2013, 10(1): 1-17. doi: 10.3934/mbe.2013.10.1

Age-structured cell population model to study the influence of growth factors on cell cycle dynamics

1. 

INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, B.P. 105, F-78153 Le Chesnay Cedex, France, France

2. 

Universite Nice-Sophia-Antipolis, Institute of Biology Valrose, CNRS, UMR 7277, INSERM, U1091, 28, avenue Valrose, F-06108, Nice Cedex 02,, France

3. 

Université Nice-Sophia-Antipolis, Institute of Biology Valrose, CNRS, UMR 7277, INSERM, U1091, 28, avenue Valrose, F-06108, Nice Cedex 02, France

4. 

School of Medicine, Université Paris V - Rene Descartes, 12, rue de l'Ecole de Medecine, F-75270, Paris Cedex 06, France

Received  April 2012 Revised  September 2012 Published  December 2012

Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations.
    Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data. Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results. This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process.
    This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.
Citation: Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences & Engineering, 2013, 10 (1) : 1-17. doi: 10.3934/mbe.2013.10.1
References:
[1]

O. Arino, A survey of structured cell population dynamics,, Acta Biotheor., 43 (1995), 3.  doi: 10.1007/BF00709430.  Google Scholar

[2]

O. Arino and M. Kimmel, Comparison of approaches to modeling of cell population dynamics,, SIAM J. Appl. Math., 53 (1993), 1480.  doi: 10.1137/0153069.  Google Scholar

[3]

O. Arino and E. Sanchez, A survey of cell population dynamics,, J. Theor. Med., 1 (1997), 35.  doi: 10.1080/10273669708833005.  Google Scholar

[4]

D. Barbolosi, A. Benabdallah, F. Hubert and F. Verga, Mathematical and numerical analysis for a model of growing metastatic tumors,, Math. Biosci., 218 (2009), 1.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[5]

B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling the flow [corrected] cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison,, Bull Math. Biol., 67 (2005), 815.  doi: 10.1016/j.bulm.2004.10.003.  Google Scholar

[6]

B. Basse and P. Ubezio, A generalised age- and phase-structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies,, Bull Math. Biol., 69 (2007), 1673.  doi: 10.1007/s11538-006-9185-6.  Google Scholar

[7]

S. Benzekry, N. André, A. Benabdallah, J. Ciccolini, C. Faivre, F. Hubert and D. Barbolosi, Modeling the impact of anticancer agents on metastatic spreading,, Mathematical Modelling of Natural Phenomena, 7 (2012), 306.  doi: 10.1051/mmnp/20127114.  Google Scholar

[8]

F. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. Ouillon and S. Saito, Synchronisation and control of proliferation in cycling cell population models with age structure,, Math. Comp. Simul., (2012).   Google Scholar

[9]

J. Clairambault, Optimizing cancer pharmacotherapeutics using mathematical modeling and a systems biology approach,, Personalized Medicine, 8 (2011), 271.  doi: 10.2217/pme.11.20.  Google Scholar

[10]

J. Clairambault, S. Gaubert and T. Lepoutre, Comparison of Perron and Floquet eigenvalues in age structured cell division models,, Mathematical Modelling of Natural Phenomena, 4 (2009), 183.  doi: 10.1051/mmnp/20094308.  Google Scholar

[11]

J. Clairambault, S. Gaubert and T. Lepoutre, Circadian rhythm and cell population growth,, Mathematical and Computer Modelling, 53 (2011), 1558.  doi: 10.1016/j.mcm.2010.05.034.  Google Scholar

[12]

J. Clairambault, B. Laroche, S. Mischler and B. Perthame, A mathematical model of the cell cycle and its control,, Technical report, (4892).   Google Scholar

[13]

A. A. Cohen, T. Kalisky, A. Mayo, N. Geva-Zatorsky, T. Danon, I. Issaeva, R. Kopito, N. Perzov, R. Milo, A. Sigal and U. Alon, Protein dynamics in individual human cells: Experiment and theory,, PLoS one, 4 (2009), 1.  doi: 10.1371/journal.pone.0004901.  Google Scholar

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M. Cross and T. M. Dexter, Growth factors in development, transformation, and tumorigenesis,, Cell, 64 (1991), 271.  doi: 10.1016/0092-8674(91)90638-F.  Google Scholar

[15]

S. Davis and D. K. Mirick, Circadian disruption, shift work and the risk of cancer: A summary of the evidence and studies in seattle,, Cancer Causes Control, 17 (2006), 539.  doi: 10.1007/s10552-005-9010-9.  Google Scholar

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S. S. Fatimah, G. C. Tan, K. H. Chua, A. E. Tan and A. R. Hayati, Effects of epidermal growth factor on the proliferation and cell cycle regulation of cultured human amnion epithelial cells,, J. Biosci. Bioeng., 114 (2012), 220.  doi: 10.1016/j.jbiosc.2012.03.021.  Google Scholar

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E. Filipski, X. M. Li and F. Lévi, Disruption of circadian coordination and malignant growth,, Cancer Causes Control, 17 (2006), 509.  doi: 10.1007/s10552-005-9007-4.  Google Scholar

[18]

E. Filipski, P. Subramanian, J. Carrière, C. Guettier, H. Barbason and F. Lévi, Circadian disruption accelerates liver carcinogenesis in mice,, Mutat. Res., 680 (2009), 95.  doi: 10.1016/j.mrgentox.2009.10.002.  Google Scholar

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D. A. Foster, P. Yellen, L. Xu and M. Saqcena, Regulation of g1 cell cycle progression: Distinguishing the restriction point from a nutrient-sensing cell growth checkpoint(s),, Genes Cancer, 1 (2010), 1124.  doi: 10.1177/1947601910392989.  Google Scholar

[20]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28 (1990), 671.  doi: 10.1007/BF00160231.  Google Scholar

[21]

J. Hansen, Risk of breast cancer after night- and shift work: Current evidence and ongoing studies in denmark,, Cancer Causes Control, 17 (2006), 531.  doi: 10.1007/s10552-005-9006-5.  Google Scholar

[22]

N. Hansen, The CMA evolution strategy: A comparing review. towards a new evolutionary computation,, in, (2006), 75.   Google Scholar

[23]

P. Hinow, S. E. Wang, C. L. Arteaga and G. F. Webb, A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor,, Theor. Biol. Med. Model, 4 (2007).  doi: 10.1186/1742-4682-4-14.  Google Scholar

[24]

K. Iwata, K. Kawasaki and N. Shigesada, A dynamical model for the growth and size distribution of multiple metastatic tumors,, J. Theor. Biol., 203 (2000), 177.  doi: 10.1006/jtbi.2000.1075.  Google Scholar

[25]

S. M. Jones and A. Kazlauskas, Connecting signaling and cell cycle progression in growth factor-stimulated cells,, Oncogene, 19 (2000), 5558.  doi: 10.1038/sj.onc.1203858.  Google Scholar

[26]

Y. Kheifetz, Y. Kogan and Z. Agur, Long-range predictability in models of cell populations subjected to phase-specific drugs: Growth-rate approximation using properties of positive compact operators,, Math. Models Methods Appl. Sci., 16 (2006), 1155.  doi: 10.1142/S0218202506001492.  Google Scholar

[27]

F. Lévi, A. Okyar, S. Dulong, P. F. Innominato and J. Clairambault, Circadian timing in cancer treatments,, Annu. Rev. Pharmacol. Toxicol., 50 (2010), 377.  doi: 10.1146/annurev.pharmtox.48.113006.094626.  Google Scholar

[28]

F. Lévi and U. Schibler, Circadian rhythms: Mechanisms and therapeutic implications,, Annu. Rev. Pharmacol. Toxicol., 47 (2007), 593.  doi: 10.1146/annurev.pharmtox.47.120505.105208.  Google Scholar

[29]

J. Massagué, How cells read TGF-$\beta$ signals,, Nat. Rev. Mol. Cell Biol., 1 (2000), 169.  doi: 10.1038/35043051.  Google Scholar

[30]

J. Massagué, S. W. Blain and R. S. Lo, TGF$\beta$ signaling in growth control, cancer, and heritable disorders,, Cell, 103 (2000), 295.  doi: 10.1016/S0092-8674(00)00121-5.  Google Scholar

[31]

A. L. Mazlyzam, B. S. Aminuddin, L. Saim and B. H. I. Ruszymah, Human serum is an advantageous supplement for human dermal fibroblast expansion: Clinical implications for tissue engineering of skin,, Arch. Med. Res., 39 (2008), 743.  doi: 10.1016/j.arcmed.2008.09.001.  Google Scholar

[32]

H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression,, Proc. Natl. Acad. Sci. USA, 31 (1997), 814.  doi: 10.1073/pnas.94.3.814.  Google Scholar

[33]

A. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Math. Soc., 54 (1926), 98.   Google Scholar

[34]

J. Mendelsohn and J. Baselga, Status of epidermal growth factor receptor antagonists in the biology and treatment of cancer,, J. Clin. Oncol., 14 (2003), 2787.  doi: 10.1200/JCO.2003.01.504.  Google Scholar

[35]

J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations,", volume 68 of Lecture Notes in Biomathematics. Springer, (1986).   Google Scholar

[36]

A. B. Pardee, A restriction point for control of normal animal cell proliferation,, Proc. Natl. Acad. Sci. USA, 71 (1974), 1286.  doi: 10.1073/pnas.71.4.1286.  Google Scholar

[37]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics series. Birkhäuser, (2007).   Google Scholar

[38]

S. I. Reed, E. Bailly, V. Dulic, L. Hengst, D. Resnitzky and J. Slingerland, G1 control in mammalian cells,, J. Cell Sci. Suppl., 18 (1994), 69.   Google Scholar

[39]

D. Resnitzky, M. Gossen, H. Bujard and S. I. Reed, Acceleration of the G1/S phase transition by expression of cyclins D1 and E with an inducible system,, Mol. Cell Biol., 14 (1994), 1669.   Google Scholar

[40]

B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J. P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents,, J. Theor. Biol., 243 (2006), 532.  doi: 10.1016/j.jtbi.2006.07.013.  Google Scholar

[41]

A. Sakaue-Sawano, H. Kurokawa, T. Morimura, A. Hanyu, H. Hama, H. Osawa, S. Kashiwagi, K. Fukami, T. Miyata, H. Miyoshi, T. Imamura, M. Ogawa, H. Masai and A. Miyawaki, Visualizing spatiotemporal dynamics of multicellular cell-cycle progression,, Cell, 132 (2008), 487.  doi: 10.1016/j.cell.2007.12.033.  Google Scholar

[42]

A. Sakaue-Sawano, K. Ohtawa, H. Hama, M. Kawano, M. Ogawa and A. Miyawaki, Tracing the silhouette of individual cells in S/G2/M phases with fluorescence,, Chem. Biol., 15 (2008), 1243.  doi: 10.1016/j.chembiol.2008.10.015.  Google Scholar

[43]

V. Shahreazaei and P. G. Swain, Analytical distributions for stochastic gene expression,, Proc. Natl. Acad. Sci. USA, 105 (2008), 17256.  doi: 10.1073/pnas.0803850105.  Google Scholar

[44]

E. Sherer, E. Tocce, R. E. Hannemann, A. E. Rundell and D. Ramkrishna, Identification of age-structured models: cell cycle phase transitions,, Biotechnol. Bioeng., 99 (2008), 960.  doi: 10.1002/bit.21633.  Google Scholar

[45]

R. Taub, Liver regeneration: from myth to mechanism,, Nat. Rev. Mol. Cell Biol., 5 (2004), 836.  doi: 10.1038/nrm1489.  Google Scholar

[46]

G. Webb, Resonance phenomena in cell population chemotherapy models,, Rocky Mountain J. Math., 20 (1990), 1195.  doi: 10.1216/rmjm/1181073070.  Google Scholar

[47]

A. Zetterberg, O. Larsson and K. G. Wiman, What is the restriction point?,, Curr. Opin. Cell Biol., 7 (1995), 835.  doi: 10.1016/0955-0674(95)80067-0.  Google Scholar

show all references

References:
[1]

O. Arino, A survey of structured cell population dynamics,, Acta Biotheor., 43 (1995), 3.  doi: 10.1007/BF00709430.  Google Scholar

[2]

O. Arino and M. Kimmel, Comparison of approaches to modeling of cell population dynamics,, SIAM J. Appl. Math., 53 (1993), 1480.  doi: 10.1137/0153069.  Google Scholar

[3]

O. Arino and E. Sanchez, A survey of cell population dynamics,, J. Theor. Med., 1 (1997), 35.  doi: 10.1080/10273669708833005.  Google Scholar

[4]

D. Barbolosi, A. Benabdallah, F. Hubert and F. Verga, Mathematical and numerical analysis for a model of growing metastatic tumors,, Math. Biosci., 218 (2009), 1.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[5]

B. Basse, B. C. Baguley, E. S. Marshall, G. C. Wake and D. J. N. Wall, Modelling the flow [corrected] cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison,, Bull Math. Biol., 67 (2005), 815.  doi: 10.1016/j.bulm.2004.10.003.  Google Scholar

[6]

B. Basse and P. Ubezio, A generalised age- and phase-structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies,, Bull Math. Biol., 69 (2007), 1673.  doi: 10.1007/s11538-006-9185-6.  Google Scholar

[7]

S. Benzekry, N. André, A. Benabdallah, J. Ciccolini, C. Faivre, F. Hubert and D. Barbolosi, Modeling the impact of anticancer agents on metastatic spreading,, Mathematical Modelling of Natural Phenomena, 7 (2012), 306.  doi: 10.1051/mmnp/20127114.  Google Scholar

[8]

F. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. Ouillon and S. Saito, Synchronisation and control of proliferation in cycling cell population models with age structure,, Math. Comp. Simul., (2012).   Google Scholar

[9]

J. Clairambault, Optimizing cancer pharmacotherapeutics using mathematical modeling and a systems biology approach,, Personalized Medicine, 8 (2011), 271.  doi: 10.2217/pme.11.20.  Google Scholar

[10]

J. Clairambault, S. Gaubert and T. Lepoutre, Comparison of Perron and Floquet eigenvalues in age structured cell division models,, Mathematical Modelling of Natural Phenomena, 4 (2009), 183.  doi: 10.1051/mmnp/20094308.  Google Scholar

[11]

J. Clairambault, S. Gaubert and T. Lepoutre, Circadian rhythm and cell population growth,, Mathematical and Computer Modelling, 53 (2011), 1558.  doi: 10.1016/j.mcm.2010.05.034.  Google Scholar

[12]

J. Clairambault, B. Laroche, S. Mischler and B. Perthame, A mathematical model of the cell cycle and its control,, Technical report, (4892).   Google Scholar

[13]

A. A. Cohen, T. Kalisky, A. Mayo, N. Geva-Zatorsky, T. Danon, I. Issaeva, R. Kopito, N. Perzov, R. Milo, A. Sigal and U. Alon, Protein dynamics in individual human cells: Experiment and theory,, PLoS one, 4 (2009), 1.  doi: 10.1371/journal.pone.0004901.  Google Scholar

[14]

M. Cross and T. M. Dexter, Growth factors in development, transformation, and tumorigenesis,, Cell, 64 (1991), 271.  doi: 10.1016/0092-8674(91)90638-F.  Google Scholar

[15]

S. Davis and D. K. Mirick, Circadian disruption, shift work and the risk of cancer: A summary of the evidence and studies in seattle,, Cancer Causes Control, 17 (2006), 539.  doi: 10.1007/s10552-005-9010-9.  Google Scholar

[16]

S. S. Fatimah, G. C. Tan, K. H. Chua, A. E. Tan and A. R. Hayati, Effects of epidermal growth factor on the proliferation and cell cycle regulation of cultured human amnion epithelial cells,, J. Biosci. Bioeng., 114 (2012), 220.  doi: 10.1016/j.jbiosc.2012.03.021.  Google Scholar

[17]

E. Filipski, X. M. Li and F. Lévi, Disruption of circadian coordination and malignant growth,, Cancer Causes Control, 17 (2006), 509.  doi: 10.1007/s10552-005-9007-4.  Google Scholar

[18]

E. Filipski, P. Subramanian, J. Carrière, C. Guettier, H. Barbason and F. Lévi, Circadian disruption accelerates liver carcinogenesis in mice,, Mutat. Res., 680 (2009), 95.  doi: 10.1016/j.mrgentox.2009.10.002.  Google Scholar

[19]

D. A. Foster, P. Yellen, L. Xu and M. Saqcena, Regulation of g1 cell cycle progression: Distinguishing the restriction point from a nutrient-sensing cell growth checkpoint(s),, Genes Cancer, 1 (2010), 1124.  doi: 10.1177/1947601910392989.  Google Scholar

[20]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28 (1990), 671.  doi: 10.1007/BF00160231.  Google Scholar

[21]

J. Hansen, Risk of breast cancer after night- and shift work: Current evidence and ongoing studies in denmark,, Cancer Causes Control, 17 (2006), 531.  doi: 10.1007/s10552-005-9006-5.  Google Scholar

[22]

N. Hansen, The CMA evolution strategy: A comparing review. towards a new evolutionary computation,, in, (2006), 75.   Google Scholar

[23]

P. Hinow, S. E. Wang, C. L. Arteaga and G. F. Webb, A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor,, Theor. Biol. Med. Model, 4 (2007).  doi: 10.1186/1742-4682-4-14.  Google Scholar

[24]

K. Iwata, K. Kawasaki and N. Shigesada, A dynamical model for the growth and size distribution of multiple metastatic tumors,, J. Theor. Biol., 203 (2000), 177.  doi: 10.1006/jtbi.2000.1075.  Google Scholar

[25]

S. M. Jones and A. Kazlauskas, Connecting signaling and cell cycle progression in growth factor-stimulated cells,, Oncogene, 19 (2000), 5558.  doi: 10.1038/sj.onc.1203858.  Google Scholar

[26]

Y. Kheifetz, Y. Kogan and Z. Agur, Long-range predictability in models of cell populations subjected to phase-specific drugs: Growth-rate approximation using properties of positive compact operators,, Math. Models Methods Appl. Sci., 16 (2006), 1155.  doi: 10.1142/S0218202506001492.  Google Scholar

[27]

F. Lévi, A. Okyar, S. Dulong, P. F. Innominato and J. Clairambault, Circadian timing in cancer treatments,, Annu. Rev. Pharmacol. Toxicol., 50 (2010), 377.  doi: 10.1146/annurev.pharmtox.48.113006.094626.  Google Scholar

[28]

F. Lévi and U. Schibler, Circadian rhythms: Mechanisms and therapeutic implications,, Annu. Rev. Pharmacol. Toxicol., 47 (2007), 593.  doi: 10.1146/annurev.pharmtox.47.120505.105208.  Google Scholar

[29]

J. Massagué, How cells read TGF-$\beta$ signals,, Nat. Rev. Mol. Cell Biol., 1 (2000), 169.  doi: 10.1038/35043051.  Google Scholar

[30]

J. Massagué, S. W. Blain and R. S. Lo, TGF$\beta$ signaling in growth control, cancer, and heritable disorders,, Cell, 103 (2000), 295.  doi: 10.1016/S0092-8674(00)00121-5.  Google Scholar

[31]

A. L. Mazlyzam, B. S. Aminuddin, L. Saim and B. H. I. Ruszymah, Human serum is an advantageous supplement for human dermal fibroblast expansion: Clinical implications for tissue engineering of skin,, Arch. Med. Res., 39 (2008), 743.  doi: 10.1016/j.arcmed.2008.09.001.  Google Scholar

[32]

H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression,, Proc. Natl. Acad. Sci. USA, 31 (1997), 814.  doi: 10.1073/pnas.94.3.814.  Google Scholar

[33]

A. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Math. Soc., 54 (1926), 98.   Google Scholar

[34]

J. Mendelsohn and J. Baselga, Status of epidermal growth factor receptor antagonists in the biology and treatment of cancer,, J. Clin. Oncol., 14 (2003), 2787.  doi: 10.1200/JCO.2003.01.504.  Google Scholar

[35]

J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations,", volume 68 of Lecture Notes in Biomathematics. Springer, (1986).   Google Scholar

[36]

A. B. Pardee, A restriction point for control of normal animal cell proliferation,, Proc. Natl. Acad. Sci. USA, 71 (1974), 1286.  doi: 10.1073/pnas.71.4.1286.  Google Scholar

[37]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics series. Birkhäuser, (2007).   Google Scholar

[38]

S. I. Reed, E. Bailly, V. Dulic, L. Hengst, D. Resnitzky and J. Slingerland, G1 control in mammalian cells,, J. Cell Sci. Suppl., 18 (1994), 69.   Google Scholar

[39]

D. Resnitzky, M. Gossen, H. Bujard and S. I. Reed, Acceleration of the G1/S phase transition by expression of cyclins D1 and E with an inducible system,, Mol. Cell Biol., 14 (1994), 1669.   Google Scholar

[40]

B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J. P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents,, J. Theor. Biol., 243 (2006), 532.  doi: 10.1016/j.jtbi.2006.07.013.  Google Scholar

[41]

A. Sakaue-Sawano, H. Kurokawa, T. Morimura, A. Hanyu, H. Hama, H. Osawa, S. Kashiwagi, K. Fukami, T. Miyata, H. Miyoshi, T. Imamura, M. Ogawa, H. Masai and A. Miyawaki, Visualizing spatiotemporal dynamics of multicellular cell-cycle progression,, Cell, 132 (2008), 487.  doi: 10.1016/j.cell.2007.12.033.  Google Scholar

[42]

A. Sakaue-Sawano, K. Ohtawa, H. Hama, M. Kawano, M. Ogawa and A. Miyawaki, Tracing the silhouette of individual cells in S/G2/M phases with fluorescence,, Chem. Biol., 15 (2008), 1243.  doi: 10.1016/j.chembiol.2008.10.015.  Google Scholar

[43]

V. Shahreazaei and P. G. Swain, Analytical distributions for stochastic gene expression,, Proc. Natl. Acad. Sci. USA, 105 (2008), 17256.  doi: 10.1073/pnas.0803850105.  Google Scholar

[44]

E. Sherer, E. Tocce, R. E. Hannemann, A. E. Rundell and D. Ramkrishna, Identification of age-structured models: cell cycle phase transitions,, Biotechnol. Bioeng., 99 (2008), 960.  doi: 10.1002/bit.21633.  Google Scholar

[45]

R. Taub, Liver regeneration: from myth to mechanism,, Nat. Rev. Mol. Cell Biol., 5 (2004), 836.  doi: 10.1038/nrm1489.  Google Scholar

[46]

G. Webb, Resonance phenomena in cell population chemotherapy models,, Rocky Mountain J. Math., 20 (1990), 1195.  doi: 10.1216/rmjm/1181073070.  Google Scholar

[47]

A. Zetterberg, O. Larsson and K. G. Wiman, What is the restriction point?,, Curr. Opin. Cell Biol., 7 (1995), 835.  doi: 10.1016/0955-0674(95)80067-0.  Google Scholar

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