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Copyright 0 1986 by the Genetics Society of America

A NUMERICAL SIMULATION OF THE ONE-LOCUS,

MULTIPLE-ALLELE FERTILITY MODEL

ANDREW G . CLARK* AND MARCUS W. FELDMAN‘

“Department of Biology, 208 Mueller Laboratory, Pennsylvania State University, University Park, Pennsylvania 16802, and tDepartment of Biological Sciences, Stanford University,

Stanford, Calijornia 94305

Manuscript received September 23, 1985 Revised copy accepted January 13, 1986

ABSTRACT

Numerical simulations were performed to determine the equilibrium behavior of the one-locus fertility model in which fitness is considered as a property of a pair of mating diploids. A series of patterns of “fertility matrices” were consid- ered for a single locus with two to six alleles. From these simulations, 19 differ- ent statistics were collected that characterize, at equilibrium, the heterozygosity, the mean fitness and the fate of populations begun at the allele-frequency cen- troid. For more than one-half of the trajectories produced by random fertility matrices, there was a decrease in the mean fitness at some time on the way to equilibrium. T h e mean number of alleles maintained at equilibrium increased only slightly with matrix dimension. Despite the potential for fertility models to display multiple stable equilibria, random fertility models maintain fewer distinct stable points than do random one-locus viability models. Pleiotropic models were also considered with fertility and viability selection operating sequentially within each generation. Most of the equilibrium statistics (with the exception of mean fertility) for the pleiotropic model were intermediate between the corresponding random viability and fertility models.

NE of the strongest criticisms of heterozygote advantage as an explanation

0

for polymorphism is the fact that the theoretical conditions on the fitness in a multiple allele viability model for maintaining a multiple allele equilibrium become exceedingly stringent as the number of alleles increases (LEWONTIN,

GINZBURG and TULJAPURKAR 1978; KARLIN 198 1). Nevertheless, empirical

observations of multiple allele polymorphisms are not uncommon (for a liter- ature review, see KARLIN 1981). Since it is extremely unlikely that viability selection alone can maintain many alleles in a stable equilibrium, it is reason- able to ask whether other modes of selection are similarly constrained. In particular, since fertility selection acts as a property of mating pairs, the mar- ginal genotypic fitnesses in effect become frequency-dependent. If the result of this frequency-dependence is to “protect” an allele when rare, then it seems reasonable that fertility selection could maintain more alleles at equilibrium.

To explore the role of fertility selection in maintaining multiple allele poly- morphism, we numerically analyzed a set of models for sexually reproducing

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162 A. G. CLARK AND M. W . FELDMAN

diploid organisms in which fitness operates on genotypic pairs. The numerical study presented here is motivated by the following questions: (1) Does the fertility model yield qualitatively different results from the classical viability model in the number of alleles at equilibrium, the heterozygosity or the mean fitness? (2) Do the equilibrium results of the fertility model depend on the structure of the fertility matrix? (3) What are some reasonable structures for fertility matrices, and what are their biological interpretations?

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How does the stability of equilibria under the fertility model compare to the viability model? (5) How often does mean fitness decrease under random and structured fertility models? (6) If pleiotropic effects are allowed, so that both the classical viability effects and fertility selection described below act on the same locus, how does this affect the equilibrium properties of the model?

T h e model we shall examine is the multiple allele extension of models ex- amined by PENROSE (1949), BODMER (1965) and HADELER and LIBERMAN

(1975). We make use of many of the analytical properties reported by FELD-

MAN, CHRISTIANSEN and LIBERMAN (1 983). In particular, the computation is greatly simplified by using the fact that, for any fertility matrix having elements

FII,kl that specify the fertility of the mating between genotypes g , and gkl, the matrix can be symmetrized, such that F I I , k l = (FII,kl

+

Fk1,,)/2. This is referred to as sexual symmetry, and it is a consequence of the assumption that the reciprocal matings between g, and g k l are equally frequent and produce the same offspring array. The recurrence equations are

where T is the sum of the right-hand expressions, and vy is the viability of genotype gq.

MATERIALS A N D METHODS

In order to save on computation time, the recursion system (I), (2) was explicitly written for each model having two through six alleles. Since the recursions are in terms of genotype frequencies, they involve three simultaneous equations for the two-allele system, six equations for three alleles, ten equations for four alleles, 15 equations for five alleles and 21 equations for the six-allele model. T h e lines of FORTRAN code that described these recursion systems were, themselves, written by another program that followed the algorithm of equations ( 1 ) and (2) (see APPENDIX A).

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FERTILITY SELECTION 163

fertility parameter and the degree of dominance. ( 5 ) partial dominance: the fertility of a mating is again determined by the fitness of the most dominant genotype, and allelic dominance is determined as in KARLIN and FELDMAN (1981). Fertilities were not or- dered with respect to degree of dominance. (6) ordered dominance: the ordering of fertilities matches the ordering of dominance, with the most dominant allele being the most fertile. This pattern is analogous t o the “both-ordered dominance” of KARLIN and FELDMAN (1981). (7) mate identity: fertility is a property of the number of shared alleles and the “distance” between them. This distance is determined by the difference in the allele indices. Each such mating class is assigned a fertility chosen at random from the uniform distribution on [0, 11. T h e three-allele mate-identity fertility matrix has the pattern shown below:

Male

Female 11 12 13 22 23 33

1 1

fi

f2

fs

P

f.

P

12

f2

fi

A

f e

fs

f.

13

fs

f2

fi

P

P

f3

22

P

f2

P

f1

f2

fi

23

P

f3

A

f2

fi

f2

33

P

f5

A

h

fp

fi

T h e mate-identity pattern does not have fertilities ordered with respect t o the simi- larity of mating genotypes. (8) ordered mate-identity: T h e fertility pattern is as listed above, but the parameters are ordered such that fi

<

f2

<

fs

.

. .

<f6. This pattern

implies that the more dissimilar the parental genotypes, the greater the fertility of the mating. (9) Random viability-fertility pleiotropy: Random parameters are chosen from a uniform distribution on [0, 11 for the fertility matrix and also for viabilities. Each generation, selection operates on the mating pairs through the recursions (1) and (Z), where genotype frequencies are weighted by their viabilities.

For each matrix corresponding to r alleles, 10r initial genotype frequency configu- rations were chosen by assigning random numbers and normalizing. For each starting condition, the recursions were iterated until the maximum change in genotype fre- quency was less than When equilibrium was attained, the allele frequencies and mean fertility were retained, and another initial vector was chosen. When all 10r starts were completed, 19 statistics were calculated for the respective fertility matrix. These statistics include the following: (1) Max(F): the maximum equilibrium value of the mean fertility, maximized over starting vectors and averaged over matrices. (2) No. x, # 0 (F): the number of alleles segregating at the equilibrium that maximized F. (3) H(F): the heterozygosity at the equilibrium with maximum mean fertility. (4) Max no. x, # 0: the maximum number of segregating alleles, maximized over initial vectors and aver- aged over matrices. (5) Min no. x, # 0: the minimum number of segregating alleles, minimized over initial conditions. (6) No. x, # 0: the average number of alleles segre- gating at equilibrium, averaged over initial vectors and matrices. (7) No. limits: the number of distinct equilibria (A pair of equilibrium allele frequency vectors X and Y were considered identical if for all i , Ix,

-

y,I (8) Max(H): the maximum equilibrium heterozygosity, defined as

H = 1 - E x : .

(9) No. x, # 0 (H): the number of alleles segregating at the equilibrium that maximized heterozygosity. (10) F(c), the mean fertility at the equilibrium arrived at when the initial vector is the centroid. (11) No. x, # 0 (c): the number of segregating alleles at the centroid-derived equilibrium. (1 2) H(c): the heterozygosity at the centroid-derived equi-

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164 A. G. CLARK AND M. W . FELDMAN

I .o

-

F

I

I

0.5 I

2 3 4 5 6

ALLELES

FIGURE 1 .-Mean fertility averaged over all initial vectors and random matrices. The solid line represents the pure fertility model, with error bars representing f 1 standard error (calculated from the mean fertilities of the 100 matrices). The dashed line is for the pleiotropic model, where the same genetic locus determines fertility and viability (parameters for both models chosen ran- domly). The dotted line represents the mean fitness for the random viability model (viability data are from KARLIN and FELDMAN 1981).

librium. (13) Limit = c: the fraction of initial vectors that yielded an equilibrium iden- tical to the centroid-derived equilibrium. (14) Limit = max(F): the fraction of initial vectors that yielded the equilibrium with the maximum mean fertility. (15) Limit = max(H): the fraction of initial vectors that yielded the equilibrium with the maximum heterozygosity. (16) F the mean fertility averaged over all initial vectors and 100 matrices. (1 7 ) No. iterations: the mean number of generations required to attain equi- librium (as defined above). (1 8) No. iterations (c), the number of generations required

to attain equilibrium from the centroid. (19) AF

<

0, the fraction of trajectories that decreased mean fertility at any time.

Simulations were performed on a PRIME 550 minicomputer in double precision (28 digit) FORTRAN-77. Details of the verification procedures appear in APPENDIX A.

RESULTS

Summary tables of equilibrium statistics appear in APPENDIX B. The moti- vation for this study closely parallels that of KARLIN and FELDMAN (1 98 l), and the most important conclusions stem from comparisons with the viability model. One of the most interesting quantities to compare is mean fitness, since it is maximized in multiallele viability models, but evidently is seen to decrease quite frequently in the fertility model (POLLACK 1978). These are difficult models to compare, however, because viability is a property of genotypes and fertility is a property of mating pairs. It is also difficult to compare structured viability and fertility models, because fertility models can have structure at the level of gene expression in the individual, or in the fertility of the mating pairs. Despite these caveats, an examination of the simulation results reveals striking parallels between the models.

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FERTILITY SELECTION 165

FIGURE 2.-Maximum mean fitness, maximized over initial vectors and averaged over random matrices. Line patterns are the same as in Figure 1 .

random fertilities and viabilities also reveals an increasing trend. Its curve falls below the others because the two components of fitness were sequentially applied, and evidently viability selection perturbs genotype frequencies in such a way that mean fertility is decreased. In particular, viability selection tends to

move genotype frequencies further from the Hardy-Weinberg surface. Figure

2

shows the maximum mean fitness, minimized over initial vectors and aver- aged over fertility matrices. T h e maximum mean fitness shows a similar in- creasing trend, and thedifference between the mean fitness and the maximum fitness is remarkably constant among random models and allele numbers.

All patterns of fertility selection exhibited an increasing trend in maximum and mean fertility as the number of alleles increased. T h e patterns varied in the magnitude of the difference between the maximum and mean fertility. This difference was the greatest for the random matrices, lower for the struc- tured matrices without ordering, and lowest for the structured and ordered models. T h e differences between the two-allele and six-allele models did not follow such a clear pattern. T h e multiplicative model showed the greatest variation among initial allele numbers, whereas the mean and maximum fit- nesses for the symmetric model were the smallest. Of course, the variation among dimensions is partly determined by the number of parameters, and the symmetric model was the most restrictive, having only three parameters for each number of alleles.

T h e number of alleles segregating at equilibrium can be directly compared across fertility and viability models (Figure 3). There is no apparent difference in the propensity of random fertility and viability models (or the pleiotropic model) to maintain multiple alleles at equilibrium. There were, however, con- siderable differences among the structured models in the number of alleles retained at equilibrium. T h e symmetric and dominant models maintained the greatest number of alleles at equilibrium, and ordering the structured models enabled them to retain more alleles. These results are consistent with what was observed in the viability model.

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166 A. G. CLARK AND M. W. FELDMAN

2.5

5

I .o

2 3 4 5 6

ALLELES

FIGURE 3.-The number of alleles segregating at equilibrium, averaged over all initial vectors and random matrices. Line patterns are the same as in Figure 1.

0.11

2 3 4 5 6

ALL EL E S

FIGURE 4.-Maximum heterozygosity at equilibrium, maximized over initial vectors and aver- aged over random matrices. Line patterns are the Same as in Figure 1 .

result was observed for all patterns of selection. Similarly, the number of alleles segregating at the equilibrium with the highest heterozygosity had more alleles than the average for all initial vectors. T h e number of alleles segregating at the equilibrium attained from the centroid was typically somewhat greater than the average for all starts, but in general, the behavior of the model can be reasonably well summarized by following the trajectory from the centroid.

Figure

4

presents the heterozygosity maximized across initial vectors and averaged across matrices. There is evidently a tendency for the fertility model to maintain a higher maximum heterozygosity than the viability model, while the pleiotropic model produces intermediate results. The symmetric and iden- tity models tended to maintain a higher heterozygosity than the random model, while dominance tended to reduce heterozygosity. T h e highest mean hetero- zygosity was found for the ordered symmetric model. T h e heterozygosity at the equilibrium with the highest mean fertility was usually lower than the mean heterozygosity for all initial vectors.

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16’7 FERTILITY SELECTION

31

ALL EL E S

FIGURE 5.-Number of distinct stable equilibria found among initial vectors, averaged over random matrices. Line patterns are the same as in Figure 1 .

equilibrium points found from the l o r initial vectors and averaged across matrices. T h e interesting result here is that, despite the potential of the two- allele fertility model to maintain more equilibria than the viability model (HAD- ELER and LIBERMAN 1975), the realized number of stable points remains lower for fertility models. This may be due either to a reduction in number of stable limits or to a reduction in the size of the domain of attraction of some of the stable equilibria. Equilibria with very small domains may be missed in the sample of 10r starts. Additional simulations were performed with the random fertility matrices with 100 initial vectors for each of 100 matrices. T h e equi- librium vectors were examined to determine the number of the starting con- dition that resulted in the last new equilibrium. Cumulative plots revealed the saturation kinetics of the numbers of distinct equilibria as the number of initial frequency vectors increased.

