Female choice can explain the fertilization pattern in oaks

We extend a two-step lottery model of Craft et al. to test the hypothesis that oak trees pursue a form of within-flower female choice to increase the diversity of fathers. Oak trees produce six ovules per flower while maturing just one acorn. When assuming a random ovule selection - which is a natural assumption in the absence of other hypotheses - observed fertilization patterns in oaks cannot be explained: long-distance fertilization is unusually common, even as nearby oak trees may be absent as pollen donors. Our model demonstrates how producing multiple ovules per flower permits selection for rare, distant fathers. The number of ovules per flower that maximizes paternal diversity increases with the number of trees. We introduce a cost function for ovule production for which six ovules per flower balance these costs with the benefits of diversifying fathers. Using data from two published field studies, 7 of 8 investigated maternal oaks had actual paternal diversity indices (average diversity index of 15.42) that fit the female choice hypothesis (estimated diversity of 14.66) significantly better than assuming a random selection from the six available ovules (estimated diversity of 7.649). A third field study permitted us to compare paternity by distance classes for two maternal trees. Both fit the female choice model better than random ovule selection.


Introduction
Alternatively, by way of bet-hedging, the plant may abort some fraction of each flower's 23 fertilized ovules based on variation in resource availability. However, then, a plant can 24 achieve the same result by aborting flowers rather than ovules within flowers, and such 25 fruit abortion occurs commonly among plant species. 26 However, these hypotheses may only partly explain observed fertilization 27 patterns [17]. It is unlikely that differences in pollen availability explain why 28 self-incompatible perennials have lower ovule fertilization rates than self-incompatible 29 annuals [18]. Other studies have found that seed-set does not necessarily increase with 30 pollen augmentation [19,20]. 31 Craft et al. proposed that pollinating several ovules while maturing just one permits 32 a form of female choice that increases the diversity of fathers among the acorns of a 33 given oak tree [21]. We assume trees benefit from diversifying the number of fathers 34 among their acorns. Due to a higher diversity of fathers, some are overall more fit than 35 others [22,23], or the increased genetic variability of acorns may be a hedge against 36 environmental contingencies [24][25][26][27]. A diversity of fathers reduces sib-sib competition 37 by increasing the likelihood that acorns are half-rather than full-sibs [28,29]. Finally, 38 fertilization by trees from far away may reduce inbreeding [30,31]. 39 Craft et al. hypothesized a two-step lottery taking place in the fertilization process 40 of oaks: In the first step, each of the six ovules becomes fertilized [21]. The likelihood of 41 being the father of an ovule is simply a random selection from the pollen grains arriving 42 at the flower. A father providing twice as much pollen to the flower as another has 43 twice the likelihood of pollinating the ovule. Each ovule becomes pollinated in this 44 manner, independent of the other ovules in the flower. In the second step of the lottery, 45 the flower selects the ovule to become the acorn that has no paternal sibs within the 46 flower. If it is not a unique ovule, a random draw from the ovules without paternal sibs 47 selects the surviving ovule. If all ovules have paternal sibs, the surviving ovule is randomly drawn from the ovules with the least number of paternal sibs within the 49 flower. In the remainder of this article, this second step of the lottery is referred to as 50 the female choice. This two-step weighted lottery system increases the odds that more 51 distant oak trees sire acorns. While in the first step, nearby oak trees are more likely to 52 fertilize ovules, their pollen becomes selected against during the second-step due to the 53 presence of more ovules fertilized by these nearby trees. While a simple model of 54 randomly selecting one of the fertilized ovules could not explain the distribution of an 55 oak tree's fathers, the female choice hypothesis can explain empirical data. 56 Here, we extend the spatially implicit model of Craft et al. in two novel directions. 57 First, we consider any number of oak trees with a spatially explicit distribution. This 58 becomes essential for comparing model output to actual data. Second, Craft et al. only 59 considered a single focal tree as the mother. Oak trees are non-selfing 60 hermaphrodites [1,2,32,33]. Thus, in our model, we let all trees contribute both pollen 61 and ovules. Our model tracks the diversity of fathers for each oak tree within a stand, 62 as well as the diversity of maternal oak trees among the acorns fathered by each oak 63 tree's pollen. We validate the model predictions to data from several published data 64 sets that explicitly map the coordinates of trees and their pollen donors. 65 In the absence of other hypotheses for why oaks produce six ovules and abort five of 66 them, we compare the female choice mechanism to randomly choosing one of the six 67 ovules to mature. 68 In what follows, we start with a description of our continuous-space model and 69 present simulation results for hypothetical scenarios supporting the understanding of 70 the mechanism. We then compare our model predictions with data from three different 71 published studies [2,34,35]  If four ovules are fertilized by father 1, one ovule by father 2 and one by father 3, then 88 either father 2 or 3 will sire the acorn with equal probability. If each father fertilizes the 89 same number of ovules then each has the same probability of siring the acorn. The 90 nearest father has the highest probability of winning step one. However, this father is 91 more likely to lose fertilized ovules during the second step. The opposite holds for more 92 distant fathers as they are favored by the second step.

93
In contrast to the model by Craft et al., in our model, the probability that tree i 94 pollinates an ovule of tree j explicitly takes into account the position of i and j and is 95 given as: where d ij is the distance between trees i and j. Oaks in our model can only fertilize 97 other trees, i.e., we do not consider selfing [1,2,32,33]. All trees function as maternal 98 trees and pollen donors to other trees. We assume that pollen flow follows an inverse 99 square law with distance between any two trees. This assumption approximates the 100 leptokurtic distribution of pollen flow from a tree that has been measured empirically 101 for wind-dispersed pollen [36][37][38]. For our spatial model, we consider both a torus field 102 with periodic boundary conditions (no edges) and a square field with discrete 103 boundaries. In the remainder of the manuscript, we will refer to these two fields as torus 104 and square, respectively. On a torus, pollen leaving the field on the left/bottom side 105 enter the field again on the right/top side. No tree experiences any boundary effects.

106
The square field scenario imagines woodlots with distinct boundaries. Trees near the 107 boundaries will, on average, be farther away from other trees than trees near the middle. 108 We use a version of Simpson's diversity index (SDI) to measure the consequence of 109 the two step lottery for diversifying fathers among an oak's acorns [39]:  selection, the probabilities q i to be father of the acorn will conform to the probabilities 123 p i to fertilize an ovule in the flower, because the expected number of ovules fertilized by 124 father i is ap i with a being the total numbers of ovules in the flower and 125 q i = api k ap k = p i . And thus it holds that p i = q i . With female choice, the father nearest 126 to the maternal tree still has the highest probability to win the first lottery step. 127 However, this father is more likely to lose fertilized ovules at the second lottery step, 128 and p i > q i for trees close to the focal tree. The situation is reversed for the farthest 129 father. He fertilizes the fewest ovules in the first lottery and sees the fewest discarded  six ovules per flower. In the area above the red dotted horizontal line, it holds that q 1 , 149 the probability for father 1 to win the second lottery step, is higher than 0.5. While the 150 red dashed line is showing the relationship between p 1 and q 1 for the random selection 151 mechanism, the blue curve shows the relationship for the female choice model.

152
Considering a random ovule selection (red dashed line), p 1 = q 1 . The blue curve, 153 resulting from calculations that can be found in Appendix A, shows the non-linearity in 154 the relationship between p 1 and q 1 for the female choice mechanism. From Fig 1 we see 155 that there are two possibilities for father 1 to have a larger chance than father 2 to win 156 the second lottery (q 1 > 0.5): either p 1 ∈ (0.1, 0.5) or p 1 > 0.9. The first range shows 157 that the rare father indeed has a higher chance of siring a flower's acorn. However, the 158 rare father needs to fertilize at least one ovule in that flower. Thus, it needs to hold 159 that p 1 > 0.1, because otherwise, father 2 as the common father has a high chance to 160 fertilize all ovules within a flower. For p 1 > 0.9, it is very likely that father 1 fertilizes 161 all ovules within that flower and thus also wins the second lottery step. Fig 1   162 furthermore shows that the female choice favors rare fathers as the blue curve lies above 163 the red curve for p 1 < 0.5. results are similar whether we assume a torus or a square field. The boundary-less torus 207 reaches a slightly higher maximum SDI, and this maximum is reached with slightly 208 fewer ovules (Fig 2). As would be expected, the variance of the SDI is smaller for the 209 torus than for a square field.   Values are averaged over 20 simulation runs and over all trees in the stand. A stand simulated on a square field achieves a lower SDI than the same stand simulated on a torus field. As expected, the SDI on both fields is highest for 20 fathers. However, the SDI follows a similar curve having only one local maximum for 5, 10 and 20 fathers.
experiences such border effects, and thus, the SDI is higher for a torus. Furthermore, 230 the variance in the optimal number of ovules per run of the simulation is higher for the 231 square than for the torus. for an SDI measure which takes these costs into account, SDI costs : Taking into account these costs for ovule production, the optimal number of ovules 241 increases with the number of trees in the stand and converges to about five to six ovules 242 per flower (see Figs. 4C and 4D).
We consider all combinations of numbers of possible fathers in the stand {1, . . . , 40}, on 248 a square or torus field, and four different numbers of ovules per flower (1, 6, or the 249 optimal number for that number of fathers). We ran 20 simulations of each combination 250 of different numbers of fathers to obtain the mean and variance of SDI r (Fig 5). Each 251 simulation re-randomizes the position of trees in the stand.

252
Regardless of the number of ovules, SDI r will always start at 100% when there is for just one ovule. In all cases, the torus generates a slightly higher SDI r than the 261 square stand of trees. Given that ovules must come at a cost, we again see six ovules as 262 greatly improving a tree's diversity of fathers. In a square field, there may be differences between trees near the boundary and those 265 more to the interior of the stand. To examine these boundary effects, we simulated a 266 stand of 1, 000 trees. We evaluated each tree's acorns SDI of fathers under the 267 assumption of one and six ovules per flower, respectively. We then plotted each tree's 268 SDI versus its shortest distance to a boundary (Fig 6)  10% of trees was 2 and 11.5, respectively, for 1 and 6 ovules per flower. 274 We used a χ 2 -test on the counts in each of the four quadrants formed by the vertical 275 (distance to boundary) and horizontal (SDI) median lines (Fig 6), to determine whether 276 there is a positive or negative association between distance to the boundary and SDI.

277
Both one ovule (χ 2 = 16.0, p < 0.01) and six ovules (χ 2 = 16.2, p < 0.001) produce a 278 positive association between distance to the boundary and SDI. This suggests that the 279 effective dispersion of nearest neighbors away from the focal tree is higher for those 280 close to the boundary than those on the interior.    Fig 11) [2]. The majority of the pollen donors were from outside the 293 stand. In these cases, the pollination distance is unknown. Thus, the data used here 294 include the within stand pollination only. From their spatial data, we extracted for each 295 October 22, 2021 11/29 maternal tree how many of the surrounding trees fertilized a particular number of 296 acorns (see Table 1). In this we did not consider a putative father because its location was nearly identical to 303 maternal tree 33W. As it furthermore did not sire acorns, we concluded that it was than for the random selection. For these two trees, the actual SDI is beyond any of the 314 30, 000 values generated by random selection (Fig 7). at different distance intervals (see column "Offspring data from field studies" in 322 Table 2). Therefore, we used the middle distance of each interval for our computations 323 (see column "Number offspring used as input for SDI calculations" in Table 2). The tree locations of the field study and using the same number of flowers per tree as acorns 328 empirically investigated (see Table 2, row "Number Acorns"). For all cases, the actual 329 SDI conforms more closely to female choice (six ovules) than random ovule selection 330 (one ovule). In two cases, the actual SDI is close to the simulated mean for female outside any of the simulated values when there is random selection (Fig 8).
334 Table 2. Extracted values from spatial data from a field study conducted by Pluess et al. [35] Offspring data from field studies of the focal tree.

343
The distribution of potential fathers (blue bars in Fig 9) for maternal tree E 344 increases with distance. This is to be expected for an evenly or randomly distributed 345 stand of trees, since the area covered by a distance ring will increase with distance (see 346 Appendix B, Fig 13). In contrast to that expectation, we see a skewed distribution of  multiple ovules within a flower [40]. When assuming six ovules per flower, we found -as 395 a showcase -that there exist costs for which the benefits of diversifying fathers and 396 costs are balanced.

397
In our case studies, we assumed that all six ovules in a flower are fertilized. However, 398 there are some studies claiming that not all ovules are fertilized [5,8]. Thus, in Thus, the real SDI in these studies is most likely -as well as our modeled SDI -higher 413 than the SDI that we used to validate our model. In the studies of Dow and Ashley,414 more than half of the pollen donors come from outside the stand [1]. From the studies of 415 Pluess et al., we used paternal data from at most 250 metres from the maternal tree [35]. 416 In Streiff et al., more than 60% of the pollination was from other than the investigated 417 oaks. In total, this makes the difference of the empirically measured SDI to the modeled 418 SDI obtained under the assumption of a random ovule selection even larger [34]. change the paternity analysis [43].

440
Our model considered the trees as points on the landscape. Actual trees occupy 441 space with varied and expansive canopies. Small or less robust trees producing less 442 pollen will be less successful in the first step of the lottery but will have more to gain 443 during the second step. Our female choice model adds another source of diminishing 444 returns to a tree from producing more pollen.

445
Proximity to different sides of a neighbor's canopy should matter. One would expect 446 the pollen of a southward neighbor to contribute disproportionately to the south rather 447 than the north side of a focal tree [44]. Data on this is sparse, and data for paternity in 448 oaks has not yet been localized to position on the maternal tree. half-sibs within the fertilized ovules of an oak. We proposed that fertilization is random 480 with respect to pollen accrual to the flower. Yet, other mechanisms exist for preventing 481 successful fertilization of ovules based on mating types or histone compatibility.

482
Self-incompatibility systems genetically prevent fertilization from the plant itself or 483 closely related plants [50][51][52]. These systems can further be distinguished between 484 gametophytic and sporophytic systems [53]. Moreover, pre-fertilization barriers might post-pollination mechanisms in the species Quercus suber [57]. Their study examined 493 pollen-pistil interactions in order to gain insights into intra-and interspecific crosses.

494
Oaks are not alone in having multiple ovules per flower while maturing just one. In 495 species of Symphoricarpos and Cornus only one out of multiple ovules mature [58]. 496 Moreover, in Erodium cieutarium, only one ovule per schizocarp develops into a seed [58]. 497 We can observe a similar behavior in the species Pongamia pinnata, that matures only 498 one of the two seeds in most of its pods [59]. Although, with just two seeds, some sort 499 of between flower selection would be required to improve paternal diversity.