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2 NBREEDNG N MENDELAN POPULATONS WTH SPECAL REFERENCE TO HUMAN COUSN MARRAGE BY J. B. S. HALDANE AND PEARL MOSHNSKY N most human populations matings between close relatives are fairly rare, some types being forbidden by law. t has been pointed out by many authors that a result of such matings is an increase of homozygosis, and in particular a great increase in the proportion of rare recessives. Among the principal theoretical discussions are those of Wright (1922), Lenz (1919), Weinberg (1909), Dahlberg (1929, 1938), and Wahlund (1928). However, none of these authors has discussed sex-linked genes, nor certain general aspects of the problem for autosomal genes. The practical importance of the question may be gauged from the work of Sjogren (1931), who investigated Swedish cases of juvenile amaurotic family idiocy, a clear-cut recessive character. He found that the parents of 15.3 % of his cases were first cousins, of 3.4 % first cousins once removed, and of 6.8 % second cousins. The following assumptions will be made, though some will later be removed: (1) The population is so large that fluctuations can be neglected. (2) No selection or mutation occurs. (3) Mating is at random except where specified. That is to say, the relationship of two mates can be completely specified in finite terms. Now consider a woman W who produces a gamete carrying the gene a. f its frequency and that of its allelomorph A are known, we ask, in the case of a given relationship, what is the probability that a gamete produced by the woman s husband or mate H will also carry a, if he is a relative. n the case of completely sex-linked genes, H will only carry such genes in his X-chromosome. n that of partially sex-linked genes, the probabilities must be specified separately for the X- and Y-chromosomes. AUTOSOMAL GENES Consider a population in which the gene-pair A, a, occurs with frequencies qa, pa. Wright (1922) first clearly stated that for every relationship the probability that an a gamete of W will be fertilized by an a gamete of H is p + fq, where f is the coefficient of inbreeding of their progeny. However, Wright was mainly concerned with the special case p = i, and employed the notion of correlation, whereas an entirely elementary proof is possible. We can express the relationship between two individuals W and H by the number m of steps in each path of relationship connecting them. Each path passes from W to H

3 322 NBREEDNG N MENDELAN POPULATONS through a latest common ancestor, unless one is an ancestor of the other. A step is the relation of parent and offspring. Thus a parent and child are connected by a path of one step, a grandparent and grandchild or a half-brother and half-sister by a path of two steps, and so on. n general there are several paths of relationship. n human pedigrees these usually occur in pairs of equal paths, owing to the practice of monogamy. Thus an uncle and niece are connected by two paths of three steps each. Where there are several paths they may coincide to a greater or less extent. n particular, if the latest common ancestor is inbred, and is thus not a random sample of the population, an extra path (or paths) of relationship runs through him or her and the latest common ancestor of his or her parents. But since one generation of outbreeding wipes out the effects of inbreeding as measured by homozygosis, such extra paths only occur when the parents of the latest common ancestor are related, and not when his grandparents or other ancestors are related. We now have the following theorem : f paths are specified as above, then if W and H are connected by paths of lengths m,, m,,..., m,,... steps, the probability that an a gamete of W will be fertilized by an a gamete of H is p + fq, where f = $x 2-mr. n particular, if p = 0, i.e. the gene is very rare, the probability is f. The proof follows. t is not, of course, rigorous. A rigorous proof would demand a set of axioms such as those developed by Woodger (1937). First consider a single path of relationship. Since W is not inbred, the probabilities that she is Aa and au are q and p respectively. That is to say, the gametes of a group of W s are BqA, (p+ Bq) a. Now consider any group of individuals whose gametes are (1- f) 4.4, (p + fq) a. f they mate with random members of the population, their progeny are Thus the gametes of their offspring are (l-&f)qa, (p+$fq) a. Similarly, the gametes which produced them occurred with frequency (1 - f) qa, (p + fq) a. But since the genes in the homologous chromosomes of their parents were in the frequency qa, pa, the total gametes of their parents were ( 1 -if) qa, (p + Bfq) a. Thus a step in either direction halves the value off. And since f = + when H = W (self-fertilization), it is clearly 2-m-1 for a path of m steps. Now consider a case of double relationship, the paths being of lengths m, and m,. Thus W and H in Fig. 1 are half-first cousins, and also half-first cousins once removed, m, = 4, m2 = 5. The values off for the parents of H are 2-mi and 2-mz. That is to say, the gametic outputs of a group of such parents are (1-2-ml) qa, (p- 2-mlq) a, and (1-2-m~) qa, (p- 2-m~q) a. Thus the zygotic frequencies of H are (1-2-%) (1-2-rnz) q2aa, And the gametes of H are therefore (1 -f 1 q2aa [fq2 + (2 -f 1 PPl Aa, (P2 +.fp4) aa. [(2-22-ml-2-m2 (p + 2-9) (p + 2 -y) au. ) pq + (2-mi + 2-mz - 21-mi-mz)l q2aa, (1-2-m m2-1 ) PA, [p+ (2-m1-1+2-mZ-1 ) P a. r

