F 1 F n kx. Keywords: entropy; independent assortment; random match; Hardy-Weinberg equilibrium; ternary tree algorithm

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1 HEREDITAS (Beijig) 007 8, 9(8): 07 0 ISSN w DOI: 0.60/yc F F kx,,,, : e k w  r, Âg k m k, gw e m k  e k  m ; k m, Âd F p F r m, k F o, F r, q k k ¼: Â; ; ; Hardy-Weiberg ; LawsofthechageigeotypeetropyfromF to F for pairs of idepedet gees of the selfig populatio LI Da-Li, CHEN Qi, WEI We-Hui, HUANG Yi Liuzhou Vocatioal & Techical College, Liuzhou , Chia Abstract: This paper costructs a mathematic model for the chage i each geeratio s geotype etropy of the selfig populatio with idepedet heterogees, ad describes a terary tree algorithm to compute the proportio of every geotype. It reveals a liear relatioship betwee the populatio geotype etropy ad the pairs of idepedet heterogees m, ad a oliear relatioship betwee the populatio geotype etropy ad the ordial umber of a selfig geeratio. As is fixed, the populatio geotype etropy with m pairs of idepedet heterogees is m times as may as that with oly oe pair. As the pairs of the idepedet heterogees m is fixed, the populatio geotype etropy icreases geeratio after geeratio from F to F, reachig the maximum value at F, ad decreases geeratio after geeratio from F, reachig equilibrium fially at the geeratio whose geotype etropy is miimum. I this paper, the sigificace of crossbreedig is also discussed. Keywords: etropy; idepedet assortmet; radom match; Hardy-Weiberg equilibrium; terary tree algorithm x k [], z,, xh x, x, ƒ, x Hardy-weiberg [], x k [] P 0.5, k [4], x Œ x,, k p x : 006 ; : : (968 ), u,,,, : t Tel: ; ldli668@6.com : (955 ), u,, t,, : ƒ t ¹, ƒ d, ƒ Š Tel: ; taiyag_cc@6.com

2 08 HEREDITAS (Beijig) x [5]» x k, [6],, x k x º [7] x kx, m F F xk, F k º F F x k u Hm ( ) m x k j, H m () = 0. x Aa, F x AA Aa aa x Š [8],, () () kx¹ [], F ( )x k H ( ) l l () x» x (, +), ( x x l x l x ) x l (l( ) l) x H ( x) 0, x log x < log, H log, H ( <0 u, < + log <, H ().096, H ().089 x, : t, F x k H (), F x k H ( ) x ; >, H ( ) x {, F 0, k x. x x ƒ x,,, u x x x F, 9 ƒ AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb : AA, AABB AABb AAbb; Aa, AaBB AaBb Aabb; aa, aabb aabb aabb, x (),,, x, x (),, j BB Bb bb 9 ƒ, () u AaBb z x, z x p, u 9 ƒ F x, x AA Aa aa BB Bb bb,, Š x ux, x¹ p»¹ x, x x AA Aa aa BB Bb bb x, () ½ u [9] x, m x x x x, ½ x Â,,,, ½ x : F F 5 Fig. The tred of the values of populatio geotype etropy for a pair of heterogees from F to F 5 AA ( ) ( ) ( ) AABB AABb AAbb ()

3 8 Š: F F kx 09 F p Table Schedule o zygotes ad probability about two pairs of idepedet heterogees i F uder idepedet assortmet χ BB Bb bb ( ) ( ) ( ) AA AABB AABb AAbb Aa ( ) 4 AaBB AaBb ( ) Aabb ( ) ( ) ( ) aa aabb aabb aabb, () AA F x AA, x F x 9 ƒ x ƒ AABB AABb AAbb () x, x z u(), x u x () x t,, F x 7 ƒ, x ƒ AABbCC AABbCc AABbcc x u()x ( ) x AABb ()x, t [9] : ( ) AABb ( ) ( ) ( ) AABbCC AABbCc AABbcc, m Aa, Aa, e, Am am, F x m ƒ, º x i ƒ xyx y x m y m ( xj, yj A j a j ) x p i m, F x m ƒ x ƒ x y x m y m A m A m, xy xm ym Amam, xy xm ym amam x ¹ : pxyx y x y i m m (4) p xy x y A A pxy x y A a pxy x y a a i m m m m i m m m m i m m m m, m= x (), u x, t, x m x m m x x, u m x F x m ƒ, z ukx [] ¹ Hm ( ), hp x x t ( ), m x x k x x kx m, Hm ( ) = mh ( )» : m x i xy xm ym p i, m = (4),, x y A A x y x y a a m m m m xm ym Amam i m m m m p i, x y p H ( ) x¾ p i l p i p i l p i = p i l + l I + I p i m

4 00 HEREDITAS (Beijig) = l = l p i + H ( ) i H = l,,, m, Œ, m m+ ( ) = m ( ( l p i )+ H ( ) p i + l p i + H ( ) p i ) m = m ( ) l H +H ( ), Hm ( ) = mh ( )» t Hm ( ) Hm () F, k Hm ( ) x m,, Hm ( ) x ml» : p p, Hm (), Hm ( )  u(), lim Hm ( ) = lim m ( l l )= ml» p m x, k p, F, k p, F 0, k x u½ x, ƒ x{,, 0 x,  m Aa, Aa, e, Am a m j m x, ( ), m ƒ x, x x m x Â, k ml u {, kx x,, k j, k x m x ¹ º. x x k x, k, Hardy-Weiberg [], k, u z x, x F F k u F F k, F Â, k,, k, kx ml x t k [0] j, x, F F xk x,» t xk []. F F k x ¹º x, º F xk F xk j, º, F x F, ¹ x x, F x x u, x u x x u z x kx {, z xk [], x x o xz,, º  xk x u,, ƒv, ¾¹, F x p {, ux, F x v F x ¹» x,,, ƒ, v xƒ, txƒ, v xƒ, F u F x ƒ tx, F F x xƒ, x u x, x, x x x, ƒ xx F ¹pºx F x, F, F Ä F, x ƒ x º t x F xk F xk, u x, º ¹ F x F xk F xkx º, x», x x pº, x z z x x [] x g x x

5 8 Š: F F kx 0 [] k x, x ox, jk x, z z, x x, ƒ, x. ƒ x ƒ x u ½v ƒ, z x v ƒ,, x x, g; vƒ, x F F k x, x ƒ u, v ƒ, ½ x, ƒ x, x, u x, ½ F, ux F, F xk k,, x F, xk, k, x  à ƒ x (Refereces): [] CHEN Qi, WEI We-Hui, LI Da-Li. Etropy icremet ad restrait i segregatio ad recombiatio of gees. Joural of Shaghai Jiaotog Uiversity (Agricultural Sciece), 007,5():7 75..,, xk ( ƒ m), 007, 5(): [] WANG Xiao-Log, YUAN Zhi-Fa, GUO Ma-Cai, SONG Shi-De, ZHANG Qua-Qi, BAO Zhe-Mi. Maximum etropy priciple ad populatio geetic equilibrium. Acta Geetica Siica, 00,9(6): ,,,,, k p., 00, 9(6): [] LIU Jia-Ju, GUO Ma-Cai, XIE Xiao-Li, ZHANG Hog-Li, ZHOU Jig-Yu, YUAN Zhi-Fa. Study o iformatio model of sex-liked equilibrium populatio. Joural of Northwest Sci-Tech Uiversity of Agriculture ad Forestry (Nat Sci ed), 00, 0(5): 4 6.,,,,, x. ƒ ( jƒ m), 00, 0(5): 4 6. [4] ZHANG Hog-Li, GUO Ma-Cai, XIE Xiao-Li, LIU Jia-Ju, ZHOU Jig-Yu, YUAN Zhi-Fa. The etropy character of a pair alleles populatio i ibreedig. Joural of Northwest Sci-Tech Uiversity of Agriculrure Ad For(Nat Sci Ed), 00,(): ,,,,, Œ Š xk ¾. ƒ ( jƒ m), 00, (): [5] ZHANG Hog-Li, ZHANG Hog-Ya. Study o the maximum etropy priciple ad populatio geetic equilibrium. Hereditas(Beijig), 006, 8(): 4 8., k p x ¹, 006, 8(0): 4 8. [6] GUO Ma-Cai, XIE Xiao-Li, LIU Jia-Ju, ZHANG Hog-Li, SONG Shi-De, ZHOU Jig-Yu, YUAN Zhi-Fa. The etropy character of alleles equilibrium populatio. Joural of Northwest Sci-Tech Uiversity of Agriculture ad Forestry (Nat Sci Ed),00,0(4): 9.,,,,,, Š kx ¾. ƒ ( jƒ m), 00, 0(4): 9. [7] ZHANG Hog-Li, GUO Ma-Cai, XIE Xiao-Li, DU Ju-Li, LIU Jia-Ju, ZHOU Jig-Yu, YUAN Zhi-Fa. The character of etropy of the associated probability distributio betwee a female geotype ad its descedat geotypes i ibreedig. Joural of Northwest Sci-Tech Uiversity of Agriculture ad Forestry (Nat Sci Ed), 00, (): 9 97.,,,,,, x k ¾. ƒ ( jƒ m), 00,(): [8] HUANG Yu-Qua, FAN Zheg-Zhog, CHEN Cai-A. Geetics. Beijig: Higher Educatio Press, 994, 5.,, : Š m, 994, 5. [9] GENG Su-Yu, QU Wa-Lig. Discrete Mathematics. Beijig: Higher Educatio Press, 00, 8. Œ, o : Š m, 00, 8.

6 0 HEREDITAS (Beijig) [0] WANG She-Li. A table of great achievemets i geetic research withi 40 years. Joural of Natural Sciece of Hua Normal Uiversity, 00,4(): jƒ, 00, 4(): [] CHEN Qi, WEI We-Hui, LI Da-Li. Reductio of etropy i plat populatio heredity. Joural of Sheyag Agricutural Uiversity, 006, 7(6): ,, º xk., 006, 7(6): [] CHEN Qi, WEI We-Hui. Secod law of thermodyamics ad quatitative aalysis about heredity. Joural of Guagxi Agricultural ad Biological Sciece, 006, 5(): 7 77., h px. t ƒ, 006, 5(): [] Istitute of Geetics of Fuda Uiversity. Geetics Dictioary. Beijig: Sciece Press,979,5,76. ¼ : ƒ m, 979, 5, 76. ˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇˆˇ q» 008 t 955 t x u ƒ ƒ t t x ƒ t ƒhg t gx t tpt ƒ t t t hg ƒ x º gx t x» º z ƒ º x ¹ º ƒ t z t 004 CAJCED.7 t 004 CSTPCD CSTPCD.5 9% ½ Œ BIOSIS CA BA AJ Ecological Abstracts Advaces i Ecology AGRIS Iteratioal ƒ ƒ º Š t º t p Š ƒ ƒ x ƒ tš z ½ mž 76 : ISSN CN-97/Q BM45 ¹ ¹ ¹ 0 t º 0 t apes@ibcas.ac.c