U among relatives in inbred populations for the special case of no dominance or

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1 PARENT-OFFSPRING AND FULL SIB CORRELATIONS UNDER A PARENT-OFFSPRING MATING SYSTEM THEODORE W. HORNER Statistical Laboratory, Iowa State College, Ames, Iowa Received February 25, 1956 SING the method of path coefficients, WRIGHT (1921) derived correlations U among relatives in inbred populations for the special case of no dominance or epistasis. FISHER (1947) described the use of generation matrix methodology in the theory of inbreeding, some of this methodology being due to earlier writers. FISHER S concern was with mating types, where the matings AA X AA and aa X aa, for instance, are of the same type. More recently, KEMPTHORNE (1955) pointed out two distinct types of problems involving correlations between relatives in inbred populations. Assume that a random mating population is inbred according to a regular system of inbreeding. In generation n we have the population where F, is the coefficient of inbreeding attained at generation n. The first problem is represented by the following: (1) what is the correlation between full sibs in generation n and (2) what is the correlation between an individual of that generation and his offspring in generation n + l? For the second problem, suppose that the above population is in equilibrium, then what is the correlation between relatives in this population? This same paper dealt with the first problem, using generation matrix methodology and assuming two alleles per locus, for a full sib inbreeding system. Kinds of matings were employed rather than types; i.e., AA X AA isa different kind of mating from aa X aa. The objective in the present paper was to obtain parent-offspring and full sib correlations in the nth generation under a parent-offspring mating system. This necessitates obtaining the frequencies of the various kinds of matings in the nth generation as a function of the frequencies in the zero generation. PARENT-OFFSPRING MATING SYSTEM The offspring of a mating is always mated to the younger parent. Considering a single locus with two alleles per locus there are nine kinds of matings, which are shown in table 1. These are arbitrarily numbered from one to nine. Mating kind frequencies will be indicated as fi, fi,..., f9 with superscripts in parenthesis indicating the appropriate generation. Thusfi3 is the frequency of the first kind of mating in the third generation. 1 Journal Paper No. J-2976 of the Iowa Agricultural Experiment Station, Ames, Iowa. Project number 1285.

2 PARENT-OFFSPRING AND FULL SIB CORRELATIONS 461 Younger parent Kind of mating Older parent Y Z AA X AA AA X Aa rla X aa ila X AA Aa X Aa Aa X aa aa X AA aa X Aa aa X aa - 3esignation requency n generation zero.il(o) fico) fy jio) jp j? ho) fio) f? TABLE 1 Kinds of matings Proportions of kinds of mating> resulting from mating of a given kind 2 - f Kind of mating KIND OF MATING FREQUENCIES IN THE Fn GENERATIOX The kind of mating frequencies in the first generation are linear functions of the kind of mating frequencies in the zero generation. These linear functions are obtained in the following manner.2 Mating kind one leads to mating kind one. The offspring from mating kind two (AA X Aa) are $AA and 4Aa. These are mated to the younger parent which in this case is AA; and which becomes the older parent in reference to the new generation. Hence a mating of kind two leads to matings of which one half are of kind one and one half are of kind four. In the right half of table 1 are shown the proportions of the various kinds of matings which result from the kind of mating indicated in the left hand column. The frequency of mating i in the first generation can be obtained from table 1 as the sum of products of corresponding elements in the frequency in generation zero column and the ith kind of mating column to the right. If we let (f(n)) = (,fl L)fi(n).. fin ) be a column matrix, then kind of mating frequencies in the first generation relate to those in the zero generation by the equationf = Af O) where A is a 9 x 9 matrix. Using generation matrix methodology, it can be shown that f( ) = C-lAnCf(O) where A is a diagonal matrix whose elements along the diagonal are the latent roots of the A matrix. In the present case these roots are 1, 1,,,, - 3, 4, E and E where E = i(1 +l/s) and E = i(l - ds). The C matrix is any matrix satisfying the equation C AP = A. Assuming that the genotypic array in the zero generation is p2(aa) 4-2pq(Aa) and q2(aa), the f matrix is * A more detailed derivation is available from the author in mimeographed form.

3 462 T. W. HORNER PARENT OFFSPRING COVARIANCE The values of the genotypes AA, Aa and aa will be represented as y, $(y - x) and. On this scale the degree of dominance is -x/y. The parent offspring covariance is COV(P,-I,,) = f?'(aa)(aa) + fi"'(aa)(aa) +. *. fi"'(aa)(aa) - pnpn-l where p, = (AA)(f!"' +fin) + f?') + (Aa)(fi"' +fp +fp) and + (aa)(fy +f? +fin') pn-1 = (AA)(f?' + fy' + f:"')+(aa)(fy +f:"' +f3+ (@a)(@ +e + fp') In the above formula genotypic values are to be inserted for genotypes. After such substitution, collection of terms in y2, xy and x2, and substitution of + - $Fn-I + F, 1955) and 3 + ~F,-I - F, for Sn+l, for S, (see KEMPTHORNE in terms of WRIGHT'S coefficient of inbreeding F. Thus we have Cov(Pn-l,OR) CO~(P,-~,,) = &($ + tfn-~+ 3Fn)y ) [($)" Fn-i + 3F,]xy + ipq(1 - F,-d [(l- 2Pl2 + 4pqFn1X2 n 21 The variance in the izth generation is U; = (AA)2(fin' +fin) +fin') + (Aa)*(fin' +fin' + fin') + (aa)2(f:n' + fin' + fp') - pl2n or U% = +Pq(1 + F,)Y - Pq(1-2P) (1 - F,)xy + +Pq[l - F, - 2Pq(l - F,)2]~2 The variance in the n - 1 generation, U;, can be obtained from U; by substitution

4 PARENT-OFFSPRING AND FULL SIB CORRELATIONS 463 of (n - 1) for n. The genotypic correlation of parents in the n - 1 generation with their offspring in the nth generation is COV(P,-.l,,) UY Uz FULL SIB COVARIANCE Each kind of mating in the nth generation gives rise to a progeny in the (n + 1) generation; the frequency of each progeny being the same as the frequency of each kind of mating in the nth generation. The covariance of full sibs in the (n + 1)th generation is the same as the variance among progeny means in the (n + 1) th generation. Thus Cov(Ful1 sibs in generation n + 1) 2 = (AA x AA)@ + (AA x Aa)$* + (aa x aa)$n - cl,+l where progeny means are to be inserted for the indicated matings. After simplifying the above equation and substitution of n - 1 for n we have Cov(Ful1 sib in generation n) = pq(+ + gon-1 - if-1 + Fn)yZ + pq(1-2p) [($In - (1- F,)IxY + P~[$PqO l + $(I - e) - Pq(1 - Fn)21~ The genotypic correlation among full sibs is Cov(Ful1 sib in generation n) 4 ADDITIVE GENETIC VARIANCE The additive genetic variance of the nth generation in the case of 2 alleles is the sum of squares attributable to regression of genotypic value upon the number of A genes of the genotype. The variance is (TA = [Cov (Genotypic Value, Number of A genes)] Variance of number of A genes In the nth generation the covariance of the numerator is

5 464 T. W. HORNER Hence In the random mating population (generation or 1) the additive genetic variance is a1 = $p(l - p) {y + (29-1)x)' PARENT OFFSPRING AND FULL SIB CORRELATIONS The relationship between these correlations and generation (n) are shown in graphs la (parent offspring) and lb (full sib) for selected degrees of dominance and X gene frequencies. Degree of dominance equals - - = a, say. Y In the case of no dominance these correlations, which in the first generation are one half, rise to one as n -+, with the parent offspring correlations being somewhat higher, particularly in the early generations, than the full sib correlations. When dominance is present these correlations are decreased as the frequency of the A allele increases, particularly in the early generations, which may have correlations considerably less than one half. This decrease is not as drastic for the full sib correlation as for the parent offspring correlation. When y =, degree of dominance is infinite and the parent offspring correlation approaches =.5559 as n --f CO and the full sib correlation approaches 3 as n +. 5: -C Q: -f -g I GENERAT I ON FIGURE la.-relationships between the parent offspring correlation and generation. Case (a) a =, all values of p; (h) a = 1, p =.1; (c) a = 1, p =.9; (d) a = 1.5, p =.3; (e) a = 1.5, p =.9; (f) a = w, p =.1 or.9, (g) a = CO, p =.5.

6 PARENT-OFFSPRING AND FULL SIB CORRELATIONS 465 z t.5 a i I I IO GENERATION FIGURE 1b.-Relationships between the full sib correlation and generation. Case (a) a =, all values of p; (b) a = 1, p =.5; (c) a = 1, p =.9; (d) a = 1.5, p =.5; (e) a = 1.5, p =.9; (f) a = m, p =.1 or.9; (8) a =, p =.5. ESTIMATION OF ADDITIVE GENETIC VARIANCE OF ORIGINAL RANDOM MATING POPULATION Parent Ojspring covariance procedure The additive genetic variance in the original random breeding population on the assumption of no dominance is +p(l - p)yz and the covariance of parent in the (n - 1) generation and offspring in the nth generation is - p) (1 + Fn-1 + 2Fn)y2 which suggests estimating the additive genetic variance in the random mating population by A 2~%1(pn-1,n) UA(P) = 1 + Fn-1 -t 2Fn This procedure is, of course, unbiased when there is no dominance. The bias for various gene frequencies and degrees of dominance is graphed against generation in figure 2a. When gene frequency is one half, the bias may be of the order of 1% in early generations for the case of complete dominance. The bias increases as the degree of dominance increases and as p deviates from one half. For example in the case of complete dominance and a gene frequency of.7, the bias using the covariance of parents in the n - 1 = 4 generation and offspring in n = 5 generation is 122%.

7 466 T. W. HORNER Full sib covariance procedure Assuming no dominance, an unbiased estimate of a: ing population is A - 2cGv (F.s. in generation n> d(f.8) - [+(I + On-') - +Fn-I + 4Fn1 in the original random mat- The bias from this procedure is illustrated in figure 2b. The curves are similar to those of figure lb in that the percentage bias approaches the same limit as n increases. :hey differ in that the bias need not be zero in the first generation and in general g:(f.s) is larger than cr:(~..~) for corresponding cases. Total genotypic variance procedure A third unbiased estimate, assuming no dominance, of the additive genetic variance in the random mating population is 27 - p=.f, p=s, a=i p=.l,.3 p =.I, a = 1.5 I IO GENERATI ON FIGURE 2a.-Percentage bias in each generation of the estimate of the additive genetic variance )c of the random mating population by vi (p. o.). oyof= -1% FIGURE 2b.-Percentage - G E N ER AT ION of the random mating population by C T ~ ( ~, ~ ). "-p=.7, -p=.5, as.5 a45 -p=:i I ; Zi.% bias in each generation of the estimate of the additive genetic variance

8 PARENT-OFFSPRING AND FULL SIB CORRELATIONS 467 cn a m w 9 z w W a 5% 4% 3% pf.7, a= t -1% I IO GENE RAT ION p=.l,.5 Figure 2c.-Percentage bias in each generation of the estimate of the additive genetic variance h of the random mating population by Q $(~.~,~). and the bias is graphed in figure 2c against generation. As n increases the bias approaches the same limit as before. In early generations the positive bias is larger (or the negative smaller) than the corresponding cases for the parent offspring and full sib procedures. a: p=.l, a=.5 b: pz.9, as.5 c p=.3, a=l d: p8.3, ~1.5 e: p=.9, a= I f: p=.9,.1.5 GENERATION FIGURE 3.-The additive genetic fraction of the total genotypic variance in the nth generation; the additive genetic variance heing defined in the nth generation.

9 468 T. W. HORNER PARTITION OF THE TOTAL GENOTYPIC VARIANCE The proportion of the additive genetic variance, when this variance is defined in the F, generation, of the total genotypic variance is graphed against generation in figure 3. As n increases the additive fraction tends to constitute, of course, the whole of the genotypic variance. In early generations for the higher gene frequencies and degrees of dominance, this fraction may be quite low. For example in the case of complete dominance and p =.9, the additive genetic variance is only (6%) of the total genotypic variance in the 3rd generation. SUMMARY Parent offspring and full sib correlations in the nth generation are derived for a parent offspring mating system using generation matrix techniques. These correlations are reduced when dominance is present and in early generations may be considerably less than one half. Biases in the estimate of additive genetic variance of the random breeding population by (a) a parent offspring procedure, (b) a full sib procedure and (c) a total genotypic variance procedure are small when gene frequency is close to one half for partial or complete dominance. When dominance is present, considerable positive or negative bias may result, depending on whether gene frequency is larger or smaller than one half, the bias becoming larger as the degree of dominance increases. LITERATURE CITED FISHER, R. A., 1947 The Theory of Inbreeding. Edinburgh: Oliver and Boyd. KEHPTHORNE, O., 1955 The correlations between relatives in inbred populations. Genetics 4 : WRIGHT, SEWALL, 1921 Systems of mating. 11. The effects of inbreeding on the genetic composition of a population. Genetics 6: