Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model D. Waxman To cite this version: D. Waxman. Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model. Journal of Theoretical Biology, Elsevier, 2009, 257 (2), pp.245. . HAL Id: hal-00554542 https://hal.archives-ouvertes.fr/hal-00554542 Submitted on 11 Jan 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Author’s Accepted Manuscript Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model D. Waxman PII: DOI: Reference: S0022-5193(08)00613-9 doi:10.1016/j.jtbi.2008.11.025 YJTBI 5378 To appear in: Journal of Theoretical Biology Received date: Revised date: Accepted date: 17 May 2008 21 November 2008 21 November 2008 www.elsevier.com/locate/yjtbi Cite this article as: D. Waxman, Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model, Journal of Theoretical Biology (2008), doi:10.1016/j.jtbi.2008.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. s u ! n a "# " $%& m !' d e t p i r c t p (# )* +, - . , * e c c A / 0 0 ! 0 "# " $%&0 !' 1 * 23 /* 455 6789 :8;;$ # . < ,, , # = = , > , ? 0 , @ , =3 = t p , # , > i r c , 0 =# # ? , > = s u , , # . ,, =30 = = n a , , , , A , ### < ,,0 =# m # ,, > , = , d e <0 ## 60 t p B 7 , < ? e c c A ' =* # C <C , <C , C B C ,, # & , < ,, # ? # =3 > # . 6. $0 # $97 , , , #= ,, A # " 0 , 1 = = , ,, ,, # t p @ , =3 = , # . i r c = = > # ,, # 6 #0 s u "D3 57 = # ,, ## n a # 6" :7 @ , =30 = 0 ,0 < m = ,,C , = , > 0 d e = ,,0 , = , <0 # t p = # 0 # 1 e c , # , . ,0 = c A ### ,,0 = 6 70 ## 0 < 1 , > ### = , , =0 , #0 # , @ , 0 , , 1 9 , , . ,0 = A , 0 ,, 0 D , > ,, 0 A ,, 6 70 = 1 E D , , A ? t p > , =3 # , i r c , # . 0 A 0 ? 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A 0 0 # 3 > ,# = , 0 = #0 ? # ,# : ,, 0 # . $$$ 1 ,, D 0 0 , & 0 ,#1 #, = ,, = ,# ,1 , )# t p > ? i r c #,0 = ,, )=0 = # , 0 = s u = > , #3 6$$0 , ;70 = , n a ? # = , 0 0 ? m > = , GH # d e . 0 = t p = 0 = - 0 , D# e c D 60 , 7 @ 0 c A # D#0 # 0 #0 A ?0 = ? # > , = #< A ? 0 # . @0 ,0 A 8 , , # &0 # 0 = ? , 0 # 6 ? # , #7 @ #< , 0 ? #,0 0 = = 0 t p # =# , 0 i r c #< , @ , =30 = ## 6 7 ? s u - ?0 = ? # # 0 ? =# # # d e m n a 67 = # = 1 t p > , # 0 = ,0 ,0 = 1 e c , A @ = = c A # . 6. $0 # $97 = , > , ? # . # , ,,0 = # 697 , A , , 0 ,1 , ,, - 0 3 , # # 0 B 0 0 t p ,,0 i r c > , ?0 0 # = s u 657 n a . 0 0 # 0 ?0 0 = m = = , d e = , > ,# t p , # # e c c A 6;7 = ,0 , ? 0 # ? > , 3 $ 6:7 = F = , 0 = 3 ? 6;7 ? # 6 0 = 1 70 = 0 , 0 # 1 0 > , t p GH0 0 , = i r c ! s u "# $ % n a @ # , , , 0 = ? 