Anales del Jardín Botánico de Madrid, Vol 72, No 1 (2015)

A Gompertz regression model for fern spores germination

Jose María Gabriel y Galán
Department of Plant Sciences (Botany), Faculty of Biological Sciences, Universidad Complutense, Spain

Carmen Prada
Department of Plant Sciences (Botany), Faculty of Biological Sciences, Universidad Complutense, Spain

Cristina Martínez-Calvo
Department of Applied Mathematics (Biomathematics), Faculty of Biological Sciences, Universidad Complutense de Madrid, Spain

Rafael Lahoz-Beltrá
Department of Applied Mathematics (Biomathematics), Faculty of Biological Sciences, Universidad Complutense de Madrid, Spain


Germination is one of the most important biological processes for both seed and spore plants, also for fungi. At present, mathematical models of germination have been developed in fungi, bryophytes and several plant species. However, ferns are the only group whose germination has never been modelled. In this work we develop a regression model of the germination of fern spores. We have found that for Blechnum serrulatum, Blechnum yungense, Cheilanthes pilosa, Niphidium macbridei and Polypodium feuillei species the Gompertz growth model describe satisfactorily cumulative germination. An important result is that regression parameters are independent of fern species and the model is not affected by intraspecific variation. Our results show that the Gompertz curve represents a general germination model for all the non-green spore leptosporangiate ferns, including in the paper a discussion about the physiological and ecological meaning of the model.


Leptosporangiate ferns; mathematical model; non-green spore; spore germination modelling

Full Text:



Angell, A. C. & Kielland, K. 2009. Establishment and growth of white spruce on a boreal forest floodplain: Interactions between microclimate and mammalian herbivory. Forest Ecology and Management 258, 2475-2480.

Ashcroft, C. J. & Sheffield, E. 2000. The effect of spore density on germination and development in Pteridium, monitored using a novel culture technique. American Fern Journal 90: 91-99.

Baskin, C. C. & Baskin, J. M. 1998. Seeds: ecology, biogeography, and evolution of dormancy and germination. San Diego, CA. Academic Press.

Bewley, J. D. 1997. Seed germination and dormancy. Plant Cell 9: 1055-1066. PMid:12237375 PMCid:PMC156979

Bosch, A., Maronna, R. A. & Yantorno, O. M. 1995. A simple descriptive model of filamentous fungi spore germination. Process Biochemistry 30: 599-606.

Bradbeer, J. W. 1988. Seed dormancy and germination. Glasgow. Blackie and Son Ltd.

Brown, R. F. & Mayer, D. G. 1988. Representing Cumulative Germination. 2. The Use of the Weibull Function and Other Empirically Derived Curves. Annals of Botany 61: 127-138.

Cheng, C. & Gordon, I. L. 2000. The Richards function and quantitative analysis of germination and dormancy in meadowfoam (Limnanthes alba). Seed Science Research 10: 265-277.

Conway, E. 1949. The autecology of bracken (Pteridium aquilinum (L.) Kuhn) The germination of the spore, and the development of the prothallus and the young sporophyte. Proceeding of the Royal Society of Edinburgh 63 B: 325-342.

Dantigny, P., Nanguy, S-M., Judet-Correia, D. & Bensoussan, M. 2011. A new model for germination of fungi. International Journal of Food Microbiology 146: 176-181. PMid:21396731

Draper, N. & Smith, H. 1998. Applied regression analysis. New York. John Wiley & Sons.

Dyer, A. 1979. The culture of fern gametophytes for experimental investigation. In: A. Dyer (ed.) The experimental biology of ferns: 254-305. London.

Finch-Savage, W. E. & Leubner-Metzger, G. 2006. Seed dormancy and the control of germination. New Phytologist 171: 501-523. PMid:16866955

Fitzhugh, H. A. 2011. Analysis of growth curves and strategies for altering their shape. Journal of Animal Science 42: 1036-1051.

Gabriel y Galán, J. M. 2010. Gametophyte development and reproduction of Niphidium macbridei Lellinger (Polypodiaceae, Pteridophyta). Nova Hedwigia 90: 513-519.

Gabriel y Galán, J. M. 2011. Gametophyte development and reproduction of Argyrochosma nivea (Poir.) Windham (Pteridaceae). Biologia 65: 50-54.

Gabriel y Galán, J. M., Migliaro, G. & Lahoz-Beltrá, R. 2011. Effect of temperature and dark pre-treatment in the germination of three species of Jamesonia (Pteridaceae, Polypodiopsida). Plant Species Biology 26: 254-258.

Gabriel y Galán, J. M., Passarelli, L. M., Prada, C. & Rolleri, C. H. 2008a. Sporophyte morphology and gametophyte development of the fern Blechnum sprucei (Pteridophyta: Blechnaceae). Revista De Biologia Tropical 56: 2027-2040. PMid:19419099

Gabriel y Galán, J. M. & Prada, C. 2009. Gametophytes of Pleurosorus papaverifolius (Kunze) Fee (Aspleniaceae) and Cheilanthes glauca (Cav.) Mett. (Pteridaceae), two South American fern. Acta Botanica Brasilica 23: 805-811.

Gabriel y Galán, J. M. & Prada, C. 2010a. Gametophyte of the Andean fern Cheilanthes pilosa Goldm. (Pteridaceae). American Fern Journal 100: 32-38.

Gabriel y Galán, J. M. & Prada, C. 2010b. Gametophyte of the Andean fern Polystichum pycnolepis (Kunze ex Klotzsch) T. Moore (Dryopteridaceae). American Fern Journal 100: 103-109.

Gabriel y Galán, J. M. & Prada, C. 2010c. Pteridophyte spores viability. In: H. Fernández, A. Kumar & M. A. Revilla (eds.), Working with ferns: issues and applications: 193-205. New York.

Gabriel y Galán, J. M. & Prada, C. 2012. Farina production by gametophytes of Argyrochosma nivea (Poir.) Windham (Pteridaceae) and its implications for cheilanthoid phylogeny. American Fern Journal 102: 191-197.

Gabriel y Galán, J. M., Prada, C. & Rolleri, C. H. 2008b. Germinación de la espora y desarrollo del gametófito del helecho americano Polypodium feuillei Bert. Gayana Botanica 65: 14-22.

Goodchild, N. A. & Walker, M. G. 1971. A method of measuring seed germination in physyiological studies. Annals of Botany 35: 615-621.

Hsu, F. H., Nelson, C. J. & Chow, W. S. 1984. A mathematical-model to utilize the logistic function in germination and seedling growth. Journal of Experimental Botany 35: 1629-1640.

Lapp, M. S. & Skoropad, W. P. 1976. A mathematical model of conidial germination and appressorial formation for Colletotrichum graminicola. Canadian Journal of Botany 54: 2239-2242.

Lloyd, R. & Klekowski, E. J. 1970. Spore germination and viability in pteridophyta: evolutionary significance of chlorophyllous spores. Biotropica 2: 129-137.

Maguire, J. D. 1962. Speed germination-aid in selection and evaluation for seedling emergence and vigour. Crop Science 2: 176-177.

Marquardt, D. 1963. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics 11: 431-441.

Migliaro, G. & Gabriel Y Galán, J. M. 2012. Gametophyte development and reproduction of the Asian fern Polystichum polyblepharum (Roem. ex Kunze) C. Presl (Dryopteridaceae, Polypodiopsida). Plant Biosystems 146: 368-373.

Neter, J., Kutner, M. H., Nachtsheim, C. J. & Wasserman, W. 1996. Applied Linear Statistical Models. Richard D. Irwin, Inc., Chicago, IL.

O'Neill, M., Thompson, P., Jacobs, B., Brain, P., Butler, R., Turner, H. & Mitakda, B. 2004. Fitting and comparing seed germination models with a focus on the inverse normal distribution. Australian & New Zealand Journal of Statistics 46: 349-366.

Myers, R. H. 1990. Classical and Modern Regression with Applications. Duxbury Press, Belmont, CA.

Prada, C. 2004. Helechos. In: J. Izco (ed.) Botánica: 385-415. Madrid.

Prada, C., Moreno, V. & Gabriel Y Galan, J. M. 2008. Gametophyte development, sex expression and antheridiogen system in Pteris incompleta Cav. (Pteridaceae). American Fern Journal 98: 14-25.[14:GDSEAA]2.0.CO;2

Raghavan, V. 1989. Developmental biology of fern gametophytes. Cambridge. University Press.

Ranal, M. A. & Garcia De Santana, D. 2006. How and why to measure the germination process? Revista Brasileira de Botânica 29: 1-11.

Roush, W. B., Dozier, W. A. & Branton, S. L. 2006. Comparison of Gompertz and neural network models of broiler growth. Poultry Science 85: 794-797. PMid:16615365

Sheffield, E. 1996. From pteridophyte spore to sporophyte in the natural environment. In: M. G. J. M. Camus, R. Johns (ed.) Pteridology in Perspective: 541-549. Kew.

Smith, D. L. & Robinson, P. M. 1975. Effects of spore age on germination and gametophyte development in Polypodium vulgare L. New Phytologist 74: 101-108.

Stat Trek. 2012. Statistics: Influential points.

Thompson, P. A. 1973. Seed germination in relation to ecological and geographical distribution. In: V. A. Heywood (ed.) Taxonomy and ecology: 93-119. London.

Torres, M. & Frutos, G. 1990. Analysis of germination curves of aged fennel seeds by mathematical-models. Environmental and Experimental Botany 29: 409-415.

Waggoner, P. E. & Parlange, J.-Y. 1973. Mathematical model for spore germination at changing temperatures. Phytopathology 64: 605-610.

Weinberg, E. S. & Voeller, B. R. 1969. Induction of fern spore germination. Proceedings of the National Academy of Sciences of the United States of America 64: 835-842. PMid:16591804 PMCid:PMC223310

Wiklund, K. & Rydin, H. 2004. Ecophysiological constraints on spore establishment in bryophytes. Functional Ecology 18: 907-913.

Wu, L. & Wang, Y. (2009). Fitting Gompertz curve using grey method. In: Proceedings of the 2009 IEEE International Conference on Systems, Man and Cybernetics. pp. 5093-5096. San Antonio.

Copyright (c) 2015 Consejo Superior de Investigaciones Científicas (CSIC)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Contact us:

Technical support: