Comparison of Seed Images with Geometric Models, an Approach to the Morphology of Silene (Caryophyllaceae)

: Seed morphological description is traditionally based on adjectives, which originated from the comparison with other shapes, including geometric ﬁgures. Nevertheless, descriptions based on this feature are not quantitative and measurements giving the percentage of similarity of seeds with reference ﬁgures are not available in the literature. Lateral views of Silene seeds resemble the cardioid and cardioid-derived ﬁgures. Dorsal views, nonetheless, resemble ellipses and derivatives, allowing seed shape quantiﬁcation by comparison with deﬁned geometric ﬁgures. In this work, we apply already-described models as well as new models to the morphological analysis of 51 Silene species. Our data revealed the existence of a link between lateral and dorsal models. Lateral models closed in the hilum region (models LM2 and LM4) were associated with those convex models of the dorsal seed views (DM1-DM4, DM10). Lateral models more open around the hilum region adjusted to seeds characterized as dorso canaliculata type better, i.e., to those geometric models with partial concavities in their dorsal views. The relationship between lateral and dorsal models, as well as between the models to their utility in taxonomy, is discussed.

As a new morphological approach, we have described the lateral and dorsal views of Silene seeds comparing the seed contour with algebraically defined geometric figures, such as the cardioid, modified cardioid, or diverse ellipses. We provided a J index, a numerical value indicating the percentage of similarity between a defined seed silhouette and a given geometric figure, taken as a model [14][15][16][17][18]. The comparison of lateral seed

Seeds
The seeds used in this work are described in Table A1 (see Appendix A). They belong to populations of 51 Silene species and the species Eudianthe coeli-rosa, which proceed from laboratories and botanical gardens through a program of international cooperation with the seed collection (Carpoespermateca) of the botanical garden at the University of Valencia. They were sent to IRNASA-CSIC in June 2022. The seed images used in the analysis are stored in Zenodo (see Supplementary Materials) and a description of their silhouettes has been given [13].
The plant nomenclature (genera, subgenera, sections, species, and subspecies) and the taxa authorities were adapted according to Plants of the World Online (POWO) [19]. The taxonomical classification in subgenera and sections of the genus Silene follows Jaffari et al. [20].

Photography
Lateral and dorsal views of these seeds used in the morphological analysis were taken with a Nikon Z6 camera equipped with an objective AF-S Micro NIKKOR 60 mm f/2.8G ED.

Image Preparation
Twenty seeds per species were selected both for the lateral and dorsal views and ordered (aligned) in a document (.PSD format of Corel Photo Paint). This document containing the aligned seeds was used to obtain average silhouettes for the lateral and dorsal views of the seeds using the method described [15,21]. A video describing the method used is available (see Supplementary Materials).
The average silhouettes for the lateral and dorsal views (L and D, respectively) of each species were assembled in image archives (.PSD). The white and black images of models were superimposed to the average silhouettes for each species in the assembled .PSD images, and the J index was calculated (see below 2.5).

Models
Some of the lateral and dorsal models used here were previously described in [14,16], as shown in Figure 1 (LM1 to LM8) and Figure 2 (DM1 to DM9). In addition, 4 new dorsal models were obtained for this work based on Fourier transform [18], to get the best adjustments between the seed contour and the J index values (see results section).

Seeds
The seeds used in this work are described in Table A1 (see Appendix A). They belong to populations of 51 Silene species and the species Eudianthe coeli-rosa, which proceed from laboratories and botanical gardens through a program of international cooperation with the seed collection (Carpoespermateca) of the botanical garden at the University of Valen cia. They were sent to IRNASA-CSIC in June 2022. The seed images used in the analysis are stored in Zenodo (see Supplementary Materials) and a description of their silhouettes has been given [13].
The plant nomenclature (genera, subgenera, sections, species, and subspecies) and the taxa authorities were adapted according to Plants of the World Online (POWO) [19] The taxonomical classification in subgenera and sections of the genus Silene follows Jaffar et al. [20].

Photography
Lateral and dorsal views of these seeds used in the morphological analysis were taken with a Nikon Z6 camera equipped with an objective AF-S Micro NIKKOR 60 mm f/2.8G ED.

Image Preparation
Twenty seeds per species were selected both for the lateral and dorsal views and ordered (aligned) in a document (.PSD format of Corel Photo Paint). This document containing the aligned seeds was used to obtain average silhouettes for the lateral and dorsa views of the seeds using the method described [15,21]. A video describing the method used is available (see Supplementary Materials).
The average silhouettes for the lateral and dorsal views (L and D, respectively) o each species were assembled in image archives (.PSD). The white and black images of models were superimposed to the average silhouettes for each species in the assembled .PSD images, and the J index was calculated (see below 2.5).

Models
Some of the lateral and dorsal models used here were previously described in [14,16] as shown in Figure 1 (LM1 to LM8) and Figure 2 (DM1 to DM9). In addition, 4 new dorsa models were obtained for this work based on Fourier transform [18], to get the best ad justments between the seed contour and the J index values (see results section).

Development of New Models
New models DM10 to DM13 were obtained by elliptic Fourier transform as described [18]. Basically, a set of points (between 60 and 100) were taken from the average silhouettes of seeds for a given species or combination of species (average silhouette of average silhouettes), and the resulting Fourier curves were used as models. Models DM11, DM12, and DM13 were specifically designed for S. chlorantha (Willd.) Ehrh, S. linicola C.C. Gmel., and S. damascena Boiss. and Gaill, respectively (see the corresponding average silhouettes in Figure A1 of Appendix A), while DM10 was designed for the seeds of the species S. aprica Turcz. ex Fisch. and C.A. Mey, S. chungtienensis (Speg.) Bocquet, S. dichotoma Ehrh., S. multiflora (Erhr.) Pers, and S. viridiflora L. The Mathematica code for these models has been stored in Zenodo (see Supplementary Materials). More details about these models are given in the results section.

Similarity of the Seed Silhouettes with Models (J Index Calculation)
The similarity of the seed images with models was evaluated as the J index. The J index was calculated both for the average silhouette and for the mean of 20 seeds representative of each species, as: where S is the area shared between the seed and the model, and T is the total area occupied by both images. The J index has a maximum value of 100, corresponding to the cases where the geometric model and the seed image areas coincide. The adjustments are considered good when J index values are superior to 90. The value of S (shared area) is obtained in Image J by the superposition of the seed silhouette with a model in white, and the value of T (total area) is obtained by the superposition of the seed silhouette with a model in black. Representative images with the models in white and in black superimposed to the silhouettes are shown in Figure 3.

Development of New Models
New models DM10 to DM13 were obtained by elliptic Fourier transform as described [18]. Basically, a set of points (between 60 and 100) were taken from the average silhouettes of seeds for a given species or combination of species (average silhouette of average silhouettes), and the resulting Fourier curves were used as models. Models DM11, DM12, and DM13 were specifically designed for S. chlorantha (Willd.) Ehrh, S. linicola C.C. Gmel., and S. damascena Boiss. and Gaill, respectively (see the corresponding average silhouettes in Figure A1 of Appendix A), while DM10 was designed for the seeds of the species S. aprica Turcz. ex Fisch. and C.A. Mey, S. chungtienensis (Speg.) Bocquet, S. dichotoma Ehrh., S. multiflora (Erhr.) Pers, and S. viridiflora L. The Mathematica code for these models has been stored in Zenodo (see Supplementary Materials). More details about these models are given in the results section.

Similarity of the Seed Silhouettes with Models (J Index Calculation)
The similarity of the seed images with models was evaluated as the J index. The J index was calculated both for the average silhouette and for the mean of 20 seeds representative of each species, as: where S is the area shared between the seed and the model, and T is the total area occupied by both images. The J index has a maximum value of 100, corresponding to the cases where the geometric model and the seed image areas coincide. The adjustments are considered good when J index values are superior to 90. The value of S (shared area) is obtained in Image J by the superposition of the seed silhouette with a model in white, and the value of T (total area) is obtained by the superposition of the seed silhouette with a model in black. Representative images with the models in white and in black superimposed to the silhouettes are shown in Figure 3.
The J index was calculated on the average silhouettes for the lateral and dorsal views first, in each species with models LM1 to LM8 and DM1 to DM9 (excluded DM7 because of low similarity), respectively. The results obtained with the average silhouettes give an orientation at which models best adjust to the seeds, so the analysis involving mean values of 20 seeds is undertaken only in the models that give higher scores on the average silhouettes. The results obtained in LM1 (the cardioid) were excluded because it is less discriminant than the other lateral models. Method to obtain the J index in the average silhouette of Eudianthe coeli-rosa. From left to right: superimposed silhouettes of 20 seeds (lateral views); average silhouette; in 2 descending rows, left to right, average silhouette with models superimposed: LM4, LM3, and LM5, in black (above) and white (below). Area quantification with Image J in the compositions above (with the models in black) gives total area (T), while quantification in the compositions below (with the models in white) gives the values of the area shared by the average silhouette and the model (S). The J index is the ratio S/Tx100. The method is identical for individual seeds excluding the first step (obtention of the average silhouette).
The J index was calculated on the average silhouettes for the lateral and dorsal views first, in each species with models LM1 to LM8 and DM1 to DM9 (excluded DM7 because of low similarity), respectively. The results obtained with the average silhouettes give an orientation at which models best adjust to the seeds, so the analysis involving mean values of 20 seeds is undertaken only in the models that give higher scores on the average silhouettes. The results obtained in LM1 (the cardioid) were excluded because it is less discriminant than the other lateral models.

Statistical Analysis
The Euclidean distance and Ward algorithm for clustering were used to calculate the dendrogram. The matrix used for the analysis combined the data for LM2/4 and LM5 (Table A7 in Appendix A). The ten best scores obtained with each model in the species were used. Statistical analysis was done with IBM SPSS statistics SPSS es 29.0.0.0 (241). Figure A1 (Appendix A) shows the average silhouettes for the lateral and dorsal views of seeds of 51 Silene species and Eudianthe coeli-rosa. The average silhouettes of the seeds reveal important aspects of their morphology, such as general symmetry, aspect ratio (equal to ratio length/width), and convexity for both the lateral and dorsal seed views. Average silhouettes in the dorsal views for the majority of seeds are convex. Nevertheless, concavities in the upper and lower seed views are notable in some species (e.g., S. damascena, S. frivaldskyana, S. roemeri, S. ruprechtii, or S. saxatilis), most of them belonging to S. subg. Silene. A detailed study of the morphological measurements and the geometry of these seeds was already presented [13]. Method to obtain the J index in the average silhouette of Eudianthe coeli-rosa. From left to right: superimposed silhouettes of 20 seeds (lateral views); average silhouette; in 2 descending rows, left to right, average silhouette with models superimposed: LM4, LM3, and LM5, in black (above) and white (below). Area quantification with Image J in the compositions above (with the models in black) gives total area (T), while quantification in the compositions below (with the models in white) gives the values of the area shared by the average silhouette and the model (S). The J index is the ratio S/Tx100. The method is identical for individual seeds excluding the first step (obtention of the average silhouette).

Statistical Analysis
The Euclidean distance and Ward algorithm for clustering were used to calculate the dendrogram. The matrix used for the analysis combined the data for LM2/4 and LM5 (Table A7 in Appendix A). The ten best scores obtained with each model in the species were used. Statistical analysis was done with IBM SPSS statistics SPSS es 29.0.0.0 (241). Figure A1 (Appendix A) shows the average silhouettes for the lateral and dorsal views of seeds of 51 Silene species and Eudianthe coeli-rosa. The average silhouettes of the seeds reveal important aspects of their morphology, such as general symmetry, aspect ratio (equal to ratio length/width), and convexity for both the lateral and dorsal seed views. Average silhouettes in the dorsal views for the majority of seeds are convex. Nevertheless, concavities in the upper and lower seed views are notable in some species (e.g., S. damascena, S. frivaldskyana, S. roemeri, S. ruprechtii, or S. saxatilis), most of them belonging to S. subg. Silene. A detailed study of the morphological measurements and the geometry of these seeds was already presented [13].

