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Survival Probabilities of Genes in A Two-Locus Diploid Partial Selfing Population:Multi-Type Branching Process Approach

(Muhamad Sabran)

81

SURVIVAL PROBABILITIES OF GENES IN A TWO-LOCUS DIPLOID PARTIALSELFING POPULATION: MULTI-TYPE BRANCHING PROCESS APPROACH

Muhamad SabranIndonesian Agency for Agriculture Research and Development, Jalan Ragunan 29, Pasar minggu Jakarta Selatan

e-mail:[email protected] (Makalah diterima, 6 Oktober 2011 – Revisi, Desember 2011)

ABSTRACT

Multitype branching processes theory was used to approximatethe survival of mutant genes in a two-locus diploid populationreproduced by partial selfing. It is concluded that in the two-locus situation, where there are initially two mutant geneswhich have epistatic effect on fitness, tight linkages betweenthe two loci is necessary for the survival of the mutant genes.If recombination is possible, a population with a high rate ofselfing is more likely to accumulate epistatically favourablegenes than one reproducing largely by random mating. Thisadvantage of selfing becomes more pronounced as the strengthof selection toward the favourable genes increases.

Key words: Survival Probabilities; partial selfing; selection.

INTRODUCTION

Most field crop species that reproduce by sexualmeans may be grouped according to usual method ofpollination as “normally self-pollinated “or “normallycross pollinated” crops. These groups are notdistinct, since slight cross pollination usually occursin the crops normally classified as self-pollinated andsome self pollination usually occurs within thenormally cross-pollinated crops. The amount ofnatural cross pollination that may occur within theself-pollinated crops and the amount of selfpollination within the cross-pollinated crops mayvary from none to 5%. Furthermore, there are somecrops that do not fit in either cross- or self-pollinatedcategories. Cotton is one of the principal crops inthis group. Cotton is predominantly self-pollinatedbut cross pollination may range from 5 to 25%. Othercrops in this group are sorghum, pigeonpea, etc. Alist of cross pollination rates in self-pollinated cropscan be found in Lande and Schemske (1984).

The fact described above show the importance ofpopulations which have a mixed system, partly by selfpollination and partly by cross pollination. We call thistype of population as partial selfing population. Theamount of selfing may vary from none to 100%, so thistype of population includes all population describedabove.

Most theoretical studies on partial selfingpopulations assume that the population size is infinite.Such studies include investigation on the equilibriumbehaviour of the population without selection (e.g.,Garber 1951; Benett and Binet, 1956; Ghai, 1964; Narain,1969), under selection (e.g., Holden, 1979;Charlesworth, Chalresworth and Strobeck, 1979;Hedrick, 1979) and linkage disequilibrium (e.g.,Christiansen, 1989; Holdsinger and Feldman, 1982). Innature, as well as in the laboratory, the population sizeis finite. As a consequence, there is random samplingof gametes. This, along with differences in thecapability among individuals to produce gametes, givesstochastic properties to the population.

One important stochastic property in a population isthe survival of genes or gametic types. Theprobabilities of these events are important from theevolutionary, as well as plant and animal breeding, pointof view. Evolutionists are often interested in the fate ofalleles that have just arisen by mutation. Plant andanimal breeders are particularly concerned with the limitto response to their selection program and the numberof generations to achieve such limit or certain fractionof it. These studies require information on the survivalprobabilities of the favorable genes which, in finitepopulation, is equivalent with fixation probabilities,since finiteness in population size result in either lossor fixation of genes.

While much research has been done on survival orfixation of genes in a population, few of them considerpartial selfing populations. Pollak (1987) studied finitepartial selfing population and calculated the survivalprobabilities of a slightly advantageous mutant geneoriginally present in one locus of heterozygousindividual, using a branching process approximation.Later Pollak and Sabran (1992) studied the sameproblem by following the reasoning of Moran (1961,1962).These studies considered only one locus indiploid population. In two-locus diploid population,Pollak and Sabran (1999) calculate numerically thesurvival probability of genes in two model of selection.

Informatika Pertanian, Vol. 20 No.2, Desember 2011 : 81 - 86

82

This paper calculates the survival probability in two-locus diploid population for special case where there iscomplete linkage and quote the numerical result ofPollak and Sabran (1999) for the general case.

THE MODEL

Consider two loci, A and B. Suppose that initially thepopulation consist of individuals with genotype ab/ab. At generation 0, an ab chromosome in oneindividual mutates to AB and no more mutation occur.Here, a and b are alleles that differ from A and B,respectively.

Let us assume that the population is large but finite,and reproduce by partial selfing. There are fivepossible genotypes under this type of mating, namelyAB/ab, Ab/aB, AB/aB, AB/Ab and AB/AB. In the earlygeneration after mutation, the number of individuals thathave an A or B in their genotype is small. Therefore, theprobability that two mutant will mate is negligible. Inother words, any mutant will almost certain to mate withindividuals with genotype ab/ab, if they do notreproduce by selfing. More generally, a line descendedfrom one mutant individual will develop independentlyof other lines. Hence, we can model the change innumbers individuals with AB approximately as a five-type branching processes.

Suppose that neither of the two mutant genes A andB exhibits fitness high enough for independentsurvival, but epitasis between the two give rise to ahigh joint fitness value. In other words, a line ofmutants descended from an individual with any of thegenotypes Ab/Ab, Ab/ab, aB/aB, aB/ab, will ultimatelybecome extinct with probability one. The relativeviabilities under this model is given in Table 1. Theinterpretation of this table is that, for example, theprobability with which zygotes with genotypes AB/Aband Ab/aB survive to adulthood are in the ratio(1+2s) : (1+s), where s>0.

Let c be the probability of recombination between theA-a and B-b loci. If there are no selection, i.e., s=0, thenthe offspring distribution would be as listed in Table 2.

If there are no selection, the expected number ofoffspring of an individual is two if it reproduces byrandom mating and one if it reproduces by selfing. Thefirst moment matrix of the five types branchingprocesses would be :

Table 2. Distribution of offspring in two-locus diploid with recombination

...(5)

000)1(24/2/0014/02/014/2/)1(2/)1(2/)1()1()2/(

4/)1(2/)1(2/)1(2/)1)(1()2/)1((222

222

)0(

cccccccccccccccc

M c

Table 1. Relative viabilities model of genotypes in two-locus diploid

*) Other genotypes are:Ab/ab, Ab/Ab,, aB/ab, aB/aB, ab/ab


(Muhamad Sabran)

83

If there is selection, the first moment matrix is )(scM where,

ss

ss

s

MM csc

3100000210000021000001000001

)0()( …….. (6)

If we denote individual with genotypes AB/ab, Ab/aB, AB/aB, AB/Ab and AB/AB, as of type 1 to 5,respectively, and Yij is the number of offspring of typej produced by an individual of type j, then theprobability of survival of an individual of type 1 (AB/ab), u1, is (Hoppe, 1992):

)()1)((2 5

10

'00

011

ovCvp

vu

iii

Provided that the branching processes is slightlysupercritical, i.e., the first moment matrix )(scM has thedominant eigenvalue equal to )( with

1)(lim 0 and the dominant eigenvalue of)0(cM is equal to 1. Here 0v and 0p are the

standardized right and left eigenvectors of , and andare their i-th elements respectively. Ci is the covariancematrix of , the number of offspring produced by anindividual of type i when there is no selection.

We have to ensure that the branching process iscritical when there is no selection, i.e., the dominanteigenvalue of is equal to 1. Unfortunately, theeigenvalue of is difficult to obtain. However we canconsider special case when c= 0, i.e., complete linkage.In this case a parent of type 1 can only produceoffspring of type 1 and 5 and so does a parent of type5. So the branching process now become a two-typebranching process with the first moment matrix become

)1(242

1)0(0M

The characteristic equation for )0( 0M is

)1)(2

(0 , so that the dominanteigenvalue equal to 1. The r ight and lefteigenvectors associated with this eigenvalue are

)),1(4(34

1'0

p

and

……………………. (7)

)1,21

(2

34'0

v

If there is selection, the first moment matrix becomes

)()( 0 sDIMs

where

2001

D . If we write the r ight and left

eigenvectors of )0( cM as dpp '0' and

evv 0 , then

evevMevsDIM c 00)0(0)0( 0 Since 00

)0(0 vvM and 0e as 0 , we

have after some algebra,

sGu )var(12

241

sGu )var(1224

1

eosDvMv 00001 ……………… (8)Premultiply both side of (8) with '0p we obtain

eosDvp 0'01The numerator of (7) now becomes

234

222 010

'0 svDvp

The denominator of (7) is equal to

)

21var()

21var()1(4

234

555115112 YYYY

If G is a random variable which denotes the number ofsuccessful gametes produced by an individual of type5, then, if there is no selection, as pointed out byCaballero and Hill (1992), G is equal to twice the numberof selfed offspring plus the number of offspring fromrandom mating. Hence,

)(412

41

21

55515551 GVarYYVarYYVar

Also, regardless of whether an individual of type 1reproduce by selfing or random mating, half of itssuccessful gametes are expected of type AB, if there iscomplete linkage and no selection. Hence, givenG=g, 1511 2YY , with no selection, will be binomiallydistributed with g trials and probability of successequal to

21

. So,


84

GYYVar var2412 1511

Since we assume that the expected number ofsuccessful gametes is equal to two. Hence thedenominator of (7) becomes

)(12

24)34(2 GVar

The survival probability of the mutant if it is initiallypresent in one individual of type 1 is, from (7),

sGu )var(12

241

…………………... (9)

If there is Poisson distribution of offspring then

22)1(24)( GVarHence ( 9) becomes

su )2(1 su )2(1 ……………. (10)

In general, when c>0, we can only do numericalcalculation of the survival probability. In whatfollows, we will quote the result that was given byPollak and Sabran (1999), and give some key step thatlead to the numerical calculation to the survivalprobability

Let qi be the probability that a line originating from anancestor of type 1, ultimately become extinct. Then if

54321 ,,,, rrrrrprp ii is the probability that anindividual of type i produce rj offspring of type j, forj=1,2,3,4,5,

5151 ............ 511 5

qqfqqrpq irr

r rii .........…...... (11)

Where fi is the probability generating function of thedistribution of numbers of offspring of various typesproduced by a parent of type i. If we assume the totalnumber of offspring produced by any individual followa Poisson distribution and that genotypes of separateoffspring are independent, then (11) reduces to

5

1

5

1exp1exp1

jjij

jjijii umqmqu

Where ijm is the mean number of offspring of type jof a parent of type 1, i.e., is the ij-th element of . Thesurvival probabilities, ui, i=1,2,3,4,5, satisfy theequation,

01ln5

1

j

jiji umu , for i=1,2,3,4,5.

Equation ( 5 ), ( 6 ) and ( 11), implies that u3= u4, and

0314

1

21112

12

1111ln

5

2

32

2

1

2

1

usc

usccusc

usc

cu

... ( 12)

0314

21112

11

211ln

5

2

32

2

1

2

2

usc

usccuscusccu

........( 13)

0314

212

111ln 5313 usususu

... (14 )

031121ln 515 usuu …… ( 15 )

Pollak and Sabran (1999) first calculated the value ofu1, by considering the special cases for c=0, â=0 or â=1.For the remaining pairs of (â,c), they evaluated thevalue of u1 by linear interpolation. The values of u1computed by this method are used as initial values in aniterative procedure on equation (12 )-(15) that lead tothe computation of u1 to any desired degree ofaccuracy. The result is given in Table 4. Only positivevalues of u1 are shown.

DISCUSSION

In two-locus model, the result on complete linkage isnot surprising, since, with complete linkage, thepopulation behaves like a one-locus population. Infact, Pollak (1987) in a corrected version in Pollak andSabran (1992), showed that, if there was a Poissonoffspring distribution, and relative viabilities of AA,Aa and aa individuals were 1+s1:1+s2:1, respectively,then the survival probability of A, if it initially presentin one heterozygote was

21 12 ssu Applying this result to two-locus case with complete

linkage, and by noting that the relative viabilities of AB/AB, AB/ab, and ab/ab are 1+3s:1+s:1, respectively weobtain

sssu 2123 .............……... (16)

which is exactly equal to (10).

When the relative viabilities of the three genotypesare 1+s:1+s:1, i.e., s1=s2; then (16) becomes

sssu 212 , which indicate thatthe survival probabilities does not necessary increaseby the increase in the rate of selfing. Pollak and Sabran


(Muhamad Sabran)

85

noted that there are two possible effects on u as âincreases. One result from it being more probable underselfing than under random mating that an offspring ofan AB/ab parent receive at least one copy of AB in itsgenotype. This favors an increase in u. But it is also thecase that when there is selfing, an AB/ab has onaverage s143 offspring with at least one AB,rather than (1+s), as it would have under random mating.This act to decrease u. Under our model, the probabilitythat an offspring survive to adulthood increases withthe number of AB’s it has in its genotype. Hence, thesurvival probability increases with the increase in therate of selfing.

In general, Table 3 indicates that the survivalprobability increases with the increase in â and is adecreasing function of c. Thus inbreeding favors thesurvival of AB, whereas crossing over disrupt thisfavorable combination and thus reduces theprobability that it will ultimately survive. This leads usto conclude that if recombination is possible, apopulation with high probability of selfing is more likelyto accumulate epistatically favorable genes than onereproducing by random mating. This advantage ofselfing becomes more pronounced as the strength ofselection in favor of the AB combination increases.

CONCLUSION

In two-locus situation, where there are initially twomutant genes which have epistatic effect on fitness,tight linkages between the two mutants is necessaryfor the survival of the two genes. If recombination ispossible, a population with high rate of selfing ismore likely to accumulate epistatically favourablegenes than one reproducing by random mating. Thisadvantage of selfing becomes more pronounced asthe strength of selection toward the favourable geneincreases.

REFERENCES

Bennett, J. H., and F. E. Binet. 1956. Association betweenMendelian factors with mixed selfing and random mating.Heredity 10:51-55.

Caballero, A, and W. G. Hill. 1992. Effects of partial inbreedingon fixation rates and varia tion of mutant genes. Genetics131:493-507

Christiansen, F. B. Linkage equilibrium in multilocus genotypicfrequencies with mixed selfing and random mating. Theoreticaland Population Biology 35:307-336

Charlesworth, D., B. Charlesworth, and C. Strobeck. 1979.Selection for recombination in partially self-fertil izingpopulation. Genetics 93:237-244.

Table 3. Survival probabilities (u1) in Two-locus diploid Model under various rates of selfing (â), recombination (c)and selection coefficient (s).

βc s

0 0.2 0.4 0.6 0.8 1 .0

0 0 . 0 1 0 .0 1 9 7 0 .0 2 1 6 0 .0 2 3 5 0 .0 2 5 4 0 .0 2 7 3 0 .0 2 9 10 . 0 2 0 .0 3 9 0 .0 4 2 6 0 .0 4 6 2 0 .0 4 9 8 0 .0 5 3 2 0 .0 5 6 60 . 0 4 0 .0 7 5 9 0 .0 8 2 6 0 .0 8 9 1 0 .0 9 5 4 0 .1 0 1 5 0 .1 0 7 40 . 0 8 0 .1 4 4 4 0 .1 5 5 6 0 .1 6 6 4 0 .1 7 6 5 0 .1 8 6 1 0 .1 9 5 1

0 .1 0 . 0 1 - - - - - 0 .0 2 4 50 . 0 2 - - - - 0 .0 1 7 2 0 .0 4 7 90 . 0 4 - - - 0 .0 2 4 4 0 .0 6 1 7 0 .0 9 1 90 . 0 8 - - - 0 .0 0 6 1 0 .0 8 0 7 0 .1 7 0 4

0 .3 0 . 0 1 - - - - - 0 .0 1 8 50 . 0 2 - - - - - 0 .0 3 6 50 . 0 4 - - - - 0 .0 1 6 1 0 .0 7 10 . 0 8 - - - 0 .0 0 6 1 0 .0 8 0 7 0 .1 3 5

0 .5 0 . 0 1 - - - - - 0 .0 1 4 90 . 0 2 - - - - 0 .0 2 9 40 . 0 4 - - - - 0 .0 5 7 70 . 0 8 - - - 0 .0 4 7 6 0 .0 1 1 2


86

Garber, M. J. 1951. Approach to genotypic equilibrium withvarying percentage of self-fertilization. Journal of Heredity42:299-300.

Ghai, G. L. 1964. The genotypic composition and variability inplant population under mixed self-fertil ization and randommating. Journal of the Indian Society of Agricultural Statistics16:94-125

Holden, L. R. 1979. New properties of the two locus partialselfing model with selection. Genetics 93:217-236.

Holsinger, K. F., and M. W. Feldman. 1983. Linkage modificationwith mixed selfing and random mating: A numerical study.Genetics 103:323-333.

Hoppe, F.M. 1992. The survival probability of a mutant in amultidimensional population. Journal of Mathematical Biology

Kimura, M. 1964. Diffusion model in population genetics. Journalof applied probability 1:177-232.

Lande, R., and D. W. Schemske. 1985. The evolution of self-fertilization and inbreeding depression in plant. Evolution39:24-40.

Moran, P. A. P. 1960. The survival of mutant gene under selection.Journal of American Mathematical Society 1:121-126.

Moran, P. A. P. 1961. The survival of mutant gene under generalcondition. Proceedings of Cambridge Philosophical Society57:304-314.

Narain, P. 1969. Effect of linkage on homozygosity of apopulaton under mixed selfing and random mating. Genetics54:304-314.

Pollak, E. 1987. On the theory of partially inbreeding populationI. Partial selfing. Genetics 117:353-360

Pollak, E. and M. Sabran. 1992. On the theory of partiallyinbreeding population III. Fixation probability under partialselfing when heterozygous are intermediate in viability. Genetics131:979-985.

Pollak, E. and M. Sabran. 1999. On the theory of partiallyinbreeding population VI. The survival probability of a two-locus allele combination when there is partial selfing.Mathemathecal Biosciences 159(2) : 97-112.

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