Abstract
In order to move in a three-dimensional extracellular matrix, the nucleus of a cell must squeeze through the narrow spacing among the fibers and, by adhering to them, the cell needs to exert sufficiently strong traction forces. If the nucleus is too stiff, the spacing too narrow, or traction forces too weak, the cell is not able to penetrate the network. In this article, we formulate a mathematical model based on an energetic approach, for cells entering cylindrical channels composed of extracellular matrix fibers. Treating the nucleus as an elastic body covered by an elastic membrane, the energetic balance leads to the definition of a necessary criterion for cells to pass through the regular network of fibers, depending on the traction forces exerted by the cells (or possibly passive stresses), the stretchability of the nuclear membrane, the stiffness of the nucleus, and the ratio of the pore size within the extracellular matrix with respect to the nucleus diameter. The results obtained highlight the importance of the interplay between mechanical properties of the cell and microscopic geometric characteristics of the extracellular matrix and give an estimate for a critical value of the pore size that represents the physical limit of migration and can be used in tumor growth models to predict their invasive potential in thick regions of ECM.
Similar content being viewed by others
References
Arduino A, Preziosi L (2015) A multiphase model of tumour segregation in situ by a heterogeneous extracellular matrix. Int J Nonlinear Mech 75:22–30
Balzer EM, Tong Z, Paul CD, Hung WC, Stroka KM, Boggs AE, Martin SS, Konstantopoulos K (2012) Physical confinement alters tumor cell adhesion and migration phenotypes. FASEB J 26(10):4045–4056
Beadle C, Assanah MC, Monzo P, Vallee R, Rosenfeld SS, Canoll P (2008) The role of myosin II in glioma invasion of the brain. Mol Biol Cell 19:3357–3368
Birk DE, Trelstad RL (1984) Extracellular compartments in matrix morphogenesis: collagen fibril, bundle, and lamellar formation by corneal fibroblast. J Cell Biol 99:2024–2033
Broers JL, Peeters EA, Kuijpers HJ, Endert J, Bouten CV, Oomens CW, Baaijens FP, Ramaekers FC (2004) Decreased mechanical stiffness in LMNA\(-\)/\(-\)cells is caused by defective nucleo-cytoskeletal integrity: implications for the development of laminopathies. Hum Mol Genet 13:2567–2580
Cavalcanti-Adam EA, Volberg T, Micoulet A, Kessler H, Geiger B, Spatz JP (2007) Cell spreading and focal adhesion dynamics are regulated by spacing of integrin ligands. Biophys J 92:2964–2974
Chaplain MAJ, Graziano L, Preziosi L (2006) Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math Med Biol 23:197–229
Coyer SR, Singh A, Dumbauld DW, Calderwood DA, Craig SW, Delamarche E, García AJ (2012) Nanopatterning reveals an ECM area threshold for focal adhesion assembly and force transmission that is regulated by integrin activation and cytoskeleton tension. J Cell Sci 125(Pt 21):5110–5123
Dahl KN, Kahn SM, Wilson KL, Discher DE (2004) The nuclear envelope lamina network has elasticity and a compressibility limit suggestive of a molecular shock absorber. J Cell Sci 117:4779–4786
Davidson PM, Denais C, Bakshi MC, Lammerding J (2014) Nuclear deformability constitutes a rate-limiting step during cell migration in 3-D environments. Cell Mol Bioeng 7(3):293–306
Deguchi S, Yano M, Hashimoto K, Fukamachi H, Washio S, Tsujioka K (2007) Assessment of the mechanical properties of the nucleus inside a spherical endothelial cell based on microtensile testing. J Mech Mater Struct 2(6):1087–1102
Evans EA, Waugh R, Melnik L (1976) Elastic area compressibility modulus of red cell membrane. Biophys J 16:585–595
Fedorchak GR, Kaminski A, Lammerding J (2014) Cellular mechanosensing: getting to the nucleus of it all. Prog Biophys Mol Biol 115:76–92
Friedl P, Brocker EB (2000) The biology of cell locomotion within three-dimensional extracellular matrix. Cell Mol Life Sci 57(1):41–64
Friedl P, Sahai E, Weiss S, Yamada KM (2012) New dimensions in cell migration. Nat Rev Mol Cell Biol 13(11):743–747
Friedl P, Wolf K (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat Rev Cancer 3(5):362–374
Friedl P, Wolf K (2010) Plasticity of cell migration: a multiscale tuning model. J Cell Biol 188(1):11–19
Friedl P, Wolf K, Lammerding J (2011) Nuclear mechanics during cell migration. Curr Opin Cell Biol 23(1):55–64
Fu Y, Chin LK, Bourouina T, Liu AQ, VanDongen AM (2012) Nuclear deformation during breast cancer cell transmigration. Lab Chip 12(19):3774–3778
Gerlitz G, Bustin M (2011) The role of chromatin structure in cell migration. Trends Cell Biol 21(1):6–11
Giverso C, Grillo A, Preziosi L (2014) Influence of nucleus deformability on cell entry into cylindrical structures. Biomech Model Mechanobiol 13(3):481–502
Giverso C, Scianna M, Grillo A (2015) Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations. Mech Res Commun 68:31–39
Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended potts model. Phys Rev Lett 69:2013–2016
Guck J, Lautenschläger F, Paschke S, Beil M (2010) Critical review: cellular mechanobiology and amoeboid migration. Integr Biol 2:575–583
Guilak F, Tedrow JR, Burgkart R (2000) Viscoelastic properties of the cell nucleus. Biochem Biophys Res Commun 269:781–786
Harada T, Swift J, Irianto J, Shin JW, Spinler KR, Athirasala A, Diegmiller R, Dingal PC, Ivanovska IL, Discher DE (2014) Nuclear lamin stiffness is a barrier to 3D migration, but softness can limit survival. J Cell Biol 204:669–682
Helfrick W (1973) Elastic properties of lipid bilayers: theory and possible experiments. Z Naturforschung 28(11):693–703
Ho CY, Lammerding J (2012) Lamins at a glance. J Cell Sci 125(Pt 9):2087–2093
Hung WC, Chen SH, Paul CD, Stroka KM, Lo YC, Yang JT, Konstantopoulos K (2013) Distinct signaling mechanisms regulate migration in unconfined versus confined spaces. J Cell Biol 202(5):807–824
Isermann P, Lammerding J (2013) Nuclear mechanics and mechanotransduction in health and disease. Curr Biol 23(24):R1113–R1121
Jain RK (1987) Transport of molecules in the tumor interstitium: a review. Cancer Res 47:3039–3051
Kaufmann A, Heinemann F, Radmacher M, Stick R (2011) Amphibian oocyte nuclei expressing lamin A with the progeria mutation E145K exhibit an increased elastic modulus. Nucleus 2:310–319
Kim M-C, Neal DM, Kamm RD, Asada HH (2013) Dynamic modeling of cell migration and spreading behaviors on fibronectin coated planar substrates and micropatterned geometries. PLoS Comput Biol 9(2):e1002926
Komai Y, Ushiki T (1991) Three-dimensional organization of collagen fibrils in the human cornea and sclera. Invest Ophthalmol Vis Sci 32(8):2244–2258
Krause M, Te Riet J, Wolf K (2013) Probing the compressibility of tumor cell nuclei by combined atomic force-confocal microscopy. Phys Biol 10(6):065002
Krause M, Wolf K (2015) Cancer cell migration in 3D tissue: negotiating space by proteolysis and nuclear deformability. Cell Adhes Migr 9(5):357–366
Lammerding J, Fong LG, Ji JY, Reue K, Stewart CL, Young SG, Lee RT (2006) Lamins A and C but not lamin B1 regulate nuclear mechanics. J Biol Chem 281:25768–25780
Lautenschläger F, Paschke S, Schinkinger S, Bruel A, Beil M, Guck J (2009) The regulatory role of cell mechanics for migration of differentiating myeloid cells. Proc Natl Acad Sci 106(37):15696–15701
Liu H, Wen J, Xiao Y, Liu J, Hopyan S, Radisic M, Simmons CA, Sun Y (2014) In situ mechanical characterization of the cell nucleus by atomic force microscopy. ACS Nano 8(4):3821–3828
Lombardi ML, Lammerding L (2011) Keeping the LINC: the importance of nucleocytoskeletal coupling in intracellular force transmission and cellular function. Biochem Soc Trans 39:1729–1734
Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Macklin P, Wise SM, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23:R1–R91
Makhija E, Jokhun DS, Shivashankar GV (2015) Nuclear deformability and telomere dynamics are regulated by cell geometric constraints. Proc Natl Acad Sci 13(1):E32–E40
Massia SP, Hubbell JA (1991) An RGD spacing of 440 nm is sufficient for integrin \(\alpha _v-\beta _3\)-mediated fibroblast spreading and 140 nm for focal contact and stress fiber formation. J Cell Biol 114(5):1089–1100
Netti PA, Jain RK (2003) Interstitial transport in solid tumours. In: Preziosi L (ed) Cancer modelling and simulation. CRC-Press, Chapman Hall, Boca Raton
Paszek MJ et al (2005) Tensional homeostasis and the malignant phenotype. Cancer Cell 8:241–254
Petrie RJ, Yamada KM (2012) At the leading edge of three-dimensional cell migration. J Cell Sci 125(Pt 24):5917–5926
Preziosi L, Tosin A (2009) Multiphase and multiscale trends in cancer modelling. Math Model Nat Phenom 4(3):1–11
Rajagopalan P, Marganski WA, Brown XQ, Wong JY (2004) Direct comparison of the spread area, contractility, and migration of balb/c 3T3 fibroblasts adhered to fibronectin- and RGD-modified substrata. Biophys J 87(4):2818–2827
Ribeiro AJ, Khanna P, Sukumar A, Dong C, Dahl KN (2014) Nuclear stiffening inhibits migration of invasive melanoma cells. Cell Mol Bioeng 7(4):544–551
Rolli CG, Seufferlein T, Kemkemer R, Spatz JP (2010) Impact of tumor cell cytoskeleton organization on invasiveness and migration: a microchannel-based approach. PLos ONE 5(1):e8726
Rowat AC, Lammerding J, Herrmann H, Aebi U (2008) Towards an integrated understanding of the structure and mechanics of the cell nucleus. BioEssays 30:226–236
Sabeh F, Shimizu-Hirota R, Weiss SJ (2009) Protease-dependent versus -independent cancer cell invasion programs: three-dimensional amoeboid movement revisited. J Cell Biol 185(1):11–19
Sahai E (2007) Illuminating the metastatic process. Nat Rev Cancer 7(10):737–749
Saidi IS, Jacques SL, Tittel FK (1995) Mie and Rayleigh modeling of visible-light scattering in neonatal skin. Appl Opt 34(31):7410–7418
Schoumacher M, Goldman RD, Louvard D, Vignjevic DM (2010) Actin, microtubules, and vimentin intermediate filaments cooperate for elongation of invadopodia. J Cell Biol 189:541–556
Shankar J, Messenberg A, Chan J, Underhill TM, Foster LJ, Nabi IR (2010) Pseudopodial actin dynamics control epithelial-mesenchymal transition in metastatic cancer cells. Cancer Res 70:3780–3790
Shaw LM (2005) Tumor cell invasion assays. In: Guan J-L (ed) Cell migration: developmental methods and protocols, vol 294. Humana Press, New York, pp 97–105
Skalak R, Tozeren A, Zarda RP, Chien S (1973) Strain energy function of red blood cell membrane. Biophys J 13:245–264
te Boekhorst V, Preziosi L, Friedl P (2016) Plastiticy of cell migration in vivo and in silico. Annu Rev Cell Dev Biol 32:491–526
Tu ZC, Ou-Yang ZC (2004) Geometric theory on the elasticity of bio-membranes. J Phys A Math Gen 37:11407–11429
Tu ZC, Ou-Yang ZC (2008) Elastic theory of low-dimensional continua and its applications in bio- and nano-structures. J Comput Theor Nanosci 5:422–448
Vargas-Pinto R, Gong H, Vahabikashi A, Johnson M (2013) The effect of the endothelial cell cortex on atomic force microscopy measurements. Biophys J 105:300–309
Vaziri A, Lee H, Kaazempur Mofrad MR (2006) Deformation of the cell nucleus under indentation: mechanics and mechanisms. J Mater Res 21:2126–2135
Verdier C, Etienne J, Duperray A, Preziosi L (2009) Review: rheological properties of biological materials. C R Phys 10(8):790–811
Versaevel M, Grevesse T, Gabriele S (2012) Spatial coordination between cell and nuclear shape within micropatterned endothelial cells. Nat Commun 3:671
Weigelin B, Bakker G-J, Friedl P (2012) Intravital third harmonic generation microscopy of collective melanoma cell invasion. Principles of interface guidance and microvesicle dynamics. IntraVital 1(1):32–43
Wiseman PW, Brown CM, Webb DJ, Hebert B, Johnson NL, Squier JA, Ellisman MH, Horwitz AF (2004) Spatial mapping of integrin interactions and dynamics during cell migration by image correlation microscopy. J Cell Sci 117:5521–5534
Wolf K, Alexander S, Schacht V, Coussens LM, von Andrian UH, van Rheenen J, Deryugina E, Friedl P (2009) Collagen-based cell migration models in vitro and in vivo. Semin Cell Dev Biol 20(8):931–941
Wolf K, Friedl P (2011) Extracellular matrix determinants of proteolytic and non-proteolytic cell migration. Trends Cell Biol 21(12):736–744
Wolf K, Te Lindert M, Krause M, Alexander S, Te Riet J, Willis AL, Hoffman RM, Figdor CG, Weiss SJ, Friedl P (2013) Physical limits of cell migration: control by ECM space and nuclear deformation and tuning by proteolysis and traction force. J Cell Biol 201(7):1069–1084
Wolf K, Wu YI, Liu Y, Geiger J, Tam E, Overall C, Stack MS, Friedl P (2007) Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion. Nat Cell Biol 9:893–904
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Giverso, C., Arduino, A. & Preziosi, L. How Nucleus Mechanics and ECM Microstructure Influence the Invasion of Single Cells and Multicellular Aggregates. Bull Math Biol 80, 1017–1045 (2018). https://doi.org/10.1007/s11538-017-0262-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-017-0262-9