Author Institution: Department of Dairy Science and the Institute of Genetics, The Ohio State University, Columbus and Department of Dairy Science, The Ohio Agricultural Experiment Station, Wooste

Fechheimer, N. S.

Laird, R.

Overton, Louise F.

The Ohio Journal of Science

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THE OHIO JOURNAL OF SCIENCEVOL. 58 MAY, 1958 No. 3INHERITANCE OF BLACK HAIR PATTERNS IN CATTLELACKING THE EXTENSION FACTOR FOR BLACK (E).III. A MULTIPLE ALLELIC HYPOTHESIS TOEXPLAIN THE INHERITANCE OF BLACKISHAND BLACKISH PATTERN1*LOUISE F. OVERTON,2 R. LAIRD^ AND N. S. FECHHEIMERDepartment of Dairy Science and the Institute of Genetics, The Ohio State University, ColumbusandDepartment of Dairy Science, The Ohio Agricultural Experiment Station, WoosterINTRODUCTIONBaldwin et al. (1954, 1956) have defined the terms "blackish" and "blackishpattern" and have concluded that the gene for blackish is dominant to its allele,the nonblackish gene. With reference to blackish pattern, Baldwin et al. (1956)concluded that its expressivity depends on the presence of the blackish gene.Further, they confirmed that blackish pattern is sex-influenced, the gene forblackish pattern acting as a dominant to its allele for blackish (nonpattern) inthe male and as a recessive in the female. From their study, Baldwin et al. (1954,1956) suggested that there are two loci involved in the inheritance of blackishand blackish pattern.In the present paper an alternative hypothesis involving only one locus willbe tested. This locus has three alleles, viz., the genes for blackish pattern, blackish,and nonblackish. The nonblackish gene is assumed to be recessive to the gene forblackish and also to the gene for blackish pattern. The blackish pattern gene isagain assumed to be sex-influenced, being dominant to the gene for blackish inthe male and recessive to it in the female. The symbols adopted for the genes areas follows: BsM, blackish pattern; Bs, blackish; and bs, nonblackish. The termi-nology is similar to that suggested by Baldwin et al. (1954, 1956). The lower caseletter b in bs is used because this gene is completely recessive to the other twoalleles; the subscript M in BSM is used because this gene is dominant to Bs in themale and recessive in the female. In the female it is assumed that the homozygousstate of the blackish pattern gene is expressed phenotypically as blackish patternwhile the heterozygous condition is expressed as blackish. The phenotypes of thevarious genotypes are shown in table 1.EXPERIMENTAL PROCEDURE AND RESULTSIn order to test this hypothesis on a given sample, it is necessary to develop amethod of statistical analysis. If a given population is panmictic with respect tothe colors involved and if there are no gene interactions with other loci, we mayuse the method of maximum likelihood to obtain estimates of gene frequency fromjournal Article No. 31-57, The Ohio Agricultural Experiment Station.2Present address: 7842 Broadview Drive, Houston 17, Texas.3Present address: Animal Husbandry Department, West of Scotland Agricultural College,Auchincruive, Ayr, Scotland.THE OHIO JOURNAL OF SCIENCE 58(3): 133, May, 1958.134 LOUISE F. OVERTON, R. LAIRD AND N. S. FECHHEIMER Vol. 58a random sample and to test the hypothesis outlined above. General descriptionsof the maximum likelihood method are presented in Mather (1951) andHogben (1946).TABLE 1The phenotypes of the genotypes of the blackish allelic seriesGenotype PhenotypeMale Femaleblackish pattern blackish patternblackish pattern blackishblackish pattern blackishblackish blackishblackish blackishnonblackish nonblackishDevelopment of FormulaeIn order to apply the maximum likelihood method of estimation to genefrequencies of the present problem, let,ai = the observed number of blackish pattern males,a2 = the observed number of blackish males,a3 = the observed number of nonblackish males,a4 = the observed number of blackish pattern females,a5 = the observed number of blackish females,a6 = the observed number of nonblackish females,i = 1, 2, 3, 4, 5, 6 classes,n =ai+a2+a3+a4+a5+a6, the total number of individuals,m = (ai+a2+as)/n, the proportion of males in the sample,1-m =(a4-|-a5+a,6)/n, the proportion of females in the sample,p = the true frequency of BsM,q =the true frequency of Bs,r =the true frequency of bs,(Note that p+q+r=l.)p = the maximum likelihood estimate of p,q = the maximum likelihood estimate of q,f =the maximum likelihood estimate of r,fi = the probability associated with the ith class,n/, = the expected number of individuals in the ith class.Table 2 shows symbolically the observed numbers and probabilities for the sixclasses.TABLE 2Algebraic symbols for observed numbers and probabilitiesObservedClass number Probability/123456(blackish pattern males)(blackish males)(nonblackish males)(blackish pattern females)(blackish females)(nonblackish females)No. 3 BLACK HAIR PATTERNS