Our study was carried out in 2007 in Brittany (western France), where sea beets colonize coastal areas (Fievet et al., 2007). Exhaustive sampling of all flowering plants was carried out at two study sites within two isolated coves separated by 30 km. The first site, MOR, extends over c. 300 m (N 48°34’168’’; E –2°34’831’’) and the second one, PAL, over c. 600 m (N 48°40’497’’; E –2°52’911’’). At both sites, individuals appeared to be strongly clustered in space (Fig. 1). In MOR, there were four geographical patches (MA, MB, MC and MD), totalling 1094 sampled individuals, plus four isolated plants growing outside these patches (see Fig. 1 and Table 1). In PAL, 592 individuals were clustered within five patches (PA, PB, PC, PD and PE), and 23 plants were geographically isolated (Fig. 1 and Table 1). No B. vulgaris individuals were found for at least 1000 m on either side of these sites.
Figure 1. Spatial distribution of Beta vulgaris ssp. maritima individuals (grey dots for potential fathers and red dots for the mother plants used for progeny analyses) and delimitation of geographical patches (grey circles) for the two study sites, MOR and PAL. The blue dashed lines show the position of the shore line.
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Table 1. Major characteristics of the different geographical patches of Beta vulgaris ssp. maritima within the two study sites MOR (a) and PAL (b)
| ||MA||MB||MC||MD||Overall|| |
|NTOTAL||69||635||281||109||1094 (4)|| |
|NF/NHCMS/NHNonCMS||0/3/62 (4)||0/0/633 (2)||17/60/195 (9)||0/0/106 (3)||17/63/996 (18)|| |
|Sex ratio||0||0||0.063||0||0.016|| |
|NMOTHER PLANTS||6||6||32||6||50|| |
|Mean PROGENY SIZE (± SD)||23.83 (± 1.33)||22.17 (± 6.94)||24.16 (± 2.85)||25 (± 0)||23.98 (± 2.29)|| |
|NF/NHCMS/NHNonCMS||0/0/142 (4)||6/6/2 (4)||61/45/185 (9)||0/0/32 (1)||3/18/71 (3)||70/69/432 (21)|
|Mean PROGENY SIZE (±SD)||24.75 (± 0.87)||–||23.80 (± 4.13)||24.63 (± 1.06)||24.53 (± 1.81)||24.38 (± 2.51)|
At the two study sites, all flowering plants were genotyped for RFLP markers diagnostic for CMS cytoplasms (see De Cauwer et al., 2012) and gynodioecy appeared to be based on a simple cytonuclear determination, with fertile cytoplasms and (except for two plants) only one CMS type (CMS E, which is the most common source of male sterility in this region; see Dufay et al., 2009). Finally, strong spatial genetic structure was detected within both sites (see De Cauwer et al., 2012).
Modelling the dispersal kernel and heterogeneity in male fecundity
The aim of the study was to disentangle the effects of various sources of variation on male fecundity (i.e. ability to produce offspring). A spatially explicit mating model was used to investigate how male reproductive success depended on (1) the physical distance between individuals, (2) the geographical patch, (3) flowering synchrony, (4) the investment in reproduction, (5) pollen production, (6) cytotype (CMS vs non-CMS) and (7) the probability of carrying restorer alleles. The method used here is related to fractional attribution of paternity, where each offspring is partially assigned to each of the possible candidate fathers on the basis of their relative likelihoods of parentage (see Oddou-Muratorio et al., 2005; Jones et al., 2010). Analyses were carried out independently for each study site (MOR and PAL).
Following Burczyk et al. (2002) and Oddou-Muratorio et al. (2005), we assumed that each offspring could be the result of either self-pollination, pollen coming from outside the study sites or pollen coming from one of the sampled individuals. A seed o sampled on a mother plant jo with genotype gjo was then expected to have the genotype go with probability:
- (Eqn 1)
(s, the selfing rate; m, the rate of incoming pollen flow; (1 –m–s), the probability that the pollen donor is inside the study site; πjk, the composition of the pollen pools (described below; Eqn 3).) The allele frequencies (AF) in the incoming pollen pool were measured independently in MOR and PAL from the estimated contribution of individuals located outside the study sites (i.e. through offspring that have no compatible father inside the study sites). The transition probabilities T (.|.,.) are the Mendelian likelihoods of observing a genotype for a seedling, conditional on the genotype of the parents (Meagher, 1986). The information carried by the genotypes of all seeds sampled in one site was combined into a log-likelihood function, assuming that all fertilization events were independent,
- (Eqn 2)
The proportion of pollen from each father k in the pollen pool of each mother j originating from all known fathers, πjk, was assumed to follow the mass-action law,
- (Eqn 3)
Disp jk is taking into account the effect of physical distance between individuals. This effect was modelled using a dispersal kernel, describing the probability density that a pollen grain lands at a given position away from the source. As suggested in previous studies (Oddou-Muratorio et al., 2005; Fénart et al., 2007), we investigated several shapes for the dispersal kernel, including the exponential-power function and the logistic function (reviewed in Austerlitz et al., 2004).
