the experiment; and the distribution of residuals (or non-heritable

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1 SOME GENOTYPIC FREQUENCIES AND VARIANCE COMPONENTS OCCURRING IN BIOMETRICAL GENETICS J. A. NELDER National Vegetable Research Station, Wellesbourne, Warwicks. Received 30.xi.5 i 1. INTRODUCTION MATHER (1948, 1949) has developed an analysis of the variation of characters which are phenotypically continuous in expression, using data derived from the crossing of two true-breeding lines. In essence the model used makes the following assumptions: the genes are additive in effect; the segregations are Mendelian; no selection operates during the experiment; and the distribution of residuals (or non-heritable components) is independent of the genotype. In the usual notation, genotypes AA, Aa, aa of a locus a produce on the character in question average effects da, ha, and da respectively, the zero point on the scale being the mid-parent value. The recombination fraction between loci a and b is written Pat, With this model, it is found that the expectations of certain mean squares, such as the variance of F2 plants, the variance of F3 family means, etc., can be expressed as known linear functions of three quantities D, H, E, where D is a function of the d's and the p's only, H is a function of the h's and the p's only, and E is an environmental variance dependent in structure on the layout of the experiment. Mather has shown that when linkage is absent, D and H are the same throughout and are equal to Ed2 and Eh2 respectively, but that when linkage is present, D takes the form Ed2 + E'f(Pab) dadb, where E denotes summation over all the loci concerned, E' summation over all pairs of loci andf(p) is a function of the recombination fractions which changes with the generations of, say, selfing or backcrossing; H is similarly modified. Mather (ig) gives expressions for the coefficients of D and H and the form off(p) for selfing up to F4 and for backcrossing for two generations. In this paper these expressions will be generalised to cover any generation of selfing or backcrossing to either parent and, in obtaining these expressions, the effect of k generations of selfing or backcrossing on the relative proportions of genotypes of two linked genes will also be derived. 2. METHODS The genetical structure of a population with regard to two linked genes is defined by ten quantities, four giving the proportions of the homozygotes AABB, AAbb, aabb, and aabb, four the proportions of the single heterozygotes, AABb, aabb, AaBB, and Aabb, and two 387

2 388 J. A. NELDER giving the proportions of the two phases, coupling and repulsion, of the double heterozygote AaBb. These ten quantities will be written as a column vector X={x1, x2,, x0} with components representing the genotypes as in table x. TABLE i AA Aa aa BB Bb 4 5C/6R 7 bb 8 g Thus, assuming Mendelian segregation with recombinatioi fraction p throughout, {o, o, o, o, i, o, o, o, o, o} represents the F1 of a cross between AABB and aabb and ' P f ' p ' P2 P the J F2 from such an F1, where q = i p. In considering the descendants of two true-breeding lines, five quantities suffice to describe the population since x1 = x10, x3 = x8, and x2 = x4 = x7 = x9 throughout. The ten components will be retained here for the sake of generality. The effect on a population of selfing or of crossing to an independent population is to replace the components of x by other components linearly dependent on them. Writing x0 for the original population and x1 for the population after one generation of; say, selfing we can put ==Tx where T is a io x 10 matrix, the transition matrix of selfing, and the components of T depend onp but not on x0. The effect of Ic generations of se4fing is to produce proportions Xk given by xk =Tkx0 The problem of finding these k-th generation proportions is thus reduced to the problem of finding the k-th power of a matrix. This is most conveniently done by using the so-called spectral matrices of T. It can be shown that, to each latent root of T, )ç say, there corresponds a matrix Q such that (i) T-EQ (ii) (iii) QQ = 0, P It follows immediately that Tc=AQ, so that Tk is expressed as a linear function of the spectral matrices Qwith coefficients equal to the k-th power of the latent roots. It should be noted that the method fails for biparental progeny because the components of x1 are then quadratic functions of the components of x0 instead of linear ones, so that a method based on linear transformations becomes inapplicable. 3. THE TRANSITION MATRIX FOR SELFING By considering the progeny derived from selfirig the various genotypes, we find as the transition matrix for selfing

3 GENOTYPIC FREQUENCIES AND VARIANCE COMPONENTS 389 a2 p2 I 0 44 o o 0 o I o 0 0 T p2 o o o 0 0 o o 0 0 o o o 2 2 o o 0 0 I 0 44 o o 0 o o I 44 / Column 4, for example, expresses the fact that selfing genotype 4 (AABb from table i) produces. of genotype i, i.e. AABB, of genotype 4, i.e. AABb, and of genotype 8, i.e. AAbb. we have, for the latent roots of T, i fourfold, fourfold, ( x 4i), and (i 2pq). The spectral matrix for A x is given by / I 2(I+2p) I+2 o 4 i 0 p-- i I+2p 2(I+2p) 4 o 4 o o o 0 1+2/) 2(I+2p) J 2(I+2p) I+2p /

4 390 for A = Q2= by / J. A. NELDER o 4- o o\ 0 I o o i I for A=(i 2p) by Q3= / i 4i 1 2p 4(1+41) 4(1+4)) ) (I+2p) 4(1-41) (1+41) 4(1+41) p (1+4)) 4(1+41) and for A=4(x 2pq) by Q4= / o o 0 0 / The transition matrix for Ic generations of selfing is given by Tk=Q1+(WQ2+(' _2P)kQ (I _2PyQ It may be noted in passing that as k-, Tk Q1. By putting x1= {o, o, o, o, i, o, o, o, o, o} and evaluating Xk= T1x1 we have the genotype frequencies for the Fk generation from two truebreeding lines for two genes in coupling. In fact,

5 GENOTYPIC FREQUENCIES AND VARIANCE COMPONENTS 391 / i i (i 2P)k (i 2Pq) 2(1 +2p) 2k+i@ +2p) 2k+1 i(i _2pq)' 2k p i (i 2?)k (i 2pq)1 I+2p2k+2k+1(I+2p)+ I _(I 2pq)' + 2k+i 2k (i 2P)l+(I 2pq)' 2k (i 2P)i+(I _2pq)k_l Xk7 = Xkl = XklXk2 XklO= To obtain the corresponding frequencies for two genes in repulsion, it is only necessary to interchange the 1st and 3rd components, the 8th and ioth components, and the 5th and 6th components. The limiting frequencies as k- cr for the coupling phase are given by Q1 x1, that is the proportions of AABB and aabb are aabb,, and the remainder are zero. 1+ 2p 2k, those of AAbb, 2(1 +2p) 4. THE BACKCROSS TRANSITION MATRIX By considering the result of crossing each genotype to AABB, we find for the transition matrix of backcrossing to AABB / I 0 q p o I 0 p T= 0 0 o o o p 0 I 0 o o 0 0 q jb 0 I Remaining five rows all zeros / The transition matrices for backcrossing to the other homozygotes may be obtained by an interchange of appropriate columns in the above matrix. The characteristic roots are o sixfold, twofold, i, and q. The spectral matrix for A =i is given by Q,= i i r i i i i i i Remainder zeros for A = by / 0 I 2 I I I 2 0 I 2 Q2= I I I I Remainder zeros /

6 39 J. A. NELDER and for A =q by / p1 i 2\ o o 0 0 I - 0 q q q q I I 2 0 0! o o q 0 qo 00 q 0 q0 I I 2 o o 0 0 I 0 q q q q o o q q q q Rema!nder zeros so that Tk=Q1+(Q2+(Q3. To obtain the frequencies resulting from the k-th backcross of an F1 in coupling to parent AABB we put x0 ={o, o, o, o, i, o, o, o, o, o} and evaluate Xk = Tkx0 = 'c r ()k i Remainder zeros / The frequencies for the backcross to the other parent, aabb, are obtained by interchanging x1 and x10, x2 and x9, and x4 and x7 in the above expression. 5. APPLICATION TO VARIANCE COMPONENTS (I) Selfing Series Consider the Fk generation following the crossing of two truebreeding lines. The plants in this generation can be divided into families of parents in the Fk_l; the parents can be divided into families of plants in the Fk_2, and so on. We shall say that plants of Fk which had a common ancestor in F1 but not in F1+1 form an i-group. We define as the component of variance of (k 2 i)-group means within (Ic 2)-groups in the total heritable variance of the Fk, with obvious modifications when I takes the extreme values i and Ic i. To fix the ideas, consider the F4 and write X.,k for the measurement of the k-th plant having grandparent i and parent j. Suppose i runs from x to r, j runs from i to s, and Ic runs from i to t. The total sum of squares (x x.. ) 2 (where a dot suffix denotes, as usual, an averaging ijk over the suffix concerned) can be split up into three parts, namely x...)2 (x,k x,.)2 + t(x5. x..)2 +st(x.. ijk ii i

7 GENOTYPIC FREQUENCIES AND VARIANCE COMPONENTS 393 corresponding to the primary analysis of variance in a double split-plot experiment. For the corresponding mean squares, we write then I V374 =,. rst I) Jk V4=(' i) V4 = (xi.. ( = VIF4) a27 + E g(v4) _a2+4+e (x1. x2..)2 S(V174) _a2+4+4+e where g denotes expectation and E is the non-heritable component of variance. The E occurring in the three expressions above is not necessarily the same in each case. In the notation of Mather (1949), V374 = V74, V2F4 =Vj and V4 = V. The form of U217k in the presence of linkage has been given by Mather (5949) as far as the F4 components. We shall now derive the general expression for 21Fk The means of any generation of plants are unaffected by linkage. In particular, a gene occurring in the homozygous state contributes to the mean in the Ic-tb generation of selfing independently of k, while a gene occurring in the heterozygous state contributes i to the same mean. Now Fk can be expressed as the difference of the total variance of (k 1 i) -group means and (Ic 1)-group means in Fk, ignoring the variation within means. In computing the variance of (k I) -group means, ignoring the variation within means, the variate is that of the mean in the (Ic 1) -th generation of selfs while the frequencies will be those of the F1. These are shown in table 2. Evaluating the variance of the means in table 2, and subtracting it from the variance obtained by substituting i x for I, we find, after some algebraic reduction, that ajfk{d2+d+2dodb(i _2P)1]± 2kii[1a2+h+2kj1b(1 2pq) (I 4pqfl For linkage in repulsion, the term in dadb has its sign changed. Noting that i 4q = (i 2p)2 and I 2pq = I 2+ 2p2, it will be found that these values for 017k are in agreement with those up to given by Mather (ii) Backcross Series In the progeny of the Ic-th backcross to one of the parents, the components of variance may be defined in an exactly analogous manner

8 394 J. A. NELDER TABLE 2 Genotype Mean of (k l)-th generation Frequency in F1 of selfs derived from genotype AABB AaBB aabb AABb da+d, + db da+do d+--- AaBb.C. + as previously derived 2 2 for Fk with I written AaBb.R. ha for Ic 2k 1 2k aabb AAbb da - Aabb aabb dadb to those of Fk. It will be found, however, that these components of variance contain cross-terms in d h. These can be eliminated if an average variance over the two k-tb backcrosses is taken. We shall denote this average variance by a and a similar argument to that in the previous section shows that p)1(i p) +h+h+2hahb(1 P)1(' 2P)] (Blc[(1a21+2dadb(I As before, the sign of the dadö term must be changed for linkage in repulsion. In Mather's notation 021B1 = 1( VB1 + VB2) ( V11 + VB22) t21b2 +22B2 = Vj+ V) where n is the number of plants per family in the second backcross. 6. SUMMARY General formulae are given for the relation between the initial genotype frequencies of two linked genes in any population and those obtaining after k generations of selfing, or of backcrossing to one of the homozygotes. These formulae are used to derive expressions for certain variance components occurring in biometrical genetics. 7. REFERENCES MATHER, K Biometrical Genetics. London: Methuen. MATHER, K The genetical theory of Continuous variation. Hereditas, Suppl. Vol., 1949.