Estimation of Recombination Frequencies and Constructionof RFLP Linkage Maps in Plants From Crosses Between Heterozygous Parents E. Ritter, C. Gebhardt and F. Salamini Max-Planck-Institut f u r Ziichtungsforschung, 0-5000Koln 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 leadsto 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 LENTJARIS et al. 1986;BERNATZKYand TANKSLEY “restriction fragment length polymorphisms” 1986). We have recently produced a RFLP map for the (RFLP), marker based selection is currently receiving potato (Solanum tuberosum ssp. tuberosum) (GEBHARDT attention and support in crop breeding (reviewed in BECKMANNand SOLLER1986).RFLP linkage maps et al. 1989). In the diploid state, potato clones are selfhave been constructed for several crop species includincompatible and characterized by a high genetic load. ing maize, tomato,lettuce, rice andpotato(HEBoth conditions preclude the possibility of obtaining LENTJARIS et al. 1986;BERNATZKYand TANKSLEY pure lines. In the case of our map, two highly heter1986; HELENTJARIS 1987; LANDRY et al. 1987; ZAMIR ozygous parents were crossed to obtain a segregating offspring. In the paper we describe the theoretical and TANKSLEY 1988; MCCOUCH et al. 1988; BONIERBALE, PLAISTED and TANKSLEY 1988;GEBHARDTet background for RFLP linkage analysis from any type al. 1989). Moreover, RFLP markers are virtually unof F1 populations, including those from heterozygous limited in numbers, the only restriction to the effiindividuals, which encompasses as special cases mapciencyof this techniquebeingthe DNA sequence ping in FP and backcross populations from homozydivergence between the genotypes tested. gous inbred lines. Restriction fragments of nuclear DNA varying in length between parental genotypes are detected by METHODS A N D RESULTS Southern blot hybridization to cloned homologous sequences as probes(SOUTHERN1975). Any single Calculation of recombination frequencies bepolymorphic restriction fragment segregates as a cotween loci defined by single restriction fragments: dominant Mendelian marker in the progeny from This situation has been considered as separated from parents being heterozygous for that fragment. The the case of loci defined by the existence ofallelic distance on the linkage map between any two RFLP restriction fragments (see later). In the case of a locus markers is determined by measuring the recombinadefined only by a single fragment A, care is not taken tion frequency. Linked markers areaggregated in to individuate possible fragments allelic to the same linkage groups. T h e linear order of markers within locus: the presence of A is scored versus its absence in each linkage group is deduced from the genetic disa progeny segregating for A. Allelic states to A are tances relative to each other in two-, three- or multitherefore scored as null (0= no alternative fragment). ple-point estimates. T h e number of linkage groups is Genotypes having the same phenotype (A present) equivalent to the chromosome number of thespecies. may be homozygous (AA) or heterozygous(AO), Most RFLP maps in plants have been obtained from where A behaves as a dominant marker. In aF1 cross segregating populations, F2 and/or backcrosses, deof the type A 0 X 00, the segregation ratio is 1: 1 rived from homozygous inbred lines (e.g., HE(presence vs. absence), while the F1 of a cross A 0 X Genetics 125: 645-654 (July, 1990)

646

E. Ritter, C. Gebhardt and F. Salamini TABLE 1

Derivation of recombination frequencies A. Single Fragment Loci I:rsglllent configuration of plrents

Mating table for an AB/OO type (coupling)

GF I-p

AB 00 X00

I-p

00

CF

GT

-p' -

AB

AB 1

00

AB AB

-P

2

2

00

00

2

-P 2

00

AB

2

1 -p' -

00

00

00

OB

OB

00

2

A0

pA' 0

AA 00

A0

OB

OB

2

p'

OB

2 Distribution of phenotypes

Absolute linkage Absence of linkage

P=P'

8

: A 0 :

:

0

OB : 00

:

0

:

8

4 : 4 : 4 : 4 Calculation table

Phenotypes

AB

AB

PJ

I-p

-1 -

2

2

P

-1 2

A

2 OB

SUI11

P 2

-1

1

0

-1 1-P

1

-1

1 2P

P -1 P

2

a

2(1 - P) b C

2=-

1

n

P(l - P ) Maximunl likelihood equation

- +-a- + - + b- = oc I-P P P

-d

1-P

A 0 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 configurationsin the fourloci available for two diploid parents,including cases of homobesides 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(192 1) and requires aspecific treatment for each of the parental fragment configurations. As anexample, the derivation of P for the configuration AB/OO X OO/OO (fragments A and B are both heterozygous and present on the same chromosome only in one parent) is shown in Table 1A. The mating table shows thegamete types (GT), their expected frequencies (GF)as functions of p and the phenotypes of the progeny resulting fromcrossing

+p=-

b+c n

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. Recombination frequencies between A and B can only beestimated based on the difference between the expectedphenotypicfrequenciesforabsolute linkage and for an independent segregation of the two markers. The Chi square test (MATHER 1938) will establish whetherthe observed numbers of phenotypes (2,) deviate significantly from those expected incaseof independent segregation. If A and B are supposed to be linked, the recombination frequency p can be estimated by solving the maximum likelihood Equation 1

RFLP Mapping in Allogamous Plants

_

~

_

~

~~

~~

~

~

647

-

~

~

B. Loci with Allelic Fragments I

Mating table for orB/ap type (coupiingr .. . .

Frapnent cwlfiguratton of Irrl-enls

AIBI AIBI X-

AyB2

1 - P

GF

2

AaBn

GF

A~IBI

GT

1 -p' -

2

I-p

-P

2

2

AzBz

A IBz

P AS,

A ,B/

AIBI

AlAeBIBy

AlBlBy

AlAnBl

AzBz

A I A'B, Ba

A2Bn

AlAyBn

AeBl Bn

A IB.L

AIA~BIB?

AIA'BIBB

AyBl

2

I-p 2

p'

AIAYB? AlBlBz

A IBZ

2

p_l

A~BIB?

&B/

AIA~BI

2 1)intr.ibution of pllrnolype\

AIB, : AIB? : AIBIBr : AsBl : AnBx : AzBIB2 : A ~ A Y B: ~AlAzBy : AlAZBlBy

Absolute linkage Absence of linkage

4

:

o

I

:

]

:

:

0 2

: :

0 1

: :

4 1

0 2

: :

: :

0

:

2

:

Calculation table

P=P' I'hrnotype\

PI

(I

BI

AlB?

AIBIB,

A~BIU? AIAYB~ A I AZBy

PI

>P 2

P'

P

-4

-2 -

1

Ul

1-P -2

1

a2

P

2

1 - 2p 2

- 2p P(1 - P)

p'

P

-2

2

( 1 - P)'

P p .

4

2

P(I - P )

1 - 2p

2

2

P(1 - P ) 2

1 - 2p 2

-P)

1 - 2p

2

- 2p

+ 2py

-1(1

- 2p)

2 0

1

SUlll

-

6P

P(1 - P )

4

8 4

-i PI

@J

2

P(l 1

6P

4

2 AIAsBIB~

-1 6PI

4

AnBl

AnBy

- p)'

6P, -

: :

(%T

-

~~

AI

0 2

(1

1

- 2P)' - e,

US

2P(l

P -2 1-P - 2p P(1 - P ) I - 2p 1

-P) 1 - 2p P ( l - P) -2(1 - 2 p ) 1 - 2 p + 2p' P(1

j =

1

a4

1

a5

( I - 2p)'

a6

2P(l - P ) ( 1 - 2P)'

a7

2PG - P ) (1 - 2P)'

a8

2P(I - P ) 2(1 - 2p)Y 1 - 2 p + 2p' 2(1 - 3 p + 3 p 7

a9 n

P ( l - P ) ( l - 2 p + 2P') M;rximunl IiLelillood equation

-2(al + a.5) I -P

+

( 1 - 2p)(a3

GF = WIletic frequeV'; CT = ganlete type; phenotypes. ('See tex1.

+ a6 + a7 + a 8 ) + 2(a2 + ~ 4 +) -2(1 - 2 p ) d = o 1 - 2 p + 2px -P) P

P(1 p,p'

= recombination frequency of male and female gametes; 2, = observed nun,bers of

E. Ritter, C. Gebhardt and F. Salamini

648

(FISHER192 1 )

where p j arethe expectedfrequencies and Zj the observed numbers of phenotypes. Here and in the following Equation 2, the terms needed for the solution are calculated as examplified in the calculation table (Table 1A). The information fuction Zp which measures the quality of the estimate P is given by (MATHER 1 9 8) 3

where n is the sample size (=number of offspring). T h e variance of P is then given by V ( P ) = l/zp

a.

(3)

and the standard deviation by The maximum likelihood estimator is a minimum variance unbiased estimator ofthe recombinationfrequency p (RAO 1952). In Table 1A the expected frequencies pj (first column) are obtainedby multiplying the gamete frequencies 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 that 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

I-P and solving for p P =

+ -b + - c+ - - -d P

-0

P

1-P gives the estimate b+c n

b+c a+b+c+d

=-

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 FISHERand BALMAKUND (1928) which is easy tq calculate (IMMER1930). Thus p is estimated by P solving the equation:

with p , as expected frequencies and Z j as observed numbers of phenotypes. If the variance is the same as with the maximum likelihood method then the product 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 configurations at the loci A and B of two diploid parents. In crosses these fragment configurations originate maxa imum of four phenotypesbecause the homozygous or heterozygous states for a fragment cannot be distinguished. 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 Aand B in one parent andabsence in the other; 2. The AB/AO with the configurations AB/OO X AO/OO (coupling) and AO/OB X AO/OO (repulsion) 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 (repulsion) and A B / 0 0 X AO/OB (coupling/repulsion) with both fragments shared by the parents. For each informative fragmentconfiguration 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 recombinationfrequency 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. T h e phenotypicfrequencies 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 equation ( 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 2-p

C "d +-1 +b p +-+" p 1-p

-0

(AB/AO coupling)

RFLP Mapping Plants in Allogamous

would in fact lead in this case to a polynomial of third order, while the application of the product formula gives aquadraticequation,the 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, then the estimate of p converges upon 0.5 and zero respectively. Similar conclusions can be drawn for the other cases, when the product formula is applied. DISTORTEDSEGREGATIONRATIOS In the F1 a deviation from the segregation ratio of 1 : 1 for fragments contributedonly by one parent and from 3:l for fragments present in both parents may result due to a reducedviability of some of the resulting 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 MATHER1938). 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

- P)/(u +

= u(l - p ) / ( u + I), poo = (1 = up/(.

+ 1) and

$10,=

P/(u

PAO

+ 1)

summingto 1 (BAILEY1961). When only one fragment shows distortedsegregation, u disappears in subsequent calculations and the estimate for p is the same as with segregating fragments without distortion. Nevertheless the variance must be specifically calculated because it is differentfromthe caseof absence of distortion. If both fragments aredistorted and the“skewing factor” forB is v then thephenotypic frequencies are expressed as

PAB = uv( 1 - p)/D; Po0 = (1 - P ) / D ; PAO= uP/D and POB = vP/D with D = U V (~p)

+ P(u + + 1 - p . V)

The estimate for p results in complex maximum likelihood equations, but using the product formula as suggested by BAILEY(1961), solutions can be found for theAB/OO type and theAB/AB type (see $ * and $* in Table 2A). For the AB/AO type, the estimation formula for p is always the same whether distorted segregation ratiosare observed or not,since the product 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

649

(p

= 0), they can betreated 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 codominant allelic fragments in the combinations AIAs, AIA4,A2As, AZA4if A I and A:! are the alleles of P1 and A: