2015, 12(5): 965-981. doi: 10.3934/mbe.2015.12.965

Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme

1. 

Department DISBEF, University of Urbino "Carlo Bo", Italy, Italy

2. 

Department DISBEF, University of Urbino "Carlo Bo", and Gran Sasso Science Institute, Italy

3. 

Department DISB, University of Urbino "Carlo Bo", Italy, Italy

Received  October 2014 Revised  April 2015 Published  June 2015

This work is the outcome of the partnership between the mathematical group of Department DISBEF and the biochemical group of Department DISB of the University of Urbino "Carlo Bo" in order to better understand some crucial aspects of brain cancer oncogenesis. Throughout our collaboration we discovered that biochemists are mainly attracted to the instantaneous behaviour of the whole cell, while mathematicians are mostly interested in the evolution along time of small and different parts of it. This collaboration has thus been very challenging. Starting from [23,24,25], we introduce a competitive stochastic model for post-transcriptional regulation of PTEN, including interactions with the miRNA and concurrent genes. Our model also covers protein formation and the backward mechanism going from the protein back to the miRNA. The numerical simulations show that the model reproduces the expected dynamics of normal glial cells. Moreover, the introduction of translational and transcriptional delays offers some interesting insights for the PTEN low expression as observed in brain tumor cells.
Citation: Margherita Carletti, Matteo Montani, Valentina Meschini, Marzia Bianchi, Lucia Radici. Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme. Mathematical Biosciences & Engineering, 2015, 12 (5) : 965-981. doi: 10.3934/mbe.2015.12.965
References:
[1]

A. Abdulle and A. Medivikov, Second order Chebyshev methods based on orthogonal polynomials, Numerische Mathematik, 90 (2001), 1-18. doi: 10.1007/s002110100292.

[2]

A. Abdulle and S. Cirilli, S-ROCK: Chebyshev methods for stiff stochastic differential equations, SIAM J. Sci. Comput., 30 (2008), 997-1014. doi: 10.1137/070679375.

[3]

U. Ala, F. A. Karreth, C. Bosia, A. Pagnani, R. Taulli, V. Léopold, Y. Tay, P. Provero, R. Zecchina and P. P. Pandolfi, Integrated transcriptional and competitive endogenous RNA networks are cross-regulated in permissive molecular environments, PNAS, 110 (2013), 7154-7159. doi: 10.1073/pnas.1222509110.

[4]

C. T. H. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315-335. doi: 10.1112/S1461157000000322.

[5]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory Regulation of Hes1: Discrete stochastic delay modelling and simulation, PLoS Comput Biol, 2006.

[6]

D. P. Bartel, MicroRNAs: Genomics, biogenesis, mechanism, and function, Cell, 116 (2004), 281-297. doi: 10.1016/S0092-8674(04)00045-5.

[7]

D. P. Bartel, MicroRNAs: Target recognition and regulatory functions, Cell, 136 (2009), 215-233. doi: 10.1016/j.cell.2009.01.002.

[8]

K. Burrage, T. Tian and P. M. Burrage, A multi-scaled approach for simulating chemical reaction systems, Progress in Biophysics and Molecular Biology, 85 (2004), 217-234. doi: 10.1016/j.pbiomolbio.2004.01.014.

[9]

M. Carletti, Stochastic Modelling of Biological Processes, PhD Thesis, The University of Queensland, Brisbane, Australia, 2008.

[10]

A. Carracedo, A. Alimonti and P. P. Pandolfi, PTEN level in tumor suppression: How much is too little?, Cancer Res., 71 (2011), 629-633. doi: 10.1158/0008-5472.CAN-10-2488.

[11]

A. de Giorgio, J. Krell, V. Harding, J. Stebbing and L. Castellano, Emerging roles of competing endogenous RNAs in cancer: Insights from the regulation of PTEN, Mol Cell Biol., 33 (2013), 3976-3982. doi: 10.1128/MCB.00683-13.

[12]

M. Figliuzzi, E. Marinari and A. De Martino, MicroRNAs as a selective channel of communication between competing RNAs: A steady-state theory, Biophys J., 104 (2013), 1203-1213. doi: 10.1016/j.bpj.2013.01.012.

[13]

P. Garcia-Junco-Clemente and P. Golshani, PTEN: A master regulator of neuronal structure, function, and plasticity, Commun Integr Biol., 2014.

[14]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[15]

D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733. doi: 10.1063/1.1378322.

[16]

J. Goutsias, Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems, J. Chem. Phys., 122 (2005), 184102. doi: 10.1063/1.1889434.

[17]

D. Hernandez and R. Spigler, Convergence and stability of implicit Runge-Kutta methods for systems with multiplcative noise, BIT Num. Math., 33 (1993), 654-669. doi: 10.1007/BF01990541.

[18]

F. A. Karreth, Y. Tay, D. Perna, U. Ala, S. Mynn Tan, A. G. Rust, G. De Nicola, K. A. Webster, D. Weiss, P. A. P. Mancera, M. Krauthammer, R. Halaban, P. Provero, D. J. Adams, D. A. Tuveson and P. P. Pandolfi, In vivo identification of Tumor-suppressive PTEN ceRNAs in an oncogenic BRAF-induced mouse model of melanoma, Cell, 147 (2011), 382-395. doi: 10.1016/j.cell.2011.09.032.

[19]

A. Leier, T. T. Marquez-Lago and K. Burrage, Generalized binomial tau-leap method for biochemical kinetics incorporating both delay and intrinsic noise, J. Chem. Phys., 128 (2008), 205107.

[20]

S. Mukherji, M. S. Ebert , G. X. Zheng, J. S. Tsang, P. A. Sharz and A. van Oudenaarden, MicroRNAs can generate thresholds in target gene expression, Nat. Genet, 43 (2011), 854-859. doi: 10.1038/ng.905.

[21]

L. Poliseno, L. Salmena, J. Zhang, B. Carver, W. Haveman and P. P. Pandolfi, A coding-independent function of gene and pseudogene mRNAs regulates tumour biology, Nature, 465 (2010), 1033-1038. doi: 10.1038/nature09144.

[22]

P. Rue, J. Villa-Freixa and K. Burrage, Simulation methods with extended stability for stiff biochemical kinetics, BMC Systems Biology, (2010), p110.

[23]

P. Sumazin, X. Yang, H. S. Chiu, W. J. Chung, A. Iyer, D. Llobet-Navas, P. Rajbhandari, M. Bansal, P. Guarnieri, J. Silva and A. Califano, An extensive MicroRNA-mediated network of RNA-RNA interactions regulates established oncogenic pathways in Glioblastoma, Cell, 147 (2011), 370-381. doi: 10.1016/j.cell.2011.09.041.

[24]

Y. Tay, L. Kats, L. Salmena, D. Weiss, S. M. Tan, U. Ala, F. Karreth, L. Poliseno, P. Provero, F. Di Cunto , J. Lieberman, I. Rigoutsos and P. P. Pandolfi, Coding Independent Regulation of the Tumor Suppressor PTEN by Competing Endogenous mRNAs, Cell, 147 (2011), 344-357. doi: 10.1016/j.cell.2011.09.029.

[25]

Y. Tay, J. Rinn and P. P. Pandolfi, The multilayered complexity of ceRNA crosstalk and competition, Nature, 505 (2014), 344-352. doi: 10.1038/nature12986.

[26]

T. Tian, K. Burrage, P. M. Burrage and M. Carletti, Stochastic delay differential equations for genetic regulatory network, J. Comp App. Math., 205 (2007), 696-707. doi: 10.1016/j.cam.2006.02.063.

[27]

T. E. Turner, S. Schnell and K. Burrage, Stochastic approaches for modelling in vivo reactions Comput. Biol. and Chem., 28 (2004), 165-178. doi: 10.1016/j.compbiolchem.2004.05.001.

[28]

J. Xu, Z. Li, J. Wang, H. Chen and J. Y. Fang, Combined PTEN Mutation and Protein Expression Associate with Overall and Disease-Free Survival of Glioblastoma Patients, Transl Oncol., 7 (2014), 196-205. doi: 10.1016/j.tranon.2014.02.004.

show all references

References:
[1]

A. Abdulle and A. Medivikov, Second order Chebyshev methods based on orthogonal polynomials, Numerische Mathematik, 90 (2001), 1-18. doi: 10.1007/s002110100292.

[2]

A. Abdulle and S. Cirilli, S-ROCK: Chebyshev methods for stiff stochastic differential equations, SIAM J. Sci. Comput., 30 (2008), 997-1014. doi: 10.1137/070679375.

[3]

U. Ala, F. A. Karreth, C. Bosia, A. Pagnani, R. Taulli, V. Léopold, Y. Tay, P. Provero, R. Zecchina and P. P. Pandolfi, Integrated transcriptional and competitive endogenous RNA networks are cross-regulated in permissive molecular environments, PNAS, 110 (2013), 7154-7159. doi: 10.1073/pnas.1222509110.

[4]

C. T. H. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315-335. doi: 10.1112/S1461157000000322.

[5]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory Regulation of Hes1: Discrete stochastic delay modelling and simulation, PLoS Comput Biol, 2006.

[6]

D. P. Bartel, MicroRNAs: Genomics, biogenesis, mechanism, and function, Cell, 116 (2004), 281-297. doi: 10.1016/S0092-8674(04)00045-5.

[7]

D. P. Bartel, MicroRNAs: Target recognition and regulatory functions, Cell, 136 (2009), 215-233. doi: 10.1016/j.cell.2009.01.002.

[8]

K. Burrage, T. Tian and P. M. Burrage, A multi-scaled approach for simulating chemical reaction systems, Progress in Biophysics and Molecular Biology, 85 (2004), 217-234. doi: 10.1016/j.pbiomolbio.2004.01.014.

[9]

M. Carletti, Stochastic Modelling of Biological Processes, PhD Thesis, The University of Queensland, Brisbane, Australia, 2008.

[10]

A. Carracedo, A. Alimonti and P. P. Pandolfi, PTEN level in tumor suppression: How much is too little?, Cancer Res., 71 (2011), 629-633. doi: 10.1158/0008-5472.CAN-10-2488.

[11]

A. de Giorgio, J. Krell, V. Harding, J. Stebbing and L. Castellano, Emerging roles of competing endogenous RNAs in cancer: Insights from the regulation of PTEN, Mol Cell Biol., 33 (2013), 3976-3982. doi: 10.1128/MCB.00683-13.

[12]

M. Figliuzzi, E. Marinari and A. De Martino, MicroRNAs as a selective channel of communication between competing RNAs: A steady-state theory, Biophys J., 104 (2013), 1203-1213. doi: 10.1016/j.bpj.2013.01.012.

[13]

P. Garcia-Junco-Clemente and P. Golshani, PTEN: A master regulator of neuronal structure, function, and plasticity, Commun Integr Biol., 2014.

[14]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[15]

D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733. doi: 10.1063/1.1378322.

[16]

J. Goutsias, Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems, J. Chem. Phys., 122 (2005), 184102. doi: 10.1063/1.1889434.

[17]

D. Hernandez and R. Spigler, Convergence and stability of implicit Runge-Kutta methods for systems with multiplcative noise, BIT Num. Math., 33 (1993), 654-669. doi: 10.1007/BF01990541.

[18]

F. A. Karreth, Y. Tay, D. Perna, U. Ala, S. Mynn Tan, A. G. Rust, G. De Nicola, K. A. Webster, D. Weiss, P. A. P. Mancera, M. Krauthammer, R. Halaban, P. Provero, D. J. Adams, D. A. Tuveson and P. P. Pandolfi, In vivo identification of Tumor-suppressive PTEN ceRNAs in an oncogenic BRAF-induced mouse model of melanoma, Cell, 147 (2011), 382-395. doi: 10.1016/j.cell.2011.09.032.

[19]

A. Leier, T. T. Marquez-Lago and K. Burrage, Generalized binomial tau-leap method for biochemical kinetics incorporating both delay and intrinsic noise, J. Chem. Phys., 128 (2008), 205107.

[20]

S. Mukherji, M. S. Ebert , G. X. Zheng, J. S. Tsang, P. A. Sharz and A. van Oudenaarden, MicroRNAs can generate thresholds in target gene expression, Nat. Genet, 43 (2011), 854-859. doi: 10.1038/ng.905.

[21]

L. Poliseno, L. Salmena, J. Zhang, B. Carver, W. Haveman and P. P. Pandolfi, A coding-independent function of gene and pseudogene mRNAs regulates tumour biology, Nature, 465 (2010), 1033-1038. doi: 10.1038/nature09144.

[22]

P. Rue, J. Villa-Freixa and K. Burrage, Simulation methods with extended stability for stiff biochemical kinetics, BMC Systems Biology, (2010), p110.

[23]

P. Sumazin, X. Yang, H. S. Chiu, W. J. Chung, A. Iyer, D. Llobet-Navas, P. Rajbhandari, M. Bansal, P. Guarnieri, J. Silva and A. Califano, An extensive MicroRNA-mediated network of RNA-RNA interactions regulates established oncogenic pathways in Glioblastoma, Cell, 147 (2011), 370-381. doi: 10.1016/j.cell.2011.09.041.

