Seed germination responses to varying environmental conditions and provenances in Crucianella maritima L., a threatened coastal species

Seed germination (effects of light, temperature, NaCl and KNO₃) of the coastal endangered species Crucianella maritima was investigated by testing seeds from three different populations. Data were analyzed by means of Generalized Linear Mixed Model (GLMM). The principal results showed that germination of C. maritima seeds was characterized by photoinhibition, absence of primary dormancy and salt-induced secondary dormancy, with no need for high nutrient availability (KNO₃). Intraspecific differences in germination pattern emerged, apparently due to a different seed mass. These results show important germination traits of C. maritima which should be taken into account in possible reintroduction attempts aimed at restoring threatened populations of this species.     

In vitro biosynthesis of the cyanogenic glucoside taxiphyllin in Triglochin maritima

When l-tyrosine and NADPH were incubated with a microsomal fraction from Triglochin maritima seedlings, 4-hydroxymandelonitrile (measured as 4-hydroxybenzaldehyde and HCN) and 4-hydroxyphenylacetonitrile were synthesized. Taxiphyllin, a cyanogenic glucoside occurring in this plant, was formed when UDPglucose and a soluble protein fraction from T. maritima, were also included in the assay. The above glucosyl transferase activity also produced taxiphyllin when incubated with R,S-4-hydroxymandelonitrile and UDPglucose. No trace of dhurrin, the enantiomer of taxiphyllin, was detected. The highest microsomal activity was found in 5- to 6-day-old etiolated seedlings. It was absolutely necessary to remove all seed coats from the seedlings in order to obtain an active microsomal fraction.     

How environmental stressors affect reproductive potential in a saltmarsh plant species Plantago maritima

We examined whether the presence or absence of different environmental stressors influenced the reproductive potential of a saltmarsh species—Plantago maritima. We focused on total seed output, seed quality, and biomass of progeny. So far, there are no studies trying to answer the question of how different saltmarsh management affects the quality of seed in saltmarsh species. For the purposes of the study, plots subjected to light mowing, light or heavy grazing, trampling, or rooting were designated in three nature reserves in Poland. On each plot, the abundance of infructescences per sq. meter was calculated. Mature infructescences were collected, and their length and number of fruit capsules were measured. The seeds obtained from fruit capsules were weighted and sown in controlled conditions. The germination rate and the final germination percentage were calculated. A representative number of sprouts were grown. After a period of 2 months, the individuals of P. maritima were harvested and their total dry mass was measured. It was found that heavy grazing had the greatest effect on all of the studied characteristics. The presence of this factor resulted in shorter infructescences with a smaller number of fruit capsules. However, this phenomenon was compensated by the higher abundance of infructescences per sq. meter. At the same time, seeds produced by grazed individuals were significantly lighter. Interestingly, intensive trampling by people affected P. maritima individuals in a similar way to heavy grazing, while mowing and rooting had less impact on the considered characteristics. Although a positive correlation between seed mass and germination success was found, the altogether lower seed mass had a negligible effect on germination parameters. Also, the differences in seed parameters did not affect dry mass of obtained progeny grown in laboratory conditions. Synthesis and applications: Different environmental stressors, such as grazing and mowing, have an effect on reproductive potential of a saltmarsh species P. maritima. In the case of habitats created anthropogenically, such as brackish saltmarshes, the role of management is crucial for their conservation. Therefore, searching for the best active protection methods is important. In light of the results obtained, extensive or rotational grazing appears to be the best form of saltmarsh management.     

High-resolution structure of shikimate dehydrogenase from Thermotoga maritima reveals a tightly closed conformation

Shikimate dehydrogenase (SDH), which catalyses the NADPH-dependent reduction of 3-dehydroshikimate to shikimate in the shikimate pathway, is an attractive target for the development of herbicides and antimicrobial agents. Structural analysis of a SDH from Thermotoga maritima encoded by the Tm0346 gene was performed to facilitate further structural comparisons between the various shikimate dehydrogenases. The crystal structure of SDH from T. maritima was determined at 1.45 Å by molecular replacement. SDH from T. maritima showed a monomeric architecture. The overall structure of SDH from T. maritima comprises the N-terminal α/β sandwich domain for substrate binding and the C-terminal domain for NADP binding. When the T. maritima SDH structure was compared with those of the SDHs from other species, the SDH from T. maritima was in a tightly closed conformation, which should be open for catalysis. Notably, α7 moves toward the active site (∼5 Å), which forces the SDH of T. maritima in a more closed form. Four ammonium sulfate (AMS) ions were identified in the structure. They were located in the active site and appeared to mimic the role of the substrate in terms of the enzyme activity and stability. The new high resolution structural information reported in this study, including the AMS binding sites as a potent inhibitor binding site of SDHs, is expected to supplement the existing structural data and will be useful for structure-based antibacterial discovery against SDHs.     

Species-specific effects of elevated CO2 on resource allocation in Plantago maritima and Armeria maritima

We investigated the effects of increased atmospheric CO₂ on the biomass, photosynthesis, protein and phenolic concentrations and content of Plantago maritima and Armeria maritima. This enabled us to test the protein competition model (PCM) for predicting C allocation to phenolics. Three contrasting responses to elevated CO₂ (600 μmol CO₂ mol⁻¹) between the two study species were observed. (1) In P. maritima, plant biomass increased and the maximum carboxylation rate of Rubisco (V_c,max) was decreased. However, in A. maritima, shoot biomass decreased and the V_c,max of Rubisco was unchanged. (2) The total phenolic content increased in P. maritima but decreased in A. maritima. (3) Protein concentrations and content decreased in P. maritima and root protein concentrations and content increased in A. maritima. We conclude that C and N allocation to phenolics and proteins is species- and organ-specific and the PCM predictions were correct when phenolics and proteins were expressed on a per plant content basis.     

Factors Influencing Seedling Establishment Rates in Zostera marina and Their Implications for Seagrass Restoration

Selection of strategies to efficiently utilize limited seed supplies in efforts to restore the seagrass Zostera marina (eelgrass) requires a better understanding of the processes that limit seedling establishment at potential restoration sites. We investigated the effect of seed distribution timing on seedling establishment and tested for interactive effects of seed burial and distribution timing. We also investigated the effect of habitat type on seedling establishment by distributing Z. marina seeds inside and outside of established Ruppia maritima (widgeongrass) patches and examined mechanisms causing habitat differences by manipulating seed position (buried or unburied) and vulnerability to seed predators (unprotected or protected in packets). Seeds distributed on the sediment surface in the summer (July or August) produced fewer seedlings than seeds distributed in fall (October) in five of six trials over 3 years. Seed burial increased success rates for seeds distributed in summer at one of two sites tested, eliminating the effect of season, but reduced success at the other site. Seeds placed in R. maritima generally produced fewer seedlings than seeds in bare sand, and although seed burial and protection in packets increased success in bare sand at three of four sites, the effect was less consistent in R. maritima. We conclude that seed predation and physical interactions were influential in reducing seedling establishment in R. maritima, contrary to hypotheses positing a nursery role for existing vegetation. Efficient restoration efforts with Z. marina seeds should target unvegetated areas after summertime sources of mortality have diminished. Direct seed burial may enhance seedling establishment rates.     

Isothiocyanates,nitriles and thiocyanates as products of autolysis of glucosinolates in Cruciferae

Glucosinolates from seventy-nine 8-week-old plant species were hydrolysed and the volatile products identified by GC-MS and related to previous published findings. Known compounds, identified in new plant sources, were 4-methylthiobutyl thiocyanate in Alyssum, 4-methylthiobutyl isothiocyanate in Diplotaxis and Eruca and isopropyl isothiocyanate and 5-vinyl-2-oxazolidinethione in Plantago.     

Protocols for Use of Potamogeton perfoliatus and Ruppia maritima Seeds in Large‐Scale Restoration

Restoration of submerged aquatic vegetation from seed has been hampered by a lack of information on the appropriate conditions for collecting, processing, and storing seeds prior to dispersal. Seeds must be processed and stored under conditions that maintain seed viability, meet dormancy requirements, and prevent premature germination. This study examined the effects of collection date, processing technique, aeration, storage and induction temperature and salinity, and storage period on seed germination of two mesohaline aquatic species, Potamogeton perfoliatus and Ruppia maritima. Collection date and processing technique were significant factors affecting seed yield from donor populations. Seeds of both species remained viable and germinated best when stored at 4°C, and then exposed to freshwater induction conditions. However, their responses to other factors differed. Aeration during storage was necessary in order to maintain viability of P. perfoliatus seeds, whereas it was unnecessary for R. maritima seeds. Storage in freshwater at 4°C prevented germination of P. perfoliatus seeds, while high salinity during cold storage was necessary to minimize premature germination of R. maritima. Mean germination time of P. perfoliatus was dependent on storage salinity; in contrast, mean germination time of R. maritima seeds was dependent on induction salinity. These differences indicate that the methods required to produce large quantities of underwater plant seed amenable to large‐scale restoration efforts must be tailored to the specific requirements of individual species and must consider the range of processes from initial harvest through seed testing prior to field establishment.     

Host-parasite relationship between colonial terns and bacteria is modified by a mutualism with a plant with antibacterial defenses

Predator–prey and host-parasite interactions and mutualisms are common and may have profound effects on ecosystems. Here we analyze the parasitic and mutualistic associations between three groups of organisms: the plant Artemisia maritima, bacteria, and a colonial seabird (the sandwich tern Sterna sandvicensis) that breeds in dense colonies covered in feces produced by both adults and chicks. A disproportionately large fraction of colonies of the sandwich tern in Denmark were located in patches covered by A. maritima. This association was specific for the densely colonial sandwich tern, but was not present for four other sympatric species of terns that breed in much less dense colonies. A. maritima reduced the abundance of pathogenic Staphylococcus on chicken eggshells in a field experiment. Recruitment by sandwich terns breeding in patches of A. maritima was 18 % higher than for sandwich terns breeding in the absence of A. maritima. A. maritima benefitted from the association with sandwich terns due to the supply of nutrients from feces and uneaten food lost by young. These findings are consistent with sandwich terns exploiting the association with A. maritima and its antimicrobial properties to improve their reproductive success, while sandwich terns and A. maritima are involved in a mutualistic interaction.     

Factors affecting the population dynamics of Suaeda maritima at initial stages of development

This study describes life history and population dynamics of the annual halophyte Suaeda maritima, at initial stages of seedling development, in salt-marshes of the Mont-Saint-Michel Bay. Effects of salinity on seed germination and seedling development were studied both in the laboratory and in the field. Salinity was measured as sediment electrical conductivity. Experiments in the laboratory showed that Suaeda maritima was relatively tolerant to salinity during the germination stage. In the field, germination occurred when soil salinity was minimal. Almost all seeds germinated in the field; this may explain the absence of a persistent seed bank. Salinity did not seem to affect the growth of seedlings either. Mortality in the field was attributed to physical factors such as anoxia during temporary immersion, burial by sediment and tidal uprooting. The intensity of these effects varied according to the location of Suaeda maritima seedlings: in hollows, along the edge of tidal creeks, in gaps among dense vegetation cover, or in pattern with Puccinellia maritima. Perennial vegetation did not restrict Suaeda maritima germination. In contrast, perennials such as Puccinellia maritima and Halimione portulacoides limited the development of Suaeda maritima seedlings since only a small number of this annual emerged beneath their canopies. The fate of seedlings depended to a great extent on the permeability of the perennial canopy to light. Thus, disturbance resulting in bare patches inside the perennial vegetation seemed essential for the development of this annual species. Suaeda maritima can grow in the presence of Puccinellia maritima so long as the latter present as an open matrix. Suaeda maritima may also benefit from protection against desiccation and tidal action where this occurs.     

Ageing effects in an iteroparous plant species with a variable life span

Background and Aims

Ageing effects may be due to dysfunction leading to decreasing reproduction and survival with age. In plants, however, other (physiological) causes, associated with size for example, may also play a role. Iteroparous plants with genetically variable life spans can be helpful in unravelling these two aspects of changes associated with growing older.

Methods

In a long-term experiment, Beta vulgaris ssp. maritima (sea beet) plants from the same set of populations but with different ages were compared for flowering date over several years. Flowering date, root growth and seed production were measured in a synthetic population and in progenies derived from reciprocal crosses over three consecutive years and analysed with respect to the number of years yet to live. Heritabilities of these three characters and of life span were estimated.

Key Results

Flowering occurred on average 1·3 d later each year over a plant''s whole lifetime. In the year before dying, plants flowered on average 3·3 d later and both root investment and seed production decreased significantly compared with plants that remained alive for at least 1 further year. The negative relationship (trade-off) between reproduction and root investment in early life became positive near the end of life, and the positive relationship between flowering date and root growth became negative.

Conclusions

Effects of ageing – in the sense of a decline in reproduction and root storage – combined with later flowering were particularly pronounced in the year before death. The gradual change in flowering phenology, observed over the whole lifetime, could have a physiological basis unrelated to dysfunction.Key words: Ageing, Beta vulgaris ssp. maritima (sea beet), flowering phenology, longevity, perennial, root investment, seed production, trade-offs, whole-plant senescence     

Note on organic dormancy of estuarine Ruppia maritima L. seeds

Information on seed dormancy is one of the primary requirements for successful seedling propagation of submerged aquatics and seagrass. Studies on Ruppia maritima seed germination have been done, but the presence, requirements, and the types of dormancy have not been well understood. A laboratory study was conducted to understand presence and types of organic dormancy of the seeds of estuarine R. maritima collected from Lake Pontchartrain, Louisiana, USA. Our study results indicate that the brackish estuarine R. maritima population produces seeds that do not have any noticeable initial morphological, physical, and physiological dormancy. Although dry stratification reduced seed viability and final germination rates, drying seems to induce an earlier germination in R. maritima. Desiccation also appears to induce an environmental dormancy that can be disrupted by exposure to water. Further study on environmental dormancy is needed to provide information to develop methods for long-term seed storage that can be employed in greenhouse seedling propagation. Handling editor: P. Viaroli     

COMPARISON OF SEED PROTEINS IN THE GENUS SUAEDA (CHENOPODIACEAE) BY MEANS OF DISC GEL ELECTROPHORESIS

Polyacrylamide gel disc electrophoresis was utilized to determine seed protein patterns in ten taxa within the genus Suaeda. The results indicated a close relationship, averaging 92.2 %, similarity between the seed protein patterns of S. maritima var. macrocarpa, S. maritima var. vulgaris, and S. maritima var. flexilis. These intraspecific forms were more similar to one another than to any of the other taxa investigated. A continental North American species, S. depressa, was related to the S. maritima complex, and its protein pattern indicates that it may have evolved from S. maritima. The technique of disc electrophoresis proved useful in assessing the degree of similarity between taxa within the genus Suaeda.     

Toxicity of Glucosinolates and Their Enzymatic Decomposition Products to Caenorhabditis elegans

An aquatic 24-hour lethality test using Caenorhabditis elegans was used to assess toxicity of glucosinolates and their enzymatic breakdown products. In the absence of the enzyme thioglucosidase (myrosinase), allyl glucosinolate (sinigrin) was found to be nontoxic at all concentrations tested, while a freeze-dried, dialyzed water extract of Crambe abyssinica containing 26% 2-hydroxyl 3-butenyl glucosinolate (epi-progoitrin) had a 50% lethal concentration (LC₅₀) of 18.5 g/liter. Addition of the enzyme increased the toxicity (LC₅₀ value) of sinigrin to 0.5 g/liter, but the enzyme had no effect on the toxicity of the C. abyssinica extract. Allyl isothiocyanate and allyl cyanide, two possible breakdown products of sinigrin, had an LC₅₀ value of 0.04 g/liter and approximately 3 g/liter, respectively. Liquid chromatographic studies showed that a portion of the sinigrin decomposed into allyl isothiocyanate. The results indicated that allyl isothiocyanate is nearly three orders of magnitude more toxic to C. elegans than the corresponding glncosinolate, suggesting isothiocyanate formation would improve nematode control from application of glucosinolates.     

Do Conditions During Dormancy Influence Germination of Suaeda maritima?

Background and Aims

Seeds of annual halophytes such as Suaeda maritima experience fluctuating salinity, hydration, hypoxia and temperature during dormancy. Germination then occurs in one flush of 2–3 weeks after about 5 months of winter dormancy during which time the seeds can remain in saline, often waterlogged soil. The aim of this study was to investigate the effect of simulated natural conditions during dormancy on germination and to compare this with germination following the usual conditions of storing seeds dry. The effects of hydration, salinity, hypoxia and temperature regimes imposed during dormancy on germination were investigated. Also looked at were the effects of seed size on germination and the interaction between salinity during dormancy and salinity at the time of germination.

Methods

Various pre-treatments were imposed on samples of seeds that had been stored dry or wet for different periods of time during the 5 months of natural dormancy. Subsequent germination tests were carried out in conditions that simulated those found in the spring when germination occurs naturally. Various salinities were imposed at germination for a test of interaction between storage salinity and salinity at germination.

Key Results

A temperature of about 15 °C was needed for germination and large seeds germinated earlier and better than small seeds. Cold seawater pre-treatment was necessary for good germination; the longer the saline pre-treatment during the natural dormancy period the better the germination. There appeared to be no effect of any specific ion of the seawater pre-treatment on germination and severe hypoxia did not prevent good germination. A short period of freezing stimulated early germination in dry-stored seed. Storage in cold saline or equivalent osmotic medium appeared to inhibit germination during the natural dormancy period and predispose the seed to germinate when the temperature rose and the salinity fell. Seeds that were stored in cold wet conditions germinated better in saline conditions than those stored dry.

Conclusions

The conditions under which seeds of S. maritima are stored affect their subsequent germination. Under natural conditions seeds remain dormant in highly saline, anoxic mud and then germinate when the temperature rises above about 15 °C and the salinity is reduced.Key words: Suaeda maritima, germination, pre-treatment, salinity, temperature     

Resistance to water stress of Alnus maritima: intraspecific variation and comparisons to other alders

Alnus maritima (seaside alder) is a rare and threatened woody species that occurs naturally as three disjunct subspecies. While its limited numbers and distribution have prompted investigations into the cause of its rarity, the aesthetic appeal and apparent flood resistance of A. maritima have led to a strong interest in its horticultural potential. The occurrence of all three subspecies of A. maritima exclusively in waterlogged soils in nature suggests that water availability may limit both the success of plants in the wild and the potential for their use in landscapes. We conducted three experiments to assess how A. maritima would respond to a wide range of root-zone moisture contents. First, we examined foliar gas exchange and growth of one clone during drought and flooding. Plants survived, grew, and avoided damage to and abscission of leaves during seven treatments that ranged from complete root-zone inundation to severe drought. Partial flooding and slight drought maximized plant dry weight, while total flooding and mild, moderate, and severe drought limited growth. High rates of photosynthesis were sustained except during severe drought. Next, we examined genotypic variation in drought resistance among the subspecies of A. maritima and compared the resistance of A. maritima to that of four other species of Alnus. Only subtle differences were detected among subspecies in how deficit irrigation influenced the physiology and development of plants. Of the five species evaluated, A. maritima and Alnus nepalensis most strongly expressed indicators of drought resistance. Lastly, a field trial with 270 seedlings of subspecies oklahomensis confirmed that A. maritima can be grown on soils with moisture contents ranging from dry (≈10% water by volume) to saturated. We conclude that, even though A. maritima thrives when a portion of its root zone is constantly inundated, it is more drought tolerant than some other species indigenous to well drained soils. The drought resistance of A. maritima can be attributed in part to its continued high rate of photosynthesis, fine control of stomatal conductance, increased specific leaf weight, and increased root:shoot ratio under conditions of drought. Considering the natural habitat to which A. maritima is restricted, our data illustrate that the niche a species occupies in nature does not necessarily represent how adapted the species may be to dissimilar environmental conditions. We also conclude that A. maritima and A. nepalensis show promise for increased use as ornamentals resistant to disparate soil water contents.     

Growth, sodium uptake and antioxidant responses of coastal plants differing in their ecological status under increasing salinity

In the present study, we compared the response to salinity of three plants from Brittany coast with contrasted ecological status: Limonium latifolium (salt marshes), Matricaria maritima (beach tops and sand dunes) and Crambe maritima (fixed dunes). Under controlled glasshouse conditions, the growth of the three plants decreased with increasing external salinity. L. latifolium and C. maritima exhibited the highest and lowest resistance to severe salt stress (400 mM), respectively. M. maritima could be considered as an intermediate species, since it tolerated salinity up to 200 mM. The same observation could be made with sodium absorption and acuumulation in plant tissues, the most tolerant species (L. latifolium being the least Na accumulator. Hydrogen peroxide (H₂O₂) and malondialdehyde (MDA), commonly produced in conditions of stress, accumulated significantly in salt treated C. maritima and M. maritima while not in the tolerant L. latifolium. The latter used glutathione reductase to maintain constant H₂O₂ levels under salt stress while peroxidases were very low and ascorbate peroxidase did not respond to salinity stimulation. The medium tolerant halophyte M. maritima used peroxidases to protect from NaCl-induced H₂O₂, while the sensitive C. maritima failed to detoxify H₂O₂ despite a sharp increase in catalase activity. Results showed that the three coastal species differ in resistance to salinity. They also suggested that the level of plant resistance to salinity could be attributed to differing mechanisms to manage the accumulation of sodium and cope with the oxidative damages.     

Effects of pH and ascorbate on benzylglucosinolate degradation in seed extracts of Lepidium sativum

The effects of pH on the enzymic degradation of benzylglucosinolate in Lepidium sativum seed autolysates were investigated both with and without addition of the enzyme co-factor ascorbic acid. Benzyl cyanide, isothiocyanate, thiocyanate and alcohol were identified in autolysates, although only traces of the alcohol were obtained. The nitrile was always the major product (80% of total glucosinolate products) even at pH 8 and 9 when the usually accepted, proton-dependent mechanism of nitrile production cannot be operative. Thiocyanate was always the second most abundant product. In the absence of added ascorbate, isothiocyanate production decreased with increasing pH, again contrary to accepted theory. L. sativum seeds thus constitute an inherently nitrile-producing system which exhibits ‘anomalous’ glucosinolate degradation. In the absence of added ascorbate, thiocyanate was the only product which was formed in approximately constant amounts, whatever the pH, so its mechanism of production is not necessarily pH-dependent. The presence of added ascorbate in general promoted enzyme activity and showed a maximum effect at ca pH 5, although minimum isothiocyanate formation was observed at that pH. At pH 4 and below, there was less glucosinolate degradation in the presence of added ascorbate than in its absence, and the conclusion is reached that at relatively high acidities the enzyme co-factor behaves as an inhibitor.     

Studies on glucosinolate degradation in Lepidium sativum seed extracts

Analysis of Lepidium sativum seeds showed the presence of allyl, 2-phenethyl and benzyl glucosinolates, the first two being reported for the first time from this source. The effects of temperature, pH of the extraction medium and the length of time allowed for autolysis were assessed on the benzyl glucosinolate degradation products in seed extracts. In particulàr benzyl thiocyanate was not produced at higher temperatures but at ambient and lower temperatures it exceeded isothiocyanate. Nitrile was always the major product under the conditions studied, ever at pH levels as high as 7.4. Five new possible benzyl glucosinolate degradation products were detected and evidence is presented that benzaldehyde and benzyl alcohol could be secondary products formed thermally from isothocyanate and thiocyanate, respectively. Benzyl mercaptan and benzyl methyl sulphide also appear to be thermally produced.     

Hydrogen production by the hyperthermophilic bacterium <Emphasis Type="Italic">Thermotoga maritima</Emphasis> Part II: modeling and experimental approaches for hydrogen production

Background

Thermotoga maritima is a hyperthermophilic bacterium known to produce hydrogen from a large variety of substrates. The aim of the present study is to propose a mathematical model incorporating kinetics of growth, consumption of substrates, product formations, and inhibition by hydrogen in order to predict hydrogen production depending on defined culture conditions.

Results

Our mathematical model, incorporating data concerning growth, substrates, and products, was developed to predict hydrogen production from batch fermentations of the hyperthermophilic bacterium, T. maritima. It includes the inhibition by hydrogen and the liquid-to-gas mass transfer of H₂, CO₂, and H₂S. Most kinetic parameters of the model were obtained from batch experiments without any fitting. The mathematical model is adequate for glucose, yeast extract, and thiosulfate concentrations ranging from 2.5 to 20 mmol/L, 0.2–0.5 g/L, or 0.01–0.06 mmol/L, respectively, corresponding to one of these compounds being the growth-limiting factor of T. maritima. When glucose, yeast extract, and thiosulfate concentrations are all higher than these ranges, the model overestimates all the variables. In the window of the model validity, predictions of the model show that the combination of both variables (increase in limiting factor concentration and in inlet gas stream) leads up to a twofold increase of the maximum H₂-specific productivity with the lowest inhibition.

Conclusions

A mathematical model predicting H₂ production in T. maritima was successfully designed and confirmed in this study. However, it shows the limit of validity of such mathematical models. Their limit of applicability must take into account the range of validity in which the parameters were established.