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Computational modelling of cancer development and growth: modelling at multiple scales and multiscale modelling. (English) Zbl 1394.92065

Summary: In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-\(\kappa \)B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-\(\kappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

MSC:

92C50 Medical applications (general)
92C37 Cell biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C42 Systems biology, networks

Software:

Wavos; URDME
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alarcón T, Byrne H, Maini P (2003) A cellular automaton model for tumour growth in inhomogeneous environment. J Theor Biol 225:257-274 · Zbl 1405.92085
[2] Alberts B, Bray D, Hopkin K, Johnson A, Lewis J, Raff M, Roberts K, Walter P (eds) (2010) Essential cell biology. Garland Publishing, Inc., New York
[3] Alcaraz, JL; Buscemi, M; Grabulosa, X; Trepat, B; Fabry, R; Farre, D; Navajas, D, Microrheology of human lung epithelial cells measured by atomic force, Biophys J, 84, 2071-2079, (2003) · doi:10.1016/S0006-3495(03)75014-0
[4] Andasari, V; Roper, R; Swat, MH; Chaplain, MAJ, Integrating intracellular dynamics using compucell 3D and bionetsolver: applications to multiscale modelling of cancer cell growth and invasion, PLoS ONE, 7, e33726, (2012) · doi:10.1371/journal.pone.0033726
[5] Anderson, ARA; Chaplain, MAJ, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull Math Biol, 60, 857-899, (1998) · Zbl 0923.92011 · doi:10.1006/bulm.1998.0042
[6] Arenzana-Seisdedos, F; Turpin, P; Rodriguez, M; Thomas, D; Hay, RT; Virelizier, JL; Dargemont, C, Nuclear localization of I\(\upkappa \)B alpha promotes active transport of NF-\(\upkappa \)B from the nucleus to the cytoplasm, J Cell Sci, 110, 369-378, (1997)
[7] Ashall, L; Horton, CA; Nelson, DE; Paszek, P; Harper, CV; Sillitoe, K; Ryan, S; Spiller, DG; Unitt, JF; Broomhead, DS; Kell, DB; Rand, DA; Sée, V; White, MRH, Pulsatile stimulation determines timing and specificity of NF-\(\upkappa \)B-dependent transcription, Science, 324, 242-246, (2009) · doi:10.1126/science.1164860
[8] Baker, AH; Falgout, RD; Kolev, TV; Yang, UM; Berry, MW (ed.); Gallivan, KA (ed.); Gallopoulos, E (ed.); Grama, A (ed.); Philippe, B (ed.); Saad, Y (ed.); Saied, F (ed.), Scaling hypre’s multigrid solvers to 100,000 cores, 261-279, (2012), Berlin · doi:10.1007/978-1-4471-2437-5_13
[9] Balagadde, FK; Song, H; Ozaki, J; Collins, CH; Barnet, M; Arnold, FH; Quake, SR; You, L, A synthetic Escherichia coli predator-prey ecosystem, Mol Syst Biol, 4, 187, (2008) · doi:10.1038/msb.2008.24
[10] Bar-On, D; Wolter, S; Linde, S; Heilemann, M; Nudelman, G; Nachliel, E; Gutman, M; Sauer, M; Ashery, U, Super-resolution imaging reveals the internal architecture of nano-sized syntaxin clusters, J Bio Chem, 287, 27158-27167, (2012) · doi:10.1074/jbc.M112.353250
[11] Barrio, M; Burrage, K; Leier, A; Tian, T, Oscillatory regulation of HES1: discrete stochastic delay modelling and simulation, PLoS Comput Biol, 2, e117, (2006) · doi:10.1371/journal.pcbi.0020117
[12] Becskei, A; Serrano, L, Engineering stability in gene networks by autoregulation, Nature, 405, 590-593, (2000) · doi:10.1038/35014651
[13] Bernard, S; Čajavec, B; Pujo-Menjouet, L; Mackey, MC; Herzel, H, Modeling transcriptional feedback loops: the role of gro/TLE1 in hes1 oscillations, Philos Trans A Math Phys Eng Sci, 15, 1155-1170, (2006) · Zbl 1152.92316 · doi:10.1098/rsta.2006.1761
[14] Bertuzzi, A; Gandolfi, A, Cell kinetics in a tumour cord, J Theor Biol, 204, 587-599, (2000) · doi:10.1006/jtbi.2000.1079
[15] Bertuzzi, A; Fasano, A; Filidoro, L; Gandolfi, A; Sinisgalli, C, Dynamics of tumour cords following changes in oxygen availability: a model including a delayed exit from quiescence, Math Comput Model, 41, 1119-1135, (2005) · Zbl 1080.92036 · doi:10.1016/j.mcm.2005.05.007
[16] Bertuzzi, A; Fasano, A; Gandolfi, A; Sinisgalli, C, Necrotic core in EMT6/ro tumour spheroids: is it caused by an ATP deficit?, J Theor Biol, 262, 142-150, (2010) · Zbl 1403.92104 · doi:10.1016/j.jtbi.2009.09.024
[17] Betzig, E; Patterson, GH; Sougrat, R; Lindwasser, OW; Olenych, S; Bonifacino, JS; Davidson, MW; Lippincott-Schwartz, J; Hess, HF, Imaging intracellular fluorescent proteins at nanometer resolution, Science, 313, 1642-1645, (2006) · doi:10.1126/science.1127344
[18] Busenberg, S; Mahaffy, JM, Interaction of spatial diffusion and delays in models of genetic control by repression, J Math Biol, 22, 313-333, (1985) · Zbl 0593.92010 · doi:10.1007/BF00276489
[19] Casciari, J; Sotirchos, S; Sutherland, R, Mathematical modelling of microenvironment and growth in EMT6/ro multicellular tumour spheroids, Cell Prolif, 25, 1-22, (1992) · doi:10.1111/j.1365-2184.1992.tb01433.x
[20] Chaplain, MAJ; Ptashnyk, M; Sturrock, M, Hopf bifurcation in a gene regulatory network model: molecular movement causes oscillations, Math Model Methods Appl Sci, 25, 1179-1215, (2015) · Zbl 1326.92019 · doi:10.1142/S021820251550030X
[21] Chen, YY; Galloway, KE; Smolke, CD, Synthetic biology: advancing biological frontiers by building synthetic systems, Genome Biol, 13, 240, (2012) · doi:10.1186/gb-2012-13-2-240
[22] Cheong, R; Hoffmann, A; Levchenko, A, Understanding NF-\(\upkappa \)B signaling via mathematical modeling, Mol Syst Biol, 4, 192, (2008) · doi:10.1038/msb.2008.30
[23] Chu, Y-S; Thomas, WA; Eder, O; Pincet, E; Thiery, JP; Dufour, S, Force measurements in E-cadherin-mediated cell doublets reveal rapid adhesion strengthened by actin cytoskeleton remodeling through rac and cdc42, J Cell Biol, 167, 1183-1194, (2004) · doi:10.1083/jcb.200403043
[24] Cytowski, M; Szymańska, Z, Large scale parallel simulations of 3-D cell colony dynamics, IEEE Comput Sci Eng, 16, 86-95, (2014) · doi:10.1109/MCSE.2014.2
[25] Cytowski M, Szymańska Z (2015a) Enabling large scale individual-based modelling through high performance computing. In: ITM Web of Conferences, vol 5, p 00014
[26] Cytowski, M; Szymańska, Z, Large scale parallel simulations of 3-D cell colony dynamics. II. coupling with continuous description of cellular environment, IEEE Comput Sci Eng, 17, 44-48, (2015) · doi:10.1109/MCSE.2015.66
[27] Cytowski M, Szymańska Z, Umiński P, Andrejczuk G, Raszkowski K (2017) Implementation of an agent-based parallel tissue modelling framework for the Intel MIC architecture. Sci Program 2017, Article ID 8721612, 11 pages. doi:10.1155/2017/8721612
[28] D’Antonio, G; Macklin, P; Preziosi, L, An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix, Math Biosci Eng, 10, 75-101, (2013) · Zbl 1259.65001 · doi:10.3934/mbe.2013.10.75
[29] Drasdo, D; Höhme, S, A single-cell-based model of tumor growth in vitro: monolayers and spheroids, Phys Biol, 2, 133-147, (2005) · doi:10.1088/1478-3975/2/3/001
[30] Drawert B, Engblom S, Hellander A (2012) URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst Biol. doi:10.1186/1752-0509-6-76
[31] Elowitz, MB; Leibler, S, A synthetic oscillatory network of transcriptional regulators, Nature, 403, 335-338, (2000) · doi:10.1038/35002125
[32] Engblom, S; Ferm, L; Hellander, A; Lötstedt, P, Simulation of stochastic reaction-diffusion processes on unstructured meshes, SIAM J Sci Comput, 31, 1774-1797, (2009) · Zbl 1190.65015 · doi:10.1137/080721388
[33] Galle, J; Loeffler, M; Drasdo, D, Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro, Biophys J, 88, 62-75, (2005) · doi:10.1529/biophysj.104.041459
[34] Geva-Zatorsky N, Rosenfeld N, Itzkovitz S, Milo R, Sigal A, Dekel E, Yarnitzky T, Liron Y, Polak P, Lahav G, Alon U (2006) Oscillations and variability in the p53 system. Mol Syst Biol. doi:10.1038/msb4100068 · Zbl 1400.92194
[35] Gibson, MA; Bruck, J, Efficient exact stochastic simulation of chemical species and many channels, J Phys Chem, 104, 1876-1889, (2000) · doi:10.1021/jp993732q
[36] Gillespie, DT, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J Comput Phys, 22, 403-434, (1976) · doi:10.1016/0021-9991(76)90041-3
[37] Glass, L; Kauffman, SA, Co-operative components, spatial localization and oscillatory cellular dynamics, J Theor Biol, 34, 219-237, (1970) · doi:10.1016/0022-5193(72)90157-9
[38] Goodwin, BC, Oscillatory behaviour in enzymatic control processes, Adv Enzyme Regul, 3, 425-428, (1965) · doi:10.1016/0065-2571(65)90067-1
[39] Griffith, JS, Mathematics of cellular control processes. I. negative feedback to one gene, J Theor Biol, 20, 202-208, (1968) · doi:10.1016/0022-5193(68)90189-6
[40] Gumbiner, BM, Regulation of cadherin-mediated adhesion in morphogenesis, Nat Rev Mol Cell Biol, 6, 622-634, (2005) · doi:10.1038/nrm1699
[41] Hanahan, D; Weinberg, RA, The hallmarks of cancer, Cell, 100, 57-70, (2000) · doi:10.1016/S0092-8674(00)81683-9
[42] Hanahan, D; Weinberg, RA, Hallmarks of cancer: the next generation, Cell, 144, 646-674, (2011) · doi:10.1016/j.cell.2011.02.013
[43] Harang, R; Bonnet, G; Petzold, LR, WAVOS: a MATLAB toolkit for wavelet analysis and visualization of oscillatory systems, BMC Res Notes, 5, 163, (2012) · doi:10.1186/1756-0500-5-163
[44] Hiersemenzel K, Brown ER, Duncan RR (2013) Imaging large cohorts of single ion channels and their activity. Front Endocrinol. doi:10.3389/fendo.2013.00114
[45] Hirata, H; Yoshiura, S; Ohtsuka, T; Bessho, Y; Harada, T; Yoshikawa, K; Kageyama, R, Oscillatory expression of the bhlh factor hes1 regulated by a negative feedback loop, Science, 298, 840-843, (2002) · doi:10.1126/science.1074560
[46] Hlatky, L; Hahnfeldt, P; Folkman, J, Clinical application of anti-angiogenic therapy: microvessel density, what it does and doesn’t tell us, J Natl Cancer Inst, 94, 883-893, (2002) · doi:10.1093/jnci/94.12.883
[47] Hoffmann, A; Levchenko, A; Scott, M; Baltimore, D, The I\(\upkappa \)B-NF-\(\upkappa \)B signaling module: temporal control and selective gene activation, Science, 298, 1241-1245, (2002) · doi:10.1126/science.1071914
[48] Jagiella, N; Müller, B; Müller, M; Vignon-Clementel, IE; Drasdo, D, Inferring growth control mechanisms in growing multi-cellular spheroids of nsclc cells from spatial-temporal image data, PLoS Comput Biol, 12, e1004412, (2016) · doi:10.1371/journal.pcbi.1004412
[49] Jensen, MH; Sneppen, J; Tiana, G, Sustained oscillations and time delays in gene expression of protein hes1, FEBS Lett, 541, 176-177, (2003) · doi:10.1016/S0014-5793(03)00279-5
[50] Lachowicz, M; Parisot, M; Szymańska, Z, Intracellular protein dynamics as a mathematical problem, Discrete Contin Dyn Syst B, 21, 2551-2566, (2016) · Zbl 1352.35205 · doi:10.3934/dcdsb.2016060
[51] Lahav, G; Rosenfeld, N; Sigal, A; Geva-Zatorsky, N; Levine, AJ; Elowitz, MB; Alon, U, Dynamics of the p53-mdm2 feedback loop in individual cells, Nature Genet, 36, 147-150, (2004) · doi:10.1038/ng1293
[52] Lee, RE; Walker, SR; Savery, K; Frank, DA; Gaudet, S, Fold change of nuclear NF-\(\upkappa \)B determines TNF-induced transcription in single cells, Mol Cell, 53, 867-879, (2014) · doi:10.1016/j.molcel.2014.01.026
[53] Lewis, J, Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator, Curr Biol, 13, 1398-1408, (2003) · doi:10.1016/S0960-9822(03)00534-7
[54] Lipniacki, T; Kimmel, M, Deterministic and stochastic models of NF\(\upkappa \)B pathway, Cardiovasc Toxicol, 7, 215-234, (2007) · doi:10.1007/s12012-007-9003-x
[55] Mackey, MC; Glass, L, Oscillation and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036 · doi:10.1126/science.267326
[56] Macklin, P; McDougall, S; Anderson, ARA; Chaplain, MAJ; Cristini, V; Lowengrub, J, Multiscale modelling and nonlinear simulation of vascular tumour growth, J Math Biol, 58, 765-798, (2009) · Zbl 1311.92040 · doi:10.1007/s00285-008-0216-9
[57] Macnamara, CK; Chaplain, MAJ, Diffusion driven oscillations in gene regulatory networks, J Theor Biol, 407, 51-70, (2016) · doi:10.1016/j.jtbi.2016.07.021
[58] Macnamara, CK; Chaplain, MAJ, Spatio-temporal models of synthetic genetic oscillators, Math Biol Eng, 14, 249-262, (2017) · Zbl 1362.92021 · doi:10.3934/mbe.2017016
[59] Mahaffy, JM, Genetic control models with diffusion and delays, Math Biosci, 90, 519-533, (1988) · Zbl 0684.92012 · doi:10.1016/0025-5564(88)90081-8
[60] Mahaffy, JM; Pao, CV, Models of genetic control by repression with time delays and spatial effects, J Math Biol, 20, 39-57, (1984) · Zbl 0577.92010 · doi:10.1007/BF00275860
[61] Mahaffy, RE; Shih, CK; McKintosh, FC; Kaes, J, Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells, Phys Rev Lett, 85, 880-883, (2000) · doi:10.1103/PhysRevLett.85.880
[62] Manley, S; Gillette, JM; Patterson, GH; Shroff, H; Hess, HF; Betzig, E; Lippincott-Schwartz, J, High-density mapping of single-molecule trajectories with photoactivated localization microscopy, Nat Methods, 5, 155-157, (2008) · doi:10.1038/nmeth.1176
[63] Marquez-Lago TT, Leier A, Burrage K (2010) Probability distributed time delays: integrating spatial effects into temporal models. BMC Syst Biol. doi:10.1186/1752-0509-4-19
[64] McDougall, SR; Anderson, ARA; Chaplain, MAJ, Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies, J Theor Biol, 241, 564-589, (2006) · Zbl 1447.92096 · doi:10.1016/j.jtbi.2005.12.022
[65] Miron-Mendoza, M; Koppaka, V; Zhou, C; Petroll, WM, Techniques for assessing 3-D cellmatrix mechanical interactions in vitro and in vivo, Exp Cell Res, 319, 2470-2480, (2013) · doi:10.1016/j.yexcr.2013.06.018
[66] Momiji, H; Monk, NAM, Dissecting the dynamics of the hes1 genetic oscillator, J Theor Biol, 254, 784-798, (2008) · Zbl 1400.92194 · doi:10.1016/j.jtbi.2008.07.013
[67] Monk, NAM, Oscillatory expression of hes1, p53, and NF-\(\upkappa \)B driven by transcriptional time delays, Curr Biol, 13, 1409-1413, (2003) · doi:10.1016/S0960-9822(03)00494-9
[68] Mueller-Klieser, WF; Sutherland, RM, Oxygen consumption and oxygen diffusion properties of multicellular spheroids from two different cell lines, Adv Exp Med Biol, 180, 311-321, (1984) · doi:10.1007/978-1-4684-4895-5_30
[69] Näthke, IS; Hinck, L; Nelson, WJ, The cadherin/catenin complex: connections to multiple cellular processes involved in cell adhesion, proliferation and morphogenesis, Semin Dev Biol, 6, 89-95, (1995) · doi:10.1016/S1044-5781(06)80018-6
[70] Nelson, DE; Ihekwaba, AEC; Elliott, M; Johnson, JR; Gibney, CA; Foreman, BE; Nelson, G; See, V; Horton, CA; Spiller, DG; Edwards, SW; McDowell, HP; Unitt, JF; Sullivan, E; Grimley, R; Benson, N; Broomhead, D; Kell, DB; White, MRH, Oscillations in NF-\(\upkappa \)B signaling control the dynamics of gene expression, Science, 306, 704-708, (2004) · doi:10.1126/science.1099962
[71] O’Brien, EL; Itallie, EV; Bennett, MR, Modeling synthetic gene oscillators, Math Biosci, 236, 1-15, (2012) · Zbl 1237.92028 · doi:10.1016/j.mbs.2012.01.001
[72] O’Dea, E; Hoffmann, A, The regulatory logic of the NF-\(\upkappa \)B signaling system, Cold Spring Harb Perspect Biol, 2, a00021, (2010)
[73] Pekalski, J; Zuk, P; Kochanczyk, M; Junkin, M; Kellogg, R; Tay, S; Lipniacki, T, Spontaneous NF\(\upkappa \)B activation by autocrine TNF\(\upalpha \) signaling: a computational analysis, PLoS ONE, 8, e78887, (2013) · doi:10.1371/journal.pone.0078887
[74] Purcell, O; Savery, NJ; Grierson, CS; Bernardo, M, A comparative analysis of synthetic genetic oscillators, J R Soc Interface, 7, 1503-1524, (2010) · doi:10.1098/rsif.2010.0183
[75] Ramis-Conde, I; Drasdo, D; Anderson, ARA; Chaplain, MAJ, Modelling the influence of the E-cadherin-\(\upbeta \)-catenin pathway in cancer cell invasion: a multi-scale approach, Biophys J, 95, 155-165, (2008) · doi:10.1529/biophysj.107.114678
[76] Ramis-Conde, I; Drasdo, D; Anderson, ARA; Chaplain, MAJ, Multi-scale modelling of cancer cell intravasation: the role of cadherins in metastasis, Phys Biol, 6, 016008, (2009) · doi:10.1088/1478-3975/6/1/016008
[77] Ritchie, T; Zhou, W; McKinstry, E; Hosch, M; Zhang, Y; Näthke, IS; Engelhardt, JF, Developmental expression of catenins and associated proteins during submucosal gland morphogenesis in the airway, Exp Lung Res, 27, 121-141, (2001) · doi:10.1080/019021401750069375
[78] Schaller, G; Meyer-Hermann, M, Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model, Phys Rev E, 71, 051910-1-051910-16, (2005) · doi:10.1103/PhysRevE.71.051910
[79] Schlüter, DK; Ramis-Conde, I; Chaplain, MAJ, Computational modeling of single cell migration: the leading role of extracellular matrix fibers, Biophys J, 103, 1141-1151, (2012) · doi:10.1016/j.bpj.2012.07.048
[80] Schlüter, DK; Ramis-Conde, I; Chaplain, MAJ, Multi-scale modelling of the dynamics of cell colonies: insights into cell-adhesion forces and cancer invasion from in silico simulations, J R Soc Interface, 12, 20141080, (2015) · doi:10.1098/rsif.2014.1080
[81] Shirinifard, A; Gens, J; Zaitlen, B; Poplawski, N; Swat, M; Glazier, J, 3D multi-cell simulation of tumor growth and angiogenesis, PLoS ONE, 4, e7190, (2009) · doi:10.1371/journal.pone.0007190
[82] Shymko, RM; Glass, L, Spatial switching in chemical reactions with heterogeneous catalysis, J Chem Phys, 60, 835-841, (1974) · doi:10.1063/1.1681157
[83] Skaug, B; Chen, J; Du, F; He, J; Ma, A; Chen, ZJ, Direct, noncatalytic mechanism of IKK inhibition by A20, Mol Cell, 44, 559-571, (2011) · doi:10.1016/j.molcel.2011.09.015
[84] Smolen, P; Baxter, DA; Byrne, JH, Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems, Am J Physiol, 277, c777-c790, (1999) · doi:10.1152/ajpcell.1999.277.4.C777
[85] Smolen, P; Baxter, DA; Byrne, JH, Modeling Circadian oscillations with interlocking positive and negative feedback loops, J Neurosci, 21, 6644-6656, (2001) · doi:10.1523/JNEUROSCI.21-17-06644.2001
[86] Smolen, P; Baxter, DA; Byrne, JH, A reduced model clarifies the role of feedback loops and time delays in the drosophila Circadian oscillator, Biophys J, 83, 2349-2359, (2002) · doi:10.1016/S0006-3495(02)75249-1
[87] Spiller, DG; Wood, CD; Rand, DA; White, MRH, Measurement of single-cell dynamics, Nature, 465, 736-45, (2010) · doi:10.1038/nature09232
[88] Sturrock, M; Terry, AJ; Xirodimas, DP; Thompson, AM; Chaplain, MAJ, Spatio-temporal modelling of the hes1 and p53-mdm2 intracellular signalling pathways, J Theor Biol, 273, 15-31, (2011) · Zbl 1405.92085 · doi:10.1016/j.jtbi.2010.12.016
[89] Sturrock, M; Terry, AJ; Xirodimas, DP; Thompson, AM; Chaplain, MAJ, Influence of the nuclear membrane, active transport, and cell shape on the hes1 and p53-mdm2 pathways: insights from spatio-temporal modelling, Bull Math Biol, 74, 1531-1579, (2012) · Zbl 1312.92021 · doi:10.1007/s11538-012-9725-1
[90] Sturrock, M; Hellander, A; Matzavinos, A; Chaplain, MAJ, Spatial stochastic modelling of the hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonic stem cell differentiation, J R Soc Interface, 10, 20120988, (2013) · doi:10.1098/rsif.2012.0988
[91] Szymańska, Z; Parisot, M; Lachowicz, M, Mathematical modeling of the intracellular protein dynamics: the importance of active transport along microtubules, J Theor Biol, 363, 118-128, (2014) · Zbl 1309.92041 · doi:10.1016/j.jtbi.2014.07.022
[92] Thompson DW (1917) On growth and form. Cambridge University Press, Cambridge · doi:10.5962/bhl.title.11332
[93] Tian, T; Burrage, K; Burrage, PM; Carlettib, M, Stochastic delay differential equations for genetic regulatory networks, J Comput Appl Math, 205, 696-707, (2007) · Zbl 1112.92029 · doi:10.1016/j.cam.2006.02.063
[94] Tiana, G; Jensen, MH; Sneppen, K, Time delay as a key to apoptosis induction in the p53 network, Eur Phys J B, 29, 135-140, (2002) · doi:10.1140/epjb/e2002-00271-1
[95] Linde, S; Löschberger, A; Klein, T; Heidbreder, M; Wolter, S; Heilemann, M; Sauer, M, Direct stochastic optical reconstruction microscopy with standard fluorescent probes, Nat Protoc, 6, 991-1009, (2011) · doi:10.1038/nprot.2011.336
[96] Walenta, S; Mueller-Klieser, WF, Oxygen consumption rate of tumour cells as a function of their proliferative status, Adv Exp Med Biol, 215, 389-391, (1987) · doi:10.1007/978-1-4684-7433-6_47
[97] Weinberg RA (2007) The biology of cancer. Garland Science, New York
[98] Won, S; Lee, B-C; Park, C-S, Functional effects of cytoskeletal components on the lateral movement of individual BK ca channels expressed in live COS-7 cell membrane, FEBS Lett, 585, 2323-2330, (2011) · doi:10.1016/j.febslet.2011.05.069
[99] Yordanov, B; Dalchau, N; Grant, PK; Pedersen, M; Emmott, S; Haseloff, J; Phillips, A, A computational method for automated characterization of genetic components, ACS Synth Biol, 3, 578-588, (2014) · doi:10.1021/sb400152n
[100] Zacharaki, E; Stamatakos, G; Nikita, K; Uzunoglu, N, Simulating growth dynamics and radiation response of avascular tumour spheroids: model validation in the case of an EMT6/ro multicellular spheroid, Comput Methods Programs Biomed, 76, 193-206, (2004) · doi:10.1016/j.cmpb.2004.07.003
[101] Zaman, MH; Trapani, LM; Sieminski, AL; MacKellar, D; Gong, H; Kamm, RD; Wells, A; Lauffenburger, DA; Matsudaira, P, Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis, Proc Natl Acad Sci, 103, 10889-10894, (2006) · doi:10.1073/pnas.0604460103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.