For models with r =

2,

3, 4, 5, 6 alleles, 10r initial vectors consistently revealed at least 85% of all equilibria that were detected by 100 starts. When the 19 summary statistics were calculated based on 100 initial vectors, no significant differences from the results with 10r initial vectors were detected. We conclude that although the sampling scheme misses some equilibria, the observed patterns are consistent with an exhaustive sampling.

There was a consistent tendency for models with more alleles to have a greater number of distinct equilibria. T h e different patterns of fertility mat- rices produced substantially different behavior in regard to the number of equilibria. With the exception of the multiplicative pattern, the other struc- tured models had more distinct equilibria (on average) than did the random model. This reached an extreme in models for which many fixation states were simultaneously stable.

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168 A . G . CLARK A N D M. W . FELDMAN

2 2

2

A I

H

FIGURE 6.-Mean fertility and heterozygosity for equilibria from random matrices. The digit

the centroid decreases as the dimension of the fertility matrix increases, and this is consistent for all patterns. T h e striking finding was that, for random fertilities, the fraction of starts that went to the centroid-derived equilibrium was always greater than the fraction of starts that maximized mean fertility or heterozygosity. In other words, the behavior from the centroid was a better predictor of the trajectory with a random start than either mean fertility or heterozygosity.

Another measure of the robustness of equilibria is the number of generations required to attain equilibrium. All models required more generations for higher dimensions, but the variation among fertility patterns in time required to attain equilibrium was large. Generally, the dominance models required the longest time, a result consistent with simulations of the viability model (KARLIN and FELDMAN, 1981).

Correlations among the 19 equilibrium statistics also revealed some interest- ing properties of the fertility model. T h e figures given below refer to the Spearman rank correlations for the five-allele random fertility model, and the correlation structure was rather similar for all patterns and dimensions. T h e relationship between the mean fertility and heterozygosity was surprising. The correlation between maximum mean fertility and the heterozygosity at the equilibrium with maximal mean fertility was -0.787 ( P

<

O.OOOl), and the correlation between the maximum mean fertility and the number of alleles segregating at the equilibrium with maximal mean fertility was -0.751 ( P

<

0.0001). This negative relationship is also clear when data are pooled over numbers of alleles, as in Figure 6. T h e correlation between F(c) and H ( c ) is -0.826 ( P 0.0001). T h e negative relationship between mean fitness and heterozygosity was also observed for viability models (KARLIN and FELDMAN

T h e fraction of starts that maximized mean fertility was positively correlated with the magnitude of the mean fertility (rho = 0.361, P = 0.0002). This implies that if an equilibrium has a high mean fertility, it is more likely to have a larger domain of attraction. T h e fraction of starts that maximized

0 1 .

0

that is displayed represents the number of alleles in the initial vector.

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FERTILITY SELECTION 169

heterozygosity was negatively correlated with the number of alleles segregating at that equilibrium (rho = 0.428, P

<

O.OOOl), indicating that equilibria with many alleles segregating have smaller domains of attraction.

T h e number of distinct equilibria was negatively correlated with the fraction of starts that ended at the centroid-derived equilibrium, the maximum fertility and the maximum heterozygosity [respective correlations and probabilities were -0.350 (P = 0.0003), -0.316 (P = 0.0013) and -0.284 (P = 0.0042)].

This implies that the greater the likelihood that an initial vector will be drawn to one of these three points, the fewer will be the number of distinct equilibria found. This may be due either to a decrease in the number of existing distinct equilibria or to the failure to find them because of their small domains of attraction.

T h e correlations between equilibrium statistics underscore the way in which the trajectory commenced at the centroid summarizes the dynamic behavior of the model. T h e correlation between the mean fertility at equilibrium (av- eraged over all starts) and the mean fertility of the centroid derived equilib- rium was 0.902 (P

<

0.0001). T h e correlation between the total number of iterations to reach equilibrium and the number of iterations from the centroid was 0.850 (P

<

0.0001). Finally, the correlation between the number of alleles segregating at equilibria arrived at from random starts and the number of segregating alleles at the centroid-derived equilibrium was 0.876 (P

<

0.000 1). That the centroid reveals properties of the model with random starting values was also observed in the case of the classical viability models.

DISCUSSION

It is important to examine the biological relevance of any theoretical model, but it is particularly necessary to examine the relevance of assumptions used in performing numerical simulations. Results of numerical simulations are often interpreted as a summary of potential behaviors of the model, and as much as one would like to claim that these results circumscribe the biologically reasonable results, it must be recalled that the entire parameter space is being sampled. Because we include patterns of fertilities and allele frequency distri- butions that may be biologically unlikely, the summary statistics may be com- promised. This is one of the reasons why it is so important to consider patterns of viabilities or fertilities. By using these patterns we restrict ourselves to por- tions of the entire parameter space. This allows us to examine not only the consequences of those assumptions but also the generality of the results from the total random parameter model.

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170 A. G . CLARK A N D M. W. FELDMAN

inance, partial dominance and ordered dominance. T h e motivation for these is described in KARLIN (1981). T h e mate-identity models are analogous to the distince models of KARLIN (1 98 1). There is ample biological evidence for this sort of phenomenon, including both pollen-stigma compatibility systems in plants and blood group compatibility in animals (OBER et aE. 1983; SCHAAP et

al. 1984) T h e ordering of identities may be such that more similar mating genotypes have a higher fertility or a lower fertility than less similar mating pairs. In the former case, polymorphism could not be maintained; therefore, we only examined the latter.

T h e role of structure in the viability matrix was stressed by KARLIN and

FELDMAN (1981) and is clearly also important in the case of fertilities. The patterns with the most structure are the symmetric, completely dominant and ordered mate-identity models, and these produce the highest levels of allelism and heterozygosity at equilibrium. These fertility models appears to behave like the distance-viability models of KARLIN and FELDMAN, in that there are many alleles at equilibrium, and these are rather evenly distributed. By the same token, the centroid is not as good a predictor in the models with more structure, probably because of the larger number of polymorphic equilibria available in these cases. T h e high values of row (13) in the models without structure probably reflect the frequent occurrence of global stability of fixa- tion.

Although mean fertility often decreases in the models treated here, it is generally more likely that the equilibrium reached will be that with the highest mean fertility, rather than that with the maximum heterozygosity. It is also worth noting that, as a class, the three “dominant” models are more likely to maintain an increase in the mean fertility than are the others. Therefore, it may be worthwhile to investigate the recursions of the models with dominance in order to clarify any structural similarity they might have to viability models. In general, the results of this study revealed a strong parallel with the clas- sical viability model. Both models showed an increase in mean fitness, the number of equilibria, the mean number of alleles, heterozygosity and number of generations required to attain equilibrium as the number of alleles in the initial vector increased. Both models also exhibited an increasing likelihood of polymorphism with more structured models. Finally, the random pleiotropic model gave results that were generally intermediate between the pure fertility and pure viability models.

Differences between viability and fertility selection are underscored by recent analytical results (FELDMAN and LIBERMAN 1985; LIBERMAN and FELDMAN

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FERTILITY SELECTION 171

two-locus model (with zero recombination) considered by LIBERMAN and FELD-

MAN (1985). Their model allows one locus to determine symmetric fertilities and another locus to determine symmetric viabilities. At r = 0, this model allows two high complementarity boundary equilibria and two high comple- mentarity polymorphic equilibria, whereas the central equilibrium is unstable. If genotypes gg and gkI have viabilities vg and U ~ I , respectively, then the pleio-

tropic model can be expressed in terms of “effective fertilities,” obtained by weighting fertility parameters by the viabilities of the two relevant genotypes (in this case, the effective fertility is vg v k l j j , k ~ ) . In light of this observation, it

is not surprising that the numerical simulations failed to reveal novel behavior of the pleitropic model.

We thank TOM and BARBARA RYAN, directors of Minitab, Inc., for their generous donation of computer time. TOM WHI-ITAM (Penn State University) provided helpful comments on the man- uscript. This work was supported by National Insitutes of Health grants HD18379 to A.G.C. and GM28106 to M.W.F.

LITERATURE CITED

BODMER, W. F., 1965 Differential fertility in population genetics models. Genetics 51: 41 1-424. FELDMAN, M. W., F. B. CHRISTIANSEN and U. LIBERMAN, 1983 On some models of fertility

A symmetric two-locus fertility model. Genetics 1 0 9

Selection models with fertility differences. J. Math. Biol.

KARLIN, S., 1981 Some natural viability systems for a multiallelic locus: a theoretical study.

KARLIN, S. and M. W. FELDMAN, 1981 A theoretical and numerical assessment of genetic varia- bility. Genetics 97: 475-493.

LEWONTIN, R. C., L. R. GINZBURC and S. D. TULJAPURKAR, 1978 Heterosis as an explanation for large amounts of genic polymorphism. Genetics 8 8 149-169.

LIBERMAN, U. and M. W. FELDMAN, 1985 A symmetric two-locus model with viability and fertility selection. J. Math. Biol. 22: 31-60.