Lateral Models
The results of J index calculations on average silhouettes for 51 Silene species were grouped into two tables, separating those species giving maximum values of J index in models LM2, LM4, and LM8 (Table A2 in Appendix A, with 43 species), from those giving maximum J index values in model LM5 (Table A3, with 8 species). Models LM6 and LM7 did not give maximum values in any species. The seeds of species associated with model LM5 (Table A3) are characterized by more open lateral views with larger concavities in the hilum zone, while those resembling models LM2, LM4, and LM8 (Table A2) are more closed, tending to be convex (higher solidity values [12]). From the 43 species in Table A2, 19 belong to S. subg. Behenantha, and 24 to S. subg. Silene. All species in Table A3 from model LM5 belong to S. subg. Silene.

Dorsal Models
The results were grouped into three tables, corresponding, respectively, to convex models DM1 to DM4 (Table A4), intermediate models with low degrees of concavity  (models DM5 and DM6; Table A5), and models with notable concavities in the poles (DM8 and DM9; Table A6). Model DM7 did not give good results with any of the species tested in this work. Maximum values of J index with models DM1 to DM4 were obtained in 10 species (Table A4), all of them included in Table A2 of species with similarity to models LM2 and LM4. Out of the nine species in Table A4, six species (66.6%) belong to Silene subg. Behenantha. Maximum values in models DM5 and/or DM6 were obtained in 23 species, of which eight (34.8%) belong to Silene subg. Behenantha. A total of 17 species gave maximum values in other models (DM7 and DM8; Table A6). Of these, five belong to Silene subg. Behenantha (31.6%), four corresponded to Silene sect. Physolychnis, and one to Silene sect. Cucubaloides (S. yunnanensis). In general, the average silhouettes of seeds in Table A6 are more elongated in their dorsal views (higher aspect ratio).

New Models for the Dorsal Views of Silene Species
The new models are shown in Figure 4. DM10 corresponds to a hyperellipse and it is a convex figure. It differs from DM1 in a certain asymmetry between the terminal sides, since the lower one is slightly broader than the upper side (Figures 1 and 4). The model DM11 also resembles DM1, but it is narrower and has small concave regions in the four sides (Figures 1 and 4). DM12 and 13 are figures with partial concavity regions for the upper and lower sides (Figure 4), designed for S. linicola and S. damascena, respectively. did not give maximum values in any species. The seeds of species associated with model LM5 (Table A3) are characterized by more open lateral views with larger concavities in the hilum zone, while those resembling models LM2, LM4, and LM8 (Table A2) are more closed, tending to be convex (higher solidity values [12]). From the 43 species in Table A2, 19 belong to S. subg. Behenantha, and 24 to S. subg. Silene. All species in Table A3 from model LM5 belong to S. subg. Silene.

Dorsal Models
The results were grouped into three tables, corresponding, respectively, to convex models DM1 to DM4 (Table A4), intermediate models with low degrees of concavity  (models DM5 and DM6; Table A5), and models with notable concavities in the poles (DM8 and DM9; Table A6). Model DM7 did not give good results with any of the species tested in this work. Maximum values of J index with models DM1 to DM4 were obtained in 10 species (Table A4), all of them included in Table A2 of species with similarity to models LM2 and LM4. Out of the nine species in Table A4, six species (66.6%) belong to Silene subg. Behenantha. Maximum values in models DM5 and/or DM6 were obtained in 23 species, of which eight (34.8%) belong to Silene subg. Behenantha. A total of 17 species gave maximum values in other models (DM7 and DM8; Table A6). Of these, five belong to Silene subg. Behenantha (31.6%), four corresponded to Silene sect. Physolychnis, and one to Silene sect. Cucubaloides (S. yunnanensis). In general, the average silhouettes of seeds in Table A6 are more elongated in their dorsal views (higher aspect ratio).

New Models for the Dorsal Views of Silene Species
The new models are shown in Figure 4. DM10 corresponds to a hyperellipse and it is a convex figure. It differs from DM1 in a certain asymmetry between the terminal sides, since the lower one is slightly broader than the upper side (Figures 1 and 4). The model DM11 also resembles DM1, but it is narrower and has small concave regions in the four sides (Figures 1 and 4). DM12 and 13 are figures with partial concavity regions for the upper and lower sides (

Mean Values of J Index with 20 Seeds per species. Correspondence between the Lateral and Dorsal Views
J index values as the mean of 20 seeds were evaluated for lateral and dorsal views of all species with the best-fitting models. The models were selected after comparison of the results with the average silhouettes. A threshold was set at 89.0, such that species giving

Mean Values of J Index with 20 Seeds per species. Correspondence between the Lateral and Dorsal Views
J index values as the mean of 20 seeds were evaluated for lateral and dorsal views of all species with the best-fitting models. The models were selected after comparison of the results with the average silhouettes. A threshold was set at 89.0, such that species giving values lower than this threshold for the mean of 20 seeds for the lateral or dorsal view were discarded. The only exception was given to S. damascena with a J index of 86.5 in the dorsal view for the model DM13, due to the high specific shape of this model, which gives much lower values than any other species. A total of 24 species did not reach the established threshold value for the J index, in either lateral or dorsal models, or in both. These species correspond to 12 out of 33 in Table A2 (models DM2 and DM4), and 12 species in Table A3 (the rest of the lateral models). Regarding dorsal models, they correspond to: (i) a unique species in Table A4 (S. perlmanii W.L.Wagner, D.R.Herbst and Sohmer), (ii) 7 species in Table A5, and (iii) 16 out of the 17 species in Table A6 (all of them except S. yunnanensis Franch). Thus, the species giving low values of J indexes as the mean of 20 seeds are concentrated in the group of those whose average silhouettes gave best scores in models DM7 and DM9, which were the dorsal models more elongated and with concave regions at the poles. The remaining 27 species gave values of J indexes superior to 89.0 in both lateral and dorsal models, and will be the object of this section. Table 1 presents the J index values obtained as mean values of 20 seeds for the lateral and dorsal models (LM and DM) in E. coeli-rosa and 27 species of Silene. The seed images of five Silene species gave good scores in LM2 and some other dorsal models (Table 1). Among them, both S. marizii and S. petersonii adjusted to LM2 and DM2 best ( Figure 5), the species S. fruticosa to LM2 and DM3 ( Figure 6); and, finally, the two species S. aprica and S. koreana to LM2 and DM10 (Figure 7). The seed images of five Silene species gave good scores in LM2 and some other dorsal models (Table 1). Among them, both S. marizii and S. petersonii adjusted to LM2 and DM2 best ( Figure 5), the species S. fruticosa to LM2 and DM3 ( Figure 6); and, finally, the two species S. aprica and S. koreana to LM2 and DM10 (Figure 7).

Figure 5.
Silene marizii: average silhouettes (left), the models giving best scores with 20 seeds (LM2 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 5.
Silene marizii: average silhouettes (left), the models giving best scores with 20 seeds (LM2 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 5.
Silene marizii: average silhouettes (left), the models giving best scores with 20 seeds (LM2 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 6.
Silene fruticosa: average silhouettes (left), the models giving best scores with 20 seeds (LM2 and DM3), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Twelve species gave the best scores in the model LM4, but the dorsal models were different for some of them ( Table 1). The seeds of four species adjusted well to convex models DM1 to DM3. The seeds of S. jeniseensis adjusted to LM4 and DM1 (Figure 8), those of S. hookerii and S. virginica to LM4 and DM2 (Figure 9), and those of S. baccifera, to LM4 and DM3 (Figure 10). Only the seeds of S. chlorifolia presented the combination of LM4 and DM5 (Figure 11). Six species (S. chungtienensis, S. fabaria, S. firma, S. nana, S. suksdorfii, and S. viridiflora) shared the same model combination of LM4 and DM10 (Figure 12), while S. yunnanensis was the only species with a combination between LM4 and DM11 ( Figure  13). Twelve species gave the best scores in the model LM4, but the dorsal models were different for some of them ( Table 1). The seeds of four species adjusted well to convex models DM1 to DM3. The seeds of S. jeniseensis adjusted to LM4 and DM1 (Figure 8), those of S. hookerii and S. virginica to LM4 and DM2 (Figure 9), and those of S. baccifera, to LM4 and DM3 ( Figure 10). Only the seeds of S. chlorifolia presented the combination of LM4 and DM5 ( Figure 11). Six species (S. chungtienensis, S. fabaria, S. firma, S. nana, S. suksdorfii, and S. viridiflora) shared the same model combination of LM4 and DM10 (Figure 12), while S. yunnanensis was the only species with a combination between LM4 and DM11 ( Figure 13).
Four Silene species gave good J index values in model LM8, and two models DM5 and DM10. Just as S. villosa showed the combination of LM8 and DM5 (Figure 18), the combination for S. dichotoma, S. integripetala, and S. multiflora was with DM10 ( Figure 19). Finally, the seeds of E. coeli-rosa adjusted well to models LM5 and DM10 (Figure 20). models DM1 to DM3. The seeds of S. jeniseensis adjusted to LM4 and DM1 (Figure 8), those of S. hookerii and S. virginica to LM4 and DM2 (Figure 9), and those of S. baccifera, to LM4 and DM3 (Figure 10). Only the seeds of S. chlorifolia presented the combination of LM4 and DM5 (Figure 11). Six species (S. chungtienensis, S. fabaria, S. firma, S. nana, S. suksdorfii, and S. viridiflora) shared the same model combination of LM4 and DM10 (Figure 12), while S. yunnanensis was the only species with a combination between LM4 and DM11 ( Figure  13).  . Silene hookeri: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 10. Silene baccifera: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM3), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 9.
Silene hookeri: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.
Taxonomy 2023, 3, FOR PEER REVIEW 10 Figure 9. Silene hookeri: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM2), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 10. Silene baccifera: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM3), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 10. Silene baccifera: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM3), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 10. Silene baccifera: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM3), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 11. Silene chlorifolia: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM5), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 11. Silene chlorifolia: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM5), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.
Taxonomy 2023, 3, FOR PEER REVIEW 11 Figure 12. Silene chungtienensis: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 13.
Silene yunnanensis: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 13.
Silene yunnanensis: average silhouettes (left), the models giving best scores with 20 seeds (LM4 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 15.
Silene chlorantha: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 16.
Silene linicola: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM12), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 14. Silene squamigera subsp. vesiculifera: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM6), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.  Figure 14. Silene squamigera subsp. vesiculifera: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM6), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 15.
Silene chlorantha: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 16.
Silene linicola: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM12), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 15. Silene chlorantha: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.  Figure 14. Silene squamigera subsp. vesiculifera: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM6), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 15.
Silene chlorantha: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM11), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 16.
Silene linicola: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM12), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Four Silene species gave good J index values in model LM8, and two models DM5 and DM10. Just as S. villosa showed the combination of LM8 and DM5 (Figure 18), the combination for S. dichotoma, S. integripetala, and S. multiflora was with DM10 ( Figure 19). Finally, the seeds of E. coeli-rosa adjusted well to models LM5 and DM10 ( Figure 20).  Four Silene species gave good J index values in model LM8, and two models DM5 and DM10. Just as S. villosa showed the combination of LM8 and DM5 (Figure 18), the combination for S. dichotoma, S. integripetala, and S. multiflora was with DM10 ( Figure 19). Finally, the seeds of E. coeli-rosa adjusted well to models LM5 and DM10 ( Figure 20). Figure 18. Silene villosa: average silhouettes (left), the models giving best scores with 20 seeds (LM8 and DM5), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 18. Silene villosa: average silhouettes (left), the models giving best scores with 20 seeds (LM8 and DM5), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.  Figure 19. Silene multiflora: average silhouettes (left), the models giving best scores with 20 seeds (LM8 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 19. Silene multiflora: average silhouettes (left), the models giving best scores with 20 seeds (LM8 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm. Figure 19. Silene multiflora: average silhouettes (left), the models giving best scores with 20 seeds (LM8 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

Figure 20.
Eudianthe coeli-rosa: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

The Relationship between Geometric Models and Taxonomic Sections
The dendrogram in Figure 21 shows the relationship between Silene species based on seed morphology. The results confirm and expand a certain relationship between the morphological groups obtained by the comparison with models and the current taxonomy of Silene. Species in S. subgen. Behenantha have more convex seeds corresponding to models LM2/LM4. In contrast, the models with large concavities, such as LM5, define better the shape of seeds in the species of S. sect. Silene in subg. Silene. Further analyses will be needed, including new models to better define the relationships between species. Figure 20. Eudianthe coeli-rosa: average silhouettes (left), the models giving best scores with 20 seeds (LM5 and DM10), and three representative images for the lateral (above) and dorsal (below) seed views. Bar represents 1 mm.

The Relationship between Geometric Models and Taxonomic Sections
The dendrogram in Figure 21 shows the relationship between Silene species based on seed morphology. The results confirm and expand a certain relationship between the morphological groups obtained by the comparison with models and the current taxonomy of Silene. Species in S. subgen. Behenantha have more convex seeds corresponding to models LM2/LM4. In contrast, the models with large concavities, such as LM5, define better the shape of seeds in the species of S. sect. Silene in subg. Silene. Further analyses will be needed, including new models to better define the relationships between species.

Discussion
The similarity of lateral views of Silene seeds with the cardioid led to the application of geometric models in the quantification of seed shape. In general, the mean value of the J index obtained with the cardioid (LM1) in seeds of species of S. subg. Behenantha was higher than in seeds of S. subg. Silene [14]. Nevertheless, the description of other, additional geometric models derived from the cardioid gave better J index values when applied to the species of S. subg. Silene (e.g., LM3 for S. gallica, LM6 for S. mellifera, LM5 for S. tridentata). The same trend applied to the species of S. subg. Behenantha (e.g., LM2 for S.

Discussion
The similarity of lateral views of Silene seeds with the cardioid led to the application of geometric models in the quantification of seed shape. In general, the mean value of the J index obtained with the cardioid (LM1) in seeds of species of S. subg. Behenantha was higher than in seeds of S. subg. Silene [14]. Nevertheless, the description of other, additional geometric models derived from the cardioid gave better J index values when applied to the species of S. subg. Silene (e.g., LM3 for S. gallica, LM6 for S. mellifera, LM5 for S. tridentata). The same trend applied to the species of S. subg. Behenantha (e.g., LM2 for S. latifolia, LM4 for S. diclinis) [14,15].
In general, the combination of models LM2 to LM8 may be more discriminant than the cardioid (LM1) itself, because they narrow better the morphological characteristics of seeds (open or closed in the hilum region). Geometric models LM2, LM4, and LM8 corresponded to those seeds with a more plane and closed region around the hilum, characterized by low partial concavities and high solidity [12]. Meanwhile, models LM3, LM5, LM6, and LM7 presented an open region (partial concavity) in the hilum region [14,15]. Thus, both groups represent alternative morphologies. In addition, the dorsal view of seeds was also analyzed to provide quantitative data about shape and morphology from this view [16]. The application of new models for the dorsal views showed that convex models adjusted better to seed images of S. subg. Behenantha [16]. On the contrary, we have seeds with partial concavities in the upper and lower sides of their dorsal views due to a channel running through the profile of the seed. These are known as dorso canaliculata [1][2][3] and correspond to non-convex models in their dorsal views, such as S. apetala, S. colorata Poir., S. inaperta, or S. ramosissima. They are frequently present in S. subg. Silene, and less common or absent in some sections of Silene subg. Behenantha, such as Silene sect. Melandrium [16].
These results suggest that the comparison with geometric figures might be a useful tool to apply morphological traits in plant taxonomy. Hence, an important question for the consideration of the taxonomic value of these morphological characters based on geometric models concerns their stability. Although these geometric models revealed levels of stability for certain complex taxonomic groups of Silene [12][13][14][15][16][17][18], more data are still needed to fully support these previous statements.
In this work, we have classified 51 species of Silene from diverse geographic provenances. The classification was according to their highest scores of J index values with specific geometric models for their lateral and dorsal views. The analysis started with the values of the J index in the average silhouettes for the lateral and dorsal view of each species. Average silhouettes gave a preliminary approximation to overall seed shape in both views, lateral and dorsal [15][16][17]. They provide information about aspects of seed morphology such as general symmetry, aspect ratio, and convexity. For example, the seeds of eight species resembling model LM5 show positive correspondence in DM5 to DM9, but it is negative in models DM1-DM4. A total of 10 species giving strong similarity in their average silhouettes for the dorsal views with convex models (DM1-DM4) belong to the group of high similarity with convex models LM2 and LM4. Likewise, non-convex models DM8 and DM9 are linked to the group of 17 species which are elongated and gave low results of J indexes in the tests based on mean values of 20 seeds. All of them, except S. yunnanensis, were discarded from further study. This might indicate that these seeds require more specific models or that their seeds were quite heterogenous. Nevertheless, elevated values of aspect ratio (elongated seeds) could be a symptom of dehydration due to prolonged storage or storage in adverse conditions [22].
We found an association between the J index in the average silhouettes and the J index as the mean value of 20 individual seeds. The group of species whose average silhouettes gave higher scores in LM2, LM4, and LM8 contains all the species giving the best scores in these models when estimated as means of 20 seeds. Similarly, most species giving elevated scores of J index as the mean value of 20 individual silhouettes in LM5 (e.g., S. linicola, S. longicilia, S. pygmaea, and S. damascena) also had high values in this model in their average silhouettes. Values obtained with average silhouettes, although indicative of seed shape in a group of seeds, could give overestimations because the analyses with the average silhouettes concentrate more on conserved regions, and hence, the shape of many seeds can differ from their corresponding average silhouette. Thus, the final analysis including J index values measured as the mean of 20 seeds results in much more precise estimations.
A protocol based on Fourier transform allows us to define models according to their shape, independently of known geometric figures [18]. The protocol has been applied here to design the new dorsal models DM10 to DM13. Among them, DM10 is convex while the other three models present regions of variable concavities. DM10 fits well for the dorsal view in 11 species. All of them combined with LM2, LM4, or LM8: S. aprica and S. koreana (LM2), S. firma, S. nana, S. chungtienensis, S. fabaria, S. suksdorfii, and S. viridiflora (LM4), and S. dichotoma, S. integripetala, and S. multiflora (LM8). This result would confirm the hypothesis that convex models in the dorsal view are associated with convex models in the lateral view [16]. Most of these 11 species, whose seeds adjust to convex models, belong to S. subg. Behenantha. Model DM11, with small concavity regions, applies also to a variety of species in combination with both convex for LM4 in S. yunnanensis, and non-convex for LM5 in S. chlorantha, S. longicilia, and S. pygmaea. In contrast to DM10 and DM11, the new models DM12 and DM13 with larger concavity regions are highly specific for S. linicola and S. damascena, both combined in LM5, a lateral model presenting a large concavity in the hilum region.
When considering J index values obtained as the mean of 20 measurements, the combination between lateral models closed in the hilum (LM2, LM4, and LM8) and convex models (DM1-DM4, and DM10) was conserved in 18 species: (i) 13  The cardioid (reniform or kidney-shaped) shape of the seeds is associated with the campylotropous ovule, with most of the morphological axis curved, allowing a great morphological variety in ovules and seeds [23], which is associated with various types of asymmetries [14]. Consequently, the similarity of seeds with the cardioid is limited to some Silene species, while others require specific models for accurate description and quantification. The application of Fourier transform allows the description of new, specific models for these seed images that do not adjust well to defined geometrical figures.
New species of Silene are regularly being described [24][25][26], and the taxonomy of this genus is being revised [27][28][29]. Having the genus Silene in mind, we present protocols based on seed morphology, which represent a new technique to contribute to this objective. These protocols may be applied with optical photographic equipment and image analysis programs.

Conclusions
The analyses of seed morphology in 51 new Silene species and the close-related species Eudianthe coeli-rosa by lateral and dorsal models revealed the existence of a robust behavior between species and geometric models. In addition, there is a remarkable correspondence between certain lateral and dorsal models. Those lateral models closed in the hilum region (LM2 and LM4) were associated with convex dorsal models (DM1-DM4, and DM10). Conversely, those lateral geometric models more open in the hilum region were well adapted to seeds with partial concavities in their dorsal views (mostly dorso canaliculata seed types). The former correspond to species in Silene subg. Behenantha, while the later are associated with particular sections in S. subg. Silene.
Supplementary Materials: A video describing the method for obtaining the average silhouette in a seed image sample is available at (https://zenodo.org/record/4478344#.YzxbmExBxD8). The seed images used in the analysis are stored available (https://zenodo.org/record/7330942#.Y3Y8Hn3 MJD8). The Mathematica code for the new models DM10-DM13 are available (https://zenodo.org/ record/7386404#.Y4ipln3MJD8). The three items were accessed on 13 December 2022.    Table A7. Data used in the construction of the dendrogram in Figure 21. In column 2, subg., B stands for Behenantha; S for Silene.