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Genetics of Human Populations

Genetics of Human Populations

Genetics of Human Populations GUNNAR DAHLBERG Head o j the Stale Institute of Human Genetics, Uppsala, Sweden CONTENTS Page I. Populations in Panmi...

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Genetics of Human Populations GUNNAR DAHLBERG Head o j the Stale Institute of Human Genetics, Uppsala, Sweden

CONTENTS

Page

I. Populations in Panmixia . . . . . . . . . . . . . . . . . . . . . 11. Mutations. . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

. . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Assortative Mating . . . . . . . . . . . . . . . . . . . . . . . .

72 78

111. Selection

V. Intermarriage . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Isolates . . . . . . . . . . . . . . . . . . . 1. Size of the Isolates . . . . . . . . . . . . . 2. bolates and Heterozygote Frequency . . . . . 3. Breaking of Isolates . . . . . . . . . . . 4. Isolates and Race . . . . . . . . . . . . . 5. Isolates, Mutations and Random SeIection . . 6. Isolates, Intermarriage and Selection . . . . . 7. The Rise of Isolates . . . . . . . . . . . . 8. Assortative Mating, Intermarriage, and Isolatcs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . VII. Summary . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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83

87 87

88 89

90 92 93 93 96 96 98

Both heredity andenvironment determine what characters an individual will have. If, however, the genes deviate too much from the normal, no individual can come into existence. The same is true if the environment varies too much. Both heredity and environment can vary within the limits set by death, giving rise to individuals of very different types. In experimental genetics the environment can be regulated, and this may be the reason why less interest has been attached to properties which are to some extent contingent on environmental factors. The primary point of departure in experimental genetics is to study which hereditary factors determine certain characters. The problems are by no means simple when human genetics is in question. In view of this I have suggested that characters of three types should be distinguished here, namely: 1) hereditary characters in the limited sense; 2) characters due to environment; 3 ) characters depending both on heredity and environment, and which I call constellational (Dahlberg, 1939). There is also the fact that characters may emerge at different ages. When making special investigations, the influence of age on the results can be eliminated by 67

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restriction to definite age groups. When an entire population is in question, however, it is often necessary to take all its constituent age-groups into account. In that cam, appropriate statistical methods have to be used. A distinction can also be made between characters which are of short duration (more or less accidental) and characters of longer duration (more or less permanent) in the above-mentioned three types. The nature of the population determines to some extent the type under which a character is to be classified. To illustrate this, we will discuss a character normally contingent on both heredity and environment, that is to say a constellational character, where the hereditary factors necessary for its occurrence are found only in part of the population, and where the same is true of the environmental factor or factors which are similarly necessary. In a population where the necessary environmental factor is present in practically all individuals, the character will be determined by heredity; conversely, in a population where the hereditary factor is present in practically everyone, the character will be determined by environment. To take an example: paralytic dementia is usually due both to infection with syphilis and to the presence of hereditary tendencies such that infection with syphilis is followed in due course by a certain injury to the brain. In a population where everyone is infected with syphilis - as is said to be the case in Mongolia, for example - paralytic dementia will be a hereditary character, while the same character in populations where syphilis is less common must be classified as a constellational character.' In other words, the nature of such a character depends on the frequency of the environmental and the hereditary factors bearing on its occurrence (fig. 1.). Obviously, the dividing-lines between the groups of characters we have discussed are to a certain extent arbitrary, but the division should Beme for practical purposes. In the following pages we shall only be discussing genes and characters that are hereditary in the limited sense of the word. From a practical viewpoint, two main questions can be applied in human genetics. 1) What will the children of any given marriage be like? We can assume here that we have a more or less extensive knowledge of the two parents and of the characters of their relatives. 2) What will be the future make-up of a population, given more or less detailed data on its present make-up and on the crossings taking place within it? It is clear that the latter question is a very important one in human genetics, and it is therefore surprising that it has not been more extensively studied. No empirical investigations along avenues of approach of the It has becn asscrted that paralytic dementia is due t o infection with a ccrtain kind of virus (syphilis h virus ncrvcux). As far aa I can see, this thcory is hardly tenable. In any case, we disregard this possibility in the above argument.

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kind mentioned above have as yet been made. The problems have been treated solely theoretically. Studies of this sort are to be found in the most varied periodicals. It is our intention here to give a short survey of the problems from a theoretical point of view, since it may be presumed

Heredity

Environment

Heredity Environment

FIG. 1 Charactors of diffcrcnt kinds. Above, to thc right: a hereditary character; to the left: an cnvironmcntal character. Bclow: a constellational character.

that human genetics will be directed more consistently along this avenue of approach in the future. I. POPULATIONS IN PANMIXIA If crossings take place at random in the population, and if individuals of different heredity have the same chance of procreation, then we say that panmixia or amphimixis obtains. In this situation the population will change character at random. If the population is at all large, we can disregard this possibility of change and consider said population t o be constant from generation to generation. If monohybrid heredity is in question, a knowledge of the frequency of the recessive and thus also of the corresponding dominant character enables us to calculate the frequency of the heterozygotes. The composition of the population is determined by the expression r2 2rd d2 (1)

+

+

where r is the frequency of the recessive gene, which we call R, and d is the frequency of the dominant gene, which we call D (fig. 2 and 3).

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An interesting point is that the relation between heterozygotes, RD, and recessive homozygotes, RR, will be

2(1 - r) r The expression shows that if r is a very small number - i.e. if those having the recessive character are very few - then the heterozygotes will be proportionately very common. That is to say, the expression approaches when r approaches 0.

RD - = - =2rd RR r2

FIG. 2 Distribution of the zygotes (RR, RD, and DD) in a population in panmixia when the gene proportion r:d = 2:3.

Even if panmixia can usually be assumed to be present in many populations, there may be deviations which are of importance particularly when a rather longer view is taken. These deviations are: 1. Mutations 2. Selection

3. Assortative mating 4. Intermarriage 5 . Effect of isolates

If a person gets married and has children at random, his genes may become changed so that the children show other hereditary factors than would otherwise have been expected (mutation). However, he may refrain from marriage and chiIdren (selection). Further, he may marry a person with a certain character (assortative mating), marry a relation (inter-

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marriage), or marry within a limited circle (effect of isolates). All the conceivable deviations from panmixia in human populations come under these heads, and we shall now discuss them briefly. IOC

0

0.2

0.4

0.6

0.0

I

0

80

- 20

60

- 40

%

-

%

40

- 60

20

- 80

100

0 I.

0.8

0.6

d

0.4

0.2

0

FIG.3 Distribution of the zygotes in a population in panmixia at decreasing frequency of a recessive gene. Cj. formula 1.

The first to point out that the heredity of a population is of a constant nature in panmixia was Hardy (1908). However, Pearson (1903) had shown the same thing earlier for a special case, namely a population with a monohybrid gene frequency = 0.5. A survey of the composition of a population in panmixia was given by Dahlberg and Hultkrantz (1927).

11. MUTATIONS As mutations are comparatively rare, they play little part in human populations when studied over rather shorter periods. We know little of the mutation frequency in man. In Sweden four or five individuals who suffer from juvenile amaurotic idiocy are born annually (Sjogren, 1931). They always die before maturity and never have children. Consequently, the genes must arise in an equal number through mutation. The rate of mutation therefore should be 1 :25000. For a special form of dwarfism

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(chondrodyshphic dwarfs) E. T. M#rch (1941) has found a rate of mutation of 1:10000--12OOO. Haldane (1935) computed a rate of mutation of 1 :5oooO for hemophilia. Andreassen (1943) arrived a t a figure of 1 :53000 for the same disease. In any case, it can doubtless be aasumed that mutations are far too rare to produce noticeable differences between near generations of a nation. Taking a longer view of the problems, however, and particularly from the viewpoint of evolution, the mutations cannot but play a very large part. The mathematical questions in this field have been handled primarily by J. B. S. Haldane, R. A. Fisher, Sewall Wright, and others. But they fall outside the scope of this study, which merely intends a short-term treatment of the conditions in human populations.

111. SELECTION Haldane (1924) has also submitted formulae for selection - that is to say the effect that results from inhibiting or preventing the propagation of character-bearers of a certain type. Now, if their propagation is prevented, this means that that of the other types will be favoured. It is customary only to think of the negative side when talking of selection, but obviously every selection has a positive side also. If the gene is inherited dominantly and total selection is in question -i.e. the person with the character is completely prevented from propagating - the character will immediately be exterminated. If, on the other hand, recessive genes are in question, total selection does not result in an instant extermination, since the gene is to a certain extent recruited from heterozygotes; instead, there will only be a larger or smaller lowering of the frequency of the character in the subsequent generation. If the recessive gene has the frequency r, and the character thus has the frequency r2,then after n generations the frequency of the persons with the character, rn2, will be determined by the following formula: r,2 =

+

rz

(3) [l (n - 1) rI2 With the help of this formula, which has another form from that given by Haldane, and which has been taken from a work by DahlbergHultkrantz (1927), we can calculate the make-up of the population with different initial frequencies of the gene (fig. 4). It is seen that if a character is common, selection is very effective. If the character has the frequency of 25010, it falls in one generation to 11.1%. Gradually, however, selection becomes less effective, when the character becomes more and more rare. When a frequency of, say, 0.1% is reached, it will take ten generations before it is reduced by not quite

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GENETICS OF HUMAN POPULATIONS

half, namely to 0.06%. This is because when a recessive character is rare, the heterozygotes are much more common for the gene than the homozygotes (see above). %

%

99.2

99.4

60 0.4

99.6

70 80 0.2

99.8

90 1 2 3 4 hneretion

6

6

7

8

9

10

20

30

40

60

100

FIG.4 Dccrcasing frequency of a recessive character in succemive generations in total negative selection. Cf.formula 3.

It may be mentioned that the above equation can also be used to calculate the frequency of dominant characters. In monohybrid hereditv we know the frequency of dominant characters when we know that of the corresponding recessive characters. It is more important to state that a favouring of the propagation of certain characters means, relatively speaking, that the ones not having the character are inhibited. In eugenics it is customary to discuss two kinds of measures, i.e. negative measures which will prevent the propagation of certain inferior types, and also measures of a positive kind, favouring the propagation of certain types. However, the effect of positive measures can be calculated with the help of the same formula as for partial negat,ive selection. If, then, we raise or diminish the propagation of persons with the recessive character by a certain proportion, k, of what it is in panmixia, the process will be determined by the following recursion formula (Dahlbcrg, 1946):

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In this formula an+l denotes the frequency of the recessive homozygotes in’the generation (n 1) and a,, their frequency in the previous generation, 2b,, is the frequency of the heterozygotes2, and c,, the frequency of the dominant homozygotes in this generation. If k = 0, this means that the’propagation is not being changed from that in panmixia. Nor is there then any change according to the above equation. If k is a negative number < 1, there is a decrease in the frequency of the recessive character. If k = - 1, then total negative selection obtains. The formula is then simplified to agree with formula 1 above. If k is a positive number, this means that the propagation of the recessive characters is favoured. The corresponding recursion formula for the heterozygotes is :

+

With the help of formulae 4 and 5 the frequency of dominant homozygotes can be computed. If the propagation of the recessive is inhibited, it means the favouring of that of the dominant, though only of those who marry persons with the dominant characters. If all the persons with the dominant characters are to be influenced, even those marrying recessive homozygotes, orwhich is the same thing -if only recessive homozygotes marrying one another are to be influenced, then the process will be according to the following recursion formula (Dahlberg, 1947), where the symbols have the same meaning as before:

The recursion relation for the heterozygotes is the following:

When this is the situation, then, propagation is favoured or inhibited in all marriages to which dominant character-bearers are a party. The frequency of the dominant homozygotes is obtained from the above formulae by subtracting the recewive homozygotes and the heterozygotes froni the total number. As stated above, measures aimed at inhibiting the propagation of a recessive character are found to have very little effect if the character has

* Haldane has given a formula for partial negative selection, valid for aquatic organisms “which shed their gametes into water.” This formula is correct for selection in human populations caused by differential mortality in younger years but not for selection due to differential fertility. However, the difference between the figures obtained with the help of Haldane’s formula and the one given here are rather small.

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a low frequency. If its frequency is high, the effect will be more noticeable provided the amount by which propagation is checked is not unduly small. In the case of partial negative selection where the reproduction rate is half that of the rest of the population (ie. k = the frequency will have fallen from 0.1% to 0.075% after 10 generations (fig. 5). The effect of

s),

0.002

0.0015

0.00I

0.0005

0.

0

2

4

8

6

10

Fro. 5 Frequency of persons with a recessive character in successive generations at partial negative selection when their procreation is half (k -%; cf. formula 4) that of the corresponding dominant character (the lower curve), and a t partial positive selection when their procreation is double (k = 1) that of persons with the correspondingdominant character (the upper curve).