OBER, C. L., A. 0. MARTIN, J. L. SIMPSON, W. W. HAUCK, D. B. AMOS, D. D. KOSTYU, M. FOTINO and F. H. ALLEN, JR., 1983 Shared HLA antigens and reproductive performance among Hutterites. Am. J. Hum. Genet. 35: 994-1004.

PENROSE, L. S., 1949 The meaning of fitness in human populations. Ann. Eugen. (Lond.) 1 4

POLLAK, E., 1978 With selection for fecundity the mean fitness does not necessarily increase.

SCHAAP, T., R. SHEMER, 2. PALTI and R. SHARON, 1984 A B 0 incompatibility and reproductive selection. Genetics 105: 1003-1010.

FELDMAN, M. W. and U. LIBERMAN, 1985

HADELER, K. P. and U. LIBERMAN, 1975 229-253.

2: 19-32.

Genetics 97: 457-473.

301-304.

Genetics 9 0 383-389.

failure. I. Prenatal selection. Am. J. Hum. Genet. 36: 143-151.

Communicating editor: M. T. CLECC

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172

A. G. CLARK AND M. W. FELDMAN

iterated as they stand, but this results in very poor efficiency. It is much faster to explicitly code the recursions for each genotype frequency. Because the six-allele model has 21 simul- taneous difference equations and each equation has more than 20 terms, this writing of the recurrence equations was itself done with a computer routine. This routine exhaustively examined mating types and wrote for each genotype the mating types that can produce that genotype in offspring, as well as the expected fraction of offspring of the mating type that have the respective genotype. T h e algorithms were then verified by hand. T h e routines were written in FORTRAN-77 and were compiled with the proprietary PRIME F77 compiler. All variables involved in calculations of allele and genotype frequencies were declared DOU- BLE PRECISION (28 decimal places).The routines were verified by simulating with no selection and by running matrices known to globally fix one allele.

Equilibrium was defined as a maximum change in genotype frequency of less than loT5 over one generation. For very weakly attracting equilibria, this may result in some discrep- ancy between the true equilibrium and the point where iterations stopped. To test the magnitude of this problem, simulations were run on 40 starts for each of 5 0 matrices with the 5-allele model. Genotype frequencies were retained for the equilibrium defined by the IOe5 cutoff, and iterations were continued until a maximum change in genotype frequencies was T h e maximum difference in genotype frequency between the two equilibria defined by these two criteria was 4.4 X In no case did the more restrictive criterion result in a divergence t o a completely different equilibrium, and the IOw5 cutoff always yielded an equilibrium in close proximity to the “true” equilibrium.

T h e equilibrium cutoff brings the simulation into the neighborhood of the true equilib- rium, and the routine that counts the number of distinct equilibria is sensitive to this neigh- borhood size. Equilibria are considered distinct if the maximum allele frequency difference is greater than If this number were lowered by a few orders of magnitude, it is apparent that the points in the neighborhood of the same equilibrium would be considered as distinct equilibria. O n the other hand, if the criterion for distinctness were much greater than we would run the risk of considering truly distinct equilibria as being in the neighborhood of the same equilibrium. T h e two cutoffs were verified by repeating simulations with an equilibrium cutoff of and a distinctness cutoff of IO-’.

APPENDIX B TABLE 1

Random fertilities

No. of alleles

Statistic 2 3 4 5 6

( 1 ) Max F

(2) No. x,#O ( F )

(4) Max no. x,#O

(5) Min no. x,#O

(6) No. x,#O

(7) No. limits (8) M a x H (9) No. x,#O

(3) H ( F )

(10) F(c)

(12) H(c)

(1 1) No. x,#O ( c )

(13) Limit = c

(14) Limit = max F

(15) Limit = max H

(17) No. iterations (18) No. iterations ( c )

(16) F

(19) AF<o

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FERTILITY SELECTION 1'73

TABLE 2

Multiplicative fertilities

No. of alleles

Statistic 2 3 4 5 6

(1) Max F 0.403 0.496 0.520 0.597 0.612 (2) NO. x ~ # O ( F ) 1.570 1.500 1.620 1.700 1.720

(3) H ( F ) 0.199 0.195 0.232 0.234 0.211

(4) Max no. x,#O 1.610 1.660 2.060 1.990 2.100

(5) Min no. x,#O 1.550 1.480 1.670 1.640 1.660

(6) No. x,#O 1.591 1.57 1 1.907 1.818 1.876

(7) No. limits 1.130 1.220 1.310 1.370 1.450 (8) Max H 0.210 0.250 0.343 0.350 0.355 (9) No. x,#O 1.610 1.660 2.030 1.980 2.070 (10) F ( 4 0.403 0.484 0.497 0.562 0.573

(1 1) NO. xi#O ( c ) 1.590 1.580 1.840 1.860 1.920

(1 2) 4 4 0.205 0.224 0.254 0.295 0.286 (13) Limit = c 0.990 0.938 0.896 0.881 0.850 (14) Limit = max F 0.929 0.892 0.82 1 0.8 16 0.783 (15) Limit = max H 0.977 0.864 0.756 0.739 0.663 (16) F 0.397 0.482 0.500 0.569 0.580 (17) No. iterations 47.930 96.770 120.800 15 1.760 149.400 (18) No. iterations (c) 47.340 94.130 123.270 159.860 146.560 (19) AFCO 0.453 0.353 0.369 0.292 0.287

TABLE 3

Symmetric fertilities

No. of alleles

Statistic

~ ~~

2 3 4 5 6

(1) Max F (2) No. x , f O ( F )

(4) Max no. x,#O

(5) Min no. x,#O

(6) No. x,#O

(7) No. limits (8) Max H (9) No. x,$O

(3) H ( F )

(10) F ( 4

(12) H(c) (1 1) No. x,+O ( c )

(13) Limit = c (14) Limit = max F (1 5) Limit = max H

(17) No. iterations (18) No. iterations (c) (19) AFCO

(16) F

(14)

174 A. G. CLARK AND M. W . FELDMAN

TABLE 4

Completely dominant fertilities

No. of alleles

Statistic 2 3 4 5 6

(1) Max F

(2) No. x,#O (F)

(4) Max n o . x,#O

(5) Min no. xi#O

(6) No. x,#O

(7) No. limits (8) Max H

(9) No. x,#O

(3) H(F)

(10) F(c)

(12) H ( 4 (1 1) No. x,#O ( c )

(13) Limit = c (14) Limit = max F (15) Limit = max H

(1 7) No. iterations (18) No. iterations (c) (16) F

(19) AFCO

0.65 1 1.490 0.055 1.490 1.490 1.490 1.010 0.055 1.490 0.651 1.490 0.055 1

.ooo

0.975 1.000 0.651 319.04 334.9 0.465 0.768 2.080 0.172 2.410 1.970 2.234 2.060 0.251 2.190 0.754 2.280 0.200 0.219 0.883 0.159 0.756 4 15.940 4 19.030 0.225 0.786 2.550 0.175 2.970 2.420 2.710 3.400 0.273 2.700 0.777 2.720 0.210 0.172 0.873 0.106 0.776 446.770 446.910 0.145 0.795 2.970 0.200 3.780 2.670 3.366 4.310 0.342 3.210 0.763 3.410 0.252 0.060 0.748 0.021 0.765 483.380 488.400 0.065 0.814 3.980 0.230 4.660 3.560 4.095 4.140 0.338 3.960 0.804 4.100 0.249 0.058 0.874 0.019 0.806 493.620 492.200 0.010

TABLE 5

Partially dominant fertilities

No. of alleles

Statistic 2 3 4 5 6

(1) Max F (2) No. x,#O (F)

(4) Max no. x*#O

(5) Min no. xi#O

(6) No. x,#O

(7) No. limits (8) Max H

(9) No. x,#O

(3) H ( F )

(10) F ( 4

(12) H(c) (1 1) No. x,#O (c)

(13) Limit = E

(14) Limit = max F

(15) Limit = max H

(16) F

(17) No. iterations (18) No. iterations (E)

(19) AF<o

(15)

FERTILITY SELECTION

TABLE 6

Ordered dominant fertilities

175

No. of alleles

Statistic

(1) Max F

(2) No. x Z 0 (F)

(4) Max no. x,#O

(5) Min no. xi#O

(6) No. x,#O

(7) No. limits (8) Max H

(9) No. x,+O

(3) H(F)

(10) F(c)

(1 2) H(c)

(1 1) No. x,#O (c)

(13) Limit = c (14) Limit = max F

(15) Limit = max H

(1 7) No. iterations (18) No. iterations (c)

(19) AF<o

(16) F

2 0.699 2.000 0.179 2.000 1 .goo 1.987 1.350 0.180 2.000 0.699 2.000 0.179 0.91 1 0.987 0.887 0.698 403.300 406.6 1 0

0.536 3 0.818 2.600 0.234 2.610 2.400 2.610 2.960 0.237 2.610 0.8 18 2.610 0.236 0.814 0.977 0.736 0.818 5 12.050 509.700 0.383 4 0.855 2.890 0.300 2.930 2.570 2.829 3.160 0.305 2.800 0.855 2.840 0.302 0.597 0.964 0.508 0.855 755.450 764.310 0.388 5 0.875 2.800 0.309 2.870 2.660 2.799 4.720 0.315 2.770 0.875 2.810 0.312 0.567 0.934 0.406 0.875 835.390 824.370 0.248 6 0.905 2.860 0.262 3.130 2.680 2.884 4.870 0.266 2.890 0.905 2.840 0.263 0.488 0.956 0.368 0.905 870.580 864.700 0.099

TABLE 7

Mate-identity fertility pattem

No. of alleles

Statistic 2 3 4 5 6

(1) Max F

(2) No. x Z 0 (F)

(4) Max no. xi#O

(5) Min no. x,+O

(6) No. x Z 0

(7) No. limits (8) Max H

(9) No. x,#O

(3) H(F)

(10) F(c)

(12) H(c) (1 1) No. x Z 0 (c)

(13) Limit = c

(14) Limit = max F

(15) Limit = max H

(16) F

(1 7) No. iterations (18) No. iterations (19) AF<o

0.578 1.670 0.284 1.740 1.670 1.707 1.380 0.321 1.740 0.494 2.000 0.485 0.425 0.940 0.943 0.571 141.7 10 7.350 0.632 0.67 1 1.970 0.371 2.1 10 1.910 2.045 2.390 0.421 2.110 0.63 1

2.380 0.464 0.467 0.875 0.818 0.551 226.320 79.940 0.650 0.674 2.140 0.45 1 2.650 1.980 2.127 3.280 0.487 2.180 0.669 2.830 0.462 0.428 0.896 0.750 0.667 256.460 133.400 0.641 0.698 2.560 0.469 2.670 2.460 2.555 3.680 0.500 2.660 0.659 3.090 0.546 0.458 0.896 0.759 0.692 282.570 155.700 0.659 0.735 2.480 0.447 2.780 2.360 2.574 3.820 0.518 2.780 0.646 3.460 0.607 0.235 0.773 0.61 1

(16)

176 A. G . CLARK AND M. W. FELDMAN

TABLE 8

Ordered mate-identity fertility pattern

No. of alleles

Statistic

(1) Max F (2) No. x,#O (F)

(4) Max no. x,#O

(5) Min no. x,ZO

( 6 ) No. x,#O

(7) No. limits

(8) Max H (3) H(F)

(9) NO. xi#O (10) F(c)

(12) H(c) (1 1) No. x,#O (c)

(13) Limit = c (14) Limit = max F (15) Limit = max H

(1 7) No. iterations (1 8) No. iterations (19) AF<o

(16) F

2 0.412 2.000 0.568 2.000 2.000 2.000 1.020 0.568 2.000 0.4 12 2.000 0.568 1 .DO0 1.000 1.000 0.412 126.0 10 8.720 0.877 3 0.373 2.680 0.693 2.700 2.680 2.697 2.200 0.697 2.700 0.371 2.700 0.697 0.973 0.982 0.971 0.371 103.110 95.510 0.869 4 0.125 3.590 0.757 3.620 3.560 3.606 2.910 0.769 3.620 0.124 3.600 0.768 0.750 0.968 0.718 0.124 182.140 182.530 0.775 5 0.086 4.110 0.769 4.110 4.110 4.1 10 4.270 0.780 4.110 0.086 4.110 0.781 0.679 0.981 0.670 0.086 168.140 143.730 0.670 6 0.069 5.260 0.803 5.340 5.260 5.297 5.280 0.808 5.300 0.069 5.320 0.808 0.466 0.985 0.455 0.069 234.960 2 1 1.440 0.637

TABLE 9

Random viability-fertility pleiotropic model

No. of alleles

Statistic 2 3 4 5 6

(1) Max F (2) No. x,#O (F)

(4) Max no. x,#O

(5) Min no. x,ZO

( 6 ) No. x,+O

(7) No. limits (8) Max H (9) No. x,#O

(3) H(F)

(10) F(c)

(1 2) H(c) (1 1) No. x,#O (c)

(13) Limit = c (14) Limit = max F (15) Limit = max H

(17) No. iterations (18) No. iterations (c)

(16) F

(19) AFCO

Figure

FIGURE from the mean fertilities of the the same genetic locus determines fertility and viability (parameters for both models chosen ran- domly)
FIGURE 2.-Maximum mean fitness, maximized over initial vectors and averaged over random matrices
FIGURE 3.-The number and random matrices. Line patterns are the same as in Figure of alleles segregating at equilibrium, averaged over all initial vectors 1
FIGURE 5.-Number random matrices. Line patterns are the same as in Figure of distinct stable equilibria found among initial vectors, averaged over 1
+7

References

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