Dynamic QTL analysis for rice blast resistance under natural infection conditions. Yibo Li, Changjun Wu, Yongzhong Xing, Huilan Chen and Yuqing He *

Transcription

1 Australian Journal of Crop Scinc Southrn Cross Journals (2):73-82 (2008) ISSN: Dynamic QTL analysis for ric blast rsistanc undr natural infction conditions Yibo Li, Changjun Wu, Yongzhong Xing, Huilan Chn and Yuqing H * National Ky Laboratory of Crop Gntic Improvmnt, National Cntr of Plant Gn Rsarch (Wuhan) and Collg of Lif Scinc and Tchnology, Huazhong Agricultural Univrsity, Wuhan , China *Corrsponding author mail: yqh@mail.hzau.du.cn Abstract Dynamic analyss of ric blast rsistanc for th assssmnt of gntic and nvironmntal ffcts wr charactrizd mploying a rcombinant inbrd lins (RILs) population. Th study was conductd at thr diffrnt dvlopmntal stags of ric using natural infction tsts in two yars. Th numbr of main-ffct quantitativ trait loci (QTL), pistatic QTL and thir nvironmntal intractions gratly diffrd across various masuring stags. Two major QTL (r11a and r11b) both on chromosom 11 could b dtctd at all stags, whras most QTL wr idntifid only at on or two stags in th population. It was suggstd that th unstabl activitis of most QTL idntifid for blast rsistanc may wll b du to ffcts of major QTL, pistatic ffcts btwn diffrnt loci, th dvlopmntal status of ric, and th nvironmnts in which thy wr grown. Comparison of QTL analysis conductd undr th conditions of natural infction and artificial inoculation was prformd and drw a nw conclusion that QTL analysis of plant rsistanc basd on natural infction would hav mor advantags than that basd on artificial inoculation. Kywords: Ric blast rsistanc, Dynamic analysis, Main-ffct QTL, Epistatic intractions, QTL-bynvironmnt intractions Introduction Ric blast, with its gntic instability (Rddy and Bonman, 1987), is still on of th most dstructiv disass of ric in both tropical and tmprat countris, dspit grat fforts toward its control. Many studis (Wang t al., 1994; Nagato and Yoshimura, 1998; Ahn t al., 2000; Wu t al., 2005; Li t al., 2007), indicatd that th gntic control of blast rsistanc is complx and involvs both major and minor gns with complmntary or additiv ffcts (H t al., 1989), as wll as thir nvironmnt intractions (Bonman, 1992). Nvrthlss, gn or QTL pyramiding rmains to b a promising mthod to provid broad-spctrum and durabl ric blast rsistanc (Tabin t al., 2002). Up to now, slction for rsistanc has bn prformd undr natural infction conditions. What is mor, nvironmntal conditions, such as tmpratur and moistur, gratly affct th pidmics of ric blast disass and hampr th brding programs for this trait. Howvr, almost all th gntic studis including ric blast rsistanc was basd on th artificial inoculation, and always focusd on plants at a spcific or a final growth stag xcpt our formr work (Li t al., 2007). Such studis could not fully captur th ral gn action during th growth of plant. Morovr, brdrs want to know whthr th rsults from artificial inoculation ar consistnt with thos obtaind undr natural infction conditions (Lübbrstdt t al., 1999). Thrfor, dynamic mapping and gn xprssion studis of plants at 73

2 Tabl 1. Dscriptiv statistics of th blast rsistanc traits in parnts and th RILs population undr natural infction conditions obsrvd in 2004 (uppr) and 2005 (lowr). Traits Parnt (man ± SD) RIL population Zhnshan97 Minghui 63 Man ± SD Rang rs 4.0± ± ± ± ±0 4.1± rt 4.0±0 2.8± ± ± ±0 3.8± rh 4.0±0 3.0± ± ± ± ± rs=rsistanc at sdling stag, rt=rsistanc at tillring stag, rh= rsistanc at hading stag diffrnt dvlopmntal stags ar ndd. In th prsnt study, w analyzd th QTL of main ffcts (which can show consistnt ffcts across nvironmnts), pistatic ffcts (whos ffcts dpnds on a scond QTL), and thir nvironmntal intractions using a RIL population at thr diffrnt stags. Matrials and mthods Exprimntal Population and Fild Planting In th xprimnt, w usd 241 F 1 0 RILs drivd by singl-sd dscnt mthod from a cross btwn two indica lins, Zhnshan 97 and Minghui 63 (Xing t al., 2002), th parnts of Shanyou 63, th most widly cultivatd hybrid in th last two dcads in China. Th RILs population and th corrsponding parnts wr grown simultanously during th ric-growing sasons of 2004 and 2005 in a blast hot-spot in Yuan an County, Hubi Provinc (Hubi sit), China. Yuan an is a mountainous ara, which has an altitud of 540m, with an avrag tmpratur of approximatly 25C and high humidity annually. Th micro-climat at th sit favors local ric blast disas dvlopmnt at pidmic proportions yar aftr yar. Th fild planting was th sam as our formr work (Li t al., 2007) xcpt th matrials. To adquatly induc blast disas infction, a highly suscptibl varity, CO39 (Intrnational Ric Rsarch Institution, Los Banos, Th Philippins), was plantd at both sids of ach row and around th population. Fild managmnt followd ssntially normal agricultural practics, with th xcption of using bactricid. Phnotypic Masurmnts Eight RIL plants in th middl of ach row was scord at sdling stag, tillring stag and hading stag rspctivly (about vry 35 days), which will b rfrrd to as thr traits for as of dscription, namd as rs, rt, rh, rspctivly. In ach plant w scord, th most sriously disasd laf of th top two or thr nw lavs at ach stag was dtrmind by using th 0 5 scal rating systm of Bonman t al., (1986), in which scors of 0 3 indicatd an incompatibl (rsistant) raction and scors of 4 and 5 indicatd a compatibl (suscptibl) raction. Th phnotypic valus (dgr of lsion) rprsntd th nt incras of rsistanc lvl at diffrnt stags and thus rflctd th dynamic rsponss to ric blast. DNA Markrs and Map Construction A total of 227 polymorphic markrs, covring th ntir ric gnom and including 168 RFLPs and 59 SSRs, wr usd to dvlop th gntic linkag map of th population. Of thm, 220 wr from th prvious work (Xing t al., 2002) and th othr 7 SSRs wr addd to fill th gaps in th map. Th gntic linkag map was constructd using Mapmakr 3.0 (Lincoln t al., 1992) at a LOD valu of

3 Tabl 2. Corrlation cofficints of th blast traits masurd in th RILs population, obsrvd in 2004 and All th corrlations ar significantly diffrnt from zro at th P 0.01 lvl (r 0.01 <0.181) Traits rs(2004) rs(2005) rt(2004) rt(2005) rh(2004) rs(2005) 0.85** rt(2004) 0.78** 0.39** rt(2005) 0.61** 0.59** 0.45** rh(2004) 0.64** 0.36** 0.81** 0.47** rh(2005) 0.50** 0.44** 0.44** 0.63** 0.50** rs=rsistanc at sdling stag, rt=rsistanc at tillring stag, rh= rsistanc at hading stag. ** Significanc at P<0.01 lvl Data Analysis and QTL mapping Th man of two rplications for ach lin in ach yar was usd as th raw valu for QTL analysis. To analyz gntic componnts of th traits, w mployd QTLMAPPER VER. 1.6 (Wang t al., 1999), which is basd on a mixd linar modl approach (Zhu and Wir, 1998) that stimats main-ffct and dignic pistatic QTL and simultanously prdicts QTL-bynvironmnt (QE) intraction ffcts (Li t al., 2007). Rsults and Discussion Variation, Corrlation and two-way ANOVA analyss of th Traits Tabl 1 prsnts th phnotypic variation of th dgr of lsions (a masur of blast rsistanc) at th thr dvlopmntal stags for th RIL population and its parnts in 2004 and Larg diffrncs wr found in th dgr of rsistanc btwn th RIL lins and thir parnts. Th RIL population xhibitd transgrssiv sgrgations in both dirctions for all traits, and th population showd approximatly normal distributions at all stags. Corrlations among th six traits masurd in 2004 and 2005 ar prsntd in Tabl 2. Th six traits in th population wr highly, significantly and positivly corrlatd with ach othr at P<0.01lvl in both yars (Tabl 2). In som sns, ths rsults suggstd th consistncy of componnt and quantity of isolats across diffrnt yars and also across diffrnt growth stags. Thus, phnotyp data ar comparabl and suitabl for QTL mapping. According to two-way ANOVA (Tabl 3), for all of th traits in RIL lins, as wll as th gnotyps, showd highly significant diffrncs btwn th two yars (nvironmnts), and what was mor, th nvironmnt factors had a highly significant ffct on ric blast. This nvironmntal ffct highlights th importanc of conducting th xprimnts undr natural infction conditions. Main-ffct QTL at th Thr Stags Svn to twlv main-ffct QTL wr idntifid using th RIL population at ach stag, jointly xplaining th phnotypic variation of 22.1%, 14.0% and 14.7% at sdling, tillring and hading stag, rspctivly (Tabls 4, 5 and 6). Only two QTL within th two rgions RG103-CDO534 and RM229- RM209 both on chromosom 11 (Fig. 1), wr simultanously dtctd at all th thr masuring stags in th RILs, corrsponding with th thr main-ffct QTL, rs11b/rt11b/rh11b and rs11d /rt11/rh11c, rspctivly. Thus, thy wr th sam QTL, dsignatd as r11a and r11b rspctivly. Two intrvals, C161-R753 on chromosom 1 and R1952b-RZ404 on chromosom 9, wr simultanously idntifid at both stags in th RIL population (Fig. 1), corrsponding with th two main-ffct QTL, rs1/ rt1 and rt9b/rh9a, rspctivly. Th othr QTL wr activ at only on stag. That only two QTL (r11a and r11b) wr dtctd at all stags suggstd that (1) diffrnt loci wr likly to b involvd in th gntic control of blast rsistanc at diffrnt ags, or (2) multipl gns in th sam gnomic rgion may hav diffrnt xprssion pattrns rlatd to th trait dvlopmnt. 75

