Seed Morphology in the Vitaceae Based on Geometric Models

Morphometric methods based on artificial vision algorithms provide measurements for magnitudes descriptive of seed images (i.e., the length, width, area, and surface circularity index). Nevertheless, their results frequently omit the resemblance of the images to geometric figures that may be used as models. A complementary method based on the comparison of seed images with geometric models is applied to seeds of Vitis spp. The J index gives the percentage of similarity between a seed image and the model. Seven new geometric models are described based on the heart-shaped and piriform curves. Seeds of different species, subspecies and cultivars of Vitis adjust to different models. Models 1 and 3, the heart curve and the water drop, adjust better to seeds of V. amurensis, V. labrusca and V. rupestris than to V. vinifera. Model 6, the Fibonacci’s pear, adjusts well to seeds of V. vinifera, in general, and better to V. vinifera ssp. vinifera than to V. vinifera ssp. sylvestris. Seed morphology in species of Cissus and Parthenocissus, two relatives of Vitis in the Vitaceae, is also analysed. Geometric models are a tool for the description and identification of species and lower taxonomic levels complementing the results of morphometric analysis.


Introduction
Grapevines belong to the genus Vitis L., which comprises about seventy species of lianes in the family Vitaceae Juss. Grape (Vitis vinifera L. ssp. vinifera) is one of the most ancient fruit-bearing plants in the Old World, and together with the olive, fig, pomegranate and date palm, it comprises the oldest groups of domesticated fruit trees [1][2][3]. Today, it is widely accepted that the modern cultivars of V. vinifera are the result of the domestication of V. vinifera ssp. sylvestris (C.C.Gmel.) Hegi (wild grape) [4][5][6]. The main feature used to distinguish between wild and cultivated Vitis is the reproductive system: the wild grape is dioecious, whereas the cultivated grapevine has hermaphroditic flowers; Other geometric models include the ovoid and the ellipse, that give high values of J index with seeds of species in the Cucurbitaceae [27] and in the Euphorbiaceae, such as Ricinus L. and Jatropha L. [28,29]. Seed shape in wheat kernels was described based on three geometric figures: (1) an ellipse of aspect ratio (AR) = 1.8 fitting the "round varieties" (T. aestivum L. ssp. aestivum cv. Zebra and Torka), (2) a lens of AR = 3.2 for the elongated kernels (T. monococcum L.), and (3) an ellipse of AR = 2.4 that adjusted well the kernels of intermediate-shaped varieties such as T. durum Desf. cv. Floradur [30]. This latter work, demonstrating the adjustment of seed shape to geometrical forms, is useful to complement the results of morphometric analysis as those reported in Vitis sp. [9,14,15,[18][19][20][21].
Our objective was to describe seed shape in species, subspecies and cultivars of Vitis by a combined geometric and statistical approach. The images of grape pips belonging to different genotypes were compared with geometrical models derived from mathematical curves that represent diverse heart shapes [31,32] or the pyriform curve [33]. This allows the description of seed shape with adjectives that may be measurable. For example, cordate means heart-like, thus defining a family of curves, responding to different equations that include different models. Among the cordate figures (heart-shaped), the cardioid is the curve defined by the trajectory of the point in a circle rolling around another circle of the same radius. This curve responds to an algebraic equation and is reproducible, thus, for any given two-dimensional figure well defined by a profile, we may calculate the percentage similarity to the cardioid. News models based on geometrical figures are defined for seed shape quantification in the species and cultivars of Vitis. In addition, the overall morphology of Cissus and Parthenocissus, genus related to Vitis in the Vitaceae [34] is described, presenting an overview of seed shape in the Vitaceae.
Seeds of C. verticillata were collected at Pastaza (Ecuador) in disturbed areas and, after being photographed, the seeds were returned into the wild. Seeds of P. tricuspidata were collected at the Faculty of Pharmacy, University of Salamanca.

Seed Images
The seeds were oriented with their chalaza downwards, such as to expose the ribs upwards (ventral orientation; Figure 1). This offers an advantage in relation to the dorsal orientation, because the seed has a more plane surface of sustentation. In the dorsal orientation, the seed rests on top of the rib and the seed is skewed. Photographs containing between 33 and 122 seeds per accession were taken with a digital camera Sony ILCE 5100, with an AF-S Micro NIKKOR 60 mm. 1:2.8 G ED objective (Nikon). Ventral views are used in all the images, because the seeds are more stable and the orientation occurs in a plane perpendicular to the line of vision. In the dorsal view, the seed tends to oscillate resting on the central rib and a flank, resulting in more variable sub-lateral orientations.
The photographs were obtained with a single-leg tripod, putting the seeds on a glass and illuminating them above and below it to avoid shadows. The resolution of the images is 13.8 pixels per mm. Image data have been deposited in Zenodo: https://zenodo.org/record/3786726#.XrVL0agzaM8.

