Nonself-recognition-based self-incompatibility can alternatively promote or prevent introgression

Traditionally, we expect that self-incompatibility alleles (S-alleles), which prevent self-fertilization, should benefit from negative-frequency dependent selection and rise to high frequency when introduced to a new population through gene flow. However, the most taxonomically widespread form of self-incompatibility, the ribonuclease-based system ancestral to the core eudicots, functions through nonself-recognition, which drastically alters the process of S-allele diversification. We analyze a model of S-allele evolution in two populations connected by migration, focusing on comparisons among the fates of S-alleles originally unique to each population and those shared among populations. We find that both shared and unique S-alleles originating from the population with more unique S-alleles were usually fitter than S-alleles from the population with fewer. Resident S-alleles were often driven extinct and replaced by migrant S-alleles, though this outcome could be averted by pollen limitation or biased migration. Nonself-recognition-based self-incompatibility will usually either disfavor introgression of S-alleles or result in the whole-sale replacement of S-alleles from one population with those from another.


22
In flowering plants, a self-incompatibility locus (S-locus) is a highly polymorphic region of the genome responsible for rejecting self pollen and is maintained by 24 negative frequency-dependent selection, a form of balancing selection. A given S-locus allele (S-allele) encodes paired pollen and pistil phenotypes, and the pis-26 til phenotype rejects pollen expressing the matching pollen phenotype, which necessarily includes self pollen. With self-incompatibility (SI), the fitness of an 3 Description 178 We modeled the evolution of the S-locus in two self-incompatible populations, the local population and the foreign population, of infinite size connected by 180 pollen migration. In the simplest versions of this model, migration was unidirectional. From the perspective of the population receiving immigrants, that 182 population is "local" while the population sending emigrants is "foreign." For consistency, we retained this terminology throughout, but the local and foreign 184 designations become arbitrary when migration is bidirectional. Each population harbored a number of S-haplotypes, some unique to that population and some 186 shared between populations. We considered both self-and nonself-recognition models, but incompatibility was always gametophytic: the pollen phenotype 188 was determined by its own haploid genotype, and the maternal phenotype was determined codominantly by its two haplotypes. Under GSI, the equilibrium 190 condition in the absence of migration is equal frequency among S-haplotypes (Nagylaki, 1975;Boucher, 1993;Steiner and Gregorius, 1994), so we set this 192 as the initial state for each population. Selection occurred only through pollen competition and maternal fecundity: there was no selection on viability. In 194 pollen competition, all pollen compatible with a given maternal genotype competed equally to fertilize individuals of that maternal genotype. Pollen had zero 196 success on maternal genotypes with which it was incompatible. We allowed for pollen limitation of maternal fecundity, in which maternal genotypes that 198 accepted more pollen enjoyed greater seed success. Two considerations suggest that pollen limitation is an important process to model. First, since carriers 200 of unique migrant haplotypes reject the majority of all pollen they receive (all resident haplotypes), they are likely to suffer severe pollen limitation in nature. 202 Second, since pollen limitation is especially unfavorable to migrant haplotypes, we expect it to act as a barrier to gene flow that might partially counteract the 204 advantage of rarity.
We use our lock and key metaphor (Harkness et al., 2019) to model non-206 self recognition. That is, the style is a door locked by two codominant RNase 5 "locks," and a pollen grain must carry the SLF "keys" to both of a seed parent's locks in order to unlock the door and proceed to fertilization. For simplicity, we imagine that each key unlocks one lock, but in reality some SLFs are com-210 plementary to two or more RNases (e.g. Sun and Kao, 2013). Pollen is always incompatible with the plant that produced it because either possible key ring it 212 could possess is missing one of the keys to one of the plant's own two locks. For self-recognition, we use the simplest model of a two-gene S-locus we could 214 conceive, though we could not invent an adequate metaphor for this system. The S-locus under self-recognition consists of one polymorphic pistil-expressed gene 216 tightly linked to one polymorphic pollen-expressed gene. Each pollen allele corresponds to one pistil allele, and corresponding alleles always exist on the 218 same haplotype. By default, all pollen is accepted, but pollen carrying the pistil allele corresponding to either of the seed parent's codominantly expressed 220 pistil alleles is rejected. Since each haplotype carries a pollen allele that would be rejected by its own pistil allele, each haplotype is self-incompatible. This is 222 similar to the programmed cell death mechanism in poppy, in which recognition between the pollen and pistil components triggers self-destruction in the pollen 224 (Franklin-Tong et al., 1993), but differs from many better-known self-recognition based systems (Hiscock, 2002). We chose this system to isolate the effects of self 226 vs nonself based recognition away from complications of dominance hierarchies among alleles in sporophytic SI systems.

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Self-and nonself-recognition lead to different behavior in inter-population crosses. A self-recognition pistil allele only rejects one pollen allele: its comple-230 ment. Non-matching pollen will always be accepted, regardless of its population of origin. In contrast, with nonself-recognition, a pistil locks out all pollen with-232 out a key. In a single population, this difference is immaterial: selection for maximal pollen success will ensure every S-haplotype is "complete" (Bod'ová 234 et al., 2018) with respect to its own population, meaning that it carries keys to all locks in that population except its own lock. However, there would be 236 little selection for a key to a lock that is rarely encountered and no selection for a key to a lock that is never encountered. Given enough time and isolation, 238 populations will come to diverge in their sets of locks (either through novel mutations or differential loss), and each population will only maintain keys to 240 locks present within that population. This leads to very different outcomes for haplotypes that are shared among populations and haplotypes unique to a pop-242 ulation. Pollen from one population will always be compatible with any other haplotype from the same population, whether shared or unique. But pollen from 244 one population will only be compatible with a haplotype from another population if that haplotype is shared with the pollen's population of origin. We 246 only explicitly model this ideal high-isolation case, but consider other plausible biological in the Discussion. 248 We classify all S-haplotypes based on their uniqueness and population of origin. This is sufficient for our model because the compatibility of pollen with 250 locally unique S-haplotypes is determined by the pollen's population of origin under nonself-recognition. S-haplotypes are either unique to the local popula-the local population (SL), or shared and originating from the foreign population (SF ). Each class may contain one or more S-haplotypes, and we assume all haplotypes within a class always occur in equal proportions. We justify this 256 assumption on the logic that, so long as it were true initially, it would remain true because haplotypes within the same class would have equal pollen and ovule 258 success.
Shared local and shared foreign haplotypes come in pairs, e.g., a local S 1 260 and a foreign S 1 . Under nonself-recognition, the members of a shared pair differ in their key rings: each has only the keys to locks originating from its own pop-262 ulation. Under self-recognition, the members of a shared pair are functionally identical, though in nature they could be characterized by different neutral mu-264 tations. We use the same classification for self-and nonself-recognition for ease of comparison, but note that population of origin only affects phenotype under 266 nonself-recognition, in which it determines an S-haplotype's key ring. Functionally, there is only a single difference between our self-and nonself-recognition 268 models: S-haplotypes originating from one population are incompatible as pollen with all haplotypes unique to another population under nonself-recognition but 270 compatible under self-recognition ( Fig. 1). This characterization allows us to compare the expected amount and pace of gene flow at the S-haplotype across 272 comparable parameters for self-and non-self based SI. We denote the frequency of each class in each population by p L and p F . E.g., in the local population, the frequency of all haplotypes initially unique to the local population (U L) is p L U L . The frequency of a haplotype class is the sum of the frequencies of all haplotypes in that class. Since all haplotypes in a population begin at equal frequencies, the initial frequencies of the haplotype classes are determined by the number of haplotypes in each class. These numbers are denoted n L (number unique to the local population) n F (number unique to the foreign population), and n S (number shared). In the local population, initially, p L U L = n L / (n L + n S ) p L SL = n S / (n L + n S ) p L U F = p L SF = 0 In the foreign population, initially, p F U L = p F SL = 0 Gene flow occurs through pollen migration such that a proportion m F L of 274 pollen in the local pollen pool is migrant pollen from the foreign population, and m LF of pollen in the foreign pollen pool of is migrant pollen from the 276 local population. When migration is unidirectional from foreign to local, the foreign population remains at its initial equilibrium, so we only track the local population. When migration is bidirectional, we track both populations.
We model pollen limitation through a root function: a given genotype's seed success is a b X , where a X is the frequency of all pollen compatible with genotype X in the available pollen pool and b is a shape parameter (with b = 0 282 corresponding to no pollen limitation). This allows seed success to saturate as compatible pollen received increases. Since the origin of a haplotype only affects 284 is behavior in pollen, we drop the origin designation from the pollen accepted by each genotype X. E.g., a U F SF = a U F SL , so we instead simply write a F S . 286 We first describe analytical results for the case of unidirectional migration, and then describe numerical results for bidirectional migration. For this case, 288 we focus entirely on frequencies within the local population and drop the superscript denoting population: e.g., the frequency in the local population of an 290 allele initially unique to the foreign population p L U F is simply labeled p U F . We use this case to lay out the three deterministic processes underlying haplotype 292 frequency change.
First, selection on ovules occurs through pollen limitation. Each diploid genotype X contributes a number of successful ovules P * X = P X a b X . Second, assuming unidirectional migration, pollen migration generates the pollen pool. So where a * denotes a frequency after migration.

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Third, pollen selection occurs through differential pollen success, resulting in new genotype frequencies. Pollen fitness is a function of the proportion of 296 stylar genotypes it can pollinate -which depends only on a pollen specificity's frequency under self-recognition, and on the frequency of styles to which it holds 298 the key under nonself recognition. These differing rules are reflected in the genotype frequency after mating under self-and nonself-recognition incompatibility 300 (Appendix XX). We note that with nonself-recognition, no individual ever carries both a unique foreign and unique local (P UFUL = 0) because neither holds 302 the key to the other's lock. We further note that these analytical results did not include pollen limitation, a biologically relevant process which we explore 304 in our numeric iterations.
We derive analytical results for a special case. Focusing on nonself-recognition and with no pollen limitation, the marginal fitnesses of the shared haplotypes are Taking the difference, the third term in each cancels out, and all remaining terms depend on the frequency of carriers for the unique haplotypes. The sign of this difference is obvious when one population harbors 308 no unique haplotypes. For example, when the local population has no unique S-haplotypes (n L = 0) and the foreign population has at least one unique S-310 haplotype (n F > 0), the fitness of shared haplotypes from the foreign population exceeds that of shared haplotypes from the local population. Since w SL < w U F 312 at all genotype frequencies, there is no non-zero equilibrium frequency of p SL . So long as p SF > 0 and p U F > 0, the SL haplotype will be eliminated at 314 equilibrium. Similarly, when n L > 0 and n F = 0, w SL > w SF at all genotype frequencies.

316
The biological interpretation of this result is straightforward. The only difference between shared haplotypes originating from different populations is their 318 collection of pollen-function keys: all L haplotypes (shared or unique) are compatible with S-haplotypes initially unique to the local population (U L), while 320 SF haplotypes are compatible with U F haplotypes. If U F haplotypes exist but U L haplotypes do not, SF haplotypes have all the pollen success of SL 322 haplotypes plus additional pollen success on U F haplotypes. In this case, each SF haplotype is always fitter than its SL counterpart, and selection will drive 324 all SL haplotypes extinct unless opposed by biased migration. That is, a population harboring no unique haplotypes will have its shared haplotypes replaced 326 by shared haplotypes from the other population. No such process occurs under self-recognition.

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This analytical result suggests an important role for the relative numbers of unique haplotypes in each population. To investigate the effect of these and 330 other parameters in more general cases, we implemented the full deterministic model in R. Using this model, we tracked evolution of haplotype frequencies 332 under scenarios varying the numbers of shared (n S ) and unique haplotypes (n L and n F ), the presence (b = 1/2) or absence of pollen limitation (b = 0), the duration of migration (continuous or a single pulse), and the form of selfincompatibility (self-or nonself-recognition). For this part of the investigation, 336 we kept migration unidirectional from foreign to local (m LF = 0). Migration persisted throughout the iterations if continuous or ceased after a single generation if pulsed, and it occurred at the same modest rate (m F L = 0.01) whether continuous or pulsed. mum value of m F L that resulted in invasion of SF for different values of n L and n F under nonself-recognition. We varied m F L from 0-0.1 in increments of 344 0.01 and varied n L and n F from 0-10. We considered values of n S = 5 or 20 and b = 0 or 1/2 but held m LF = 0 constant. Invasion was considered to have 346 occurred if p U F + p SF > p U L + p SL in the local population at generation 1000. Note that this definition of invasion does not imply positive selection on the 348 SF haplotypes: a neutral haplotype would also invade under this unidirectional migration scheme.

350
Finally, we tracked haplotype frequencies for several bidirectional migration scenarios. We considered four scenarios in which migration and the num-352 ber of shared haplotypes were held constant (m LF = m F L − 1, n S = 5), in which one (n L = 0 and n F = 1) or both populations possessed a unique allele 354 (n L = n F = 1), and in which pollen limitation was either present (b = 1/2) or absent (b = 0). Additionally, we more closely investigated the haplotype 356 freqeuncy trajectories in another four scenarios, which had resulted in coexistence of foreign and local haplotypes at equilibrium. In these scenarios, the 358 numbers of shared and unique haplotypes and the strength of pollen limitation were constant (n S = 20, n L = n F = 1, b = 1/2), and migration

Results
Self-recognition-based SI: The consistent outcome of self-recognition was 366 for rare pistil specificities to rise to higher frequencies regardless of their origin (Fig. 2). All unique haplotypes always coexisted at equilibrium for both ongoing 368 unidirectional migration ( Fig. 2A  Under self-recognition, pollen limitation did not severely impact the spread of foreign S-haplotypes (c.f. the right hand columns of Fig SL and SF haplotypes was determined entirely by migration pressure. Under continuous unidirectional migration, shared haplotypes originating in the 376 donor population globally displaced shared haplotypes originating in the recipient population (Fig. 2). But under a single pulse of unidirectional migration, 378 when only selection was ongoing, the initially low-frequency unique haplotypes migrating from the foreign population (U F ) rose to match the frequency of the 380 haplotypes unique to the local population (U L), while shared haplotypes (SF ) remained at their initial low frequency (Fig. 2). 382 We note that the equilibrium frequency of all shared haplotypes relative to all unique haplotypes increased as the number of shared haplotypes increased 384 (compare dashed and full lines in Fig. 2), as would be expected from increasing the number of haplotypes in a category and counting them collectively. A 386 similar result also occurs with nonself-recognition, as seen in Fig. 3. Because results from the case of self recognition are straightforward and confirmed cur-388 rent understanding, we spend the remainder of our efforts on the results from nonself-recognition SI.

390
Nonself-recognition-based SI: Under nonself-recognition, the results are sensitive to parameter values, and many outcomes were possible.

392
Without pollen limitation and with ongoing unidirectional gene flow, foreign unique S-haplotypes can often invade. For example, with one foreign and no 394 local unique S-haplotypes, the unique foreign S-haplotype rapidly establishes and reaches its equilibrium frequency (Bottom row, first column of Fig. 3A).

396
This process is somewhat slower when the local population also has a unique S-haplotype (Bottom row, second column of Fig. 3A). Unlike the case of self-398 recognition, with nonself-recognition, the foreign unique S-haplotype replaces the local unique S-haplotype (when green lines in e.g. the bottom panel of Fig.   400 3A are above zero, pink lines head towards zero). Additionally, because pollen of both local shared and local unique S-haplotypes is incompatible with styles 402 with unique foreign locks, its fitness decreased as the unique foreign S-haplotype rose in frequency, consistent with our analytical predictions (Eq. 3). In contrast, 404 the shared foreign S-haplotype enjoyed rapidly increasing pollen fitness as the unique foreign haplotype rose in frequency (compare the slower rise in frequency 406 of shared foreign haplotypes in the bottom row of Fig. 3A to that in Fig. 2A). As a result, foreign shared haplotypes replaced local shared haplotypes, just as 408 foreign unique haplotypes replaced local unique haplotypes.
In the absence of pollen limitation, increasing the number of shared haplo-410 types from 5 to 20 did not qualitatively affect the outcome of migration but modified the equilibrium frequency of shared and unique haplotypes (compare To quantify the effects of shared and unique haplotypes, as mediated by the extent of gene flow, we calculated the minimum value of m F L for which 430 p U F + p SF > p U L + p SL at generation 1000 (Fig. 4). Without pollen limitation, invasion was possible for a broad range of n L and n F values, though it 432 could occur even at low migration rates when n F ≥ n L . With pollen limitation and n S = 5 shared haplotypes, invasion was only possible for m F L ≤ 0.1 when 434 n F ≥ n L . However, the exact effect of pollen limitation depended on n S . At n S = 5, pollen limitation consistently raised the threshold migration for inva-436 sion. At n S = 20, pollen limitation usually raised the threshold when n L < n F but lowered it when n L > n F . When pollen is limiting and there is some migra-438 tion, the seed success of unique haplotypes is reduced. When there are many haplotypes, the pollen advantage of rarity is smaller. These disadvantages com-440 bined may be sufficient to eliminate rare unique haplotypes, thereby eliminating a barrier to the invasion of foreign haplotypes whether shared or unique. We 442 demonstrate such a process for a case of bidirectional migration below.
When migration was equal in both directions, the equilibrium state depended  In some cases, pollen limitation could drive unique haplotypes from both populations extinct (Fig. 7). Since unique haplotypes could not accept immi-468 grant pollen, they suffered reduced seed success compared to shared haplotypes.
Once the unique haplotypes were gone, the distinction between shared haplotypes from different populations became irrelevant to selection: each kind carried a key to a different extinct lock. The shared haplotypes could then coexist 472 in the absence of unique haplotypes.

474
Existing theory predicts that rare advantage at an S-locus will elevate gene flow by favoring rare immigrant S-haplotypes (e.g., Pickup et al., 2019). We confirm this prediction for self-recognition systems, but find that nonself-recognition can either enhance or reduce gene flow depending on the S-haplotypes present 478 in each population. The crucial parameter is the number of S-haplotypes unique to each population because pollen originating from one population is incompat-  Importantly, biased introgression under at the S-locus under non-self recognition applied to shared haplotypes as well as unique haplotypes. As a unique 496 migrant haplotype invades, the advantage of compatibility with that haplotype increases. Thus, a unique haplotype can invade by the advantage of rarity while 498 the shared haplotypes from the same population ride its coattails. A common pattern we observed was that the population with more unique haplotypes or 500 higher emigration rates essentially swamped the other population, replacing both unique and shared haplotypes.

502
The role of introgressed locks in facilitating the invasion of keys originating from the same population is analogous to the surprising effect of introgressed 504 female preferences on male traits in sexual selection predicted by Servedio and Bürger (2014). Sexual selection has classically been viewed as a barrier to intro-506 gression because, if two populations have divergent female preferences, migrant males with out-of-place traits will suffer reduced reproductive fitness. However, 508 female preferences may themselves introgress if they not under direct selection. As a preference introgresses, the reproductive fitness of the corresponding male 510 trait increases, allowing the trait to increase in frequency. This facilitated introgression can counteract the fitness advantage of so-called "magic traits" that are 512 13 both adaptive in the local environment and initially preferred by local females. Pistil-expressed self-incompatibility locks are analogous to strong female prefer-514 ences for a given pollen-expressed key, analogous to a male trait. The difference is that, instead of one male trait with different values in each population, pollen 516 keys are a collection of many binary traits. Thus, pollen with more keys may be preferable to many seed parents in both populations, resulting in a directional 518 asymmetry absent from pure Fisherian sexual selection.
Pollen limitation and the number of shared haplotypes also affected intro-520 gression of foreign S-haplotypes. Pollen limitation typically reduced introgression and could result in the maintenance of two effectively isolated populations 522 with their own sets of haplotypes. In contrast, introgression increased as the number of shared haplotypes increased. Shared haplotypes, unlike unique hap-524 lotypes, could accept immigrant pollen. When there were many shared haplotypes, immigrant pollen was rarely rejected and had high fitness.
526 Surprisingly, when migration was bidirectional and roughly symmetric, pollen limitation could instead eliminate unique haplotypes in both populations, which 528 reduced the overall number of haplotypes but allowed shared haplotypes to introgress freely. Unique haplotypes were lost because they rejected migrant 530 pollen and thus suffered greater pollen limitation than shared haplotypes, which did not. This is a consequence of the model's square root function of pollen lim-532 itation, in which seed success monotonically increases with compatible pollen and never fully flattens. Therefore, accepting more pollen always increases seed 534 success somewhat, and unique haplotypes are disadvantaged as seed parents. If seed success instead plateaus, unique haplotypes should still be disadvantaged 536 unless both shared and unique haplotypes accept enough pollen to reach the plateau.

538
From a genomic perspective, each invasion of a unique migrant haplotype under nonself-recognition should reduce among-population neutral divergence for 540 all haplotypes at the S-locus. In contrast, invasion of a migrant haplotype under self-recognition should only reduce divergence among copies of a single haplo-542 type. We therefore predict greater among-population neutral divergence at the S-locus in species with self-recognition than in those with nonself-recognition.
544 Surprisingly, a greater rate of S-haplotype diversification might therefore lead to reduced divergence between populations at the S-locus. Data on S-allele overlap 546 and divergence across populations is available, at least for self-recognition. In Arabidopsis halleri and A. lyrata, which possess the self-recognition-based S-548 locus receptor kinase (SRK) incompatibility system, Castric et al. (2008) found 18 pairs of SRK alleles diverging by less than or equal to 12 substitutions. The 550 remaining 12 alleles in A. halleri and 20 alleles in A. lyrata had no such counterparts in the other population. The low divergence within each pair could be 552 explained by elevated gene flow of these alleles but not by the mere reduced effective population size of a single S-allele. This pattern is consistent with our 554 expectation for self-recognition that migrant S-alleles will rise to high frequency without greatly reducing diversity in the S-locus as a whole. However, we also 556 predict that any pairs of functionally identical S-alleles that were initially shared by the two species would have persisted in each population. It is possible that 558 14 these originally shared alleles have since been lost in one or both populations or that they have since accumulated enough neutral or functional divergence that 560 they are no longer recognizably shared. We also note that the SRK system is sporophytic rather than the gametophytic self-recognition system we modeled.

562
More comprehensive theory already exists for the interaction between haplotype number and migration in the case of self-recognition. In a model combining 564 migration and balancing selection (either symmetrical overdominance or SI), Muirhead (2001) predicted for a given migration rate both the expected total 566 number of alleles and the distribution of the proportion of alleles shared between two populations, three populations, etc. For the total number, she found 568 a non-monotonic relationship in which allele number is minimized for intermediate migration rates. This non-monotonic relationship was previously observed 570 in simulations by Schierup (1998). For the proportions of shared alleles, she found that increasing the migration rate skewed the distribution towards alleles 572 shared among many populations. In the SI version of this model, all unlike S-haplotypes were assumed to be cross-compatible. This is equivalent either to 574 nonself-recognition in which all haplotypes are complete or to self-recognition. However, it is not equivalent to nonself-recognition in which some haplotypes 576 are more complete than others.
Theory on both self-and nonself-recognition has revealed hurdles to S-578 haplotype diversification and suggested that gene flow can help populations overcome these hurdles (Uyenoyama et al., 2001;Gervais et al., 2011;Harkness 580 et al., 2019). Our results constrain how S-haplotype diversification could occur in a subdivided population or metapopulation. Uyenoyama et al. (2001) found 582 that local turnover, replacement of old haplotypes without increasing the total number, was possible under a much broader set of parameter values than local 584 diversification. They therefore hypothesized that diversification occurs through local turnover followed by introgression of the new haplotype into the metapop-586 ulation and reintroduction of the lost haplotype from the metapopulation. We predict under this process that novel S-haplotypes from the same population 588 can easily spread simultaneously, but novel haplotypes from different populations may interfere and eliminate each other. Reintroduction of a haplotype 590 lost to turnover would require the reintroduced haplotype to become compatible with the novel haplotype. Until cross-compatibility is restored, the novel 592 haplotype may continue to replace the locally lost haplotype in every subpopulation. Gene conversion could restore cross-compatibility (Kubo et al., 2015; reduce pollen limitation, and these self-compatible mutants might introgress more easily than self-incompatible haplotypes. In contrast, a haplotype with 606 the key to its own lock would only affect the pollen phenotype: self pollen would be compatible, but the maternal plant would still reject the same subset 608 of nonself pollen. While acquiring a key (which offers new siring opportunities) would typically increase fitness more than losing a lock, the substantial pollen 610 limitation suffered by migrant haplotypes might reverse this inequality: losing a lock would completely eliminate pollen limitation by accepting all pollen, while 612 gaining a key would only mitigate pollen limitation by accepting self pollen. If accepting self pollen is insufficient to achieve full seed set, losing a lock should 614 provide greater benefit to seed set than gaining a key. We assumed that all haplotypes were incompatible with haplotypes unique 616 to other populations. But if the difference in S-haplotypes between populations was caused by recent differential loss, it is likely that each population would 618 temporarily retain cross-compatibility with locally lost haplotypes. That is, the loss of a pistil-function lock does not imply immediate loss of the corresponding

630
We have only modeled the S-locus, but any S-haplotype in nature comes as part of a whole parental genome. That genome might not be as well adapted to 632 the local environment as resident genomes. In this case, linkage disequilibrium between the S-locus and loci involved in environmental adaptation might slow . Similarly, we show that gene flow is more complex under nonself-than under self-recognition. The S-locus, an ancient and widespread feature controlling the breeding system of many flowering plants, seems only to get stranger the more it is investigated.

Acknowledgments
This work was supported by NSF DEB #1754246 to Y. B.

648
A. H. was funded through the University of Minnesota Doctoral Dissertation Fellowship. We would like to thank Dr. Shelley Sianta and Dr. Emma Goldberg, 650 whose especially thorough and insightful feedback greatly improved the rigor, clarity, and focus of this manuscript.

Nonself-recognition
After pollen migration, the frequency of each haplotype in the pollen pool is changed from p to p * . For shared haplotypes, population of origin only affects pollen behavior. Therefore, there are effectively only three kinds of haplotypes 754 from the perspective of maternal plant phenotype: local unique haplotypes, foreign unique haplotypes, and all shared haplotypes collectively. We drop the 756 population of origin when denoting the pollen accepted by a genotype: e.g., both U F SF and U F SL accept the same proportion a F S of pollen rather than 758 distinct values of a U F SF and a U F SL . Assuming no pollen limitation, mating with nonself-recognition produces the genotype frequencies:

Self recognition
Assuming no pollen limitation, after migration and mating with self recognition, 764 genotype frequencies are:  Figure 1: Pollen compatibility under self-and nonself-recognition. The focal S 3 haplotype is shared and originates from the local population. a shows the consequences of self-and nonself-recognition for compatibility with other haplotypes. Under self-recognition, S 3 pollen is only incompatible with S 3carrying plants. Under nonself-recognition, S 3 pollen is still incompatible with S 3 -carrying plants, but it is also incompatible with carriers for haplotypes unique to another population (S 4 ). b details the mechanistic basis of rejection for a local S 3 pollen grain on a foreign S 1 S 4 plant for self-and nonself-recognition. Under self-recognition, neither S 1 nor S 4 matches S 3 , so the pollen is not rejected. Under nonself-recognition, the local S 3 haplotype's key ring contains keys to all locks present in the local population (allowing it to unlock S 1 ) but not the keys to locks unique to the foreign population (S 4 ), so it is rejected. We place both rejection mechanisms in the style to match the RNase-based system, but rejection could occur at the stigma (as it does in some taxa) without affecting the model.

No Pollen Limitation
Pollen Limitation x Figure 4: Effect of unique haplotypes on invasion threshold. The minimum rate of unidirectional migration (m F L ) needed for the frequency of all foreign haplotypes to exceed the frequency of all local haplotypes (p U F + p SF > p U L + p SL ) in the local population at generation 1000. Invasion could occur at low migration rates when the number of foreign unique haplotypes (n F ) exceeded the number of local unique haplotypes (n L ) but required high migration rates when n L > n F . Many shared haplotypes (n S = 20) lowered the invasion threshold compared to few shared haplotypes (n S = 5). Pollen limitation raised the invasion threshold when n S = 5, but when n S = 20, the effect of pollen limitation was to erase partly the effect of n F . Seed set for a genotype was a square root function of the proportion of pollen compatible with that genotype. The y = x line is shown in faint red. Frequency of foreign haplotypes Figure 6: Equilibrium frequency of foreign haplotypes in each population with bidirectional migration. Self-incompatibility functions through nonselfrecognition in all panels. Each population harbors one unique haplotype (n L = n F = 1), and many (n S = 20) or few (n S = 5) shared haplotypes.

Continuous Migration
In the absence of pollen limitation, foreign and local haplotypes usually coexist unless migration rates are extremely biased. With pollen limitation, moderately biased migration rates result in the loss of foreign or local haplotypes, and coexistence is only possible for a narrower band of more nearly equal migration rates.  ) and asymmetric (C, D) migration rates, with asymmetric rates altering the equilibrium frequencies of the shared alleles. There were nS = 20 shared haplotypes, n L = 1 unique local haplotype, and n F = 1 unique foreign haplotype, and seed success was a square root function of compatible pollen.