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A Cellular Potts model simulating cell migration on and in matrix environments
1.  Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino 
2.  Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
3.  Department of Cell Biology, Radboud University Nijmegen Medical Centre, 6500 HB Nijmegen, Netherlands 
References:
[1] 
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, "Molecular Biology of the Cell," $3^{rd}$ edition, Garland Science, 1994. 
[2] 
M. Arnold, V. C. HirschfeldWarneken, T. Lohmüller, P. Heil, J. Blümmel, E. A. CavalcantiAdam, M. LópezGarcía, P. Walther, H. Kessler, B. Geiger and J. P. Spatz, Induction of cell polarization and migration by a gradient of nanoscale variations in adhesive ligand spacing, Nano Lett., 8 (2008), 20632069. 
[3] 
M. Bajénoff, J. G. Egen, L. Y. Koo, J. P. Laugier, F. Brau, N. Glaichenhaus and R. N. Germain, Stromal cell networks regulate lymphocyte entry, migration, and territoriality in lymph nodes, Immunity, 25 (2006), 9891001. doi: 10.1016/j.immuni.2006.10.011. 
[4] 
A. Balter, R. M. Merks, N. J. Poplawski, M. Swat and J. A. Glazier, The GlazierGranerHogeweg model: Extensions, future directions, and opportunities for further study, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 157167. 
[5] 
A. L. Bauer, T. L. Jackson and Y. Jiang, A cellbased model exhibiting branching and anastomosis during tumorinduced angiogenesis, Biophys. J., 92 (2007), 31053121. doi: 10.1529/biophysj.106.101501. 
[6] 
C. Beadle, M. C. Assanah, P. Monzo, R. Vallee, S. Rosenfeld and P. Canoll, The role of myosin ii in glioma invasion of the brain, Mol. Biol. Cell., 19 (2008), 33573368. doi: 10.1091/mbc.E08030319. 
[7] 
J. Behring, R. Junker, X. F. Walboomers, B. Chessnut and J. A. Jansen, Toward guided tissue and bone regeneration: Morphology, attachment, proliferation, and migration of cells cultured on collagen barrier membranes. A systematic review, Odontology, 96 (2008), 111. 
[8] 
A. O. Brightman, B. P. Rajwa, J. E. Sturgis, M. E. McCallister, J. P. Robinson and S. L. VoytikHarbin, Timelapse confocal reflection microscopy of collagen fibrillogenesis and extracellular matrix assembly in vitro, Biopolymers, 54 (2000), 222234. doi: 10.1002/10970282(200009)54:3<222::AIDBIP80>3.0.CO;2K. 
[9] 
A. Brock, E. Chang, C. C. Ho, P. LeDuc, X. Jiang, G. M. Whitesides and D. E. Ingber, Geometric determinants of directional cell motility revealed using microcontact printing, Langmuir, 19 (2003), 16111617. 
[10] 
B. T. Burgess, J. L. Myles and R. B. Dickinson, Quantitative analysis of adhesionmediated cell migration in threedimensional gels of RGDgrafted collagen, Ann. Biomed. Eng., 28 (2003), 110118. 
[11] 
R. M. Capito and M. Spector, Scaffoldbased articular cartilage repair, IEEE Eng. Med. Biol. Mag., 22 (2003), 4250. 
[12] 
E. A. CavalcantiAdam, T. Volberg, A. Micoulet, H. Kessler, B. Geiger and J. P. Spatz, Cell spreading and focal adhesion dynamics are regulated by spacing of integrin ligands, Biophys. J., 92 (2007), 29642974. 
[13] 
J. Condeelis and J. E. Segall, Intravital imaging of cell movement in tumours, Nat. Rev. Cancer, 3 (2003), 921930. 
[14] 
M. W. Conklin, J. C. Eickhoff, K. M. Riching, C. A. Pehlke, K. W. Eliceiri, P. P. Provenzano, A. Friedl and P. J. Keely, Aligned collagen is a prognostic signature for survival in human breast carcinoma, Am. J. Pathol., 178 (2011), 12211232. doi: 10.1016/j.ajpath.2010.11.076. 
[15] 
R. B. Dickinson, S. Guido and R. T. Tranquillo, Biased cellmigration of fibroblasts exhibiting contact guidance in oriented collagen gels, Ann. Biomed. Eng., 22 (1994), 342356. doi: 10.1007/BF02368241. 
[16] 
P. A. DiMilla, J. A. Stone, J. A. Quinn, S. M. Albelda and D. A. Lauffenburger, Maximal migration of human smoothmuscle cells on fibronectin and typeIV collagen occurs at an intermediate attachment strength, J. Cell. Biol., 122 (1993), 729737. doi: 10.1083/jcb.122.3.729. 
[17] 
A. D. Doyle, F. W. Wang, K. Matsumoto and K. M. Yamada, Onedimensional topography underlies threedimensional fibrillar cell migration, J. Cell. Biol., 184 (2009), 481490. 
[18] 
N. Dubey, P. C. Letourneau and R. T. Tranquillo, Neuronal contact guidance in magnetically aligned fibrin gels: effect of variation in gel mechanostructural properties, Biomaterials, 22 (2001), 10651075. 
[19] 
G. A. Dunn and T. Ebendal, Contact guidance on oriented collagen gels, Exp. Cell. Res., 111 (1978), 475479. 
[20] 
G. A. Dunn, Characterizing a kinesis response: Timeaveraged measures of cell speed and directional persistence, Agents Actions Suppl., 12 (1983), 1433. 
[21] 
A. Engler, L. Bacakova, C. Newman, A. Hategan, M. Griffin and D. Discher, Substrate compliance versus ligand density in cell on gel responses, Biophys. J., 86 (2004), 617628. 
[22] 
P. Friedl, F. Entschladen, C. Conrad, B. Niggemann and K. S. Zänker, CD4+ Tlymphocytes migrating in threedimensional collagen lattices lack focal adhesions and utilize $\beta$1 integrinindependent strategies for polarization, interaction with collagen fibers and locomotion, Eur. J. Immunol., 28 (1998), 23312343. 
[23] 
P. Friedl and E. B. Brocker, The biology of cell locomotion within threedimensional extracellular matrix, Cell. Mol. Life Sci., 57 (2000), 4164. doi: 10.1007/s000180050498. 
[24] 
P. Friedl and K. Wolf, Tumourcell invasion and migration: Diversity and escape mechanisms, Nat. Rev. Cancer, 3 (2003), 362374. 
[25] 
P. Friedl, K. Maaser, C. E. Klein, B. Niggemann, G. Krohne and K. S. Zänker, Migration of highly aggressive MV3 melanoma cells in 3dimensional collagen lattices results in local matrix reorganization and shedding of alpha2 and beta1 integrins and CD44, Cancer Res., 57 (1997), 20612070. 
[26] 
P. Friedl, K. Wolf and J. Lammerding, Nuclear mechanics during cell migration, Curr. Opin. Cell. Biol., 23 (2011), 253. 
[27] 
P. Friedl and B. Weigelin, Interstitial leukocyte migration and immune function, Nat. Immunol., 9 (2008), 960969. 
[28] 
P. Friedl and K. Wolf, Plasticity of cell migration: A multiscale tuning model, J. Cell. Biol., 188 (2009), 1119. 
[29] 
P. Friedl and D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell. Biol., 10 (2009), 445457. 
[30] 
C. Gaudet, W. Marganski, S. Kim, C. T. Brown, V. Gunderia, M. Dembo and J. Wong, Influence of type I collagen surface density on fibroblast spreading, motility, and contractility, Biophys. J., 85 (2003), 33293335. 
[31] 
G. Gerlitz and M. Bustin, The role of chromatin structure in cell migration, Trends Cell. Biol., 21 (2011), 611. 
[32] 
C. Giverso, M. Scianna, L. Preziosi, N. Lo Buono and A. Funaro, Individual cellbased model for invitro mesothelial invasion of ovarian cancer, Math. Model. Nat. Phenom., 5 (2010), 203223. 
[33] 
J. A. Glazier, A. Balter and N. J. Poplawski, Magnetization to morphogenesis: A brief history of the GlazierGranerHogeweg model, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 79106. 
[34] 
J. A. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 47 (1993), 21282154. 
[35] 
S. L. Goodman, G. Risse and K. Vondermark, The E8 subfragment of laminin promotes locomotion of myoblasts over extracellular matrix, J. Cell. Biol., 109 (1989), 799809. 
[36] 
F. Graner and J. A. Glazier, Simulation of biological cell sorting using a twodimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 20132016. doi: 10.1103/PhysRevLett.69.2013. 
[37] 
B. A. Harley, H. Kim, M. H. Zaman, I. V. Yannas, D. A. Lauffenburger and L. J. Gibson, Microarchitecture of threedimensional scaffolds influences cell migration behavior via junction interactions, Biophys. J., 95 (2008), 40134024. doi: 10.1529/biophysj.107.122598. 
[38] 
B. A. Harley, M. H. Spilker, J. W. Wu, K. Asano, H. P. Hsu, M. Spector and I. V. Yannas, Optimal degradation rate for collagen chambers used for regeneration of peripheral nerves over long gaps, Cells Tissues Organs, 176 (2008), 153165. 
[39] 
O. Ilina and P. Friedl, Mechanisms of collective cell migration at a glance, J. Cell Sci., 122 (2009), 32033208. 
[40] 
E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik., 31 (1925), 253. 
[41] 
R. M. Kuntz and W. M. Saltzman, Neutrophil motility in extracellular matrix gels: mesh size and adhesion affect speed of migration, Biophys. J., 72 (1997), 14721480. doi: 10.1016/S00063495(97)787939. 
[42] 
S. Kurosaka and A. Kashina, Cell biology of embryonic migration, Birth Defects Res. C. Embryo Today, 84 (2008), 102122. 
[43] 
E. Lamers, R. Van Horssen, J. Te Riet, F. C. Van Delft, R. Luttge, X. F. Walboomers and J. A. Jansen, The influence of nanoscale topographical cues on initial osteoblast morphology and migration, Eur. Cell. Mater., 9 (2010), 329343. 
[44] 
D. A. Lauffenburger and J. J. Lindermann, "Receptors: Bodels for Binding, Trafficking, and Signaling," Oxford University Press, London, 1996. 
[45] 
D. A. Lauffenburger and A. F. Horwitz, Cell migration: a physically integrated molecular process, Cell, 84 (1996), 359369. doi: 10.1016/S00928674(00)812805. 
[46] 
D. Lehnert, B. WehrleHaller, C. David, U. Weiland, C. Ballestrem, B. A. Imhof and M. Bastmeyer, Cell behaviour on micropatterned substrata: limits of extracellular matrix geometry for spreading and adhesion, J. Cell. Sci., 117 (2004), 4152. doi: 10.1242/jcs.00836. 
[47] 
M. P. Lutolf, J. L. LauerFields, H. G. Schmoekel, A. T. Metters, F. E. Weber, G. B. Fields and J. A. Hubbell, Synthetic matrix metalloproteinasesensitive hydrogels for the conduction of tissue regeneration: Engineering cellinvasion characteristics, Proc. Natl. Acad. Sci. U. S. A., 100 (2003), 54135418. 
[48] 
K. Maaser, K. Wolf, C. E. Klein, B. Niggemann, K. S. Zänker, E. B. Bröcker and Friedl, Functional hierarchy of simultaneously expressed adhesion receptors: Integrin $\alpha$2$\beta$1 but not CD44 mediates MV3 melanoma cell migration and matrix reorganization within threedimensional hyaluronancontaining collagen matrices, Mol. Biol. Cell., 10 (1999), 30673079. 
[49] 
A. W. Mahoney, B. G. Smith, N. S. Flann and G. J. Podgorski, Discovering novel cancer therapies: A computational modeling and search approach, in "IEEE conference on Computational Intelligence in Bioinformatics and Bioengineering," (2008), 233240. 
[50] 
A. F. M arée, V. A. Grieneisen and P. Hogeweg, The Cellular Potts Model and biophysical properties of cells, tissues and morphogenesis, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 107136. 
[51] 
R. M. Merks, E. D. Perryn, A. Shirinifard and J. A. Glazier, Contactinhibited chemotaxis in de novo and sprouting blood vessel growth, PLoS Comput. Biol., 4 (2008), e1000163, 16 pp. 
[52] 
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 10871092. 
[53] 
A. Mogilner and G. Oster, Polymer motors: Ppushing out the front and pulling up the back, Current Biology, 13 (2003), R721733. 
[54] 
F. J. O'Brien, B. A. Harley, I. V. Yannas and L. Gibson, Influence of freezing rate on pore structure in freezedried collagenGAG scaffolds, Biomaterials, 25 (2004), 10771086. 
[55] 
L. H. Olde Damink, P. J. Dijkstra, M. J. Luyn, P. B. Wachem, P. Nieuwenhuis and J. Feijen, Glutaraldehyde as a crosslinking agent for collagenbased biomaterials, J. Mater. Sci. Mater. Med., 6 (1995), 460472. 
[56] 
J. M. Orban, L. B. Wilson, J. A. Kofroth, M. S. ElKurdi, T. M. Maul and D. A. Vorp, Crosslinking of collagen gels by transglutaminase, J. Biomed. Mater. Res. A, 68 (2004), 756762. 
[57] 
S. P. Palecek, J. C. Loftus, M. H. Ginsberg, D. A. Lauffenburger and A. F. Horwitz, Integrinligand binding properties govern cell migration speed through cellsubstratum adhesiveness, Nature, 385 (1997), 537540. 
[58] 
E. D. Perryn, A. Czirok and C. D. Little, Vascular sprout formation entails tissue deformations and VEcadherindependent cellautonomous motility, Dev. Biol., 313 (2008), 545555. doi: 10.1016/j.ydbio.2007.10.036. 
[59] 
S. R. Peyton and A. J. Putnam, Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion, J. Cell. Physiol., 204 (2005), 198209. doi: 10.1002/jcp.20274. 
[60] 
T. D. Pollard and G. G. Borisy, Cellular motility driven by assembly and disassembly of actin filaments, Cell, 112 (2003), 453465. doi: 10.1016/S00928674(03)00120X. 
[61] 
R. B. Potts, Some generalized orderdisorder transformations, Proc. Camb. Phil. Soc., 48 (1952), 106109. 
[62] 
G. P. Raeber, M. P. Lutolf and J. A. Hubbell, Molecularly engineered PEG hydrogels: a novel model system for proteolytically mediated cell migration, Biophys. J., 89 (2005), 13741388. 
[63] 
C. B. Raub, V. Suresh, T. Krasieva, J. Lyubovitsky, J. D. Mih, A. J. Putnam, B. J. Tromberg and S. C. George, Noninvasive assessment of collagen gel microstructure and mechanics using multiphoton microscopy, Biophys. J., 92 (2007), 22122222. doi: 10.1529/biophysj.106.097998. 
[64] 
A. J. Ridley, M. A. Schwartz, K. Burridge, R. A. Firtel, M. H. Ginsberg, G. Borisy, J. T. Parsons and A. R. Horwitz, Cell migration: Integrating signals from front to back, Science, 302 (2003), 17041709. 
[65] 
B. A. Roeder, K. Kokini, J. E. Sturgis, J. P. Robinson and S. L. VoytikHarbin, Tensile mechanical properties of threedimensional type I collagen extracellular matrices with varied microstructure, J. Biomech. Eng., 124 (2002), 214222. doi: 10.1115/1.1449904. 
[66] 
C. G. Rolli, T. Seufferlein, R. Kemkemer and J. P. Spatz, Impact of tumor cell cytoskeleton organization on invasiveness and migration: A microchannelbased approach, PLoS ONE, 5 (2010), e8726. 
[67] 
B. M. Rubenstein and L. J. Kaufman, The role of extracellular matrix in glioma invasion: A Cellular Potts Model approach, Biophys. J., 95 (2006), 56615680. 
[68] 
F. Sabeh, R. ShimizuHirota and S. J. Weiss, Proteasedependent versus independent cancer cell invasion programs: Threedimensional amoeboid movement revisited, J. Cell. Biol., 185 (2009), 1119. 
[69] 
E. Sahai, Illuminating the metastatic process, Nat. Rev. Cancer, 7 (2007), 737749. 
[70] 
M. Scianna, A multiscale hybrid model for proangiogenic calcium signals in a vascular endothelial cell, Bull. Math. Biol., 76 (2011), 12531291. 
[71] 
M. Scianna and L. Preziosi, Multiscale developments of the Cellular Potts Model, Multiscale Model. Simul., 10 (2012), 342382. 
[72] 
M. T. Sheu, J. C. Huang, G. C. Yeh and H. O. Ho, Characterization of collagen gel solutions and collagen matrices for cell culture, Biomaterials, 22 (2001), 17131719. doi: 10.1016/S01429612(00)00315X. 
[73] 
S. Schmidt and P. Friedl, Interstitial cell migration: Integrindependent and alternative adhesion mechanisms, Cell. Tissue Res., 339 (2010), 8392. doi: 10.1007/s0044100908929. 
[74] 
R. C. Siegel, Collagen crosslinking. Synthesis of collagen crosslinks in vitro with highly purified lysyl oxidase, J. Biol. Chem., 251 (1976), 57865792. 
[75] 
C. Spadaccio, A. Rainer, S. De Porcellinis, M. Centola, F. De Marco, M. Chello, M. Trombetta, M. and J. A. Genovese, A GCSF functionalized PLLA scaffold for wound repair: An in vitro preliminary study, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 843846. 
[76] 
M. S. Steinberg, Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation, Science, 141 (1963), 401408. 
[77] 
M. S. Steinberg, Does differential adhesion govern selfassembly processes in histogenesis? Equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells, J. Exp. Zool., 171 (1970), 395433. 
[78] 
D. G. Stupack, The biology of integrins, Oncology, 21 (2007), 612. 
[79] 
T. A. Ulrich, E. M. De Juan Pardo and S. Kumar, The mechanical rigidity of the extracellular matrix regulates the structure, motility, and proliferation of glioma cells, Cancer Res., 69 (2009), 41674174. 
[80] 
M. C. Wake, C. W. Patrick and A. G. Mikos, Pore morphology effects on the fibrovascular tissue growth in porous polymer substrates, Cell. Transplant., 3 (1994), 339343. 
[81] 
K. Wolf, Y. I. Wu, Y. Liu, J. Geiger and E. Tam, Multistep pericellular proteolysis controls the transition from individual to collective cancer cell invasion, Nat. Cell. Biol., 9 (2007), 893904. 
[82] 
K. Wolf, S. Alexander, V. Schacht, L. M. Coussens, U. H. Von Andrian, J. Van Rheenen, E. Deryugina and P. Friedl, Collagenbased cell migration models in vitro and in vivo, Semin. Cell. Dev. Biol., 20 (2009), 931941. 
[83] 
K. Wolf and P. Friedl, Extracellular matrix determinants of proteolytic and nonproteolytic cell migration, Trends Cell. Biol., 21 (2011), 736744. doi: 10.1016/j.tcb.2011.09.006. 
[84] 
K. Wolf, I. Mazo, H. Leung, K. Engelke, U. H. Von Andrian, E. I. Deryugina, A. Y. Strongin, E. B. Bröcker and P. Friedl, Compensation mechanism in tumor cell migration mesenchymalmoeboid transition after blocking of pericellular proteolysis, J. Cell. Biol., 160 (2003), 267277. 
[85] 
Y. L. Yang and L. J. Kaufman, Rheology and confocal reflectance microscopy as probes of mechanical properties and structure during collagen and collagen/hyaluronan selfassembly, Biophys. J., 96 (2009), 15661585. 
[86] 
I. V. Yannas, E. Lee, D. P. Orgill, E. M. Skrabut, G. F. Murphy, Synthesis and characterization of a model extracellular matrix that induces partial regeneration of adult mammalian skin, Proc. Natl. Acad. Sci. U. S. A., 86 (1989), 933937. doi: 10.1073/pnas.86.3.933. 
[87] 
M. H. Zaman, P. Matsudaira and D. A. Lauffenburger, Understanding effects of matrix protease and matrix organization on directional persistence and translational speed in threedimensional cell migration, Ann. Biomed. Eng., 35 (2007), 91100. 
[88] 
M. H. Zaman, L. M. Trapani, A. L. Sieminski, D. Mackellar, H. Gong, R. D. Kamm, A. Wells, D. A. Lauffenburger and P. Matsudaira, Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cellmatrix adhesion and proteolysis, Proc. Natl. Acad. Sci. USA, 103 (2006), 1088910894. 
show all references
References:
[1] 
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, "Molecular Biology of the Cell," $3^{rd}$ edition, Garland Science, 1994. 
