Genetic Drift in Poultry Control Strains

PULLORUM DISEASE

boring bacterium pullorum. J. Med. Res. 27: 481-495. W. R. H., 1928. New England Conference of Laboratory Workers in Bacillary White Diarrhea Control. J. Am. Vet. Med. Assn. 73: 263-264. Schaffer, J. M., A. D. MacDonald, W. J. Hall and H. Bunyea, 1931. A stained antigen for the rapid whole blood test for pullorum disease. J. Am. Vet. Med. Assn. 79: 236-240.

873

Simms, B. T., 1946. Poultry improvement plans aid efficiency of production. Ann. Rep. Chief, BAI: 26-27. Rettger, L. F., W. F. Kirkpatrick and R. E. Jones, 1915. Bacillary white diarrhea of young chicks: its eradication by the elimination of infected breeding stock (Fifth Report). Storrs Agric. Exp. Sta. Bull. 85: 149-167.

Genetic Drift in Poultry Control Strains R.

JARDINE

Department of Agriculture, Melbourne, Australia

G

OWE, Robertson and Latter (1959) have recently discussed the problem of designing poultry control strains so as to minimize, amongst other things, the possible drift of some character under investigation with respect to its mean genetic value. Such genetic drift is an unavoidable consequence of a breeding group being finite in size, for this then enforces some degree of inbreeding and thereby enhances genetic variation. Gowe et al. discuss the relative merits of two breeding systems, random breeding (R) and pedigree breeding (P), in reducing drift and the following paragraphs refer, in addition, to what is here designated as restricted random breeding (RR). This latter is, in fact, very little different from P unless substantial differences in reproductive ability intervene. While Gowe et al. discuss genetic drift from the viewpoint of the gene sampling process involved, the present treatment considers these breeding systems as inbreeding processes. The conclusions reached do not, of course, conflict with those of the above authors, but, since control flocks are a valuable adjunct to random sample laying tests, there may be some advantage in considering this alterna-

tive, more detailed approach to the problem. The breeding systems to be discussed are—• R. Random breeding, where the group consists of M males and Mn females and each individual is the progeny of parents selected wholly at random from the previous generation. RR. Restricted random breeding, where the group consists of M males each of which is mated to a specific set of n females, 1 male and n female progeny then being selected from each of the M such sire groups. This is equivalent to M separate random matings, as in R above, with random reconstitution of the M such sire groups. This is equivalent P. Pedigree breeding, where the group is as in RR above, except that the n female progeny from any sire group are selected so that each dam contributes to one, and only to one, of those progeny. This differs from RR in that female progeny cannot be full sib to each other as they may in the latter system. Suppose, now, that coefficients of inbreeding (F) and relationship (G) are defined as by Malecot (1948) {i.e., as probabilities of appropriate corresponding loci

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(Received for publication October 6, 1959)

874

R. JAEDINE

being derived from some single locus at a previous meiotic division) and that in generation O F° = G° = 0 for all individuals. Suppose, further, that any generation pxy°, pxy1, Pxy2 denote the probabilities of any two individuals selected at random from sex groups x, y {i.e., x may refer to the m(ale) or f(emale) group and y likewise) having 0, 1, 2 parents in common. Under the P system the actual relative frequencies in the group are equal to these probabilities, while, in the others where a random choice of parents is involved, frequencies

[G*]=[

P t

=

p t _ r ;

(jXy = axy (jrmm

+ (? t 1

t-1
*• "xy) *Jf f

t 2

+iG m £ - +b x y G m f - +b x y . For the R system, where G m m t =Gff t =G m f *=G t , the above relation simplifies to Gt=(l-2b)Gt-1+bG'-2+b. In more explicit terms one has, for the above systems, the following matrix equations—

(4Mn-n-l) (n+1) (n+1) 4Mn 8Mn 8Mn .

G t-2

• 1 1 1 0 — 0 4 4 r o t-i 2 (n2-l) (M-l)n (Mn2-2n+l) 1 (n2-l) G„* RR= 4(Mn-l) 4(Mn-l)n 2 8(Mn-l)n 8(Mn-l)n _ Gmf*. (M-l) (Mn-1) 1 (n+1) (n+1) 2 8Mn 8Mn J 4Mn . 4M •

G * GH* G mf *

1

4 (M-l)n

1 — 4

1 — 2 1

0

and probabilities only agree on the average. Defining aXy=(Pxy°)/4 bxy=(Pxy1+2Pxy2)/8

it may be noted that these coefficients, being dependent upon the p xy are determined by the breeding system and are constant from one generation to the next. Then, for a randomly selected pair of individuals of generation t from sex groups x, y the follow-

1

0

1 (n-D 4(Mn-l) 4 2 8(Mn-l) 8(Mn-l) (M-l) (Mn-1) 1 (n+1) (n+1) 4M 4Mn 2 8Mn 8Mn J (n-1)

mm

GH*-1

Gf,'-1 Gn,,'-1

G mf '- 2 1

together with initial conditions Gxy° = 0, GXyx = bxy, where the appropriate b xy is given by the final column of an above matrix. Then, from the values of M and n determining the size and composition of the breeding group, the coefficients of relationship may be calculated successively so as to yield, finally, those pertaining to generation t. The effect of these inbreeding processes

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R

ing recurrence relations hold—

875

GENETIC DRIFT

may now be assessed by assuming the usual independent factor/noninteraction genetic model (no linkage, dominance, epistasis, genetic-environment interaction), each individual being considered as a member of an infinite hypothetical population in which phenotypic value y (with respect to some character) is given by— y=y+gi+gi'+g2+g2/+ • • •

+gk+g/+e.

Vg.

Then, firstly, for a metric character such as egg production, the mean genetic value of Mn females will have an expected value

IAV

=[

(y±2-v/Vo) pertaining to generation 0. The difference between these limits then indicates the extent of possible genetic drift due to unavoidable inbreeding. Alternatively, the finite group of generation 0 may be considered as the population from whose mean, y, the possible drift is required, and, since this is formally equivalent to the case where Mn is infinitely large and the coefficients have values as above, that drift is given by, 2V2G„*-VE. Secondly, for the allele A of an arbitrary factor, the gene frequency in the group of M(n + 1) individuals will have an expected value q, and in generation 0 a variance, V0=[l/2M(n+l)]-q(l-q). By generation t, setting and FAV*= G„

( M - 1 ) G m m *+n(Mn-1) G ff t +2MnG mf t (n+l)[M(n+l)-l]

y and in generation 0, since Ff° =G f f 0 = 0, a variance, V 0 =(l/Mn)-V«. By generation t, F f *(= G.nf'-1) and G,f* are as calculated above, so that, rl+Gmft-1-2Gfft 1 Vt = ™ + 2 G « 4 •v« Mn and the actual mean genetic value is expected to lie within approximate 95% limits (y+2VV0,

]•

one has,

V

r l + FAvt-2GAvt

-[

1

IMM-I) + H - 1 C - *

Genetic drift is again indicated by the difference between limits formulated as previously, and, for the case where the finite group of generation 0 is regarded as the relevant population (with gene frequency q), that drift is given by, 2VGAT'-q(l-q). A final point to be noted is that the treatment of the inbreeding processes given here involves theoretically calculated val-

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The population mean is y; g, and g / denote random variables describing genetic variation due to the i th factor (which comprehends 2 alleles ai and A t , these being represented in the population by genes of frequencies pi and q i; respectively); and e a random error variable. These random variables are all of mean zero, uncorrected (except as genetic relationship intervenes) and total genetic variation is denoted by

rather than the limits

876

R. JARDINE

ues for coefficients of relationship which, in any actual finite population, may only be realised approximately. Moreover, differences in reproductive ability could produce deviations substantially greater than those due to chance and it is evident that, ideally, the whole complex pattern of relationship should be recorded, and the necessary coefficients calculated for that pattern. This is not, of course, a practicable proposition, but, nevertheless, if the breeding system does not conform with reason-

able accuracy to the theoretical ideal, then observation must, in some manner, provide estimates of the actual coefficients of relationship pertaining to the relevant generations. REFERENCES Gowe, R. S., A. Robertson and B. D. H. Latter, 1959. Environment and poultry breeding problems. 5. The design of poultry control strains. Poultry Sci. 38:462-471. Malecot, G., 1948. Les Mathematiques de l'Heredite. Masson et cie. Paris.

LLOYD R. CHAMPION

Department of Poultry Science, Michigan State University, East Lansing (Received for publication October 7, 1959) INTRODUCTION

I

T HAS been theorized that the effects of deleterious sex-linked genes may be responsible for differential embryonic mortality between the sexes in mammals (Lenz, 1923; Huxley, 1924). Since the female is the heterogametic sex in Aves, the expectation under this hypothesis is that greater female embryonic mortality would occur. Later, Riddle (1931) proposed the metabolic theory of sex. According to this theory, males are more apt to suffer a greater embryonic mortality rate than females. The viewpoints of Landauer and Landauer (1931) and MacArthur and Baillie (1932) support this latter theory. In chickens, Mussehl (1924), Lambert and Knox (1926), MacArthur and Baillie (1932), Dudley and Hindhaugh (1939), and others, observed sex ratios2 which favored males in some stocks, whereas Cal1

Published with the approval of the Director as Journal Article No. 2S09 from the Michigan Agricultural Experiment Station. 2 The sex ratio is usually presented as one figure, the percentage of males.

lenbach (1929), Byerly and Jull (1935), and Hazel and Lamoreux (1946) found slight deficiencies of male chicks at hatching in other stocks. Landauer and Landauer (1931) combined the figures reported in 14 different publications and came up with a secondary sex ratio3 of 48.77. The proportion of male chicks was found to be higher in Rhode Island Red chicks than in White Leghorn chicks by Callenbach (1929), Byerly and Jull (193S) and Crew (1938), while the sex ratio was found to be higher in White Leghorns than in Rhode Island Reds by Hays (1945) and Wilcox (1959). Mussehl (1924) found that different sires produced progeny which differed significantly in sex ratio, and Christie and Wriedt (1930) observed that some dams produced aberrant sex ratios. Further, Dudley and Hindhaugh (1939) reported strain differences in secondary sex ratios. On the other hand, Hazel and Lamoreux (1946) Japanese vent-sexed 8,355 chicks from 464 full-sib families and 3

The proportion of males at hatching.

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Sex Ratios in Two Strains of Leghorns and Their Reciprocal Crosses1