SELECTION WITH PARTIAL SELFING. I. MASS SELECTION

<:opyriglit 0 1985 by the Genetics Society of America

SELECTION WITH PARTIAL SELFING. I . MASS

SELECTION

A. J. WRIGHT* AND C. CLARK COCKERHAM+

*Plant Breeding Institute, Trumpington, Cambridge CB2 2 L Q England, and +Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695-8203

Manuscript received June 1 1, 1984 Revised copy accepted October 3 1, 1984

ABSTRACT

The expected responses to mass selection carried out before or after repro- duction in a population whose members all have a fixed probability of self pollination ( 5 ) are formulated using covariances of relatives and their compo-

nent quadratic functions for a model with arbitrary additive and dominance effects. The response measured in the first generation offspring after selection (immediate gain) can differ from that retained when the population has re- gained equilibrium (permanent gain). The population mean behaves in a pre- dictable manner during the return to equilibrium, and its value at any time can be predicted from earlier generations. The permanent gain from selection after reproduction is always (1

+

s ) / 2 times as large as that from selection before reproduction, but the relationship of the immediate gains depends on the genetic model assumed.-Numerical analysis applied to a model with two alleles per locus and varying allele frequencies, dominance ratios and numbers of loci showed that the proportion of the immediate gain retained at equilib- rium was reduced with the large inbreeding depression associated with increas- ing dominance levels and numbers of loci and was generally lower for selection after reproduction than before. In the absence of information as to the mag- nitude of genetic variances and inbreeding depression in species reproducing by partial selfing, the importance of this phenomenon is unknown.

EVERAL economically important crop species have mating systems that

S

involve a mixture of self- and cross-pollination. A naturally reproducing population of such a species differs from those produced under either strict autogamy or allogamy in that it consists of a mixture of individuals with dif- ferent histories of selfing and, therefore, different coefficients of inbreeding (WEIR and COCKERHAM 1973). Since these differences contribute to the phen- otypic variance of the population, the effect of artificial selection is more complex than in either of the above cases.

Such work as has been done on this system has been restricted to just one or two loci (HAYMAN 1953; WORKMAN and JAIN 1966; HUHN 1980). T h e covariances among certain relatives stemming from a population in equilibrium with respect to inbreeding, and with a constant probability of selfing, were given by COCKERHAM and WEIR ( 1 984) for an arbitrary number of nonepistatic loci. These statistics, in the company of others which will be derived in this paper, allow the formulation of the expected response to mass selection in such a population.

586 A. J. WRIGHT AND C. C. COCKERHAM

BACKGROUND AND NOTATION

A population in which all members reproduce with a constant probability of selfing, s, approaches an equilibrium state following several generations of reproduction without selection at which the frequency of individuals with an immediate history of t generations of selfing and an inbreeding coefficient of 1

-

(1/2)f is (1

-

s)s‘ for all integer values oft. The description of this complex genotypic structure involves novel quadratic functions not necessary with com- plete selfing or outcrossing [COCKERHAM and WEIR 1984, hereinafter referred to as (84)].

For a model excluding epistasis, the following quadratic functions are re- quired, where the a’s and d ’ s are additive and dominance effects, respectively, in the noninbred reference population indexed for the ith locus and the j, kth allele.

Single locus Sum over loci

Additive variance _2a2, = 2

2

p,,a;

j

Dominance variance _{a% =}

₁

₁

_{p J l p ~ , ( d i , ) 2}

J k

Inbreeding depression h, =

1

pJ,d:,

J

Covariance a, and d: d l , =

1

p,lajld:l Di = d i l

Variance of d ; d g = pJt(d;,)*

-

h?

D $

= d g

3 1

I 1

With only two alleles at each locus, H* = a%, and with all gene frequencies of one-half,

D 1

= D b = 0. These functions are closely related to those used by PEDERSON (1 969a,b) to examine selection response in a self-fertilizing spe- cies.

With only two alleles at each locus,

H*

= U,: and with all gene frequencies

of one-half,

D1

=

DZ

= 0.

In addition to the equilibrium inbreeding coefficient, F, other coefficients are required, including the two-locus inbreeding coefficient $, which is the probability of identity by descent at both loci. Linkage increases

p ,

but it was argued (84) that the effect will generally be small. Under the assumption of no linkage (84),

2

= F ( l

+

2 F ) / ( 2

+

F ) = s(2

+

s)/(2

-

s)(4

-

s), since F =

s/(2

-

s). All coefficients can be phrased in terms of F or equivalent functions of s.

SELECTION RESPONSE

SELECTION WITH PARTIAL SELFING 587

ardized selection differential, i , and the phenotypic standard deviation, up, of the parents

A = iCp/u,.

There are two parent-offspring covariances to be reckoned with: C p o , be- tween parent and outbred offspring, and C p s , between parent and selfed off- spring. There are also two selection methods that may be applied. T h e more common method for annual crops involves selection after reproduction so that the outcross pollen comes from unselected individuals. T h e other possible practice, more common in perennial crops in which individuals can be main- tained after evaluation, is for selection to precede reproduction so as to allow only selected individuals to reproduce. These two systems will be designated 1 and

2,

respectively.

With selection after reproduction, the required covariance is obtained by averaging the two parent-offspring covariances according to the probabilities of outcrossing (1

-

s) and selfing (s) so that

C p l = (1

-

s)CPo

+

SCPS, A1 = i C p l / u p .

With selection before reproduction, both gametes in the outcrossed progeny come from selected parents so that the contribution of Cpo is doubled:

C p 2 = 2(1

-

s)Cpo

+

SCPS, A2 = iCp2/up.

These covariances may be identified as CPO = C1 and CPS = C:! in table 2 (84) and are included in Table 1 here, along with C p l , Cp2 and u:, the total genotypic variance of the parents.

T h e original equilibrium of the population with respect to inbreeding classes may have been disturbed by this selection. Since it is known that subsequent generations of reproduction without selection will restore equilibrium, it is of interest to inquire into the permanent selection response measured at this time. More formally, this can be defined as the difference in mean value between equilibrium populations derived from unselected and selected parents.

For selection before reproduction, it is necessary to consider four covariances among ancestors and descendants as follows:

1 2

C”

= - U ;

CNI = CIN =

CII = 2 ~ ;

+

401

+

0:

+

D1

SELECTION WITH PARTIAL SELFING 589

inbreeding coefficient of a descendant at equilibrium to be independent of those of its ancestors:

Cp2m = 2(1

-

F d ) ( ( l

-

F)CNN

+

F C I N )

+

Fd((1

-

F)CNI

+

FCrrJ

= ( 1

+

F ) u ~ -I- (2F

+

Fd

+

FFd)Di

+

FFdD?

and

A2m = iCpZm/up.

Here, the factor of two applied to CNN and C,N allows for the fact that the nonidentical descendant alleles have come from two different selected ances- tors. Fd, the mean inbreeding coefficient in the descendants, has been kept distinct from that of the ancestors.

T o find the permanent gain resulting from selection after reproduction, it is only necessary to recognize that ( 1

+

s ) / 2 of the alleles in the offspring generation came from selected parents and that these are randomly distributed among the descendants at equilibrium, irrespective of their inbreeding status. T h e covariance Cp2m is, therefore, simply diluted to give

Cplm = CP2m( 1

+

s)/2.

Provided that no artificial constraints are imposed on the mating system and that it has not been influenced by the selection process itself, then Fd = F =

s / ( 2

-

s). Table

2

gives the expectations of the four covariances of interest in terms of s.

By the same argument as is used above, the response to selection measured in any chosen generation, say the tth, is a function of the covariance of the original ancestor and descendants in that generation. T h e direct evaluation of the covariance for any but the very early generations is algebraically tedious, as the possible pathways via selfing and outcrossing at each intervening gen- eration soon become very numerous. However, inductive reasoning based on the formulas for the early generations and for t = 03 leads to the conclusion

that these covariances are always simple functions of those already given:

Cp2t = Cp2m

+

(Cp2

-

Cp2m)(~/2)'-'

CPlt = CPlm

+

(CP1

-

CPlm)(s/2)"-'

This result is not unexpected since the transient gain is due to dominance effects in the parents, and the probability of retaining a specific allelic combi- nation through t generations is ( s / 2 ) ' . Because of the orderly behavior of the population, its equilibrium mean can be predicted from those of any two other generations after selection. Using the first two generations, for either selection met hod:

y m = ( 9 2 - y 1 ) / ( 2

-

s),

where yt is the population mean in the tth generation.

For selection before reproduction, this relationship among the covariances can be extended back to include the parental generation, in which U,$ can be

SELECTION WITH PARTIAL SELFING 591

C p 2 i = C p g m

+

(U:

-

C p 2 m )(S/2Y.

This means that C P 2 m and permanent gain can be predicted from statistics

calculated from parents and offspring only:

C p 2 m = ( 2 C p 2

-

sai)/(2

-

s).

T h e prediction can be cast in an alternative form that uses the observed first generation response:

A y m = (2A21

-

sA20)/(2

-

S)

where Azo = i a ; / a p is the mean deviation of the selected parental genotypes. C p l m and AI, are estimated by multiplying these formulas by (1

+

s)/2.

ANALYSIS OF RESPONSE

General properties: As s approaches 0, the response degenerates into that appropriate for an outbred population with

U: = U:

+

U,; C p 2 = CPZm = U: and C p l = C p l m = a f / 2 .

As s approaches 1, the response approaches that for a population of homozy- gotes with

U: = C p z = C p z m = C p 1 = C p l m = 2 ~ :

+

401

+

0 ; .

All responses now utilize the entire genotypic variance, as selection is simply among fixed entities that reproduce exactly. It should be noted that this de- scribes a situation different from that of most interest for self-fertilizing species, in which an initial generation of crossing destroys inbreeding equilibrium prior to selection (PEDERSON 1969a).

It is only with intermediate values of s that the immediate and permanent selection responses are not equal ( C P 2 m # C p 2 , C P l m # C P I ) , and, as Table 2

shows, this is due to differences in contributions from all components except

U:. With inbreeding depression, selection will tend to favor the less inbred

individuals, and since their offspring will also be less inbred than the average, the immediate gains are due partly to changes in F and, therefore, involve the components of variance for inbreeding depression, H* and H 2 . As the inbreed- ing coefficient is only temporarily affected, these contributions are not present in the covariances for permanent gains. T h e initial contribution from a; is due to a similar transient effect of selection for heterozygosity within inbreed- ing groups. T h e disturbance of the inbreeding equilibrium is what causes the contributions of

D1

and 0 ; in immediate and permanent responses to differ. At all intermediate values of s, the coefficients of D1 rank in the order C P 2

>

C p z m

>

C p l

>

C p I m and those for

0 2

as C p z = C p l

>

C p 2 m

>

C p l m . Since these

are broadly in line with the rankings for U: (Cpz = CPzm

>

C p l = C p l m ) or a i ,

H* and H' ( C p 2 = C p l

>

C P p m = C p l m = 0), they are not likely to play a major

592 A. J. WRIGHT AND C. C. COCKERHAM

Quantitative analysis: Very few general conclusions about the expectations of response can be drawn from the formulations in Table 2 . Selection before reproduction can be expected to be the preferred method whenever s

<

1 , and because of the larger variance generated, response will increase with in- creasing s in the presence of error variation.

More detailed evaluation requires the application of specific genetic models and numerical analysis. T h e comparison of responses from different genetic models also requires the specification of error variance, but since all response formulas from a particular genetic model depend on the same population of observed values with standard deviation up, and can be practiced with the same selection intensity, i , then the relative sizes of any two responses can be ob- tained simply as the ratio of the appropriate covariances.

A two-allele model will be used, with

Genotype AA Aa aa Value 1 d -1

and

p

being the frequency of allele A, with n such loci initially considered to have the same effects and allele frequencies. The quadratic functions are then

CT: = n 2 p ( l

-

p ) [ l

+

( 1

-

2p)dI2

U; = H* = n4p2(1

-

p)'d2

D1

= -n2p(l

-

p)[l

+

( 1

-

2 p ) d ] ( l

-

2p)d

D?

= n 4 p ( l

-

p ) [ ( l

-

2p)dI2

H 2 = [ n 2 p ( l

-

p ) d I 2

T h e resulting quadratic components are given in Table 3 for varying values of

p

and d and with n = 100. With this model, H 2 = nu; and can be very large relative to other components when there are many loci. T h e values of the components for n = 1 can be derived by dividing the values given in Table 3 by 100 but setting H 2 equal to U;. Three components, u;, H 2 and

D?

are symmetrical around an allele frequency of 0.5, this representing a maximum for u; and H 2 but a minimum for 02. T h e other unfamiliar component,

D 1 ,

is negative for

p

<

0.5 and positive d .

Table 3 also contains the expectations for a model in which the allele fre- quencies at each of the 100 dominant loci are allowed to vary uniformly from 0 to 1 (model A) and for the same model when the dominance effects of 50 loci are positive and those of the remaining 50 are negative. T h e first of these gives rise to U ; ,

D $

and H 2 values similar to those for a complete dominance model with all allele frequencies in the range 0.1 to 0.3 but with much smaller absolute values for C T ~ and D1. T h e effect of introducing balanced dominance contributions (model B) is simply to remove H 2 .

SELECTION WITH PARTIAL SELFING 593

T A B L E 3

Quadratic genetic components with n = 100 for selected values of p and d and f o r

models A and B

0.1 1 .0

0.6 0.2 58.3 39.4 24.2 3.2 1.2 0.1 -25.9 -12.8 -3.3 23.0 8.3 0.9 324 117 13

0.5 1 .0

0.6 0.2 50.0 50.0 50.0 25.0 9.0

1 .0

0.0 0.0 0.0 0.0 0.0 0.0 2500 900 100

0.9 1 .0

0.6 0.2 0.7 4.9 12.7 3.2 1.2 0.1 2.9 4.5 2.4 23.0 8.3 0.9 324 117 13

Model A Model B

43.6 43.6 14.7 14.7 -7.1 -7.1 14.2 14.2 1467 0

T A B L E 4

Percentages 100Cp~oolC~~forselectedvaluesofn,p,dand sandformodelsAandB

s = 0.1 s = 0.5 s = 0.9 n = l n = 1 0 0 n = l n = 1 0 0 n = 1 n = 100

P d

0.1 1 .0

0.6 0.2

99.7 99.0

99.9 99.5

100.0 99.9

98.9 100.0 100.4 82.9 91.5 99.0

99.3 74.3

100.3 86.0

100.4 98.8

0.5 1

.o

0.6 0.2

97.7 92.3

99.2 97. I

99.9 99.7

92.3 97.1 99.7 44.0 69.0 95.2

96.1 45.4

98.5 68.7

99.8 95.4

0.9 1 .0

0.6 0.2

85.0 67.0

98.5 96.1

99.9 99.8

75.7 92.7 98.9 35.4 73.0 97.0

92.1 61.8

96.5 82.7

99.2 97.5

Model A Model B

98.4 94.6

98.5 98.5

94.0 94.8

51.5 94.8

96.6 50.9

97.6 97.6

5 94 A. J. WRIGHT AND C. C. COCKERHAM

TABLE 5

Percentages 100 Cplio/Cpl for selected values of n, p, d and s and for models A and B

s = 0.1 s = 0.5 s = 0.9

n = l n = 1 0 0 n = l n = 1 0 0

P d n = l n = 100

0.1 1

.o

0.6 0.2 101.5 101.2 100.6 100.2 100.5 100.5 104.5 104.2 102.1 82.1 92.3 100.2 102.3 102.3 101.1 75.0 89.5 99.4

0.5 1

.o

0.6 0.2 95.9 98.5 99.8 86.9 94.8 99.4 90.0 96.2 99.6 37.5 62.5 93.8 95.9 98.5 99.8 44.1 68.7 95.2

0.9 1

.o

0.6 0.2 68.4 93.8 99.0 49.1 90.0 98.8 62.8 84.6 96.7 27.6 63.6 94.1 88.4 94.3 98.5 59.1 80.4 96.7

Model A

Model B

98.1 98.2 91.4 98.2 93.1 94.2 44.6 94.2 96.6 97.5 49.6 97.5

TABLE 6

Values of 100 CpI/Cpl for selected values of n, p, d and s and for models A and B

~ ~ ~~ -~ ~ ~

s = 0.1 s = 0.5 s = 0.9

P d n = l n = 1 0 0 n = 1 n = 100 n = l n = 1 0 0

0.1 1

.o

54.1 54.4 71.0 75.7 92.2 94.2

0.6 54.3 54.4 72.0 74.4 93.2 94.0

0.2 54.7 54.7 73.7 74.1 94.3 94.4

0.5 1 .0 56.0 58.5 76.9 88.9 95.2 97.7

0.6 55.4 56.3 75.7 82.8 95.1 96.5

0.2 55.0 55.2 75.1 76.2 95.0 95.2

0.9 1

.o

68.4 75.1 91.9 96.2 98.9 99.2

0.6 57.7 58.8 82.3 86.1 97.3 97.7

0.2 55.5 55.5 76.8 77.2 95.7 95.8

Model A 55.2 56.9 75.8 86.7 95.1 97.4

Model B 55.2 55.2 75.5 75.5 95.0 95.0

(1 + 412 55.0 55.0 75.0 75.0 95.0 95.0

SELECTION WITH PARTIAL SELFING 595

low

p

values, some percentages in excess of 100 occur, implying that the population mean increases during the return to equilibrium after selection. This is due to the large negative values of D1 in this range (Table 3), and can be considered unlikely in practice, and emphasizes that models with extreme allele frequencies can give misleading results. With uniformly distributed allele frequencies (model A), values as low as 51% are obtained, but with no inbreed- ing depression, only 5% of the gain may be transient, showing that components other than H 2 generally have small effects.

T h e parallel calculations for selection after reproduction (Table 5 ) give a similar picture. There are now more values in excess of 100, but all those for

p

>

0.5 are smaller than their counterparts in Table 4, many of them substan- tially so. T h e minimum value, found with the same combination of parameters as before, is now only 27.6% and for model A is 44.6%. T h e reason for these lower values is that this selection method places more emphasis on selection among inbreeding groups and, therefore, causes greater disequilibrium. For both methods, the rate of loss of transient gain is rapid. Even with s as high as 0.9, the population will lose 80% of its transient gain by the third generation and much sooner with smaller s.

As pointed out, Table 6 carries no new information but merely presents the same results from a different viewpoint. T h e values progress from 0.5 when s = 0 to 1.0 when s = 1, but for intermediate s, the majority of the ratios are larger than the corresponding values of (1

+

s)/2, particularly with high n , s and

p .

DISCUSSION

T h e most striking feature revealed by this analysis is the magnitude of the response following either selection method which may be transient and is not due to a change in the breeding system, such as occurs in a pedigree selection scheme. This may be considered a special property of the partially selfing mating system, as it is likely to be much more important than any transient gain that can follow selection with epistasis in purely allogamous species (GRIFF-

INC 1960). T h e results obtained depend to a large extent on the importance of inbreeding depression in relation to other quadratic components, and this is in turn determined by the number of loci and dominance ratios introduced into the genetic model used. Since these are unknown quantities in practice, a more realistic approach could be made if estimates of the quadratic compo- nents from real data were available, so that the genetic modeling step could be bypassed. Autogamous species tend to show less depression than allogamous species, and its importance in partial selfers is likely to be intermediate and related to s. However, as reliable information about the likely magnitude of H 2 and other components is lacking for these species, it can only be noted here that the potential for losses of selection response during subsequent re- production may be important.

596

U& = (1

+

F ) c ~

+

FD1

+

( 2 F 2 / ( 1

+

F ) ) D $

A . J. WRIGHT AND C. C. COCKERHAM

= t;2p(1

-

P)[(l

+

F )

+

(1

-

F)(1

-

2 p ) d ] ?

T h e expectation of C P ~ ~ for the two-allele model can be factorized in a similar way:

Cp2m ( 1

+

F ) u ~

+

(2F

+

F d

+

FFd)D1

+

FFdD$

= 2 2 p ( l

-

P)[(l

+

F )

+

(1

-

F)(1

-

2 9 ) d ] [ l

+

(1

-

Fd)(l

-

2 # ) d ] .

T h e first term in square brackets here relates to the rate of change of the allele frequency

p

under selection, whereas the second relates to the rate of change of the population mean with changes in

p .

These two terms are equiv- alent to the average excess and average effect of alleles as defined by FISHER

(1941).

Apart from allowing a characterization of the properties of mass selection with partial selfing, the formulas developed in this paper and (84) now open the way to methods of numerical prediction from experimental data collected from specific populations. T h e prediction of immediate responses can be achieved by estimation of Cpo and Cp, by controlled pollinations or by esti- mation of C p l directly as the covariance of parents and naturally produced progeny. As they involve only the t w o generations that are actually measured, these predictions are wholly statistical in nature and rely on no genetic reason- ing. They, therefore, depend only on knowledge of s and are independent of any assumptions as to genetic control or population equilibrium.

T h e prediction of permanent response can be achieved in either of two ways. T h e quadratic components can be separately estimated using the parents and the progeny of controlled pollinations as described elsewhere (84) and the covariances CP2- and Cplm calculated from these. T h e chief problem is the

necessity to combine estimates of H 2 based on comparisons among the means of different generations with those of other components that are derived from covariances. Alternatively, the relationship between parent-descendant covari- ances for different generations can be used to estimate the equilibrium co- variance directly, as described earlier. C P ~ ~ can be estimated using just the parents and their first generation offspring and Cplm derived by multiplying this by ( 1

+

s ) / 2 . Errors in this method of prediction follow from the extrap- olation of the relationship of covariances with generations, and the use of later generations would improve accuracy. Both methods depend on the absence of epistasis and on gametic equilibrium, as well as inbreeding equilibrium in the parents.

SELECTION WITH PARTIAL SELFING 597

described as a nonlinearity of parent-offspring regression with a reduced gra- dient at the upper end where selection is operative. T h e magnitude of this effect is difficult to predict without analysis or simulation taking into account the selection differential as well as the genetic model. However, it can be regarded as a second possible source of overestimation of permanent response which can augment that due to the use of parent-offspring instead of ancestor- descendant covariances.

In a practical breeding context, selection cycles are often applied recurrently with no intervening unselected generations. In this system, equilibrium is per- manently destroyed, and, although the covariances Cp l m and CP2m are not af-

fected by disequilibrium, all other covariances as well as the population vari- ance are changed, and the response formulas given here cannot be strictly applied to the second and subsequent cycles. When s is large, the population is largely a collection of inbred lines, so that the rate of response may be reduced after the first cycle and may be exhausted after very few. With small s, on the other hand, the population will contain a large proportion of non- inbred members, equilibrium will suffer little disturbance and response may continue over many generations. It may be noted, however, that estimates of the parent-offspring covariances, Cp1 and Cpz, and of the phenotypic variance made in any generation, are always appropriate for the prediction of the im- mediate response in that cycle.

Paper 9380 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, North Carolina. This investigation was supported in part by National Institutes of Health research grant GM 11 546 from the National Institute of General Medical Sciences.

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