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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

Urszula Foryś 1, and  Beata Zduniak 2,

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

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 and 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), 33-53.

[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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