FlowerMate: Multidimensional reciprocity and inaccuracy indices for style-polymorphic plant populations

Received: 12 December 2023 | Accepted: 7 June 2024 DOI: 10.1002/aps3.11618 SOFTWARE NOTE FlowerMate: Multidimensional reciprocity and inaccuracy indices for style‐polymorphic plant populations Violeta Simón‐Porcar 1,2 |A. Jesús Muñoz‐Pajares 3,4 |Juan Arroyo 1 | Steven D. Johnson 2 1 Department of Plant Biology and Ecology, University of Seville, Seville E‐41080, Spain 2 Centre for Functional Biodiversity, School of Life Sciences, University of KwaZulu‐Natal, Pietermaritzburg 3209, South Africa 3 Department of Genetics, University of Granada, Granada E‐18071, Spain 4 Research Unit Modeling Nature, University of Granada, Granada E‐18071, Spain Correspondence Violeta Simón‐Porcar, Department of Plant Biology and Ecology, University of Seville, E‐41080 Seville, Spain. Email: [email protected] Abstract Premise: Heterostyly in plants promotes pollen transfer between floral morphs, because female and male sex organs are located at roughly reciprocal heights within the flowers of each morph. Reciprocity indices, which assess the onedimensional variation in the height of sex organs, are used to define the phenotypic structure of heterostyly in plant populations and to make inferences about selection. Other reciprocal stylar polymorphisms (e.g., enantiostyly) may function in a similar manner to heterostyly. In‐depth assessment of their potential fitwith pollinators requires accounting for the multidimensional variation in the location of sex organs. Methods and Results: We have adapted the existing reciprocity indices used for heterostylous plant populations to incorporate multidimensional data. We illustrate the computation of the adapted and original indices in the freely available R package FlowerMate. Conclusions: FlowerMate provides fast computation of reliable indices to facilitate understanding of the evolution and function of the full diversity of reciprocal polymorphisms. KEYWORDS FlowerMate, heterostyly, inaccuracy, reciprocal polymorphism, reciprocity, stylar polymorphism Assessing phenotypic variation in natural populations is a fundamental component of evolutionary ecology (Fox et al., 2001), and the development of biological metrics that appropriately represent such variation is thus crucial. Although fitness measures are key for testing natural selection, reduced phenotypic variation has also been regarded as an indication of the strength of longterm directional or stabilizing selection (Fisher, 1958; Lande and Arnold, 1983;Fenster,1991; Cresswell, 1998). Heterostylous plants are a classical model system to test evolution by natural selection (Darwin, 1877; Barrett, 2019). Typical heterostylous plants present polymorphic populations with two (distyly) or three (tristyly) floral morphs bearing stigmas and anthers at different and roughly reciprocal heights. The polymorphism may function for efficient transfer of pollen between morphs on different parts of a pollinator's body (Darwin, 1877; e.g. Massinga et al., 2005), and this may depend on the exact reciprocity in the height of sex organs (Ferrero et al., 2011a;BrysandJacquemyn,2020;Simón‐Porcar et al., 2022). In consequence, morph ratios may be directly related to reciprocity, particularly in heterostylous plants without heteromorphic incompatibility (Thompson et al., 2012). Reciprocity indices have been a common approach for defining the phenotypic structure of heterostyly in plant populations (e.g., Richards and Koptur, 1993;Eckertand Barrett, 1994;Faife‐Cabrera et al., 2014;Oliveira et al., 2022) and for analyzing their correlates with plant fitness to test natural selection (e.g., Faife‐Cabrera et al., 2018; Jacquemyn et al., 2018; Ganguly and Barua, 2021; Ganguly et al., 2021). Two indices that have been used Appl. Plant Sci. 2024;12:e11618. wileyonlinelibrary.com/journal/AppsPlantSci | 1of10 https://doi.org/10.1002/aps3.11618 This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made. © 2024 The Author(s). Applications in Plant Sciences published by Wiley Periodicals LLC on behalf of Botanical Society of America. extensively in the past decade to assess the extent of reciprocity in populations are the reciprocity index of Sánchez et al. (2008,2013) and the inaccuracy index of Armbruster et al. (2017). Calculating these indices has involved manual or spreadsheet computations, which may be prone to error and time consuming when handling large databases. Reciprocityindicescanbeextendedtoinclude three‐dimensional forms of heterostyly (Armbruster et al., 2006;Turkettietal.,2012;Rechetal.,2020)and other stylar polymorphisms such as enantiostyly (Barrett, 2002), flexistyly (Li et al., 2001), inversostyly (Pauw, 2005), or resupinate dimorphy (Harley et al., 2017). These polymorphisms, with a relatively constant fitofpollinators seeking nectar, can result in disassortative pollen transfer in a similar manner to heterostyly (e.g., Minnaar and Anderson, 2021;Johnsonetal.,2023)andarethus appropriate for the application of reciprocity indices (e.g., Braga et al., 2022). In these cases, reciprocity indices must account for the multidimensional variation in the location of sex organs, as this ultimately determines their contact with pollinator bodies. We adapted the reciprocity (Sánchez et al., 2008,2013) and inaccuracy (Armbruster et al., 2017) indices for multidimensional data. Both the adapted and the original indices are implemented in the R package FlowerMate (Muñoz‐Pajares and Simón‐Porcar, 2024), which also includes various automated functions for data handling in a flexible environment that accommodates virtually any type of stylar polymorphism. METHODS AND RESULTS Adaptation of the reciprocity index for heterostylous populations to multidimensional data Below, we replicate and complete the mathematical formulation for the reciprocity index for distylous plant populations of Sánchez et al. (2008,2013)and show the modifications applied to incorporate multidimensional data. Symbols and all abbreviations used in the following equations are summarized and described in Table 1. First, Sánchez et al. (2008) averaged the heights of all sex organs (stigmas and anthers) in the population. To avoid over‐representation of anthers in comparison with stigmas (of which there is frequently only one), this value was calculated as the average of the mean number of anthers and the mean number of stigmas.               XEe np Ss mq =1 2 ∑∑( + ) ++∑∑( + ) + i n k pik j mv qjv (1) Second, the authors averaged the height gaps (stigma–anther distances) of all possible crosses (i.e., individual mating pairs) in the population at each height level (where L represents the upper level and S the lower level), using the X value to standardize the stigma–anther distances across populations, to obtain the mean relative reciprocities at each level: TABLE 1 List of abbreviations used in the mathematical formulations. Abbreviation Definition E Stamen height at level L (long), i.e., in flowers of morph‐ A plants S Stigma height at level L, i.e., in flowers of morph‐Bplants e Stamen height at level S (short), i.e., in flowers of morph‐ A plants s Stigma height at level S, i.e., in flowers of morph‐Bplants n Total number of all single stamens (i) at level L m Total number of all single stigmas (j) at level L p Total number of all single stamens (l) at level S q Total number of all single stigmas (k) at level S ra Mean relative reciprocity at level a (Sánchez et al., 2008)  X Weighted average of heights of all organs (stigmas and stamens) in the population R Reciprocity index, following Sánchez et al. (2013) r Overall reciprocity of the population r LRelative reciprocity at level L r SRelative reciprocity at level S s d r Overall standard deviation of the height gaps in the population s dr a Standard deviation of the height gaps at level a s dr L Standard deviation of height gaps at level L s dr SStandard deviation of height gaps at level S r a D 3 Mean relative 3D reciprocity at level a  X D3 Average of 3D heights of all organs (stigmas and stamens) in the population  A Average height of anthers at upper level  S Average height of stigmas at upper level  a Average height of anthers at lower level  s Average height of stigmas at lower level V A Variance in the height of anthers at upper level VS Variance in the height of stigmas at upper level Va Variance in the height of anthers at lower level Vs Variance in the height of stigmas at lower level MST I 2Mean 2 ‐standardized total inaccuracy P AD O Population average distance to the origin T Total number of sex organs measured 2of10 | FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License                               ∑∑ r nm ES X =1|−| L i n j mij (2a)                           ∑∑ r pq es X =1|−| S k p v qkv (2b) and the overall reciprocity in a population as the Euclidean distance from zero to the population's location in a bivariate space defined by the two relative reciprocity indices: r rr=()+() LS 22 (3) Third, they calculated the average standard deviation of height gaps among sex organs at each level:                                                                                     ∑∑ s dr nm ES Xr=1|−| − L i n j mij L 2 (4a)                                                                                 ∑∑ s dr pq es Xr=1|−| − S k p v qkv S 2 (4b) and averaged them to obtain the overall standard deviation: s dr sdr sdr=1 2(+ ) LS (5) Finally, the reciprocity index, following the modification of Sánchez et al. (2013) was calculated as: Rrsdr=1−(10 × ) (6) Requals one in situations of perfect reciprocity and decreases as variance and mismatches between corresponding sex organs increase. Remarkably, the index can potentially take on negative values, although Sánchez et al. (2013) did not discuss this. The basis of the adaptation of the reciprocity index to multidimensional data was substituting sex organ height (i.e., distances along the y‐axis) with distances in a virtual three‐dimensional (3D) space; i.e., xx yy zz( − )+( − )+( − ) 21 221 221 2(7) Hence, the heights of sex organs were substituted with their distances to the origin in a 3D space (e.g., for a stamen at level L): xyz()+()+() EEE 222 iii (8) and the distance for a stigma–stamen pair was calculated (e.g., at level L) as: () () () xx yy zz−+−+− ES ES ES 222 ijijij(9) Although we only provide formulation for three dimensions, setting coordinates to zero in one or two dimensions would lead to the corresponding index in two‐dimensional or one‐dimensional space (the original index of Sánchez et al., 2008,2013), respectively. For the sake of clarity, we maintain the “L”and “S”nomenclature of sex organ levels of distylous species, although as explained above these indices may apply to other types of stylar morphs. To adapt the reciprocity index, we first substituted sex organ height in Eq. 1with their distances to the origin (Eq. 8).                                                                 ( ) ( ) X xyz xyz np xyz xyz mq ¯=1 2 ∑∑ ( )+( )+( ) +()+()+() + + ∑∑ ( )+( )+( ) +()+()+() + D i n k pEEE eee j mv qSSS sss 3 222 222 222 222 iii kkk jjj vvv (10) Second, we computed the mean relative reciprocity at each level, following Eq. 2a,2b, and 9:                                                               () () () ∑∑ rnm xx yy zz X =1−+−+− ¯ LD i n j mES ES ES D 3 222 3 ijijij (11a)                                                           ()()() ∑∑ r pq xx yy zz X =1−+−+− ¯ SD k p v qes es es D 3 22 2 3 kvkvkv (11b) and the overall 3D reciprocity in the population, following Eq. 3: r rr=( )+( ) DLDSD33 232(12) Third, we computed the average standard deviation of 3D distances among sex organs at each level, following Eq. 4a,4b, and 9: FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES | 3of10 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License                                                                                                                                                                 () () () ∑∑ sdr nm xx yy zz Xr=1−+−+− − LD i n j mES ES ES D LD 3 222 3 3 2 ijijij (13a)                                                                                                                                                 ()()() ∑∑ sdr pq xx yy zz Xr=1−+−+− − SD k p v qes es es D SD 3 22 2 3 3 2 kvkvkv (13b) and the overall average standard deviation following Eq. 5: s dr sdr sdr=1 2(+ ) DLDSD333 (14) Finally, the 3D reciprocity index, following Eq. 6, was defined as: Rrsdr=1−(10 × ) DDD333 (15) Sánchez et al. (2013) applied their reciprocity index to tristylous plant populations by computing r L ,r S , and r M (for the mid‐level sex organs) as in Eq. 2, and by computing the overall reciprocity in a population as the mean value of the Euclidean distances from zero to each value of relative reciprocity in each one of the three two‐dimensional spaces defined by the three levels: r rr rr r r=1 3(()+() + ()+( ) + ( )+() ) LS LM MS 22 2 2 22 (16) Similarly, they computed sdr L ,sdr S , and sdr M as in Eq. 4, and the overall standard deviation as their average value: s dr sdr sdr sdr=1 3(+ + ) LSM (17) Their final reciprocity index for tristylous populations was calculated as in Eq. 6, using the values for rand sdr provided in Eq. 16 and 17. We applied the same modifications in Eq. 11 to 15 to compute a multidimensional reciprocity index for trimorphic populations (notably, so far only reported in heterostylous systems and not in other reciprocal stylar polymorphisms). Computation of the inaccuracy index for multidimensional data Adaptive inaccuracy is a comprehensive conceptual framework for estimating the fitness consequences of population or individual plant‐level deviations from morphological fitness optima (Armbruster et al., 2009a,2009b; Pélabon et al., 2012). We acknowledge that this framework is beyond the concept of “index,”as applied here for the sake of simplicity. Armbruster et al. (2017)defined the inaccuracy for heterostylous flowers in the context of the adaptive inaccuracy of a morph on a fitness scale. Below, we replicate and complete the mathematical formulation for the inaccuracy index of Armbruster et al. (2017) and show the modifications applied to incorporate multidimensional data. Symbols and all abbreviations used in the following equations are summarized and described in Table 1. Armbruster et al. (2017)defined inaccuracy of heterostylous populations giving the same weight to both the maladaptive bias (mean departure from optimum) and imprecision (within‐population variance) of the phenotypic values in the population. Accordingly, they defined the following parameters: AS V V I naccuracy = ( − ) + + AS high organs 2(18a) as VV I naccuracy = ( − ) + + a s low organs 2(18b) Total Inaccuracy = Inaccuracy + Inaccuracy high organs low organ s (19) ^Mean 2 − Standardized Total Inaccuracy (M STI) = Total Inaccuracy Population average organ height 2 2 (20) Inaccuracy at various dimensions is calculated as a linear deviation from the optimum in a two‐or threedimensional Euclidean space, making measurements comparable to one‐dimensional inaccuracies. Hence, we adapted the inaccuracy index to multidimensional data by computing the Euclidean distance between average xyz coordinates of stigmas and stamens at each level in Eq. 18a and 18b: xx yy zz V V 3Dinaccuracy = ( − )+( − )+( − )+ + AS AS A S DA DS high organs 222 33 (21a) xx yy zz V V 3Dinaccuracy = ( − )+( − )+( − )+ + as as as Da Ds low organs 222 33 (21b) In these equations, the variance of each group of sex organs is defined from their Euclidean distances in the 3D space from their centroid (i.e., the mean of variances 4of10 | FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License for each dimension; for example, for stamens at level L, where Nequals the number of individuals with stamens at level L):             ∑ V Nxx yy zz=1( − )+( − )+( − ) DA i N iA iA iA3 =1 222 (22) Finally, total inaccuracy and mean 2 ‐standardized total inaccuracy in a 3D space are calculated as: 3Dtotal Inaccuracy = 3Dinaccuracy + 3Dinaccuracy high organs low organ s (23) 3 DM STI = 3Dtotal Inaccuracy PADO 2 2 (24) where PADO is the population average distance to the origin of all pooled sex organs, calculated as:             ∑ P ADO Txyz=1++ i T iii =1 22 2 (25) where T e quals the total number of sex organs measured . In trimorphic populations, inaccuracy is also calculated for mid‐level sex organs, and this contributes to total inaccuracy. 3 Dtotal Inaccuracy3M = 3Dinaccuracy + 3Dinaccuracy + 3Dinaccuracy high organs mid organs low organ s (26) 3 DM STI3M = 3Dtotal Inaccuracy3M PADO 2 2(27) Reciprocity vs. inaccuracy indices Although both indices are highly correlated for distylous populations, the inaccuracy index of Armbruster et al. (2017) has to some extent replaced the reciprocity index of Sánchez et al. (2008,2013) in recent years, based on two major criticisms. As Armbruster et al. (2017) depicted, (i) Rincorporates measures of variation twice (both in rand in sdr), and (ii) the arbitrary multiplication of rby sdr (Eq. 6) implies that “even a large deviation from perfect reciprocity will have almost no effect on the total reciprocity if the standard deviation is close to zero.”Furthermore, in contrast to variances, standard deviations are not additive and averaging them sensu Eq. 4and 15 is problematic. We note, however, that Sánchez's r(sensu Eq. 3for dimorphic populations and Eq. 16 for trimorphic) overcomes both shortcomings of Sánchez's Rand is comparable across dimorphic and trimorphic populations. Using Sánchez's rmay also be appropriate in cases of polymorphic plants bearing intrafloral variation in the location of sex organs (e.g., Ferrero et al., 2011b; Turketti et al., 2012). Moreover, the multidimensional rindex can be fairly applied to stylar polymorphisms including spatial and temporal dimensions (e.g., flexistyly; Li et al., 2001). Due to its additive nature, inaccuracy is not comparable across dimorphic and trimorphic populations and cannot directly apply to plants with intra‐individual variation, but on the other hand, it allows the components of inaccuracy to be explored in depth. The R package FlowerMate provides tools for the most appropriate approach in each particular case, which will depend on the study system and objectives. The FlowerMate R package The R package FlowerMate v1.0 (Muñoz‐Pajares and Simón‐Porcar, 2024) is freely available in CRAN (see Data Availability Statement) and includes a function inaccuracy that computes either the one‐, two‐, or three‐dimensional versions of the reciprocity and inaccuracy indices, either for dimorphic or trimorphic populations, depending on the data input and options chosen. The input format includes one row per measured sex organ and the following eight columns: population code, floral morph (L, S, M, which may also be used for nonheterostylous morphs), individual ID number, sex organ ID number, sex organ type (stigma or anther), and x,y,zcoordinates. The user can select the coordinates to compute, thereby obtaining either uni‐,bi‐,orthree‐dimensional indices. In multidimensional systems, this option also allows comparisons of reciprocity between various dimensions, which might provide insights into the differential strength of phenotypic selection. Some stylar polymorphic species present the same sex organs at strikingly different locations (e.g., unusual style‐length polymorphic species with two stamen whorlsinsteadofone,suchasinNarcissus [Arroyo et al., 2002], or enantiostylous flowers with both left‐and right‐facing stamens [Johnson et al., 2023]), and for such cases inaccuracy offers the possibility to select different data subsets for computation. For polymorphisms that are not associated with heteromorphic incompatibility systems, it may also be interesting to test reciprocity within morphs to assess the potential contribution of intra‐morph cross‐mating (Johnson et al., 2023), and inaccuracy also offersthispossibility.Itmay also be the case that missing data existed for some sex organs for at least one dimension. In such case, inaccuracy calculates both indices (in contrast to the Microsoft Excel macro of Sánchez et al., 2008,2013), warning that the values may not be comparable across populations (e.g., if the distribution of missing data is strongly biased across morphs or dimensions). The output of inaccuracy consists of a single table including the indices and their requested components for each population. Although we encourage the use of Sánchez's r(sensu Eq. 3,12,and16), FlowerMate also computes Sánchez's R(sensu Eq. 6and 15), as it may be potentially FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES | 5of10 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License useful for researchers, for instance, for those interested in comparing their values to those published in previous studies. FlowerMate also includes functions that simulate stylepolymorphic populations with the average coordinates and variance specified for each sex organ level and morph. These functions are SimDimor and SimTrimor for simulating dimorphic and trimorphic data, respectively. Testing inaccuracy for one‐dimensional data First, we tested inaccuracy by replicating the onedimensional reciprocity index retrieved with the Microsoft Excel macro of Sánchez et al. (2013) in two sets of 20 simulated populations, dimorphic and trimorphic, respectively. We used the functions SimDimor and SimTrimor in the FlowerMate package to simulate 20 dimorphic and 20 trimorphic populations with variable height mean differences between sex organs at each level and standard deviations. There was a perfect match between the reciprocity values retrieved with both approaches for both sets of populations, except for some very slight deviations that were attributed to the handling of decimal values (Appendices S1 and S2; see Supporting Information with this article). The computation of FlowerMate was notably faster than that of the Excel macro. From the values retrieved by FlowerMate, Sánchez's r(sensu Eq. 3; Sánchez et al., 2008) showed a stronger correlation than Sánchez's R(sensu Eq. 6; Sánchez et al., 2013) with both total inaccuracy and mean 2 ‐standardized total inaccuracy (sensu Eq. 19 and 20; Armbruster et al., 2017) in dimorphic and trimorphic populations (Pearson's r> 0.53, P< 0.05; Table 2). Second, we tested inaccuracy using the large database published by Jacquemyn et al. (2018), who calculated the inaccuracy index for 40 populations from nine distylous species of Pulmonaria (we excluded one duplicated population from their database [i.e., Crailsheim]). The value for all the components of the inaccuracy index retrieved with our code perfectly matched those published for most populations (Appendix S3). We attribute the mismatches found in a few populations to the improved handling of large amounts of data using our automated approach. For the values retrieved by FlowerMate, both Sánchez's Rand rwere significantly correlated with total inaccuracy and mean 2 ‐standardized total inaccuracy (r> 0.85, P< 0.001; Table 2). Testing inaccuracy for multidimensional data First, we simulated 12 sets of five dimorphic populations that embodied different scenarios of increasing phenotypic variation in two or three dimensions. These population sets increased mismatch and/or variance in one or more dimensions while keeping the other components constant (Table 3,AppendicesS4 and S5). Reciprocity and inaccuracy indices increased in all cases as expected (Figure 1). Second, we tested inaccuracy using morphological data from 10 natural populations of Linum suffruticosum L. (Appendix S6), the first reported example of three‐dimensional TABLE 2 Pearson's rcorrelation coefficients between the different components of the reciprocity index of Sánchez et al. (2008,2013; Sánchez's rand Sánchez's R) and the inaccuracy index of Armbruster et al. (2017; total inaccuracy and mean 2 ‐standardized inaccuracy) for dimorphic and trimophic populations simulated with FlowerMate and in Pulmonaria populations (Jacquemyn et al., 2018). Simulated dimorphic populations Simulated trimorphic populations Pulmonaria populations Sánchez's rSánchez's RSánchez's rSánchez's RSánchez's rSánchez's R Total inaccuracy 0.95*** −0.79*** 0.91*** −0.86*** 0.88*** −0.85*** Mean 2 ‐STI 0.95** −0.62** 0.74*** −0.53* 0.95*** −0.99** Note: STI = standardized total inaccuracy. Significance: ***P< 0.001; **P< 0.01; *P< 0.05. TABLE 3 Summary of dimorphic and trimorphic population sets representing different scenarios of increased phenotypic variation simulated to test the multidimensional variants of the reciprocity (Sánchez et al., 2008,2013) and inaccuracy (Armbruster et al., 2017) indices. Set Description Dimorphic populations Set 2D1 Increasing mismatch at x‐axis, keeping total accuracy at y‐axis Set 2D2 Increasing variance at x‐axis, keeping total accuracy at y‐axis Set 2D3 Increasing mismatch and variance at x‐axis, keeping total accuracy at y‐axis Set 2D4 Increasing mismatch at x‐and y‐axes, with variances equal to 0 Set 2D5 Increasing variance at x‐and y‐axes Set 2D6 Increasing mismatch and variance at x‐and y‐axes Set 2D7 Increasing mismatch at x‐and y‐axes, with variances equal to 0.4 Set 2D8 Increasing mismatch and variance at x‐and y‐axes only at upper level Set 2D9 Increasing mismatch and variance at x‐and y‐axes, x coordinates = ycoordinates/2 Trimorphic populations Set 3D1 Increasing mismatch and variance at x‐axis, keeping total accuracy at y‐and z‐axes Set 3D2 Increasing mismatch and variance at x‐and y‐axes, keeping total accuracy at z‐axis Set 3D3 Increasing mismatch and variance at x‐,y‐, and z‐axes 6of10 | FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License heterostyly (Armbruster et al., 2006). Notably, although the locations of sex organs in this species occupy a threedimensional space, the radial symmetry effectively entails twodimensional functioning based on vertical and horizontal herkogamy, as pollinators visiting the flowers from any direction would contact sex organs in the same way. Previous authors (e.g., Armbruster et al., 2006;Pérez‐Barrales and Armbruster, 2023) have analyzed reciprocity in this species by comparing the length of styles and filaments, but given their bending, such measurements do not correspond to the actual positioning of sex organs and may have little functional significance. Hence, xy coordinates are adequate to define the functional position of sex organs in this species and in other similar examples, and these can be easily obtained through measuring the height (ycoordinate) and the diameter (2xcoordinate) of each sex whorl in fresh flowers. These measures can be obtained from scaled photos (Figure 2A)or, more easily, by using calipers. We used data for Linum suffruticosum to compare the reciprocity and inaccuracy indices retrieved with the xy coordinates and either the yor x coordinate. For the populations analyzed, we found a much higher variation along the y‐axis than along the x‐axis, which translated into a higher correlation between the values computed for xy and ydata (r> 0.94, P<0.001)thanforxy and x data (r< 0.51, P> 0.1; Appendix S6). These results indicate that in this study system the height of sexual organs is a good predictor of two‐dimensional reciprocity, which may not be thecaseinallstudysystems. Finally, we tested inaccuracy with three‐dimensional data using morphological data from one natural population FIGURE 1 Components of the inaccuracy and reciprocity indices calculated with FlowerMate for 12 series of five populations representing different scenarios of increased phenotypic variation in the location of sex organs at two (A) or three (B) dimensions (see Table 3for details). FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES | 7of10 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License of the enantiostylous plant Wachendorfia paniculata L. (Haemodoraceae). Three‐dimensional data are appropriate for zygomorphic flowers, which pollinators always approach from the same direction, and enantiostylous species present a typical case. Enantiostylous flowers in the Haemodoraceae bear one style and one stamen facing one side (either left‐or right‐handed) and two stamens, one longer that the other, facing the opposite side. We used front‐and top‐scaled photographs of freshly opened flowers of W. paniculata to retrieve the xyz coordinates of each sex organ in each morph (Figure 2B, Appendix S7) using the nectaries as the origin. We used this dataset to compare the three‐dimensional reciprocity and inaccuracy indices using each of the three existing anthers (Figure 2B). As expected, the reciprocity was highest for anther 1 (Sánchez's r 3D = 0.37, 3DM 2 STI = 0.173), intermediate foranther2(Sánchez'sr 3D = 0.606, 3DM 2 STI = 0.426), and lowestforanther3(Sánchez'sr 3D = 1.326, 3DM 2 STI = 1.817). Sánchez's r 3D and 3DM 2 STI showed a high correlation across the three calculations (Pearson's r= 0.995, P=0.061). CONCLUSIONS Multidimensional variants expand the range of applications of the existing reciprocity (Sánchez et al., 2008,2013) and inaccuracy (Armbruster et al., 2017)indicesthathave been applied to heterostylous species. We have illustrated their potential application to various kinds of reciprocal stylar polymorphisms, and moreover, we advocate for their potential use in the characterization of other types of reciprocal polymorphisms maintained through disassortative mating (Armbruster et al., 2017). These may also include a temporal dimension (e.g., heterodichogamy; Renner, 2001) and may even be applied to animal populations (e.g., Schilthuizen et al., 2007; Takahashi and Hori, 2008). FlowerMate provides easy and fast computation of unidimensional and multidimensional indices. This new tool is designed to facilitate understanding of the evolution and function of the full diversity of reciprocal polymorphisms. FIGURE 2 Representation of the coordinate calculations for sex organs in two‐dimensional (A; e.g., Linum suffruticosum, short‐styled morph) and three‐dimensional (B; e.g., Wachendorfia paniculata, left‐handed morph) systems. White lines represent the reference system, with the origin established at the level of the nectaries. Stigma coordinates are represented in each system by red lines, and anther coordinates by light green, dark green, and blue lines. 8of10 | FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License AUTHOR CONTRIBUTIONS V.S.‐P. conceived the idea, developed the mathematical formulation, and prepared the first version of the manuscript. A.J.M.‐P. and V.S.‐P. developed and tested the R code. A.J.M.‐P. packaged the R code. S.D.J. collected data on Wachendorfia paniculata. All authors contributed to and approved the final version of the manuscript. ACKNOWLEDGMENTS This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 897890. This work was also supported by funding from the Ministry of Science and Innovation of the Spanish Government (MICIN; PGC2018‐099608‐B‐I00 and PID2021‐122715NB‐I00). We are grateful to Hans Jacquemyn (and collaborators), Letícia Rodrigues‐Novaes, and Bruna Braga for sharing phenotypic data from natural populations of Pulmonaria,Linum suffruticosum, and Senna rugosa, respectively, for code testing, and Scott Armbruster for helpful insights about the computation of adaptive accuracy. DATA AVAILABILITY STATEMENT The data used to test the code are available as Supporting Information or from the source articles. The R package is available on CRAN (https://CRAN.R‐project.org/package= FlowerMate). ORCID Violeta Simón‐Porcar http://orcid.org/0000-0003- 4024-2824 REFERENCES Armbruster,W.S.,R.Pérez‐Barrales, J. Arroyo, M. E. Edwards, and P. Vargas. 2006. Three‐dimensional reciprocity of floral morphs in wild flax (Linum suffruticosum): A new twist on heterostyly. New Phytologist 171: 581–590. Armbruster, W. S., T. F. Hansen, C. Pélabon, R. Pérez‐Barrales, and J. Maad. 2009a. The adaptive accuracy of flowers: Measurement and microevolutionary patterns. 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FLOWERMATE: MULTIDIMENSIONAL RECIPROCITY AND INACCURACY INDICES | 9of10 21680450, 2024, 6, Downloaded from https://bsapubs.onlinelibrary.wiley.com/doi/10.1002/aps3.11618 by Readcube (Labtiva Inc.), Wiley Online Library on [25/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License