On the orientation of plane tensegrity cytoskeletons under biaxial substrate stretching

ABSTRACT

Two different simple cases of plane tensegrity cytoskeleton geometries are presented and investigated in terms of stability. The tensegrity frames are used to model adherent cell cytoskeletal behaviour under the application of plane substrate stretching and describe thoroughly the experimentally observed reorientation phenomenon. Both models comprise two elastic bars (microtubules), four elastic strings (actin filaments) and are attached on an elastic substrate. In the absence of external loading shape stability of the cytoskeleton is dominated by its prestress. Upon application of external loading, the cytoskeleton is reorganized in a new direction such that its total potential energy is rendered a global minimum. Considering linear constitutive relations, yet large deformations, it is revealed that the reorientation phenomenon can be successfully treated as a problem of ma- thematical stability. It is found that apart from the magnitude of contractile prestress and the magnitude of extracellular stretching, the reorientation is strongly shape–dependent as well. Numerical applications not only justify laboratory data reported in literature but such experimental evidence as the concurrent appearance of two distinct and symmetric directions of orientation, indicating the cellular coexistence of phases phenomenon, are clearly detected and incorporated in the proposed mathematical treatment.

Two different simple cases of plane tensegrity cytoskeleton geometries are presented and investigated in terms of stability. The tensegrity frames are used to model adherent cell cytoskeletal behaviour under the application of plane substrate stretching and describe thoroughly the experimentally observed reorientation phenomenon. Both models comprise two elastic bars (microtubules), four elastic strings (actin filaments) and are attached on an elastic substrate. In the absence of external loading shape stability of the cytoskeleton is dominated by its prestress. Upon application of external loading, the cytoskeleton is reorganized in a new direction such that its total potential energy is rendered a global minimum. Considering linear constitutive relations, yet large deformations, it is revealed that the reorientation phenomenon can be successfully treated as a problem of ma- thematical stability. It is found that apart from the magnitude of contractile prestress and the magnitude of extracellular stretching, the reorientation is strongly shape–dependent as well. Numerical applications not only justify laboratory data reported in literature but such experimental evidence as the concurrent appearance of two distinct and symmetric directions of orientation, indicating the cellular coexistence of phases phenomenon, are clearly detected and incorporated in the proposed mathematical treatment.

Cite this paper

nullPirentis, A. and Lazopoulos, K. (2010) On the orientation of plane tensegrity cytoskeletons under biaxial substrate stretching.*Advances in Bioscience and Biotechnology*, **1**, 12-25. doi: 10.4236/abb.2010.11003.

nullPirentis, A. and Lazopoulos, K. (2010) On the orientation of plane tensegrity cytoskeletons under biaxial substrate stretching.

References

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[2] Dartsch, P.C. and Haemmerle, H. (1986) Orientation response of arterial smooth muscle cells to mechanical stimulation. European Journal of Cell Biology, 41, 339-346.

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[4] Neidlinger-Wilke, C., Grood, E.S., Wang, J.H.–C., Brand, R.A. and Claes, L. (2001) Cell alignment is induced by cyclic changes in cell length: studies of cells grown in cyclically stretched substrates. Journal of Orthopaedic Research, 19, 286-293.

[5] Wang, J.H.-C., Goldschmidt-Clermont, P., Wille, J. and Yin, F.C.-P. (2001) Specificity of endothelial cell reorientation in response to cyclic mechanical stretching. Journal of Biomechanics, 34, 1563-1572.

[6] Takakuda, K. and Miyairi, H. (1996) Tensile behavior of fibroblasts cultured in collagen gel. Biomaterials, 17, 1393-1397.

[7] Eastwood, M., Mudera, V.C., Mcgrouther, D.A. and Brown, R.A. (1998) Effect of precise mechanical loading on fibroblast populated collagen lattices: morphological changes. Cell Motility and the Cytoskeleton, 40, 13-21.

[8] Collinsworth, A.M., Torgan, C.E., Nagada, S.N., Rajalingam, R.J., Kraus, W.E. and Truskey, G.A. (2000) Orientation and length of mammalian skeletal myocytes in response to unidirectional stretch. Cell and Tissue Research, 302, 243-251.

[9] Ignatius, A., Blessing, H., Liedert, A., Kaspar, D., Kreja, L., Friemert, B. and Claes, L. (2004) Effects of mechanical strain on human osteoblastic precursor cells in type I collagen matrices. Orthopade, 33, 1386-1393.

[10] Iba, I. and Sumpio, B.E. (1991) Morphological response of human endothelial cells subjected to cyclic strain in vitro. Microvascular Research, 42, 245-254.

[11] Wang, J.H.-C., Goldschmidt-Clermont, P. and Yin, F.C.-P. (2000) Contractility affects stress fiber remodelling and reorientation of endothelial cells subjected to cyclic mechanical stretching. Annals of Biomedical Engineering, 28, 1165-1171.

[12] Sipkema, P., van der Linden, J.W., Westerhof, N. and Yin, F.C.-P. (2003) Effect of cyclic axial stretch or rat arteries on endothelial cytoskeletal morphology and vascular reactivity. Journal of Biomechanics, 36, 653-659.

[13] Wille, J.J., Ambrosi, C.A. and Yin, F.C.-P. (2004) Comparison of the effects of cyclic stretching and compression on endothelial cell morphological responses. ASME Journal of Biomechanical Engineering, 126, 545-551.

[14] Kaunas, R., Nguyen, P., Usami, S. and Chien, S. (2005) Cooperative effects of Rho and mechanical stretch on stress fiber organization. Proceedings of the National Academy of Sciences of the USA, 102, 15895-15900.

[15] Kurpinski, K., Chu, J., Hashi, C. and Li, S. (2006) Anisotropic mechanosensing by mesenchimal stemm cells. Proceedings of National Academy Sciences, USA, 103, 16095-16100.

[16] Roth, B. and Whiteley, W. (1981) Tensegrity frameworks. Transactions of the American Mathematical Society, 265, 419-446.

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[18] Connelly R. and Back, A. (1998) Mathematics and tensegrity, American Scientist, 86, 142-151.

[19] Skelton, R.E., Helton, J., Adhikari, R., Pinaud, J. and Chan, W. (2002) An introduction to the mechanics of tensegrity structures (chapter 17), CRC Press.

[20] Williams, W.O. (2007) A primer on the mechanics of ten- segrity structures. http://www.math.cmu.edu/users/wow/ papers/newprimer.pdf

[21] Stamenović, D. (2006) Models of cytoskeletal mechanics based on tensegrity. In Mofrad, M.R.K., Kamm, R.D. (Eds.), Cytoskeletal mechanics: models and measurements, Cambridge University Press, Manhattan, 103-128.

[22] Ingber, D.E. (1993) Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. Journal of Cell Science, 104, 613-627.

[23] Ingber, D.E. (1998) The architecture of life. American Science, 48(1), 75-83.

[24] Ingber, D.E. (2008) Tensegrity-based mechanosensing from macro to micro. Progress in Biophysics and Molecular Biology, 97(2-3), 163-179.

[25] Gilmore, R. (1981) Catastrophe theory for scientists and engineers. Wiley and Sons, New York.

[26] Thompson, J.M.T. and Hunt, G.W. (1984) Elastic instability phenomena. Wiley and Sons, Chichester.

[27] Lazopoulos, K.A. and Pirentis, A. (2007) Substrate stretching and reorganization of stress fibers as a finite elasticity problem. International Journal of Solids and Structures, 44, 8285-8296.

[28] Pirentis, A.P. and Lazopoulos, K.A. (2009) On stress fibre reorientation under plane substrate stretching. Archives of Applied Mechanics, 79, 263-277.

[29] Stamenović, D., Lazopoulos, K.A., Pirentis, A. and Suki, B. (2009) Mechanical stability determines stress fiber and focal adhesion orientation: a mathematical model. Cellular and Molecular Bioengineering, 2(4), 475-485.

[30] Lazopoulos, K.A. and Stamenovic, D. (2006) A mathematical model of cell reorientation in response to substrate stretching. Molecular Cellular and Biomechanics, 3, 43-48.

[31] Kamm, R.D. and Mofrad, M.R.K. (2006) Introduction, with the biological basis for cell mechanics. In: Mofrad,

[32] M.R.K., Kamm, R.D. (Eds.), Cytoskeletal mechanics: models and measurements. Cambridge University Press, Manhattan, 1-17.

[33] Stamenović, D., Fredberg, J.J., Wang, N., Butler, J.P. and Ingber, D.E. (1996) A microstructural approach to cytoskeletal mechanics based on tensegrity. Journal of Theoretical Biology, 181, 125-136.

[34] Ogden, R.W. (1997) Non–linear elastic deformations. Dover, New York.

[35] Holzapfel, G.A. (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley and Sons, Chichester.

[36] Ericksen, J.L. (1991) Introduction to the thermodynamics of solids. In: Knops, R.J., Morton, K.W. (Eds.). Chapman & Hall, London, 39-61.

[37] Pitteri, M. and Zanzotto, G. (2003) Continuum models for phase transitions and twinning in crystals. Chapman, et al., Boca Raton, etc.

[38] Hayakawa, K., Sato, N. and Obinata, T. (2001) Dynamic reorientation of cultured cells and stress fibers under mechanical stress from periodic stretching. Experimental Cell Research, 268, 104-114.

[39] Deguchi, S., Ohashi, S. and Sato, M. (2006) Tensile properties of single stress fibers isolated from cultured vascular smooth muscle cells. Journal of Biomechanics, 39, 2603-2610.

[1] Suresh, S. (2007) Biomechanics and biophysics of cancer cells. Acta Biomaterialia, 3, 413-438.

[2] Dartsch, P.C. and Haemmerle, H. (1986) Orientation response of arterial smooth muscle cells to mechanical stimulation. European Journal of Cell Biology, 41, 339-346.

[3] Takemasa, T., Sugimoto, K. and Yamashita, K. (1997) Amplitude - dependent stress–fiber reorientation in early response to cyclic strain. Experimental Cell Research, 230, 407-410.

[4] Neidlinger-Wilke, C., Grood, E.S., Wang, J.H.–C., Brand, R.A. and Claes, L. (2001) Cell alignment is induced by cyclic changes in cell length: studies of cells grown in cyclically stretched substrates. Journal of Orthopaedic Research, 19, 286-293.

[5] Wang, J.H.-C., Goldschmidt-Clermont, P., Wille, J. and Yin, F.C.-P. (2001) Specificity of endothelial cell reorientation in response to cyclic mechanical stretching. Journal of Biomechanics, 34, 1563-1572.

[6] Takakuda, K. and Miyairi, H. (1996) Tensile behavior of fibroblasts cultured in collagen gel. Biomaterials, 17, 1393-1397.

[7] Eastwood, M., Mudera, V.C., Mcgrouther, D.A. and Brown, R.A. (1998) Effect of precise mechanical loading on fibroblast populated collagen lattices: morphological changes. Cell Motility and the Cytoskeleton, 40, 13-21.

[8] Collinsworth, A.M., Torgan, C.E., Nagada, S.N., Rajalingam, R.J., Kraus, W.E. and Truskey, G.A. (2000) Orientation and length of mammalian skeletal myocytes in response to unidirectional stretch. Cell and Tissue Research, 302, 243-251.

[9] Ignatius, A., Blessing, H., Liedert, A., Kaspar, D., Kreja, L., Friemert, B. and Claes, L. (2004) Effects of mechanical strain on human osteoblastic precursor cells in type I collagen matrices. Orthopade, 33, 1386-1393.

[10] Iba, I. and Sumpio, B.E. (1991) Morphological response of human endothelial cells subjected to cyclic strain in vitro. Microvascular Research, 42, 245-254.

[11] Wang, J.H.-C., Goldschmidt-Clermont, P. and Yin, F.C.-P. (2000) Contractility affects stress fiber remodelling and reorientation of endothelial cells subjected to cyclic mechanical stretching. Annals of Biomedical Engineering, 28, 1165-1171.

[12] Sipkema, P., van der Linden, J.W., Westerhof, N. and Yin, F.C.-P. (2003) Effect of cyclic axial stretch or rat arteries on endothelial cytoskeletal morphology and vascular reactivity. Journal of Biomechanics, 36, 653-659.

[13] Wille, J.J., Ambrosi, C.A. and Yin, F.C.-P. (2004) Comparison of the effects of cyclic stretching and compression on endothelial cell morphological responses. ASME Journal of Biomechanical Engineering, 126, 545-551.

[14] Kaunas, R., Nguyen, P., Usami, S. and Chien, S. (2005) Cooperative effects of Rho and mechanical stretch on stress fiber organization. Proceedings of the National Academy of Sciences of the USA, 102, 15895-15900.

[15] Kurpinski, K., Chu, J., Hashi, C. and Li, S. (2006) Anisotropic mechanosensing by mesenchimal stemm cells. Proceedings of National Academy Sciences, USA, 103, 16095-16100.

[16] Roth, B. and Whiteley, W. (1981) Tensegrity frameworks. Transactions of the American Mathematical Society, 265, 419-446.

[17] Motro, R. (1992) Tensegrity systems: the state of art. Int. Journal of Space Structure, 7(2), 75-83.

[18] Connelly R. and Back, A. (1998) Mathematics and tensegrity, American Scientist, 86, 142-151.

[19] Skelton, R.E., Helton, J., Adhikari, R., Pinaud, J. and Chan, W. (2002) An introduction to the mechanics of tensegrity structures (chapter 17), CRC Press.

[20] Williams, W.O. (2007) A primer on the mechanics of ten- segrity structures. http://www.math.cmu.edu/users/wow/ papers/newprimer.pdf

[21] Stamenović, D. (2006) Models of cytoskeletal mechanics based on tensegrity. In Mofrad, M.R.K., Kamm, R.D. (Eds.), Cytoskeletal mechanics: models and measurements, Cambridge University Press, Manhattan, 103-128.

[22] Ingber, D.E. (1993) Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. Journal of Cell Science, 104, 613-627.

[23] Ingber, D.E. (1998) The architecture of life. American Science, 48(1), 75-83.

[24] Ingber, D.E. (2008) Tensegrity-based mechanosensing from macro to micro. Progress in Biophysics and Molecular Biology, 97(2-3), 163-179.

[25] Gilmore, R. (1981) Catastrophe theory for scientists and engineers. Wiley and Sons, New York.

[26] Thompson, J.M.T. and Hunt, G.W. (1984) Elastic instability phenomena. Wiley and Sons, Chichester.

[27] Lazopoulos, K.A. and Pirentis, A. (2007) Substrate stretching and reorganization of stress fibers as a finite elasticity problem. International Journal of Solids and Structures, 44, 8285-8296.

[28] Pirentis, A.P. and Lazopoulos, K.A. (2009) On stress fibre reorientation under plane substrate stretching. Archives of Applied Mechanics, 79, 263-277.

[29] Stamenović, D., Lazopoulos, K.A., Pirentis, A. and Suki, B. (2009) Mechanical stability determines stress fiber and focal adhesion orientation: a mathematical model. Cellular and Molecular Bioengineering, 2(4), 475-485.

[30] Lazopoulos, K.A. and Stamenovic, D. (2006) A mathematical model of cell reorientation in response to substrate stretching. Molecular Cellular and Biomechanics, 3, 43-48.

[31] Kamm, R.D. and Mofrad, M.R.K. (2006) Introduction, with the biological basis for cell mechanics. In: Mofrad,

[32] M.R.K., Kamm, R.D. (Eds.), Cytoskeletal mechanics: models and measurements. Cambridge University Press, Manhattan, 1-17.

[33] Stamenović, D., Fredberg, J.J., Wang, N., Butler, J.P. and Ingber, D.E. (1996) A microstructural approach to cytoskeletal mechanics based on tensegrity. Journal of Theoretical Biology, 181, 125-136.

[34] Ogden, R.W. (1997) Non–linear elastic deformations. Dover, New York.

[35] Holzapfel, G.A. (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley and Sons, Chichester.

[36] Ericksen, J.L. (1991) Introduction to the thermodynamics of solids. In: Knops, R.J., Morton, K.W. (Eds.). Chapman & Hall, London, 39-61.

[37] Pitteri, M. and Zanzotto, G. (2003) Continuum models for phase transitions and twinning in crystals. Chapman, et al., Boca Raton, etc.

[38] Hayakawa, K., Sato, N. and Obinata, T. (2001) Dynamic reorientation of cultured cells and stress fibers under mechanical stress from periodic stretching. Experimental Cell Research, 268, 104-114.

[39] Deguchi, S., Ohashi, S. and Sato, M. (2006) Tensile properties of single stress fibers isolated from cultured vascular smooth muscle cells. Journal of Biomechanics, 39, 2603-2610.