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Two-stage model of carcinogenic mutations with the influence of delays
1. | University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw |
2. | Warsaw University of Life Science, Faculty of Applied Informatics and Mathematics, Nowoursynowska 159, 02-776 Warsaw, Poland |
References:
[1] |
R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, Electron. J. Diff. Eqns., 10 (2003), 33-53. |
[2] |
P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[3] |
C. M. Beauséjour, A. Krtolica, F. Galimi, M. Narita, S.W. Lowe, P. Yaswen and J. Campisi, Reversal of human cellular senescence: roles of the p53 and p16 pathways, EMBO J, 22 (2003), 4212-4222. |
[4] |
K. Camphausen, M. A. Moses, C. Ménard, M. Sproull, W. Beecken, J. Folkman and M. S. O'Reilly, Radiation abscopal antitumor effect is mediated through p53, Cancer Res., 63 (2003), 1990-1993. |
[5] |
Z. Chen, L. C. Trotman, D. Shaffer, H. Lin, Z. A. Dotan, M. Niki, J. A. Koutcher, H. I. Scher, T. Ludwig and W. Gerald, Crucial role of p53-dependent cellular senescence in suppression of Pten-deficient tumorigenesis, Nature, 436 (2005), 725-730.
doi: 10.1038/nature03918. |
[6] |
K. L. Cooke and P. van den Driessche, On Zeroes of Some Transcendental Equations, Funkcj. Ekvacioj, 29 (1986), 77-90. |
[7] |
J. Coppé, C. K. Patil, F. Rodier, Y. Sun, D. P. Muñoz, J. Goldstein, P. S. Nelson, P. Desprez and J. Campisi, Senescence-associated secretory phenotypes reveal cell-nonautonomous functions of oncogenic RAS and the p53 tumor suppressor, PLoS Biol., 6 (2008) e301.
doi: 10.1371/journal.pbio.0060301. |
[8] |
E. R. Fearon and B. Vogelstein, A genetic model for colorectal tumorigenesis, Cell, 61 (1990), 759-767.
doi: 10.1016/0092-8674(90)90186-I. |
[9] |
U. Foryś, Biological delay systems and the Mikhailov criterion of stability, J. Biol. Sys., 12 (2004), 45-60. |
[10] |
U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, J. Appl. Anal., 11 (2005), 200-281.
doi: 10.1515/JAA.2005.283. |
[11] |
U. Foryś, Multi-dimensional Lotka-{Volterra} system for carcinogenesis mutations, Math. Meth. Appl. Sci., 32 (2009), 2287-2308.
doi: 10.1002/mma.1137. |
[12] |
U. Foryś, Influence of time delays on a two-stage mutations model, in Proceedings of the XIX National Conference Applications of Mathematics in Biology and Medicine, (2013), Gdańsk University of Technology. |
[13] |
J. S. Fridman and S. W. Lowe, Control of apoptosis by p53, Oncogene, 22 (2003), 9030-9040.
doi: 10.1038/sj.onc.1207116. |
[14] |
M. S. Greenblatt, W. P. Bennett, M. Hollstein and C. C. Harris, Mutations in the p53 tumor suppressor gene: clues to cancer etiology and molecular pathogenesis, Cancer Res., 54 (1994), 4855-4878. |
[15] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[16] |
B. Hat, K. Puszyński and T. Lipniacki, Exploring mechanisms of oscillations in p53 and nuclear factor-$\kappa$B systems, Systems Biology, IET, 3 (2009), 342-355.
doi: 10.1049/iet-syb.2008.0156. |
[17] |
E. Michalak, A. Villunger, M. Erlacher and A. Strasser, Death squads enlisted by the tumour suppressor p53, Biochem. Bioph. Res. Co., 331 (2005), 786-798.
doi: 10.1016/j.bbrc.2005.03.183. |
[18] |
M. J. Piotrowska, U. Foryś, M. Bodnar and J. Poleszczuk, A simple model of carcinogenic mutations with time delay and diffusion, Math. Biosci. Eng., 10 (2013), 861-872.
doi: 10.3934/mbe.2013.10.861. |
[19] |
K. Puszyński, B. Hat and T. Lipniacki, Oscillations and bistability in the stochastic model of p53 regulation, Journal of Theoretical Biology, 254 (2008), 452-465. |
[20] |
L. D. Wood, D. W. Parsons, S. Jones, J. Lin, T. Sjöblom, R. J. Leary, D. Shen, S. M. Boca, T. Barber, J. Ptak, N. Silliman, S. Szabo, Z. Dezso, V. Ustyanksky, T. Nikolskaya, Y. Nikolsky, R. Karchin, P. A. Wilson, J. S. Kaminker, Z. Zhang, R. Croshaw, J. Willis, D. Dawson, M. Shipitsin, J. K. Willson, S. Sukumar, K. Polyak, B. H. Park, C. L. Pethiyagoda, P. V. Pant, D. G. Ballinger, A. B. Sparks, J. Hartigan, D. R. Smith, E. Suh, N. Papadopoulos, P. Buckhaults, S. D. Markowitz, G. Parmigiani, K. W. Kinzler, V. E. Velculescu and B. Vogelstein, The genomic landscapes of human breast and colorectal cancers, Science, 318 (2007), 1108-1113.
doi: 10.1126/science.1145720. |
show all references
References:
[1] |
R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, Electron. J. Diff. Eqns., 10 (2003), 33-53. |
[2] |
P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[3] |
C. M. Beauséjour, A. Krtolica, F. Galimi, M. Narita, S.W. Lowe, P. Yaswen and J. Campisi, Reversal of human cellular senescence: roles of the p53 and p16 pathways, EMBO J, 22 (2003), 4212-4222. |
[4] |
K. Camphausen, M. A. Moses, C. Ménard, M. Sproull, W. Beecken, J. Folkman and M. S. O'Reilly, Radiation abscopal antitumor effect is mediated through p53, Cancer Res., 63 (2003), 1990-1993. |
[5] |
Z. Chen, L. C. Trotman, D. Shaffer, H. Lin, Z. A. Dotan, M. Niki, J. A. Koutcher, H. I. Scher, T. Ludwig and W. Gerald, Crucial role of p53-dependent cellular senescence in suppression of Pten-deficient tumorigenesis, Nature, 436 (2005), 725-730.
doi: 10.1038/nature03918. |
[6] |
K. L. Cooke and P. van den Driessche, On Zeroes of Some Transcendental Equations, Funkcj. Ekvacioj, 29 (1986), 77-90. |
[7] |
J. Coppé, C. K. Patil, F. Rodier, Y. Sun, D. P. Muñoz, J. Goldstein, P. S. Nelson, P. Desprez and J. Campisi, Senescence-associated secretory phenotypes reveal cell-nonautonomous functions of oncogenic RAS and the p53 tumor suppressor, PLoS Biol., 6 (2008) e301.
doi: 10.1371/journal.pbio.0060301. |
[8] |
E. R. Fearon and B. Vogelstein, A genetic model for colorectal tumorigenesis, Cell, 61 (1990), 759-767.
doi: 10.1016/0092-8674(90)90186-I. |
[9] |
U. Foryś, Biological delay systems and the Mikhailov criterion of stability, J. Biol. Sys., 12 (2004), 45-60. |
[10] |
U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, J. Appl. Anal., 11 (2005), 200-281.
doi: 10.1515/JAA.2005.283. |
[11] |
U. Foryś, Multi-dimensional Lotka-{Volterra} system for carcinogenesis mutations, Math. Meth. Appl. Sci., 32 (2009), 2287-2308.
doi: 10.1002/mma.1137. |
[12] |
U. Foryś, Influence of time delays on a two-stage mutations model, in Proceedings of the XIX National Conference Applications of Mathematics in Biology and Medicine, (2013), Gdańsk University of Technology. |
[13] |
J. S. Fridman and S. W. Lowe, Control of apoptosis by p53, Oncogene, 22 (2003), 9030-9040.
doi: 10.1038/sj.onc.1207116. |
[14] |
M. S. Greenblatt, W. P. Bennett, M. Hollstein and C. C. Harris, Mutations in the p53 tumor suppressor gene: clues to cancer etiology and molecular pathogenesis, Cancer Res., 54 (1994), 4855-4878. |
[15] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[16] |
B. Hat, K. Puszyński and T. Lipniacki, Exploring mechanisms of oscillations in p53 and nuclear factor-$\kappa$B systems, Systems Biology, IET, 3 (2009), 342-355.
doi: 10.1049/iet-syb.2008.0156. |
[17] |
E. Michalak, A. Villunger, M. Erlacher and A. Strasser, Death squads enlisted by the tumour suppressor p53, Biochem. Bioph. Res. Co., 331 (2005), 786-798.
doi: 10.1016/j.bbrc.2005.03.183. |
[18] |
M. J. Piotrowska, U. Foryś, M. Bodnar and J. Poleszczuk, A simple model of carcinogenic mutations with time delay and diffusion, Math. Biosci. Eng., 10 (2013), 861-872.
doi: 10.3934/mbe.2013.10.861. |
[19] |
K. Puszyński, B. Hat and T. Lipniacki, Oscillations and bistability in the stochastic model of p53 regulation, Journal of Theoretical Biology, 254 (2008), 452-465. |
[20] |
L. D. Wood, D. W. Parsons, S. Jones, J. Lin, T. Sjöblom, R. J. Leary, D. Shen, S. M. Boca, T. Barber, J. Ptak, N. Silliman, S. Szabo, Z. Dezso, V. Ustyanksky, T. Nikolskaya, Y. Nikolsky, R. Karchin, P. A. Wilson, J. S. Kaminker, Z. Zhang, R. Croshaw, J. Willis, D. Dawson, M. Shipitsin, J. K. Willson, S. Sukumar, K. Polyak, B. H. Park, C. L. Pethiyagoda, P. V. Pant, D. G. Ballinger, A. B. Sparks, J. Hartigan, D. R. Smith, E. Suh, N. Papadopoulos, P. Buckhaults, S. D. Markowitz, G. Parmigiani, K. W. Kinzler, V. E. Velculescu and B. Vogelstein, The genomic landscapes of human breast and colorectal cancers, Science, 318 (2007), 1108-1113.
doi: 10.1126/science.1145720. |
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