vavilovВавиловский журнал генетики и селекцииVavilov Journal of Genetics and Breeding2500-3259Institute of Cytology and Genetics of Siberian Branch of the RASvavilov-196Research ArticleСтатьиArticlesМОДЕЛИРОВАНИЕ БИОМЕХАНИКИ И МОРФОДИНАМИКИ РАСТЕНИЙ В ПАКЕТЕ COMSOLMODELING OF PLANT BIOMECHANICS AND MORPHODYNAMICS IN THE COMSOL PACKAGEНиколаевС. В.NikolaevS. V.nikolaev@bionet.nsc.ruФедеральное государственное бюджетное научное учреждение «Федеральный исследовательский центр Институт цитологии и генетики Сибирского отделения Российской академии наук», Новосибирск, РоссияРоссияInstitute of Cytology and Genetics SB RAS, Novosibirsk, RussiaRussian Federation201311012015174/1724737Copyright © Николаев С.В., 20152015Николаев С.В.Nikolaev S.V.This work is licensed under a Creative Commons Attribution 4.0 License.https://vavilov.elpub.ru/jour/article/view/196

В статье дан краткий обзор пакета COMSOL Multiphysics; показаны способы построения модели и спецификации задачи в пакете COMSOL; продемонстрированы возможности пакета при изучении нескольких конкретных проблем биомеханики и морфодинамики растений.

This is a short review of the COMSOL Multiphysics package. Model building in COMSOL and methods of biophysical problem specification are demonstrated. Examples of the investigation of several problems in plant biomechanics and morphodynamics are considered.

COMSOL Multiphysicsметод конечных элементовматематическое моделированиебиомеханикаморфодинамикаатомная силовая микроскопияАСМклетки растенийстенка растительной клеткиCOMSOL Multiphysicsfinite element methodmathematical modelingbiomechanicsmorphodynamicsatomic force microscopyplant cellplant cell wallСуперкомпьютерный центрй Новосибирского государственного университета; гранты РФФИ-НИСИ № 11-04-91397-а, РФФИ № 11–04–01748-аReferencesБате К., Вилсон Е. Численные методы анализа и метод конечных элементов. M., Стройиздат, 1982. 448 с.Бате К., Вилсон Е. Численные методы анализа и метод конечных элементов. M., Стройиздат, 1982. 448 с.Николаев С.В., Колчанов Н.А., Голушко С.К. и др. Моделирование морфодинамики на ранних стадиях эмбриогенеза растения // Вавилов. журн. генет. и селекции. 2012. Т. 16. Вып. 4/1. С. 805–815.Николаев С.В., Колчанов Н.А., Голушко С.К. и др. Моделирование морфодинамики на ранних стадиях эмбриогенеза растения // Вавилов. журн. генет. и селекции. 2012. Т. 16. Вып. 4/1. С. 805–815.Тимошенко С.П. Сопротивление материалов. Т. 1. Элементарная теория и задачи. М.: Наука, 1965.Тимошенко С.П. Сопротивление материалов. Т. 1. Элементарная теория и задачи. М.: Наука, 1965.Фейнман Р., Лейтон Р., Сэндс М. Фейнмановские лекции по физике. Т. 7. Физика сплошных сред. М.: Мир, 1967.Фейнман Р., Лейтон Р., Сэндс М. Фейнмановские лекции по физике. Т. 7. Физика сплошных сред. М.: Мир, 1967.Atchley W.R., Hall B.K. A model for development and evolution of complex morphological structures // Biol. Rev. 1991. V. 66. P. 101–157.Atchley W.R., Hall B.K. A model for development and evolution of complex morphological structures // Biol. Rev. 1991. V. 66. P. 101–157.Barreira R., Elliott C., Madzvamuse A. The surface finite element method for pattern formation on evolving biological surfaces // J. Mathemat. Biol. 2011. V. 63. P. 1095–1119.Barreira R., Elliott C., Madzvamuse A. The surface finite element method for pattern formation on evolving biological surfaces // J. Mathemat. Biol. 2011. V. 63. P. 1095–1119.Boudaoud A. An introduction to the mechanics of morphogenesis for plant biologists // Trends Plant Sci. 2010. V. 15. P. 353–360.Boudaoud A. An introduction to the mechanics of morphogenesis for plant biologists // Trends Plant Sci. 2010. V. 15. P. 353–360.Braybrook S.A., Hofte H., Peaucelle A. Probing the mechanical contributions of the pectin matrix: insights for cell growth // Plant Signal Behav. 2012. V. 7. P. 1037–1041.Braybrook S.A., Hofte H., Peaucelle A. Probing the mechanical contributions of the pectin matrix: insights for cell growth // Plant Signal Behav. 2012. V. 7. P. 1037–1041.Bruce D.M. Mathematical modelling of the cellular mechanics of plants // Philos. Trans. Roy. Soc. Lond. Series B: Biological Sciences. 2003. V. 358. P. 1437–1444.Bruce D.M. Mathematical modelling of the cellular mechanics of plants // Philos. Trans. Roy. Soc. Lond. Series B: Biological Sciences. 2003. V. 358. P. 1437–1444.Campbell G.S., Norman J.M. An Introduction to Environmental Biophysics. 2nd ed. Springer, 2000. 286 p.Campbell G.S., Norman J.M. An Introduction to Environmental Biophysics. 2nd ed. Springer, 2000. 286 p.Chatziprodromou I., Tricoli A., Poulikakos D., Ventikos Y. Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model // J. Biomech. 2007. V. 40. P. 412–426.Chatziprodromou I., Tricoli A., Poulikakos D., Ventikos Y. Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model // J. Biomech. 2007. V. 40. P. 412–426.Dyson R., Band L., Jensen O. A model of crosslink kinetics in the expanding plant cell wall: Yield stress and enzyme action // J. Theor. Biol. 2012. V. 307. P. 125–136.Dyson R., Band L., Jensen O. A model of crosslink kinetics in the expanding plant cell wall: Yield stress and enzyme action // J. Theor. Biol. 2012. V. 307. P. 125–136.Fernandes A.N., Chen X., Scotchford C.A. et al. Mechanical properties of epidermal cells of whole living roots of Arabidopsis thaliana: an atomic force microscopy study // Phys. Rev. E. 2012. V. 85. Р. 021916.Fernandes A.N., Chen X., Scotchford C.A. et al. Mechanical properties of epidermal cells of whole living roots of Arabidopsis thaliana: an atomic force microscopy study // Phys. Rev. E. 2012. V. 85. Р. 021916.Geitmann A. Mechanical modeling and structural analysis of the primary plant cell wall // Curr. Opin. Plant Biol. 2010. V. 13. P. 693–699.Geitmann A. Mechanical modeling and structural analysis of the primary plant cell wall // Curr. Opin. Plant Biol. 2010. V. 13. P. 693–699.Hamant O., Traas J. The mechanics behind plant development // New Phytol. 2010. V. 185. P. 369–385.Hamant O., Traas J. The mechanics behind plant development // New Phytol. 2010. V. 185. P. 369–385.Hansen S.L., Ray P.M., Karlsson A.O. et al. Mechanical properties of plant cell walls probed by relaxation spectra // Plant Physiol. 2011. V. 155. P. 246–258.Hansen S.L., Ray P.M., Karlsson A.O. et al. Mechanical properties of plant cell walls probed by relaxation spectra // Plant Physiol. 2011. V. 155. P. 246–258.Heisler M.G., Hamant O., Krupinski P. et al. Alignment between PIN1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport // PLoS Biol. 2010. V. 8. e1000516.Heisler M.G., Hamant O., Krupinski P. et al. Alignment between PIN1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport // PLoS Biol. 2010. V. 8. e1000516.Kraft R.H., Mckee P.J., Dagro A.M., Grafton S.T. Combining the finite element method with structural connectomebased analysis for modeling neurotrauma: connectome neurotrauma mechanics // PLoS Comput Biol. 2012. V. 8. Р. e1002619.Kraft R.H., Mckee P.J., Dagro A.M., Grafton S.T. Combining the finite element method with structural connectomebased analysis for modeling neurotrauma: connectome neurotrauma mechanics // PLoS Comput Biol. 2012. V. 8. Р. e1002619.Kwiatkowska D. Structural integration at the shoot apical meristem: models, measurements, and experiments // Amer. J. Bot. 2004. V. 91. P. 1277–1293.Kwiatkowska D. Structural integration at the shoot apical meristem: models, measurements, and experiments // Amer. J. Bot. 2004. V. 91. P. 1277–1293.Laux T., Jürgens G. Embryogenesis: a new start in life // Plant Cell. 1997. V. 9. P. 989–1000.Laux T., Jürgens G. Embryogenesis: a new start in life // Plant Cell. 1997. V. 9. P. 989–1000.Milani P., Gholamirad M., Traas J. et al. In vivo analysis of local wall stiffness at the shoot apical meristem in Arabidopsis using atomic force microscopy // Plant J. 2011. V. 67. P. 1116–1123.Milani P., Gholamirad M., Traas J. et al. In vivo analysis of local wall stiffness at the shoot apical meristem in Arabidopsis using atomic force microscopy // Plant J. 2011. V. 67. P. 1116–1123.Mirabet V., Das P., Boudaoud A., Hamant O. The role of mechanical forces in plant morphogenesis // Annu. Rev. Plant Biol. 2011. V. 62. P. 365–385.Mirabet V., Das P., Boudaoud A., Hamant O. The role of mechanical forces in plant morphogenesis // Annu. Rev. Plant Biol. 2011. V. 62. P. 365–385.Missel P. Finite element modeling of diffusion and partitioning in biological systems: the infinite composite medium problem // Ann. Biomed. Eng. 2000. V. 28. P. 1307–1317.Missel P. Finite element modeling of diffusion and partitioning in biological systems: the infinite composite medium problem // Ann. Biomed. Eng. 2000. V. 28. P. 1307–1317.Nakielski J. The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions // Planta. 2008. V. 228. P. 179–189.Nakielski J. The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions // Planta. 2008. V. 228. P. 179–189.Ortega J.K. Augmented growth equation for cell wall expansion // Plant Physiol. 1985. V. 79. P. 318–320.Ortega J.K. Augmented growth equation for cell wall expansion // Plant Physiol. 1985. V. 79. P. 318–320.Peaucelle A., Braybrook S.A., Le Guillou L. et al. Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis // Curr. Biol. 2011. V. 21. P. 1720–1726.Peaucelle A., Braybrook S.A., Le Guillou L. et al. Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis // Curr. Biol. 2011. V. 21. P. 1720–1726.Proseus T.E., Ortega J.K., Boyer J.S. Separating growth from elastic deformation during cell enlargement // Plant Physiol. 1999. V. 119. P. 775–784.Proseus T.E., Ortega J.K., Boyer J.S. Separating growth from elastic deformation during cell enlargement // Plant Physiol. 1999. V. 119. P. 775–784.Rayfield E.J. Using finite-element analysis to investigate suture morphology: A case study using large carnivorous dinosaurs // Anat. Rec. Part A: Discoveries in Molecular, Cellular, and Evolutionary Biology. 2005. V. 283A. P. 349–365.Rayfield E.J. Using finite-element analysis to investigate suture morphology: A case study using large carnivorous dinosaurs // Anat. Rec. Part A: Discoveries in Molecular, Cellular, and Evolutionary Biology. 2005. V. 283A. P. 349–365.Rayfield E.J. Finite element analysis and understanding the biomechanics and evolution of living and fossil organisms // Annu. Rev. Earth Planet. Sci. 2007. Book Series: Annu. Rev. Earth Planet. Sci. V. 35. P. 541–576.Rayfield E.J. Finite element analysis and understanding the biomechanics and evolution of living and fossil organisms // Annu. Rev. Earth Planet. Sci. 2007. Book Series: Annu. Rev. Earth Planet. Sci. V. 35. P. 541–576.Richmond B.G., Wright B.W., Grosse I. et al. Finite element analysis in functional morphology // Anat. Rec. Part A: Discoveries in Molecular, Cellular, and Evolutionary Biology. 2005. V. 283A. P. 259–274.Richmond B.G., Wright B.W., Grosse I. et al. Finite element analysis in functional morphology // Anat. Rec. Part A: Discoveries in Molecular, Cellular, and Evolutionary Biology. 2005. V. 283A. P. 259–274.Routier-Kierzkowska A.-L., Weber A., Kochova P. et al. Cellular force microscopy for in vivo measurements of plant tissue mechanics // Plant Physiol. 2012. V. 158. P. 1514–1522.Routier-Kierzkowska A.-L., Weber A., Kochova P. et al. Cellular force microscopy for in vivo measurements of plant tissue mechanics // Plant Physiol. 2012. V. 158. P. 1514–1522.Schopfer P. Biomechanics of plant growth // Am. J. Bot. 2006. V. 93. P. 1415–1425.Schopfer P. Biomechanics of plant growth // Am. J. Bot. 2006. V. 93. P. 1415–1425.Spatz H., Kohler L., Niklas K. Mechanical behaviour of plant tissues: composite materials or structures? // J. Experim. Biol. 1999. V. 202. P. 3269–3272.Spatz H., Kohler L., Niklas K. Mechanical behaviour of plant tissues: composite materials or structures? // J. Experim. Biol. 1999. V. 202. P. 3269–3272.Suslov D., Verbelen J.-P., Vissenberg K. Onion epidermis as a new model to study the control of growth anisotropy in higher plants // J. Experim. Bot. 2009. V. 60. P. 4175–4187.Suslov D., Verbelen J.-P., Vissenberg K. Onion epidermis as a new model to study the control of growth anisotropy in higher plants // J. Experim. Bot. 2009. V. 60. P. 4175–4187.Tang Y., Cao G., Chen X. et al. A finite element framework for studying the mechanical response of macromolecules: application to the gating of the mechanosensitive channel MscL // Biophys. J. 2006. V. 91. Р. 1248–1263.Tang Y., Cao G., Chen X. et al. A finite element framework for studying the mechanical response of macromolecules: application to the gating of the mechanosensitive channel MscL // Biophys. J. 2006. V. 91. Р. 1248–1263.Thompson D.S. How do cell walls regulate plant growth? // J. Experim. Bot. 2005. V. 56. P. 2275–2285.Thompson D.S. How do cell walls regulate plant growth? // J. Experim. Bot. 2005. V. 56. P. 2275–2285.Thompson D.S. Extensiometric determination of the rheological properties of the epidermis of growing tomato fruit // J. Experim. Bot. 2001. V. 52. P. 1291–1301.Thompson D.S. Extensiometric determination of the rheological properties of the epidermis of growing tomato fruit // J. Experim. Bot. 2001. V. 52. P. 1291–1301.Volokh K.Y. A simple phenomenological theory of tissue growth // Mech. Chem. Biosyst. 2004. V. 1. P. 147–160.Volokh K.Y. A simple phenomenological theory of tissue growth // Mech. Chem. Biosyst. 2004. V. 1. P. 147–160.You T.J., Harvey S.C. Finite element approach to the electrostatics of macromolecules with arbitrary geometries // J. Computat. Chem. 1993. V. 14. P. 484–501.You T.J., Harvey S.C. Finite element approach to the electrostatics of macromolecules with arbitrary geometries // J. Computat. Chem. 1993. V. 14. P. 484–501.

The authors declare that there are no conflicts of interest present.