[24]

Y. Tay, L. Kats, L. Salmena, D. Weiss, S. M. Tan, U. Ala, F. Karreth, L. Poliseno, P. Provero, F. Di Cunto , J. Lieberman, I. Rigoutsos and P. P. Pandolfi, Coding Independent Regulation of the Tumor Suppressor PTEN by Competing Endogenous mRNAs, Cell, 147 (2011), 344-357. doi: 10.1016/j.cell.2011.09.029.

[25]

Y. Tay, J. Rinn and P. P. Pandolfi, The multilayered complexity of ceRNA crosstalk and competition, Nature, 505 (2014), 344-352. doi: 10.1038/nature12986.

[26]

T. Tian, K. Burrage, P. M. Burrage and M. Carletti, Stochastic delay differential equations for genetic regulatory network, J. Comp App. Math., 205 (2007), 696-707. doi: 10.1016/j.cam.2006.02.063.

[27]

T. E. Turner, S. Schnell and K. Burrage, Stochastic approaches for modelling in vivo reactions Comput. Biol. and Chem., 28 (2004), 165-178. doi: 10.1016/j.compbiolchem.2004.05.001.

[28]

J. Xu, Z. Li, J. Wang, H. Chen and J. Y. Fang, Combined PTEN Mutation and Protein Expression Associate with Overall and Disease-Free Survival of Glioblastoma Patients, Transl Oncol., 7 (2014), 196-205. doi: 10.1016/j.tranon.2014.02.004.

[1]

William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041

[2]

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic and Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001

[3]

Jiayuan Yan, Ding-Xue Zhang, Bin Hu, Zhi-Hong Guan, Xin-Ming Cheng. State bounding for time-delay impulsive and switching genetic regulatory networks with exogenous disturbance. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1749-1765. doi: 10.3934/dcdss.2022004

[4]

Roberto Serra, Marco Villani, Alex Graudenzi, Annamaria Colacci, Stuart A. Kauffman. The simulation of gene knock-out in scale-free random Boolean models of genetic networks. Networks and Heterogeneous Media, 2008, 3 (2) : 333-343. doi: 10.3934/nhm.2008.3.333

[5]

Jesse Berwald, Marian Gidea. Critical transitions in a model of a genetic regulatory system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 723-740. doi: 10.3934/mbe.2014.11.723

[6]

Kristin R. Swanson, Ellsworth C. Alvord, Jr, J. D. Murray. Dynamics of a model for brain tumors reveals a small window for therapeutic intervention. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 289-295. doi: 10.3934/dcdsb.2004.4.289

[7]

Somkid Intep, Desmond J. Higham. Zero, one and two-switch models of gene regulation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 495-513. doi: 10.3934/dcdsb.2010.14.495

[8]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[9]

Ö. Uğur, G. W. Weber. Optimization and dynamics of gene-environment networks with intervals. Journal of Industrial and Management Optimization, 2007, 3 (2) : 357-379. doi: 10.3934/jimo.2007.3.357

[10]

Erik Kropat, Gerhard-Wilhelm Weber, Erfan Babaee Tirkolaee. Foundations of semialgebraic gene-environment networks. Journal of Dynamics and Games, 2020, 7 (4) : 253-268. doi: 10.3934/jdg.2020018

[11]

Lijin Wang, Pengjun Wang, Yanzhao Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 819-836. doi: 10.3934/dcdss.2021095

[12]

Mahdi Jalili. EEG-based functional brain networks: Hemispheric differences in males and females. Networks and Heterogeneous Media, 2015, 10 (1) : 223-232. doi: 10.3934/nhm.2015.10.223

[13]

Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058

[14]

Ingenuin Gasser, Marcus Kraft. Modelling and simulation of fires in tunnel networks. Networks and Heterogeneous Media, 2008, 3 (4) : 691-707. doi: 10.3934/nhm.2008.3.691

[15]

Kunwen Wen, Lifang Huang, Qiuying Li, Qi Wang, Jianshe Yu. The mean and noise of FPT modulated by promoter architecture in gene networks. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2177-2194. doi: 10.3934/dcdss.2019140

[16]

Qi Yang, Lei Wang, Enmin Feng, Hongchao Yin, Zhilong Xiu. Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture. Journal of Industrial and Management Optimization, 2020, 16 (2) : 579-599. doi: 10.3934/jimo.2018168

[17]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573

[18]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625

[19]

Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105

[20]

Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (2)

[Back to Top]