Influence of the Nuclear Membrane, Active Transport, and Cell Shape on the Hes1 and p53–Mdm2 Pathways: Insights from Spatio-temporal Modelling

Abstract

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53–Mdm2, both of which are implicated in cancer.

In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15–31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity.

Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53–Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

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References

  1. Agrawal, S., Archer, C., & Schaffer, D. (2009). Computational models of the notch network elucidate mechanisms of context-dependent signaling. PLoS Comput. Biol., 5, e1000390.

    Article  Google Scholar 

  2. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2008). Molecular biology of the cell. Garland science (5th ed.). Oxford: Taylor and Francis.

    Google Scholar 

  3. Bancaud, A., Huet, S., Daigle, N., Mozziconacci, J., Beaudouin, J., & Ellenberg, J. (2009). Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin. EMBO J., 28, 3785–3798.

    Article  Google Scholar 

  4. Barik, D., Baumann, W. T., Paul, M. R., Novak, B., & Tyson, J. J. (2010). A model of yeast cell-cycle regulation based on multisite phosphorylation. Mol. Syst. Biol., 6, 405.

    Article  Google Scholar 

  5. Barik, D., Paul, M. R., Baumann, W. T., Cao, Y., & Tyson, J. J. (2008). Stochastic simulation of enzyme-catalyzed reactions with disparate timescales. Biophys. J., 95, 3563–3574.

    Article  Google Scholar 

  6. Barrio, M., Burrage, K., Leier, A., & Tian, T. (2006). Oscillatory regulation of Hes1: discrete stochastic delay modelling and simulation. PLoS ONE, 2, e117.

    Google Scholar 

  7. Baserga, R. (2007). Is cell size important? Cell Cycle, 6(7), 814–816.

    Article  Google Scholar 

  8. Batchelor, E., Loewer, A., & Lahav, G. (2009). The ups and downs of p53: understanding protein dynamics in single cells. Nat. Rev., Cancer, 9(5), 371–377.

    Article  Google Scholar 

  9. Batchelor, E., Mock, C., Bhan, I., Loewer, A., & Lahav, G. (2008). Recurrent initiation: a mechanism for triggering p53 pulses in response to DNA damage. Mol. Cell, 30, 277–289.

    Article  Google Scholar 

  10. Beck, M., Forster, F., Ecke, M., Plitzko, J. M., Melchoir, F., Gerisch, G., Baumeister, W., & Medalia, O. (2004). Nuclear pore complex structure and dynamics revealed by cryoelectron tomography. Science, 306, 1387–1390.

    Article  Google Scholar 

  11. Bernard, S., Čajavec, B., Pujo-Menjouet, L., Mackey, M., & Herzel, H. (2006). Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations. Philos. Trans. R. Soc. A, 364, 1155–1170.

    MATH  Article  Google Scholar 

  12. Brown, G., & Kholodenko, B. (1999). Spatial gradients of cellular phospho-proteins. FEBS Lett., 457, 452–454.

    Article  Google Scholar 

  13. Busenberg, S., & Mahaffy, J. (1985). Interaction of spatial diffusion and delays in models of genetic control by repression. J. Math. Biol., 22, 313–333.

    MathSciNet  MATH  Article  Google Scholar 

  14. Cangiani, A., & Natalini, R. (2010). A spatial model of cellular molecular trafficking including active transport along microtubules. J. Theor. Biol., 267, 614–625.

    Article  Google Scholar 

  15. Carbonaro, M., O’Brate, A., & Giannakakou, P. (2011). Microtubule disruption targets HIF-1α mRNA to cytoplasmic P-bodies for translation repression. J. Cell Biol., 192, 83–99.

    Article  Google Scholar 

  16. Caspi, A., Granek, R., & Elbaum, M. (2000). Enhanced diffusion in active intracellular transport. Phys. Rev. Lett., 85, 5655–5658.

    Article  Google Scholar 

  17. Chahine, M. N., & Pierce, G. N. (2009). Therapeutic targeting of nuclear protein import in pathological cell conditions. Pharmacol. Rev., 61, 358–372.

    Article  Google Scholar 

  18. Ciliberto, A., Novak, B., & Tyson, J. (2005). Steady states and oscillations in the p53/Mdm2 network. Cell Cycle, 4, 488–493.

    Article  Google Scholar 

  19. Cole, C., & Scarcelli, J. (2006). Transport of messenger RNA from the nucleus to the cytoplasm. Curr. Opin. Cell Biol., 18, 299–306.

    Article  Google Scholar 

  20. Cole, N., & Lippincott-Schwartz, J. (1995). Organization of organelles and membrane traffic by microtubules. Curr. Opin. Cell Biol., 7, 55–64.

    Article  Google Scholar 

  21. Davidson, M. W. (2011). Micromagnet website. http://micro.magnet.fsu.edu/primer/techniques/fluorescence/gallery/cells/u2/u2cellslarge8.html (accessed 26th April 2011)

  22. Diller, L., Kassel, J., Nelson, C., Gryka, M., Litwak, G., Gebhardt, M., Bressac, B., Ozturk, M., Baker, S., Vogelstein, B., & Friend, S. (1990). p53 functions as a cell cycle control protein in osteosarcomas. Mol. Cell. Biol., 10, 5772–5781.

    Google Scholar 

  23. Dinh, A., Theofanous, T., & Mitragotri, S. (2005). A model of intracellular trafficking of adenoviral vectors. Biophys. J., 89, 1574–1588.

    Article  Google Scholar 

  24. Feldherr, C., & Akin, D. (1991). Signal-mediated nuclear transport in proliferating and growth-arrested BALB/c 3T3 cells. J. Cell Biol., 115, 933–939.

    Article  Google Scholar 

  25. Ferrari, S., & Palmerini, E. (2007). Adjuvant and neoadjuvant combination chemotherapy for osteogenic sarcoma. Curr. Opin. Oncol., 19, 341–346.

    Article  Google Scholar 

  26. Finlay, C. (1993). The mdm-2 oncogene can overcome wild-type p53 suppression of transformed cell growth. Mol. Cell. Biol., 12, 301–306.

    Google Scholar 

  27. Gasiorowski, J. Z., & Dean, D. (2003). Mechanisms of nuclear transport and interventions. Adv. Drug Deliv. Rev., 55, 703–716.

    Article  Google Scholar 

  28. Geva-Zatorsky, N., Rosenfeld, N., Itzkovitz, S., Milo, R., Sigal, A., Dekel, E., Yarnitzky, T., Liron, Y., Polak, P., Lahav, G., & Alon, U. (2006). Oscillations and variability in the p53 system. Mol. Syst. Biol., 2, E1–E13.

    Article  Google Scholar 

  29. Giannakakou, P., Sackett, D., Ward, Y., Webster, K., Blagosklonny, M., & Fojo, T. (2000). p53 is associated with cellular microtubules and is transported to the nucleus by dynein. Nat. Cell Biol., 2(10), 709–717.

    Article  Google Scholar 

  30. Glass, L., & Kauffman, S. (1972). Co-operative components, spatial localization and oscillatory cellular dynamics. J. Theor. Biol., 34, 219–237.

    Article  Google Scholar 

  31. Gordon, K., Leeuwen, I. V., Laín, S., & Chaplain, M. (2009). Spatio-temporal modelling of the p53–mdm2 oscillatory system. Math. Model. Nat. Phenom., 4, 97–116.

    MathSciNet  MATH  Article  Google Scholar 

  32. Hamstra, D., Bhojani, M., Griffin, L., Laxman, B., Ross, B., & Rehemtulla, A. (2006). Real-time evaluation of p53 oscillatory behaviour in vivo using bioluminescent imaging. Cancer Res., 66, 7482–7489.

    Article  Google Scholar 

  33. Hirata, H., Yoshiura, S., Ohtsuka, T., Bessho, Y., Harada, T., Yoshikawa, K., & Kageyama, R. (2002). Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science, 298, 840–843.

    Article  Google Scholar 

  34. Johansson, T., Lejonklou, M., Ekeblad, S., Stålberg, P., & Skogseid, B. (2008). Lack of nuclear expression of hairy and enhancer of split-1 (HES1) in pancreatic endocrine tumors. Horm. Metab. Res., 40, 354–359.

    Article  Google Scholar 

  35. Jordan, M. A., & Wilson, L. (2004). Microtubules as a target for anticancer drugs. Nat. Rev., Cancer, 4, 253–265.

    Article  Google Scholar 

  36. Kar, S., Baumann, W. T., Paul, M. R., & Tyson, J. J. (2009). Exploring the roles of noise in the eukaryotic cell cycle. Proc. Natl. Acad. Sci. USA, 106, 6471–6476.

    Article  Google Scholar 

  37. Kau, T., Way, J., & Silver, P. (2004). Nuclear transport and cancer: from mechanism to intervention. Nature, 4, 106–117.

    Google Scholar 

  38. Kavallaris, M. (2010). Microtubules and resistance to tubulin-binding agents. Nat. Rev., Cancer, 10, 194–204.

    Article  Google Scholar 

  39. Kherlopian, A., Song, T., Duan, Q., Neimark, M., Po, M., Gohagan, J., & Laine, A. (2008). A review of imaging techniques for systems biology. BMC Syst. Biol., 2, 74–92.

    Article  Google Scholar 

  40. Kholodenko, B. (2006). Cell-signalling dynamics in time and space. Nat. Rev. Mol. Cell Biol., 7, 165–174.

    Article  Google Scholar 

  41. Kim, I., Kim, D., Han, S., Chin, M., Nam, H., Cho, H., Choi, S., Song, B., Kim, E., Bae, Y., & Moon, Y. (2000). Truncated form of importin alpha identified in breast cancer cells inhibits nuclear import of p53. J. Biol. Chem., 275, 23139–23145.

    Article  Google Scholar 

  42. Klonis, N., Rug, M., Harper, I., Wickham, M., Cowman, A., & Tilley, L. (2002). Fluorescence photobleaching analysis for the study of cellular dynamics. Eur. Biophys. J., 31, 36–51.

    Article  Google Scholar 

  43. Lahav, G., Rosenfeld, N., Sigal, A., Geva-Zatorsky, N., Levine, A., Elowitz, M., & Alon, U. (2004). Dynamics of the p53–Mdm2 feedback loop in individual cells. Nat. Genet., 36, 147–150.

    Article  Google Scholar 

  44. Lane, D. (1992). p53, guardian of the genome. Nature, 358, 15–16.

    Article  Google Scholar 

  45. Lewis, J. (2003). Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator. Curr. Biol., 13, 1398–1408.

    Article  Google Scholar 

  46. Lightcap, E., McCormack, T., Pien, C., Chau, V., Adams, J., & Elliott, P. (2000). Proteasome inhibition measurements: clinical application. Clin. Chem., 46, 673–683.

    Google Scholar 

  47. Locke, J., Southern, M., Kozma-Bognár, L., Hibberd, V., Brown, P., Turner, M., & Mllar, A. (2005). Extension of a genetic network model by iterative experimentation and mathematical analysis. Mol. Syst. Biol., 1, 1–9.

    Article  Google Scholar 

  48. Loewer, A., Batchelor, E., Gaglia, G., & Lahav, G. (2010). Basal dynamics of p53 reveal transcriptionally attenuated pulses in cycling cells. Cell, 142(1), 89–100.

    Article  Google Scholar 

  49. Lomakin, A., & Nadezhdina, E. (2010). Dynamics of nonmembranous cell components: role of active transport along microtubules. Biochemistry (Moscow), 75(1), 7–18.

    Article  Google Scholar 

  50. Ma, L., Wagner, J., Rice, J., Hu, W., Levine, A., & Stolovitzky, G. (2005). A plausible model for the digital response of p53 to DNA damage. Proc. Natl. Acad. Sci. USA, 102, 14266–14271.

    Article  Google Scholar 

  51. Mahaffy, J. (1988). Genetic control models with diffusion and delays. Math. Biosci., 90, 519–533.

    MathSciNet  MATH  Article  Google Scholar 

  52. Mahaffy, J., & Pao, C. (1984). Models of genetic control by repression with time delays and spatial effects. J. Math. Biol., 20, 39–57.

    MathSciNet  MATH  Article  Google Scholar 

  53. Maki, C., Huibregtse, J., & Howley, P. (1996). In vivo ubiquitination and proteasome-mediated degradation of p53. Cancer Res., 56(11), 2649–2654.

    Google Scholar 

  54. Marfori, M., Mynott, A., Ellis, J. J., Mehdi, A. M., Saunders, N. F. W., Curmi, P. M., Forwood, J. K., Boden, M., & Kobe, B. (2011). Molecular basis for specificity of nuclear import and prediction of nuclear localization. Biochim. Biophys. Acta, Mol. Cell Res., 1813(9), 1562–1577.

    Article  Google Scholar 

  55. Masamizu, Y., Ohtsuka, T., Takashima, Y., Nagahara, H., Takenaka, Y., Yoshikawa, K., Okamura, H., & Kageyama, R. (2006). Real-time imaging of the somite segmentation clock: revelation f unstable oscillators in the individual presomitic mesoderm cells. Proc. Natl. Acad. Sci. USA, 103, 1313–1318.

    Article  Google Scholar 

  56. Matsuda, T., Miyawaki, A., & Nagai, T. (2008). Direct measurement of protein dynamics inside cells using a rationally designed photoconvertible protein. Nat. Methods, 5, 339–345.

    Google Scholar 

  57. Mayo, L., & Donner, D. (2001). A phosphatidylinositol 3-kinase/akt pathway promotes translocation of mdm2 from the cytoplasm to the nucleus. Proc. Natl. Acad. Sci. USA, 98(20), 11598–11603.

    Article  Google Scholar 

  58. Mendez, V., Fedotov, S., & Horsthemke, W. (2010). Reaction-transport systems. Berlin: Springer.

    Google Scholar 

  59. Meyers, J., Craig, J., & Odde, D. (2006). Potential for control of signaling pathways via cell size and shape. Curr. Biol., 16, 1685–1693.

    Article  Google Scholar 

  60. Michalet, X., Pinaudand, F., Bentolila, L., Tsay, J., Doose, S., Li, J., Sundaresan, G., Wu, A., Gambhir, S., & Weiss, S. (2005). Quantum dots for live cells, in vivo imaging, and diagnostics. Science, 307, 538–544.

    Article  Google Scholar 

  61. Mihalas, G., Neamtu, M., Opris, D., & Horhat, R. (2006). A dynamic P53–MDM2 model with time delay. Chaos Solitons Fractals, 30, 936–945.

    MathSciNet  MATH  Article  Google Scholar 

  62. Momiji, H., & Monk, N. (2008). Dissecting the dynamics of the Hes1 genetic oscillator. J. Theor. Biol., 254, 784–798.

    Article  Google Scholar 

  63. Monk, N. (2003). Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays. Curr. Biol., 13, 1409–1413.

    Article  Google Scholar 

  64. Muller, M., Klumpp, S., & Lipowsky, R. (2008). Tug-of-war as a cooperative mechanism for bidirectional cargo transport of molecular motors. Proc. Natl. Acad. Sci. USA, 105, 4609–4614.

    Article  Google Scholar 

  65. Nelson, D., Ihekwaba, A., Elliott, M., Johnson, J., Gibney, C., Foreman, B., Nelson, G., See, V., Horton, C., Spiller, D., Edwards, S., McDowell, H., Unitt, J., Sullivan, E., Grimley, R., Benson, N., Broomhead, D., Kell, D., & White, M. (2004). Oscillations in NF-κB signaling control the dynamics of gene expression. Science, 306, 704–708.

    Article  Google Scholar 

  66. Neves, S., Tsokas, P., Sarkar, A., Grace, E., Rangamani, P., Taubenfeld, S., Alberini, C., Schaff, J., Blitzer, R., Moraru, I., & Iyengar, R. (2008). Cell shape and negative links in regulatory motifs together control spatial information flow in signaling networks. Cell, 133, 666–680.

    Article  Google Scholar 

  67. Norvell, A., Debec, A., Finch, D., Gibson, L., & Thoma, B. (2005). Squid is required for efficient posterior localization of oskar mRNA during drosophila oogenesis. Dev. Genes Evol., 215, 340–349.

    Article  Google Scholar 

  68. O’Brate, A., & Giannakakou, P. (2003). The importance of p53 location: nuclear or cytoplasmic zip code? Drug Resist. Updat., 6(6), 313–322.

    Article  Google Scholar 

  69. Orlowski, R., & Kuhn, D. (2008). Proteasome inhibitors in cancer therapy: lessons from the first decade. Clin. Cancer Res., 14, 1649–1657.

    Article  Google Scholar 

  70. Ouattara, D., Abou-Jaoudé, W., & Kaufman, M. (2010). From structure to dynamics: frequency tuning in the p53–Mdm2 network. II. Differential and stochastic approaches. J. Theor. Biol., 264, 1177–1189.

    Article  Google Scholar 

  71. Perkins, N. (2007). Integrating cell-signalling pathways with NF-κB and IKK function. Nat. Rev. Mol. Cell Biol., 8, 49–62.

    Article  Google Scholar 

  72. Pincus, Z., & Theriot, J. (2007). Comparison of quantitative methods for cell-shape analysis. J. Microsc., 227(2), 140–156.

    MathSciNet  Article  Google Scholar 

  73. Pommier, Y., Sordet, O., Antony, S., Hayward, R., & Kohn, K. (2004). Apoptosis defects and chemotherapy resistance: molecular interaction maps and networks. Oncogene, 23, 2934–2949.

    Article  Google Scholar 

  74. Prinz, H. (2010). Hill coefficients, dose-response curves, and allosteric mechanisms. J. Chem. Biol., 3, 37–44.

    Article  Google Scholar 

  75. Proctor, C., & Gray, D. (2008). Explaining oscillations and variability in the p53–Mdm2 system. BMC Syst. Biol., 2(75), 1–20.

    Google Scholar 

  76. Puszyński, K., Bertolusso, R., & Lipniacki, T. (2009). Crosstalk between p53 and NF-κB systems: pro- and anti-apoptotic functions of NF-κB. IET Syst. Biol., 3, 356–367.

    Article  Google Scholar 

  77. Puszyński, K., Hat, B., & Lipniacki, T. (2008). Oscillations and bistability in the stochastic model of p53 regulation. J. Theor. Biol., 254, 452–465.

    Article  Google Scholar 

  78. Rangamani, P., & Iyengar, R. (2007). Modelling spatio-temporal interactions within the cell. J. Biosci., 32, 157–167.

    Article  Google Scholar 

  79. Robati, M., Holtz, D., & Dunton, C. J. (2008). A review of topotecan in combination chemotherapy for advanced cervical cancer. Ther. Clin. Risk. Manag., 4, 213–218.

    Google Scholar 

  80. Rodriguez, M. S., Dargemont, C., & Stutz, F. (2004). Nuclear export of RNA. Biol. Cell, 96, 639–655.

    Article  Google Scholar 

  81. Roth, D., Moseley, G., Glover, D., Pouton, C., & Jans, D. (2007). A microtubule-facilitated nuclear import pathway for cancer regulatory proteins. Traffic, 8(6), 673–686.

    Article  Google Scholar 

  82. Ryan, K., Phillips, A., & Vousden, K. (2001). Regulation and function of the p53 tumor suppressor protein. Curr. Opin. Cell Biol., 13, 332–337.

    Article  Google Scholar 

  83. Sang, L., Coller, H., & Roberts, J. (2008). Control of the reversibility of cellular quiescence by the transcriptional repressor HES1. Science, 321, 1095–1100.

    Article  Google Scholar 

  84. Seksek, O., Biwersi, J., & Verkman, A. (1997). Translational diffusion of macromolecule-sized solutes in cytoplasm and nucleus. J. Cell Biol., 138, 131–142.

    Article  Google Scholar 

  85. Shahrezaei, V., & Swain, P. (2008). The stochastic nature of biochemical networks. Curr. Opin. Biotechnol., 19, 369–374.

    Article  Google Scholar 

  86. Shankaran, H., Ippolito, D., Chrisler, W., Resat, H., Bollinger, N., Opresko, L., & Wiley, H. (2009). Rapid and sustained nuclear-cytoplasmic ERK oscillations induced by epidermal growth factor. Mol. Syst. Biol., 5, 322.

    Article  Google Scholar 

  87. Shymko, R., & Glass, L. (1974). Spatial switching in chemical reactions with heterogeneous catalysis. J. Chem. Phys., 60, 835–841.

    Article  Google Scholar 

  88. Smith, D., & Simmons, R. (2001). Model of motor-assisted transport of intracelullar particules. Biophys. J., 80, 45–68.

    Article  Google Scholar 

  89. Sturrock, M., Terry, A., Xirodimas, D., Thompson, A., & Chaplain, M. (2011). Spatio-temporal modelling of the Hes1 and p53–Mdm2 intracellular signalling pathways. J. Theor. Biol., 273, 15–31.

    Article  Google Scholar 

  90. Terry, A., & Chaplain, M. (2011). Spatio-temporal modelling of the NF-κB intracellular signalling pathway: the roles of diffusion, active transport, and cell geometry. J. Theor. Biol., 290, 7–26.

    Article  Google Scholar 

  91. Terry, A., Sturrock, M., Dale, J., Maroto, M., & Chaplain, M. (2011). A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics. PLoS ONE, 6, e16980.

    Article  Google Scholar 

  92. Thut, C., Goodrich, J., & Tjian, R. (1997). Repression of p53-mediated transcription by MDM2: a dual mechanism. Genes Dev., 11, 1974–1986.

    Article  Google Scholar 

  93. van Zon, J., Morelli, M., Tănase-Nicola, S., & ten Wolde, P. (2006). Diffusion of transcription factors can drastically enhance the noise in gene expression. Biophys. J., 91, 4350–4367.

    Article  Google Scholar 

  94. Wachsmuth, M., Waldeck, W., & Langowski, J. (2000). Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially-resolved fluorescence correlation spectroscopy. J. Mol. Biol., 298, 677–689.

    Article  Google Scholar 

  95. Wang, B., Xiao, Z., Ko, H. L., & Ren, E. C. (2010). The p53 response element and transcriptional repression. Cell Cycle, 9(5), 870–879.

    Article  Google Scholar 

  96. Weis, K. (2003). Regulating access to the genome: nucleocytoplasmic transport throughout the cell cycle. Cell, 112, 441–451.

    Article  Google Scholar 

  97. Weiss, M., Hashimoto, H., & Nilsson, T. (2004). Anomalous protein diffusion is a measure for cytoplasmic crowding in living cells. Biophys. J., 87, 3518–3524.

    Article  Google Scholar 

  98. Xirodimas, D., Stephen, C., & Lane, D. (2001). Cocompartmentalization of p53 and Mdm2 is a major determinant for Mdm2-mediated degradation of p53. Exp. Cell Res., 270, 66–77.

    Article  Google Scholar 

  99. Zeiser, S., Muller, J., & Liebscher, V. (2007). Modeling the Hes1 oscillator. J. Comput. Biol., 14, 984–1000.

    MathSciNet  Article  Google Scholar 

  100. Zhang, P., Yang, Y., Zweidler-McKay, P. A., & Hughes, D. (2008). Critical role of notch signaling in osteosarcoma invasion and metastasis. Clin. Cancer Res., 14, 2962–2969.

    Article  Google Scholar 

  101. Zhang, T., Brazhnik, P., & Tyson, J. (2007). Exploring mechanisms of the DNA-damage response: p53 pulses and their possible relevance to apoptosis. Cell Cycle, 6, 85–94.

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, “M5CGS—From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”.

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Correspondence to Marc Sturrock.

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Appendix

Appendix

A.1 Non-dimensionalisation of Hes1 Model

We summarise our non-dimensionalisation of the extended Hes1 model (described in Sect. 2.2). The original Hes1 model (described in Sect. 2.1) is non-dimensionalised in a similar way; for details, see Sturrock et al. (2011).