The exponential-power kernel used to model the effect of distance on mating probability was given by:
- (Eqn 4)
(djk, the distance between mother j and father k; Γ, the Gamma function; a, a scale parameter for distance; b, a shape parameter (Clark, 1998).) The average pollen dispersal distance (δ) is then given by δ = [aΓ(3/b)/Γ(2/b)].
The logistic function was given by:
- (Eqn 5)
Pop k in (0,+∞) takes into account the differences in individual male fecundities resulting from potential microenvironmental variations among the different geographical patches within each study site (see Fig. 1).
Pheno jk takes into account the effect of flowering synchrony and is given by a Gaussian-like curve,
- (Eqn 6)
(Δjk, the difference in phenology between mother j and father k; Δopt, the optimal difference in phenology; σΔ², the variance of the Gaussian curve.) A positive Δopt means that fathers flowering earlier than a given mother have a higher reproductive success than males flowering synchronously.
Fec jk takes into account the effects of pollen production (PPk) and investment in reproduction (RIk) through an exponential relation,
- (Eqn 7)
(bPP and bRI, selection gradients describing the effect of pollen production and investment in reproduction on male fecundity.) For the individuals that were not scored for pollen production or investment in reproduction, the corresponding exponential term was replaced by a parameter NSPP (different in the different patches) or NSRI (independent of the patch).
Finally, the effect of the cytotype (CMS vs non-CMS) and the effect of the probability of carrying restorer alleles were also taken into account. To assess the effect of the cytotype, we set Sexk = F = 0 for females, Sexk = HCMS for restored CMS hermaphrodites, Sexk = HNonCMS = 1 for non-CMS hermaphrodites and Sexk = NT for non-typed individuals (i.e. individuals for which the sexual phenotype was not scored).
The effect of the probability of carrying restorer alleles was considered through the local CMS restoration frequency around individual k (PFreq,k), using the following equation:
- (Eqn 8)
where bFreq,k is bHCMS if individual k is a restored CMS hermaphrodite, bHNonCMS for a non-CMS hermaphrodite, and 0 for females and non-typed individuals.
By maximizing the log-likelihood logL (Eqn 2), we jointly estimated the following parameters: levels of selfing (s) and of incoming pollen flow (m), the dispersal parameters (a and b), the effect of geographical patches (Pop1, Pop2…Popmax), the effect of flowering phenology (Δopt and σΔ), the effect of investment in reproduction (bRI and NSRI), the effect of pollen production (bPP, and NSPP1, …NSPPmax), the effect of the cytotype (HCMS and NT) and the effect of the probability of carrying restorer alleles (bHCMS and bHNonCMS). The log-likelihood function was maximized numerically using the quasi-Newton algorithm in Mathematica 7.1 (Wolfram Research, Champaign, Illinois, United States). Several contrasted initial values for the maximization of the parameters were used to confirm that we reached a global maximum (when the initial values actually led to the different maxima, we kept the one that reached the higher log-likelihood). Parameters describing the relative male reproductive success of non-scored individuals (NSRI and NSPP1, …NSPPmax) and of non-genotyped individuals (NT) were incorporated in the models to improve the fit to data, but these results will not be described further in the paper.
The significance of each effect was tested with a Type III likelihood-ratio test (LRT). For each test, the log-likelihood of a model without the tested effect was computed. The deviance (i.e. twice the difference between the log-likelihood obtained for the complete model and the log-likelihood obtained for the model without the tested effect) was then compared to a χ2 distribution, with the number of degrees of freedom equal to the difference in the number of parameters between the two models.
The confidence intervals for the parameters were obtained through a bootstrap procedure using mother plants as sampling units. For each replication, we successively sampled mother plants with replacement and equal probabilities and kept all their seedlings’ genotypes, until we reached the same total number of seedling genotypes in the bootstrapped data set as in the real data set. To reach exactly the same total number of seedlings, we sampled without replacement the correct number of seedlings among those of the last mother drawn. We then estimated the parameters on the bootstrapped data set by maximizing the log-likelihood (Eqn 2). For each study site, we derived 95% symmetric confidence intervals for all parameters from 250 bootstrapped data sets.