Relationship Between Phenotypic and Marker Distances: Theoretical and Experimental Investigations

Heredity 79(1997) 477—483 Received 26 October 1996

Relationship between phenotypic and marker distances: theoretical and experimental investigations

JUDITH BURSTIN & ALAIN CHARCOSSET* INRA-UPS-INAPG, Station de Génétique Végétale, Ferme du Moulon, 91190 Gif/Yvette, France

Numerous studies have aimed at assessing the relationships between (i) distances computed from phenotypic data, (ii) distances computed from marker data and (iii) heterosis, for pairs of individuals or populations. The conflicting results obtained illustrate that these relationships are far from simple. In this paper, we investigate the effect on these relationships of (i) the polygenic inheritance of phenotypic traits and (ii) the structure of linkage disequilibrium between genetic markers and the loci involved in the variation of quantitative traits (QTL5). Both theoretical and experimental results showed that the relationship between marker distances and phenotypic distances computed from quantitative traits displays a triangular shape: low marker distances are systematically associated with low phenotypic distances, whereas high marker distances correspond to either low or high phenotypic distances. Because of this property, the linear coefficient of correlation between both distances decreases as the number of QTLs involved in the variation of the traits considered for phenotypic distance computation increases. Similar properties are expected for the relationship between heterosis and phenotypic distances.

Keywords: genetic distances, heterozygosity, markers, polygenic inheritance.

Introduction Based on these reasons, the relationship between diversity at marker loci and morphological differen- Geneticdifferentiation between individuals or popu- tiation has been investigated in several studies. lations can be evaluated at different levels: quantita- Some significant correlations between the two tively inherited phenotypic traits, monogenic traits distances were reported (Atchley et al., 1988), but in submitted to selection pressure (e.g. disease resist- most cases no significant correlation was found ance traits), neutral molecular markers, etc. Infor- mation about the relationships that exist between (Wayne & O'Brien, 1986; Moser & Lee, 1994; Schmitt et al., 1995). Similarly, very few experi- these different levels is significant for several mental studies have demonstrated a relationship reasons. From an evolutionary standpoint, the between heterosis (which can be considered as a investigation of the relationship between genetic particular distance; see Falconer, 1981) and distance diversity and morphological differentiation has been parameters based on quantitative variations of expected to give clues to the forces that are possibly responsible for this differentiation. From a genetic phenotypic traits (Lefort-Buson, 1985). This lack of a clear relationship has been interpreted as resulting resources conservation point of view, it may be from irrelevant choices of the phenotypic traits useful to know whether or not two individuals or taken into account for the calculation of the quanti- populations that are phenotypically similar display tative distance (Siiddiqui et a!., 1977; Partap et a!., similar gene combinations. From an applied breed- 1980) or of the genotypes crossed (Peter & Rai, ing point of view, phenotypic or genetic distances 1978), or from bad appreciations of the genotype as have been expected to provide predictors for related to genotype x environment interaction (Singh heterosis. & Ramanujam, 1981; Ghaderi et a!., 1984). More recently, the need for a linkage disequilibrium *Correspondence E-mail: [email protected] between the genes involved in the calculation of the

1997 The Genetical Society of Great Britain. 477 478 J. BURSTIN & A. CHARCOSSET different distances and in heterosis has been empha- R=(Y—Y1)2 (2) sized (Charcosset et Charcosset & Essioux, al., 1991; or: 1994; Burstin eta!., 1995). The aim of this paper is to consider the effect of the polygenic inheritance of the traits used to R11=Jy—Y. (3) compute phenotypic distances on the relationship Marker loci When fl1, marker loci are available, between these distances and heterosis or marker distances between inbred lines i and jcanbe distances. We show that the relationship between computed using a well-known formula such as distances computed from quantitative phenotypic MRD2 (Rogers, 1972). For homozygous inbred lines, trait(s) and distances computed at individual loci or this distance is an estimate of the average heterozy- heterosis is not a linear one and discuss the param- gosity of the hybrid between lines i and j,andis eters that affect the magnitude of the correlation. designated MD (for marker distance). For a given This is exemplified by experimental results on the marker locus (p), O, takes the value + 1, —1 for relationship between distances computed from protein quantitative variations revealed by two- genotypesp1p1 and P2P2, respectively. Following that dimensional electrophoresis and a genetic distance notation, computed from marker polymorphism among 21 (1 -O(9) maize inbred lines. The choice of protein quantities as the phenotypic characters used to estimate quan- (4) titative distances was based on major advantages 2ii, As emphasized by Charcosset & Essioux (1994), over morphological traits: (i) they provide a high number of quantitative traits (190 in the present this model is also adapted to the case where more than two alleles are detected at marker loci [which is study), (ii) the complexity of genetic mechanisms involved in their variation ranges from oligogenic to a general case for restriction fragment length poly- polygenic and (iii) they are not affected by environ- morphisms (RFLPs) in many species]. mental fluctuations because they are assessed on 8-day-old seedlings grown undercontrolled Heterosis The heterosis expressed in the cross conditions. between inbred lines i and jisrelated to the heterozygosity of the hybrid i xj at the QTLs that display dominance effects (see Charcosset et al., 1991, for a Theory formal expression). Results presented further on for Basis of the model the relationship between quantitative distances and marker distances can therefore be readily extrapo- Quantitative traits. We consider a biallelic model lated to the relationship between quantitative describing the phenotypic value of homozygous inbred lines, following the notations used in Char- distance and heterosis if all QTLs involved in cosset et a!. (1991). The genotype of inbred line i at heterosis have dominance effects of the same magnitude. locus 1 (with alleles 11 and /2)isrepresented by the variable O, which takes the value + 1, —1for genotypes ll and 12/2, respectively. The single-locus Reference population We assume that the homo- model for the phenotypic value of inbred line i is zygous inbred lines belong to a reference population written as: (i.e. a set of lines of infinite size). We define w1 as the mean of (9 in this population. The frequency of = c1+a1O, the allele 11 in the population is: f= (1+w1)/2. where c1 is the average value of homozygotes 1111 Genetic diversity at locus 1 (H1) is proportional to and 1212, and a1 is half the difference between homo- the variance of (): H1 Var(19)/2 =(1—w)/2.In zygous l1 and /2/2 phenotypes. If the trait is this paper, the linkage disequilibrium between two controlled by n, loci acting independently (no epista- loci I and k is supposed to be either null or maximal sis), the phenotype of inbred line i (Y,) is (with (so that in this last situation, 0 =Ofor all inbred C=7ci): lines i, or O =— O,for all inbred lines i). In particular, we consider that (i) marker loci are = C+ a,O. (1) independent, (ii) QTLs are independent, and (iii) /=1 among the n. marker loci and the n1 QTLs, n1ploci are common to both sets, i.e. being markers and The phenotypic distance between i and jwas defined as: QTLs. The Genetical Society of Great Britain, Heredity, 79, 477—483. RELATIONSHIP BETWEEN PHENOTYPIC AND MARKER DISTANCES 479

Analytical results Table 1 Correlation between marker distance (MD) and quantitative distances R and R2: p (MD, R) and p (MD, Underprevious hypotheses concerning the magni- R2), for n1,.,loci(see text for hypotheses). N indicates the tude of linkage disequilibrium: number of pairs of genotypes involved in the correlation computation 1 1r Var(MD,1)=—(1—w,) (5) n11, 2 3 4 5 6 7 8 np N 16 64 256102440961638465536 (Charcosset et a!., 1991). For simplicity, the calcula- p(MD, R) 0.530.410.350.310.28 0.26 0.24 tion of the variance of phenotypic distance was only p(MD, R2)0.580.450.380.330.30 0.28 0.26 performed for R. It can been demonstrated (see Appendix) that: defined as previously) between the 2" x 2" possible Var(R) =4 a(1-w) pairs of genotypes at the n1 loci. We checked that under these conditions the

7?, 7I numerical values for the correlation between MD,1 aa(1—w)(1—w') (6) and R were consistent with those obtained from the lI analytical approach (Table 1). The correlation between MD, and R,1 was slightly lower than and between MD,1 and R. The correlation between the marker distance MD,1 and the quantitative distance 1 I parameters R and R decreased as the number of Cov(R,MD,1) =— a(1—wi). "7' lociconsidered increased (Table 1). For n,= 2to 8, k= I triangular relationships were observed (Fig. 1): low We have considered in particular the case of MD,1 values were systematically associated with low equal allelic frequencies at each locus (for all loci 1, fi, =f', =0.5),further considering that all QTLs have the same contribution to the variation of the trait of interest (for all loci 1, a1 = In this case, the corre- a). MD lation coefficient between phenotypic and marker 5 distances is:

ni 0 0 (8) \/2fl! —fl1 3 0 0 n1 1 — (9)

