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Estimation of Recombination Frequencies and Construction of RFLP Linkage Maps in Plants From Crosses Between Heterozygous Parents

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Estimation of Recombination Frequencies and Construction of RFLP Linkage Maps in Plants From Crosses Between Heterozygous Parents
  Estimation of Recombination Frequencies and Construction f RFLP Linkage Maps in Plants From Crosses Between Heterozygous Parents E. Ritter, C. Gebhardt and F. Salamini Max-Planck-Institut fur Ziichtungsforschung, 0-5000 oln 30 West Germany Manuscript received January 5, 1990 Accepted for publication March 26, 1990 ABSTRACT The construction of a restriction fragment length polymorphism (RFLP) linkage map is based on the estimation of recombination frequencies between genetic loci and on the determination of the linear order of loci in linkage groups. RFLP loci can be identified as segregations of singular or allelic DNA-restriction fragments. From crosses between heterozygous individuals several allele (fragment) configurations are possible, and this leads o a set of formulas for the evaluation ofp, the recombination frequency between two loci. Tables and figures are presented illustrating a general outline of gene mapping using heterozygous populations. The method encompasses as special cases the mapping of loci from segregating populations of pure lines. Formulas for deriving the recombination frequencies and information functions are given for different fragment configurations. Information functions derived for relevant configurations are also compared. A procedure for map construction is presented, as it has been applied to RFLP mapping in an allogamous crop. ITH the discovery of a new marker class termed “restriction fragment ength polymorphisms” (RFLP), marker based selection is currently receiving attention and support in crop breeding (reviewed in BECKMANN and OLLER 986). RFLP linkage maps have been constructed for several crop species includ- ing maize, tomato, ettuce, rice and potato HE- LENTJARIS et al. 1986; BERNATZKY and TANKSLEY 1986; HELENTJARIS 987; LANDRY t al. 1987; ZAMIR and TANKSLEY 988; MCCOUCH t al. 1988; BONIER- BALE, PLAISTED and TANKSLEY 988; GEBHARDT t al. 1989). Moreover, RFLP markers are virtually un- limited in numbers, the only restriction to the effi- ciency of this technique being he DNA sequence divergence between the genotypes tested. Restriction fragments of nuclear DNA varying in length between parental genotypes are detected by Southern blot hybridization to cloned homologous sequences as probes SOUTHERN 1975). Any single polymorphic restriction fragment segregates as a co- dominant Mendelian marker in the progeny from parents being heterozygous for hat fragment. The distance on the linkage map between any two RFLP markers is determined by measuring the recombina- tion frequency. Linked markers are aggregated in linkage groups. The linear order of markers within each linkage group is deduced from the genetic dis- tances relative to each other in two-, three- or multi- ple-point estimates. The number of linkage groups is equivalent to the chromosome number of the species. Most RFLP maps in plants have been obtained from segregating populations, F2 and/or backcrosses, de- rived from homozygous inbred lines (e.g., HE- Genetics 125: 645-654 (July, 1990) LENTJARIS et al. 1986; BERNATZKY and TANKSLEY 1986). We have recently produced a RFLP map for the potato (Solanum tuberosum ssp. tuberosum) (GEBHARDT et al. 1989). In the diploid state, potato clones are self- incompatible and characterized by a high genetic load. Both conditions preclude the possibility of obtaining pure lines. In the case of our map, two highly heter- ozygous parents were crossed to obtain a segregating offspring. In he paper we describe the theoretical background for RFLP linkage analysis from any type of F1 populations, including those from heterozygous individuals, which encompasses as special cases map- ping in FPand backcross populations from homozy- gous inbred lines. METHODS AND RESULTS Calculation of recombination requencies e- tween loci defined by single restriction fragments: This situation has been considered as separated from the case of loci defined by the existence of allelic restriction fragments (see later). In the case of a locus defined only by a single fragment A, care is not taken to individuate possible fragments allelic to the same locus: the presence of A is scored versus its absence in a progeny segregating for A. Allelic states to A are therefore scored as ull (0 = no alternative fragment). Genotypes having the same phenotype (A present) may be homozygous (AA) or heterozygous AO), where A behaves as a dominant marker. In a F1 cross of the type A0 X 00, the segregation ratio is 1: 1 (presence vs. absence), while the F1 of a cross A0 X  646 E. Ritter, C. Gebhardt and F. Salamini TABLE 1 Derivation of recombination frequencies A. Single Fragment Loci I:rsglllent configuration Mating table for an of plrents AB/OO type (coupling) B 00 00 00 X- GF I p p - P CF GT 00 0 1 -p AB AB B B B 2 - 2 2 2 - 2 1 -p p - 2 2 00 00 00 A A0 0 0 0 p OB OB OB OB B 2 Distribution of phenotypes AB : A0 : OB : 00 Absolute linkage 8:0:0:8 Absence of linkage 4:4:4:4 P=P Calculation table Phenotypes PJ AB I p -1 -1 - 2 2 1-P A <> OB P 2 - 2 - 1 2 2 - - P - P 1 a 2(1 - P) 2P b C SUI11 1 0 1 n 2=- P(l - P) Maximunl likelihood equation -a b c -d b+c -+-+-+-=o +p=- I-P P P 1-P n A0 will segregate 3: 1. Segregation at a second RFLP locus B can be defined accordingly. Presence and absence of fragments A and B can be arranged in 2H configurations n the four oci available for two diploid parents, ncluding cases of homo- besides those of heterozygosity. The best estimate P for the recombination frequency p between A and B can be obtained by use of the maximum likelihood method of FISHER (1 92 1 ) and requires a pecific treat- ment for each of the parental fragment configura- tions. As an example, the derivation of P for he configuration AB/OO X OO/OO (fragments A and B are both heterozygous and present on the same chro- mosome only in one parent) is shown in Table 1A. The mating table shows the gamete types (GT), their expected frequencies (GF) as functions of p and the phenotypes of the progeny resulting from rossing the parent AB/OO with OO/OO. In case of absolute linkage between A and B (p = 0) the two parental phenotypes are expected in the F1 progeny with a frequency of 50% each, whereas in the absence of linkage (p = 0.5) four phenotypes (two parental, two recombinant) with 25% frequency each appear. Re- combination frequencies between A and B can only be estimated based on the difference between the expected phenotypic requencies or absolute link- age and for an independent segregation of the two markers. The Chi square test (MATHER 1938) will establish whether he observed numbers of phenotypes (2,) deviate significantly from those expected in case of independent segregation. If A and B are supposed to be linked, the recombination frequency p can be esti- mated by solving the maximum likelihood Equation 1  RFLP Mapping in Allogamous Plants 647 ~ __~~ ~ ~ ~ ~ ~ B. Loci with Allelic Fragments Frapnent cwlfiguratton of Irrl-enls orB/ap type (coupiingr Mating table for .. .. IBI AIBI AyB2 AaBn - GF -P 2 I p - 2 P 2 GF GT A~IBI AzBz A IBz AS, 1 -p A ,B/ AIBI AlAeBIBy AlBlBy AlAnBl I p AzBz A A'B, Ba A2Bn AlAyBn AeBl Bn 2 p' A IBZ 2 AlBlBz IAYB? AI .L AIA~BIB? p l &B/ AIA~BI ~BIB? AIA'BIBB AyBl 2 1)intr.ibution of pllrnolype\ AIB, : AIB? : AIBIBr : AsBl : AnBx : AzBIB2 : A~AYB~ AlAzBy : AlAZBlBy Absolute linkage 4:o: 0 :0:4: 0: 0: 0: 8 Absence of linkage I:]: 2 :1:1: 2: 2: : 4 P=P Calculation table ~~ I'hrnotype\ PI - P, 6P -- 6PI PI @J i (%T PI 6P -4 AI BI AlB? AIBIB, AnBl AnBy A~BIU? I - p) - 4 4 P(1 - P) 2 p 1 - P) 4 4 P(I - ) 2 P 2 - 2 1 - 2p 2 P 2 Pp. 1 - 2p 2 2 -2 - 1-P 2 P 1 - 2p P(1 - P) 2 P -2 - - - 1-P 1 - 2p P(1 - P) 1 1 1 - 2P)' 2P(l - e, 1 1 I - 2p)' 2P(l - P) Ul a2 US a4 a5 a6 AIAYB~ P(1 - P) 1 - 2p I - 2p 1 - 2P)' a7 P(l - P) 1 - 2p 1 - 2p (1 - 2P)' a8 2 2 P(1 - ) 2PG - ) 2 2 P(l - P) 2P(I - P) 2 1 - 2p + 2p' 1 - 2p + 2p' A AZBy AIAsBIB~ 1 - 2p 2py -1 1 - 2p) -2(1 - 2p) 2 1 - 2p)Y a9 SUlll 1 0 2(1 - 3p + 3p7 n j= P(l - )(l - 2p + 2P') M;rximunl IiLelillood equation -2(al + a.5) + 1 - 2p) a3 + a6 + a7 + a8) 2(a2 ~4 2(1 - 2p)d I -P P(1 - P) + P 1 - 2p + 2px + =o GF = WIletic frequeV'; CT = ganlete type; p,p = recombination frequency of male and female gametes; 2, = observed nun,bers of ( See ex1. phenotypes.  648 E. Ritter, C. Gebhardt and F. Salamini (FISHER 92 1) where pj are he expected requencies and Zj the observed numbers of phenotypes. Here and in the following Equation 2, the terms needed for the solu- tion are calculated as examplified in the calculation table (Table 1A). The information fuction Zp which measures the quality of he estimate P is given by ( MATHER 1 9 3 ) where n is the sample size (=number of offspring). The variance of P is then given by V(P) = l/zp (3) and the standard deviation by a he maximum likelihood estimator is a minimum variance unbiased estimator of he recombination requency p (RAO 1952). In Table 1A the expected frequencies pj (first col- umn) are obtained y multiplying the gamete frequen- cies giving rise to a specific phenotype and summing up the products over the mating table. For example pAB = 2 with the male frequency of recombinant gametes (p) equal to hat of the female (p'). The other terms (columns 2, 3, and 4) are derivatives of PI. Using the calculation table, Equation 1 is formulated as -a b c -d +-+-+-- -0 I-P P P 1-P and solving for p gives the estimate b+c b+c P= =- a+b+c+d n with C(P) = P(l - P) n from Equations 2 and 3. If in a cross only four phenotypes are present, as it is with single fragment loci, it may be convenient (see below) to estimate p with the product formula of FISHER nd BALMAKUND (1928) hich is easy tq calculate (IMMER 1930). Thus p is estimated by P solving the equation: with p, as expected frequencies and Zj as observed numbers of phenotypes. If the variance is the same as with the maximum likelihood method then the prod- uct formula gives a fully efficient estimate ofp (BAILEY 196 1). Similar as shown in Table lA, mating tables can be assembled for all the 2' possible fragment configura- tions at the loci A and B of two diploid parents. In crosses these fragment configurations srcinate a ax- imum of four phenotypes because the homozygous or heterozygous states for a fragment cannot be distin- guished. However, out of the 256 configurations, only a few have expected phenotypic frequencies differing between absolutely linked and unlinked fragments A and B and these are therefore useful for linkage analysis. They are combined in three types: 1. The AB/OO-type with the configurations AB/ 00 X OO/OO (coupling, see Table 1 A) and AO/OB X OO/OO (repulsion), characterized by the presence of both fragments A and B in one parent and absence in the other; 2. The AB/AO with the configurations AB/OO X AO/OO (coupling) and AO/OB X AO/OO (repul- sion) in which one fragment is present in both parents and the other only in one; 3. The AB/AB type with the configurations AB/ 00 X AB/OO (coupling), AO/OB X AO/OB (repul- sion) and AB/00 X AO/OB (coupling/repulsion) with both fragments shared by the parents. For each informative fragment configuration as defined above, a calculation table can be developed by expressing the expected phenotypic frequency PI as a function of p, obtained from the mating table as exemplified in Equation 4, and by calculating the partial derivatives and the other terms necessary for solving the maximum likelihood Equation 1. In doing this, three assumptions are made: 1. The recombination requency during gamete formation is the same in both parents (p = p ); 2. Reciprocal crosses result in the same phenotypic frequencies (P1 X P2 = P2 X Pl); 3. The phenotypic frequencies are identical and independent of which homologous chromosomes are paired (e.g., AB/OO = OO/AB). Table 2A summarizes the formulas necessary to CalcuJate the recombination frequency estimators P and P for the seven usable fragment configurations of two single fragment loci. The formulas for the AB/ 00 and AB/AB type were derived by solving equa- tion (l), while for the AB/AO type equation (5) was used due to its lower computational complexity. The solution of the maximum likelihood equation: -a b C d +- +-+ -0 2-p 1+p p 1-p (AB/AO coupling)  RFLP Mapping in Allogamous lants 649 would in fact lead in this case to a polynomial of third order, while the application of the product formula gives a quadratic equation, he variance being the same in both cases. Analogous results can be obtained for repulsion. With the product formula the value of X, as defined in Table 2A, is always larger or equal to one. The formula is not defined for X = 1 (ad = bc) or the denominator being equal to zero [bc = 0 (COW pling), ad = 0 (repulsion)]. If, however, X approaches one or the denominator approaches zero, hen he estimate of p converges upon 0.5 and zero respec- tively. Similar conclusions can be drawn for the other cases, when the product formula is applied. DISTORTED SEGREGATION RATIOS In the F1 a deviation from the segregation ratio of 1 : 1 for fragments contributed only by one parent and from 3:l for fragments present in both parents may result due to a reduced viability of some of the result- ing phenotypes (reduced viability of certain gametes is not considered here). Significant deviations from the normal ratios are detected with the Chi square test (summarized in MATHER 1938). If the “skewing factor” u (ratio of the phenotypes with and without a fragment A) is considered, the phenotypic frequencies in the mating table of Table 1A can be expressed as PAB = u(l - p)/(u + I), poo = (1 - P)/(u + PAO = up/ . + 1) and 10, = P/(u + 1) summing o 1 (BAILEY 961). When only one frag- ment shows distorted egregation, u disappears in subsequent calculations and the estimate for p is the same as with segregating fragments without distor- tion. Nevertheless the variance must be specifically calculated because it is different rom he case of absence of distortion. If both fragments are distorted and the skewing factor” for B is v then the phenotypic frequencies are expressed as PAB = uv 1 - p)/D; Po0 = (1 - P)/D; PAO = uP/D and POB = vP/D with D = UV(~ p) + P(u + V 1 - . The estimate for p results in complex maximum like- lihood equations, but using the product formula as suggested by BAILEY 1961), solutions can be found for the AB/OO type and the AB/AB type (see * and $* in Table 2A). For the AB/AO type, the estimation formula for p is always the same whether distorted segregation ratios are observed or not, ince the prod- uct formula is used in all cases. CALCULATION OF RECOMBINATION FREQUENCIES BETWEEN LOCI DEFINED BY ALLELIC RESTRICTION FRAGMENTS If two fragments AI and A: are detected with the same probe and if they are linked 100% in repulsion (p = 0), they can be reated as allelic fragments (although they do not have to be so in the molecular sense). In a progeny of heterozygous parents, a locus may therefore be represented by up to four codomi- nant allelic fragments in the combinations AIAs, AIA4, A2As, AZA4 f AI and A: are the alleles of P1 and A:< and A4 of P2. If only two alleles are present in both parents (both parents have, for instance, AIA: ) and knowing these alleles based on their electropho- retic pattern, the homozygous or heterozygous state of a locus can be deduced. Recombination frequencies between two such loci are derived by the procedure described for single fragment loci. As an example, he fragment configuration AIBI/A: B: X AIBI/A: B: , with AI, A: and BI, B: being allelic fragments of two loci A and B, is shown in Table 1B. As seen in the mating table, nine phenotypes can be distinguished and their frequencies vary according to the linkage intensity between the loci A and B. The terms neces- sary to formulate Equations 1 and 2 are given in the calculation table. The maximum likelihood equation is a polynomial of higher order, that can be solved iteratively using, for example, Newton’s approxima- tion method. Table 2B summarizes the formulas and maximum likelihood equations (where a universal so- lution is not possible) for the estimation of p between two loci with allelic fragments (Nos. 1-3) or for mixed situations where linkages between a single fragment locus and a locus with allelic fragments are onsidered (Nos. 4-6). Allelic configurations at loci with allelic fragments are here indicated ntroducing he addi- tional letters a and (a = A,/A: , a’ = AI/As, p = Bl/ B: , /3’ = BI/B2), and their configurations in a cross are defined in terms of allelic states in Table 3. The configurations aP/OO, aB/OO (Nos. la and 4b) and aP/aO, aB/aO, aB/BO (Nos. lb, 4a and 5 have similar solutions for P as AB/OO ad AB/AO respec- tively (Table 2A). For the configurations ap/aB, a/3/ a’@‘ and aB/aB, respectively (Nos. 2, 3 and 6) the three cases of coupling, repulsion, and coupling/re- pulsion have been considered. In the three allelic configuration a/3/a’P’ (No. 3) the sixteen genotypes can be distinguished, allowing a very precise estimate of p (see also Figure 1). A configuration where our different ragments are found at a ocus is treated as the ap/a’@’ onfiguration by attaching corresponding genotypes. In a similar way as described in Table 1 further mating and calculation tables could be set up consid- ering three and more loci, where several parameters have to be estimated. INFORMATIVITY OF P DEPENDS ON THE FRAGMENT CONFIGURATION The information function Zp or its reciprocal, the variance V(P) Equations 2 and 3), is a measure of the precision of the estimated recombination frequency P (MATHER 938). Table 4 lists the information func-
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