6;7 ? @ # 0 m ? = 0 3 ?1 d e ? 6;70 = # 67 > t p # ? = , ,, , ,3# e c I # c A = ? J J = G H > F # ,, D 0 = , ?0 = 687 - # = 0 # 0 0 , 1 3 > 0 = , , 0 D # ,0 , = . ,0 = 6 7 = " 0 = 0 = < 3 t p 0 = 0 i r c , > # 0 0 , 3# s u F = # # > # > 3 n a , ,0 = # < 60 < F7 ,< @ ,0 m 0 F , ,, = d e = J 0 < > # t p 6 , 0 0 0 7 - ,0 = A ,, D 0 e c = F = # c A > 0 # , > > # 0 0 ? # # = , 0 @ , 3 ,, , =0 ? 0 0 = # # = ? = # !# , > = 0 0 0 > # = 67 t p i r c 6$7 s u ,< = # , n a ? 6$7 0 = 0 > 0 0 # d e t p e c m 67 = 0 0 , c A 67 = ? = 0 0 = # 0 , ? = - = -,, "0 ? 6;70 ?0 ,, 0 = = = 0 67 > = ? 6;7 , ? = 0 t p i r c 697 s u > ? , ? # B* = 0 d e m n a 657 3# = 0 = B 0 t p = # ? ,, ? 67 #0 ,1 e c ,0 ? J 0 0 = c A ? & $ , =3 = ? 657 ? 0 # =# 9 , # , & 0 = , <#0 0 # ,0 6;7 t p ) 0 0 0 = i r c 6:7 s u 0 D = n a D , A 0 m - = = =0 , ,1 d e ,# ? 6;7 67 # ,, t p # ### # <0 , , 67 , G##H 0 0 1 e c , = A ,, c A ## > = ? 6;7 0 , 3 ? 6;70 = 5 = 687 =* 67 = # , ,, # # , # 0 0 t p # < i r c 67 = # ? ,,1 = , ### # s u @ = <0 , # , ,, 67 ### ,, n a # # ### < ,, =1 # # 67 < ,, # ,# < m ,,0 # * d e t p < e c c A < ### ### < > ? 6;7 ? 6870 , 67 = E = , E= ### ,, ; < ,,0 , ,, = < ?0 = , < ,, > 0 = 1D 0 ,# ? 0 . 0 ? 67 = t p s u ' i r c 6$7 67 n a $ ( m @ -,, = , , ? d e , 0 67 , < # , 0 67 t p ,, ## 67 3 # , < - # 6<7 e c ?0 0 c A 67 = D 6 D 7 @ ,,# 0 = 6 7 = = : 9 )=0 # A 0 = K # , ? 6;7 ? 687 @ ,0 = 0 = 0 = = " 6 ? 6770 0 0 t p # 0 i r c # 0 = # 0 s u = <* 67 > < , d e n a m 67 67 > ### ,, 6 B 7 t p ? 0 # ? = # e c c A 697 67 > < 0 # # ? # < 657 = < , ## #0 # ,, ? < # 8 ) * > , , =3 ,# = @ -,, = , # , ? t p #0 i r c > -,, , = = 6 , s u 70 # = = n a " , < ,1 0 , , # = # 1 m , , . , # d e ? ;0 = 6? 6770 = t p , 6? 670 697 6577 = D e c ? , c A # ? 6:70 = ? 67 . 0 = A ? 6;7 > D ## , , @ D 6= D < , 70 # ,3# 6#0 +7 ? 67 @ D #0 0 #= = ,# = ? , ?0 # = > = ## , 6#0 , 7 =3 = ,0 = = # #= t p > # , 3 #0 <0 = i r c 0 , = I s u &)D , = , &" (-+0 # ,3# + 3 ,, n a 6# 7 > 6 70 0 , # m # , 0 - # d e ,, D 0 = , t p 60 # 7 ? ,, + > , = 0 = ? e c 687 = 0 , ,< 0 = c A # +0 6,# = 7 ? ,, , $ + $ @ =3 = # . , # = > , # . = , 0 # 0 # > , A ? t p ,,0 6## i r c A , 7 > , > ? s u @ , =3 , , n a 0 , 0 > m # # ,3# d e " 0 =0 = 0 , ? 0 t p # , # . 0 , e c > = , c A 30 0 ## , ,,0 , ### < ,,0 ,, ##0 < ,, < > # . = # , 3 ?0 ## 6= B 70 < , <0 = ,,0 , @ ,,0 , =30 D # . 0 , #= = ,, !# = 0 ,, D 0 3 0 0 t p ,, > = A i r c , A , s u @ = 3 3 = = n a #< , ,, d e t p e c c A m @0 t p i r c s u " n a > = # = ,, D m 0 0 , A d e , ,, # t p # > # ,< # = 3 e c < 6 ,0 70 , c A ,, < @ > 0 , ,, < 0 , > # # ? 6870 0 0 > @ -,,0 = ? 67 # 1 ?0 = ,, D A < . #0 = , = 3 '3 0 0 = 3 = = = .=# #3 6$$70 = # 0 #, 6= 7 ? = > ? > , # >3# 6= 70 ? # # > ? #,0 t p i r c s u n a # m d e t p 6;7 = #, > ,# ? # 0 e c 0 # ? 6;70 , , ? c A , #0 # = 9 @ -,,0 = ? 1 A , , ,,0 ? = - 0 # ? 687 ? F !# ? 67 670 = = ? 6;7 s u i r c t p 6:7 > , = > , n a 0 ? = !# ? 67 , d e m = 0 = = ? # t p e c . ,0 = c A 5 " @ -,,0 = , 67 , < # ,0 67 ,, ## 67 3 < 3 '3 0 0 = 3 0 = = = < = t p ? i r c ;0 = ,,, , ? 6;7 ? 687 > s u = 0 = n a C = 6 m ? 677 @ =0 d e # 0 = # 3# , ,, t p = < ?0 0 = = e c , < ,, > # , c A , ,, , ? $ ; 687 >3# < ? 67 = 67 6$7 t p D 0 i r c > < , = , <#0 # ? 0 0 s u < @ = ? 687 67 n a < d e m 697 - 0 = , 0 = ? t p 6? 6$770 = # , <#0 = , # e c 6? 6770 = # > # c A ? 6;7 , # # # # > # < # 0 . : > '3 0 0 = = > ? 697 ,# # = 1 ? ? > ### ? 0 # ? # ? 6$7 = !# ? 687 6$7 = = t p i r c s u ## ? n a 697 m > ### 1< ? d e t p < e c 6997 c A ( . ? 67 0 = 3 , , , = * 67 ## # 67 < F 8 # > = , , < # , # = <0 , #0 # , , 0 , < #0 ## > = = , * ## #0 t p # < 0 < 0 = i r c ? > 0 < n a > < # s u < d e m 6957 69;7 . ? ? 6970 = e c t p < < c A 69:7 , ### 1 > ,0 < ?0 0 < !# = = , # 0 ,0 = 0 = , , > 0 # ? 6;70 = = +,# # ? " ? 69970 ? ### 1< ?0 < t p i r c s u n a d e t p e c c A $ m / @ -,,0 = ,, =30 ,# = > , ? 0 ? , #0 > ? 3=0 ,0 0 ,, D0 t p 6#3 $$C , $70 = 6 ## 7 i r c ? 6 7 s u > #, 3 n a , 0 = # , , # # 6 7 .=# #3 6$$C , $7 = < d e ! t p e c m ! ! ! ! c A ! ! ! ! ! 6987 697 69$7 = '3 6 =7 > , ,# ? # 1 9 , =30 = ? 6;70 = 6= 7 , 1 , # , , ?0 ,# ? 67 # > 0 , 0 ? 67 # ! ! ! ! t p ! ! ! i r c 657 s u . , ?0 = = = n a = F , !# ? 670 = 1 m d e 657 t p @ =0 # e c c A 657 , , , # 0 # 9 ## @ 0 = ,0 =3 A , t p i r c s u n a d e t p e c c A 9 m 0 "0 & L0 "0 ( -0 0 0 +'0 - L0 : ! ## # / ( 89 5: "D30 & -0 '0 . -0 I#0 - M0 0 0 '0 - 0 5 / , , J # 5$0 ;;1;: t p .0 ( -0 $ I / ( 50 9195 i r c .0 L 0 +0 @ +0 + $$$ L .F + 1 / )0 !,, (0 L s u &N0 ( 0 $8$ - , 1 A n a 0 - -,, / 0 919; )#0 L0 / + ,#1O# m ' 0 +0 $;; , # ? d e ,# ) , % " 0 99J;9 t p ' 0 +0 $;; ( # 1 $0 5$159; ' 0 +0 $;: ( # 1 C = e c " 0 ;81:: c A 0 ( -0 $8; " ;0 ;1:9 < ,,0 + 0 ( - &0 - L0 $8 -#0 0 < ,# , A = > /, " 90 51; #30 > $$ @ > /, &0 ,#0 " 99 #0 $9 + ,, &0 :0 $81;$ #0 0 $5; > A ? # ? / - !- 90 9J9$ t p i r c s u n a d e t p e c c A 95 m
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