C Roux

J R Pannell
email: john.pannell@unil.ch

The opposing effects of genetic drift and Haldane's Sieve on floral morph frequencies in tristylous metapopulations

Keywords: Eichhornia paniculata, dominance drive, frequency-dependent selection, disassortative mating, tristyly

Tristyly is a genetic polymorphism in which three floral morphs are maintained at equal frequencies by negative frequency-dependent selection resulting from disassortative mating. Trimorphism is governed by two epistatically interacting diallelic loci with dominant and recessive allelic expression. Because dominant alleles are at lower frequency than their recessive counterparts at the 1:1:1 equilibrium, they are more prone to loss from tristylous populations by founder events and genetic drift. Accordingly, the short-styled morph tends to be missing in small populations of several tristylous species, notably Eichhornia paniculata, a species characterised by metapopulation dynamics. We used computer simulations to test the hypothesis that dominant alleles are more capable of establishing following migration into established demes in a metapopulation of demes subject to population turnover, due to a process similar to Haldane's Sieve. We found that Haldane's Sieve can reverse the order of morph frequency bias, particularly in metapopulations in which populations grow rapidly to large size after colonization. Our results suggest that the loss of dominant alleles in small tristylous populations of tristylous species may be ameliorated by their advantage as migrants.

Many flowering plants have evolved sexual systems that involve floral polymorphisms, with populations comprised of two or three classes of individual that produce flowers differing in morphology, sex allocation, or compatibility type (Barrett, 2002a). These polymorphic species have long fascinated botanists, and indeed were the subject of one of Charles Darwin's books about plants [START_REF]The Different Forms of Flowers on Plants of the Same Species[END_REF][START_REF] Barrett | Darwin's legacy: the forms, function and sexual diversity of flowers[END_REF]. They continue to attract the attention of researchers interested in understanding how and why polymorphism evolves from an ancestral monomorphic state, how the different floral types contribute to male and/ or female components of fitness, and how and at what proportions they are maintained or lost by various evolutionary processes (e.g., [START_REF] Lewis | Male sterility in natural populations of hermaphrodite plants[END_REF][START_REF] Ross | Evolution of dioecy from gynodioecy[END_REF][START_REF] Lloyd | Breeding systems in Cotula L. (Compositae, Anthemedae). I. The array of monoclinous and diclinous systems[END_REF][START_REF] Charlesworth | A model for the evolution of dioecy and gynodioecy[END_REF][START_REF] Lloyd | Evolution towards dioecy in heterostylous populations[END_REF]Lloyd & Webb, 1992b;Lloyd & Webb, 1992a;[START_REF] Jesson | A theoretical investigation of the evolution and maintenance of mirror-image flowers[END_REF][START_REF] Vamosi | Phylogenetic analysis of the ecological correlates of dioecy in angiosperms[END_REF][START_REF] Wolf | Pollen limitation and the evolution of androdioecy from dioecy[END_REF][START_REF] Pannell | The evolution of gender specialization from dimorphic hermaphroditism: Paths from heterodichogamy to gynodioecy and androdioecy[END_REF][START_REF] Torrices | Where do monomorphic sexual systems fit in the evolution of dioecy? Insights from the largest family of Angiosperms[END_REF][START_REF] Goldberg | Macroevolutionary synthesis of flowering plant sexual systems[END_REF]. Thus, floral polymorphisms provide useful model systems for testing general theories in evolutionary ecology and ecological and population genetics, going back to the early work of Fisher and Haldane on heterostyly (e.g., [START_REF] Fisher | On the selective consequences of East's (1927) theory of heterostylism in Lythrum[END_REF][START_REF] Haldane | Heterostylism in natural populations of the Primrose, Primula acaulis[END_REF].

Floral polymorphism also offer diverse opportunities for investigating the fitness consequences of different allocation strategies [START_REF] Lloyd | Evolutionary stable sex ratios and sex allocations[END_REF][START_REF] Ashman | Determinants of sex allocation in a gynodioecious wild strawberry: implications for the evolution of dioecy and sexual dimorphism[END_REF][START_REF] Sakai | Sexual dimorphism and the genetic potential for evolution of sex allocation in the gynodioecious plant, Schiedea salicaria[END_REF], interaction between flowers and pollinators and the corresponding implications for pollen transfer and mating [START_REF] Harder | Mating cost of large floral displays in hermaphrodite plants[END_REF][START_REF] Barrett | The evolution of polymorphic sexual systems in daffodils (Narcissus)[END_REF] and for addressing questions concerning the evolutionary forces that regulate the maintenance of genetic variation in natural populations, particularly negative frequency-dependent selection, which maintains variation, and drift, which allows it to be lost. Tristyly is particularly interesting in all these contexts, and has been the focus of a great deal of Spencer Barrett's research on the evolution and implications of plant reproductive strategies over many decades.

Tristylous populations are typically comprised of three floral morphs that differ reciprocally in stigma and anther height [START_REF]The Different Forms of Flowers on Plants of the Same Species[END_REF][START_REF] Barrett | The evolutionary biology of tristyly[END_REF]. In any given morph, stigmas and anthers occupy three positions within flowers, with the stigma presented at one position and the anthers at the other two. The three floral morphs are referred to as the long-, mid-and short-styled morphs (hereafter L-, M-and S-morph). The reciprocal correspondence of anther and stigma positions in the three morphs (a condition referred to as reciprocal herkogamy, Lloyd & Webb 1992), promotes disassortative pollen transfer among the floral morphs; more pollen is transferred by pollinators between anthers and stigmas at the same position than between those at different positions as a result of segregated pollen deposition on the bodies of pollinators (Barrett & Glover 1985;Costa et al., 2017) This pattern of pollen transfer reduces self-pollination, particularly via geitonogamy [START_REF] Harder | Mating cost of large floral displays in hermaphrodite plants[END_REF], and promotes cross pollination between morphs [START_REF]The Different Forms of Flowers on Plants of the Same Species[END_REF]Lloyd & Webb, 1992b). In many tristylous species, selfing and intra-morph mating is prevented by a trimorphic incompatibility system (Barrett & Cruzan, 1994), in which incompatible ('illegitimate') pollen transferred between flowers of the same morph, or between anthers and styles occupying different positions within the floral tube, is unable to fertilize ovules in the flower.

Disassortative mating in tristylous populations [START_REF] Barrett | Disassortative mating in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF] is the basis of strong negative frequency-dependent selection on the three morphs, such that individuals of a particular morph that happens to be rare in a population will have more prospective mates than those that are common. Tristyly is thus a 'protected' polymorphism, in which morph frequencies should be maintained at roughly equal frequencies (isoplethy) in large populations [START_REF] Heuch | Equilibrium populations of heterostylous plants[END_REF][START_REF] Barrett | Disassortative mating in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]. This is especially the case for species in which the reciprocal placement of floral organs (reciprocal herkogamy) is associated with self-and intra-morph incompatibility [START_REF] Heuch | Genotype frequencies associated with incompatibility systems in tristylous plants[END_REF]. In tristylous species that are selffertile or that possess multi-allelic self-incompatibility not associated with the floral phenotype (reviewed in Barrett & Cruzan 1994), morph frequencies may deviate from isoplethy for a variety of ecological and reproductive reasons that cause deviations from symmetrical disassortative mating. For example, in Narcissus triandrus which possesses a late-acting self-incompatibility system not associated with reciprocal herkogamy that permits intramorph mating [START_REF] Sage | Differential ovule development following self-and cross-pollination: The basis of self-sterility in Narcissus triandrus (Amaryllidaceae)[END_REF], populations in the Iberian Peninsula show a consistent L-morph bias as a result of greater levels of assortative mating in this morph compared with the othe two morphs [START_REF] Barrett | Correlated evolution of floral morphology and mating-type frequencies in a sexually polymorphic plant[END_REF]Hodgins & Barrett, 2008 ). In Eichhornia paniculata, a fully self-and intra-morph compatible tristylous species native to N.E. Brazil and the Caribbean, large populations in which mates are abundant tend to be isoplethic, but in geographic regions of the species' range in which populations are small and prone to large fluctuations in population size, or where they are likely the result of longdistance colonisation in the past, morph frequencies often deviate from isoplethy as a result of genetic drift, even to the extent that one or two morphs are absent locally or regionally [START_REF] Barrett | Ecological genetics of breakdown in tristyly[END_REF][START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF].

A remarkable feature of biases in floral morph ratios in populations of E. paniculata [START_REF] Barrett | Evolution of breeding systems in Eichhornia (Pontederiaceae): a review[END_REF][START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF][START_REF] Husband | Genetic drift and the maintenance of the style length polymorphism in tristylous populations of Eichhornia paniculata (Pontederiaceae)[END_REF], as well other tristylous systems such as Lythrum salicaria [START_REF] Heuch | Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria[END_REF][START_REF] Eckert | Stochastic loss of style morphs from populations of tristylous Lythrum salicaria and Decodon verticillatus (Lythraceae)[END_REF][START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF]Balogh & Barrett, 2016) which experience frequent cycles of colonization and extinction is that they tend to be biased in favour of the L-and M-morphs, with the S-morph often absent from populations. These biases are interesting for a number of reasons. For instance, because the loss of the S-morph from populations alters the balance of pollen transfer among anthers and stigmas, e.g., pollen from short-level anthers no longer has a corresponding target in the population, selection on mating strategies has favoured the evolution of a modified selfing M-morph in which the low-level anthers are placed at the mid-level position [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]Vallejo-Marin & Barrett, 2009). This change in position of sex organs confers a siring advantage to the M-morph over the L-morph, notably via self-fertilization, so many populations become fixed for the modified M-morph.

Interestingly, this transition from tristyly to a state in which populations are monomorphic for a modified M-morph has occurred more than once independently, with different causal mutations selected in different parts of the species' range [START_REF] Husband | Multiple origins of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae) -inferences from style morph and isozyme variation[END_REF]Fenster & Barrett, 1994;[START_REF] Ness | Mating-system variation, demographic history and patterns of nucleotide diversity in the tristylous plant Eichhornia paniculata[END_REF]Arunkumar et al., 2017). The evolutionary breakdown of tristyly initially by genetic drift and the subsequent evolution of a new floral strategy in E. paniculata has been cited as one of the few examples of two of the three phases of Wright's 'shifting balance' theory [START_REF] Coyne | Perspective: A critique of Sewall Wright's shifting balance theory of evolution[END_REF].

The biased deflection of morph frequencies from isoplethy in many tristylous populations also illustrates an interesting way in the interaction of genetic drift with natural selection depends on the underlying genetic architecture of a trait, such that the outcome of the stochastic process of drift becomes non-random and thus, surprisingly, predictable.

Tristyly in E. paniculata is the result of the segregation of alleles at two loci that have epistatic effects on the phenotype [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]Arunkumar et al., 2017 ). Possession of a dominant S allele at the S-locus underlies expression of the S-morph, irrespective of the genotype at the M-locus (i.e., genotypes S-/--). Individuals lacking the dominant S allele (and thus homozygous for the s allele) produce either L-morph flowers if they are also homozygous for the recessive m allele at the M-locus (i.e., genotype ss/mm), or M-morph flowers if they possess the dominant M allele at the M-locus (i.e., genotype ss/M-).

Remarkably, this mode of inheritance of tristyly also occur in other tristylous species in Lythraceae and Oxalidaceae (reviewed in [START_REF] Lewis | The genetics of heterostyly[END_REF], so that it is not only the phenotypes that have evolved convergently in different lineages, but also their underlying genetics.

The genetic architectureof tristyly implies that the frequency of dominant and recessive alleles will differ at the isoplethic equilibrium, with the dominant S and M alleles maintained at substantially higher frequencies than their recessive counterparts s and m, respectively. Because low-frequency alleles are more susceptible to being lost as a result of genetic drift in small populations than higher-frequency alleles, we expect the S and M alleles to be missing from populations more frequently than the s and m alleles, as observed in several tristylous species [START_REF] Heuch | Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria[END_REF][START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF][START_REF] Barrett | Tristyly in Eichhornia crassipes (Mart.) Solms (Water hyacinth)[END_REF]Balogh & Barrett, 2016). Indeed, [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF] used computer simulations to model the combined effect of drift and frequency-dependent selection, and were able to attribute the biased loss of the S-morph in Eichhornia paniculata to the greater role played by genetic drift in geographic regions in which populations are smaller and more subject to population turnover at the metapopulation level. Similar reasoning was invoked to explain deviations in morph frequencies in E. crassipes [START_REF] Barrett | Tristyly in Eichhornia crassipes (Mart.) Solms (Water hyacinth)[END_REF] and L. salicaria [START_REF] Heuch | Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria[END_REF][START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF]Balogh& Barrett, 2016).

Simulation modelling of the biased loss of the S-morph in tristylous populations while motivated to some extent by the metapopulation structure of species and the importance of population turnover due to the balance between local population extinction and colonisation [START_REF] Husband | Spatial and temporal variation in population size of Eichhornia paniculata in ephemeral habitats: implications for metapopulation dynamics[END_REF], has focused largely on the effects of drift in single populations [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF], although see [START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF]. The concordance between the results of these simulations and observed frequencies of tristylous morphs in the field is satisfying, but the question remains as to whether morph frequencies at the regional, or metapopulation, level might be subject to metapopulation-level effects that go beyond, or differ from, the effects of drift locally. The question is potentially important, because metapopulation processes involve not only fluctuations in population size, which augment drift locally, but also the movement of individuals among habitat patches, both through colonization as well as through migration among extant populations or demes [START_REF] Slatkin | Gene flow and genetic drift in a species subject to frequent local extinction[END_REF]. Although colonisation tends to amplify the effect of drift within the resulting population, with effects likely similar to those explored by [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF] for small populations, migration among demes might alter the fate of dominant versus recessive alleles in ways that differ from the effect of drift in established populations.

The fate of dominant versus recessive alleles following their immigration into a population in which the immigrant allele is not yet present should be influenced by a phenomenon similar to 'Haldane's Sieve' [START_REF] Haldane | A mathematical theory of natural and artificial selection, part I[END_REF][START_REF] Orr | Haldane's sieve and adaptation from the standing genetic variation[END_REF], which accounts for differences in the fate of positively selected new mutations that differ in their dominance [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF]. Theory predicts that the probability of ultimate fixation of a new advantageous mutation is approximately equal to twice the product of its selective advantage (or selection coefficient, s) and its dominance coefficient, h, a prediction that apples for 1/N << s << 1, i.e., for weak selection [START_REF] Haldane | A mathematical theory of natural and artificial selection, part I[END_REF][START_REF] Crow | An Introduction to Population Genetics Theory[END_REF]. This means that an advantageous dominant allele that is at least partially dominant will fix with a greater probability than a purely recessive allele, because dominant or partly dominant alleles are immediately expressed and positively selected, whereas the fate of a recessive mutation while it is still rare in a population (and thus carried only by heterozygotes and thus not expressed) is determined almost entirely by genetic drift and is very likely to be lost.

Although this prediction applies to the fixation probability of new mutations subject to weak directional positive selection [START_REF] Haldane | A mathematical theory of natural and artificial selection, part I[END_REF][START_REF] Whitlock | Fixation probability and time in subdivided populations[END_REF], the principle should also apply to alleles that migrate into a population and are then subject to negative frequencydependent (or other modes of balancing selection) that ultimately maintain them at an intermediate frequency [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF]. Thus, dominant alleles under frequencydependent selection should be more likely to spread towards an expected equilibrium frequency than their recessive counterparts. In tristylous metapopulations, specifically, we predict that the dominant S allele would have an advantage as an immigrant allele into populations that lack them than the recessive s allele (with a similar prediction made for the M and m alleles). If so, then the phenomenon of Haldane's Sieve should contribute to a bias in morph frequencies in precisely the opposite direction to that predicted for small single populations subject to the local effects of drift [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF].

The idea that Haldane's Sieve might apply to migrant alleles has both theoretical and empirical support. Referring to this effect as 'dominance drive', [START_REF] Mallet | Hybrid zones of Heliconius butterflies in Panama and the stability and movement of warning color clines[END_REF] modelled the frequencies of dominant versus recessive alleles underlying wing-colour variation in a hybrid zone between butterfly mimics subject to frequency-dependent selection. He found that '[e]ven if the two phenotypes are equally fit 'dominance drive' tends to increase the area in which the dominant allele is present' [START_REF] Mallet | Hybrid zones of Heliconius butterflies in Panama and the stability and movement of warning color clines[END_REF]. [START_REF] Schierup | Evolutionary dynamics of sporophytic self-incompatibility alleles in plants[END_REF][START_REF] Schierup | The effect of subdivision on variation at multi-allelic loci under balancing selection[END_REF] modelled the frequency of sporophytic self-incompatibility alleles with different levels of dominance and showed that population fragmentation and migration biased allele frequencies in ways consistent with expectations for Haldane's Sieve or dominance drive. [START_REF] Pannell | The maintenance of gynodioecy and androdioecy in a metapopulation[END_REF] considered the maintenance of sexual systems in a metapopulation in which hermaphrodites co-occurred with individuals expressing either male-or female-sterility mutations and found that the equilibrium frequency of these mutations and the resulting phenotypes depended on whether they were recessive or dominant, with dominant sterility mutations more likely to be maintained globally when rare, implying an effect of Haldane's Sieve. Finally, [START_REF] Craze | The fate of balanced, phenotypic polymorphisms in fragmented metapopulations[END_REF] presented a computational model of these effects for polymorphisms maintained by frequency-dependent selection in metapopulations generally, though with a focus on the effects of local drift, as in [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]. [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF] raised the possibility that the effect of Haldane's Sieve on plant reproductive polymorphisms in a metapopulation might oppose the effect of local drift. They specifically speculated that Haldane's Sieve should favour dominant alleles in a metapopulation as long as local populations grow rapidly to a large size following colonization, but not when population sizes remain small. They proposed two reasons why this may occur; first, homozygosity should be increased by inbreeding in small populations, so that advantageous recessive alleles would be immediately selected, reducing the advantage that dominant alleles might otherwise have [START_REF] Whitlock | Fixation probability and time in subdivided populations[END_REF]; and second, dominant alleles will be lost more often from established populations, as explained above, because their local equilibrium frequencies should be lower [START_REF] Bateman | Self-incompatibility systems in angiosperms[END_REF][START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]. As a result, the effect of Haldane's Sieve might be relaxed or even reversed in metapopulations with small subpopulations.

Here, we investigate the opposing forces of genetic drift in established populations versus Haldane's Sieve during migration in metapopulations of tristylous plants, drawing inspiration from studies of the effects of drift on tristylous morph frequencies [START_REF] Heuch | Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria[END_REF][START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF][START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF] and broadening its focus from the fate of alleles in single finite populations to metapopulations governed by population turnover and inter-deme migration. Our model is especially pertinent to the case of Eichhornia paniculata, in that we assume the same genetics (see Methods) as well as self-and intramorph compatibility such deviations disassortative mating can occur. We first apply our specific model to the case of individual populations, establishing a basis for comparison with earlier work and confirming that dominant alleles are lost more frequently from small populations than recessive alleles. We then test the prediction that dominant migrant alleles at the S-and M-loci should be more frequently established in individual demes than recessive alleles as a result of Haldane's Sieve than their recessive counterparts. Next, we compare the establishment success of dominant versus recessive alleles as migrants into an entire metapopulation, as might occur through dispersal linking geographical regions of a species in which population maintain tristyly with those in which populations have lost the polymorphism. Finally, we explore the combined effects on morph frequencies of drift, frequency-dependent selection and Haldane's Sieve in a dynamic metapopulation with population turnover and inter-deme migration. Here, we specifically test the prediction that the effect of Haldane's Sieve might be relaxed or even reversed in metapopulations with small subpopulations compared with those whose demes are larger or grow very rapidly [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF]. We conclude by discussing the general implication that natural selection and genetic drift can act jointly to change mating type frequencies of plant reproductive polymorphisms in ways that more traditional models that do not take into account the underlying inheritance of the polymorphisms might predict.

Methods

Overview of approach taken

We assumed a metapopulation with 100 demes, linked by gene flow via seed dispersal (and not pollen dispersal), and subject to population turnover as a result of stochastic local extinctions and recolonization. Whereas our ultimate goal was to assess the evolution of style morphs in a dynamic metapopulation with many demes under the influence of both drift within demes and the effects of Haldane's Sieve during the migration and establishment phases, we also focused attention on the effects of drift and migration on the composition of single demes in which mating followed the same rules as for simulations of the metapopulation as a whole. Here, we considered both the loss of morphs from an ancestral tristylous population at both the deme and metapopulation levels, to replicate previous work [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF], or the invasion of dominant or recessive alleles into populations fixed for the alternative allele, to examine the effects of Haldane's Sieve.

Genetics and within-deme mating

We assumed a tristylous metapopulation composed of habitat patches populated by individuals belonging to one, two or three of the tristylous morphs, with style morph determined by the expression of genes at two loci according to the genetics established for Eichhornia paniculate [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF], except that the two loci are completely unlinked in our model. With this genetic system of morph determination, the following phenotype-genotype map applies, where '-' can be either allele: S-morph: genotypes S-, --; L-morph: ss, mm; M-morph: ss, M-.

We assumed mating in each deme was disassortative, with the probability of intra-morph mating in general being lower than that expected under random mating.

We generated progeny for the next generation in two ways. In most simulations, we assumed stochastic reproduction, with parents chosen at random from within a given deme, constrained by rules for disassortative mating, governed by parameter P. Here, we used the following protocol (see the Supplementary Information for further details):

(1)We sampled a potential female and male parent at random from the population (with the possibility of choosing the same individual as dam and sire).

(2)If the chosen dam and sire pair were both of the same morph (or were the same individual) in a polymorphic deme, we accepted the pair with a probability P; if the pair was rejected, we resampled a new pair by returning to (1). In monomorphic demes, all pairs were accepted, i.e., we did not allow demes to become extinct because of a local lack of mates. This scheme rules out complete disassortative mating, which is accommodated in simulations in which P > 0 (see below).

(3)We created a progeny individual by sampling an ovule from the dam and a pollen grain from the sire with haplotypes determined by a fair meiosis at each locus and a 50% probability of recombination between loci. Thus, steps (1) and (2) were repeated until a suitable pair was chosen. Steps (1) to (3) were repeated until the deme was populated to carrying capacity N. A scenario with P = 0, which we did not simulate, would corresponded to complete disassortative mating, as one would expect for a population in which intra-morph mating is not possible because of strong trimorphic incompatibility. For 0 < P < 1, disassortative mating was intermediate, with morphs expected to be maintained by (weaker) frequency-dependent mating. P = 1 (simulations not shown) corresponds to a population in which mating is entirely random, and in which allele frequencies simply drift and fix with a probability given by their starting frequency.