4 Tabl 3. Summary of ffcts rsolvd by two-way ANOVA of th thr traits of RILs population masurd in two nvironmnts Traits Variation a SS df MS F P F crit rs G < E < G E < Error rt G < E < G E < Error rh G < E < G E < Error a G, gnotyp; E, nvironmnt; G E, gnotyp-by-nvironmnt intraction rs=rsistanc at sdling stag, rt=rsistanc at tillring stag, rh= rsistanc at hading stag Epistatic and QE Intractions at th Thr Stags Th numbr of pistatic intractions (two to 10) and th dgr of th pistatic ffcts (0.36% to 5.12%) obsrvd in th RIL lins also varid considrably among diffrnt stags, although th total ffcts of most pistatic QTL wr somwhat smallr than thos of th main-ffct QTL (Tabls 4, 5 and 6). Ths rsults indicat that various intractions btwn diffrnt loci ar likly to b involvd in th gntic control of blast rsistanc at crtain dvlopmntal stags. Most QTL for rt and rh showd intractions with th nvironmnt (Tabls 5 and 6), dmonstrating th complx inhritanc of blast rsistanc and th nvironmnt dpndnc of th gn xprssion at ths loci. In addition, th numbr (0 to 8) and th ffcts (0.01% to 4.36%) of significant QE intractions diffrd at various priods. It is intriguing to not that th QTL for rs showd no obvious intraction with nvironmnt. This phnomnon could b xplaind by two possibl rasons: (1) QTL with strong major ffct (totally 22.1% of th phnotypic variation) and strong pistatic ffcts (collctivly 18.4%) rarly intract with nvironmnt (2) th intractions with wak ffct btwn nvironmnt and QTL wr not dtctd du to th limitation of th statistical mthod. Comparativ analyss of QTL for ric blast rsistanc undr th conditions of natural infction and artificial inoculation Th rsults of QTL dtctd through natural infction tsts could b compard with thos by artificial inoculation tsts conductd by Chn t al. (2003) in our lab, who usd th sam RILs as in our study. Th numbr and ffcts of QTL dtctd btwn th two conditions wr compltly diffrnt. Six QTL intrvals, rs1 or rt1, rh1b, rt2a, rs7, rs8 and rt9a dtctd by this papr, corrsponding to th six QTL rbr1a, rbr1d, rbr2, rbr7a, rbr8 and rbr9c dtctd by artificial inoculation rspctivly, wr simultanously idntifid undr both conditions. Howvr, th QTL ffcts dtctd by natural infction wr much smallr than that by artificial inoculation (data publishd by Chn t al., 2003), possibly bcaus of lacking singl licitors of pathotyps in th natural nvironmnt. Morovr, thr wr mor 76

5 Tabl 4. Main ffcts, pistatic ffcts and nvironmntal intractions of QTL dtctd by two-locus analysis using QTLMappr1.6 for ric blast rsistanc at th sdling stag using RIL population with th LOD thrshold 4.03 (qual to a chi-squar valu for df=6 at P=0.005). Gnral contributions. additiv (A), h 2 a=22.06%; pistasis, h 2 aa=18.43%; QTL-by- nvironmnt intractions, h 2 a=0%; h 2 aa=0 Ch-Ini a b Flanking markrs QTL Ch- Flanking markrs QTL LOD a i Inj a 1-1 C161-R753 rs CDO127-R C567-C clon4-g RZ599-R R1245-RM G200-RZ C148-RM RG528-RG128 rs7 7-5 RG678-RZ RM25-RG333 rs C1633-C R1506-MP12 rs11a 11-8 clon1-c405b rs11c h 2 a i a j b h 2 a j aa ij c h 2 aa ij h 2 total f RG103-CDO534 rs11b RM229-RM209 rs11d a Ch-Ini and Ch-Inj rprsnt th chromosom numbr-intrval of th points bing tstd in th analysis. i and j man two diffrnt intrvals on chromosom (s). b a i and a j ar th additiv ffcts of th tst points i and j, rspctivly. Positiv valus of a i and a j imply that th Minghui 63 gnotyp has a positiv ffct on that trait c aa ij is th ffct of additiv-by-additiv intraction btwn points i and j; a positiv valu indicats that th parntal two-locus gnotyps hav a positiv ffct on th traits and that th rcombinants had a ngativ ffct h 2 a i, h 2 a j, h 2 aa ij, h 2 a i and h 2 a j ar th prcntags of th phnotypic variations xplaind by a i, aj, aa ij, a i, and a j, rspctivly f h 2 total is th phnotypic variation xplaind by th gntic componnts includd in th modl 77

6 Tabl 5. Main ffcts, pistatic ffcts and nvironmntal intractions of QTL dtctd by two-locus analysis using QTLMappr1.6 for ric blast rsistanc at tillring stag using RIL population with th LOD thrshold 4.03 (qual to a chi-squar valu for df=6 at P=0.005). Gnral contributions. Additiv (A). h 2 a=14.03%; Epistasis. h 2 aa=9.86%; QE intractions. h 2 a=5.08%; h 2 aa=0 Ch- Ini a Flanking markrs QTL Ch- Inj a Flanking markrs QTL LOD b a i h 2 a i b a j h 2 a j c aa ij h 2 aa ij d a i h 2 a i d a j h 2 a j h 2 total f 1-1 C161-R753 rt1 9-8 R1952b-RZ404 rt9b RG532-RM R2549-C C2340-C RM25-RG RM213-RM208 rt2a 2-5 RM48-RG520 rt2b C944-R321 rt RM227-R R1925-RM C820-C933 rt R1925-RM R2174-C909A R78-C R3203-RM20a rt11c C1447-RM RZ588-P RM25-RG C677-RM RG333-R RM20a-C104 rt11d C483-C C87-R RM215-R1952b rt9a 10-2 RM222-R R543a-Y6855R rt11a 11-6 MP12-RM RG103-CDO534 rt11b RM229-RM209 rt a-f S footnots of Tabl 4 for xplanations d a i and a j ar ffcts of th nvironmntal intraction of locus i and j, rspctivly; a positiv valu implis that th ffct in 2005 is largr than in

7 Tabl 6. Main ffcts, pistatic ffcts and nvironmntal intractions of QTL dtctd by two-locus analysis using QTLMappr1.6 for ric blast rsistanc at hading stag using RIL population with th LOD thrshold 4.03 (qual to a chi-squar valu for df=6 at P=0.005). Gnral contributions. Additiv (A). h 2 a=14.71%; Epistasis. h 2 aa=15.54%; QE intractions. h 2 a=3.02%; h 2 aa=0 Ch- Flanking markrs QTL Ch-Inj a Flanking markrs QTL LOD b a i Ini a 1-13 RM5a-RM237 rh1a 6-26 C751A-RG RM237-C RM242-RG570 rh9b RM212-C567 rh1b 5-7 C624-C C567-C2340 rh1c 5-8 C246-RM C2340-C86 rh1d CDO127-R C777-RZ RM213-RM RM213-RM R78-C G102-RM255 rh4 4-8 R78-C R830-R R496-C909B RZ66-G clon4-g4001 rh11b R1952b-RZ404 rh9a 10-2 RM222-R RG103-CDO534 rh11a RM229-RM209 rh11c a-f S footnots of Tabl 4 for xplanations d a i and a j ar ffcts of th nvironmntal intraction of locus i and j, rspctivly; a positiv valu implis that th ffct in 2005 is largr than in 2004 h 2 a i a j b h 2 a j aa ij c h 2 aa ij a i d h 2 a i h 2 total f 79

8 C161 R753 G359 RG532 RM259 RM243 RG173 RM81A G1128b C904 R2632 C39 RM5a RM237 C922 RG101 G393 R2201 RM212 C567 C2340 C86 RG236 C112 R2510 RM211 RG634 R1738 RM53 RZ599 R712 RZ324 RM29 R1843 C777 RZ386 G1314a RM240 RM213 RM208 RM207 RM48 RG520 C1176 C316 C63 RM232 G144 RG393 C1087 RZ403 R19 C746 QTL for Sdling stag C944 R321 RM55 QTL for Tillring stag QTL for Hading stag 1 cm RM200 RM227 R1925 RM148 8 RZ467 C2807 C513 RM241 G102 RM255 G235 R78 C1016 C107 RG620 C56 C820 C933 C751B RM25 RG333 R902 C1121 R1629 C483 C347 RG978 R1394 G2132 R727 L363A RM223 RZ66 G RG528 RG128 C1023 R1440 RG678 RZ471 RM70 R1245 RM234 R1789 C472 R2638 RM257 RM242 RG570 RG667 RM215 R1952b RZ404 TEL RZ536 R543a Y6855R R1506 MP12 RM224 clon1 C405b Y6854L L1044 clon3 G181 clon2 Y2668L G389 RM254 clon4 G4001 C1003B RG103 CDO534 RM21 RG2 RM229 RM209 G257 G44 C1237 RG118 C794 R2918 CDO127 R3203 RM20a C104 R2272 RM201 Fig1. Distribution of main-ffct QTL at thr dvlopmntal stag of ric on th RIL map as dtctd by QTL Mappr 80

9 QTL (totally 29) dtctd undr natural infction condition in our study than that (totally 12) undr artificial inoculation condition. It may b th rason that thr ar mor unknown and complx pathotyps of P. grisa with thir gntic instability in th natural blast nursry simultanously involvd in th inducing activitis of plant rsistanc QTL xprssion than that of artificial inoculation (Chn t al., 2001; Bilgic t al., 2006). Thrfor, QTL dtctd undr natural infction condition hav broad spctrum and durabl rsistanc, which has profound implications for practical ric rsistanc brding. Morovr, QTL analysis by a natural infction procss could fully rflct th ral gn actions in a mor natural way bcaus of lack of artificial factors. Thus, in a sns, QTL analysis of plant rsistanc basd on natural infction has mor advantags ovr that basd on artificial inoculation. In conclusion, th most important outcom of th prsnt study is th dynamic charactrization of th main-ffct and pistatic QTL as wll as thir nvironmntal intractions for ric blast rsistanc undr natural fild conditions. Our prsnt rsults ar clarly in agrmnt with our formr rsults (Li t al., 2007) and th othr study (Atchly and Zhu, 1997). Compard with thos studis involvd only th main ffcts of individual QTL at a singl stag, this findings and analytical mthod might hlp to unravl mor important information for molcular mchanism of plant rsistanc and for crating disas-rsistant plants and cloning of QTL. Acknowldgmnts This work was supportd by grants from th National High Tchnology Rsarch and Dvlopmnt Program of China (863 program) and th National Natural Scinc Foundation of China. Rfrncs Ahn SN, Kim YK, Hong HC, Han SS, Kwon SJ, Choi HC, Moon HP, McCouch SR (2000) Molcular mapping of a nw gn for rsistanc to ric blast (Pyricularia grisa Sacc.). Euphytica 116: Atchly WR, Zhu J (1997) Dvlopmntal quantitativ gntics, conditional pigntic variability and growth in mic. Gntics 147: Bilgic H, Stffnson BJ, Hays PM (2006) Molcular Mapping of Loci Confrring Rsistanc to Diffrnt Pathotyps of th Spot Blotch Pathogn in Barly. Phytopathology 96: Bonman JM (1992) Durabl rsistanc to ric blast disas-nvironmntal influncs. Euphytica 63: Bonman JM, Vrgl D, Khin MM (1986) Physiologic spcialization of Pyricularia oryza in th Philippins. Plant Dis. 70: Chn DH, Vina M, Inukai T, Mackill DJ, Ronald PC, Nlson RJ (1999) Molcular mapping of th blast rsistanc gn, Pi44(t), in a lin drivd from a durably rsistant ric cultivar. Thor. Appl. Gnt. 98: Chn HL, Chn BT, Zhang DP, Xi YF, Zhang QF (2001) Pathotyps of Pyricularia grisa in ric filds of cntral and southrn China. Plant Dis. 85: Chn HL, Wang SP, Xiong LZ, Xu CG, Hays PM, Zhang QF (2003) Comparativ analyss of gnomic locations and rac spcificitis of loci for quantitativ rsistanc to Pyricularia grisa in ric and barly. Proc. Natl. Acad. Sci. USA.100: H YY, Li J, Li C, Kiyosawa S, Higashi T, Horisu N (1989) Gn analysis of blast rsistanc of Yunnan upland ric varity, Haonaihuan. Oryza 26: Iwata N (1996) Rport of th committ on gn symbolization and linkag map. Ric Gnt. Nwsl. 13: Landr ES, Botstin D (1989) Mapping Mndlian Factors Undrlying Quantitativ Traits Using RFLP Linkag Maps. Gntics 121: Lincoln S, Daly M, Landr E (1992) Constructing gntics maps with MAPMAKER/EXP 3.0. Whithad Institut Tchnical Rport, Whithad Institut, Cambridg, MA, USA. Li YB, Wu CJ, Jiang GH, Wang LQ and H YQ (2007) Dynamic analyss of ric blast rsistanc for th assssmnt of gntic and nvironmntal ffcts. Plant Brding 126: Lübbrstdt T, Xia XC, Tan G., Liu X, Mlchingr AE (1999) QTL mapping of rsistanc to Sporisorium riliana in maiz. Thor. Appl. Gnt. 99: Nagato Y, Yoshimura A (1998) Rport of th committ on gn symbolization, nomnclatur and linkag group. Ric Gnt. Nwsl. 15: Rddy APK, Bonman JM (1987) Rcnt pidmics 81

10 of ric blast in India and Egypt. Plant Dis. 71: 850. Tabin RE, Li Z, Patrson AH, Marchtti MA, Stansl JW, Pinson SRM (2002) Mapping QTLs for fild rsistanc to th ric blast pathogn and valuating thir individual and combind utility in improvd varitis. Thor. Appl. Gnt. 105: Wang DL, Zhu J, Li ZK, Patrson AH (1999) Mapping QTLs with pistatic ffcts and QTL nvironmnt intractions by mixd linar modl approachs. Thor. Appl. Gnt. 99: Wang GL, Mackill DJ, Bonman JM, McCouch SR, Champoux MC, Nlson RJ (1994) RFLP mapping of gns confrring complt and partial rsistanc to blast in a durably rsistant ric cultivar. Gntics 136: Wu JL, Fan, YY, Li DB, Zhng KL, Lung H, Zhuang JY (2005) Gntic control of ric blast rsistanc in th durably rsistant cultivar Gumi 2 against multipl isolats. Thor. Appl. Gnt. 111: Xing YZ, Tan YF, Hua JP, Sun XL, Xu CG, Zhang QF (2002) Charactrization of th main ffcts, pistatic ffcts and thir nvironmntal intractions of QTLs on th gntic basis of yild traits in ric. Thor. Appl. Gnt. 105: Yan JQ, Zhu J, H CX, Bnmoussa M, Wu P (1998) Molcular dissction of dvlopmntal bhavior of plant hight in ric (Oryza sativa L.). Gntics 150: Zng ZB (1993) Thortical basis for sparation of multipl linkd gn ffcts in mapping quantitativ trait loci. Proc. Natl. Acad. Sci. USA. 90: Zhu J, Wir BS (1998) Mixd modl approachs for gntic analysis of quantitativ traits. In.L.S. Chn, S.G. Ruan, and J. Zhu (ds.). Advancd Topics in Biomathmatics. Procding of th Intrnational Confrnc on Mathmatical Biology, World Scintific, Singapor. 82