8
Predicting the ESS seed number per flower for two species with effects of
sex allocation and selective seed abortion.
Summary
In hermaphroditic plants, the ESS allocation of investments to male and female function is likely to be affected by selective seed abortion. Current models of sex-allocation do not include the effect of selective abortion, although it has been shown in several species. We modeled allocation to male and female function, both with and without including selective seed abortion. We used an existing ESS model, and parameters were based on data for Echium vulgare and Cynoglossum officinale. To examine the relative importance of (1) sex-allocation and (2) selective seed abortion on the ESS allocation to seeds and flowers, we analyzed the results for both hermaphrodite and male sterile individuals. Male sterile individuals do not produce pollen, therefore sex allocation does not play a role. Both species can produce four seeds per flower. However, under natural conditions, for both species seed set per flower is close to one. The simplest function that describes offspring quality as a function of number of seeds per flower is a linear decline. A linear relation might not be biologically realistic however, and therefore we modeled also a relation between offspring quality and number of seeds per flower that is based on the assumption of a Poisson distribution of deleterious alleles. This assumption yielded inverse J-shaped curves. One such curve was fitted through the experimental data of offspring quality and number of seeds per flower.
Both the effect of (1) sex-allocation (in hermaphrodites without selective seed abortion), and (2) selective seed abortion (in male sterile individuals) make the ESS number of seeds per flower drop far below its maximum of four. The combined effects of (1) and (2) lead to a relatively small further decrease (in hermaphrodites with selective seed abortion). In hermaphrodites, sex allocation seems to provide the strongest selective pressure because the reduction in ESS number of seeds per flower is the strongest for this factor.
C. Melser, T.J. de Jong & P.G.L. Klinkhamer
Introduction
During their reproductive stage, hermaphroditic plants can direct resources to more flowers or to fill developing seeds. In this way they can emphasize male fitness by producing pollen (flowers) or female fitness by producing seeds (e.g. Willson & Price 1977, Queller 1983, Sutherland and Delph 1984). Theoretical models (Spalik 1991, Morgan 1993, Rademaker &
de Jong 2000) showed that the ESS (Evolutionary Stable Strategy) allocation of resources to flower and seed production depends on the shape of the fitness gain curves (the curves that relate investment in male or female reproduction and the gain in fitness achieved). In outcrossing plants, the ESS allocation to flowers and seeds is proportional to the exponents of the gain curves. If both gain curves are linear, plants should invest 50% of their resources in flowers and 50% in seeds (Charnov 1982). The number of flowers and seeds produced then depends on the relative costs of flowers and seeds. If one flower is less costly than the seeds it would contain with full seed set, sex-allocation theory predicts that the plant should not fill all flowers with the maximum of seeds and explains in this way an apparent overproduction of flowers.
Sex-allocation theory predicts that a lower selfing rate leads to a higher investment of male function (i.e. flower production). In the theoretical models mentioned, seed quality only depends on whether seeds are selfed or outcrossed. However, extra flower production may affect seed quality also in another way. Selective abortion of a "surplus" of offspring of relatively low quality can free resources for higher quality offspring and thereby increase the female fitness of the maternal plant (e.g. Darwin 1883, Janzen 1977, Stephenson 1981, Marshall & Ellstrand 1988, Marshall & Folsom 1991). An increase in offspring quality by selective abortion has been shown for various species, e.g. the hermaphrodite self-compatible Lotus corniculatus (Stephenson & Winsor 1986), Cassia fasciculata (Lee & Bazzaz 1986), Cryptantha flava (Casper 1988), and the species studied in this thesis (Chapter 3, 6 and 7), and the self-incompatible Raphanus sativus (Marshall & Ellstrand 1988) and Phaseolus vulgaris (Rocha & Stephenson 1991).
In a previous study (Chapter 3), we modeled the effects of selective seed abortion and
the seed quality on the female fitness of plants. We calculated for plants with different
allocation to seeds and flower and thus different abortion levels the required quality of their
seeds that would produce an equal number of surviving offspring (=female fitness) for all
plants. Plants with a lower seed number per flower require seeds of a relatively high quality to
produce an equal amount of offspring in the F 1 generation in comparison to plants with a
higher seed number per flower. A decrease in seed production per flower from 4 to 0.66 (a
decrease of 84%) is compensated by an increase of offspring quality of 38% in our example,
due to selective seed abortion (see Chapter 3). This implies that a small increase in relative
fitness per seed can give rise to selection for a considerable decrease in the seed to flower
ratio. In this previous model we neglected the male function of flowers and the effects of
flower number on geitonogamous selfing. It is most likely that including male fitness would
push seed to flower ratios even lower, because a low seed to flower ratio is accompanied by
the production of more flowers, which enhances male fitness per plant if levels of inbreeding
depression are relatively low (δ<0.5). So both sex-allocation and selective seed abortion push
the ESS-seed to flower ratio downward. The relative strength of both effects and their combination has never been evaluated.
The aim of the present study is twofold. Firstly, we want to see if our data from Echium vulgare and Cynoglossum officinale support the hypothesis that the overproduction of ovules in these species can be explained by selective seed abortion. Secondly, we want to compare the relative selective strength of selective seed abortion and sex allocation in these species for the optimal division of resources over flowers and seeds. For that we will use the parameter set of these two species and analyze how male sterile individuals (as occur in the gynodioecious E. vulgare) should allocate their resources to flowers and seeds. These indivi- duals do not produce pollen, and the hypothesis of sex allocation is thus not applicable, while the hypothesis of selective seed abortion applies to all seed bearing individuals.
Materials and methods
Equations describing plant fitness Hermaphrodite, self compatible individuals
The formulation of the model follows Rademaker & de Jong (2000). The total amount of resources of a plant (T) is measured as the number of flowers (without seeds) that can potentially be formed, so the cost of one flower is 1. It is assumed that a plant cannot vary the amount of pollen (male function) within a flower. All plants are of the same size and can invest their resources in seeds or flowers. Plants allocate a fraction r of the T resources to form n flowers (with pollen), so that n = rT. The plant then has T-n resources left for the production of seeds. It is assumed that costs for seeds are fixed and independent of the number of seeds produced. If the cost per seed equals that of c flowers, a plant with n flowers produces (T-n)/c seeds. Consequently it makes (T-n)/nc seeds per flower.
Selfed seeds (fraction S) suffer from an inbreeding depression δ. Inbreeding depression is the proportion of selfed seeds that germinate and grow to reproduction, relative to outcrossed seeds. Fitness is defined as the number of copies of the (haploid) genome trans- mitted to the next generation, counted at the same life stage. In a large population of plants producing n flowers, the individual fitness W n of a hermaphrodite, self-compatible individual is described as (Charlesworth and Charlesworth 1981, Lloyd 1987, de Jong et al. 1999 and Rademaker & de Jong 2000):
c n S T
c n S T
c n S T c
n S T
W
n n n n n( )
) 1 ( ) 2 ) (
1 ) ( ) (
1 ( ) 2 ) (
1
( − − + − − + − − = − −
= δ δ (1).
The first term in equation (1) describes the outcrossed seeds of mother n. The second denotes that the plant is the mother of the selfed seeds S n , which suffer from inbreeding depression (δ) and that the parent is also the pollen donor of these selfed seeds. The last term denotes the number of seeds sired on other plants of the population.
With a different number of flowers m, a mutant will have a different level of seed
abortion, and therefore a different average quality of the remaining offspring. The individual
fitness W m of a rare mutant of a hermaphrodite, self-compatible species producing m flowers
and consequently (T-m)/c seeds with a relative quality (compared to the resident) K m /K n is then
c n S T
E E K K c
m S T
K K c
m S T
W
nn m n m m
n m m
m
( )
) 1 ) (
) ( 1 ( ) 2
) ( 1
( − − + − − + − −
= δ (2).
Here E m /E n denotes the ratio of pollen export for mutant and resident types. Note that for the resident type relative seed quality is set to K n /K n = 1. In equations (1) and (2) female fitness is linearly related to seed production. However, local resource competition (LRC) (Lloyd and Bawa 1984) among seedlings can play a role. For E. vulgare and C. officinale the female gain curves appeared to be linear and LRC is either constant or absent (Rademaker 1998) and we do not include this factor.
Male sterile individuals
Male sterile individuals or female plants of dioecious species will only produce outcrossed seeds, and do not serve as a pollen donor. Their selfing rate is 0. In E. vulgare, male sterile individuals produce yellow pollen that is infertile. Under the assumption that a female flower (without production of viable pollen) is equally costly as a hermaphrodite flower, the fitness of a female plant with n flowers is found by taking only the first term of equation (1):
c n
W
n= ( T − ) (3).
The individual fitness W m of a female plant producing m flowers and consequently (T-m)/c seeds with relative quality K m /K n is then
n m m
K K c
m
W = ( T − ) (4).
Note that the optimal flower number (coupled to optimal number of seeds per flower) of the female plant is independent of the frequency of female plants in the population or the strategy of hermaphrodite plants.
Finding the ESS
For the male sterile individual where K m has a linear relation with the number of seeds per flower, the ESS can be simply calculated analytically (see Appendix A). For other cases, the fitness of the common, resident phenotype with n flowers was compared numerically with the fitness of a mutant with m flowers. We calculated whether mutants with slightly different numbers of seeds per flower and consequently a change in offspring quality could invade in the population (fitness mutant > fitness resident) or not? If there is a number of flowers n* in which no mutant can invade, the resident strategy of producing n* flowers is evolutionary stable (ESS). In our model, we assume that there is one decisive moment in which the allo- cation to flowers and seeds is determined. In reality, abortion rate of individual plants may vary along the flowering season (Chapter 2).
Estimating parameters The species
We used the hermaphroditic, self-compatible, monocarpic perennials Echium vulgare and
Cynoglossum officinale, for which we could use most parameter estimations of the model from
Rademaker and de Jong (2000). The model of sex-allocation without selective seed abortion has been extensively analyzed in Rademaker and de Jong (2000) and this we took as a starting point for adding the effects of selective seed abortion in the model. Both species have herma- phrodite flowers, produce both pollen and seeds and are mainly pollinated by bumblebees.
Both species can act as a model for a hermaphroditic, self-compatible, monocarpic species that is insect-pollinated. The main difference between the two species is the estimated selfing rate.
Depending on flower numbers, the selfing rate for C. officinale is estimated between 0 and 70% (Vrieling et al 1999) and is much lower for E. vulgare, for which the estimation ranged between 0 and 30% (Rademaker 1998). In E. vulgare self-pollination is also limited by protan- dry and spatial separation of anthers and style within one flower. Male-sterile individuals occur in natural populations of E. vulgare (Klinkhamer et al. 1994). These male-steriles have an average seed set that is 23% higher than the hermaphrodite individuals in the same popula- tion (Klinkhamer et al. 1994). Both E. vulgare and C. officinale may produce up to four seeds per flower, but as a rule less than one seed matures per flower in the hermaphroditic indivi- duals (de Jong & Klinkhamer 1989, Klinkhamer et al. 1994). Both species have a similar flowering pattern. Each individual plant may have one to twenty flowering stems. Each flowe- ring stem may produce up to thirty cymes and each cyme carries up to 30 flowers. Flowers on a cyme that are closest to the main stem open first. A flower remains open for about 2-3 days and nectar is produced continuously. Only a proportion of the possibly hundreds of flowers are open at the same time. In our study site, Meijendel, close to The Hague in the Netherlands, pollination is mainly by bumblebees.
Resource availability
The parameters and equations describing i) resources available for reproduction, ii) trade off between flower and seed production, iii) selfing rates as a function of flower number, and iv) pollen export as a function of flower number, are obtained from Rademaker and de Jong 2000.
Resources are measured in units equivalent to the cost of producing a complete flower. The total amount of resources available for reproduction in E. vulgare (T) is set at 9111 units and in C. officinale at 1003 units. We assumed that all flowers produce four ovules, which is in the model equivalent to assume that ovule production is not costly.
Trade-off between flowers and seeds
The mass of a seed is independent of the number of seeds per flower in the flower in which it
is produced (Melser, unpubl. data). Therefore we assumed that the costs to produce a seed
relative to the costs of flowers were constant and independent of the number of seeds per
flower. The amount of resources to produce one seed (c) in E. vulgare is equivalent to the cost
of producing 1.78 flowers. The amount of resources of producing one seed (c) in C. officinale
is equivalent to 3.6 flowers (Rademaker & de Jong 2000). As explained before, female fitness
is described by the number of seeds produced times offspring quality (equation 2), male fit-
ness is described by the seeds sired times offspring quality. A shift in sex-allocation will thus
change the number of flowers and seeds, and not the allocation to pollen and ovules within
one flower. The maximum number of seeds per flower is fixed at 4.00.
Selfing rates
The selfing rate depends on the number of flowers visited in a sequence (y) and the carry-over of pollen between flowers (1-k). The geitonogamous selfing rate was estimated for E. vulgare as S n =1-(1-(1-k) y )/(y*k) with y being the number of flowers visited in a sequence (Crawford 1984, Robertson 1992, Barrett et al. 1994) and k estimated as 0.045 (Rademaker & de Jong 2000). The number of flowers visited (y) is expressed as ln(y)=0.57 + 0.31ln(0.1*n) where n is the total number of flowers of a plant and 0.1*n the number of flowers that are open simultaneously (Rademaker & de Jong 2000). For C. officinale, the selfing rate was estimated as S n =1-(1-(1-k) y )/(y*k) with y being the number of flowers visited in a sequence and k estimated as 0.10 (Rademaker & de Jong 2000). The number of flowers visited (y) was expressed as ln(y)=-0.18 + 0.65ln(0.125*n) with n as total number of flowers of a plant and 0.125*n the number of flowers that are open simultaneously (Rademaker & de Jong 2000).
Pollen export
Pollen export (E) has been estimated indirectly with using the selfing rate (see previous para- graph) and pollen discounting (λ). While visiting more flowers within the same plant, the pollinator induces more selfing and also looses part of the pollen on the body. The resulting negative relation between selfing and the efficiency of pollen export is referred to as pollen discounting (λ). For a given number of flowers, the pollen export is determined by the level of selfing rate and pollen discounting. Pollen export is estimated as E n =n-λS n n. For E. vulgare λ=1, for C. officinale λ=0.27 (Rademaker & de Jong 2000).
fr act ion survival (124 days)
0 0.50 0.75 1
0.25
fr act ion survival
00 1 2 3
selection level ovule removal average of the points
4 0.2
0.4 0.6
A
B
Figure 1:
Mean offspring quality (expressed as survival) in relation to seed number per flower
A: E. vulgare . Linear relation:
K = -0.0851 * x + 0.59. Parameters for the inverse J-shaped relation: µ=7, β=0.05, q(0)=0.699 (See Appendix B).
Open dots are results from the expe- riment with different levels of seed set and offspring quality (Chapter 7).
B: C. officinale. Linear relation:
K = -0.084 * x + 0.408. Parameters for
the inverse J-shaped relation: µ=12,
β=0.0475, q(0)=0.688 (See Appendix
B). Closed squares are results from
Chapter 3; closed diamonds are results
from Chapter 6; open dots are the mean
values of the two experiments from
Chapter 3 and Chapter 6.
Inbreeding depression
In the embryonal stage, Melser et al. (Chapter 4) detected no inbreeding depression in E.
vulgare. Pollen from selfing and outcrossing on the stigma do not differ in ability to produce a seed in both E. vulgare and C. officinale (Chapter 4, 6 and 7). Nevertheless, inbreeding effects were found later in life. These effects were incorporated in the model. The parameter for life- time inbreeding depression, δ, is obtained from Melser and Klinkhamer (Chapter 6 and 7).
Germination, survival, growth and both male reproduction (siring seeds) and female reproduction (producing seeds) were measured for offspring from self-pollination and offspring produced after cross-pollination. For E. vulgare, the offspring from selfing had lower survival (11%), and lower reproductive success (56%) compared to offspring from outcrossing. Reproductive success was measured for both male and female function, which appeared to be almost equal (0.58 and 0.54 respectively). One minus lifetime inbreeding depression is thus estimated as (1-0.11) * (1-0.56) =0.39. Lifetime inbreeding depression is then 1-0.39=0.61. For C. officinale, the offspring from selfing had a 15% lower survival compared to offspring from outcrossing while the offspring from the two pollination types reproduced equally well, which gives the value of 1-δ = (1-0.15)=0.85. Lifetime inbreeding depression is then 1-0.85=0.15.
Offspring quality in relation to abortion level
From an experiment with differential seed set during the flowering season (Chapter 7) with E.
vulgare, we have two different time periods with different numbers of seeds per flower, and a differential survival of the resulting offspring (Fig. 1A). From the offspring of the time period with the highest seed set (0.837 seeds per flower) 51.9% survived after 124 days in contrast to 54.9% from the offspring of the time period with the lowest seed set (0.489 seeds per flower).
The function that describes the relation between offspring quality and seed number per flower should fit through these two data points. The slope of the line indicates the effectivity of selective seed abortion on offspring quality.
For C. officinale two experiments were available to estimate the relation between off- spring quality and number of seeds per flower. From an experiment with an increasing abor- tion level during the flowering season (Chapter 6) with C. officinale, we derived two time periods with a decreasing number of seeds per flower along the flowering season (Fig. 1B).
From the offspring of the time period with the highest seed set (1.311 seeds per flower) 18.7%
survived after one year in contrast to 25.9% from the offspring of the time period with the
lowest seed set (0.466 seeds per flower). In an experiment in which we removed three of the
four ovules in all flowers of experimental plants (Chapter 3), the seed set in the remaining
ovule was 0.493 seed per ovule. By multiplying this by four, we obtain the number of seeds
per flower, if the flowers would have all their ovules intact and a similar abortion rate: 1.972
seeds per intact flower. Thirty-five days after the start of germination, 35.4% of the offspring
survived, in contrast to 44.7% survival of offspring from the control plants (0.852 seeds per
flower) (Fig. 1B). To obtain one function that would describe the relation between offspring
quality and mean number of seeds per flower, we took the mean value (for both the x and y
axis) of the two points (one from each experiment) with the low number of seeds per flower
and the mean value of the two points (one from each experiment) with the high number of
seeds per flower (Fig. 1B). The function that describes the relation between offspring quality and seed number per flower should fit through these two data points.
For both species we have thus two data points that relate offspring quality to seed number per flower. The simplest function that fits through these points is a straight line. How- ever, a linear relation between offspring quality and abortion rate is not based on biological arguments about the processes that lie behind. In addition to the straight line, we made a numerical calculation to obtain the shape of the function of average offspring quality in relation to number of seeds per flower. The calculation was based on four assumptions: i) the quality of an individual is determined by the number of deleterious alleles present. Those dele- terious alleles have a negative, additive effect on quality; ii) the strength of the effect on offspring quality is equal for all deleterious alleles; iii) the offspring shows a Poisson distri- bution of deleterious alleles; and iv) the seeds of lowest quality are aborted without error (truncated selection). i) is consistent with our earlier results on abortion and quality (Chap- ter 4), ii) is to start with the simplest case, iii) and iv) are assumptions frequently used in other population genetic models (e.g. Kondrashov 1995, Schultz & Willis 1995, Kozlowski &
Stearns 1989). We calculated the relative mean quality of the non-aborted offspring as a func- tion of number of seeds per flower of the maternal plant.
Results
Analytical calculations
We calculated two different relations between average offspring quality (K) and number of seeds per flower (x). Here we present a linear relation: K=ax+b with a as slope and b as intercept. The number of seeds per flower (x) is directly related to n (total number of flowers) as x = (T-n)/nc. With selective abortion, a<0. Without selection on the offspring, a=0. Off- spring quality is fixed at zero for negative values for the quality. For E. vulgare we found K=- 0.0851 * x + 0.59 (Figure 1A). The linear function for C. officinale is described as: K = -0.084
* x + 0.408 (Figure 1B). For male sterile individuals with a linear relation between offspring quality and number of seeds per flower, the optimal number of seeds can then be calculated analytically by comparing the fitness of the mutant with m flowers and related quality Km to the fitness of the resident with n flowers and related quality Kn (derivation of the equation in Appendix A).
Optimal number of seeds per flower =
( a bc ) c
a c
1 1 −
− (5).
with c being the relative cost to produce a seed, and a and b describing the slope and intercept, respectively, of the linear relationship between seed per flower (x) and offspring quality (K).
With c=0, seeds are infinitely cheap to produce and the optimal number of seeds per flower
approaches infinity. With c infinitely positive, seeds are infinitely expensive to produce com-
pared to the production of a flower and the optimal number of seeds per flower will be 0. For
the values of c that we have for E. vulgare (c=1.78) and C. officinale (c=3.6), the relation be-
tween the optimal number of seeds per flower and a and b is presented in resp. Figure 2A and
2B. With a=0 the seed quality is not affected by selective abortion and the optimal number of
seeds per flower approaches infinity. Our model species have a fixed number of ovules per
flower and the maximum seed set per flower is fixed at four. With an infinitely negative value for a, selective abortion has a very strong effect on the offspring quality, and the optimal number of seeds per flower will approach zero. With an increasing value for the intercept b, the average quality is higher, the effects of selective seed abortion will be weaker, and the optimal number of seeds per flower increases.
Only selective seed abortion: male sterile individuals
In natural populations of E. vulgare also male sterile individuals occur. Average seed set per flower is 1.12 (Klinkhamer et al. 1994). These individuals do not have a male function and all reduction in seeds per flower is due to selective seed abortion. With a linear relation between offspring quality and number of seeds per flower, the analytically calculated optimal number of seeds per flower for E. vulgare is 1.44 (equation 5, Table 1). For hypothetical male sterile individuals of C. officinale (equation 5, Table 2), the optimal number of seeds per flower is 0.916.
Simulations
The second relation between average offspring quality and number of seeds per flower did not assume a linear relationship, but used the four assumptions about offspring quality and the number of deleterious alleles present. Different parameter values (the average number of dele- terious alleles (µ), the strength of the deleterious effect of an allele (β) and the offspring quality without deleterious alleles (q(0)), all gave an inverse J-shaped function. In Appendix B we summarized the procedure of the calculation. The values of the parameters are given in Table 1 and 2.
We firstly calculated the ESS number of seeds per flower, for the full model (compa- ring fitnesses from eq. 1 and 2) for a population consisting of all hermaphroditic individuals of
4
3
2
1
0
1 0.5
0 -1
-0.5 0
Slope (a)
Intercept (b) 4
3
2
1
0
1 0.5
0 -1
-0.5 0 ESS number
of seeds per flower
Intercept (b)
A B
Figure 2:
The relation between ESS number of seeds, a (slope of the linear relation between offspring quality
K and seed number per flower x) and b (intercept of the linear relation between offspring quality K
and seed number per flower x), for male sterile individuals with a linear relation between offspring
quality and number of seeds per flower. A. E. vulgare; c=1.78. B. C. officinale; c=3.6.
either E. vulgare or C. officinale, with and without selective seed abortion. When abortion is non-selective, K m =K n for all values of m and n (which corresponds with a slope of 0 for the relation between offspring quality and number of seeds per flower in figure 1), and the results of the model resembles the ones from Rademaker and de Jong (2000). The only deviation that appears in their results and ours, is due to the different parameters of inbreeding depression
Table 1:
ESS number of flowers, seeds and selfing rate for E. vulgare for two different relations between average offspring quality and seed number per flower.
Condition flowers seeds seeds / flower estimated
selfing rate
abortion non-selective (Km=Kn) 3933 2769 0.704 0.202
linear relation*
hermaphrodite, self-compatible species 4378 2491 0.569 0.208 male sterile individuals (analytically
derived - see Appendix A)
2507 2302 1.440 0
inverse J-shaped relation**
hermaphrodite, self-compatible species 4299 2573 0.599 0.207
male sterile individuals 2087 3756 1.800 0
* Quality individual seeds = -0.0851 * (seed per flower) + 0.59
** µ=7, β=0.05, q(0)=0.699
Table 2:
ESS number of flowers, seeds and selfing rate for C. officinale with two different relations between average offspring quality and seed number per flower.
Condition flowers seeds seeds / flower estimated
selfing rate
abortion non-selective (Km=Kn) 373 175 0.468 0.353
linear relation***
hermaphrodite, self-compatible species 420 162 0.385 0.375 male sterile individuals (analytically
derived - see Appendix A)
272 203 0.747 0
inverse J-shaped relation****
hermaphrodite, self-compatible species 438 157 0.358 0.383
male sterile individuals 223 217 0.972 0
*** Quality individual seeds = -0.084 * (seed per flower) + 0.408
**** µ=12, β=0.0475, q(0)=0.688
that we used (derived from Chapter 6 and 7). Secondly, we calculated the ESS allocation to flowers and seeds including selective seed abortion with both the linear and the inverse J- shaped relationship. Thirdly we studied the impact that selective seed abortion can have on the ESS allocation to flowers and seeds of male sterile individuals (comparing fitnesses from eq. 3 and 4). We present the results for both species separately below.
Full model: hermaphrodite individuals
If abortion is assumed to be non-selective (Km=Kn), and all selection effects on reduction in seed per flower is due to sex allocation, the ESS number of seeds per flower is 0.704 in E.
vulgare (after Rademaker & de Jong 2000). When abortion was assumed to be selective, the ESS number of seeds per flower is further reduced. With a linear relation between offspring fitness and number of seeds per flower, the ESS number of seeds per flower for E. vulgare is reduced to 0.569. With an inverse J-shaped relation, the ESS number of seeds per flower for E.
vulgare is 0.599 (equation 1 and 2, Table 1). To examine the global stability of the optimum, we plotted the fitness of possible rare mutants with the resident in the ESS (Figure 3A). The ESS is locally and globally stable.
For C. officinale, the ESS number of seeds per flower with only sex allocation is 0.468 (after Rademaker & de Jong 2000). With a linear relation between offspring fitness and number of seeds per flower, the predicted ESS number of seeds per flower is 0.385. With an inverse J-shaped relation, the ESS number of seeds per flower for C. officinale is 0.358 (equa- tion 1 and 2, Table 2). To examine the global stability of the optimum, we plotted the fitness
Fitness
0 3000 6000 9000
0 1 2 3
total fitness female outcross self (x2) male outcross ESS
4
Fitness
0 400 800 1200
0 1 2 3
Number of seeds per flower
total fitness female outcross self (x2) male outcross ESS
4
A
B
Figure 3:
Partial and total fitness functions in relation to the seed number per flower of the mutant in the ESS of the resident.
A: E. vulgare
B: C. officinale
of the possible mutants in the ESS (Figure 3B). The ESS appears to be locally and globally stable.
Only selective seed abortion: male sterile individuals
With an inverse J-shaped relation, the predicted ESS number of seeds per flower for male sterile individuals of E. vulgare is 1.800. This shows that male sterile individuals are expected to produce less than half of the maximum of four seeds per flower, solely due to the effect of selective seed abortion.
With an inverse J-shaped relation between offspring quality and seed number per flower, the predicted ESS number of seeds per flower for C. officinale is 0.972.
Robustness of the simulation model
Trade off between flower and seed production
Without selective seed abortion there is a linear relation between the production cost of one seed and ESS number of seed per flower (Rademaker and de Jong 2000). This means that doubling the value of c, results in half the number of seeds per flower. When selective seed abortion is included, this relation is not linear anymore, but the deviation is small (Table 3).
When lowering the cost of a seed, the ESS number of seeds increases and consequently the number of seeds per flower increases as well.
Offspring quality as a function of abortion level
Offspring quality as a function of number of seeds per flower can be described linearly or with an inverse J-shaped relation. The predicted ESS number of seeds for both species does not deviate much between these two approaches. A linear function deviated less than 8% in predicted ESS number of seeds per flower from the simulation with deleterious alleles in most cases. Only for male sterile individuals of E. vulgare the linear relation predicted a 20% lower number of seeds per flower compared to the simulation of deleterious alleles.
We present the sensitivity of the model for the assumed values of µ (average number of deleterious alleles present) and β (effect of the deleterious alleles) in Table 3. For a lower value of the number of deleterious alleles in E. vulgare we took the integer 4, because frac- tions were not possible in our model with a Poisson distribution (a class of offspring with 3.5 deleterious alleles can not be calculated while this would be necessary in our calculation - see Appendix B). When decreasing the average number and effect of the deleterious alleles, the ESS number of seeds per flower hardly increased. In contrast, with increasing the average number or the effect of the deleterious alleles, the ESS number of seeds decreased considerably.
Discussion
Plants without male function
In male sterile individuals the predicted ESS number of seeds per flower is below half of the
maximum of four. This decrease in number of seeds per flower cannot be caused by sex allo-
cation, because the flowers lack a male function. Our results therefore show a strong potential
selective effect of embryo abortion on the ESS allocation to seeds and flowers in plants. This predicted decrease in number of seeds per flower is supported by the data from natural condi- tions of E. vulgare, where male sterile individuals indeed produced less seeds per flower than half of the maximum of four. In the gynodioecious thyme (Thymus vulgaris), female plants also produce fewer seeds per ovule than 0.5 in general (Couvet et al. 1985). We expect that in female plants of dioecious species, where a male function of the flowers is also absent, selec- tive embryo abortion reduces the ESS number of seeds per flower too. The effect of selective seed abortion on optimal number of seeds per flower should perhaps be estimated by using dioecious species.
Sex allocation and selective seed abortion
By comparing the effects in hermaphrodites and male sterile individuals, we could separate the effects of sex allocation and selective seed abortion. If both male fitness and selective seed abortion are absent, plants should produce the maximum number of seeds per flower, which in the case of our study species equals four. Both the effects of allocation to male function and selective seed abortion on their own make the ESS number of seeds per flower drop far below its maximum of four. The combined effects of allocation to male function and selective seed abortion give rise to only a relatively small further decrease. Sex allocation seems to provide
Table 3:
Sensitivity analysis for trade-off between flower and seed production (c), average number of deleterious alleles (µ) and effect of the deleterious alleles (β). ESS number of seeds per flower are calculated for hermaphroditic individuals with the inverse J-shaped relation between offspring quality and number of seeds per flower.
Echium vulgare Cynoglossum officinale parameter ESS number of seeds
per flower
parameter ESS number of seeds per flower
c (1.78) 0.599 c (3.6) 0.358
c*0.5 1.120 c*0.5 0.623
c*2 0.321 c*2 0.184
µ (7) 0.599 µ (12) 0.358
µ=4
10.623 µ*0.5 0.403
µ*2 0.438 µ*2 0.074
β (0.05) 0.599 β (0.0475) 0.358
β*0.5 0.660 β*0.5 0.397
β*2 0.348 β*2 0.183
1 Because a class of offspring with 3.5 deleterious alleles does not exist (see Appendix B for
program methods), we present here the results with the integer 4.
the strongest selective pressure because the reduction in ESS number of seeds per flower is the strongest for this factor.
Relation with another model of selective seed abortion
A paper from Burd (1998) models selective seed abortion in relation to the overproduction of ovules. This model analyses the increase in offspring quality when excess flowers are produ- ced. In this way, the excess flowers virtually add offspring of an average quality, while the equivalent amount of offspring of the lowest quality is aborted. With different possible distri- butions of offspring quality, a small surplus of flowers is a strategy that increases the plant's fitness. In this way, there is a saturating relation between excess flower number and mean off- spring quality. Because of the diminishing returns of investment in more extra flowers, plants should turn to produce more seeds rather than increase the surplus of flowers extremely. How- ever, with including a trade-off between flower and seed production, the outcome of his model would have been different. With e.g. our trade-off between flower and seed production in C.
officinale (a seed costs 3.6 flowers), decreasing the seed production with one seed, increases the (empty) flower production with 3.6 and thus counts for 3.6 * 4 ovules in the produced empty flowers = 14.4 extra seed abortions. Moreover, any difference in allocation to flowers and seeds has then an augmented effect on the number of seeds per flower (because both flowers and seeds appear in the ratio of seeds per flower) and the predicted surplus of flowers that is advantageous for the plant would have been larger. In addition, the male function of flowers is not taken into account, his model holds true only for purely female flowers of a plant, i.e. the female plants of a dioecious species or male sterile individuals.
Deviations of the model
Under natural conditions, seed set per flower is higher than the model predicts for E. vulgare and C. officinale. The model of Rademaker and de Jong (2000) predicted already a lower seed number per flower than is found for these species under natural conditions in the field. The predicted number of seeds per flower when both the effects of sex allocation and selective seed abortion are included, is even lower (Klinkhamer et al. 1994, de Jong & Klinkhamer 1989). Our model might lack factors that would increase the number of seeds per flower. We assumed e.g. that the costs of seed abortion are neglectable. Considerable costs of aborting immature seeds would decrease the advance of aborting seeds and would increase the predic- ted ESS number of seeds per flower. Furthermore, bet hedging is not included in our model.
This theory predicts that an overproduction of ovules serves as a guarantee in seed production after unexpected losses, or the possibility to increase seed production under unpredictable ad- vantageous conditions (e.g. Stephenson 1980, 1981). Including an effect of bet hedging in our model would decrease the predicted number of seeds per flower even more. Besides the exclu- sion of this factor, the estimated parameters of the model could have been subject to errors.
The discussion of possible deviations in most parameters that were included in the model of
Rademaker and de Jong has already been held elsewhere (Rademaker & de Jong 2000). The
trade-off between flower and seed production e.g. has been estimated in one single experiment
for each species by comparing flower production after hand-pollinating or after excluding all
pollination in plants and comparing the flower and seed production (de Jong & Klinkhamer
1989, Rademaker & de Jong 2000). Any error in this estimation might have considerable
effects in predicted number of seeds per flower, as indicated by the sensitivity of the model for this factor. Also, in this situation most flowers remain unfertilized so that compared to natural conditions there is less abortion and there are lower costs for supporting the developing embryos that are aborted later. However, even a two-fold difference in the trade-off between flower and seed production in the full model keeps the predicted ESS number of seeds per flower still below the number found under natural conditions for C. officinale. Another assumption was that male sterile flowers are equally costly to hermaphroditic flowers. It might be that male sterile flowers are less costly relative to seeds (i.e. a higher value for c in equation 3, 4 and 5), and consequently we estimated the seed to flower ratio too low. The shape of the curve that related offspring quality to number of seeds per flower did not have major effects on the ESS number of seeds per flower. One might argue that both lines fit through the same experimental points that are close to the ESS and that that may cause the small deviations in the results. However, other possible relations between offspring quality (e.g. logarithmic) caused large differences because of the differences in the tail of the curve. The seed produc- tion per flower hardly increased when the average number and the effect of the deleterious alleles decrease in our model. With decreasing the number or the effect of deleterious alleles (e.g. dividing µ or β by two; Table 3), the model would predict only slightly higher values for the ESS number of seeds per flower. However, the seed production per flower decreases strongly when the number or effect of the deleterious alleles would be higher than we included in our model. With increasing the average number and the effect of the deleterious alleles (e.g.
doubling µ or β; Table 3), the model predicts a considerably lower ESS number of seeds per flower. If our actual estimates of the number and effect of the deleterious alleles are deviating from reality, the calculated ESS that we presented here is rather too low than too high.
Acknowledgements
We thank Marielle Rademaker for providing her model program. Hans Metz gave conceptual programming advice. Inti Suárez helped considerably with the extension of the program.
Martin Brittijn kindly drew the figures. Eddy van der Meijden and Peter van Tienderen gave
useful comments on earlier versions of the manuscript.
Appendix A
Derivation of equation 5 - ESS number of seeds per flower for male sterile individuals if the relation between offspring quality and number of seeds per flower is linear. The ESS can be found by differentiating the fitness equation (4) to m and equal it to zero.
The fitness equation W m of a male sterile individual is:
n m
m
K
K c
m
W = ( T − ) (4).
with T as the total amount of resources for reproduction, m as the number of flowers, c as the trade-off between seeds and flower production, K m as the offspring quality with m number of flowers, K n as the offspring quality with n number of flowers. We differentiate to m and equal it to zero:
( ) 0
'
' = − 1 + − =
c m T K K K K W c
n m n m m
And this leads to: − K
m+ K
m'( T − m ) = 0 (A1)
Under the assumption that offspring quality is related linearly to number of seeds per flower:
b flower per
seeds a
K
m= * ( ) +
with a as the slope of the line and b as the intercept with the Y-axis. (A2)
Seeds per flower = ( )
mc m
T − (A3)
Substitute A3 in A2: K m = ( ) b
c a cm b aT cm
m T
a − + = − + (A4)
K' m =
2cm
− aT (A5)
Substitute A4 and A5 in A1: ç ö + −
2( − ) = 0 è
æ − +
− T m
cm b aT c a cm
aT (A6)
Multiplying with m 2 and rewriting leads to: m
2= ( a aT − bc
2) (A7)
m*(=ESS) = T a ( a − bc ) (A8)
ESS number of seeds per flower = ( )
c c m
T c m
m
T 1
*
*
* = −
− (A9)
Substituting A8 in A9:
ESS number of seeds per flower =
( a bc ) c
a c
1 1 −
− (5)
Appendix B
Derivation of the inverse J-shaped function.
The calculation is based on four assumptions:
i) the quality of an individual is determined by the number of deleterious alleles present. Those deleterious alleles are assumed to have a negative, additive effect on quality;
ii) the strength of the effect on offspring quality is equal for all deleterious alleles iii) the offspring shows a Poisson distribution of deleterious alleles; and
vi) the seeds of lowest quality are aborted without error. We calculated the relative mean quality of the non-aborted offspring as a function of number of seeds per flower of the maternal plant.
The chance of carrying 0 or n deleterious alleles according to a Poisson distribution (assumption iii) is:
!
0
e n p
e p
n n
µ
µ
µ
−
−
=
=
(B1)
in which µ is the average number of deleterious alleles present in the population.
First we have to calculate which individuals are included in the offspring, given a fraction of surviving embryos (a). To do this, we calculated the cumulative fraction of individuals present in the classes of the poisson distribution with an increasing number of deleterious alleles as P k = p (k-1) + p k . We wanted to find a point where P k = a, when the cumulative distribution of individuals P k (as a fraction of one) is equal to the fraction surviving offspring a (assumption iv). If P k > a, then all individuals of P (k-1) were included, and also a fraction of p k to meet the condition P k = a. We redefined the fraction of p k that is included as the new p k .
The quality of offspring carrying n deleterious alleles (assumption i and ii) is defined as:
n q
q
n=
0− β * (B2)
with β as the strength of the effect of a deleterious allele and q(0) as the quality of an individual without deleterious alleles. 0<q n <1, so that in the case of negative quality, the quality is set at zero (no survival at all for this population class of deleterious alleles). Without selective abortion, the average quality of the surviving offspring population Q=q 0 -βµ.
The average quality of the surviving offspring population with selective abortion is then defined as:
( )
a
n q
p
a q p Q
k
n n k
n
n n
=
=
−
=
=
10 1
