Statistical analysis on the cnidome of genus Hydra using Generalized Linear Models model evolutionary biology.

. In the systematics of cnidarians, the different types of cnidocysts are considered an important taxonomic character. In Hydra, the four types of cnidocysts found in the ectoderm, concentrated in tentacles and their measurements, together with other morphological and reproductive characteristics, are very important for the taxonomy of the species. In this study, we explore in detail the biometric and statistical characteristics of the cnidome of three species of Hydra collected in three different environments for each climate season. A total of 17,378 capsules were measured. We used ANOVA test and Generalized Linear Model to analyze the distribution and differences reflected in each cnidome, considering the factors “individuals”, “season”, “lagoon” and “species”. The results were clear: the cnidome keep specific information that, together with other taxonomic characteristics, allows us to discern between species of different groups. The same happens with cnidome of the same species but from different lagoons or climatic seasons: we observed a variation of parameters for each type of cnidocyst that could differentiate “ecological races”, since these differences are not enough to declare different species.


INTRODUCTION
Cnidocysts are the sine qua non condition of the phylum Cnidaria; all cnidarians and only they produce them (Fautin, 2009). In the systematics of this group, the different types of cnidocysts have been considered an important taxonomic character and the cnidome have been defined as the inventory of all the cnidocysts or cnidae present in a species (Weill, 1934a). Cnidocysts are found in the ectoderm of hydrozoans and in Hydra, 95% of them are concentrated in the tentacles, growing in number from the base towards the tips (Bode & Flick, 1976). In this genus, we found four types: stenoteles, desmonemes, holotrichous isorhiza and atrichous isorhiza. The first two types are involved in feeding. Stenoteles pierce prey like a harpoon (even those with strong exoskeletons), injecting a certain amount of venom. Desmonemes evert their filament that strongly adheres to the surface of the prey (Burnett, 1973). The atrichous isorhiza are involved in the walking movements, responsible for adhering the tentacles and pedal disc in each "step" (Toppe, 1910;Ewer, 1947). Finally, holotrichous isorhiza are discharged in the presence of organisms that are not normally ingested by hydras and that do not cause a nutritional stimulus, such as Paramecium. This would indicate that the function of the holotrichous isorhiza could be one of the first defense reactions in nature (Burnett, 1973).
Cnidocysts are organized in bundles within the ectodermal epithelial cells (Slautterback, 1967). Ten to twenty cnidocysts are assembled in a single epithelial cell forming "batteries" (Gelei, 1927;Kanaev, 1969). Each battery is made up of at least one stenotele surrounded by several desmonemes and atrichous isorhiza (Ewer, 1947). Over time these structures have acquired different degrees of relevance. In Hydrozoa they have been examined in detail and many studies emphasize their usefulness in taxonomy (Itô & Inoue, 1962;Östman, 1979a, b, 1982, 1983, 1988Bouillon, 1985;Watson, 1985;. Deserti et al. (2010) and Deserti & Zamponi (2012) initiated a biometric and statistical approach in the study of the cnidome of three Hydra species and reported different morphotypes of the same cnidocyst, for the same species from different environments. However, there are still few studies that exploit their statistical qualities. In other groups, such as Anthozoa, certain studied investigate their nature. Acuña et al. (2004) and Garese et al. (2016) per-formed an exhaustive statistical study with a significant amount of data, applying GLM that become independent of the assumptions of normality, without losing information and power, as in non-parametric tests.
In this study, we investigated the biometric and statistical nature of the Hydra cnidome, expanding previous knowledge and adding a much larger amount of data, using the GLM as the main tool for the analysis and comparison of data. We used three Hydra species collected at three different sampling sites for each season of the year.
Hydra vulgaris, of medium size, presented a brown coloration, separate sexes and long tentacles with a discontinuous growth pattern. The four types of cnidocysts with sizes within the ranges of the species were observed. Hydra vulgaris pedunculata, also brown and of medium size, presented the same general morphological and reproductive characteristics of H. vulgaris, but with a clearly distinguishable permanent peduncle. Hydra viridissima, small in size, presented a bright green coloration with short tentacles and the four types of cnidocytes present, the latter with sizes within the normal ranges cited for the species.
A total of 162 specimens were collected seasonally during two years: from autumn 2013 until summer 2014 and, from autumn 2014 until summer 2015. For the extraction of hydras, some substrates were collected: floating vegetation (different species depending on the season) and submersed macrophyte Ceratophyllum demersum L. The samples were packed and transported to the laboratory and conditioned in separated aquariums with water from the site and kept with aeration. The samples were maintained at 20° ± 3℃ with a photoperiod of 12 hours of light and 12 hours of dark.

Cnidocyst sampling, description and typing
The width and length of 30 intact capsules of each type of nematocyst were measured by squash of entire and live polyps using a Zeiss Axiolab light microscope with a magnification of 1000X (immersion objective). The choice was random and with these sizes the data matrix was constructed for statistical analyses. In total 17,378 capsules were measured (average ~ 30 capsules × 4 types of cnidocysts × 162 specimens × 3 species). In some cases given the low abundance of some types, all of the cnidocysts found were considered.
For each species, the descriptive statistical parameters of each type of cnidocyst were detailed, such as the average length and width and the minimum and maximum of both measurements.

Relative abundance
To estimate the relative abundance, four different optical fields were randomly focused in the squash made for each polyp, counting in each field the different types of perfectly recognizable cnidocysts. The data were grouped and the average relative abundances for each species were calculated.

Statistical analysis
All statistical analysis were carried out using the R software (https://www.r-project.org). Only the length variable was used for each cnidocyst of each type (dependent or explained variable). This variable was grouped under three different factors: 1. "Individual" factor: the values of the length variable were compared between individuals of the same species, to detect a possible intraspecific variation. 2. "Season" factor: the values of the length variable were compared between seasons, of the same lagoon and species, to detect possible variations between the different climatic seasons. 3. "Lagoon" factor: the values of the length variable of the same species were grouped by lagoons, to detect possible intraspecific variations by environment. 4. "Species" factor: the values of the length variable were grouped entirely by species in order to detect interspecific variations.
The following data packages were used: (a) H. vulgaris and H. vulgaris pedunculata 1 from La Brava, (b) H. vulgaris and H. vulgaris pedunculata from Los Padres, (c) H. vulgaris and H. vulgaris pedunculata from Nahuel Rucá and (d) H. viridissima 1 .
The normality of the capsule sizes was tested by a Shapiro-Wilks test (α = 0.05) on the residuals of a Linear Model with normal distribution.
Comparisons of the sizes grouped under the four aforementioned factors were made using an ANOVA, in those cases where normality was accepted.
For the rejection cases, GLM with gamma distribution for the errors were applied. The model was:

g (length) = ß₀ + ß₁ factor + Ɛ
In addition, for the GLM, t tests were applied to compare the coefficients of the model (ß1). The fit of these models was graphically evaluated using Q-Q Plots of the residuals vs. the theoretical quantiles of the model and scatter diagrams of the deviation of the residuals vs. the fitted values of the model.

Cnidocyst sampling, description and typing
The cnidome of the species H. viridissima, H. vulgaris and H. vulgaris pedunculata are respectively present-ed in Figs. 1-3. The different holotrichous isorhiza morphotypes, found in each species are shown in Fig. 4. Additional details of the measurements (sizes of each cnidocyst of each species for each lagoon) are presented in the tables of Appendix A as supplementary material.

Relative abundances
In the three species, the most abundant type was the desmoneme, followed by the stenotele, atrichous isorhiza and finally, holotrichous isorhiza (Fig. 5).

Statistical analysis
Normality: for the "length" variable of the four types of cnidocysts in each species, normality (α = 0.05) was rejected ( Table 1).
1. "Individuals" factor: The existence of significant differences between individuals was evaluated using a t test for the coefficients of each model. For the species H. viridissima, 25% of the individuals presented differences in the sizes of their isorhizas and desmonemes and 100% of the individuals presented differences in the sizes of their stenoteles. For the subspecies H. vulgaris pedunculata, 47.6% of the individuals presented differences in the sizes of their stenoteles, 71.4%   For H. vulgaris, all cnidocysts showed differences between the three lagoons. In all cases, the p-values obtained by the t test for the coefficients (ß1) of the different models were < 0.001. Figures 6 and 7 show, for the species H. vulgaris and H. vulgaris pedunculata, respectively, the adjustment of each model for each type of cnidocyst, by means of the Q-Q Plots graphs of the residuals vs. the theoretical quantiles of the model and scatter diagrams of the deviation of the residuals vs. the fitted values of the model. 4. "Species" factor: Table 3 lists the p-values of the t test for the coefficients (ß1) of each model for the comparison of each pair of species. Figure 8 shows    winter 13 autumn 13 *** ** ** *** autumn 14 *** *** spring 13 *** spring 14 *** * summer 14 *** *** summer 15 ** Signification levels: "***" 0.001, "**" 0.01, "*" 0.05 Table 3. Variations between species. P-values of the t-test for the coefficients (ß1) of the GLM of each type of cnidocyst. The species Hydra vulgaris pedunculata is not shown as it is the one taken for comparisons.

DISCUSSION AND CONCLUSION
The taxonomic value of the cnidome has already been much discussed (Weill, 1934a, b;Itô & Inoue, 1962;Kubota, 1976;Östman, 2000). Nowadays it is mandatory the use of the statistical distribution of the data in taxonomic issues. For H. viridissima, H. vulgaris and H. vulgaris pedunculata, at least two different morphotypes of holotrichous isorhiza have been recorded. However, despite varying in their shape, different from the classic shape of paramecium or shoe sole, their average measurements do not vary between the studied species. Especially the length value, which is the usual measurement used for statistical analysis. Deserti et al. (2010) reported two different morphotypes for H. vulgaris from Los Padres and Nahuel Rucá. In Nahuel Rucá they described morphotype I, in a classic form, and II in the form of a seed. The latter morphotype is similar to that recorded for H. vulgaris pedunculata from Los Padres and H. vulgaris from Nahuel Rucá. Other authors also described more than one holotrichous isorhiza capsules (Cordero, 1941a, b;Dioni, 1968;Hemmrich, et al., 2007;Wang et al., 2009). In these studies the shapes and sizes were described and included within the cnidome of the species. In the case of the cnidome reported by Dioni (1968) for the green hydra H. plagiodesmica, the forms are clearly different from those detailed here for H. viridissima. Although there is no theory that explains the appearance of different morphotypes, it is likely that these differences respond to variations during cnidogenesis, in response to physiological changes. This could be explained if we consider that the capsules, once formed, do not present a certain degree of plasticity that could respond to pressures within the tissues or even during their migration through the body. However, for the same species collected in Nahuel Rucá by Deserti et al. (2010), only two morphotypes were found, while in this study, an additional third morphotype is added. Taking into account that the structure of holotrichous isorhiza is in many cases of importance for the taxonomic status of a species (for example, the differentiation between H. oligactis from H. pseudoligactis), it is imperative to pay special attention and care in the typification of its different forms. Despite the fact that many times the average sizes remain constant beyond their morphology, the perfect distinction between one and the other is necessary.
Regarding the relative abundances, the same pattern was observed in all species. Although the percentages yielded different values, the abundance was, in decreasing order, desmonemes, stenoteles, atrichous and holotrichous isorhiza. These abundances agree with that recorded by Deserti et al. (2010) and  for H. vulgaris from Los Padres and Nahuel Rucá and  for H. vulgaris pedunculata from Los Padres. Other authors report different abundances; for example Bode & Flick (1976) and Zumstein (1973) for the North American H. attenuata (= H. vulgaris) place the atrichous isorhiza in second place and the stenoteles in third. Although these differences may be real, based on physiological responses to different environments, they could also respond, in part, to the methodology used for counting the cnidocysts. While we used here the rel-ative abundance based on visual counts, the aforementioned authors perform solutions in sodium duodecyl sulfate, physically separating one capsule from another and counting all the cnidocysts present in a polyp. This technique would maximize the visualization of atrichous isorhiza, which turns out to be the most difficult type to individualize, since the structure of their capsule quickly melts with the surrounding medium.
The analysis of the distribution of the data yielded several results. On the one hand, although the most frequent is the absence of normality. Many studies have shown that the normality of the data can vary. The statistical tests frequently assumed a normal distribution of the lengths of the capsules, a concept that has currently been refuted by several authors (Acuña et al., 2003(Acuña et al., , 2004(Acuña et al., , 2007Acuña & Garese, 2009;Garese, 2013), at least for several species of sea anemones. The progress in the studies at the level of the data distribution, in principle, was overturned in the use of non-parametric tests, for the case of rejected normality. These tests, in certain situations, waste information and are less powerful, characteristics that are much more minimized in the GLMs developed by Nelder & Wedderburn (1972).
In particular, GLMs with gamma errors tend to fit very well to data from biological studies and are more appropriate for analyzing skewed data with constant coefficient of variation, so that it is not necessary to resort to transformations that force us to work within unnatural scales. Furthermore, in these cases, the equality of variances is not a requirement and the link between the dependent variable and the linear combination of independent variables (in the classic case it is the identity), can take different forms (Hastie & Tibshirani, 1990). In other cnidarians, the use of these GLMs was very appropriate for quantitative comparisons (Allcock et al., 1998;Watts et al., 2000;Ardelean & Fautin, 2004;Acuña et al., 2004;Garese, 2013). Either by parametric tests, when normality was accepted or by GLM, when it was rejected, the sizes of the cnidocysts vary between individuals of the same species, in the three species studied. Of the totality of the cases analyzed, the holotrichous isorhiza were the ones that presented the higher variation, with a value of 76.8%. They are followed by atrichous isorhiza, with 74.4%, stenoteles in third place, with 38.7% and finally, desmonemes, with a 21.2% of variation in all cases. If we observe carefully, this order in the percentage of variation is exactly opposite to the order of the relative abundances. This indicates that the holotrichous isorhiza despite being the least abundant type, is the most variable. This is another characteristic that reinforces the importance that its analysis requires.
When we analyze the difference in sizes between individuals of the same species, collected in the same lagoon, we note that variations are recorded between some climatic seasons. In H. vulgaris, again the holotrichous isorhiza were the ones that presented the higher variations between seasons, however in H. vulgaris pedunculata, the first place was for the desmonemes. When analyzing these results as a whole, the holotrichous occupy the second place in terms of variation with a percentage of 54% below the atrichous isorhiza with 61%. Deserti, M.I. & Acuña, F.H.: Statistical analysis on the cnidome of Hydra Pap. Avulsos Zool., 2022;v.62: e202262023 9/13 The analysis by species at the lagoon level showed that the sizes of all the cnidocysts showed variations, with the exception of the H. vulgaris pedunculata desmoneme, which only presented differences between La Brava and Los Padres. This result may be representing changes at the physiological level related to the different limnological variables that characterize each lagoon. We could assume, that the cnidome of a species has certain flexibility when it comes to environments. This situation is similar to that recorded by Acuña & Zamponi (1997) in sea anemones belonging to different intertidal zones. The authors observed that some of their nematocysts present significant variations in their sizes according to the area they inhabit, describing these species from different areas as "ecological races", since these differences are not enough to declare different species.
The last of the comparisons, and perhaps the most relevant and most applicable, is the one that allows us to discern between different species. Although we already know that a characteristic alone does not determine one species or another, it is important to know if the cnidome has sufficient statistical power to be used as a taxonomic feature. Since the cnidome is a fundamental requirement when describing a cnidarian species, these data can become an important taxonomic argument and not simply be a numerical description.
In our analysis, the three species were different from the point of view of their cnidocysts. When we compare H. viridissima with the remaining species, we see that all its cnidocysts show variations. However, when we compare H. vulgaris with H. vulgaris pedunculata, they only differ at the level of their holotrichous isorhiza.
These results reinforce the fact that H. viridissima belongs to viridissima group, while H. vulgaris and H. vulgaris pedunculata belong to vulgaris group. When we analyze different groups we note that the variations intensify, while in those of the same group they decrease. For H. vulgaris and H. vulgaris pedunculata we are analyzing species and subspecies, so a much smaller variation in their cnidocysts makes even more sense. It is not surprising that, once again, holotrichous isorhiza are the exception in the latter case. The results show that the subspecies retains, for most of its cnidocysts, the biometric identity of the species that it is derived from.  carry out a similar study, using a smaller amount of data and different statistical methods. However, the final results were the same: species from different groups showed differences in their cnidocysts, while those belonging to the same group did not. It is likely that in the analysis presented here, the increase in the number of measurements has allowed us to detect the difference of the holotrichous isorhiza for H. vulgaris and H. vulgaris pedunculata.
It seems clear that cnidomes of different species have well-defined identities that can be used to distinguish, at least, between groups. The measurement of cnidocysts, although often not very seductive due to the difficulty represented by the small sizes and the structural complexity in the Hydrozoa (David et al., 2008), seems to have its reward; gives us a powerful statistical result to be used in a taxonomy. However, knowledge of cnidocysts at the level of data distribution and intraspecific variations is of vital importance. Such knowledge will prevent us from believing that certain "ecological races" are, in fact, different species. As we can see, there is a large amount of information contained within the cnidome of a species. As the exploration of its statistical characteristics progresses in greater depth, we can choose more suitable methodologies for its analysis that yield more robust results. Many times, the amount of data used in a study is not what gives robustness to the results, but rather the application of a correct methodology based on the intrinsic characteristics of the measurements. As we advance in this knowledge, the use of the cnidome in taxonomic issues will be more reliable.

CONFLICTS OF INTEREST:
Authors declare that there is no conflict of interest.

FUNDING INFORMATION:
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.