4 1- J. B. S. HALDANE AND PEARL MOSHNSKY 32 3 Hence f = &(2-m1+ 2-mz). The argument is precisely similar if the paths meet in some other individual than W or H, and if there are more than two paths. n case W, H, or one of their ancestors is inbred, as in the pedigree of Fig. 2, the frequency of zygotic genotypes is altered. However, since inbreeding of a group does not affect the frequency of genes within it, the argument above is unaffected. Thus in Fig. 2, where W and H are connected by four paths, two passing through each of their latest common ancestors, we have m, = m2 = m3 = m, = 6. Thus f = 4 x 4 x 2-6 =.F5. Applying this formula to a number of relationships, we have the results of Table, most of which have already been published by others. t may be remarked that f = 1 in the case of haploid parthenogenesis followed by doubling, such as was demonstrated by O i QW Fig. 1. f d 8 X x Q 1? i Fig. 2. Nabours (1919) in Apotettix. Here all the progeny are homozygotes. On the other hand, in clonal parthenogenesis all the progeny of a heterozygote are heterozygotes, so f = 0. The probability that one of W s gametes should carry a is p. The probability in this case that one of H s should carry a is p + fq. Hence the frequency of aa progeny from W x H unions is p2 + fpq. And the array of expected progeny from such unions is (q2 +fpd AA, 2(1 -f) P@, (P2 +fpd aa. d Table. Coeficients of autosomal inbreeding Relationship of H to W ~ Self Father, son, whole brother Grandfather, grandson, half-brother, uncle, nephew, double-first cousin First cousin, half-uncle, half-nephew First cousin once removed, half-first cousin Second cousin rth cousin s times removed f 3 t 8 1 li 3 i 24 +r-s-2

5 324 NBREEDNG N MENDELAN POPULATONS n particular, if a is recessive the frequency of recessives among the progeny of relatives is p + fpq, or f p + (1 - f) p2. t may be noted that the existing law of England forbids certain unions of relatives. Now such unions fall into various classes as regards the frequency with which rare recessives may be expected in their progeny. And the evil effects of inbreeding are one reason, though not the only one, why certain unions are prohibited. A community which based its marriage laws on eugenical considerations would certainly not adopt the law of England, or any other existing state. Thus in England the union of double first cousins is legal, that of uncle and niece illegal but not criminal, and that of half-brother and half-sister criminal. But from the eugenic point of view each appears to be equally dangerous. f W is a homozygote, say aa, it is clear that her parents each produced an a gamete. Thus the number of steps in each path is reduced by unity, and the probability that one of H s gametes should carry a is p + 2fq. The case is similar if H is aa. So the frequency of recessives among the progeny of relatives, one of whom is a recessive, is p+2fq, or 2f + (1-2f) p. Thus slightly over & of the progeny of albinos who marry their first cousins should be albinos. SEX-LNKED GENES t is assumed throughout that the male is heterogametic, as in man. Where the female is so, the alterations to be made are obvious. t is clear that inbreeding can only affect the daughters of consanguineous unions, since the sons do not depend on their father s genotype except in cases of non-disjunction. The conventions for reckoning m, the number of steps in a path of relationships, are as follows, where p is the frequency of a sex-linked recessive gene s : 1. For the step father-daughter or mother-daughter in either direction add 1 to m. 2. For the step mother-son add 0 to m. 3. For the step father-son add 00 to m. That is to say, paths involving this step are omitted. 4. f the latest common ancestor on any path of relationship is a male, subtract 1 from m. Given these conventions we have the following results : 5. f =&cz-m. r 6. The frequency of recessive males to be expected from any union of relatives (if no information concerning their genotypes is available) is p. 7. The frequency of recessive females expected is p2+flpq. 8. The frequency of recessives, both male and female, expected from unions of recessive fathers with relatives is p +frq. Rule 3 is obvious. The fact that a father carries s does not alter the probability p that his son should do so, nor conversely, provided that the mother is unrelated. t is also clear that for every s gene in a father or mother there are, on an average, (p+bq) s genes

6 ... J. B. S. HALDANE AND PEARL MOSHNSKY 325 in their daughters, and conversely, whilst the frequency of s genes in a group of mothers and their sons is the same. Hence rules 1 and 2 follow. To derive rule 4 we note that a male can only be a latest common ancestor in an effective path if the path runs through two of his daughters. Consider the case of the daughters of a male by two different females. f one daughter has a s gene, she derived it from her father in one-half of all cases, that is to say, her sister must have it. Thus the relationship between these two half-sisters is as close as between mother and daughter, that is to say, it involves an addition of 1 to m. The results of Table 1 follow without difficulty, and it can be extended if desired. No figures are given for double first cousins, for in each of the two possible types two paths of relationship are through H s father, and thus irrelevant,. Hence in the type where two brothers marry two sisters, their children are effectively first cousins of type A (f =A), Table 1 Relationship of H to W Father, son Sister s son Whole brother, maternal half-brother, mother s father, daughter s son, father s brother or maternal half-brother, paternal half-sister s son Mother s sister s son (first cousin A) Mother s brother or maternal half-brother, maternal half-sister s son, father s sister s son (first cousin B), mother s paternal half-sister s son (half-first cousin), father s maternal half-sister s son (half-first cousin) Mother s maternal half-sister s son (half-first cousin) Through H s father only Father s paternal half-sister s son (half-first cousin) t 1 T -~ 10 1 s L lfl 0 0 and where a brother and sister marry a brother and sister their children are effectively first cousins of type B (f = t). t is striking that, from this point of view, a man is more closely related to his maternal aunt than to his sister, or a woman to her sister s son than to her brother. The results for fist cousins of the two types shown in Fig. 3 are the most readily tested, and alternative proofs will be given for them. t will be sufficient to prove the values off in the limiting case of a very rare gene, where p is approximately zero, but H is known to be recessive. n this case the frequency of recessives of both sexes is f. Consider the union of type A (Fig. 3). H is s. Hence his mother was 8s. n half of all cases she got s from her father, so W s mother was certainly Ss. n half of all cases she got s from her mother, so W s mother was as likely as not to be 8s. Thus the probability that W s mother was Ss is 2, and the probability that W was Ss is g. Hence & of the gametes of a. group of W s are s, and f =A. Similarly, in a type B union, H s mother was Ss. n half of all cases she got s from her father. f so W is not 8s. n half of all cases she got s from her mother, and in half of these W s father was s. That is to say, a of all cases W s father was s, and W is 8s. Thus Q of the gametes of a group of W s are s, and f = +. t may be remarked that in all cases both for sex-linked and autosomal genes, this is the easiest method of obtaining the value off or f.

7 326 NBREEDNG N MENDELAN POPULATONS t is worth noting that in some primitive societies cousins of type A would be classed with the father, and marriage with them would be forbidden, whilst cousins of type B would be legitimate spouses. n an investigation carried out for another purpose, ninety-six English first-cousin marriages were classified into the four possible types. Ore1 (1932) has published data on 830 first-cousin marriages in his investigation on consanguineous marriages in the archdiocese of Vienna. The numbers are given in Table 111. The excess of marriages of the types leading to the production of daughters homozygous for sex-linked recessives is clearly significant in Orel s material, but the difference between the numbers of types A and B is barely so. The differences may be partly due to the fact that women are less mobile than men. A Fig. 3. B Table 111. Prequency of Jirst cousin marriages Relationship of husband to wife A. Mother s sister s son B. Father s sister s son C. Mother s brother s son D. Father s brother s son Unknown England -~ : 274 2; Austria NBREEDNG AND HUMAN SEX-LNKED RECESSVES We have collected in Table V from the pedigrees of haemophilia and colour-blindness in the Treasury of Human nheritance (Bulloch & Fildes, 1911; Bell, 1926) all those cases in which an affected male married a relative whose relation to him is specified. Children dying in infancy have been omitted in the case of colour-blindness. The haemophilic woman of pedigree 493 is of course doubtful. The expected number in the last column is f + (1 - f ) p, where p is the gene frequency. However, a complication arises in the case of colour-blindness, where two alleles are concerned whose frequencies are about 0.01 and (see Bell & Haldane, 1937). Calling these frequencies p, and p$ the probability that

8 ~ ~ ~ ~ -_ ~ ~ J. B. S. HALDANE AND PEARL MOSHNSKY 327 a normal mother will have a colour-blind son is pl+pz or Protanopes OCCUT with frequency pl) and a fraction p1 of their daughters by normal unrelated women are protanopes. Similarly for deuteranopes. Hence the probability that a colour-blind man of unspecified type should beget a colour-blind daughter is P +P, ~ or Pl + Pz Similarly, if a protanopic man marries a relative, the probabilities of one of her gametes being protanopic, deuteranopic, or normal are f + (1 -f ) pl, (1 -f ) (1 -pl) p,, and (1 -j ) (1 -pl) (1 -pz) respectively. Hence f + (1 -f ) (pl +pz -plp2) of his sons, and f + (1 -f ) p1 of his daughters should be colour-blind. n the case of a colour-blind man of unspecified type the corresponding probabilities are f + (1 -f ) (pl +p2 -plpz), or f (1 -f ), and P: + P? f + PlfPZ Pedigree (1 -f ), or f ( 1 -f ). The last column is calculated on this basis. Table V. Offspring of men with sex-linked recessive abnormalities who married relatives Disease Haemophilia 9, Colour- blindness,>,, Colour-blindness 39,, Relation First cousin A 9, 9, First cousin 3 Half-first cousin Half-first cousin once removedl Second cousin once removed First cousin D First cousin C Second cousin Progeny Normal Affected 1 f 3? H was also W s mother s paternal half-brother Total affected? ? EXpected affected ? All but one of the affected progeny are from wives related to their husbands in such a way that f is not zero. The one exceptional daughter may be explained by a not very remote chance, or by the fact that the parents were doubly related. For many of the examples of inbreeding come from rural or other communities where inbreeding is high, and hence double relationship frequent. Thus one of the two related wives of colourblind men in pedigree 419 was both her husband s half-niece and his half-second cousin once removed. The total affected ( out of 31) is well above the expectation in the cases where.f is not zero. This, however, need not surprise us. There can be litte doubt that a pedigree 0

9 328 NBREEDNG N MENDELAN POPULATONS where a colour-blind or haemophilic man has one or more similarly affected progeny is much more likely to be reported than one where this unusual event does not occur. t is possible that one or two of the sons may be explicable by non-disjunction (Levit & Serebrovski, 1929; cf. Haldane, 1932). To sum up, the results are in qualitative agreement with expectation, and the quantitative agreement is probably as good as could be expected. NCOMPLETELY SEX-LNKED GENES f the suggestion of Haldane (1936) is confirmed that genes exist in the homologous segment of the X- and Y-chromosomes, their behaviour in connexion with inbreeding is of interest. n any case the calculation holds for Lebistes and other fish with incomplete sex-linkage. Consider a gene-pair and i occurring with frequencies q and p, and so located that the frequency of recombination with the differential segments is c. That is to say, X X Xi the gametes of an -; $2 are $X, ixi, of an - 6, agx, frcxi, &Y, igyi, and of an - $ X% Yi Y $cx, $gxi, +gy, &Yi, where g = 1 -c. The calculation is decidedly more complicated, but is of interest if only because the expressions dealing with autosomal and ordinary sex-linked genes are both special cases of those for incompletely sex-linked genes. f W is known to have produced a gamete carrying i, we can calculate the probabilities that H will carry i in his somatic X- and Y -chromosomes respectively, and later the probabilities that one of his gametes carrying X and Y respectively will carry i. These are not identical, on account of crossing-over. The probabilities of finding i in H s somat,ic X and Y will be called h, and h,, those of finding them in his gametic X and Y will be called f (c) and #(c). These are coefficients of inbreeding. When c = $, the gene becomes effectively autosomal. That is to say,,f($) = #(+) =f. When c = 0 the gene becomes an ordinary sex-linked gene; so f(0) =f, #(O) = 0. Since the probabilities of a somatic X gene being found in a gametic X and Y are g and c respectively, and similarly for a somatic Y gene, it is clear that f(c) = gh, + chu, #(c) = C h, + gh,. To calculate hz and h, we have to consider eighteen possible steps. Parent to offspring QX to QX 3 qx to gx $ QX to $Y 0 to QX 9 $X to $X 0 $X to SY c $Y to qx c $Y to $X 0 $Yto $Y 9 Offspring to parent qx to qx $?X to gx &g QX to $Y &c $X to qx 1 gx to $X 0 gx to 6Y 0 $Y to qx 0 $Y to $X c 6Yto $Y 9

10 J. B. S. HALDANE AND PEARL MOSHNSKY 329 The entries are to be interpreted as shown by the following examples.?x to $Y (parent to offspring) = O, means that if a mother has a gene in her X-chromosome, the chance of its being found in her son s somatic Y as a result of inheritance is zero.?x to $Y (offspring to parent) = 4, means that if a daughter has a gene in one of her X-chromosomes the probability that it is derived from her father s Y is &. To simplify calculations we give the corresponding step coefficients for whole sibs. gx to?x p-cg $X to $Y 0 gx to gx & gy to gx 2cg gx to $Y cg $Y to $X 0 $X to?x 4 &Y to $Y 1-2cg $X to $X 8 The following is an example of the calculation. For the paths?x to?x there are three pairs of steps, by way of QX, $X, and $Y. Hence the total coefficient for the paths is 4. B + +g. g + ic. c = $ - cg. We can now calculate Table V, and other coefficients, e.g. for the eight different types of half-first cousin, may be calculated. The total progeny from a group of W x H is clearly [a2 +f(c) Pal 11, [1-2f (41 Pdi7 b2 +fw Prrl ii?, [q2 + #(c) Prr1 11, [- 2#(c)l Pdi7 [P ii 6. The frequencies with which various types of union occur may also be calculated from Table V7 provided only one parent of W is related to only one parent Table V. nbreeding coeflcients for incompletely sex-linked genes Relationship ofhto W Father Son Whole brother Maternal half-brother Paternal half-brother Mother s father Father s father Daughter s son Son s son Maternal uncle Paternal uncle Sister s son Brother s son First cousin A First cousin B First cousin C First cousin D Double first cousin AD Double first cousin BC QC + icg2 pcg - c2g2 tc - W g ca3

11 330 NBREEDNG N MENDELAN POPULATONS of H. f this is not so either H or W is inbred, which naturally alters the frequencies with which genotypes occur, or the probabilities represented by h, and h, are not independent, so that, for example, if W is ii the probability that H should be ii is greater than (p + hxq) (p + h,q). However, these difficulties do not occur in the case of ordinary ix first cousins. For types A and B the frequency of i x - unions, when p is very small, Y are P - (3-4cg) and P - ( 1 + 4c2) respectively, the expected frequencies of ii in a segregating 8 4 sibship being gg among the daughters and $c among the sons. While for types of C and D the frequencies are P - c( and P - cg(3-4cg) respectively, the frequencies being +c 4 2 among the daughters and +g among the sons. Thus the sex ratio of recessives should be if all types are equally frequent, and greater if A and B are commoner than C and D, as was the case in Vienna. For c = t this ratio is only 1.116, while for c =A it rises to 2.049, and for c = to 221. Thus for ordinary incompletely sex-linked genes in man there should be a moderate excess of affected daughters among the progeny of cousin marriages. But for genes like bobbed in Drosophila where c is very small, inbreeding should give a huge excess of affected daughters. THE MEAN COEFFCENTS OF NBREEDNG Bernstein (1930) defined a value a as follows. Let p be the frequency of the gene a, p' the probability that a gamete carrying a will fuse with another carrying a. Let q, q' be the corresponding values for its allelomorph A. Then the frequency of Aa zygotes can be expressed as 2p(l -p') or 2q(l-4'). So these expressions must be equal. Let 1 -p' 1 -q' -=---- q P = 1-a. Then p' = p + olq, q' = q + ap, and the population occurs in the proportions (!2 + app), 20 -a) pqaa, (p2 + apq) aa. t is at once clear that cc is the mean value of Wright's coefficient f, averaged over all matings, weighted by their fertility. Hence, as Bernstein assumed, but did not prove, it is the same for all pairs of autosomal genes. Clearly a = 0 if mating is at random, and a = 1 for complete inbreeding, that is to say, for any system of inbreeding which ultimately leads to homozygosis. t can readily be calculated for any population where the frequencies of various kinds of mating are known. Examples are given later. n the same way we can define a coefficient a' which is the weighted mean of the coefficients f' of sex-linked inbreeding. Clearly it only influences the distribution of female

12 J. B. 8. HALDANE AND PEARL MOSHNSKY 331 genotypes. 111 most cases a' is of the same order of magnitude as a. For example, if the four types of first-cousin marriage each occur with frequency A, they make a contribution of i A to a, and of,5,a to a'. n the same way mean values off(c) and $(c) may be calculated in the case of an incompletely sex-linked gene. A real difficulty arises in practice for the following reason. We have been considering unions within a homogeneous population, that is to say, one in which the gene frequencies of different fractions, picked on a basis of locality or occupation, differ as the result of sampling only. Such a population may be due to race mixture, provided that after a certain epoch mating is at random in so far as concerns the genes in respect of which the races differ. Bernstein (1930) showed that a number of populations were effectively homogeneous as regards the blood group genes. Actually, no large human population can be regarded as fully homogeneous. The differences may be due on the one hand to the fact that different partially endogamous groups differ in respect to their racial origins and the selective influences to which they have been exposed, and on the other hand to recent mutations. So unless we can refer back to a homogeneous population in the not too remote past, the evaluation of coefficients of inbreeding becomes quite artificial. For example, if all the human race were descended from Noah and his wife, they would be homozygous for a large fraction of the genes responsible for variation in antediluvian man. And in fact it is likely that existing European populations trace much of their ancestry to very small and highly inbred tribes of the glacial periods. f this is so a may approach unity for all modern European unions if we allow for common ancestors in the sufficiently remote past. To do so would, of course, be entirely incorrect if we want to use a for any practical purpose. And in particular if we are concerned with rare recessive genes we may be fairly sure that most of them have originated since the ice age (see Haldane, 1939). The relevant fractions of the mean coefficients a and a' can be roughly estimated in several populations. The largest set of data known to us is that of Ore1 (1 932) based on the catholic population of the archdiocese of Vienna, in which dispensations had to be obtained from the church for marriage of near relatives. Orel's data, based on records of 117,431 marriages, give 1 a= ,431 (&~ '-x ll+j-~796+&~3+&~ &~ ). so a > 0~ Of this value 72 o/o is contributed by marriages of first cousins, which amount to 0.68 yo of all marriages. n the above sum we have not included marriages of relatives more remote than second cousins, of which a few are recorded, as such records are obviously incomplete. An investigation now being conducted by the Medical Research Council on English hospital patients, which we are very kindly permitted to quote, leads to a figure of rather over 0.6% for English hospital patients. Thus the inbreeding is of the same order of magnitude in English towns and the Vienna. archdiocese. EUGENCS X, V 22

13 332 NBREEDNG N MENDELAN POPULATONS Dahlberg (1938) gives figures for cousin marriages in Prussia, Bavaria, and France. Between 1875 and 1926 the percentage of first cousin marriages in Prussia fell from 0.71 to Between 1876 and 1933 it fell from 0.87 to 0.20 in Bavaria. But in France the figure remained steady about 1 yo, falling in Paris but rising in other districts. Hence Orel s figures are probably fairly representative for Western Europe. Wulz s (1 925) data for a Bavarian catholic peasant population, which are limited in the same way as Orel s, give a > n these populations it may reasonably be assumed that the contribution of recent marriages to the value of a does not greatly exceed Orel s data are less adequate for an estimation of a, as some of the classified cousins were double or half-first cousins, both first cousins and second cousins, and so on, and full details are not always given. Neglecting these complications, we find that the first cousins contribution to a is 1.56 times their contribution to a. t would thus seem that a is somewhat larger than a but of the same order of magnitude. Besides these, a number of highly inbred populations (all non-catholic) have been investigated. Spindler s (1922) data for Wurtemberg peasants, based on 435 marriages, give CL > Auf der Nollenburg s (1932) records of an isolated protestant community in the Rhineland during the years , based on 376 marriages, give a>0*0050. Finally Reutlinger s (1922) data on Jews in Hohenzollern, based on only 117 marriages, gave a > The total values of a due to recent marriages may be as high as 0.05 in such aberrant populations, and the intense inbreeding and therefore high homozygosity may account for some at least of the genetical differences which appear to exist between Jews and other European populations. THE GRAPHCAL REPRESENTATON OF A POPULATON de Finetti (1927) was the first to use homogeneous co-ordinates to represent a population. f we are concerned with the absolute numbers of n different genotypes we naturally represent a population by a point in n-dimensional space, and use Cartesian co-ordinates. But if we are concerned with the frequencies of n genotypes we take our representative point within a regular simplex in (n - 1)-dimensional space, and use the perpendiculars on its n bounding flats as co-ordinates whose sum is unity. n particular, if we are dealing with a single gene pair we can represent any population by a point P in an equilateral lxiangle XYZ (Fig. 4), where x, y, z are the perpendiculars from P on YZ, ZX, XY, respectively, and x + y + z = 1. Then if the population consists of ZAA, yau, xua, P represents the population. Further, p = x + iy. Thus the representative points of populations with a given gene ratio lie on the straight line PQ, where Q is the projection of P on ZX, whose equations are x + $y = p, or 2qx + (4-p) y - 2pz = 0. Further, if the gene pair is sex-linked, P represents the female sex and Q the male sex.

14 J. B. S. HALDANE AND PEARL MOSHNSKY 333 f mating is at random, x = p2, y = 2pq, x = q2, so as de Finetti pointed out, the points representative of random mating populations lie on the parabola y2-422 = 0, whose vertex is the mid-point of the perpendicular from Y on ZX, and which touches XY and YZ and X and Z respectively. f a group of the population is derived from the unions of Fig. 4. The point P represents a population consisting of i ~ $Aa,. &AA. These are the lengths x, y, z of the perpendiculars from P on YZ, ZX, and XY respectively, the unit of length being the altitude of X YZ. Q represents the gene frequency. QZ = p = f, the unit of length being XZ. The parabola through P is y2-4x2 = 0, the locus of representative points of random mating populations. The parabola through P' is y2-4xz + 2g = 0, the locus of representative points of children of first cousins. Thus P and P' represent 7 the frequencies of the genotypes determined by the same autosomal gene pair, according as the parents are unrelated or are first cousins. Note that the parabola through P' intersects XY and YZ, whilst that through P touches them. relatives of a given type, x = p2 + fpq, y = 2( 1 - f) pq, x = q2 + fpq. Hence 4x2 - y2 = 2fY 1-f' or 4x2 - y2 =f(2x + y) (2z + y). This is the equation of a parabola whose axis is the perpendicular from Y on ZX, and which passes through X and Z, but of which all the points in the triangle lie inside the parabola y2-4x2 = 0. Similarly, for a population whose mean 22-2

15 334 NBREEDNG N MENDELAN POPULATONS 2aY coefficient of inbreeding is a, 4x2- y2 =. n the case of incomplete sex-linkage the 1-a male and female sexes are represented by points on different parabolae but the same perpendicular. We may regard a change in the mating system as a force which compels P to move vertically. Similarly, if the mating system is unaltered, P is constrained to move along one of the parabolae, and mutation and selection may be represented by forces acting on it. Whilst mutation alters the gene ratio, and may be regarded as a force acting horizontally, selection may act in any direction (though in general varying with the gene ratio). And it will cause P to move either from X to 2 or Z to X, or else to or from a point of stable or unstable equilibrium on the locus. The locus will not be exactly followed unless mating is at random or selection slow, since a population does not at once reach equilibrium under inbreeding when the gene ratio is changed. SYSTEMS OF MATNG Before calculating the values of a and a' for certain mating systems, e.g. for a population where 10% of all matings are between brother and sister, we first prove two theorems which are true for all populations in equilibrium, and in which there is no selection or mutation. The absence of selection implies that each genotype mates with equal frequency, and that all types of mating are equally fertile. Consider an autosomal gene-pair Aa. Let x, y, z be the frequencies of aa, Aa, and AA zygotes, and let the six mating types occur with frequencies b(aa x AA), d(aa x Aa), g(aa x m), c(aa x aa), e(aa x Aa), h(aa x Aa). Reciprocal crosses are grouped together. Then from a consideration of parental genotypes : b++d+&g=z, &d++e+h= y, c+$e+&g=x. From a consideration of the offspring: b++d+$h=x, Jd++e+g+$h= y, c+$e+ah=x. Hence h = 29, i.e. Aa x Aa matings are twice as frequent as AA x au and aa x AA together. Similarly, in the case of a sex-linked gene-pair S, s, let p and q be the frequencies of s arid S, while x, y, x, are those of ss, Ss, and SS. Let the six mating types occur with the frequencies b(aa x A), d(aa x A), g(aa x a), Then from a consideration of the parents: c(aa x a), e(aa x a), h(aa x A). b+g=z, d+e=y, c+h=x, b+d+h=q, c+e+g=p.

16 J. B. S. HALDANE AND PEARL MOSHNSKY 335 And from a consideration of the off spring : b+id=z, $d+qe+g+h= y, c+&e=x, b+qd+$e+g=g, c + $d + he + h = p. Hence d = 29, e = 2h, i.e. Aa x A matings are twice as frequent as AA x a, and Aa x a twice as frequent as aa x A. f a fraction h of all matings are of a type with a given value off (e.g. brother-sister matings with f = a), it would appear at first sight that a = fh. But this is only true in the limit when h is very small. Some of the parents have a further relationship due to inbreeding of remoter ancestors. Hence a contains terms involving h2, h3, and so on. f a fraction h of all matings are self-fertilizations, the rest being random matings, then g = (1-A) h = (1-A) 2x2 = (l-a) y2+hy. y2+h(&l+&e+g+gh) For among the 1 - A of the population which mates at random the frequencies of AA x m and Aa x Aa matings are clearly 2x2 and y2. Among the fraction h derived from selfing the first mating is impossible, and the second occurs with frequencies Q, &, 1, and $ among the progeny of AA x Aa, aa x Aa, AA x aa, and Aa x Aa respectively. Hence, since h = 29, 4x2 - y2 hy h = ~ era=- as repeatedly shown (e.g. by Haldane, 1924). 1-A 2-A Similarly, if a fraction h of all matings are brother-sister matings, then so Thus g = (1-A) 2xz+h.gh, h= (1-A) y2+h($d+$e+g+$h). (1 -A) (4x2- y2) = h($d+ ie+g) = Acid + &e+ $h) = hhy. h 4Xz-t~~ hy = ~ and a = ~ 2(1-A) 4-3h t can readily be shown that the same formulae hold for parent-offspring mating. n these three cases 4x2 - y2 2fhY = andcc = fh t is, however, doubtful whether 1 -A l-h+fh these formulae hold for all types of consanguineous union. n all cases a approximates to fh when h is small, but according to Wright (1921) it does not in general become 1 when h = 1. n the case of a sex-linked gene-pair S, s, 01 can readily be calculated when a fraction h of all matings are between brother and sister. Here, using the expressions of line 3 of this page, it follows that d= (1-A) qy+h(&d+ie+g), g= (l-h)p+&. h So since d = 29, 4x2 - y2 hy = and a = -~ 2(1-~y 4-3h

17 336 NBREEDNG N MENDELAN POPULATONS n the above cases the actual progeny of the consanguineous unions can easily be calculated. Thus in the case of an autosomal gene pair and brother-sister mating, all members of the population whose parents are not sibs have f = 0. Hence the mean value 1 off where it is not zero is - and the frequency of recessives among the progeny of 4-3h' p + 3( 1 - A) p2 brother-sister unions is 4-33h Fig. 5. Abscissa: frequency z = p2 of a recessive condition among children of unrelated parents. Ordinate : frequency y =,$p( p) of a recessive condition among children of first cousins. The scale is logarithmic. The dotted line is the asymptote z = 2 56~~. THE FRACTON OF AUTOSOMAL RECESSVES DUE TO COUSN MARRAGE We can divide the recessive zygotes in a population into three groups: (1) Those due to random mating. (2) Those due to first cousin marriage. (3) Those due to other unions of relatives. From Orel's data we should expect (2) and (3) to be of the same order of magnitude, and can compare the expectednumbers ingroups (1) and (2). n Fig. 5, log y = log[&p(l + 15p)], the logarithm of the frequency among children of cousins, is plotted against log x = log [p2], the logarithm of the frequency among the children of unrelated parents. y becomes twice x when p =A, x = Hence, unless the frequency of cousin marriages is very

18 J. B. S. HALDANE AND PEARL MOSHNSKY 337 accurately known, no serious evidence of the effects of inbreeding will be obtained in the case of recessive characters whose frequency in the general population exceeds 0.1 yo. and the frequency in the general population, taking a = 0.001, y = l0ox when p =&=, is one per million. For this value of p, about one quarter of all cases should occur among the children of first cousins, and as many more aniong the children of other relatives Thus of the three recorded cases of congenital steatorrhoea, Garrod (1923) reported that two were the children of the same pair of first cousins, whilst the parents of the other are not known to have been related. f a character is determined by two unlinked autosomal recessive genes with frequencies p1 and p,, the frequency of abnormal zygotes should be p,p,(p, + fq,) (p, +,fq2), or nearly f 2p1p2 when p, and p, are small, as Dahlberg (1930) pointed out in the case of first cousins. Thus marriages of more distant relatives would be of little importance. f pl and p2 are both decidedly less than &, and if x and y are the frequencies in the children of unrelated Xi parents and first cousins respectively, then y = '7. Thus if x = 10-6, y = 4 x 10-6 approxi- 256 mately. f p, = p2, the exact value of y is 4-97 x and this increases as p, and p, diverge from one another. Thus there would be little chance of discovering the effect of inbreeding on such characters if they are much commoner than one per million, as pointed out by Hogben (1932). Wibaut (1931) has discussed the case where several different recessive genes each produce a very similar effect. This is certainly true for waltzing and head-shaking in mice, and for "rex" fur in rabbits. t is probably true for congenital deaf mutism and retinitis pigmentosa in man, though here dominant and sex-linked genes are also concerned. Let the frequencies of two genes be p, and p,. Then x =pf+pi -pfpi = pf+pi approximately, and Y = Pf +Pi +f (Pl% + P2g2) - (Pf +fplql) (Pi +fp2q2) -- p; +pi + f (p,q, +p2q2) approximately. Hence f p, = xt cos 8, p2 = X+ sin 8, then!!- 1-f+ f(cosb+sino) - 1 X X+ where $77 > B > 0. Hence y/x can only vary between 1 - f + fx-k and 1 - f + 42 fx-*. So it is very doubtful whether, as Wibaut suggested, an unduly high incidence of first-cousin marriages among the parents of a given set of abnormals can be taken as an indication that several different loci are concerned. Even if there were n loci in all of which gene frequencies were exactly equal, y/x could only be raised from 1 -f+ fx-* to 1 - f + nyz-l.

19 338 NBREEDNG N MENDELAN POPULATONS DSCUSSON The values off, f, f(c) and $(c) here found can be used to form correlation tables. Wright (1921) showed that the coefficient of correlation between gametes of W and H for autosomal genes was f. Fisher (1918) further showed that the correlation for zygotic characters would be 2f if certain simplifying assumptions were made. The value should fall below 2f on account of dominance, epistasy, and other non-additive interactions of genes. Assortative mating would raise it. Environmental effects could raise it or lower it. t would, for example, in general be lowered by inhomogeneity of environment, but might be raised if the correlation between relatives for a determining condition in the environment (e.g. supply of vitamin D) were higher than the genetic correlation 2f. The correlations between first cousins should therefore be in the neighbourhood of 2f or 0 125, as those between parents and offspring, and between sibs, are in the neighbourhood of 0.5. But Elderton s (1907) results are very much higher. Seventy correlation tables, from which measures of resemblance were worked out in some cases by mean square contingency and in others by (n x n)-fold division, gave a mean value of f 0.008, and a standard deviation of f these values are correct, the argument of this paper breaks down. There are, however, several reasons for doubting their validity. n the first place they are no lower than those for uncle and nephew, which, if true, would be unintelligible on any theory of heredity. The high value can partly be explained by the fact that most of the characters recorded were subjectively estimated (e.g. health, intelligence, and success). As each pair was described by the same informant the correlation was increased if some informants had higher standards than others. Similarly, for the metrical characters the correlation was raised if some observers were grossly inaccurate (as it appears that they were). There is, however, a possible biological reason which would raise the correlation between cousins substantially. t may be that, for certain characters, the correlation between the wives of two brothers is higher than that between husband and wife. The same may hold for sisters. t is hardly likely to hold to the same extent for a brother and sister. As Elderton s results do not show a consistently higher correlation for the children of likesexed than of unlike-sexed sibs, this probably means that the above source of resemblance between cousins is not very important. However, it is certainly well worthy of investigation, if only because the theory of sexual selection postulates not merely the inheritance of pahterns and modes of behaviour, but of the appreciation of them. f the high correlations between cousins are explicable on this hypothesis, then it is unlikely that they hold for recessive genes. Thus two brothers might both prefer blondes, but it is hardly plausible that they would both prefer brides heterozygous for ichthyosis foetalis. So even if Elderton s results are confirmed, those of this paper may hold for rare recessive conditions.

20 J. B. S. HALDANE AND PEARL MOSHNSKY 339 Wahlund (1928) and Dahlberg (1938) have introduced the idea of an isolate, that is to say, a subpopulation within which matings may be regarded as occurring at, random. This is clearly only an approximation. n so far as it is true it implies that the contributions of remote inbreeding to a are large. t also implies that if N is the number of adults in the isolate, p cannot have any non-zero value less than 1/2N. Dahlberg shows that if N = 200, then at most 25 yo of all recessives will be the offspring of first-cousin marriages. We regard Dahlberg s method as complementary to our own. Both are based on abstractions. We neglect the contribution of remote ancestors to a, while Dahlberg neglects unions outside the isolate. Each abstraction will probably prove appropriate to a different set of problems, and doubtless a theory will ultimately be reached which combines the advantages of both methods. t can, however, be seen that a consideration of isolates would yield the same results as those obtained by Haldane (1939) in an accompanying paper. He finds that if a is diminished, as it has been during recent centuries, the frequency of unfit recessive phenotypes must fall, and only rise to its former value in a period depending mainly on the mutation rates, and measured in hundreds of generations. The same would be true were the size of the isolate largely increased. The possible eugenic applications of our work await a demonstration that socially desirable characters are non-existent or rare. This is plausible, but very far from certain, even though the work of Russell (1930) suggests that it is true as regards the inborn factors in intellectual ability. t may also be that for some gene pairs heterozygosis is an advantage. nbreeding reduces the probability that a given gene pair will be heterozygous to a fraction 1 - f of that found in a random-mating population. The mean fitness will therefore be a monotone-increasing function of 1 - f, and if it is approximately linear, brother-sister unions will be four times as harmful as first-cousin unions, and so on. We have seen that a community which based its marriage laws on eugenic considerations alone would not adopt those of any existing state. The order of undesirability based on the theory that heterozygosis is an advantage is of course the same as that based on the probability of producing rare recessives. The slight extra disadvantage of unions with maternal rather than paternal relatives of the same type shown by Tables V and V is probably negligible. We have to thank Prof. Lancelot Hogben for suggesting to us the problem of inbreeding where sex-linked genes were concerned, and for pointing out an error in a calculation. SUMMARY The expected progeny is calculated for various unions of relatives, with special reference to the frequency of rare recessives. The calculation for completely and incompletely sexlinked genes is now made for the first time. n the case of sex-linked genes the published pedigrees are shown to agree with our theory. The mean coefficients of inbreeding for populations are defined, and roughly estimated for certain European communities.

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