RANDOM MATING WITH THE EXCEPTION OF SISTER BY BROTHER MATING

I?Ar\l’DORl ALATIKG WITH

THE

EXC EP TIOK OF SISTER B Y B R O T H E R MATIKG

(Received March 2 3 , 19181

In tracing the fate of a Rleiidelian character, tlie results for random mating a re well known. The results for brother and sister mating have been published. I t is at once seen to be of interest to know what will be the results f or mating a t random excepting that brothers shall not

mate with sisters. For

the reader who does not care to follow the details of the discussion, the principal results are included in a summary.

We assume that all families have the same number of individuals, equally divided between males and females and that the division as to dominant

(A)

and recessive ( a ) characters is independent of tlie di- vision as to sex.

Let each family consist of 4 k male and 4 it. female children, 4 k

being an integer. I t is necessary to consider the types of families which arise from different crosses.

This paper deals with rlie problem just stated.

These are tabulated below.

TJ-pe of crohs

- -~

A A X ‘4A

,4A

x

Aa 1

A A

x

aa

Aa X An

,

A n X an

aa X aa

~ ~~~~

Letter indicating type of family and number of families of this type

I t seems essential t o express the numbers of families of the various types in one generation in terms of the corresponding numbers for the preceding generation. For this purpose we will use capital letters when

RA4NDO?VI M A T I N G E X C E P T S I S T E R BY B R O T H E R ₃₉₁

referring to one generation and small letters when referring to the pre- ceding generation. If we let

a study of the possible crosses enables us to write down the following recurrence relations :

f = p + q + r + s + t + t h ,

I t is from these equations that we must derive whatever information we get regarding the outcome of the type of mating under consideration. The complicated nature of the equations makes a direct solution of them seem hopeless. However, certain combinations of them lead to hopeful simplifications and finally yield much desired information. If we add the six equations just as they stand and use the notation

we have

A moment's thought shows that this must be the case, i.e., if each family has 8 k children, the number of families in one generation will be 4 k times the number in the preceding generation, it being assumed that one male mates with one female. If fa is the number of families in

the nth generation we have a t once

F = P

+

Q

+

R + S + T

+

U ,

F = 4 k f

392 RAINARD B. ROBBISS

2 ) f n = f o ( 4

k)".

Let a, b, c be proportional respectively to the number o f AA, A a and The table giving the composition of the

aa individuals in a generation.

various types of families enables us to choose a, b. c as follows: a = 2

p

+

q

+

s/2,

3)

i

b = q + 2 r + s + t ,

c=s/2

+

t

+

2 U .

W e note that a

+

b

+

ic = 2 f . By using equations 3 ) we are able to put equations I ) in a simpler form :

Q=- [ 2 a b - - 2 + 4 p ] ,

I

f--I

[2 a c - s / 2 ] ,

I

k

k

. I

f-1

4,

1

s

=-&-

[b' - b

-

2r],

<

If one wishes to use recurrence relations for the purpose of calculat- ing the numbers for one g-eneration from those for the preceding gen- eration, equations 4 ) are much more easily used than equations I ). By thinking of equations 3 ) as written in capital letters, we can use equa- tions 4 ) to obtain A , B, C in terms of the small letters thus:

[ ( b

+

2 a ) ( b

+

2 c ) - 3 b

+

2r--1, k

f-1 5 )

{

B =

[

( b

+

2 ~ ) ' - SC

-

b

-

2 I

+

S I .

k C =

I

Equations

5)

are more compact than equations 4 ) , but give us less information; in fact we could not use equations

5)

in calculating the numbers f o r succeeding generations because they do not give us R and

RANDOM MATING EXCEPT SISTER BY BROTHER ₃₉₃

[ z ac

-

s/2],

.

k

R=-

f-1

we have a set better adapted for calculating the numbers for successive generations than is set 4).

6 ,

I

2

S

+

Q

+

T = 4

k b.

Here we have a set of recurrence relations solved for the variables which, if we were using subscript notation, would have lower subscripts. The equations are useless for direct computation but they will soon become important in an indirect way.

From equations

5 )

we find that

B + 2 A = 4 k ( b f z a ) ,

')

{

B + z C = q k ( b + 2 c ) ,

Whence,

B + 2 A b + 2 a

53)

iI

B + Z C b + 2 C

T h e ratio of dominant to recessive gametes in a particular generation is ( b

+

z a ) / ( b

+

z c ) .

Equation 8) shows that this ratio is the same in one generation as in the preceding generation; from this we deduce immediately that the ratio of dominant to recessive gametes combining to form the individuals of the various generations is constant. This is a significant fact in that it is also true for complete random mating and for brother and sister mating. Using subscript notation, the solutions of system

7 )

are

Combining equations 4) we find that

z P + Q + R = 4 k a ,

2

U

+

T

+

R

=

4

k

C,

-

-

bn+2an= ( b o + 2 a o ) ( 4 k ) " . 9,

c

bn

+

2 ~ n = ' ( b ,

+

2 ~ 0 ) ( 4 k ) " .

By means of equations 3 ) , equations

6)

may be rewritten with sub- scripts thus :

rvL

-

4 2= 4 k a,-l

-

an,

Y,, - ~ , / 2 4 k ( ~ ~ - 1 - Cn,

-

2 r,

+

s, = 4 k b,,-l

-

b,. I O >

[

I n the light of 9) and I O ) , equations

5 )

may be written with subscripts as follows:

KAIN.IRD E. K O E B I S S 391

The solutions of this set of equations would give us U,, b,,

c,. in terms of the original data.

Consider first the case in which there are just two children in each family; in this case, 4k = I . Then equations I I ) simplify considerably, giving us,

I t is well now to distinguish two cases.

(

S(f,--r)a,,

t

.+a,l-l

+

2a,-, = ( b o

+

2 u d (b,,

+

za,,-~

1 %

4(fe-1)b,,

+

~ b , , _ ~

+

b,-2 = ( b o

+

2a0) ib,,

+

z c , ) ,

1 2 )

1

8 ( fo-1.) e,

+

4 ~ , , - ~

+

2c,_, = (bo

+

2c0) ( b o

+

2cO---I

1 .

The solutions are

(0,

+

L a o ) ( b o

+

2Uo-I

1

,

S f 0-2 a,, = p " [ k , cos n ~

+

k , sin n 61

+

I

2 (b,,

+

2 a " ) (bo

+

2co)

1

13)

/

b,L = - 2pn[kl cos n 0

+

k , sin 1% 01

+

?

I 8 f 0-2

I

- -- in which ~ / p =

zvf0--I

and tan 0 =

-

v4fo-j.

Aside from the value of these equations for purposes of computation, they give us the limiting values of a,, b,, c, as 1% increases indefinitely.

RANDOM M A T I N G E X C E P T S I S T E R BY B R O T H E R ₃₉₅

Fo r completely random mating, the proportions of the three types a re fixed after the first random mating and are'

a ' : b': c': : ( b o

+

z a 0 ) ' : 2 ( b o

+

z a o ) ( b o

+

2c0) : ( b o

+

zco)'.

I n brother and sister mating, the heterozygous type tends to disappear and for a limiting population we have (ROBBINS 1917)

Uy comparing these three sets of results we note that omitting the brother and sister mating has had the effect of increasing the propor- tion of heterozygous individuals over what it is in completely random mating, w h e n there are orcly two offspring in a family.

Consider now the case in which 4 k > I , i.e., the case in which there

are more than two offspring to a family. Returning to equations 11), it would be desirable to solve these equations without restrictions on k

if possible. However, the nature of the solutions of the simple case in which 4 k = I seems to indicate that probably the solution of the gen- eral case would be of little use for purposes of computation even if we had it. W e can get all desirable information about the limiting popula- tion without solving equations I I ) . W e can calculate immediately from equations I I ) that if 4 k > I ,

limit a,

(4 Iz)" 8 f o

b"

( 4 k ) "

( 4 IC)" 8 f o

a': c': : ( b o

+

zao)

: ( b o

+

2c0).

( b o

+

2ao)'

2 (bo

+

Z C O ) (bo

+

zao)

8 f o

(bo

+

2 c o ) 2

- _, n=co

-- -

limit n=co limit n=co 9 - c7l

--

Then using a', b', c' for the limiting values as above,

This is the same result as in random mating. Thus in case a typical family consists of more than two offspring, the effect of omitting brother and sister mating from otherwise random mating is only temporary, the final results being the same as for completely random mating.

a ' : b ' : c ' : : ( b o + 2 a o ) 2 : z ( b + 2 a o ) ( b o

+

2 c o ) : ( b o

4-

zc0)'.

SUM M A R Y

This paper is a discussion of the problem of random mating with the exception that brothers shall not mate with their sisters, a single pair of Mendelian $actors being considered. Three principal results are ac- complished. First, formulae are developed from which we can calculate

See for instance WENTWORTH and REMICK (1916).

396 RAINARD B. RORBINS

the distribution of the offspring for succeeding generations ( a ) type of family,-equations 3) and 4)-and ( b ) as to type of indiv -equations

5).

problem is solved completely in equations I 3). Third, the distributiot. of the limiting population is given in case there are two offspring in a family and also in case there are more than two offspring in a family. The most important conclusions may be stated thus :

W h e n brother and sister m t i n g

is

omitted from otherwise random

mating, the progeny in succeeding generations approachs a fixed distri-

bution as the number of generastions increases, in " k F , pure dontinants,

heterozygotes and recessives all appear. ( a ) I n case there are but two

"#spring

in

a. typicad family, the proportion of lzeterozygotes in the limit-

ing population is greater than f o r completely random mating. ( b ) I n

case there w e more than two offspring

in

a typicad family, the linzitifig

fiopudation wdl be the same as

i f

the mating had been conapletely random.

Second, in case a family consists of two offspring t.

L I T E R A T U R E C I T E D

WENTWORTH, E. N., and REMICK, B. L., 1916

eralized Mendelian population.

RCBBINS, R. B., 1917

netics 2 : 489-504.

Some breeding properties of the gen-

Ge-

Genetics 1 : 608-616.