 12 θ 1 θ 2) = 0 θ 1 Q = 4f 12(y 12 θ 1 θ 2) 4f 22(y 22 2θ 2)=0 θ 2 S5 (5a) (5b) pθ 1 + qθ 2 =0 (6) θ = θ 1 θ 2 (7) θ 1 = qθ θ 2 = pθ (8a) (8b) θ A 1 A 2 A 1 A 2 p = q a A 1A 1, A 1A 2, A 2A 2 2θ 1, θ 1 + θ 2,2θ 2 A 1 δ 11 = y 11 2θ 1 δ 12 = y 12 θ 1 θ 2 δ 22 = y 22 2θ 2 (9a) (9b) (9c) A 1A 1, A 1A 2, A 2A 2 population genetics A 1 A 2 p,q A 1A 1, A 1A 2, A 2A 2 p 2,2pq, q 2 HW HW HW

S6 V g V g = f 11y11 2 +2f 12y12 2 + f 22y22 2 (10) V a = f 11(2θ 1) 2 +2f 12(θ 1 + θ 2) 2 + f 22(2θ 2) 2 = p 2 (2θ 1) 2 +2pq(θ 1 + θ 2) 2 + q 2 (2θ 2) 2 =2pqθ 2 (6) (11) 2.2 1932 The genetical interpretation of statistics of the third degree in the study of quantitative inheritance F. R. Immer Olof Tedin d,h, d d h F 2 F 3 F 3 9 2 F 2 F 3 11 3 d h F 2 2

S7 2 Kenneth Mather DNA g e p p = g + e (12) g e 0 σ 2 a i d i e 2 P 1 P 2 F 1,F 2,F 3 2 (12) A 2 A 1 A 2 3 A 1A 1 A 1A 2 A 2A 2 g 11, g 12, g 22 (g 11 g 22)/2 A a A A 1 A 2 A 1 A 2

S8 2.. a d g 11 = 100, g 12 = 90, g 22 =60. u = 80, a = 20, d =10 2 A 1 A 1 A 2 a i A 1 A 2 a i g 12 1 (g11 + g22) A 2 d A 0 d A =0 A d A < a A d A = a A d A > a A 2 P 1 P 2A, B 2 A, B a A, a B d A, d B F 1 A 1A 1B 1B 1 u + a A + a B (13a) A 1A 2B 1B 2 u + d A + d B (13b) A 2A 2B 2B 2 u a A a B (13c) u u

S9 u k (a)= (d)= (13a,b,c) kx i=1 kx i=1 a i d i (14a) (14b) A 1A 1B 1B 1 u +(a) (15a) A 1A 2B 1B 2 u +(d) (15b) A 2A 2B 2B 2 u (a) (15c) A B (k =2)F 2 9 A 1A 1B 1B 1, A 1A 1B 1B 2, A 1A 1B 2B 2, A 1A 2B 1B 1, A 1A 2B 1B 2, A 1A 2B 2B 2, A 2A 2B 1B 1, A 2A 2B 1B 2, A 2A 2B 2B 2 1:2:1:2:4:2:1:2:1 M g[f 2]=(1/16)A 1A 1B 1B 1 +(1/8)A 1A 1B 1B 2 + +(1/16)A 2A 2B 2B 2 =(1/16)(u + a A + a B)+(1/8)(u + a A + d A)+ +(1/16)(u a A a A) = u +(1/2)d A +(1/2)d B = u +(1/2)(d) (16) (a) (d) (12) V P = V G + E (17) V G E F 2 A (1/4)A 1A 1 :(1/2)A 1A 2 :(1/4)A 2A 2 V G[F 2]=(1/4)a 2 +(1/2)d 2 +(1/4)( a) 2 ((1/4)a +(1/2)d (1/4)a) 2 =(1/2)a 2 +(1/4)d 2 (18)

S10 i a i d i 2 A,D A = D = F 2 V P [F 2] kx i=1 kx i=1 a 2 i d 2 i (19a) (19b) V P [F 1 2]= 2 A + 1 4 D + E (20) E A D F 3 P 1 BC 1 P 2 BC 2 V P [F 3]=(3/4)A +(3/16)D + E (21) V P [BC 1]+V P [BC 2]=(1/2)A +(1/2)D +2E (22) F 2 F 3 F 2 F 3 F 2 P 1,P 2 F 1 V G =0, V P = E A,D E 3 A D V P A,D,E V P = c 1A + c 2D + E (23) c 1, c 2 F 2 c 1 =1/2, c 2 =1/4 h 2 B =(c 1A + c 2D)/(c 1A + c 2D + E) (24) h 2 N = c 1A/(c 1A + c 2D + E) (25) 2002

S11 Anderson and Kempthorne 1954 1972 3. 3.1 3.2 1918 1932 294 2 1949 Biometrical Genetics 2 17 polygenic system 1

S12 2 3 Jay L. Lush Animal Breeding Plans 1937 Oscar Kempthorne Introduction to Genetic Statistics 1957Douglas Scott Falconer Introduction to Quantitative Genetics1961 Hayman (1954a,b) Griffing (1956) Finlay Wilkinson (1963) 1950 60 3.3 1930 50 1 1970 4. DNA QTL 4.1 2

S13 4.2 AABB aabb F 1 AB ab Ab ab 0 0.5 A B 2 DNA 1920 1960 4.3 DNA QTL 1980 RFLPRAPD AFLP SSR DNA DNA DNA DNA DNA DNA 1989 Lander Botstein Interval Mapping Quantitative Trait Locus QTL QTL QTL QTL DNA QTL QTL QTL QTL

S14 QTL QTL 4.4 QTL QTL 2000 2 F 2 QTL QTL 1 Q 2 A, B A, Q, B 3 AAQQBB, aaqqbb F 1 AQB/aqb / AQB aqb A Q Q B r 1, r 2 A B r 1+2 r 1 r 2 r 1 F 2 AABB, AABb, AAbb, AaBB, AaBb, Aabb, aabb, aabb, aabb 9 1 9 Q QQ, Qq, qq 1 3 F 2 i Q j p ij 1 r 1, r 2, r 1+2 Q a,d QQ,Qq,qq 1. 2 i QTL Q 1 Q 1, Q 1 Q 2, Q 2 Q 2 p ij (F 2 ) i: Q 1 Q 1 (p i1 ) Q 1 Q 2 (p i2 ) Q 2 Q 2 (p i3 ) 1: A 1 A 1 B 1 B 1 q1 2 2 q 1 q 2 q2 2 2: A 1 A 1 B 1 B 2 q 1 q 3 q 1 q 4 + q 2 q 3 q 2 q 4 3: A 1 A 1 B 2 B 2 q3 2 2 q 3 q 4 q4 2 4: A 1 A 2 B 1 B 1 q 1 q 4 q 1 q 3 + q 2 q 4 q 2 q 3 5: A 1 A 2 B 1 B 2 z 1 q 1 q 2 + z 2 q 3 q 4 z 1 (q1 2 + q2 2 )+z 2(q3 2 + q2 4 ) z 1 q 1 q 2 + z 2 q 3 q 4 6: A 1 A 2 B 2 B 2 q 2 q 3 q 1 q 3 + q 2 q 4 q 1 q 4 7: A 2 A 2 B 1 B 1 q4 2 2 q 3 q 4 q3 2 8: A 2 A 2 B 1 B 2 q 2 q 4 q 1 q 4 + q 2 q 3 q 1 q 3 9: A 2 A 2 B 2 B 2 q2 2 2 q 1 q 2 q1 2 q 1 =(1 r 1 r 2 + r 12 )/(1 r 1+2 ) q 2 = r 12 /(1 r 1+2 ) q 3 =(r 2 r 12 )/r 1+2 (r 12 =2r 1+2 r 1 r 2 ) q 4 =(r 1 r 12 )/r 1+2 z 1 =(1 r 1+2 ) 2 /{(1 r 1+ ) 2 + r 2 1+2} z 2 =1 z 1 p i1 + p i2 + p i3 =1 (i =1, 2,...,9)

S15 u + a, u + d, u a u e e 0 σ 2 QQ,Qq,qq y QQ: φ 1 = 1 e (y u a)2 2σ 2 (26a) 2π Qq : φ 2 = 1 e (y u d)2 2σ 2 2π qq: φ 3 = 1 e (y u+a)2 2σ 2 2π (26b) (26c) i Q 1 QQ y p i1ϕ 1 i y Q QQ,Qq,qq z 1,z 2,z 3 z 1 + z 2 + z 3 =1 QQ : z 1 = p i1ϕ 1/(p i1ϕ 1 + p i2ϕ 2 + p i3ϕ 3) Qq : z 2 = p i2ϕ 2/(p i1ϕ 1 + p i2ϕ 2 + p i3ϕ 3) qq : z 3 = p i3ϕ 3/(p i1ϕ 1 + p i2ϕ 2 + p i3ϕ 3) (27a) (27b) (27c) z 1,z 2,z 3 n 9Y Y i L (p i1φ ij1) z ij1 (p i2φ ij2) z ij2 (p i3φ ij3) z ij3 (28) i=1 j=1 z ij1 i j j =1,...,n iq QQ p i1φ ij1,p i2φ ij2,p i3φ ij3 i j QTL Q 1Q 1, Q 1Q 2, Q 2Q 2 (28) Q 3 n 9Y Y i L (p i1φ ij1 + p i2φ ij2 + p i3φ ij3) (29) i=1 j=1 QTL u a d σ 2 4 u QTL 1 QTL QTL QTL L û,â, ˆd, ˆσ 2 log L 1 H 0 :û = u 0,â =0, ˆd =0, ˆσ 2 = σ0 2 log L 0

S16 3. QTL LOD 68cM 17 A Q LOD 33cM LOD =log 10 L log 10 L 0 (30) EM QTL 1 cm LOD LOD QTL 3 QTL QTL QTL DNA QTL 5. DNA Anderson, V.L. and Kempthorne, O. (1954). A model for the study of quantitative inheritance. Genetics, 39: 883 898. Falconer, D.S. (1961). Introduction to Quantitative Genetics. Oliver and Boyd, Edinburgh.

S17 Finlay, K.W. and Wilkinson, G.N. (1963). The analysis of adaptation in a plant-breeding programme. Australian Journal of Agricultural Research, 14: 742 754. Fisher, R.A. (1918). The correlation between relatives on the supposition of Mendelian inheritance. Transactions of the Royal Society of Edinburgh: Earth Sciences, 52: 399 433. Fisher, R.A, Immer, F.R. and Tedin, O. (1932). The genetical interpretation of statistics of the third degree in the study of quantitative inheritance. Genetics, 17: 107 124. Griffing, B. (1956). Concept of general and specific combining ability in relation to diallel crossing systems. Australian Journal of Biological Sciences, 9: 463 493. Hayman, B.I. (1954a). The analysis of variance of diallel tables. Biometrics, 10: 235 244. Hayman, B.I. (1954b). The theory and analysis of diallel crosses. Genetics, 39: 789 809. Kempthorne, O. (1957). Introduction to Genetic Statistics. Iowa State Univ. Press. Iowa. (1960)... Lander, E.S. and Botstein, D. (1989). Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics, 121: 185 199. Lush, J.L. (1937). (1 st ), (1943) (2 nd ). Animal Breeding Plans. Iowa State Univ. Press. Ames, Iowa. (1972). 2., 22: 147 152. (2000)... (2002)...

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