Metapopulation dynamics

For multi-deme simulations, we assumed a metapopulation of n demes, each occupied by N individuals at the local carrying capacity. Each generation, each deme could become extinct with a probability E. Extinct demes were immediately recolonised by k individuals chosen randomly from elsewhere in the metapopulation. At this point, all other demes received I immigrants similarly sampled randomly from the metapopulation at large, such that the I immigrants replaced the same number of local individuals in the deme, also chosen randomly. We assumed that I was Poisson-distributed, with parameter M = Nm, i.e., m here is the migration rate, or the probability that a given individual was an immigrant the previous generation. Following extinction, colonisation and/or migration, all demes underwent a round of reproduction, with the same mating protocol applying to all demes. Reproduction maintained extant demes at their carrying capacity, whereas it returned recolonized demes to their local carrying capacity in a single generation.

Recorded parameters

In each generation we record the number of trimorphic demes, the number of demes that losing only the S-, M-and L-morph, and the number of demes that had fixed the S-, M-and L-morph . The global frequencies of the different morphs were also recorded each generation. The model was implemented in C, using the GNU Scientific Library for all functions related to random sampling, and is freely available at https://github.com/ popgenomics/triStyle.

Results

Morph loss from individual demes

Our simulations of single finite populations confirm previous results that genetic drift leads to the destabilization of tristyly, and that this loss can be rapid for small N.

Importantly, our results confirm that the dominant S allele is more rapidly lost than the recessive s allele, such that the S-morph is most frequently lost first, followed by the frequent fixation of the L-morph (Fig. 1). As expected, the details of these trajectories are sensitive to the degree of disassortative mating. The tristylous polymorphism was more strongly maintained in small populations when strong disassortative mating (P = 0.1) largely prevents within-morph mating and selfing, with very slow loss of tristyly in populations of N = 100 (Fig. 1e), and the L-morph or, less frequently, the M-morph fixed only in very small populations (N = 10; Fig. 1a). While the equilibrium was reached rapidly for N = 10 (0.3% short fixation, 10.7% mid fixation, 89% long fixation), the simulations for N = 40 and N = 100 tended towards the same equilibrium, but over periods too long to be explored exhaustively by our forward individual-based approach.

However, the dynamics of frequency change for the three morphs is expected to occur over ecological time scales (over a duration well below 10N generations) than over longer evolutionary periods (i.e., > 1,000N generations), so the long equilibration times in these simulations are probably not biologically insightful.

Invasion of a monomorphic population by a new morph

We compared the ability of a dominant allele (S-allele, determining the S-morph, and M-allele, determining the M-morph) versus a recessive allele (alleles s and m, respectively) to invade populations monomorphic for the S-morph (genotypes SSMM or SSmm), the M-morph (genotype ssMM) and the L-morph (genotype ssmm). Here, we permitted mutations to arise at a heterozygous state, and to have an immediate effect on morphology depending on their level of dominance. New S alleles occurring in a ssmm or ssMM genomic background should thus immediately generate dimorphism in the population, with the S-morph enjoying a strong early benefit of frequency-dependent selection. Accordingly, we found that the S-morph was maintained 6N generations after the introduction of the S allele in about 95% of the simulations for small population of N = 10 (Table 1), and in all simulations for N = 40 and N = 100. Similarly, the introduction of an M allele into a population monomorphic for the L-morph (ssmm) led to its long-term maintenance with the L-morph in 93.4% of simulations for N = 10 and all simulations for the larger populations we explored (Table 1).

The immigration of recessive alleles into a monomorphic population showed a very different pattern. The introduction of the s allele into a population monomorphic for the S-morph (genotype SSMM) led to the maintenance of dimorphism for the S-and Mmorph in only 22% of the simulations for N = 10 (Table 1). Also, in contrast with the immigration of dominant alleles, the rate of establishment of recessive-allele migrants decreased with population size, e.g., polymorphism was maintained in only 11% of simulations for N = 100. A very similar pattern was observed for the migration of the s allele into a population monomorphic for the S-morph, but with genotype SSmm. Comparable establishment rates were observed for migration of an m allele into a ssMM population, e.g., the maintenance of a mid/long dimorphism in 21% (N = 10), 12% (N = 40) and 10% (N = 100) of the simulations.

Effect of metapopulation dynamics on morph frequencies

As predicted, isoplethy was maintained effectively indefinitely in all simulations of a metapopulation in the absence of local deme extinction (E = 0), irrespective of the rate of migration assumed (Fig. 2). However, population turnover via extinction and recolonization dynamics tended to reduce the relative frequency of the S-morph over the entire metapopulation. Thus, for E = 0.1, the S-morph was completely lost from the entire metapopulation when the average number of immigrants per generation < 0.5 (Fig. 2A). A greater migration pressure should maintain trimorphism for longer, while maintaining the S-morph at frequencies < 0.33. For the same extinction rate, the Mmorph was also lost from the metapopulation for low migration rates (e.g., I = 0 and I = 0.1), i.e., the metapopulation became monomorphic for the L-morph. Accordingly, population turnover generally favoured the L-morph, with frequencies at the metapopulation level always > 0.47 for E = 0.1 for all parameter values explored.

Migration of a new morph in a monomorphic metapopulation

The fate of migrants invading a monomorphic metapopulation (or network of populations within a larger metapopulation) were to some extent qualitatively similar to their fate as migrants into individual demes. We illustrate the immigration of a dominant S allele as an SsMM heterozygote (and thus the S-morph) into a metapopulation that was ancestrally monomorphic for the M-morph (fixed for genotype ssMM), and compare it to a scenario in which the recessive s allele migrates into a population, again as an SsMM heterozygote, that is monomorphic for the S-morph (fixed for genotype SSMM). The results are presented in Fig. 3 in terms of the rate of establishment of the immigrant allele and the median time of its ultimate establishment as a function of the extinction rate E and the average number of immigrants I per deme and per generation. We arbitrarily define establishment as its presence in at least 75% of the metapopulation's demes. For each explored combination of E and I, the dominant S allele never established less frequently in a metapopulation initially fixed for the genotype ssMM than the s allele into its counterpart metapopulation fixed for genotype SSMM (Fig. 3A;3B).

The establishment rates of the S and s alleles were zero for E = 0 and I = 0, as expected (for there was no possible gene flow among demes). For non-zero migration values, and for each combination of E and I explored, the difference in the establishment rate between the S and s alleles varied from 0.126 (establishment rate of 0.0454 for the s allele, and 0.1719 for the S allele for E = 0.1 and I = 0.2 ) to 0.944 (establishment rate of 0.056 for the s allele, and 1 for the S allele for E = 0 and I = 1; Fig. 3). The establishment of the dominant S allele was favoured by migration rather than by extinction and recolonization dynamics, with an establishment rate that decreases with E for the same value of I.

The median time taken for a new mutation to be present in at least 75% of demes decreased with increases in migration between demes, as expected. Also, as expected, this establishment time was shorter for greater E (because population turnover indirectly brought about a further component of inter-deme migration; Fig. 3D). Interestingly, greater population turnover (greater E) tended to slow the invasion of a metapopulation by a dominant S allele, whereas greater population turnover increased the rate of establishment of the recessive s allele (Fig. 3C). This pattern of easier and faster establishment of the dominant S compared to the recessive s allele was also found for the corresponding alleles at the M locus (Figure S1).

Effect of population carrying capacity on morph frequencies in a metapopulation

We finally assessed the effects of both drift within local demes and the potential establishment advantage of dominant versus recessive migrant alleles in a dynamic metapopulation with population turnover through local extinctions and recolonizations. In simulations with low migration pressure (I = 0.1), we found that metapopulation dynamics tended to maintain the L-morph at a higher frequency than the other two morphs for a wide range of local population carrying capacities, with the M-morph maintained at an intermediate frequency and the S-morph rare (Fig. 4A). In simulations of a metapopulation with high migrant pressure (I = 1), the morph frequencies were less strongly skewed, being closer to isoplethy across the metapopulation (Fig. 4C).

However, although the frequency ranking observed for simulations with a low migrant pressure (i.e., L > M > S) was conserved for metapopulations with demes of small carrying capacity, the ranking was reversed metapopulations in which the carrying capacity of demes was higher (Fig. 4C). This pattern of the reversal of style morph frequency rankings was accentuated in simulations with an intermediate number of immigrants (I = 0.5), with the L-morph maintained at substantially lower frequencies that that other two morphs, < 0.33, the M-morph most frequent, and the S-morph at a frequency > 0.33 (Fig. 4B).

Discussion

Under near disassortative mating in tristylous populations, we expect the three style morphs to be maintained at equal frequency (isoplethy) as a result of strong frequencydependent selection [START_REF] Heuch | Equilibrium populations of heterostylous plants[END_REF][START_REF] Heuch | Genotype frequencies associated with incompatibility systems in tristylous plants[END_REF][START_REF] Barrett | Disassortative mating in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF]. However, previous work has shown that because dominant alleles for style length will be maintained at a lower frequency than recessive alleles at the same locus, they are more susceptible to being lost from small populations by drift. In the case of tristyly, as exemplified by Eichhornia paniculata [START_REF] Barrett | Evolution of breeding systems in Eichhornia (Pontederiaceae): a review[END_REF][START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF][START_REF] Husband | Genetic drift and the maintenance of the style length polymorphism in tristylous populations of Eichhornia paniculata (Pontederiaceae)[END_REF], but also in a number of other tristylous species with essentially the same genetic architecture [START_REF] Heuch | Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria[END_REF][START_REF] Eckert | Stochastic loss of style morphs from populations of tristylous Lythrum salicaria and Decodon verticillatus (Lythraceae)[END_REF][START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF], this tendency for the loss by drift of dominant alleles means that, within local demes, the Smorph is more likely to be missing than the L-or M-morphs. At the metapopulation level, this biased loss of the S-morph locally will mean that the effects of drift will tend to reduce the frequency of the S-morph below its isoplethic expectation of 0.33. Our simulations largely confirm this intuition.

Beyond confirming previous results, our chief aim was to test the hypothesis that tristylous morph frequencies should be affected not only by the biased effects of local drift, but also by a process that favours the establishment within local demes of dominant versus recessive alleles upon their migration into local demes from elsewhere in a metapopulation; this process is similar to the action of Haldane's Sieve on new mutations [START_REF] Haldane | A mathematical theory of natural and artificial selection, part I[END_REF] or on standing genetic variation in single populations [START_REF] Orr | Haldane's sieve and adaptation from the standing genetic variation[END_REF]. In the case of the tristylous system modelled here, we tested the hypothesis of [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF] that the effect of Haldane's Sieve in a metapopulation of demes linked by migration might thus oppose the biased loss of dominant alleles by drift shown previously. Several of our results confirm this intuition, at least in principle.

First, our simulations exploring the fate of dominant versus recessive migrant tristyly alleles into single populations demonstrate that the favoured establishment of dominant over recessive alleles that immigrate via seed dispersal. This result should not be immediately surprising, because migration into a deme is in important respects no different from mutation. Nevertheless, the result demonstrates that a process invoked to understand the biased fixation of new mutations under positive directional selection is also relevant to alleles under negative frequency-dependent selection that are not destined to fix in a population but rather to be maintained at an equilibrium frequency.

The reason is that Haldane's Sieve still operates on such mutations that end up under balancing selection because, when they are rare (after immigration), they are positively selected. Haldane's Sieve helps to protect dominant alleles from stochastic loss in the early stages of establishment in a deme, whereas recessive alleles are not protected from loss until they become sufficiently numerous to be found in homozygotes.

Our simulations have not only demonstrated the hypothesised effect of Haldane's Sieve on dominant migrant alleles into demes locally, but also on their establishment across a metapopulation as a whole. Again, here the dominant allele for the S-morph of the tristylous sexual system is evidently more likely to invade and spread into a metapopulation in which it has previously been lost than the recessive allele for the Lmorph. In principle, this finding runs counter to the idea that metapopulations composed of demes with small effective population sizes are more likely to be missing the S-than the L-(or M-) morphs. The fact that in tristylous metapopulations it is in fact the Smorph that tends to be lost from entire regions of a metapopulation would seem to suggest that the effect we have observed here does not occur commonly in the wild, at least not in the metapopulations of Eichhornia paniculata or, for example, Lythrum salicaria, that has also been studied so extensively by Barrett and colleagues (see Introduction).

In the absence of evidence for the favoured invasion of the short morph into populations or metapopulations, we might still ask under what circumstances the positive effect of Haldane's Sieve on the dominant alleles could overwhelm the negative effects of local population drift in causing its biased loss? We addressed this question by simulating tristylous metapopulations in which both processes would be occurring togther. For a wide range of the parameter values that we assessed, the effects of Haldane's Sieve were indeed unable to overwhelm those of local drift. Thus, for a wide range of parameter values, population turnover, which tends to reduce the effective size of both the metapopulation and local demes [START_REF] Maruyama | Genetic variability and effective population size when local extinction and recolonization of subpopulations are frequent[END_REF][START_REF] Lande | Neutral theory of quantitative genetic variance in an island model with local extinction and colonization[END_REF][START_REF] Whitlock | The effective size of a subdivided population[END_REF][START_REF] Pannell | Neutral genetic diversity in a metapopulation with recurrent local extinction and recolonization[END_REF], ultimately overwhelms the smaller effect of Haldane's Sieve by bringing about the loss of the S-morph more frequently than the L or M-morphs. In this broad sense, the original suggestions of [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF] were perhaps overly optimistic about the effect that Haldane's Sieve might have on the composition of a floral polymorphism such as tristyly in a metapopulation. Nevertheless, [START_REF] Pannell | Haldane's Sieve in a metapopulation: sifting through plant reproductive polymorphisms[END_REF] also put forward the novel and narrower hypothesis that the balance between Haldane's Sieve in a metapopulation and the effect of local drift within demes should be most sensitive to the local demes' carrying capacity, as well as on their growth rate. Our simulations confirm this intuition.

Whereas the local effects of drift appear to hold sway in metapopulations in which local demes remain small, with the biased loss of the S-morph and an elevated metapopulation-wide frequency of the L-morph, this pattern was reversed in metapopulations in which the local carrying capacity was high. The reason for this result is that, in metapopulations with large demes that nonetheless suffer stochastic extinction, particularly if population growth is rapid, drift will be minimized locally, removing the likelihood that the S-morph will be lost, but Haldane's Sieve still operates and indeed operates more effectively because inbreeding and homozygosity are reduced.

Our most important conclusion is that Haldane's Sieve is likely to rescue dominant alleles from loss through local drift only under conditions in which demes grow rapidly to large size after their establishment by colonists. We expect that these conditions will rarely be met in plant metapopulations, except during biological invasion where introduced populations of tristylous species can in fact be much larger than the native range (e.g. Lythrum salicaria - [START_REF] Eckert | Genetic drift and founder effect in native versus introduced populations of an invading plant, Lythrum salicaria (Lythraceae)[END_REF]. The importance of metapopulation dynamics for plants has been the focus of heated disagreement in the past [START_REF] Freckleton | Large-scale spatial dynamics of plants: metapopulations, regional ensembles and patchy populations[END_REF][START_REF] Ehrlén | Large-scale spatial dynamics of plants: a response to Freckleton & Watkinson[END_REF][START_REF] Pannell | Probing the primacy of the patch: what makes a metapopulation?[END_REF][START_REF] Alexander | Metapopulations and metacommunities: combining spatial and temporal perspectives in plant ecology[END_REF]. A review of the literature by [START_REF] Freckleton | Large-scale spatial dynamics of plants: metapopulations, regional ensembles and patchy populations[END_REF] concluded that while most plants claimed to be subject to metapopulation dynamics were better described in terms of other spatial structures, Eichhornia paniculata was one of the most likely systems to conform to metapopulation expectations. Importantly for the current discussion, regions of the geographical range of E. paniculata in N. E. Brazil most prone to population turnover were also those in which populations tended to be small [START_REF] Barrett | The dissolution of a complex genetic polymorphism: the evolution of self-fertilization in tristylous Eichhornia paniculata (Pontederiaceae)[END_REF][START_REF] Husband | Spatial and temporal variation in population size of Eichhornia paniculata in ephemeral habitats: implications for metapopulation dynamics[END_REF]. Similarly, metapopulations of the plant Mercurialis annua subject to elevated rates of local population extinction (and presumably turnover; [START_REF] Dorken | Small-scale and regional spatial dynamics of an annual plant with contrasting sexual systems[END_REF]) also tended to have smaller demes [START_REF] Eppley | Sexual systems and measures of occupancy and abundance in an annual plant: testing the metapopulation model[END_REF]. According to the results of our study, these are evidently not conditions optimal for an overwhelming effect of Haldane's Sieve.

The conditions under which Haldane's Sieve is likely to overwhelm local drift appear quite restrictive in the context of the maintenance of dominant versus recessive alleles underlying wild tristylous metapopulations. Nevertherless, the process might be important in reducing morph-ratio biases from what they might be in its absence.

Unfortunately, this conjecture is very difficult to test with the sorts of data that might be available from field studies. However, the processes that we have investigated illustrate the importance that the details of a trait's genetic architecture might have on its frequency in natural populations. Phenotypic selection models will not always be sufficient to predict phenotype frequencies when these are effectively the result of a sum over evolutionary trajectories that depend on the underlying genetics of polymorphism.

The importance of the details of inheritance was already clear in the case of tristyly from previous demonstrations cited above of the biased loss of the S-morph through drift in small populations. Our study has now demonstrated for the first time the extent to which phenotype frequencies in plant sexual polymorphisms may be sensitive to the interaction between genetics and demography, particularly involving migration and population turnover in metapopulations.

Table 1. The polymorphic state of populations initially monomorphic for the given ancestral genotypes following migration of either dominant or recessive alleles at the S and M loci.

  Ancestral genotypes were ssmm (L-morph), SSmm (S-morph), ssMM (M-morph) or SSMM (S-morph). Proportions are measured in terms of the number of 1,000 replicate simulations in which the stated polymorphism still persisted after 6N generations from the time of introduction of the immigrant individual. Simulations are shown for populations with N = 10, 40 and 100 individuals.

Figure 1 .

 1 Figure 1. Expected morph composition in demes subject to drift as a function of time from starting under isoplethy. Curves show the proportion of populations at a given time point that had fixed or lost particular morphs, as indicated in the inset legend. The ancestral population contains all three morphs in similar proportions. The effects of drift and frequency-dependent selection are recorded at each generation over a period of 1,000N generations, for populations of size (A) N = 10, (B) N = 40 and N = 100 (C).All simulations assumed P = 0.1 (i.e., strong disassortative mating).

Figure 2 .

 2 Figure 2. Effect of population turnover and rates of immigration on the frequency of morphs across a metapopulation. The metapopulation comprised 100 demes of 500 individuals. All simulations were started with all demes in a state of isoplethy. Metapopulation-wide morph frequencies are plotted for a range of different rates of migration (in terms of the mean number of immigrants per generation), and for three different extinction rates : E = 0 (panel A), E = 0.01 (panel B) and E = 0.1 (panel C).Results are shown after 10,000 generations. The dashed line indicates the expected frequency of the three morphs at equilibrium (isoplethy).

Figure 3 .

 3 Figure 3. Migration of dominant versus recessive alleles into a metapopulation. The establishment rate (panels A and B) corresponds to the proportion of 1,000 replicate simulations in which the migrant allele because established in at least 75% of the demes

Figure 4 .

 4 Figure 4. Effect of population carrying capacity on morph frequencies in a dynamic metapopulation. The morph frequencies maintained in a metapopulation averaged over 1000 replicate simulations, each run for 10N generations. The metapopulation comprised 100 demes liable to extinction at a probability E = 0.05, with immediate recolonization by a single founder sampled randomly from the rest of the metapopulation. Mating assumed strong (but incomplete) disassortative mating (P = 0.1), and an average number of immigrants per generation and demes of 0.1 (panel A), 0.5 (panel B) and 1 (panel C).

  Figure 1.

  Figure 2.

Figure

  Figure 4.

establishment rate of dominant versus recessive alleles at the S locus. In Fig.S1, weshow that results for the S locus are similar for the M locus, too.

Acknowledgements

We thank the organisers of the special issue for their invitation to contribute this article.

JRP acknowledges Spencer Barrett for his scientific inspiration over the years, not least in prompting thoughts about tristyly in metapopulations when he was a postdoc in the Barrett lab. Numerical results presented in this paper were carried out using the cluster from the HPC Méso-centre of Lille University. JRP is supported by funding from the Swiss National Science Foundation, and the University of Lausanne.

Author contributions JRP conceived of the project. CR performed the simulations. JRP and CR wrote the manuscript.

Supplementary Information

Simulation protocol for reproduction within the demes.

The protocol outlined in the main text describes the effective process of reproduction in our simulations. We realised the assumed process using a short-hand approach that was computationally more efficient. Thus, we assumed that the target deme contains N hermaphroditic diploid individuals. Individuals all had the same probability of transmitting their gametes by female function, but the probability of reproducing by male function depended on the frequency of the male parent considered in the deme. To produce N descendants from the potential parents in a deme, potential maternal parents were sampled randomly from the set of all individuals in the current generation, with replacement. For a chosen maternal parent , we calculated the probability that her potential partner would be of the same or of a different morph. This probability F depended on the frequency f of the maternal morph in the deme and the extent to which intra-morph mating was assumed to be possible in the run, given by parameter 1 -P, such that F = q 1 /(q 1 +q 2 ), where q 1 = f(1 -P) and q 2 = 1 -f, P is thus equivalent to the probability of disassortative mating. Under this scheme, all pairings would be heteromorphic, and mating would thus be completely disassortative, in a polymorphic deme for P = 1.

A binomial draw of probability F was then carried out for each maternal parent to determine whether the future paternal parent should be drawn among individuals of the same morph as the maternal parent, or among individuals of a different morph.

Following determination of the homomorphic or heteromorphic nature of the crossing, we randomly selected a future potential paternal parent from among all individuals of the appropriate morphological class and genotype. The progeny was finally produced by randomly sampling a parental allele from each the S and M loci from both the chosen maternal parent and the chosen paternal parent , effectively assuming free recombination between the two loci. To avoid the possibility of deme extinction for monomorphic demes, we did not run simulations for P = 0, i.e., we assumed that there was always a possibility for reproduction via within-morph mating (or selfing). This scheme is in some respects similar to an assumption of delayed selfing in the absence of inter-morph mates. We assumed non-overlapping generations, with the progeny generation replacing the parental generation.

Data from runs for the establishment of a migrant allele in a metapopulation

We provide here the median establishment rates (i.e., the proportion of simulations showing at least once a presence of the migrant allele at least 75% of the demes) as well as the median establishment times measured over 1,000 replicates of each combination of parameters. These values were used to produce Fig. 3 in the main text, where the plotted times were log-transformed. In Table S1, the reported median times are in generations, untransformed. In the main text, we detailed the differences in the