Two-stage model of carcinogenic mutations with the influence of delays

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October 2014, 19(8): 2501-2519. doi: 10.3934/dcdsb.2014.19.2501

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

Received November 2013 Revised April 2014 Published August 2014

Abstract
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In the paper we make an attempt to study the influence of time delays combined with diffusion on the dynamics of two-stage carcinogenic mutations model. Included delays represent time needed for transformation from one type of cells to the other one. In the presented analysis we focus on possible stability switches due to increasing delays and diffusion driven instability. It occurs that diffusion has no significant impact on asymptotic behaviour of the model solutions, while one of the present delays has destabilising effect in most of cases we study. Analytical results are illustrated by numerical examples of the model dynamics.

Keywords: carcinogenic mutations., stability, Delay equations, diffusion, stability switches.

Mathematics Subject Classification: Primary: 34K20, 34K28, 37G35, 37N25; Secondary: 92B05, 92B25, 92C5.

Citation: Urszula Foryś, Beata Zduniak. Two-stage model of carcinogenic mutations with the influence of delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2501-2519. doi: 10.3934/dcdsb.2014.19.2501

References:

[1]	R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations,, Electron. J. Diff. Eqns., 10 (2003), 33. Google Scholar
[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. doi: 10.1137/S0036139997325497. Google Scholar
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[6]	K. L. Cooke and P. van den Driessche, On Zeroes of Some Transcendental Equations,, Funkcj. Ekvacioj, 29 (1986), 77. Google Scholar
[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). doi: 10.1371/journal.pbio.0060301. Google Scholar
[8]	E. R. Fearon and B. Vogelstein, A genetic model for colorectal tumorigenesis,, Cell, 61 (1990), 759. doi: 10.1016/0092-8674(90)90186-I. Google Scholar
[9]	U. Foryś, Biological delay systems and the Mikhailov criterion of stability,, J. Biol. Sys., 12 (2004), 45. Google Scholar
[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. doi: 10.1515/JAA.2005.283. Google Scholar
[11]	U. Foryś, Multi-dimensional Lotka-{Volterra} system for carcinogenesis mutations,, Math. Meth. Appl. Sci., 32 (2009), 2287. doi: 10.1002/mma.1137. Google Scholar
[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). Google Scholar
[13]	J. S. Fridman and S. W. Lowe, Control of apoptosis by p53,, Oncogene, 22 (2003), 9030. doi: 10.1038/sj.onc.1207116. Google Scholar
[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. Google Scholar
[15]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar
[16]	B. Hat, K. Puszyński and T. Lipniacki, Exploring mechanisms of oscillations in p53 and nuclear factor-$\kappa$B systems,, Systems Biology, 3 (2009), 342. doi: 10.1049/iet-syb.2008.0156. Google Scholar
[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. doi: 10.1016/j.bbrc.2005.03.183. Google Scholar
[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. doi: 10.3934/mbe.2013.10.861. Google Scholar
[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. Google Scholar
[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. doi: 10.1126/science.1145720. Google Scholar

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References:

[1]	R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations,, Electron. J. Diff. Eqns., 10 (2003), 33. Google Scholar
[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. doi: 10.1137/S0036139997325497. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.1038/nature03918. Google Scholar
[6]	K. L. Cooke and P. van den Driessche, On Zeroes of Some Transcendental Equations,, Funkcj. Ekvacioj, 29 (1986), 77. Google Scholar
[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). doi: 10.1371/journal.pbio.0060301. Google Scholar
[8]	E. R. Fearon and B. Vogelstein, A genetic model for colorectal tumorigenesis,, Cell, 61 (1990), 759. doi: 10.1016/0092-8674(90)90186-I. Google Scholar
[9]	U. Foryś, Biological delay systems and the Mikhailov criterion of stability,, J. Biol. Sys., 12 (2004), 45. Google Scholar
[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. doi: 10.1515/JAA.2005.283. Google Scholar
[11]	U. Foryś, Multi-dimensional Lotka-{Volterra} system for carcinogenesis mutations,, Math. Meth. Appl. Sci., 32 (2009), 2287. doi: 10.1002/mma.1137. Google Scholar
[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). Google Scholar
[13]	J. S. Fridman and S. W. Lowe, Control of apoptosis by p53,, Oncogene, 22 (2003), 9030. doi: 10.1038/sj.onc.1207116. Google Scholar
[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. Google Scholar
[15]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar
[16]	B. Hat, K. Puszyński and T. Lipniacki, Exploring mechanisms of oscillations in p53 and nuclear factor-$\kappa$B systems,, Systems Biology, 3 (2009), 342. doi: 10.1049/iet-syb.2008.0156. Google Scholar
[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. doi: 10.1016/j.bbrc.2005.03.183. Google Scholar
[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. doi: 10.3934/mbe.2013.10.861. Google Scholar
[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. Google Scholar
[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. doi: 10.1126/science.1145720. Google Scholar

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