[2] 
M. Arnold, V. C. HirschfeldWarneken, T. Lohmüller, P. Heil, J. Blümmel, E. A. CavalcantiAdam, M. LópezGarcía, P. Walther, H. Kessler, B. Geiger and J. P. Spatz, Induction of cell polarization and migration by a gradient of nanoscale variations in adhesive ligand spacing, Nano Lett., 8 (2008), 20632069. 
[3] 
M. Bajénoff, J. G. Egen, L. Y. Koo, J. P. Laugier, F. Brau, N. Glaichenhaus and R. N. Germain, Stromal cell networks regulate lymphocyte entry, migration, and territoriality in lymph nodes, Immunity, 25 (2006), 9891001. doi: 10.1016/j.immuni.2006.10.011. 
[4] 
A. Balter, R. M. Merks, N. J. Poplawski, M. Swat and J. A. Glazier, The GlazierGranerHogeweg model: Extensions, future directions, and opportunities for further study, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 157167. 
[5] 
A. L. Bauer, T. L. Jackson and Y. Jiang, A cellbased model exhibiting branching and anastomosis during tumorinduced angiogenesis, Biophys. J., 92 (2007), 31053121. doi: 10.1529/biophysj.106.101501. 
[6] 
C. Beadle, M. C. Assanah, P. Monzo, R. Vallee, S. Rosenfeld and P. Canoll, The role of myosin ii in glioma invasion of the brain, Mol. Biol. Cell., 19 (2008), 33573368. doi: 10.1091/mbc.E08030319. 
[7] 
J. Behring, R. Junker, X. F. Walboomers, B. Chessnut and J. A. Jansen, Toward guided tissue and bone regeneration: Morphology, attachment, proliferation, and migration of cells cultured on collagen barrier membranes. A systematic review, Odontology, 96 (2008), 111. 
[8] 
A. O. Brightman, B. P. Rajwa, J. E. Sturgis, M. E. McCallister, J. P. Robinson and S. L. VoytikHarbin, Timelapse confocal reflection microscopy of collagen fibrillogenesis and extracellular matrix assembly in vitro, Biopolymers, 54 (2000), 222234. doi: 10.1002/10970282(200009)54:3<222::AIDBIP80>3.0.CO;2K. 
[9] 
A. Brock, E. Chang, C. C. Ho, P. LeDuc, X. Jiang, G. M. Whitesides and D. E. Ingber, Geometric determinants of directional cell motility revealed using microcontact printing, Langmuir, 19 (2003), 16111617. 
[10] 
B. T. Burgess, J. L. Myles and R. B. Dickinson, Quantitative analysis of adhesionmediated cell migration in threedimensional gels of RGDgrafted collagen, Ann. Biomed. Eng., 28 (2003), 110118. 
[11] 
R. M. Capito and M. Spector, Scaffoldbased articular cartilage repair, IEEE Eng. Med. Biol. Mag., 22 (2003), 4250. 
[12] 
E. A. CavalcantiAdam, T. Volberg, A. Micoulet, H. Kessler, B. Geiger and J. P. Spatz, Cell spreading and focal adhesion dynamics are regulated by spacing of integrin ligands, Biophys. J., 92 (2007), 29642974. 
[13] 
J. Condeelis and J. E. Segall, Intravital imaging of cell movement in tumours, Nat. Rev. Cancer, 3 (2003), 921930. 
[14] 
M. W. Conklin, J. C. Eickhoff, K. M. Riching, C. A. Pehlke, K. W. Eliceiri, P. P. Provenzano, A. Friedl and P. J. Keely, Aligned collagen is a prognostic signature for survival in human breast carcinoma, Am. J. Pathol., 178 (2011), 12211232. doi: 10.1016/j.ajpath.2010.11.076. 
[15] 
R. B. Dickinson, S. Guido and R. T. Tranquillo, Biased cellmigration of fibroblasts exhibiting contact guidance in oriented collagen gels, Ann. Biomed. Eng., 22 (1994), 342356. doi: 10.1007/BF02368241. 
[16] 
P. A. DiMilla, J. A. Stone, J. A. Quinn, S. M. Albelda and D. A. Lauffenburger, Maximal migration of human smoothmuscle cells on fibronectin and typeIV collagen occurs at an intermediate attachment strength, J. Cell. Biol., 122 (1993), 729737. doi: 10.1083/jcb.122.3.729. 
[17] 
A. D. Doyle, F. W. Wang, K. Matsumoto and K. M. Yamada, Onedimensional topography underlies threedimensional fibrillar cell migration, J. Cell. Biol., 184 (2009), 481490. 
[18] 
N. Dubey, P. C. Letourneau and R. T. Tranquillo, Neuronal contact guidance in magnetically aligned fibrin gels: effect of variation in gel mechanostructural properties, Biomaterials, 22 (2001), 10651075. 
[19] 
G. A. Dunn and T. Ebendal, Contact guidance on oriented collagen gels, Exp. Cell. Res., 111 (1978), 475479. 
[20] 
G. A. Dunn, Characterizing a kinesis response: Timeaveraged measures of cell speed and directional persistence, Agents Actions Suppl., 12 (1983), 1433. 
[21] 
A. Engler, L. Bacakova, C. Newman, A. Hategan, M. Griffin and D. Discher, Substrate compliance versus ligand density in cell on gel responses, Biophys. J., 86 (2004), 617628. 
[22] 
P. Friedl, F. Entschladen, C. Conrad, B. Niggemann and K. S. Zänker, CD4+ Tlymphocytes migrating in threedimensional collagen lattices lack focal adhesions and utilize $\beta$1 integrinindependent strategies for polarization, interaction with collagen fibers and locomotion, Eur. J. Immunol., 28 (1998), 23312343. 
[23] 
P. Friedl and E. B. Brocker, The biology of cell locomotion within threedimensional extracellular matrix, Cell. Mol. Life Sci., 57 (2000), 4164. doi: 10.1007/s000180050498. 
[24] 
P. Friedl and K. Wolf, Tumourcell invasion and migration: Diversity and escape mechanisms, Nat. Rev. Cancer, 3 (2003), 362374. 
[25] 
P. Friedl, K. Maaser, C. E. Klein, B. Niggemann, G. Krohne and K. S. Zänker, Migration of highly aggressive MV3 melanoma cells in 3dimensional collagen lattices results in local matrix reorganization and shedding of alpha2 and beta1 integrins and CD44, Cancer Res., 57 (1997), 20612070. 
[26] 
P. Friedl, K. Wolf and J. Lammerding, Nuclear mechanics during cell migration, Curr. Opin. Cell. Biol., 23 (2011), 253. 
[27] 
P. Friedl and B. Weigelin, Interstitial leukocyte migration and immune function, Nat. Immunol., 9 (2008), 960969. 
[28] 
P. Friedl and K. Wolf, Plasticity of cell migration: A multiscale tuning model, J. Cell. Biol., 188 (2009), 1119. 
[29] 
P. Friedl and D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell. Biol., 10 (2009), 445457. 
[30] 
C. Gaudet, W. Marganski, S. Kim, C. T. Brown, V. Gunderia, M. Dembo and J. Wong, Influence of type I collagen surface density on fibroblast spreading, motility, and contractility, Biophys. J., 85 (2003), 33293335. 
[31] 
G. Gerlitz and M. Bustin, The role of chromatin structure in cell migration, Trends Cell. Biol., 21 (2011), 611. 
[32] 
C. Giverso, M. Scianna, L. Preziosi, N. Lo Buono and A. Funaro, Individual cellbased model for invitro mesothelial invasion of ovarian cancer, Math. Model. Nat. Phenom., 5 (2010), 203223. 
[33] 
J. A. Glazier, A. Balter and N. J. Poplawski, Magnetization to morphogenesis: A brief history of the GlazierGranerHogeweg model, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 79106. 
[34] 
J. A. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 47 (1993), 21282154. 
[35] 
S. L. Goodman, G. Risse and K. Vondermark, The E8 subfragment of laminin promotes locomotion of myoblasts over extracellular matrix, J. Cell. Biol., 109 (1989), 799809. 
[36] 
F. Graner and J. A. Glazier, Simulation of biological cell sorting using a twodimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 20132016. doi: 10.1103/PhysRevLett.69.2013. 
[37] 
B. A. Harley, H. Kim, M. H. Zaman, I. V. Yannas, D. A. Lauffenburger and L. J. Gibson, Microarchitecture of threedimensional scaffolds influences cell migration behavior via junction interactions, Biophys. J., 95 (2008), 40134024. doi: 10.1529/biophysj.107.122598. 
[38] 
B. A. Harley, M. H. Spilker, J. W. Wu, K. Asano, H. P. Hsu, M. Spector and I. V. Yannas, Optimal degradation rate for collagen chambers used for regeneration of peripheral nerves over long gaps, Cells Tissues Organs, 176 (2008), 153165. 
[39] 
O. Ilina and P. Friedl, Mechanisms of collective cell migration at a glance, J. Cell Sci., 122 (2009), 32033208. 
[40] 
E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik., 31 (1925), 253. 
[41] 
R. M. Kuntz and W. M. Saltzman, Neutrophil motility in extracellular matrix gels: mesh size and adhesion affect speed of migration, Biophys. J., 72 (1997), 14721480. doi: 10.1016/S00063495(97)787939. 
[42] 
S. Kurosaka and A. Kashina, Cell biology of embryonic migration, Birth Defects Res. C. Embryo Today, 84 (2008), 102122. 
[43] 
E. Lamers, R. Van Horssen, J. Te Riet, F. C. Van Delft, R. Luttge, X. F. Walboomers and J. A. Jansen, The influence of nanoscale topographical cues on initial osteoblast morphology and migration, Eur. Cell. Mater., 9 (2010), 329343. 
[44] 
D. A. Lauffenburger and J. J. Lindermann, "Receptors: Bodels for Binding, Trafficking, and Signaling," Oxford University Press, London, 1996. 
[45] 
D. A. Lauffenburger and A. F. Horwitz, Cell migration: a physically integrated molecular process, Cell, 84 (1996), 359369. doi: 10.1016/S00928674(00)812805. 
[46] 
D. Lehnert, B. WehrleHaller, C. David, U. Weiland, C. Ballestrem, B. A. Imhof and M. Bastmeyer, Cell behaviour on micropatterned substrata: limits of extracellular matrix geometry for spreading and adhesion, J. Cell. Sci., 117 (2004), 4152. doi: 10.1242/jcs.00836. 
[47] 
M. P. Lutolf, J. L. LauerFields, H. G. Schmoekel, A. T. Metters, F. E. Weber, G. B. Fields and J. A. Hubbell, Synthetic matrix metalloproteinasesensitive hydrogels for the conduction of tissue regeneration: Engineering cellinvasion characteristics, Proc. Natl. Acad. Sci. U. S. A., 100 (2003), 54135418. 
[48] 
K. Maaser, K. Wolf, C. E. Klein, B. Niggemann, K. S. Zänker, E. B. Bröcker and Friedl, Functional hierarchy of simultaneously expressed adhesion receptors: Integrin $\alpha$2$\beta$1 but not CD44 mediates MV3 melanoma cell migration and matrix reorganization within threedimensional hyaluronancontaining collagen matrices, Mol. Biol. Cell., 10 (1999), 30673079. 
[49] 
A. W. Mahoney, B. G. Smith, N. S. Flann and G. J. Podgorski, Discovering novel cancer therapies: A computational modeling and search approach, in "IEEE conference on Computational Intelligence in Bioinformatics and Bioengineering," (2008), 233240. 
[50] 
A. F. M arée, V. A. Grieneisen and P. Hogeweg, The Cellular Potts Model and biophysical properties of cells, tissues and morphogenesis, in "SingleCellBased Models in Biology and Medicine, Mathematics and Biosciences in Interactions" (eds. A. R. A. Anderson, M. A. J. Chaplain and K. A. Rejniak), Birkhaüser, (2007), 107136. 
[51] 
R. M. Merks, E. D. Perryn, A. Shirinifard and J. A. Glazier, Contactinhibited chemotaxis in de novo and sprouting blood vessel growth, PLoS Comput. Biol., 4 (2008), e1000163, 16 pp. 
[52] 
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 10871092. 
[53] 
A. Mogilner and G. Oster, Polymer motors: Ppushing out the front and pulling up the back, Current Biology, 13 (2003), R721733. 
[54] 
F. J. O'Brien, B. A. Harley, I. V. Yannas and L. Gibson, Influence of freezing rate on pore structure in freezedried collagenGAG scaffolds, Biomaterials, 25 (2004), 10771086. 
[55] 
L. H. Olde Damink, P. J. Dijkstra, M. J. Luyn, P. B. Wachem, P. Nieuwenhuis and J. Feijen, Glutaraldehyde as a crosslinking agent for collagenbased biomaterials, J. Mater. Sci. Mater. Med., 6 (1995), 460472. 
[56] 
J. M. Orban, L. B. Wilson, J. A. Kofroth, M. S. ElKurdi, T. M. Maul and D. A. Vorp, Crosslinking of collagen gels by transglutaminase, J. Biomed. Mater. Res. A, 68 (2004), 756762. 
[57] 
S. P. Palecek, J. C. Loftus, M. H. Ginsberg, D. A. Lauffenburger and A. F. Horwitz, Integrinligand binding properties govern cell migration speed through cellsubstratum adhesiveness, Nature, 385 (1997), 537540. 
[58] 
E. D. Perryn, A. Czirok and C. D. Little, Vascular sprout formation entails tissue deformations and VEcadherindependent cellautonomous motility, Dev. Biol., 313 (2008), 545555. doi: 10.1016/j.ydbio.2007.10.036. 
[59] 
S. R. Peyton and A. J. Putnam, Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion, J. Cell. Physiol., 204 (2005), 198209. doi: 10.1002/jcp.20274. 
[60] 
T. D. Pollard and G. G. Borisy, Cellular motility driven by assembly and disassembly of actin filaments, Cell, 112 (2003), 453465. doi: 10.1016/S00928674(03)00120X. 
[61] 
R. B. Potts, Some generalized orderdisorder transformations, Proc. Camb. Phil. Soc., 48 (1952), 106109. 
[62] 
G. P. Raeber, M. P. Lutolf and J. A. Hubbell, Molecularly engineered PEG hydrogels: a novel model system for proteolytically mediated cell migration, Biophys. J., 89 (2005), 13741388. 
[63] 
C. B. Raub, V. Suresh, T. Krasieva, J. Lyubovitsky, J. D. Mih, A. J. Putnam, B. J. Tromberg and S. C. George, Noninvasive assessment of collagen gel microstructure and mechanics using multiphoton microscopy, Biophys. J., 92 (2007), 22122222. doi: 10.1529/biophysj.106.097998. 
[64] 
A. J. Ridley, M. A. Schwartz, K. Burridge, R. A. Firtel, M. H. Ginsberg, G. Borisy, J. T. Parsons and A. R. Horwitz, Cell migration: Integrating signals from front to back, Science, 302 (2003), 17041709. 
[65] 
B. A. Roeder, K. Kokini, J. E. Sturgis, J. P. Robinson and S. L. VoytikHarbin, Tensile mechanical properties of threedimensional type I collagen extracellular matrices with varied microstructure, J. Biomech. Eng., 124 (2002), 214222. doi: 10.1115/1.1449904. 
[66] 
C. G. Rolli, T. Seufferlein, R. Kemkemer and J. P. Spatz, Impact of tumor cell cytoskeleton organization on invasiveness and migration: A microchannelbased approach, PLoS ONE, 5 (2010), e8726. 
[67] 
B. M. Rubenstein and L. J. Kaufman, The role of extracellular matrix in glioma invasion: A Cellular Potts Model approach, Biophys. J., 95 (2006), 56615680. 
[68] 
F. Sabeh, R. ShimizuHirota and S. J. Weiss, Proteasedependent versus independent cancer cell invasion programs: Threedimensional amoeboid movement revisited, J. Cell. Biol., 185 (2009), 1119. 
[69] 
E. Sahai, Illuminating the metastatic process, Nat. Rev. Cancer, 7 (2007), 737749. 
[70] 
M. Scianna, A multiscale hybrid model for proangiogenic calcium signals in a vascular endothelial cell, Bull. Math. Biol., 76 (2011), 12531291. 
[71] 
M. Scianna and L. Preziosi, Multiscale developments of the Cellular Potts Model, Multiscale Model. Simul., 10 (2012), 342382. 
[72] 
M. T. Sheu, J. C. Huang, G. C. Yeh and H. O. Ho, Characterization of collagen gel solutions and collagen matrices for cell culture, Biomaterials, 22 (2001), 17131719. doi: 10.1016/S01429612(00)00315X. 
[73] 
S. Schmidt and P. Friedl, Interstitial cell migration: Integrindependent and alternative adhesion mechanisms, Cell. Tissue Res., 339 (2010), 8392. doi: 10.1007/s0044100908929. 
[74] 
R. C. Siegel, Collagen crosslinking. Synthesis of collagen crosslinks in vitro with highly purified lysyl oxidase, J. Biol. Chem., 251 (1976), 57865792. 
[75] 
C. Spadaccio, A. Rainer, S. De Porcellinis, M. Centola, F. De Marco, M. Chello, M. Trombetta, M. and J. A. Genovese, A GCSF functionalized PLLA scaffold for wound repair: An in vitro preliminary study, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 843846. 
[76] 
M. S. Steinberg, Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation, Science, 141 (1963), 401408. 
[77] 
M. S. Steinberg, Does differential adhesion govern selfassembly processes in histogenesis? Equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells, J. Exp. Zool., 171 (1970), 395433. 
[78] 
D. G. Stupack, The biology of integrins, Oncology, 21 (2007), 612. 
[79] 
T. A. Ulrich, E. M. De Juan Pardo and S. Kumar, The mechanical rigidity of the extracellular matrix regulates the structure, motility, and proliferation of glioma cells, Cancer Res., 69 (2009), 41674174. 
[80] 
M. C. Wake, C. W. Patrick and A. G. Mikos, Pore morphology effects on the fibrovascular tissue growth in porous polymer substrates, Cell. Transplant., 3 (1994), 339343. 
[81] 
K. Wolf, Y. I. Wu, Y. Liu, J. Geiger and E. Tam, Multistep pericellular proteolysis controls the transition from individual to collective cancer cell invasion, Nat. Cell. Biol., 9 (2007), 893904. 
[82] 
K. Wolf, S. Alexander, V. Schacht, L. M. Coussens, U. H. Von Andrian, J. Van Rheenen, E. Deryugina and P. Friedl, Collagenbased cell migration models in vitro and in vivo, Semin. Cell. Dev. Biol., 20 (2009), 931941. 
[83] 
K. Wolf and P. Friedl, Extracellular matrix determinants of proteolytic and nonproteolytic cell migration, Trends Cell. Biol., 21 (2011), 736744. doi: 10.1016/j.tcb.2011.09.006. 
[84] 
K. Wolf, I. Mazo, H. Leung, K. Engelke, U. H. Von Andrian, E. I. Deryugina, A. Y. Strongin, E. B. Bröcker and P. Friedl, Compensation mechanism in tumor cell migration mesenchymalmoeboid transition after blocking of pericellular proteolysis, J. Cell. Biol., 160 (2003), 267277. 
[85] 
Y. L. Yang and L. J. Kaufman, Rheology and confocal reflectance microscopy as probes of mechanical properties and structure during collagen and collagen/hyaluronan selfassembly, Biophys. J., 96 (2009), 15661585. 
[86] 
I. V. Yannas, E. Lee, D. P. Orgill, E. M. Skrabut, G. F. Murphy, Synthesis and characterization of a model extracellular matrix that induces partial regeneration of adult mammalian skin, Proc. Natl. Acad. Sci. U. S. A., 86 (1989), 933937. doi: 10.1073/pnas.86.3.933. 
[87] 
M. H. Zaman, P. Matsudaira and D. A. Lauffenburger, Understanding effects of matrix protease and matrix organization on directional persistence and translational speed in threedimensional cell migration, Ann. Biomed. Eng., 35 (2007), 91100. 
[88] 
M. H. Zaman, L. M. Trapani, A. L. Sieminski, D. Mackellar, H. Gong, R. D. Kamm, A. Wells, D. A. Lauffenburger and P. Matsudaira, Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cellmatrix adhesion and proteolysis, Proc. Natl. Acad. Sci. USA, 103 (2006), 1088910894. 
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