This last expression illustrates that the magnitude 2 0 0 of p(R, MD,J) depends on two factors: (i) the association between the marker loci and the QTLs (ni/i9),and (ii) the number of loci involved in 0 the variation of the quantitative trait (1/J7i). 0 Numericalresults Accordingto the assumptions used for the analytical developments, we considered n, loci involved with 1 2 3 4 5 equal contributions in the variation of the trait of R interest, each locus being biallelic with equal Fig.I Marker distance (MD, ordinate) vs. quantitative frequencies (0.5) for the two alleles. We further distance (R, abscissa) for a four-loci model (see text for considered that n1 =n,.,=n1,.For n1,,, =2to n1 =8, model hypotheses). The size of each circle is proportional we computed distance parameters (MD,J, R,1, R,, to its frequency.

The Genetical Society of Great Britain, Heredity, 79, 477—483. 480 J. BURSTIN & A. CHARCOSSET

quantitative distances, whereas high MD values 'vii) corresponded to either low or high quantitative 0.60 distance. With equal and independent effects of the 0.58 increasing and the decreasing alleles at the different 0.56 loci, two differentgenotypes, for example 0.54 (1.52 (+ + ——) and(—— ++), can have the same . - :• 0.50I—. • phenotype although they are different at each locus, ((.48 whereas identical genotypes necessarily have the 0.46 same phenotype. 0.44 0.42r 0.40 Materialsand methods (1.38 0.36 Twenty-onemaize inbred lines have been character- 0.34 ized as described in Burstin et at. (1994) for 142 0.32 markers resulting from the analysis of enzyme, 0.30 RFLP and anonymous protein polymorphisms, and 0.28 for the relative quantities of 190 proteins revealed (1.26 (1.24 by two-dimensional electrophoresis. The protein 0.22 quantities were determined by the KEPLER 2-DGel 0.20 analysis Software, as described in Burstin et a!. o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 (1993). For each pair of lines, we computed the Rk multilocus Rogers's distance for the 142 marker loci Fig. 2 Relationship between a marker distance computed (MD), and a distance defined for each of the 190 from 142 marker loci (MD, ordinate) and a quantitative proteins as follows: Rk =(Q,k—Qk)2with Q,,and distance computed for protein S65 (Rk, abscissa). QJkthestandardized quantities of protein k in lines i and j,respectively. Discussion results Experimentalresults reported in this study under- Experimental lined that, for a number of proteins, low marker Asexpected, the correlations between MD and the distances were associated with low phenotypic 190 phenotypic distances corresponding to the 190 distances, whereas high marker distances were assoc- protein quantities were generally small. They ranged iated with a large range of phenotypic distances. from —0.24 to 0.48, with 97 per cent of the correla- Thus, a clear tendency towards a triangular relation- tion coefficients being between —0.25 and 0.25. The ship was observed between the two distances (Fig. highest correlation coefficient (0.48) corresponded 2). This result is consistent with other studies. to the protein S207, which variation can be Burstin et a!. (1995) found a similar relationship considered as oligogenic because QTL analysis between a marker distance and Hanson and Casas demonstrated that it is mostly determined by three distances computed for yield and early vigour on 210 loci (Damerval et al., 1994). This protein has been pairs of maize inbred lines. A triangular relationship identified by microsequence comparisons as a was also found between a marker distance computed gluthatione-S-transferase (Touzet eta!., 1995). from 222 marker loci and a Mahalanobis distance Moreover, a trend towards a triangular relationship computed on 10 phenotypic traits, for 10440 pairs of clearly appeared on the plots of MD vs. the pheno- lines (Bar-Hen & Charcosset, 1995; Dillmann et at., typic distances computed on the standardized 1997). Similar results were also reported by Chanter- phenotypic of proteins. For example, Fig. 2 shows eau (1993) for sorghum inbreds. Triangular relation- the relationship observed between MD and the ships were also observed between heterosis and phenotypic distance computed from the variation of phenotypic distances (Charcosset et at., 1990, and protein S65. This protein has been identified by our unpublished data). However, Leonardi et at. amino acid composition comparisons as a chaper- (1991) reported a linear relationship between the onin hsp6o (Touzet et a!., 1996). Consistently with protein quantitative distance and heterosis for five numerical results, low MD values were associated agromorphological characters in maize. This result with low phenotypic distances, whereas high MD could be related to (I)thesmaller number of inbred values corresponded to either low or high quantita- lines that were considered in that study (five vs. 21 tive distance. in the present study), which may have led to samp!-

The Genetical Society of Great Britain, Heredity, 79, 477—483. RELATIONSHIP BETWEEN PHENOTYPIC AND MARKER DISTANCES 481 ing problems, and (ii) different scoring methods for This effect of kinship on the triangular shape of the the protein quantities (discussed in Burstin et al., relationship was illustrated experimentally in the 1995). present study: pairs of related lines corresponded to The triangular shape of the relationship between small distances, of either MD or R, whereas pairs of the marker distance or heterosis and the distance unrelated lines corresponded to large values of MD computed for phenotypic traits can be explained by and a large range of values for R. Thus, polygenic the polygenic inheritance of these traits, because a inheritance and linkage disequilibrium properties given quantitative value can be obtained with associated with kinship lead to a triangular relation- different gene combinations. Polygenic inheritance ship between marker distance and phenotypic has been demonstrated for most of the traits gener- distance. The relationship between phenotypic ally considered for phenotypic distance estimation. distances and heterosis depends in a similar way on For example, QTL mapping in maize has revealed the linkage disequilibrium between QTLs involved that numerous chromosome regions are involved in in heterosis and QTLs involved in the traits plant height variation (e.g. Beavis et a!., 1991). considered for phenotypic distance estimation. Damerval et a!. (1994) have demonstrated that indi- The results reported in this study illustrate that vidual protein quantity variations were often poly- prediction of heterosis based on quantitative genically inherited in an F2 progeny of maize. Thus, distances between parents has to be considered with the number of QTLs involved in the variation of a caution. As also discussed in Charcosset et al. phenotypic trait in a sample of diverse genotypes is (1990), a hybrid combination between two parents generally expected to be large. Our theoretical should not be discarded a priori on the basis of their results demonstrate that the correlation between morphological similarity, because similar phenotypes phenotypic distances and marker distances or can be observed for different genetic combinations. heterosis necessarily decreases with the number of The relationship between marker and phenotypic loci involved in the variation of the trait(s) of distances also has several practical applications. interest. This result is consistent with the small First, the phenotype of a given line can be predicted correlations observed between MD and the distances if it displays a high similarity at the marker level computed from protein quantities, and the fact that with a line that was characterized phenotypically. the highest correlation is observed for a protein with Secondly, among a set of lines with similar pheno- quantity controlled by a restricted number of loci. types, marker analysis allows the identification of the The polygenic inheritance of the quantitative traits lines that are likely to share similar alleles at the taken into account in the phenotypic distance QTLs. This can be extremely interesting for the computation has a similar influence on the relation- protection of owner's rights as well as for genetic ship between the distance and heterosis. resources conservation. However, because of the In addition to the polygenic inheritance of quanti- linkage disequilibrium properties, two inbreds may tative traits, one has to consider that markers gener- have similar phenotypes, share common alleles at ally have no direct effect on the quantitative traits of the QTLs and display a relatively high marker interest (i.e. they are neutral). Thus, the relationship distance at the same time. These applications would between quantitative distances and marker distances be broadened by the identification of the genes is affected by the linkage disequilibrium between involved in the variation of the traits of interest, marker loci and the QTLs involved in the traits which would allow an allelic comparison at the QTL considered for quantitative distance estimation. level. Equation (9) illustrates that a poor association between both types of loci leads to a low correlation between distances. If there is no linkage disequi- Acknowledgements librium, the two distances vary independently: high Weare grateful to C. Damerval, P. Dubreuil, A. and low marker distances can correspond to similar Gallais, M. Lefort and D. de Vienne for helpful morphological distances. If there is linkage disequi- discussions, to P. Dubreuil and P. Dufour for their librium, which is, for example, the case when inbred contribution to RFLP results and to M. Grenèche lines are related by pedigree (Charcosset and for isozymic results. Essioux, 1994), a strong relationship is expected. Because a high marker similarity is necessarily associated with kinship (e.g. Smith et a!., 1990), two lines References that are similar at marker loci will share common ATCHLEY,W. R., NEWMAN, S. AND COWLEY, D. E. 1988. alleles at the QTLs and thus be phenotypically close. Genetic divergence in mandible form in relation to

The Genetical Society of Great Britain, Heredity, 79, 477—483. 482 J.BURSTIN & A. CHARCOSSET

molecular divergence in inbred mouse strains. Genetics, yield and morphological traits in dry edible bean and 120, 239—253. faha bean. Crop Sci., 24, 37—42. BAR-HEN, A. AND CHARCOSSET, A. 1995. Relationship LEFORT-BUSON, M. 1985. Distance génétique et hétérosis 4. between molecular and morphological distances in a Utilisation de critères biometriques. In: Lefort-Buson, maize inbred lines collection —applicationfor breeders' M. and de Vienne, D. (eds) Les Distances Génétiques. rights protection. In: van Ooijen, J. W. and Jansen, J. INRA, Paris. pp. 143—157. (eds) Biometrics in Plant Breeding: Applications of LEONARDI, A., DAMERVAL, C., HEBERT, Y., GALLAIS, A. AND Molecular Markers, pp. 57—66. Proceedings of the Ninth DE VIENNE, D. 1991. Association of protein amount Meeting of the EUCARPIA Section Biometrics in polymorphism (PAP) among maize lines with perform- Plant Breeding, Wageningen. ance of their hybrids. Theo,: App!. Genet., 82, 552—560. BEAVIS, W. D., GRANT, D., ALBERTSEN, M. AND FJNCHER, M. MOSER, H. AND LEE, M. 1994. RFLP variation and genea- 1991. Quantitative trait loci for plant height in four logical distance, multivariate distance, heterosis, and maize populations and their associations with qualita- genetic variance in oats. Theor App!. Genet., 87, tive genetic loci. Theor AppI. Gene!., 83, 141—145. 947—95 6. BURSTIN, J., ZIVY, M., DE VIENNE, D. AND DAMERVAL, C. PARTAP, P. S., DI-IANKHAR, B. S., PANDITA, M. L. AND DUDI, 1993. Analysis of scaling methods to minimize experi- B. S. 1980. Genetic divergence in parents and their mental variations in two-dimensional electrophoresis hybrids in Okra (Ahelinoschus esculentus (L.) Moench). quantitative data. Applications to the comparison of Genet. Agr., 34, 323—330. maize inbred lines. Electrophoresis, 14, 1067—1073. PRIER, K. V. AND RAI, B. 1978. Heterosis as a function of BURSTIN, J., DE VIENNE, D., DUBREUIL, P. AND DAMERVAL, genetic distance in tomato. Indian .1. Genet. Plant c. 1994. Molecular markers and protein quantities as Breed., 38, 173—178. genetic descriptors in maize. I. Genetic diversity among ROGERS, J. S. 1972. Measures of genetic similarity and 21 maize inbred lines. Theo,: Appi. Genet., 89, 943—950. genetic distance. In: Wheeler, M. R. (ed.) Studies in BURSTIN, J., CHARCOSSET, A., BARRIERE, Y.,HEBERT,Y., DE Genetics VII, pp. 145—173. University of Texas Pubi. VIENNE, D. AND DAMERVAL, c. 1995. Molecular markers 7213. and protein quantities as genetic descriptors in maize. II. Prediction of performance of hybrids for forage SCHMITT, L. H., KITCHENER, D. J. AND HOW, R. A. 1995. A genetic perspective of mammalian variation and evolu- traits. P1. Breed., 114, 427—433. tion in the Indonesian archipelago: biogeographic 1993.Etude de l'hétérosis chez Ic sorgho CHANTEREAU, j. correlates in the fruit bat genus Cynopterus. Evolution, (Sorghum bicolor L. Moench) par l'exploiration d'éco- types et l'analyse de leurs divergences. Ph.D. Thesis, 49, 399—412. Université Paris Sud. SIIDDIOUI, 1. A., PRASAD, R. C. AND MEHRAR, n. 1976. CHARCOSSET, A. AND ESSIOUX, L. 1994. The effect of popu- Hybrid performance in relation to genetic divergence in lation structure on the relationship between heterosis some varieties of Gossypiwn hirsutum. Pfianzenzucht., and heterozygosity at marker loci. Theo,: App!. Genet., 77, 215—221. 89, 336—343. SINGH, S. P. AND RAMANUJAM, s. 1981. Genetic divergence CHARCOSSET, A., LEFORT-BUSON, M. AND GALLAIS, A. 1990. and hybrid performance in Cicer arietinum L. indian J. Use of top-cross designs for predicting performance of Genet., 41, 268—276. maize single cross hybrids. Maydica, 35, 23—27. SMITH, 0. S., SMITH, J. S. C., BOWEN, S. L., TENBORG, R. A. CHARCOSSET, A., LEFORT-BUSON, M. AND GALLAIS, A. 1991. AND WALL, S. j.1990.Similarities among a group of Relationship between heterosis and heterozygosity at elite maize inbreds as measured by pedigree, F grain marker loci: a theoretical computation. Theor App!. yield, grain yield, heterosis and RFLPs. Theor Appi. Genet., 89, 571—575. Genet., 80, 833—840. DAMERVAL, C., MAURICE, A., JOSSE, J. M. AND DE VIENNE, D. TOUZET, P., MORIN, C., DAMERVAL, C., LE GUILLOUX, M., 1994. Quantitative trait loci underlying gene product ZIVY, M. AND DE VIENNE, D. 1995. Characterizing allelic variation —anovel perspective for analysing regulation proteins for genome mapping in maize. E!ectrophoresis, of genome expression. Genetics, 137, 289—301. 16, 1289—1294. DILLMANN, C., BAR-HEN, A., GUERIN, D., CHARCOSSET, A. TOUZET, P., DE VIENNE, D., HUET, J. C., OUALI, C., BOUET, F. AND MURIGNEUX, A. 1997. Comparison of RFLP and AND ZIVY, M. 1996. Amino acid analysis of proteins morphological distances between maize Zen mays L. separated by two-dimensional electrophoresis in maize: inbred lines. Consequences for germplasm protection Isoform detection and function identification. Electro- purposes. Theor App!. Genet. (in press). phoresis, 17, 1393—1401. FALCONER, D. s. 1981. Introduction to Quantitative Genet- WAYNE, R. K. AND O'BRIEN, S. .i. 1986. Empirical demon- ics, 2nd edn. Longman, London. stration that structural genes and morphometric varia- GI-IADERI, A., ADAMS, M. W. AND NASSIB, A. M. 1984. Rela- tion of mandible traits are uncoupled between mouse tionship between genetic distance and heterosis for strains. J. Mammal., 67, 441—449.

TheGenetlcalSociety ofGreatBritain, Heredity, 79, 477—483. RELATIONSHIP BETWEEN PHENOTYPIC AND MARKER DISTANCES 483

Appendix Variance of R2 Var(Ri) =Var a, (9 —

— = Cov(E aia,(O—O)(U akak'(O—Ok)(O' _O))

= aaa,a/ llkk', / 1 k k wherec,lkk= Cov((O—O)(O— O);(O — — Basedon the elementary covariance property (Coy (AB; C) =E(A)Cov (B ;C)) if A is independent of B and C) and the specific property (E(O —O)=0)),o,,'kk'= 0if one locus is independent from the three other loci. In addition, 1/'kk'= 0if 1 =1',k =k'and 1 and k are independent. If one further assumes that the linkage disequilibrium between two loci can only be total or null, i1,'kk'0in the two following situations. • 1 =k,1' =k'and 11' (or 1 =k',1' =kand 1 In this situation, 1Ikk'= E((O— O)2(9 — O)2) — E2((O— O)(O— Ok)) = E((2—29O)(2—2O6))—o = 4(1—w)(1 —wi). • 1 =k=1'=k'.In this situation, 1I'kk'= Cov(—2OO;—2O6) =4(E(OO)2)—E2(OO) =4(1—wi). Thus: Var(R) =4a(1 —w)+8 aa(1 —w)(1 —w). I / kI

Covariance between R2 and MD

Cov(R;MDV) = I Cov((ai(o_o)); (1_UO))

=I a,a,.(O— O)(O. —O); (1— Cov(

=I a,a,.Cov((9—U)(O—O);(1 —OO)). Following previous assumptions concerning the possible magnitude of linkage disequilibrium, (1—OO)) =0, unless 1=1' =k. For this last situation, it can be noted that (O —e)2 = 2(1—6O)and so: Coy ((O —O)2;(1 —OO))=2Var(1— OO). So, 1 Cov(R; MDIJ) =— a(1 —wi). 1=1

The Genetical Society of Great Britain, Heredity, 79, 477—483.