American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A multiple time-scale computational model of a tumor and its micro environment
  • MBE Home
  • This Issue
  • Next Article
    An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix
2013, 10(1): 103-120. doi: 10.3934/mbe.2013.10.103

Parameter space exploration within dynamic simulations of signaling networks

Cristina De Ambrosi 1, , Annalisa Barla 1, , Lorenzo Tortolina 1, , Nicoletta Castagnino 2, , Raffaele Pesenti 3, , Alessandro Verri 1, , Alberto Ballestrero 4, , Franco Patrone 4, and  Silvio Parodi 4,

1. 

DIBRIS Department of Informatics, Bioengineering, Robotics and Systems Engineering, Università degli Studi di Genova - Via Balbi, 5 - 16126 Genova, Italy, Italy, Italy, Italy
2. 

DIBRIS Department of Informatics, Bioengineering, Robotics and Systems Engineering, Università degli Studi di Genova - Via Balbi, 5 - 16126 Genov, Italy
3. 

DiMa - Department of Management, Università Ca' Foscari - Dorsoduro 3246 - 30123 Venezia, Italy
4. 

Di.M.I - Department of Internal Medicine, A.O.U. IRCCS San Martino IST, Italy, Italy, Italy

Received  April 2012 Revised  July 2012 Published  December 2012

We started offering an introduction to very basic aspects of molecular biology, for the reader coming from computer sciences, information technology, mathematics. Similarly we offered a minimum of information about pathways and networks in graph theory, for a reader coming from the bio-medical sector. At the crossover about the two different types of expertise, we offered some definition about Systems Biology. The core of the article deals with a Molecular Interaction Map (MIM), a network of biochemical interactions involved in a small signaling-network sub-region relevant in breast cancer. We explored robustness/sensitivity to random perturbations. It turns out that our MIM is a non-isomorphic directed graph. For non physiological directions of propagation of the signal the network is quite resistant to perturbations. The opposite happens for biologically significant directions of signal propagation. In these cases we can have no signal attenuation, and even signal amplification. Signal propagation along a given pathway is highly unidirectional, with the exception of signal-feedbacks, that again have a specific biological role and significance. In conclusion, even a relatively small network like our present MIM reveals the preponderance of specific biological functions over unspecific isomorphic behaviors. This is perhaps the consequence of hundreds of millions of years of biological evolution.
Keywords: parameter space exploration., dynamic simulations, Molecular Interaction Maps, Systems biology, signaling networks.
Mathematics Subject Classification: Primary: 92C42; Secondary: 92C4.
Citation: Cristina De Ambrosi, Annalisa Barla, Lorenzo Tortolina, Nicoletta Castagnino, Raffaele Pesenti, Alessandro Verri, Alberto Ballestrero, Franco Patrone, Silvio Parodi. Parameter space exploration within dynamic simulations of signaling networks. Mathematical Biosciences & Engineering, 2013, 10 (1) : 103-120. doi: 10.3934/mbe.2013.10.103
References:
[1]

J. Kendrew, "The Encyclopedia of Molecular Biology,", Blackwell Science Ltd. Reprinted, (1995).   Google Scholar

[2]

N. Trun and T. Trempy, "Fundamental Bacterial Genetics,", Blackwell Publishing Company, (2004).   Google Scholar

[3]

T. Ideker, T. Galitski and L. Hood, A new approach to decoding life: Systems biology,, Annu Rev Genomics Hum Genet., 2 (2001), 343.   Google Scholar

[4]

H. Kitano, Computational systems biology,, Nature, 420 (2002), 206.   Google Scholar

[5]

L. Hood, Systems biology: Integrating technology, biology, and computation,, Mech Ageing Dev., 124 (2003), 9.   Google Scholar

[6]

M. Cassman, Systems biology: International research and development,, World Technology Evaluation Center. SpringerLink, (2007), 1.   Google Scholar

[7]

F. J. Bruggeman and H. V. Westerhoff, The nature of systems biology,, Trends Microbiol., 15 (2007), 45.   Google Scholar

[8]

N. Barkai and S. Leibler, Robustness in simple biochemical networks,, Nature, 387 (1997), 913.   Google Scholar

[9]

B. N. Kholodenko, Cell-signalling dynamics in time and space,, Nat. Rev. Mol. Cell Biol., 7 (2006), 165.   Google Scholar

[10]

N. Borisov, E. Aksamitiene, A. Kiyatkin, S. Legewie, J. Berkhout, T. Maiwald, N. P. Kaimachnikov, J. Timmer, J. B. Hoek and B. N. Kholodenko, Systems-level interactions between insulin-EGF networks amplify mitogenic signaling,, Mol Syst Biol., 5 (2009).   Google Scholar

[11]

L. Tortolina, N. Castagnino, C. De Ambrosi, E. Moran, F. Patrone, A. Ballestrero and S. Parodi, A multi-scale approach to colorectal cancer: From a biochemical-interaction signaling network level, to multi-cellular dynamics of malignant transformation. Interplay with mutations and onco-protein inhibitor drugs,, Current Cancer Drug Target (CCDT), 12 (2012), 339.   Google Scholar

[12]

D. Segré, D. Vitkup and G. M. Church, Analysis of optimality in natural and perturbed metabolic networks,, Proc Natl Acad Sci U S A., 99 (2002), 15112.   Google Scholar

[13]

D. Segré, A. Deluna, G. M. Church and R. Kishony, Modular epistasis in yeast metabolism,, Nat Genet., 37 (2005), 77.   Google Scholar

[14]

W. Materi and D. S. Wishart, Computational systems biology in drug discovery and development: methods and applications,, Drug Discov Today, 12 (2007), 295.   Google Scholar

[15]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. Phys.Chem., 81 (1977), 2340.   Google Scholar

[16]

D. J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems,, Nat Rev Genet., 10 (2009), 122.   Google Scholar

[17]

T. Sjöblom, S. Jones, L. D. Wood, D. W. Parsons, J. Lin, T. D. Barber, D. Mandelker, R. J. Leary, J. Ptak, N. Silliman, S. Szabo, P. Buckhaults, C. Farrell, P. Meeh, S. D. Markowitz, J. Willis, D. Dawson, J. K. Willson, A. F. Gazdar, J. Hartigan, L. Wu, C. Liu, G. Parmigiani, B. H. Park, K. E. Bachman, N. Papadopoulos, B. Vogelstein, K. W. Kinzler and V. E. Velculescu, The consensus coding sequences of human breast and colorectal cancers,, Science, 314 (2006), 268.   Google Scholar

[18]

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.   Google Scholar

[19]

C. H. Yeang, F. McCormick and A. Levine, Combinatorial patterns of somatic gene mutations in cancer,, FASEB J., 22 (2008), 2605.   Google Scholar

[20]

M. I. Aladjem, S. Pasa, S. Parodi, J. N. Weinstein, Y. Pommier and K. W.Kohn, Molecular interaction maps-a diagrammatic graphical language for bioregulatory networks,, Sci STKE., 222 (2004).   Google Scholar

[21]

K. W. Kohn, M. I. Aladjem, J. N. Weinstein and Y. Pommier, Molecular interaction maps of bioregulatory networks: A general rubric for systems biology,, Mol. Biol. Cell, 17 (2006), 1.   Google Scholar

[22]

K. W. Kohn, M. I. Aladjem, S. Kim, J. N. Weinstein and Y. Pommier, Depicting combinatorial complexity with the molecular interaction map notation,, Mol Syst Biol., 2 (2006).   Google Scholar

[23]

A. Luna, E. I. Karac, M. Sunshine, L. Chang, R. Nussinov, M. I. Aladjem and K. W. Kohn, A formal MIM specification and tools for the common exchange of MIM diagrams: An XML-Based format, an API, and a validation method,, BMC Bioinformatics, 12 (2011).   Google Scholar

[24]

D. Joyner, M. Van Nguyen and N. Cohen, "Algorithmic Graph Theory,", Version 0.5 2010 November 30., (2010).   Google Scholar

[25]

a, GLOBOCAN project, , ().   Google Scholar

[26]

G. A. Colditz, S. E. Hankinson, D. J. Hunter, W. C. Willett, J. E. Manson, M. J. Stampfer, C. Hennekens, B. Rosner and F. E. Speizer, The use of estrogens and progestins and the risk of breast cancer in postmenopausal women,, N Engl J Med., 332 (1995), 1589.   Google Scholar

[27]

a, COSMIC 2012: Catalogue of somatic mutations in cancer,, , ().   Google Scholar

[28]

M. Mukherji, L. M. Brill, S. B. Ficarro, G. M. Hampton and P. G. Schultz, A phosphoproteomic analysis of the ErbB2 receptor tyrosine kinase signaling pathways,, Biochemistry, 45 (2006), 15529.   Google Scholar

[29]

N. R. Leslie and C. P. Downes, PTEN function: how normal cells control it and tumour cells lose it,, Biochem. J., 382 (2004), 1.   Google Scholar

[30]

E. Tokunaga, E. Oki, Y. Kimura, T. Yamanaka, A. Egashira, K. Nishida, T. Koga, M. Morita, Y. Kakeji and Y. Maehara, Coexistence of the loss of heterozygosity at the PTEN locus and HER2 overexpression enhances the Akt activity thus leading to a negative progesterone receptor expression in breast carcinoma,, Breast Cancer Res. Treat., 101 (2007), 249.   Google Scholar

[31]

N. Castagnino, L. Tortolina, A. Balbi, R. Pesenti, R. Montagna, A. Ballestrero, D. Soncini, A. Nencioni and S. Parodi, Dynamic simulations of pathways downstream of ERBB-family, including mutations and treatments: Concordance with experimental results,, Current Cancer Drug Targets (CCDT), 10 (2010), 737.   Google Scholar

[32]

B. N. Kholodenko, J. B. Hoek and H. V. Westerhoff, Why cytoplasmic signalling proteins should be recruited to cell membranes,, Trends Cell Biol., 10 (2000), 173.   Google Scholar

[33]

J. Wolf, S. Dronov, F. Tobin and I. Goryanin, The impact of the regulatory design on the response of epidermal growth factor receptor-mediated signal transduction towards oncogenic mutations,, FEBS J., 274 (2007), 5505.   Google Scholar

[34]

B. N. Kholodenko, O. V. Demin, G. Moehren and J. B. Hoek, Quantification of short term signaling by the epidermal growth factor receptor,, J Biol. Chem., 274 (1999), 30169.   Google Scholar

[35]

N. I. Markevich, G. Moehren, O. V. Demin, A. Kiyatkin, J. B. Hoek and B. N. Kholodenko, Signal processing at the Ras circuit: what shapes Ras activation patterns?,, Syst Biol (Stevenage), 1 (2004), 104.   Google Scholar

[36]

A. Kiyatkin, E. Aksamitiene, N. I. Markevich, N. M. Borisov, J. B. Hoek and B. N. Kholodenko, Scaffolding protein Grb2-associated binder 1 sustains epidermal growth factor-induced mitogenic and survival signaling by multiple positive feedback loops,, J. Biol. Chem., 281 (2006), 19925.   Google Scholar

[37]

M. R. Birtwistle, M. Hatakeyama, N. Yumoto, B. A. Ogunnaike, J. B. Hoek and B. N. Kholodenko, Ligand-dependent responses of the ErbB signaling network: experimental and modeling analyses,, Mol. Syst. Biol., 3 (2007).   Google Scholar

[38]

W. W. Chen, B. Schoeberl, P. J. Jasper, M. Niepel, U. B. Nielsen, D. A. Lauffenburger and P. K. Sorger, Input-output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data,, Mol. Syst. Biol., 5 (2009).   Google Scholar

[39]

T. Nakakuki, M. R. Birtwistle, Y. Saeki, N. Yumoto, K. Ide, T. Nagashima, L. Brusch, B. A. Ogunnaike, M. Okada-Hatakeyama and B. N. Kholodenko, Ligand-specific c-Fos expression emerges from the spatiotemporal control of ErbB network dynamics,, Cell., 141 (2010), 884.   Google Scholar

[40]

G. Ernst and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems,", Springer-Verlag, (1996).   Google Scholar

[41]

J. J. Tyson, B. Novak, G. G.M. Odell, K. Chen and C. D. Thron, Chemical kinetic theory: understanding cell-cycle regulation,, Trends Biochem. Sci., 21 (1996), 89.   Google Scholar

[42]

S. S. Ng, T. Mahmoudi, E. Danenberg, I. Bejaoui, W. de Lau, H. C. Korswagen, M. Schutte and H. Clevers, Phosphatidylinositol 3-kinase signaling does not activate the Wnt cascade,, J Biol Chem., 284 (2009), 35308.   Google Scholar

[43]

D. Voskas, L. S. Ling and J. R. Woodgett, Does GSK-3 provide a shortcut for PI3K activation of Wnt signalling?,, F1000 Biol Rep., 2 (2010).   Google Scholar

show all references

References:
[1]

J. Kendrew, "The Encyclopedia of Molecular Biology,", Blackwell Science Ltd. Reprinted, (1995).   Google Scholar

[2]

N. Trun and T. Trempy, "Fundamental Bacterial Genetics,", Blackwell Publishing Company, (2004).   Google Scholar

[3]

T. Ideker, T. Galitski and L. Hood, A new approach to decoding life: Systems biology,, Annu Rev Genomics Hum Genet., 2 (2001), 343.   Google Scholar

[4]

H. Kitano, Computational systems biology,, Nature, 420 (2002), 206.   Google Scholar

[5]

L. Hood, Systems biology: Integrating technology, biology, and computation,, Mech Ageing Dev., 124 (2003), 9.   Google Scholar

[6]

M. Cassman, Systems biology: International research and development,, World Technology Evaluation Center. SpringerLink, (2007), 1.   Google Scholar

[7]

F. J. Bruggeman and H. V. Westerhoff, The nature of systems biology,, Trends Microbiol., 15 (2007), 45.   Google Scholar

[8]

N. Barkai and S. Leibler, Robustness in simple biochemical networks,, Nature, 387 (1997), 913.   Google Scholar

[9]

B. N. Kholodenko, Cell-signalling dynamics in time and space,, Nat. Rev. Mol. Cell Biol., 7 (2006), 165.   Google Scholar

[10]

N. Borisov, E. Aksamitiene, A. Kiyatkin, S. Legewie, J. Berkhout, T. Maiwald, N. P. Kaimachnikov, J. Timmer, J. B. Hoek and B. N. Kholodenko, Systems-level interactions between insulin-EGF networks amplify mitogenic signaling,, Mol Syst Biol., 5 (2009).   Google Scholar

[11]

L. Tortolina, N. Castagnino, C. De Ambrosi, E. Moran, F. Patrone, A. Ballestrero and S. Parodi, A multi-scale approach to colorectal cancer: From a biochemical-interaction signaling network level, to multi-cellular dynamics of malignant transformation. Interplay with mutations and onco-protein inhibitor drugs,, Current Cancer Drug Target (CCDT), 12 (2012), 339.   Google Scholar

[12]

D. Segré, D. Vitkup and G. M. Church, Analysis of optimality in natural and perturbed metabolic networks,, Proc Natl Acad Sci U S A., 99 (2002), 15112.   Google Scholar

[13]

D. Segré, A. Deluna, G. M. Church and R. Kishony, Modular epistasis in yeast metabolism,, Nat Genet., 37 (2005), 77.   Google Scholar

[14]

W. Materi and D. S. Wishart, Computational systems biology in drug discovery and development: methods and applications,, Drug Discov Today, 12 (2007), 295.   Google Scholar

[15]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. Phys.Chem., 81 (1977), 2340.   Google Scholar

[16]

D. J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems,, Nat Rev Genet., 10 (2009), 122.   Google Scholar

[17]

T. Sjöblom, S. Jones, L. D. Wood, D. W. Parsons, J. Lin, T. D. Barber, D. Mandelker, R. J. Leary, J. Ptak, N. Silliman, S. Szabo, P. Buckhaults, C. Farrell, P. Meeh, S. D. Markowitz, J. Willis, D. Dawson, J. K. Willson, A. F. Gazdar, J. Hartigan, L. Wu, C. Liu, G. Parmigiani, B. H. Park, K. E. Bachman, N. Papadopoulos, B. Vogelstein, K. W. Kinzler and V. E. Velculescu, The consensus coding sequences of human breast and colorectal cancers,, Science, 314 (2006), 268.   Google Scholar

[18]

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.   Google Scholar

[19]

C. H. Yeang, F. McCormick and A. Levine, Combinatorial patterns of somatic gene mutations in cancer,, FASEB J., 22 (2008), 2605.   Google Scholar

[20]

M. I. Aladjem, S. Pasa, S. Parodi, J. N. Weinstein, Y. Pommier and K. W.Kohn, Molecular interaction maps-a diagrammatic graphical language for bioregulatory networks,, Sci STKE., 222 (2004).   Google Scholar

[21]

K. W. Kohn, M. I. Aladjem, J. N. Weinstein and Y. Pommier, Molecular interaction maps of bioregulatory networks: A general rubric for systems biology,, Mol. Biol. Cell, 17 (2006), 1.   Google Scholar

[22]

K. W. Kohn, M. I. Aladjem, S. Kim, J. N. Weinstein and Y. Pommier, Depicting combinatorial complexity with the molecular interaction map notation,, Mol Syst Biol., 2 (2006).   Google Scholar

[23]

A. Luna, E. I. Karac, M. Sunshine, L. Chang, R. Nussinov, M. I. Aladjem and K. W. Kohn, A formal MIM specification and tools for the common exchange of MIM diagrams: An XML-Based format, an API, and a validation method,, BMC Bioinformatics, 12 (2011).   Google Scholar

[24]

D. Joyner, M. Van Nguyen and N. Cohen, "Algorithmic Graph Theory,", Version 0.5 2010 November 30., (2010).   Google Scholar

[25]

a, GLOBOCAN project, , ().   Google Scholar

[26]

G. A. Colditz, S. E. Hankinson, D. J. Hunter, W. C. Willett, J. E. Manson, M. J. Stampfer, C. Hennekens, B. Rosner and F. E. Speizer, The use of estrogens and progestins and the risk of breast cancer in postmenopausal women,, N Engl J Med., 332 (1995), 1589.   Google Scholar

[27]

a, COSMIC 2012: Catalogue of somatic mutations in cancer,, , ().   Google Scholar

[28]

M. Mukherji, L. M. Brill, S. B. Ficarro, G. M. Hampton and P. G. Schultz, A phosphoproteomic analysis of the ErbB2 receptor tyrosine kinase signaling pathways,, Biochemistry, 45 (2006), 15529.   Google Scholar

[29]

N. R. Leslie and C. P. Downes, PTEN function: how normal cells control it and tumour cells lose it,, Biochem. J., 382 (2004), 1.   Google Scholar

[30]

E. Tokunaga, E. Oki, Y. Kimura, T. Yamanaka, A. Egashira, K. Nishida, T. Koga, M. Morita, Y. Kakeji and Y. Maehara, Coexistence of the loss of heterozygosity at the PTEN locus and HER2 overexpression enhances the Akt activity thus leading to a negative progesterone receptor expression in breast carcinoma,, Breast Cancer Res. Treat., 101 (2007), 249.   Google Scholar

[31]

N. Castagnino, L. Tortolina, A. Balbi, R. Pesenti, R. Montagna, A. Ballestrero, D. Soncini, A. Nencioni and S. Parodi, Dynamic simulations of pathways downstream of ERBB-family, including mutations and treatments: Concordance with experimental results,, Current Cancer Drug Targets (CCDT), 10 (2010), 737.   Google Scholar

[32]

B. N. Kholodenko, J. B. Hoek and H. V. Westerhoff, Why cytoplasmic signalling proteins should be recruited to cell membranes,, Trends Cell Biol., 10 (2000), 173.   Google Scholar

[33]

J. Wolf, S. Dronov, F. Tobin and I. Goryanin, The impact of the regulatory design on the response of epidermal growth factor receptor-mediated signal transduction towards oncogenic mutations,, FEBS J., 274 (2007), 5505.   Google Scholar

[34]

B. N. Kholodenko, O. V. Demin, G. Moehren and J. B. Hoek, Quantification of short term signaling by the epidermal growth factor receptor,, J Biol. Chem., 274 (1999), 30169.   Google Scholar

[35]

N. I. Markevich, G. Moehren, O. V. Demin, A. Kiyatkin, J. B. Hoek and B. N. Kholodenko, Signal processing at the Ras circuit: what shapes Ras activation patterns?,, Syst Biol (Stevenage), 1 (2004), 104.   Google Scholar

[36]

A. Kiyatkin, E. Aksamitiene, N. I. Markevich, N. M. Borisov, J. B. Hoek and B. N. Kholodenko, Scaffolding protein Grb2-associated binder 1 sustains epidermal growth factor-induced mitogenic and survival signaling by multiple positive feedback loops,, J. Biol. Chem., 281 (2006), 19925.   Google Scholar

[37]

M. R. Birtwistle, M. Hatakeyama, N. Yumoto, B. A. Ogunnaike, J. B. Hoek and B. N. Kholodenko, Ligand-dependent responses of the ErbB signaling network: experimental and modeling analyses,, Mol. Syst. Biol., 3 (2007).   Google Scholar

[38]

W. W. Chen, B. Schoeberl, P. J. Jasper, M. Niepel, U. B. Nielsen, D. A. Lauffenburger and P. K. Sorger, Input-output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data,, Mol. Syst. Biol., 5 (2009).   Google Scholar

[39]

T. Nakakuki, M. R. Birtwistle, Y. Saeki, N. Yumoto, K. Ide, T. Nagashima, L. Brusch, B. A. Ogunnaike, M. Okada-Hatakeyama and B. N. Kholodenko, Ligand-specific c-Fos expression emerges from the spatiotemporal control of ErbB network dynamics,, Cell., 141 (2010), 884.   Google Scholar

[40]

G. Ernst and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems,", Springer-Verlag, (1996).   Google Scholar

[41]

J. J. Tyson, B. Novak, G. G.M. Odell, K. Chen and C. D. Thron, Chemical kinetic theory: understanding cell-cycle regulation,, Trends Biochem. Sci., 21 (1996), 89.   Google Scholar

[42]

S. S. Ng, T. Mahmoudi, E. Danenberg, I. Bejaoui, W. de Lau, H. C. Korswagen, M. Schutte and H. Clevers, Phosphatidylinositol 3-kinase signaling does not activate the Wnt cascade,, J Biol Chem., 284 (2009), 35308.   Google Scholar

[43]

D. Voskas, L. S. Ling and J. R. Woodgett, Does GSK-3 provide a shortcut for PI3K activation of Wnt signalling?,, F1000 Biol Rep., 2 (2010).   Google Scholar

[1]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091
[2]

Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571
[3]

Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies, Dirk Roose. Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 855-874. doi: 10.3934/dcdsb.2009.11.855
[4]

Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i
[5]

Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193
[6]

Mingxing Zhou, Jing Liu, Shuai Wang, Shan He. A comparative study of robustness measures for cancer signaling networks. Big Data & Information Analytics, 2017, 2 (1) : 87-96. doi: 10.3934/bdia.2017011
[7]

Hirotada Honda. On a model of target detection in molecular communication networks. Networks & Heterogeneous Media, 2019, 14 (4) : 633-657. doi: 10.3934/nhm.2019025
[8]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139
[9]

Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842
[10]

Alessia Marigo, Benedetto Piccoli. A model for biological dynamic networks. Networks & Heterogeneous Media, 2011, 6 (4) : 647-663. doi: 10.3934/nhm.2011.6.647
[11]

Peter Hinow, Edward A. Rietman, Sara Ibrahim Omar, Jack A. Tuszyński. Algebraic and topological indices of molecular pathway networks in human cancers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1289-1302. doi: 10.3934/mbe.2015.12.1289
[12]

Monique Chyba, Benedetto Piccoli. Special issue on mathematical methods in systems biology. Networks & Heterogeneous Media, 2019, 14 (1) : ⅰ-ⅱ. doi: 10.3934/nhm.20191i
[13]

Krzysztof Fujarewicz, Marek Kimmel, Andrzej Swierniak. On Fitting Of Mathematical Models Of Cell Signaling Pathways Using Adjoint Systems. Mathematical Biosciences & Engineering, 2005, 2 (3) : 527-534. doi: 10.3934/mbe.2005.2.527
[14]

Adil Khazari, Ali Boutoulout. Flux reconstruction for hyperbolic systems: Sensors and simulations. Evolution Equations & Control Theory, 2015, 4 (2) : 177-192. doi: 10.3934/eect.2015.4.177
[15]

Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016
[16]

Raul Borsche, Anne Meurer. Interaction of road networks and pedestrian motion at crosswalks. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 363-377. doi: 10.3934/dcdss.2014.7.363
[17]

N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59
[18]

Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895
[19]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[20]

Remi Sentis. Models and simulations for the laser-plasma interaction and the three-wave coupling problem. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 329-343. doi: 10.3934/dcdss.2012.5.329

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences