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. 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. 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. 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. 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. doi: 10.1016/S0092-8674(04)00045-5.

[7]

D. P. Bartel, MicroRNAs: Target recognition and regulatory functions,, Cell, 136 (2009), 215. 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. doi: 10.1016/j.pbiomolbio.2004.01.014.

[9]

M. Carletti, Stochastic Modelling of Biological Processes,, PhD Thesis, (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. 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. 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. 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. doi: 10.1021/j100540a008.

[15]

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

[16]

J. Goutsias, Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems,, J. Chem. Phys., 122 (2005). 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. 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. 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).

[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. 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. 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).

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0092-8674(04)00045-5.

[7]

D. P. Bartel, MicroRNAs: Target recognition and regulatory functions,, Cell, 136 (2009), 215. 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. doi: 10.1016/j.pbiomolbio.2004.01.014.

[9]

M. Carletti, Stochastic Modelling of Biological Processes,, PhD Thesis, (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. 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. 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. 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. doi: 10.1021/j100540a008.

[15]

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

[16]

J. Goutsias, Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems,, J. Chem. Phys., 122 (2005). 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. 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. 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).

[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. 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. 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).

[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. 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. 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. 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. 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. 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. doi: 10.1016/j.tranon.2014.02.004.

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