-

negative selection is, in other words, insignificant. If equally strong negative selection works against a dominant character with a frequency of 0.1%, the effect will be that the frequency of the character will be reduced to O.OOOl% (fig. 6). Here, in other words, the effect is very much greater. Positive selection, on the other hand, has a comparatively strong effect which in some cases is rather astonishing. If a recessive character has a frequency of 0.1% and the reproduction of all the individuals is double that of the rest of the population, the character will have risen only to nearly 0.2% after 10 generations, but in the case of a11 the dominant characters being favoured to the same extent, we get, after an equal number of generations, a rise to 34% (fig. 7). If, finally, a character determined by two or more factors is in question, it is easy to show that the process is slower than in monohybrid recessivity with the corresponding frequency of the said character. That is to say,

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there are proportionately fewer of the genes present in the homozygotes in recessive polyhybridism than in monohybridism, so that the greatest effect from selection is obtained with an ordinary monohybrid character. From a practical point of view it is important that the selection has so small an effect on rare characters in human populations. For this reason no great effect can be expected from negative eugenic measures, such as

”.

0

2

4

6

8

10

FIQ.6 Frequency of persons with a dominant character at partial negative selection when their procreation is half (k = 1; cf. formulse 6 and 7) that of tho corrmponding recessive character. The curve happens to be the same aa the one which is obtained at total negative selection for a sex-linked character. Cj. formula 9.

sterilisation. It must also be remembered that a “character” against which such action is taken, as a rule, actually consists of several different hereditary characters. Thus, the imbecile, the blind, etc. are built up of different hereditary types of imbecility and blindness, and also of characters of the kind due to environment. The effect of selection on such groups of individuals will naturally be far weaker than if the group had been homogeneous and consisted of a single character. Practically speaking, continued negative selection must lead to elimination of the character and the gene against which it is directed. There is, however, the contingency that the gene and the character may arise by mutation. A state of balance will therefore gradually come to be established between mutations and selection. The author has given this limit at which selection ceases to have effect the name of the least heterozygote frequency. If this is reached, the mutation frequency can be computed by investigating how many persons with a character in each generation are prevented from propagating, since the mutation frequency

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corresponds t o the number of genes that are eliminated. If a gene arises only once through mutation it must reach a certain frequency-i.e. appear in a certain number of heterozygotes - in order to occur in homozygote form via intermarriage. The frequency of the gene can never be pushed below this limit. In recessive sex-linked heredity, selection will act on the male sex as in dominance, and on the female as in recessivity. When selection has lowered the frequency of the character, so that in practice all the women can be accounted heterozygotes, the frequency of the character will, with total selection, be diminished by half in every generation. Hemophilia is an example of this. Sweden contains about 100 individuals suffering from this disease, all of them males. If all these are prevented from propagating, the Character will be maintained by the number of heterozygotes who are present among the women, and who number the same. Provided they are of average fertility and that the population is constant from generation to generation, the number of hemophilics in the next generation will be 50. That is to say, the 100 female heterozygotes have twice their own number of offspring. Half of the children are boys, and half of them in their turn will suffer from the disease. The general formula for selection in monohybrid sex-linked heredity, when the gene frequency is r, may be written in the following manner: I I

I

(8) 2,-(2"-1) r When the character is rare, the formula can be simplified approximately to: r (9) rn = r, =

The character is then eradicated in a few generations (Skold, 1944). T h a t hemophilia nevertheless does occur must be because the character arises comparatively often by mutation. As mentioned above one can, in point of fact, calculate the mutation frequency by investigating the number of persons with the character eradicated per generation. Provided that the frequency of the character is constant, the number of eliminated individuals must be met by mutations. With regard to the actual occurrence of selection in human populations, we may mention that it has been strong in the case of defective individuals, particularly a t an earlier date. They lived on the brink of starvation and had little chance of procreation. It can therefore be supposed that their frequency would be down a t the least heterozygote limit. Moreover, I should like t o remind readers that two big selection experiments have been made in human populations, one in a negative and one in a positive direction. The one is the celibacy of the Roman Catholic Church, which can

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be said td have been an unintentional attempt to exterminate talent. Since talent is a rare oharacter, the effect has probably not been great. If Catholic and Protestant countries are compared, e.g. England and France, no very perceptible difference emerges.

FIG. 7 Frequency of persons with a dominant character in successive generations at positive selection when their procreation is double (k = - W ;cj. formulae 6 and 7) that of the corresponding recessive character.

The other great experiment is the polygamy found in Mohammedan countries in particular. Obviously, in practice it was primarily the wealthy who were able to indulge in the luxury of several wives. In as much as hereditary talent can be gauged by social success and riches, the propagation of these individuals has been favoured. It is a debatable point, what characters, if any, actually are favoured by this institution. Another debatable point, however, is whether it has had any very great effect. IV. ASSORTATIVE MATING While selection and mutations alter the gene content in a population, the other deviations from panmixia only occasion a re-sorting of the genes, without changing their frequency. Assortative mating denotes the situation when persons with a character have a tendency to marry each other to a greater or a lesser extent than chance dictates. The effect of total assortative mating was computed for the first time by Robbing (1917 and 1918). Koller and Geppert (1938) have submitted recursion formulae for partial positive assortative mating. Proceeding from other assumptions, the present author has given more general for-

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mulae for partial positive assortative mating, which formulae can also be used to calculate the effect of negative assortative mating (Dahlberg, 1943). If we call the number of generations n (the first generation is denoted 0) and the gene frequency r, the frequency of recessive homozygotes rn2in total assortative mating in the nth generation is determined by the following formula: l)r2 (n rn2= 1 nr and the frequency of heterozygotes in the nth generation, 2rndn,by: 2nr2d 2r,dn = 2rd - 1 nr Such a process gradually leads to the disappearance of the heterozygotes and the population comes to consist only of the two different types of homozygotes (cf. figs. 8 and 9). The process begins quickly, and then goes more slowly towards this state of balance.

+

+

+

0

2

Generation

4

6

8

10

16

20

60

100

1000

FIQ.8 Frequency of heterozygotesat total positive assortative mating in successive generations when a recessive gene has the frequency r =0.5. Cf.formulae 10 and 11.

Now, total assortative mating can hardly exist in human populations; we need only concern ourselves with partial assortative mating. In positive partial assortative mating, the process is determined by the following recursion formula for the RR homozygotes: 2kd2rn2 rn+? = r2 1 - rn2

+

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where k is half of the marriages which do not occur. The total number of marriages not contracted thus is = 2k of the marriages between persons with the character which would take place in panmixia. rn2is the frequency of recessive homozygotes found after the process has been going on for n generations, rn+I2 is their frequency in the next generation, d2 and r2 is the frequency of the character a t the beginning of the process. The above recursion formula can be used to calculate the frequency of persons with the character in consecutive generations.

0

2

Generation

4

6

8

10

16

20

50

100 1000

FIQ. 9 Frequency of heterozygotes at total positive assortativc mating in successive generations when a reeessivc gcnc has the frequency r = 0.1. Cf.formulae 10 and 11.

The following line of thought can be used to calculate the state of balance which has been reached (Dahlberg, 1943). We assume that one and the same process takes place when starting both from panmixia and from a population in which the heterozygotes have been eradicated by, for instance, total assortative mating. In the former case the heterozygotes mill show a successive decrease, and in the latter a successive increase. The curves meet after a relatively small number of generations. In other words, it is possible in this may to show that a state of balance is soon reached (figs. 10 and 11). Negative assortative mating implies that the persons with the character marry one another to a lesser estent than chance dictates. We can, for example, imagine (or hope) that obstinate persons intermarry less than is called for by chance. This character means, of course, that the

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parties have less prospect of agreeing. That is to say, if obstinacy were a hereditary character, negative assortative mating would apply. The above formula 10 is used to calculate the effect, but k must naturally then have a negative sign. RR

0

1

Generation

2

3

4

5

6

7

FIQ. 10 Effect of positive partial assortative mating in successive generations with the intensity k = 0.30,when the frequency of thc recessive gene is r = 0.5. In the calculations we have started from pnnmixia and also assumcd that because of total assortative mating at the start there arc no hcterozygotcs in the population. Cf.formula 12.

k =o.so 0

1

2

Generation

3

4

5

6

7

FIQ. 11 Effect of positive partial wortative mating in successive generations with the intensity k = 0.30,when the frequency of the recessive gene is r = 0.1. In the calculations we have started from panmixia and also assumed that because of total assortative mating at the start there are no heteroaygotes in the population. Cf.formula 12.

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It may also be pointed out that the formula for negative total assortative mating is obtained by the introduction into formula 10 of:

Here, too, the process approaches a state of balance very rapidly. To calculate the effect of partial assortative mating, however, we can also assume that the marriages contracted by persons with the character form a constant proportion, h, of the marriages which would take place in every generation if marriages between persons with the character were contracted a t random (Dahlberg, 1943). If rn2is the frequency of recessive homowgotes in a certain generation, r,+12 their frequency in the next generation, and r the gene frequency a t the beginning of the process in panmixia, the general recursion formula will look like this:

If h = 3,there will be no change. The composition of the population is constant. If h>r2, then there is positive assortative mating. If h = 1, there is total positive assortative mating. If h <9, there is negative assortative mating, and if h = 0, there is total negative assortative mating (fig. 12). RR

0

Om.

1

9

4

6

6

FIQ. 12 Frequency of a recessive character at total negative assortative mating when the frequency of the recessive gene ia 0.5. Cf.formula 14.

This formula, too, shows that a state of balance is soon reached both in positive and negative assortative mating. The effect of such processes is very moderate, even if there is assortative mating to a comparativeIy

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high per cent. We do not at present know enough about assortative mating to be able to decide if the assumptions behind the last-mentioned formula are more in accordance with reality than those on which formula 12 is based. Our empirical knowledge of processes of this kind in human populations is very limited. It has been found, for example, that tall persons are not inclined to marry short ones. As Pearson and Lee (1903) have shown there is a faint tendency towards positive assortative mating as regards stature. Characters such as musical talent may, in so far as they are due to heredity, be assumed to make for positive assortative mating, as also all hereditary characters whatsoever that can be thought to direct the interests in a given direction, since these interests form a bond between the respective parties which should promote their marriage. It can also be assumed that criminal individuals, for example, who may thus suffer from an inherited lack of judgment and balance, tend to marry one another. In so far as we are dealing here with rare characters, the possibility of assortative mating should have been on the increase up to the present, due to the development of communications, removal to the towns, etc. - in brief, by enlargement of the isolate. For that matter, assortative mating proceeds to a certain extent via isolates of special kinds. We shall be returning to this question.

V. INTERMARRIAGE Intermarriage has the same effect as assortative mating. Extreme intermarriage, i.e. marriage between siblings, which is continued through several generations in crossing experiments, gradually eradicates the heterozygotes; this was first shown by Mendel and later mathematically worked out in more detail by Sewall Wright. A process of such intensity does not occur in human populations, however, though inbreeding of this kind was practised in the royal line of the Ptolemies in ancient Egypt and in the Inca of Peru. A certain frequency of blood marriage occurs in panmixia, so that in calculating the effect of intermarriage, regard must be paid to the size of the population and to the degree of intermarriage which is to be expected from chance. I n small populations and large families we should expect a certain frequency of marriages between siblings, which is not the case in human populations. It can, however, be shown that the deviation this causes is but slight. The fact that evolution has taken place implies that all marriages are blood marriages, even though very distant. As already pointed out, panmixia presupposes in all cases a certain degree of intermarriage. Intermarriage can be said to take place if more blood marriages are found than

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are to be expected in panmixia. (If the frequency of the blood marriages is lower than was to be expected, we could call this “outbreeding.”) In very large populations, the frequency of blood marriages to be expected from random mating is extremely small. I n practice, however, marriage between cousins in such populations must be expected to a greater extent than chance would suggest. If such marriages are assumed to take place to loo%, i.e. if the contracting parties are always cousins, the deviation from panmixia after one generation can be calculated according to the following formula: 1 - r2) = 2rd -(r 16 32

-

which means that the heterozygotes diminish by %, and that this diminution is compensated by an increase of the two kinds of homozygotes (Dahlberg, 1929). The extent of this increase, then, depends on the gene In this case the recessive homozygotes frequency. It is greatest if r = in panmixia will constitute 25%. In one hundred per cent cousin marriages, their frequency will be 26.56%. In other words, the increase is 1.56%. The increase obtained in different gene frequencies appears from fig. 13.

x.

0.ot

0.01

r’O

0.1

0.0

0.S

0.4

0.6

0.6

0.7

0.8

0.0

1.0

FIG.13 Increase of the frequency of a monohybrid recessive character in a population where only marriages between cousins are contracted compared with panmixia at different frequencies of a recessive gene. To make the curve visible the scale is ten timcs larger in vertical than in horizontal direction. Cf,formula 15.

As is seen, it is very moderate. Now, there is no population with one hundred per cent marriages between cousins. The normal frequency for such marriages in the larger civilised countries is probably less than ?&. We can therefore say that the increase in a character’s frequency to be expected in practice in human populations has no important bearing on the degree and frequency of intermarriage. (However, intermarriage does not affect only one gene in a population; it affects them all and from this viewpoint is by no means negligible. See below.) We have IIOW examined the effect of intermarriage on a monohybrid character from a populational viewpoint, If we look a t this effect from

GENETICS OF HUMAN POPULATIONS

85

the viewpoint of the persons with the character, we get another result. If a gene is very rare, that is to say, has sprung from a single mutation in a population, it can come together in homoaygote form only by a blood marriage. If the character is a usual one, intermarriage proves to be of no consequence. If, on the other hand, it is rare, intermarriage means everything, but the character then has no importance for the population. For an idea of the importance of intermarriage from the viewpoint of the persons with the character, it will be simplest to calculate the proportion of persons with the character whose parents were cousins. An approximate formula for this has been given b y Lena (1919). The correct formula has been given by Dahlberg (1929), and runs as follows, where k implies the proportion of persons with the character whose parents are cousins, r the gene frequency, c1 the actual frequency of marriages between cousins, and cz the frequency of such marriages to be expected in panmixia:

k= c1 - cz

+

c1 lGr(1 - c1 CZ) 1 15r

+

+

(16)

The formula shows that the rarer the gene in the population, the greater the number of persons with the character springing from marriages between cousins (fig. 14). Everything else being equal, this proportion will naturally be higher in a population where such marriages are common, than in one where they are rare. One consequence of this is that if characters are rare, they are often found together among the children of parents who are cousins. This concurrence may be wrongly taken to express linkage. If kl is the proportion of persons with the character springing from married cousins in respect to the one character, kz the proportion in respect to the other, and c the frequency of marriages between cousins in the population, the relation between the real and the expected frequency of the coincidence of two characters in one and the same individual in a population (y) will be given by the following formula (Dahlberg, 1943) :

Y =

(1

- kl) (1 - kz) +-kikz

1-c C The formula shows that this mechanism has a noticeable effect only on very rare characters. Other circumstances being equal, the effect is naturally greater when the marriages between cousins are common than when they are rare. This effect of blood marriages also leads one to expect that siblings of persons exhibiting a rare hereditary defect will show this and other rare defects to a somewhat higher frequency than is the case in an average population.

86

QUNNAR DAHLBERQ

As intermarriage somewhat increases the frequency of homozygotes but decreases the frequency of heterozygotes, this means that negative selection has a greater effect in a population showing a high intermarriage frequency than in one with a low one. It also means that more recessive mutations become manifest in a population with a high intermarriage k 3.8

3.4

3.0

2.0

2.9

1.8

s= 1 yo 1.4

1.0

c=o.a:/. 0.0

c =0.5

0.0

O.OOOI n.ool

yo

c=O.196 o.00~~

0.005

0.01 RR

FIQ. 14 Percentage of persons with a recessive character deriving from marriages between cousins (k) in populationa where the frequency of marriages between cousins (c) and the frequency of a recessive character is varying. Cf.formula 16.

frequency than in one with low. In other words, if the intermarriage frequency of a population is lowered, the state of balance between negative selection and mutations will be upset. The Ieast heteroeygote limit is raised when intermarriage falls, which means that the gene content will rise very slowly, due to mutations] before equilibrium is again reached. In principle, of course, intermarriage works in the same way as assortative mating.

GENETICS OF HUMAN POPULATIONS

87

VI. ISOLATES

A population can be said to consist of partial populations of isolates, in which there is panmixia. The situation here is naturally very different in different populations. Isolates can be of two kinds, geographic and social. Geographic isolates were specially common in European populations of earlier ages, where the population of a country consisted of settlements and villages separated by forests, etc. It may be remembered that the people usually lived by water-courses, and that lakes and rivers formed links of communication, whereas forests and mountains formed dividing barriers. The social isolates may be conditioned by economic factors, but religious elements and traditions of other kinds may cause a certain group of individuals to intermarry and keep apart from the rest of the nation. 1. Size of the Isolates If marriages between cousins are contracted a t random, their frequency will obviously come to depend partly on the average size of the isolates and partly on the average size of the family. Conversely, then, the size of the isolates can be reckoned if we know the frequency of the cousin marriages and the average size of the families. If the size of the isolate is called n, the average number of children becoming adult and marrying b, and the frequency of cousin marriages c, we have the following formula given by Dahlberg (1938):

If we assume that the numbers of the population remain fairly constant, b must be equal to 2. The average size of the isolates in different frequencies of cousin marriages is seen in Table 1. Needless to say, the values there TABLE I Size of the Isolates ( = n), when Marriages between Cousins are Contracted at Random in Ditrerent Frequencies (= c). Cf.Formula 18

-_

C

0.0001 0.001 0.0025 0.005

0.01

0.05 0.1

--

n 40 000 4000 1 600

800 400 80 40

88

GUNNAR DAHLBERG

are highly approximate. An isolate very seldom has clear-cut boundaries. We can often say that it merges gradually into the population, and that every person has his or her own isolate. The isolate calculated by the above formula is in point of fact often fictitious: the calculated isolate is a sharply demarcated one, which would give the same frequency of marriages between cousins as the real and less clear-cut isolate. The above formula presupposes that marriage between cousins is contracted a t random and is solely dependent on the size of the isolate in relation to the size of the family. In populations where the parents arrange the marriage of the children (as was once specially common in France, for example), we can expect marriage between cousins to show a far greater frequency, and this frequency to remain more constant than would otherwise have been the case. In many populations, however, the frequency of such marriages has decreased up to the present time, due to the breaking of isolatcs and to smaller families. 2. Isolates and Heterozygote Frequency Obviously, the heteroaygote frequency obtained if panmixia is assumed in-a population is far greater than that found if the population is divided up into isolates. In other words, we get a wrong gene frequency if we calculate it with the help of the known frequency of recessive characters and assume panmixia. The correct frequency, rt, is obtained from the following formula, submitted by Wahlund (1928) :

rt = drz where

+ qz

(19)

- rl)* + i2(r - r,)z + i3(r - r3)2 + . i,(r -2),r In this formula ili& . . . i, denotes the size of the different isolates, and u$ = il(r

their gene frequency is shown by r1r2ra . . . r,, while r gives the frequency of the R-gene in the total population obtained by the formula for panmixia. If an ordinary character is in question, it is hardly likely that the gene content in the different isolates would cause greater differences from those to be found in panmixia throughout the entire population. I n the question of rare characters, OR the other hand, it is likely that the relevant gene only occurs in a few isolates and is completely absent from the others. Such rare genes become manifest-i.e. occur in homozygote formprimarily in small isolates, e.g. remote villages, in which the frequency of intermarriage is relatively high. The appearance of these rare characters is often laid a t the door of intermarriage and the population is called degenerate. It is, however, more correct to say that lack of communications is the real cause. In any case too high heterozygote frequency and too

QENETICS OF HUMAN POPULATIONS

89

high gene frequency are obtained if panmixia is assumed for a rare character which only occurs in certain isolates.

3. Breaking of Isolates If we envisage a single isolate in a population where there is a certain frequency of a gene causing a defect, lacking in other isolates, it is plain that marriage beyond the isolate boundaries is harmless, since it is not likely that the gene is present in the neighbouring isolates. For this reason, a breaking of the isolate will lead to a lowering of the homozygotes -in other words, a lowering of the frequency of rare hereditary defects, and an increase of the heterozygote frequency. If the gene in question has the frequency r, and if the isolate is enlarged by a part,, a, of itself, the frequency of the persons with the character will be: r2 RR= l+a In other words, the frequency of the character diminishes in proportion to the expansion of the isolate, so that if the isolate is doubled, the number of persons with the character decreases to half their original number (Dahlberg, 1938). Now we know that the frequency of marriages between cousins was formerly much higher than now. In Bavaria, for example, it was 0.20% in 1926-33, while it was 0.87% 50 years earlier. This decrease is due partly to the fact that fertility and size of families has fallen somewhat, but above all to the fact that the development of communications, industrialisation, and migration to the towns has caused a breaking of the isolates. This process must materially have lowered the frequency of hereditary defects in civilized countries up to our own time. As far as the frequency of hereditary defects is concerned, the attempt to keep the population on the land is naturally of doubtful value. If, then, the frequency of marriages between cousins prompts us to assume that the breaking of isolates has caused, say, a doubling of the size, this should have led to a halving of the frequency of the rare defects, such as hereditary blindness, mental deficiency of different kinds, etc. If there is only a certain interchange beyond the boundaries of the isolate, the difference between isolates will take time to level out. The formula for the speed a t which this levelling-out takes place has been given by Wahlund (1928), who has also given the formula for the process in polyhybridism. If we assume that the isolate limits are abolished, so that free crossing took place, we would at once - i.e. in the next generation -get that composition of the population for monohybrid characters which is to be expected in panmixia. When polyhybrid charactersare concerned,

90

QUNNAR DAHLBERQ

however, the genes will accompany one another to a certain extent. The connection will only gradually be dissolved, giving the composition to be expected in panmixia. This process takes many generations. To take a practical example, it may be mentioned that the increase in stature which has been noticed in several countries, and which in Sweden amounts to 9 cm in 100 years, can probably be mainly attributed to isolate breaking (Broman, Dahlberg and Lichtenstein, 1942). 4. Isolates and Race It is customary to define a race as a group of individuals who show

deviating hereditary characters from other groups. Now this definition needs clarifying in several respects. For the moment we can only dwell on what is meant by a group in this connection. In a population, tall and short individuals can be differentiated according to some more or less arbitrary norm. But we do not consider the tall ones to constitute a separate race, provided they cross freely with the short ones. It is only if the tall individuals keep to themselves, i.e: form an isolate or a group of isolates, that we may be inclined to discuss whether they should not be regarded as a separate race. The isolate conception therefore in some ways lays the foundations for the conception of race (Dahlberg, 1942). If the members of an isolate or a group of isolates are to be regarded as a separate race, they must naturally show deviations from the members of other isolates, Strictly speaking, one cannot talk of racial characters, but only of racial differences. A character which is not a racial character today may become so tomorrow if a race is discovered without it. It is true that two races can be so completely different that it is possible to say with certainty whether an individual belongs to one or the other (e.g. negroes and Mongolians). When speaking of races which partly have a common range of variation, however, it is necessary to find a gauge for the degree of the difference. A suitable method is first to study the variability of the characters to be used in the two groups to be compared. The individuals of both groups are then imagined mixed together, and the knowledge one has of the separate groups is then used in trying to sort them out again correctly; one can, of course, relegate to the one group those individuals falling outside the other’s range of variation. The sorting out is done, first according to single characters, and then according to combinations of characters (Dahlberg, i841). The percentage of individuals correctly sorted out can be used as a measure of the size of the race difference. Another way is to use the information obtained for making the best possible dividing up of the mixed groups. It must then be remembered that in random sorting 50% can be expected to be correctly sorted out. The amount by which the empirical percentage exceeds 50 must therefore

GENETICS OF HUMAN POPULATIONS

91

be a gauge of the racial difference. The racial difference coefficient drawn up by Dahlberg therefore has the following form:

k=- 50 - p

(21)

100 where p is the percentage which can be differentiated with certainty. We may point out that when separating individuals of different races, the chances are better if several characters are used. Not much is gained, however, by increasing the number of investigated characters, if there is correlation between them. The smallest number, N, of characters required to sort out all individuals is obtained from the following fgrmula: 9(1 - r) N = (22) a* - 9r In this formula, r is the average correlation and a the average difference between the means of the races for the different characters (fig. 15). If there are racial diffcrences within a people, this must cause a

7,

4

6

8

10

12

14

18

18

20

22

24

Number of Characters

FIG. 15 The smallest number of characten required to differentiate each individual from two amaljamated groups at different degrees of average correlation (r) and average difference between the means (a). Cf. formula 22.

92

GUNNAR DAHLBERG

correlation between characters. If we find that there is no correlation between physical characters, it therefore means that there are no grounds for assuming different racial groups within a population, even if it originally sprang from different races. It is homogeneous. Naturally, we must here build on characters that are not occasioned by the same genes. For example, there is a correlation between the colour of the hair of the hezd and that of the body, which is contingent on the same genes; needless to say, such characters cannot be used in this connection. The correlation between genes diminishes one-half per generation, as has been shown by Wahlund (1932). This implies that racial differences are very quickly smoothed out by free crossing, i.e. by isolate breaking. 5. Isolates, Mutations and Random Selection

Recessive mutations will come to the fore more quickly in a population built up of small isolates than in one composed of large ones. When a recessive gene arises by mutation, it will only after some time occur in a “double dose” by means of intermarriage - soonest by a marriage of cousins. In other words, when the character appears, the gene has acquired a certain frequency in heterosygote form. That is to say, the frequency of the gene can never be pushed below this frequency (least heterosygote frequency) by negative selection - always assuming that the isolate in question is not very small or the frequency of the gene extremely low, since in that case the gene may disappear a t random. There is always a certain likelihood of a mutation disappearing, after once arising. The person or persons with the new gene may not procreate, and then of course the gene disappears from the isolate. A risk of this kind may also extend to future generations. On the other hand, a gene may be lucky and, by chance, have a very large circulation. It is easier in small isolates than in large for the gene to increase a t random until it is found in all the individuals there. These possibilities were first pointed out by Hagedoorn (1921). As previously mentioned, the two main investigators to treat this problem from a mathematical aspect are Sewall Wright and R. A. Fisher. It is plain that the mechanism indicated here will cause differences between the isolates making up a population. The random selection mechanism also means that, when polyhybrid characters are concerned, the frequency of persons with the character diminishes. If a character is contingent on several pairs of genes, it will be easier for some of the pairs, and therewith the prospects of the character, to disappear if the isolate is small. Different pairs of genes can in this way be expected to disappear a t random in different isolates. The genes occasioning the character can then only come together after a breaking of the isolate. The same is true of polymeric characters. Some of the

GENETIC8 OF HUMAN POPULATIONS

93

polymeric genes will disappear, others will increase, until they embrace the entire isolate. This process will diminish the difference between the isolates. When the isolate is broken, the polymerically conditioned character will have greater variability, while the mean will remain unchanged. 6 . Isolates, Intermarriage and SeZection

It has already been stated that if a recessive character is rare, the

individuals showing this character will to a relatively large extent be the children of cousins. If the character is very rare and the frequency of cousin marriages high, this situation will be particularly marked. When the population is divided up into isolates, however, it will mean a restriction of the possible differences. Very rare genes cannot occur in small isolates, since there must be at least two heterozygotes possessing the gene for the character to appear. It is therefore only in large isolates that a gene can be very rare. In large isolates, e.g. a capital, however, we cannot expect a high frequency of marriages between cousins. For this reason, the two factors, the gene frequency and the frequency of marriages entailing parents who are cousins will balance one another, so that the frequency of cousin marriages to be expected among the parents will be between 15% and 30% (Dahlberg,1938). This theoretical result has been verified empirically. The frequency of marriages between cousins among parents of individuals with rare defects keeps at the calculated per cent. In juvenile amaurotic idiocy, for example, it is 15%, in retinitis pigmentosa 17%) in Friedreich’s ataxia 9%) in albinism 17%, etc. This means that by forbidding marriage between cousins, the frequency of persons with the character can at best be lowered by 15%, but hardly by much more. The gain thus achieved is reached in a generation: there will be no change after that. On the other hand the process affects all rare characters, in the population, above all those with monohybrid recessive heredity. It does not act so strongly on the polyhybrid characters.

7. The Rise of Isolates While the gene content of a population is changed by mutations and selection, assortative mating and intermarriage - as already mentioned cause only a decrease of the heterosyygotism, without changing the gene content. The appearance of isolates has the same effect in regard to inherited characters which appear only in a few of the isolates. Conversely, isolate-breaking in this situation leads to an increase of the heterosygotism up to the level characterizing panmixia. Isolates of different make-up can arise in several ways. The cause may be selective emigration. When a new country is populated, e.g. America after about 1500, the immigration on which the population is

94

GUNNAR DAHLBERG

contingent is doubtless often of a selective nature. Moreover, the new cuuntry gives rise to isolates which are bound frequently to be of different natures. If the new isolates are small, they may happen to acquire different characters a t random. We know nothing of the extent to which populational differences between European countries are occasioned by prehistoric immigration of a similar kind. In regard to isolationism in the U.S.A., we know very little about where it is going to lead us. Another possible cause of differences between isolates is the process by whicfi social classes arise in a population. It is stated that the upper class is entitled to its favourable position by birth - that is to say, the upper class has arisen by a selective process, and contains a greater number of particularly able persons than the lower classes. However, we know very little of the origins of the upper class, but it is interesting to make a theoretical analysis of an assumed selection process of this kind. The simplest way is to assume that the population consists of only two isolates, isolate A (the upper class), which is comparatively small, and isolate B (the lower class) which is comparatively large. We will further assume that, initially, the isolates have the same frequency of a recessive gene, r, and that a certain proportion, k, of those possessing the character are transferred from B to A in each generation. The frequency which the character will have in isolate A is determined by the following recursion formula (Dahlberg, 1943). If the frequency in a given generation is xn2, the frequency in the following generation will be:

where rn2is the frequency of the character in isolate B. This frequency is determined by the previously given formula for partial selection (formula 4). That is to say, there is partial selection here, which under all circumstances should have very little effect, as the isolate is relatively large. The significance this selection process will have for isolate A depends on the intensity of the process, i.e. of the constants included in the above formula. Now, let us assume isolate A (the upper class) comprises, say, 5% of the population, that the character subjected to selective transference from one isolate to the other has the frequency of 1% a t the time the isolate arose, and that the selection covers 10% (k=0.1) of the persons with the character (i.e. 10% are transferred to isolate A in each generation), then in 10 generations the frequency will rise to 2.4% in A. If the selection only covers 1% (k = 0.01) of the persons with the character, the increase will be infinitesimally small, namely 1.1% per 10 generations. The corresponding effect in isolate B (lower class) with a selection of 1% (k = 0.01) will be a fall in the frequency of the character from 1% to 0.99%. If the selec-

95

GENETICS O F HUMAN POPULATIONS

tion is 10%(k=0.1), the frequency of persons with the character will fall in the same time to 0.94%. These figures shorn that the selection has little effect on the lower class, but may sometimes but not always noticeably affect the make-up of the upper class (fig. 10). k- I

1'0.024

0.022

0.020

0.018

0.016

0.014

-

0.012

k 0.50 0.010

0.008

k

-

k

-

0.W6

0.25

0.001

0.002

0

Qen.

1

2

3

4

5

6

7

8

0

0.10

10

FIG. 16 Composition of the upper class in successive generations when it embraces 5% of the population and the persons with the character at the start have the frequency of lo/* and when the transference to the upper class has the frequency 10% (k = 0.1) to 100% (k = 1) according to formula 23.

As previously mentioned, however, we have no empirical constants for the class circulation, so that calculations have as yet a solely theoretical interest, They might conceivably be thought of importance in that they show the impossibility of making any pronouncement about class differences that is based on real knowledge of what happened and is happening. Even if the upper class originated from selection, there is naturally no reason why we should be satisfied with the effect achieved. If an upper class of particularly high quality is wanted, the simplest thing should be

96

GUNNAR DAHLBERG

to give everyone the same chance of working up to a leading position. Genetically speaking, this would mean a gain, since the selection would then have a wider scope than if it were made only from a smdler part of the population.

8. Assortative Mating, Intermarriage and Isolates As has already been mentioned, assortative mating, intermarriage and isolates change the frequency of a gene, in that the genes are given another distribution than in panmixia. The deviation is due to the fact that the marriages which are contracted are determined in a certain way. It is therefore clear that it may sometimes be a matter of taste whether one talks of one process rather than another. As regards negroes in the U.S.A., for example, we can talk of the effect of isolates, but we can equally well say that there is assortative mating, caused by a special psychological attitude in the population. When dealing with certain defectives, e.g. deaf-mutes, blind, etc., we can similarly assume the effect of isolates, but we can also claim a certain degree of assortative mating. The defectives are brought together by societies, so that they have contact and marry to a particularly large extent. Their physical disabilities put difficulties in the way of marriage with others. In other words, the defectives form an isolate which has the peculiarity of being newly formed for each generation. Formulae have been drawn up for defective isolates of this kind, but they build on very uncertain empirical foundations (see Dahlberg, 1943). As the defectives are undoubtedly less fertile than the average population, we can say that there is selection in combination with isolates which are formed by nssortative mating. It is naturally also a matter of taste whether we attribute the heterozygote increase taking place in present-day populations to the breaking of isolates or to less intermarriage. The view taken in individual cases must depend on such things as the empirical points of departure we have at our disposal. VII. SUMMARY Human genetics have arisen from animal and plant genetics. In Germany this branch of research came to be influenced by ideas about race, as can be seen, for example, in the German term Rassenbiologie. I n that country it gradually came to enter into, and be influenced by, the Nazi ideology. In England, human genetics came to be linked with Galton’s eugenics (the doctrine of the well-born), and was therefore extensively directed

GENETICS OF HUMAN POPULATIONS

97

towards investigations of families considered to be of a particularly high quality. Thus, with human genetics focussed on population problems connected with politics as it was in England and Germany, no very important research was done. The main result from this point of view was the subjective coloured propaganda literature without any scientific value worth the name. However, both Germany and England produced many valuable detail investigations bearing on the heredity of normal and pathological characters, though there was perhaps too great a tendency to imitate the crossing experiments that had been made in animal and plant genetics. Now, human subjects cannot be crossed in order to see what the result will be. Instead, family investigations were made, from which were evolved pedigrees which, formally, bear a certain similarity to crossing experiments. But this is beginning at the wrong end - that is to say, the start is made not from the so-called first ancestors but from individuals now living, who are related and who show an accumulation of a certain character. From these, one proceeded backwards in time. The result a certain crossing would have on the human subject was not studied; it was to a certain extent a foregone conclusion, thanks to the selected initial material. In spite of this, important results have beyond any doubt been obtained, and a fund of knowledge collected as to how different characters, in particular more or less rare diseases, are inherited. Meanwhile, a few investigations into the mechanism behind changes of population have been made. They are theoretical and mathematical in character, and are to be found in the most varied periodicals. I have tried to review these investigations in the present paper, since it seems to me that such studies will have particular importance for human genetics in the future. Before attacking the problems practically, however, it is necessary to have a firm theoretical point of departure for the avenues of approach that are laid down. Mathematical investigations are also of importance, as they counteract the more or less incongruous constructions that are universally circulated in this field. But necessary though it may be to develop the theoretical-cummathematical side of the problems, the primary need is for empirical investigations of the processes taking place in human populations. We require both knowledge of the frequency of intermarriage, assortative mating, the formation of isolates, etc., and also investigations of the actual frequency of individual characters in populations. We have, however, still very little possibility of comparing the make-up of a population a t different junctures, or of comparing different populations at the same juncture. A great deal must be done to achieve an empirical foundation

98

GUNNAR DAALBERQ

for the assessment of populations from the viewpoint of heredity. But this must be regarded as a very important task for human genetics to carry out. VIII. REFERENCES Andreassen, M., Haemofili i Danmark. Ejnar Munksgaard, K~benhavn(1943). Broman, B., Dahlberp, G., and Lichtenstein, A., Acta pzediatr., Stoekh. SO, 1-66 (1942). Dahlberg, G., Genetics 14, 421-454 (1929). Proc. roy. Soc. Edinb. 68, 213-232 (1938). Die Vererbung dcr allergischen Disposition. P. Kallbs; Fortschritte dcr Allergielehre. s. Karger, Base1 and New York. (1939). The conception of race and a new mcthod of delimiting races, demonstrated on a material of Swedes and Lapps. G. Dahlbcrg and S. Wahlund: The race biology of the Swedish Lapps. Part 11. Almqvist and Wiksell, Uppsala. (1941). Hum. Biol.1 4 , 3 7 2 4 5 (1942). Acla med. scand. Suppl. 148 (1943). 2001.Bidr. Uppsala 26, 21-32 (1947) Mathematical Methods for Population Genetics. S. Karger, New York (1947). Dahlberg, G., and Hultkrantz; J. V., Arch. Ram.-u. Ges. Biol. 19,129-165 (1927). Hegedoorn, A. L., and Hagedoorn-Vonthcuvcl La Brand, A. C., The relative value of the processes causing evolution. Martinus Nijhoff, The Hague (1921). Haldtme, J. B. S.,Trans. Camb. phil. SOC.23, 19-41 (1924). J . Genet. 31, 317-326 (1935). Hardy, 0. H., Science 28, 49-50 (1908). Koller, S., and Geppert, H., Erbmathematik. Quellc und Meyer, Leipzig (1938). Lenr, F., Miineh. med. Wschr. 66, 2 (1919). M$rch, E. T., Chondrodystrophic dwarfs in Denmark. Ejnar Munksgaard, K$benhavn (1941).

Pearson, K., Philos. Trans. 203, 53-86 (1903). Pearson, K., and Lee, Alice, Biometrika 2, 357-462 (1903). Robbins, R. B., Genetics 2, 000 (1917). Genetics 3, 000 (1918). Sjhgren, T., Die juvenile amaurotischc Idiotic. Hereditas, Lund 14, 197-425 (1931). Skold, E., Acta med. scund. Suppl. (1944). Wahlund, S., Hereditas, Lund 11, 65-106 (1928). Demographicstudies in the nomadic and the settled population of northern Lapland, Uppsala (1932).