General Morphological Description by Image Analysis
Photographs were used to obtain data of the area (A), perimeter (P), length of the major axis (L), length of the minor axis (W), aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R). All these magnitudes were calculated with the Image J program [35]. Length and width are the Feret diameters of the seed images. Feret, or calliper diameters, are the distance between two parallel planes restricting the object. The seeds where oriented vertically, so length and width correspond respectively with the vertical and horizontal Feret diameters. The models were adjusted to fit the size of the seed images. Circularity Iindex and roundness were calculated as described [36][37][38].

Comparison with Geometric Models
The morphological description of seed shape in Vitis accessions is based on the comparison with geometric figures used as models. The seven models are original from this work. They are shown in Figure 2 and described below.

Description of the Geometric Models Used
Model 1 (M1 in Figure 2) has been obtained inverting the heart curve in Weisstein [31]; then, it is defined by the equation: Model 2 (a rounded heart curve, M2 in Figure 2) results from a slight modification of Equation (1): Model 3 (M3 in Figure 2; the water drop) is obtained from M1, adapting the basis in the heart curve to a circumference overlapping the maximum width of the curve. It is the joint graphical representation in the interval [-6,6] of the function f defined by: and g derived from the circumference with centre (0,−6/11) and radius 6: Model 4 (M4 in Figure 2) results from the elongation of M3 in the y-axis by a factor of 123/100; then, it corresponds to the joint graphical representation of y = 123/100 f (x) and y = 123/100 g(x), with f and g defined in (3) and (4), respectively.
Model 5 (M5 in Figure 2) results from the elongation of M1 in the y-axis by a factor of 123/100; then, it is determined by the equation: To obtain Model 6 (M6 in Figure 2; the Fibonnacci's pear), we shrink the pear curve with r = 2 in [30], so that it adjusts to a Fibonacci's rectangle. The proposed curve is given by the equation: 49x 2 + 25y 2 117, 649x 6 + 15, 625 y 3 − 2y 2 + y − 1 2 + 60, 025x 4 3y 2 − 4y + 2 +30, 625x 2 3y 4 − 8y 3 + 8y 2 + 2y − 3 = 1, 562, 500 Model 7 (M7 in Figure 2) derives from the modification of an ellipse of equation: It is given by the combined representation of the functions:  Figure 3 shows examples of the adjustment between seeds and models and Figure 4 two examples for the calculation of the J index.

Calculation of the J Index
Images containing the seed photographs and the geometric model were composed with Corel PHOTO-PAINT X7.
Areas of seed images were calculated with ImageJ program. To obtain the J index, the areas in two regions were obtained: the region shared by the model and the seed image (common region, C) and the region not shared between both areas (D). The J index is defined by [22,32]: where C represents the common region and D the regions not shared ( Figure 4). Note that J is a measure of seed shape, not of its area. It ranges between 0 and 100 decreasing when the size of the not-shared region grows and equals 100, when the geometric model and the seed image areas coincide; that is, when area (D) is zero. Similarity was considered when J index values were over 90.

Statistical Analysis
One way ANOVA was used to show significant differences between populations for the measured variables, followed by Scheffé or Tukey's tests to provide specific information on which means were significantly different from one another. The former is used if the cases have different number of samples and the latter if the cases have the same or similar number of samples. The analysis was done with software IBM SPSS statistics v25. Table 1 shows the mean values of the area (A), perimeter (P), length of the major axis (L), length of the minor axis (W), aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R) obtained for the images of the four species of Vitis analysed. Table 1. Mean values of the area (A), perimeter (P), length of the major axis (L), length of the minor axis (W), aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R) in the seeds of four species of Vitis analyzed. In relation to seed size (area of images), V. labrusca seeds were larger and a group composed by V. amurensis and V. vinifera ssp. vinifera, smaller. The seeds of V. rupestris and V. vinifera ssp. sylvestris had intermediate values.

Comparison of Vitis Species and Subspecies
The seeds of V. vinifera ssp. vinifera had higher AR values, corresponding to lower circularity and roundness, characteristics of the elongated or piriform seed morphology [5]. The seeds of V. vinifera ssp. sylvestris had intermediate values of AR. The seeds of V. amurensis, V. labrusca and V. rupestris had lower values of AR and higher values of roundness, corresponding to their more circular morphology. Smaller differences between circularity and roundness are due to the irregularity of the perimeter of the images. Table 2 presents the mean values of the morphometric magnitudes measured in six cultivars of V. vinifera ssp. vinifera. Table 2. Mean values of the area (A), perimeter (P), length of the major axis (L), length of the minor axis (W), aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R), obtained for the images of the seeds of six cultivars of V. vinifera ssp. vinifera). When the cultivars are classified by their values of roundness, the order is inverse to the classification by aspect ratio. Smaller differences between circularity and roundness in some cultivars are due to the irregularity of the perimeter of the seed images.

Comparison of Vitis Species and Subspecies
The seeds of V. amurensis, V. labrusca and V. rupestris adjusted better to M1 than the seeds of V. vinifera ssp. sylvestris and Vitis vinifera ssp. vinifera. V. amurensis and V. rupestris adjusted better to M3 than the other species, with values of J index of 90.6 and 90.0, respectively (Table 3; Figure 5). Seeds of V. vinifera ssp. sylvestris and V. vinifera ssp. vinifera adjusted better to M6 and M7 than the seeds of the other species. V. vinifera ssp. sylvestris gave higher values than V. vinifera ssp. vinifera with M3, while the reverse was observed with M6 (Table 3; Figure 5).

Comparison of Vitis vinifera Cultivars
In general, the seeds of all cultivars of V. vinifera ssp. vinifera tested adjusted better to Models 5, 6 and 7 than to Models 1, 2, 3 or 4 (not shown). Model 5, the elongated heart curve, gave the highest values of J index with Meserguera (Table 4, Figure 6). The Fibonnacci's pear, M6, gave similar high values with all cultivars tested, being the model that better adjusted the shape of the seeds of Carignano (Table 4, Figure 6). Differences with M7 were observed between Carignano and the other cultivars, with lower values of J index in the former (Table 4, Figure 6).

Seed Size and Shape of Cissus and Parthenocissus, Two Relatives of Vitis in the Vitaceae
C. verticillata seeds are smaller and more elongated than P. tricuspidata, having lower values of circularity (C) and roundness (R) ( Table 5; Figures 7 and 8). Seeds of both C. verticillata and P. tricuspidata are contained in berry fruits. While in C. verticillata there is only one seed per fruit, in P. tricuspidata the number of seeds oscilates between one and three. Figure 8 contains representative images of seeds obtained from berries having one, two and three seeds obtained from the same plant. Table 5. Mean values of the area (A), perimeter (P), length of the major axis (L), length of the minor axis (W), aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R) for seeds in the species C. verticillata and P. tricuspidata.   Figure 8. From left to right: Model 2 (three rounded heart curves), three figures composed with the silhouettes of 20 seed images, and five representative images of seeds of P. tricuspidata. The first row contains seeds from one-seed fruits; the second from two-seed fruits and the third, from three seed-fruits. The bar is equal to 1 cm.
Seeds of C. verticillata adjust well to M4 (mean J index values = 91.2 with N = 28), while seeds of P. tricuspidata adjust to M2 with different values of J index, depending on the number of seeds per fruit (Table 6, Figure 7). Table 6. Mean values of the area (A), perimeter (P), length of the major axis (L), length of the minor axis , aspect ratio (AR is the ratio L/W), circularity (C) and roundness (R) and J index values in seeds of P. tricuspidata (Siebold and Zucc.) Planch has one, two or three seeds per fruit. J index values were obtained with M4 (an elongated water drop).

Discussion
Seven geometric models have been described for the first time and applied to the morphological description of seeds in the Vitaceae. M1, a heart curve defined by Weinstein [31], is useful for the quantification of seed shape in species of Vitis and for the differentiation between the species V. amurensis, V. labrusca and V. rupestris, in one side, and V. vinifera, in the other. M1 adjusts better to seeds of V. amurensis, V. labrusca and V. rupestris than V. vinifera. M6, the Fibonacci's pear, adjusts well to seeds of V. vinifera, in general, and better to V. vinifera ssp. vinifera than to V. vinifera ssp. sylvestris. This way, V. vinifera ssp. sylvestris can be differentiated from V. vinifera ssp. vinifera, because the values of J index obtained with M6 (the Fibonacci's pear) are higher in the former. The water drop, a rounded heart curve (M3) corresponds better to the rounded seeds of species V. amurensis and V. rupestris than those of V. labrusca, also giving high values of J index with the seeds of Vitis vinifera ssp. sylvestris. Thus, M3 permits to differentiate between V. vinifera ssp. vinifera and V. vinifera ssp. sylvestris, the former giving higher values of J index with this model. The combination of M3, M6 and M7 is useful for the differentiation of V. vinifera ssp. sylvestris from other species of Vitis.
The models proposed are related to previous seed shape descriptions of the Euvitis and Muscadinia taxonomic groups. Models 1, 2 and 3 resemble the Euvitis type, while Models 4, 5, 6 and 7 fit better the Muscadinia shape [9,10]. The seeds of wild species are smaller, robust and with a rounded outline resulting in a heart-like shape, often termed as "cordate", with short stalks and a flat ventral side with sharp angles and a strongly developed chalaza, while those of cultivars are large, elongated, oval or pyriform, with an elongated stalk [9].
The identification of geometric curves resembling the outline of seed images complements the results of morphometric analysis and allows more exactitude in the description of species and cultivars and for their comparison. Among the two subspecies of grape, subspecies sylvestris adjusts better to M3, while subspecies vinifera adjusts better to M6, thus concerning seed shape, V. vinifera ssp. sylvestris resembles more other species of Vitis than V. vinifera ssp. vinifera. This demonstrates a great capacity of variation in the shape of seeds of cultivars, where significant changes in seed shape are due to selection during cultivation [28].
Concerning cultivars, Meserguera gave higher values with the elongated heart curve (M5), while Malvasia seeds adjusted better to M7. New models based on geometrical curves can be useful for the description of seed shape in other cultivars.
Seeds of Cissus verticillata adjust well to M2, while seeds of the species Parthenocissus tricuspidata adjust better to M4. The quantification of seed shape with M4 shows differences between seeds depending on the number of seeds per fruit. Fruits with one or two seeds have their seeds more rounded and with higher values of J index than the seeds from fruits with three seeds per fruit.
Both C. verticillata and P. tricuspidata seeds resemble more the seeds of V. amurensis, V. labrusca and V. rupestris than those of cultivars of V. vinifera, suggesting that the primitive seed type, associated with a reduced number of seed per fruit, resembles more Model 1.
The seeds in the Vitaceae present a very peculiar shape, for which there is no adaptation to a single geometric model, like the ellipse, the ovoid or the cardioid [30,32], but to more complex figures. This is in contrast with other families, where the ellipse, the ovoid or the cardioid are the models defining seed shape for most species and genera [22][23][24][25][26][27][28][29][30]32].
The magnitudes used in morphology have different information about the shape of a plane figure. Magnitudes such as the area, perimeter, length or width do not give much information on shape, and thus, are of little value on their own to discriminate between cultivars or species. Other measures, such as circularity or roundness are more informative when they are close to the unity, but their usefulness depends on the similarity of the figure with the circle. If the figure is different to a circle, low values are of no informative value. In contrast, J index compares the image of a plane figure with a similar geometric figure selected as a model. In the case of grape seeds, J index is a way to provide valuable information in a single measure and, in consequence, useful in ampelography studies.
The current edition of the OIV Descriptor List for Grape Varieties and Vitis species [39] concerning seeds includes only three variables: length, weight and the presence of transversal ridges on dorsal side of seeds. J index is a new way of describing seeds that allows one to relate the variety studied to its genetic origin. The longer seed varieties are of eastern origin and generally table grapes, as seen in the descriptor, and the varieties of western origin are shorter and rounded linked to the origin of the domestication of most wild vines, as, for example, varieties of French and Spanish origin [40,41]. The work described in this article allows an approximation to elucidate if the variety is of eastern or western origin. Intermediate varieties such as Pinot and Merlot are of mixed origin. Further study of Spanish varieties where there is an Arabic influence would allow us to test this methodology.
The description of seed images by comparison with geometric figures will contribute to the validation of results obtained by automated artificial vision methods, improving the description and identification of species and varieties. It is a new, low-cost and relatively simple technique that can help growers, producers or breeders in seed identification.

Conclusions
New geometric models for seed shape quantification in the Vitaceae are described. Model 1, a heart curve, adjusts well the shape of the seeds of species V. amurensis, V. labrusca and V. rupestris. Model 2, a rounded heart curve, defines well the shape of the seeds of P. tricuspidata. Model 3, the water drop, adjusts well the shape of the seeds of V. amurensis and V. rupestris. Model 4 gives a good adjustment with the seeds of C. verticillata. A combination of the models is useful for the differentiation of the seeds of cultivars of V. vinifera, ssp. vinifera. Geometric models provide a new method for the description and classification of grape species and cultivars. Future studies will be directed to the identification of cultivar-specific models.