IN CATTLE 135Then, the exact probability w of obtaining the observed sample is,A simplified logarithmic likelihood expression L may be derived from T byfactoring, by using the relationship (p-\-q-\-r)2=l, and by omitting constants:The maximum likelihood method requires that L be maximized, by takingthe partial derivatives of L with respect to p, q, and r, setting the resulting threeequations equal to zero, and solving them simultaneously for p, q, and f. Inthe present problem the variables are mutually dependent under the restrictionsuch that the partial derivatives of L with respect to each of the three variablesbecome complicated. The method of Lagrange multipliers provides a tool forsolving such a system. A discussion of this method may be found in Widder (1947),page 113.By the method of Lagrange, we introduce the Lagrange multiplier X and forma new function V, where,Then we may treat p, q, and r as if they were independent variables and we setWe may solve this system of equations simultaneously withWe will thus obtain values of X, p, q, and r. The estimates of p, q, and r will bethe maximum likelihood estimates p, q, and f, which we seek. Since we areinterested in X only as a tool, we may discard it.Following the plan outlined above, we obtain the equations,Eliminating q by substituting q—l — p — r, we obtain136 LOUISE F. OVERTON, R. LAIRD AND N. S. FECHHEIMER Vol. 58(7) minus (6) produces the equation,and (5) minus (6) gives, • •By eliminating fractions in (8) and (9), we may obtain equations of estimationof p and r:and,Since there exists no easy algebraic approach to the simultaneous solution ofthese two equations, they may be solved for any specific example by iteration.A discussion of this iterative method will appear in the examples of the subsequentsection.Test of HypothesisThe data used to test the hypothesis in this study are those of Baldwin et al.(1954, 1956). Only males of 1 year of age and above and females of 2 years ofage and above were used. Animals carrying much white {Iw'lw) were omittedbecause the presence or absence of the blackish pattern on such animals could notbe determined. The animals used in this study were 229 Jerseys (61 males and168 females) and 372 Ayrshires (66 males and 306 females). The numbers ineach of the 6 phenotypic classes are indicated in table 3.TABLE 3Observed phenotypic frequencies of Jerseys and AyrshiresPhenotype Jerseys AyrshiresBlackish pattern males 46 36Blackish males 14 23Nonblackish males 1 7Blackish pattern females 50 51Blackish females 112 200Nonblackish females 6 55Jersey data.—By formulae (10) and (11) the two equations of estimation ofp and r for the Jersey case are,and,No. 3 BLACK HAIR PATTERNS IN CATTLE 137Since there is no easy algebraic method of solution, these equations must besolved simultaneously by iteration. This method involves successive approxima-tions of p and r, until the values of f(p,r) and g(p,r) become as close to zero asthe desired degree of accuracy. Three decimal accuracy was chosen in the presentproblem.In the Jersey case the value of r was narrowed to the range .171 to .173; andp, to the range .531 to .533. Table 4 represents the values of f(p,r) and g(p,r)under the various combinations of p and r in the above ranges.The best simultaneous solution of the two equations is p = .532 and r=.172;therefore, these values are designated as the maximum likelihood estimates p and f.Through the relationship q=l — p — t, we obtain # = .296. It will be noted thatf does not differ appreciably from the value of .19 found by the method of Baldwinet al. (1954).From p, q, and r theoretically expected phenotypic frequencies may be computedby means of the / ; denned in table 2. Table 5 presents observed and calculatednumbers for the six phenotypes.TABLE 4Values o/f(p,r) and g(p,r) corresponding toapproximations for p and r1—123456P.531.532.533.531.532.533.531.532.533r.171.171.171.172.172.172.173.173.173f(P,r).013Oil.008.001- .001- .003-.010- .012- .015TABLE 5g(P,r).107.039- .029.096.029- .039.085.018-.050Quantities from Jersey data necessary for calculation of y}Classi(blackish pattern males)(blackish males)(nonblackish males"I(blackish pattern females)(blackish females)(nonblackish females)Observednumberen46141501126Calculatednumbernfi47.57211.5401.80247.572115.541, 4.973The calculated frequencies were compared with the observed frequencies bythe chi-square test. In the Jersey case the calculated numbers in classes 3 and 6are so small that the two classes were combined for purposes of %2 calculation;therefore, we must consider that we have only five classes. Since two of the genefrequencies were calculated independently (the third by difference from 1), thedegrees of freedom appropriate to the chi-square test are the number of classesminus the number of independent parameters estimated from the data minus138 LOUISE F. OVERTON, R. LAIRD AND N. S. FECHHEIMER Vol. 581, i.e., 5 — 2—1=2. The chi-square value for the Jerseys is .816 for 2 degreesof freedom with a probability of about 96%, indicating that the suggested hypoth-esis fits the data very well.Ayrshire data.—The Ayrshire data in table 3 lead to equations of estimation(14) and (15):f(p,r)=285 r4-347 r2-177 r2/>2+524 r2p-62 £4+124 p^-124 £+62 = 0 (14)and,g(p,r) = 138+138 r4-276 r2-87 r4£+496 r2/>+157 r2p2-177 r ^ - 4 0 9 p-157 £2+1299 p*-U81 £4+310 pb = 0. (15)Solution of these equations gives ^=.378 and f=.400; therefore, # = .222.It will be noted that the value for f is similar to the corresponding value of .42found by Baldwin et al. (1954). Observed and calculated frequencies necessaryfor calculation of x2 are presented in table 6.TABLE 6Quantities necessary for calculation of x2for Ayrshire caseObserved CalculatedClass numbers numbers12345636237512005540.37014.93910.53543.745213.42648.985In this case %2 = 8.795 with 3 degrees of freedom. The suggested hypothesisis therefore rejected at the 5 percent level of significance but accepted at the2 percent level.DISCUSSIONThe above analysis has shown that the proposed hypothesis of gene expressionis accepted in the Jersey case but is rejected at the 5 percent level in the Ayrshirecase. If the genes are the same in the two breeds, an assumption which seemsreasonable, the results are inconclusive regarding the validity of the hypothesis.This may be due to the fact that either or both of the sexes in either or both of thebreeds are not random samples of the population in question. Further samplesmight help to elucidate this point. There is also the possibility of non-randommating, but as far as is known, there is no discrimination for or against blackishpattern or blackish.The results of matings could also distinguish between the present hypothesisand that of Baldwin et al. (1954, 1956). If a mating between a blackish bull and anonblackish cow produced a male blackish pattern offspring, then the multipleallelic hypothesis would be rejected; since, under this hypothesis, the bull wouldbe either BsBs or Bsbs and the cow bsbs, no blackish pattern offspring could resultfrom this mating. Under the hypothesis of Baldwin and his associates the cowcould be bsbs BpM-', when mated with a blackish bull Bs- BpBp she could producea blackish pattern male (BsBs BpMBp). Another distinguishing mating couldbe a blackish pattern cow with a nonblackish bull. By the multiple allelichypothesis only blackish pattern male offspring and blackish female offspringNo. 3 BLACK HAIR PATTERNS IN CATTLE 139can result. On the other hand, under the hypothesis of Baldwin et al. (1954)males may be blackish pattern or nonblackish, while females may be of all threephenotypes. Such matings might be undertaken to test the multiple allelichypothesis.Progeny resulting from matings of the two kinds suggested have not beenobserved for two reasons: 1) The data used for the calculations in this paperwere collected by Baldwin et al. (1954, 1956) for a different purpose and did notinclude information on mating structure; 2) these matings occur relatively infre-quently and therefore resulting progeny are quite scarce.SUMMARY AND CONCLUSIONSA multiple allelic hypothesis is suggested to explain the inheritance of blackishand blackish pattern in Jersey and Ayrshire cattle. The three alleles are blackishpattern (BsM), blackish (Bs) and nonblackish (bs). Blackish pattern is sex-influenced.Sample data were used to test this hypothesis. The hypothesis was accepted—with a high degree of probability—in the Jersey data "but was rejected at the5 percent level of significance in the Ayrshire data. The results for the two breedsare conflicting and are inconclusive regarding the validity of the hypothesis.Matings are suggested for further testing this hypothesis.LITERATURE CITEDBaldwin, C. S., L. O. Gilmore, and N. S. Fechheimer. 1954. Inheritance of black hair patternsin cattle lacking the extension factor for black (E). I. Blackish. Jour. Dairy Sci. 37:1368-1375., , , and D. C. Rife. 1956. Inheritance of black hair patterns in cattlelacking the extension factor for black (E). II. Blackish pattern. Jour. Dairy Sci. 39:435-439.Hogben, L. 1946. An Introduction to Mathematical Genetics. W. W. Norton and Co.,New York. 260 pp.Mather, K. 1951. The Measurement of Linkage in Heredity, 2nd edition. John Wiley andSons, New York. 149 pp.Widder, D. V. 1947. Advanced Calculus. Prentice-Hall, Inc., New York. 432 pp.

Inheritance of Black Hair Patterns in Cattle Lacking the Extension Factor for Black (E.). III, A Multiple Allelic Hypothesis to Explain the Inheritance of Blackish and Blackish Pattern

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Inheritance of Black Hair Patterns in Cattle Lacking the Extension Factor for Black (E.). III, A Multiple Allelic Hypothesis to Explain the Inheritance of Blackish and Blackish Pattern

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