American Institute of Mathematical Sciences

Advanced
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 & 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

[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

show all references

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

[1]

Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861
[2]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[3]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[4]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[5]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[6]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[7]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[8]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[9]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[10]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[11]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[12]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[13]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[14]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[15]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[16]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[17]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[18]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[19]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[20]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences