,I. Genet., Vol. 66, No. 3, December 1987, pp. 177-185. �9 Printed in India.

A note on the analysis of reciprocal effects in diallel crosses

WIM E C R U S I O Institut ftir Hnmangenetik und Anthropologie, Universitfit Heidelberg, Im Neuenbeimer Feld 328, D-6900 Heidelberg, FRG

MS received 11 June 1987; revised 14 September 1987

Abstract. Maternal effects and sex-linkage give rise to differences between reciprocal crosses. In diallel-cross analyses, the presence of these effects will cause biases in the estimates of the genetical components of the variation. A method of analysis is described in which this bias is removed. Also, a worked example demonstrates the analysis for a case where malea only are available.

Keywords. Quantitative genetics.; diallel cross; reciprocal effects; maternal effects; sex linkage.

1. Introduction

The diallel cross is commonly used by plant and animal breeders for evaluating the genetical structure of a population of pure-bred lines. An accurate analysis of this mating design is therefore not only of theoretical but also of economical importance and has been dealt with by many authors (e.g. Eisen et al 1983; Virk et al 1985). A common assumption underlying most analyses of the diallel cross is the absence of any reciprocal differences, which are primarily caused by sex-linkage and maternal effects. Although the effects of such differences on the analysis of diallel crosses have been examined by Wearden (1964), Durrant (1965), Eisen et al (1966), Topham (1966), and Mather and Jinks (1982), tl]ese investigators were primarily concerned with testing the significance of possible effects and with determining the probable cause of reciprocal differences. Little attention was paid to the estimation of variance components along the lines of Hayman (1954b) and Jinks and Hayman (1953) if reciprocal differences are in fact present. Killick (1973) addressed the problem of estimating the genetic components of the phenotypical variance in the presence of sex-linkage, but maternal effects were not considered.

As will be shown below, the presence of reciprocal effects will bias the estimates of the additive-genetic variance, and thus of the heritability in the narrow sense, in an unpredictable direction. This quantity is of great practical importance for breeders and in this note a method is outlined through which unbiased estimates of genetic components may be obtained even if sex-linkage and maternal effects are present simultaneously.

t77

178 W i r e E C r u s i o

2. Mode l

Mather and Jinks (1982) describe two major sources of reciprocal differences: sex-linkage and. maternal effects. Possible causes of the latter effects may be cytoplasmic inheritance, maternal nutrition either via the egg or via pre- and postnatal supplies of food, transmission of pathogens and antibodies through the prenatal blood supply or by postnatal feeding, imitative behaviour, and/or intera'ction between sibs either directly with another or through the m~)ther (Mather and Jinks 1982, p. 30]). The model these authors used to describe these effects is shown in table 1, limited to those generations that are used in diallel crosses (inbreds and their F~'s). Here , d m stands for the materna l effect of homozygous genes. (In the normal diallel situation, i .e . using inbred lines as parents, we need not consider the maternal effects o fhe t e rozygous genes). The additive-genetic and dominance effects of sex-linked genes in the homogametic sex, are symbolized by dx and h x , respectively. In the heterogametic sex the sex-linked gene will be hemizygous and its effect need not equal the additive-genetic effect in the homozygous state. Hence , this effect is indicated by another symbol: d x ' . As is evident from the table, in diallel crosses the occurrence of sex-linkage is indicated if a reciprocal difference found in the heterogametic progeny of a set of reciprocal crosses is absent in the homogamet ic progeny. Maternal effects are indicated by reciprocal differences occurring not only in the heterogametic progeny of a set of reciprocal crosses, but in the homogametic progeny as well. If both effects are present simultaneously, then the magnitude of reciprocal differences will differ between the homo- and the heterogametic progenies. Whether these differences will be larger in the fo rmer or in the latter will depend on the magnitude and direction of the individual effects.

In partitioning theophenotypical variance found in a diallel cross, it is common usage to average reciprocal Ft 's before estimating the different components of variation (e.g. Mather and Jinks 1982; Crusio et a 1 1 9 8 4 ) . As is evident from table 11

T a b l e 1. The contribution of an autosomal (A) and a heterosomal (B) single-gene difference to the means of male (heterogametic) and female (homogametic) progeny families produced by crossing in- bred lines in all possible combinations.

Paternal parent

AAB - AAb - aaB - aab -

Maternal parent

AABB O" d, ,+(lx{ ,+dmO d, ,+dx~,+dm, haq-dx~+dm a h.+dx{,+dm~, d~ + dXb +dm~, d. + hXb + dm~, h. + d x b + dm~ h~, + hxb + din.

AAbb O" d , , - d x { , + d m , d~-dx( .+dm, , h.-dx{,-t-dm~, h~,-dx(.+&n,, f~ da + hXb + din,, d. - dXb + dm~, ha + hXb + dtn,~ tz. - ,dr b + dm,~

aaBB O ~ tl,, + dx ~, - d i n , , h l , + dx ~, - d i n , , - d~, + dx ~ - d m ~ , - d . + dx{, - d m ~ ,

h,, + dXb -- dm~, h,, q- h x b - d m a - d. + dx b - dm~, - d. + hxu - din.

aabb C~ h , , - dx(, - d i n . h . - dx{, - & n ~ - d~, - dx~, - d i n , , - d~, - d x ~ - d i n , ,

f~ h~ + hxu - din. h,, - dx b - dm,~ - d. + hXb -- dm~ - d~, - dXb -- din.

*See text for explanation.

Reciprocal effects in diallel crosses 179

such a procedure removes the effects of reciprocal differences, but only so in Fl's. Thus, the family means of the inbreds (the diagonal entries) will still contain a contribution of reciprocal effects. This will cause a bias in the estimates of Vp (the variance of inbred family means), Vr (the mean variance of arrays), V? (the vari- ance of array means), and Wr (the mean parent-offspring covariance of arrays), re- suiting in a bias in the estimates of genetic parameters. However, if both male and female progeny of a diallel set of matings is available, all sources of variation can adequately be accounted for as follows.

3. Analysis

Suppose we have a situation with k autosomal genes, l sex-linked genes (both with only two alleles per gene), no epistatic interactions, and independent distribution of alleles. Let's suppose further that k' of the k autosomal genes cause a maternal effect. Now we may write

k

D = ~ 4u iv id2 , i = 1

k

Ht = ~. 4uivi h2 , i = 1

k

H 2 = ~/, 16uZv~h~, i = 1

k

F = ~ 8tlivi(Lli--vi)dihi, i = 1

and, analogously,

i Dx = ~, 4U i V i dx 2

i = l

O x t =

Hx 1 = i = 1

/

Hx2 = i = l

/

F x =

l

Z 4UiVi dX'2, i = 1

l

Y. 4u~ v, hx ~,

16u 2i v ~ hx 2,

Z 81tiVi(tli-- vi)dxihxi i = 1

180 Wire E Crusio

k'

D i l l = Z i = 1

k'

Dill d

i=

k'

Drab ~ i=

4 . i " i elm 7 ,

4Ui Vi d i d m i , 1

8l l i Vi (Lti - - l'i )h i d m i .

The above parameters, plus E (tile environmental contribution to the phenotypical variance), can be used to describe the total variance occurring ill a diallel set of crosses.

If no averaging over reciprocal Fl's takes place, V,.,. W,., and Vi (see above) can be calculated in two different ways: (i) over arrays that have the maternal parent in common, and (ii) over arrays that have the paternal parent in common. Addition- ally, these statistics can also be calculated after averaging over reciprocal Fi's. Finally, V~, can be estimated, giving 10 statistics in total for each of the two progeny sets available from tim diallel cross (one for the males, one for the females). Now we have 20 equations from which to solve 12 parameters (table 2). As no single, unique, perfect-fit solution is available, a least-squares procedure may be used to find the expe'ctations of the different genetical and reciprocal-effect parameters. Unfortunately, as was already stated by Mather and Jinks (1982, p. 275), no good estimate of the errors of these components is available. Nevertheless, a replicated experiment may offer some clues. If only few replications are available, the magnitude of the different estimates may be compared over blocks, whereas if some more blocks have been raised, an en)pirical standard error may be calculated.

Of course, one ,night wonder about the consequences of ignoring significant reciprocal effect:s. In such a case, fhe formulae presented by Mather and Jinks (1982, p. 275) and by Crusio et al (1984) to estimate D, HI , H2, and F (after averaging over reciprocal families) will estimate D +Dm + 2Drag+ Dx', Hi, H2, and F+ Dmh, respectively, in the heterogametic sex. In the homogametic sex this becomes D+ Dm+2Dmd+DX, H~ + H x l , H2+Hx2 , ancl F+Fx+Dmh. Thus, with sex-linkage present, males and females render different estimates for the genetic parameters. If maternal effects only are present, the estimates of males and females will agree, but the estimates for D and F are biased. The magnitude of this bias not only depends on the magnitude of Din, Drag, and Drab, but also on the sign of the latter two parameters. If Drag and/or Drab are negative, it is conceivable that D and F are biased downwards. Such a situation may arise in cases where the maternal effects of genes oppose, on average, their additive-genetic or dominance effects. Although this possibility may, at first sight, seem improbable, it must be stressed, that a negative correlation between additive-genetic and additive- maternal effects is not only theoretically possible but has indeed been found (Fulker 1972).

Often, only one sex of the progeny is measured. If this is the homogametic sex, no information about [dx'] would be available. Further, Dx, ILvi, Hx2, and Fx cannot be separated from D, Hi , He, and F, respectively. However, we can still extract the w/riance caused by maternal effects, Din. If only data from the

Reciprocal ef[ects in dial~el crosses 181

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Reciprocal effects in diallel crosses 183

heterogametic sex are available, then D x ' cannot be separated from Din, but the contributions to the phenotypical variance of reciprocal effects can be removed from the " t rue" genetical components, D, HI , H2, and F. In this situation we may refer to [dx'] + [dm] as [dr] and to Dx ' + D m as Dr. Although a situation in which both sexes are measured is obviously preferable over a situation in which only one sex is available for measurement, correcting our estimates of genetic parameters for reciprocal differences will also render more reliable estimates of important population parameters if one sex only is sampled.

As far as testing of the assumptions underlying diallel-cross analyses is concerned, averaging over reciprocal families leaves the coiwentional W: V graph unbiase d by maternal effects and/or sex-linkage (except for a bias in the intercept). Mather and Jinks (1982, p. 302 et seq.) investigate the consequences of deviations from the simple maternal-effects model used here.

4. Worked example

To illustrate my case, I present an example from behaviour genetics in which male mice only have been used. To my knowledge, no complete dataset, comprising both sexes, is available from the ,literature. However, the present example demonstrates how the method might profitably be used in this situation.

The phenotype studied was locomotor activity displayed in an open-field. The experimental design used was a five times replicated 4 x 4 diallel cross, with a nested replication of the leading diagonal. All data and procedural details have been presented previously by Crusio et al (1984). Thus; for the sake of brevity, only the statistics that are needed for the present analysis are here presented in table 3. The previous analysis (Crusio et al 1984), by means of Hayman's ANOVA (Hayman 1954a), showed the presence of significant additive-genetic effects and ambi- directional dominance. Further, Hayman's c-item was significant, indicating the

Table 3. Locolnotor activity of male mice in a'n open-field: second-degree statistics and estimates of genetic and reciprocal-effects components of variation.

Statistic Vp Vrl Vr2 V/q Vf2 Wrl Wr2 Vrl2 VI=I2 1Nrl2 E 12990.1 2226-8 4595.8 3317.0 949.8 3488.7 6128.9 3152.4 1917.8 4808.0 1979.3

Parameter estimates with model

Including reciprocal Excluding reciprocal effects effects

D 8090-0 12792.2 Hi 5483.7 5572.8 H 2 4163.8 4344.6 F 6004.1 6547.1 Dr 578"l - - Dma 2062.1 -- Dtnt, 544.2 -- h ~,,) 0.22 0.49 h (~,) 0.36 0.64 HiD) 112 0.82 0.66

J

184 W i r e E C r u s i o

presence of reciprocal differences. The W : V graph did not indicate any deviations from the simple model (without epistatic interactions and/or correlated allele distributions).

The estimates of the genetic components of variation obtained by Crusio et al

(1984), ignoring these significant reciprocal effects, are also shown in table 3,. together with the estimates of genetic and reciprocal-effects components as obtained by the present method. The consistency of the relative magnitudes of the estimates over blocks (values not shown), means that we can interpret our results with some confidence.

As can be seen, rather small estimates are obtained for Dr and Dmh, but the estimate for Dm~t seems to be substantial. The estimates of the genetic parameters D, Hi , H2, and F are differentially influenced. Clearly, the estimate of D is affected rather dramatically, whereas the effects on the estimate of F are relatively minor. As pointed out earlier, the estimate of F that is obtained when reciprocal differences are ignored equals F + Dmh. The latter parameter is estimated to be rather small, which may indicate either that u - v is rather small for the genes concerned or that there is ahnost no covariation between dm and h. In the light of the ambidirectional dominance that had been found in the earlier analysis, the rather low estimate for Dmh seems acceptable. As expected (see above), the estimates of Hi and H2 are hardly influenced at all. The positive estimate for Dm~/ indicates that maternal effects tend to reinforce additive-genetic effects in this case.

In consequence, the estimated degree of dominance becomes higher and the heritabilities in the narrow and the broad sense are estimated to be nmch lower than in the original analysis ignoring reciprocal effects.

Acknowledgements

I thank Profs. F Vogel and W Buselmaier (Heidelberg) for their interest in my work and for allowing me to work as a guest at their Institute. Thanks are due to Dr. J H F van Abeelen (Nijmegen) for critically reading an earlier version of the manuscript. Dr N D Henderson (Oberlin, Ohio) first drew nay attention to the problem discussed in this article. This study was supported by a NATO Science Fellowship awarded by the Netherlands Organisation of Pure Research ( zwo) and an Alexander-von-Humboldt stipend.

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Reciprocal effects irz diallel crosses 185

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