A Mathematical Study of the Founder Principle

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A Mathematical Study of the Founder Principle of Evolutionary Genetics

Author(s): P. Holgate

Source: Journal of Applied Probability, Vol. 3, No. 1, (Jun., 1966), pp. 115-128

Published by: Applied Probability Trust

Stable URL: http://www.jstor.org/stable/3212041

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J. Appl.Prob.3, 115-128(1966)Printed n Israel






A MATHEMATICAL STUDY OF THE FOUNDER PRINCIPLE

OF EVOLUTIONARY GENETICS

P. HOLGATE, The Nature Conservancy,London

Smmary

Some comparisons are made between various characteristicsof the geneticstructures of populations of the same size and age, which have (i) evolvedfrom a small founder population, and (ii) evolved from a population whichhas been of constant size throughout the period considered.

1. Introduction

Consider a gene having two alleles A and a. In the absence of selection, and if

there is no mutation, it is possible that one of these will become fixed in a popula-tion as a result of the random fluctuations in its frequency from generation to

generation. This is generally known as the Sewall Wright effect, and its possiblecontribution to evolution has been the subject of considerablediscussion. A sum-

mary of Wright's own views and a survey of his mathematical work on the subjectare given in Wright (1964).

Mathematical work on the tendency to homozygosity of a finite populationhas been surveyed by Moran (1962), and Kimura (1964). Roughly speaking, for

given initial gene frequencies, the probability that the populations will become

homozygous within a given number of generations may be appreciablefor small

populations, but becomes negligible as the population size increases.

Suppose however that a small founder population colonises an area, and inthe course of several generations multiplies to a much greater size. Fixation mayoccur at a given locus when the population is small, and the effect of populationgrowth will then lead to a homozygous population of such a size that fixationas a result of random fluctuation would be inconceivable in a population whichhad always had that same, large size. This phenomenon is known as the founder

principle, associated with Mayr (1942, 1963). Experimentalwork in relation to itis reportedby Dobzhansky and Pavlovsky (1957), and it has recentlybeen invokedin the controversy about the origins of geographical variations between snailcolonies in Southern England, (Goodhart 1962, 1963;Cain and Currey 1963a, b).

Despite its interest, little mathematicalwork has appearedon geneticfluctuations

Receivedin revisedform 6 May 1965.

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A mathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary enetics

this question for the growing population. If Zt is the number of heterozygotes

in Ft then{Zo

= 1, Z1, Z2, "*-} is a branching stochastic process of Galton-

Watson type. (For the theory of these processes see Bartlett (1960), Harris(1963)).

The probability generating function of the number of heterozygous offspring

of a heterozygote is

(1) f(s)=(2+f s).

Since the mean of the distribution given by (1) is 1, the ultimate elimination of

heterozygotes is certain. The p.g.f. of Zt is obtained by functional iteration which

may be carried out explicitly for small t. For instance, the probability that F3

will contain j heterozygotes is the coefficient of sJ in

(2) f3(s)= 2( 2(s+2 + 2 +2 '

This distribution is tabulated in the first column of Table I.

The elimination of heterozygotes in a population where each individual con-

tributed a single offspringto the next generation was discussed by Bartlett(1937).

Suppose, for comparison with the above case, that a population had been initiated

by 8 heterozygotes,and had undergone 3 generations of selfing,with each individ-

ual contributing a single offspring to the next generation. The probability that a

given line would remain heterozygous is 1/8, and the p.g.f of the number ofheterozygotes in F3 is therefore

(3) g3(s) = ( 1 + 7)8

This binomial distribution is tabulated in the second column of Table I.

TABLE

Distribution of number of heterozygotes

after three generations of selfing

ProbabilityGrowing Stationary

Population Population

0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000


in Ft then{Zo





(1) f(s)=(2+f s).





(2) f3(s)= 2( 2(s+2 + 2 +2 '








(3) g3(s) = ( 1 + 7)8


TABLE





0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000


in Ft then{Zo





(1) f(s)=(2+f s).





(2) f3(s)= 2( 2(s+2 + 2 +2 '








(3) g3(s) = ( 1 + 7)8


TABLE





0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000


in Ft then{Zo





(1) f(s)=(2+f s).





(2) f3(s)= 2( 2(s+2 + 2 +2 '








(3) g3(s) = ( 1 + 7)8


TABLE





0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000


in Ft then{Zo





(1) f(s)=(2+f s).





(2) f3(s)= 2( 2(s+2 + 2 +2 '








(3) g3(s) = ( 1 + 7)8


TABLE





0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000


in Ft then{Zo





(1) f(s)=(2+f s).





(2) f3(s)= 2( 2(s+2 + 2 +2 '








(3) g3(s) = ( 1 + 7)8


TABLE





0 .483459 .343609J

1 .217285 .392696

2 .176514 .196348

3 .077637 .056099

4 .032593 .010018

5 .009277 .001145

6 .002686 .000082

7 .000488 .0000038 .000061 .000000

1171717171717



Despitethe smallnumberof generations,TableI illustrateshefactthat boththe

probabilityof completeelimination of heterozygotes,and the probability hattheir numberwill be large,aregreateror thegrowing population.The variancesare1.5 for the growingpopulationand0.875 for the stationary ne.

Thevarianceof Z, forgeneral maybe obtainedby letting -2 inFormula7a)in Bartlett 1937),or by substitutingor a2 inFormula 5.3)inHarris 1963,p. 6),eitherof which ead to:

(4) Zt 1, varZ,= t.

The probabilityhat therewould be no heterozygotes fter t generationss given

by the asymptotic ormulaof Kolmogorov 1938)as

(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t

In fact, the p.g.f. (1) and result 5)arementionedby Kolmogorovas an exampleat the end of Section4 of his paper,where heyarise n the solutionof adifferent

problemn populationgenetics. t maybe notedthat t has to be quite arge or(5)to give good numericalresults.For t = 40 when (5) gives 0.9, the correct alueobtainedbyiterations 0.9142.However hederivation ivenbyHarris1963,p. 20

ff.)suggests /(4+ t) as a betterapproximation, ndfor t = 40 it gives0.9091.For

comparisonconsider an Ft which had arisenfrom an Fo of 2t heterozygotes,evolving throught t generationsof selfingatconstant population size, and letZt be the numberof heterozygotesn Ft in this population. It shouldbe bornein mind that the sequence{Z'},althougha stochasticprocess n themathematical

sense,does not represent he evolutionof a definitepopulation.)By reasoningsimilar to that given in the case t = 3, the distributionof Zt is binomial withn = 2t, p = (1/2)t, and hence

(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.

Comparison f thevariances iven n (4)and(6), and theprobabilitieshat hetero-

zygoteswill be absent,given in (5) and (7), togetherwith the numericalresultsof TableI, illustrate he fact that selfingpopulationswhichhavereacheda largesize by repeateddoubling will be more variable in respectto proportionsof

heterozygotes, nd in particularwill be morelikelyto containnone, thanthose




(4) Zt 1, varZ,= t.



(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t






(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.







(4) Zt 1, varZ,= t.



(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t






(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.







(4) Zt 1, varZ,= t.



(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t






(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.







(4) Zt 1, varZ,= t.



(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t






(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.







(4) Zt 1, varZ,= t.



(5) Pr(Z, =0) 1- 2 I 4- 1tf(1) t






(6) Z;=l1, varZt'=(1- ())

(7) Pr(Zf=0)== (1-( ) e- =0.367879.




1181818181818 P. HOLGATE. HOLGATE. HOLGATE. HOLGATE. HOLGATE. HOLGATE



A mathematicaltudyof the ounderrinciple f evolutionaryeneticsmathematicaltudyof the ounderrinciple f evolutionaryeneticsmathematicaltudyof the ounderrinciple f evolutionaryeneticsmathematicaltudyof the ounderrinciple f evolutionaryeneticsmathematicaltudyof the ounderrinciple f evolutionaryeneticsmathematicaltudyof the ounderrinciple f evolutionaryenetics

of the same size and age that have always been that size. The stochastic process

describing the number of individuals in Ft which are homozygous for a given

allele, say A, is not a branching process, nor even a Markov process. However,

the probability P, that Ft will consist entirely of AA's, has a simple recurrence

relation. The probabilities that the offspring of a heterozygote are (AA,AA),

(AA,Aa) or (Aa,Aa) are 1/16,1/4,1/4 respectively. It is then required that the

offspring should consist entirely of AA's after (t- 1) generations, which leads to

11 1 2Pt = + Pt-1 + Pt2-1

from which P3 = 0.083840. Since the Pt are obviously increasing, lim0 Pt = P,

the probability that the population ultimately consists entirely of AA's satisfies

11 1 1 2P=- +-p +P2

16 4 4

of which the relevant root is P = 1 1/2 - V2 = 0.085786. For the comparable

stationary population the chance that all its members are AA is given by

(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.

In the doubling population the elimination of heterozygotes occurs with

probability one. When it has occurred the population which will then consist

entirely of AA's and aa's will continue to increase, but with the numbersof these

types, and hence of the two alleles, in a constant proportion. The remainderof this section is devoted to obtaining some propertiesof Y,the randomproportion

of AA's when the heterozygotes have been eliminated. Clearly, Y can take only

those values whose expression as a binary fraction can be written in terminating

form, and every such value in the interval [0,1] is taken with positive probability.

It is therefore an example of a random variable whose distributionfunction has

jumps at a set of points everywheredense (in [0,1]), although this peculiarity is a

mathematicalconsequence of insisting on the population doubling exactly at each

generation. If Ytis the proportion of the limiting population belonging to lines

which became AA at exactly the tth generation, then Y = t 1 Yt. t is interesting

to note that the stochastic process

2Yt, j=1,2,-.,t= I

is such that almost every realisation is monotonically increasing, the form of the

'remainder' term in the submartingale decomposition theorem (Loeve, 1963,

p. 389). Some properties of Y may be obtained by enumerating the possible

structures of F1, and equating the unconditional expectations of certain variables,

to the appropriate expectations obtained after conditioning by the outcome of F1.

The possible F1 's are listed in column 2 of TableII,preceded by their probabilities.








11 1 2Pt = + Pt-1 + Pt2-1



11 1 1 2P=- +-p +P2

16 4 4



(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.














2Yt, j=1,2,-.,t= I














11 1 2Pt = + Pt-1 + Pt2-1



11 1 1 2P=- +-p +P2

16 4 4



(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.














2Yt, j=1,2,-.,t= I














11 1 2Pt = + Pt-1 + Pt2-1



11 1 1 2P=- +-p +P2

16 4 4



(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.














2Yt, j=1,2,-.,t= I














11 1 2Pt = + Pt-1 + Pt2-1



11 1 1 2P=- +-p +P2

16 4 4



(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.














2Yt, j=1,2,-.,t= I














11 1 2Pt = + Pt-1 + Pt2-1



11 1 1 2P=- +-p +P2

16 4 4



(1/4 + 1/2 *1/4 + 1/2 *1/2 *1/4)8 = 0.001342.














2Yt, j=1,2,-.,t= I







1191919191919



Column 3 givesthe conditionalexpectationof Y,columns4 and 5 give the con-

ditionalvarianceof Y and the probability hat Y = 0, wherev and P denote he

unconditionalvalues of these quantities.Column6 gives the conditional xpec-tation of ez, wherem(z) is the unconditionalvalue.

TABLE I

Probabilitiesof possibleF1's,and values of various haracteristicsf populationstructure onditionalon these,for the growing,selfingpopulation

Pr. F,1 S{Y\F1} var{Yr|FI Pr{Y=OIF,} ?{ez\F,}

1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA




TABLE I



1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA




TABLE I



1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA




TABLE I



1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA




TABLE I



1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA




TABLE I



1AA,AA 1 0 0 ez16

- AA,Aa 4 ^ 0 e '2m( z)

AA,aa1

0 0 ezI

Aa,Aa 2 \2

(z

aa,aa 0 0 1 116

AA

For instance to obtain v we have on writingdown the two expressionsor the

secondmomentaboutzero:

-v+4T6H+4+4v+8(1+1 4 + )14(16+4)whichgivesv = 1/12. For the probabilityof the eliminationof A,

1 1 1p 2P=- +-P + -p2

whichwas derivedaboveby a more or lessequivalentmethod.An identityor the

moment generating unction written down from the last column simplifieso

4m(z)= {(l+ez/2)+m( z)}2

Clearly,Yis distributedymmetricallybout1/2.If (z)is the momentgenerating




1 1 1p 2P=- +-P + -p2



4m(z)= {(l+ez/2)+m( z)}2





1 1 1p 2P=- +-P + -p2



4m(z)= {(l+ez/2)+m( z)}2





1 1 1p 2P=- +-P + -p2



4m(z)= {(l+ez/2)+m( z)}2





1 1 1p 2P=- +-P + -p2



4m(z)= {(l+ez/2)+m( z)}2





1 1 1p 2P=- +-P + -p2



4m(z)= {(l+ez/2)+m( z)}2





A mathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary eneticsmathematical tudyof thefounderprincipleof evolutionary enetics

function of Y - 1/2, then on writing m(z) = e'Z2 /(z), and then replacing z by

2z for convenience,+(z)

is found to satisfy the functional relation

(9) 4+(z)= cosh z + (z) .

Since Y is bounded, all its moments exist, and expansion of the m.g.f. in (9)

provides a recurrencerelation for the even ordercentralmoments. The first few are

1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'

of which the first value confirms the solution of (8). Since the probabilities of

the end points are known exactly, the moments of the distribution conditionalon fixation not occurring can be calculated from the formula

(10) 12k -2P 2k-2P )

from which we obtain

12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.

The coefficient of kurtosis of the conditional distribution is f2 = 2.3071.

3. RandommatingNotation used in this section is not everywherecomparable with that of the last.

Since a random mating population cannot be initiated by a single individual,

in order to make the situations in this section and the last as similar as possible,

I have assumed that the population begins with an F1 consisting of 2 animals,

each of whose 4 genes are A or a with probability 1/2. This means that if the

mathematical definition were extrapolated backwards, independently of its

interpretation, we could imagine an Fo consisting of a single heterozygote. As in

studies of random mating populations of constant size, only the number of genes

in each generation is considered, and they are supposed to be determined inde-

pendently, being A or a with probabilities equal to the proportions in the previousgeneration. Moran (1962, pp. 12-20) has discussed the implications for the form

of the offspring distribution of the constant population model and analogous

considerations apply here. Let Zt be the number of A genes in F,. The distribution

of Zt conditional on Z- = z,_ is binomial with n =2'+ p = (1/2)tzti.

The distribution may be computed explicitly for small t, and for t = 3 it is given

in the first column of Table III. For a comparable population of constant size,

suppose that Fo had consisted of 8 heterozygotes, which had then evolved through

3 generations of random mating. The distribution of Z3, the number of A's in

F3 is given by post-multiplying the 1 x 17 row vector with 1 in its ninth column

and zeros elsewhere, three times by the matrix (ail) with elements




(9) 4+(z)= cosh z + (z) .



1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'



(10) 12k -2P 2k-2P )


12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.
























(9) 4+(z)= cosh z + (z) .



1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'



(10) 12k -2P 2k-2P )


12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.
























(9) 4+(z)= cosh z + (z) .



1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'



(10) 12k -2P 2k-2P )


12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.
























(9) 4+(z)= cosh z + (z) .



1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'



(10) 12k -2P 2k-2P )


12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.
























(9) 4+(z)= cosh z + (z) .



1 11 121 61663#2 2= , P4 =

720 6 = 36288' P8 79315200'



(10) 12k -2P 2k-2P )


12 = 0.048816, /14= 0.005498, 16 = 0.000789, % = 0.000129.





















1212121212121



al= 16- i1

iJ{ HJ)

( )6 16)

(Feller, 1951).The resulting distribution is given in the second column of Table III.

In this case the probability of elimination of A can be seen to be greater for the

growing population. The variances of the proportion of A genes are 0.0962 and

0.0440 for the growing and stationary populations respectively.

TABLE II

Distribution of number of A's after three generations of random mating

(The distribution is symmetric about 8.)

Probability

Growing Stationaryj Population Population

0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096

Now consider the sequence of random variables Y = Z/2t+1, the proportions

of A genes in successive generations.

It will be shown that Yt ends to a limit Y as t -+ oo, and some of the propertiesof Y obtained.

A recurrencerelation can be derived for the second moments of the Zt about

the origin as follows: let P(z,t) = Pr(Zt = z),

2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)

al= 16- i1

iJ{ HJ)

( )6 16)





TABLE II



Probability


0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096






2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)

al= 16- i1

iJ{ HJ)

( )6 16)





TABLE II



Probability


0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096






2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)

al= 16- i1

iJ{ HJ)

( )6 16)





TABLE II



Probability


0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096






2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)

al= 16- i1

iJ{ HJ)

( )6 16)





TABLE II



Probability


0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096






2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)

al= 16- i1

iJ{ HJ)

( )6 16)





TABLE II



Probability


0 .0996 .0068

1 .0284 .0155

2 .0423 .0290

3 .0505 .0457

4 .0562 .0639

5 .0607 .0816

6 .0738 .0964

7 .0655 .1062

8 .0661 .1096






2t"I

He(t)

= 2x2P(x,

t)

=t

x 2y(2X

=O 0

= ]P(y,t - 1) X2y

y=0

2= P(yt - 1) 2y 1 +(2 -I1) - -

H;( -)+( 2-^l )82( -)




A mathematicaltudyof thefounderprincipleof evolutionary eneticsmathematicaltudyof thefounderprincipleof evolutionary eneticsmathematicaltudyof thefounderprincipleof evolutionary eneticsmathematicaltudyof thefounderprincipleof evolutionary eneticsmathematicaltudyof thefounderprincipleof evolutionary eneticsmathematicaltudyof thefounderprincipleof evolutionary enetics

The step from the thirdto the fourth line follows from the fact that the secondsummation s the second moment about zero of a binomialdistributionwithn = 2+, p = y/2t. On writing 4 = u2

+ ,u2 a recurrence elationfor the var-iancesof Z, is obtained

82(t) = (4-

1 2(t-1) + 2t-.

If v2(t) = #2(t)/4t+l denotesthe variances f the Yt heysatisfy herecurrenceela-tion

(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )

Since the variancesof all distributions n [0,1] areboundedaboveby 1/4, (11shows that the sequence tends monotonically, as t -> ooto a limit

_

1/4.On writing 11)in the form

I I 1( V2(t)) ()(--2(t-))

an explicitsolution can be obtained or any t,

(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).

More lengthyalgebra eads to the followingrecurrencerelation for the fourthmomentsv4(t)

{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .

With nitial valuesv2(0) = v4(0)=0, computationusing(11) and(12) leadsto thelimits

(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),



82(t) = (4-

1 2(t-1) + 2t-.


(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )


_


I I 1( V2(t)) ()(--2(t-))


(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).


{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .


(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),



82(t) = (4-

1 2(t-1) + 2t-.


(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )


_


I I 1( V2(t)) ()(--2(t-))


(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).


{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .


(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),



82(t) = (4-

1 2(t-1) + 2t-.


(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )


_


I I 1( V2(t)) ()(--2(t-))


(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).


{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .


(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),



82(t) = (4-

1 2(t-1) + 2t-.


(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )


_


I I 1( V2(t)) ()(--2(t-))


(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).


{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .


(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),



82(t) = (4-

1 2(t-1) + 2t-.


(11) v2(t)= v2(t- 1) +V-4T ~- v2(t 1) )


_


I I 1( V2(t)) ()(--2(t-))


(--.(,): (1_4) (,-)..( __ ) -(4 0(0)).


{IV 1 I 2t3 1 t

V4(t) =4(t() )

d 2 )( , - , ) { ~ ) ) _ ( T ) .


(13) v2 = 0.105606, V4= 0.020182.

Foreveryk,

V2k(t)= Yt- )

-tPr (Yt1 = q)

t

2 Yt-=

>_ Pr(y,t = ) (?-

= V20(t-1),

1232323232323



on applyingJensen'stheoremto the convexfunction {(Yt- )2k Yt-1=t }, andthen usingthe symmetry f Yt-1about ?. Hencethe sequencev2k(t) is increasingandit is bounded aboveby(1/2)2k.Hence he momentsequences f the Yt end to

a limiting sequenceand since this must be the moment sequenceof a boundedrandomvariable, t definesit uniquely,and the convergenceof the sequence

{Yt} follows from the secondlimit theoremof the probabilitycalculus.(Seee.g. Rao and Kendall(1950)).Alternatively,{&Yt Yt_-I Yt-} = Yt-1 and so

{Yt}is a martingale,and the fact that its random variables are bounded is

easily enough seen to ensureconvergence y themartingale onvergenceheorem

(Doob, 1953,Chapter7).

In thisparagraph nd the next I obtainupperand lowerbounds ortheproba-

bilityof ultimatehomozygosity, howing n particular hat it is less thanone. Inthe presentproblem,actual computation or the early generations ollowedbynumerical extrapolationenables the exact value to be calculatedaccurately,but the methods of obtaining he bounds are of interest,andmaybe of valuein

parallelstudies, since it is clearlyimportantto know whetherthe probabilityof absorptionis boundedaway from one. The coefficientof kurtosis of Y is

P2 = 1.809738.Henceusingthe generalisation f Tchebychef'snequalityquoted

e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.

The probability hat a givenallele,say A, will be fixedis less than half this, i.e.0.1511.

Let yt be the probability hat the populationbecomeshomozygousat exactlythe tth generationconditionalon it beingheterozygousat the (t - 1)th,y, the

probabilitythat it becomeshomozygous at or before the tth generation,and

y = lim, . Vt hat it ultimatelybecomeshomozygous.The chance that fixationwill occur preciselyat time t can be shown to be greatestwhen Ft-_ contains

only one representative f one of the alleles,and least whentheyarepresentn

equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -











e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.






equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -











e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.






equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -











e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.






equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -











e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.






equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -











e.g. by Cram6r1946, p. 256)

Pr ( |ry - 1.5385997)

(P2 1)/{(1.5385992 - 1)2 + P2 1}

= 0.3022.






equalnumbers.Hence

1 )t 2t ( )t 2t+1

^(~~~~~-a'^

) ) " -




A mathematicaltudy of thefounderprincipleof evolutionary eneticsmathematicaltudy of thefounderprincipleof evolutionary eneticsmathematicaltudy of thefounderprincipleof evolutionary eneticsmathematicaltudy of thefounderprincipleof evolutionary eneticsmathematicaltudy of thefounderprincipleof evolutionary eneticsmathematicaltudy of thefounderprincipleof evolutionary enetics

Suppose hat for some k, yk is known, then

Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -

By computation, heprobabilitieshat a givenallele,sayA will be fixedbythe kth

generation avebeenobtainedupto k = 5, and aregiven n TableIV.

TABLE V

Probabilitiesof the fixationof A aftert generationsof randommating,for the

growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536

Takingk = 5 in (14)leads to

1y > 0.105598,

which n thepresentcaseis only slightly arger hanI 75. Using the 62 technique

(Aitken,1926) a numericalapproximation o ly may be obtainedfrom the

termsgiven

in Table IV.Applying

the method toI72,

i73 andI74 gives

Iy = 0.10669, and applying it to I73, I74 and Iy5 gives Iy = 0.10672, which

is thereforeprobablycorrectto 5 decimalplaces.Acceptingthis the moments

of Y conditionalon absorptionnot havingtakenplace may be calculated rom

(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.

For a populationof 2t individualswhichhas evolved through t generationsof

randommating at constant size, the probabilityof fixation is approximately

1-(1-(1/2)t+1)' (Moran, 1962, Chapter 4), and this approaches zero very

quicklyas t increases.


Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -



TABLE V


growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536


1y > 0.105598,



termsgiven


the method toI72,

i73 andI74 gives




(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.






Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -



TABLE V


growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536


1y > 0.105598,



termsgiven


the method toI72,

i73 andI74 gives




(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.






Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -



TABLE V


growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536


1y > 0.105598,



termsgiven


the method toI72,

i73 andI74 gives




(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.






Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -



TABLE V


growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536


1y > 0.105598,



termsgiven


the method toI72,

i73 andI74 gives




(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.






Y = Yk+ (1 - yk)Yk+ + (1 - Y)(l - Yk+I)7k+2 +

[1 (1 )k+i 1 \ ( k+

>= +-) +(l-

-e-)

(14) ( 1 k+2

Y-k + (Y) [ ) 1 1([2eI)}] -



TABLE V


growingpopulation

k 1 2 3 4 5

i Yk .062500 .088997 .099550 .103808 .105536


1y > 0.105598,



termsgiven


the method toI72,

i73 andI74 gives




(10)and(13),

t2 = 0.06643, r/4= 0.00982, P2 = 2.23.





1252525252525



4. A fullystochasticmodel

A simplefully stochasticmodelis obtainedby lettingthe numberof offspringof each individual be, not a constant,but a Poisson variable.Bartlett(1937,Formula 7b))obtained he variances f thenumbersof heterozygotesn successive

generations,(without conditioningon the population as a whole remaining

non-extinct).For the case where i) the Poissonmean s unityand hencethe mean

populationsize is constant, ii)the numberof generations ufficientlyargefor the

probability hat the heterozygousoffspringof a given individualwill not have

been eliminatedto be given by the asympoticformuladerivede.g. by Harris

(1963, p. 18),Formula9.5), and (iii) the founderpopulationso large that the

chanceof completeextinction s negligible,Bartlettalso showedhow the proba-

bility that the population should consist entirelyof homozygotes could becalculated.

In the remainder f this sectionI present ome numerical aluesrelating o the

resultsof threegenerationsof mating,for a population nitiatedby a singlein-

dividualandhavinga mean rate of increaseof 2, and for a population nitiated

by 8 individuals,and havinga mean rate of increaseof unity.If the numberof

offspringof each individuals a Poisson variablewith mean2, andthe offspringof a heterozygote re AA, Aa, aa withprobabilities1/4, 1/2, 1/4. Thetrivariate

stochasticprocessgiving he numbers f the three ypes n successive enerationss

a multi-type Galton-Watson process (Harris, 1963, Ch. 2) with generatingfunctions

f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).

Takingthe generatingunctionfor Fo as s2, that for Ft may be obtainedbymultivariateunctional teration.Fromit, the probabilitieshatvariouscombina-

tions of classes will have no representativesmay be computed.The probabilityof completeextinction urns out to be 0.192975.Onmultiplyingup theremaining

probabilities,he probability hat a population,giventhat it is still in existence

after threegenerations,consists entirelyof AA's, or entirelyof homozygotes,

may be calculated.The comparable tationarypopulation s obtainedby takingthe mean numberof offspringas one instead of two, and startingwith eight

heterozygotes.n this case the chance of total extinction n F3 is much smaller,0.023558.In Table V a numberof probabilities elating o the F3's of various

models consideredare collectedtogether.


















f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).

























f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).

























f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).

























f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).

























f '(S, S2,S3)= exp2(sl - 1)

f2(sl,s2,s3) = exp2 s + + 3-

f3 (s1, s2,s3) = exp2s3- 1).











A mathematicaltudyof thefounderprincipleof evolutionary enetics 127

Acknowledgements

I am grateful o Mr. J. G. Skellam or severalhelpfulsuggestionsmadein thecourse of this work,andto Mr. K. Lakhaniand Mr. D. Spalding or assistancein computation.

TABLEV

Some fixationand eliminationprobabilitiesor variousmodelsconsidered.Forstochasticgrowth models probabilitiesare conditionalon completeextinction

not havingoccurred.

ProbabilityhatF3 consistsentirelyof:(i) AA's (ii) homozygotes

Growingpopulation

Deterministiceproduction 0.0838 0.4835

Stochastic eproduction 0.1580 0.4555

Stationary opulation

Deterministiceproduction 0.0013 0.3436Stochastic eproduction 0.0645 0.3222


Acknowledgements


TABLEV


not havingoccurred.


Growingpopulation






Acknowledgements


TABLEV


not havingoccurred.


Growingpopulation






Acknowledgements


TABLEV


not havingoccurred.


Growingpopulation






Acknowledgements


TABLEV


not havingoccurred.


Growingpopulation






Acknowledgements


TABLEV


not havingoccurred.


Growingpopulation





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BARTLETT,. S. (1937) Deviations from expectedfrequency n the theory of inbreedingJ. Genet. 35, 83-87.

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CRAMER,. (1946) MathematicalMethodsof Statistics. PrincetonUniv. Press.

DOBZHANSKY, . AND PAVLOVSKY, . (1957) An experimental study of the interactionbetweengenetic drift and natural selection. Evolution11, 311-319.

DOOB, . L. (1953) StochasticProcesses. Wiley, New York.

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A Mathematical Study of the Founder Principle