To non-dimensionalise the extended Hes1 model given by (1)–(4) and (15), subject to the conditions in (8)–(14), we first define re-scaled variables by dividing each variable by a reference value. Re-scaled variables are given overlines to distinguish them from variables that are not re-scaled. Thus we can write:

(47)

where the right-hand side of each equation is a dimensional variable divided by its reference value. From (47), we can write variables in terms of re-scaled variables and then substitute these expressions into (1)–(4) and (15), and into the conditions in (8)–(14). This gives a model defined in terms of re-scaled variables which has the same form as the dimensional model, but now the parameters are all non-dimensional. Denoting the non-dimensional parameters with an asterisk, they are related to dimensional parameters as follows:

(48)

We solve the non-dimensional model using the method described in Sect. 2.1. We simulate the model in COMSOL 3.5a, finding non-dimensional parameter values that yield oscillatory dynamics. We chose the same values as in Eq. (25) in Sturrock et al. (2011) except for those parameters which were new because of our extension to the model. These latter values were chosen as follows: \(D^{*}_{m} = D^{*}_{i_{j}}/5\), \(D^{*}_{p} = D^{*}_{i_{j}}/15\), d =0.01, a =0.03, l =0.63.

Finally, we calculated the dimensional parameter values. To do this, we needed to estimate the reference values. Since Her1 in zebrafish and Hes1 in mice are both pathways connected with somitogenesis, we used the reference concentrations for Her1 protein and her1 mRNA in Terry et al. (2011) as our reference concentrations for Hes1 protein and hes1 mRNA. Thus, we chose [m 0]=1.5×10−9 M and [p 0]=10−9 M. We assumed a cell to be of width 30 μm. But from Figs. 2 and 4, the cell width is equal to three non-dimensional spatial units or 3L-dimensional units (using (47)). Hence we set 3L=30 μm, so that L=10 μm. The experimentally observed period of oscillations of Hes1 is approximately 2 hours (Hirata et al. 2002). Our simulations of the non-dimensionalised model gave oscillations with a period of approximately 300 non-dimensional time units or 300 τ-dimensional units (using (47)). Hence we set 300τ=2 h=7200 s, so that τ=24 s. Using our references values and non-dimensional parameter values, we found dimensional parameter values from (48).

Note that we chose our reference time τ=24 s based on simulations of the extended Hes1 model since this was our most realistic Hes1 model. For the original Hes1 model and for all special cases of the Hes1 model (for example, setting active transport rates to zero), we retained the reference time τ=24 s.

A.2 Non-dimensionalisation of p53–Mdm2 Model

We non-dimensionalised the p53–Mdm2 model defined in Sect. 3.1, and the extended p53–Mdm2 model defined in Sect. 3.2, using the technique described above for non-dimensionalising the extended Hes1 model. We give brief details for our non-dimensionalisation of the extended p53–Mdm2 model.

To non-dimensionalise the extended p53–Mdm2 model given by (19)–(26) and (44)–(45), subject to conditions (27) and (32)–(43), we define re-scaled variables (denoted by overlines) by dividing each variable by a reference value:

(49)

Substituting the scaling in (49) into the extended p53–Mdm2 model gives a non-dimensionalised model with non-dimensional parameters (which we denote with asterisks) that are related to dimensional parameters as follows:

(50)

We solve the non-dimensional model using COMSOL 3.5a, finding non-dimensional parameter values that yield oscillatory dynamics. We chose the same values as in Eq. (60) in Sturrock et al. (2011) except for those parameters which were new because of our extension to the model. These latter values were chosen as follows: θ =1, ζ =0.35, \(D^{*}_{m} = D^{*}_{i_{j}}/5\), \(D^{*}_{p} =D^{*}_{i_{j}}/15\), d =0.01, a =0.03, l =0.63.

Finally, we calculated the dimensional parameter values. To do this, we had to estimate the reference values. As in Sturrock et al. (2011), we chose the following reference concentrations: [p530]=0.5 μM, [Mdm2m 0]=0.05 μM, [Mdm20]=2 μM. In addition, we chose [p53m 0]=0.025 μM (in keeping with relative concentrations of mRNA and protein revealed by simulation of the non-dimensionalised model). As with the Hes1 model, we assumed a cell to be of width 30 μm, which again leads to the reference length L=10 μm. Our simulations of the non-dimensionalised model gave oscillations with a period of approximately 360 non-dimensional time units or 360 τ-dimensional units (using (50)), and the experimentally observed period is approximately 3 hours (Monk 2003). Hence we set 360τ=3 h=10800 s, so that τ=30 s. The reference time τ=30 s was based on simulations of the extended p53–Mdm2 model since this was our most realistic p53–Mdm2 model. For all variants of this model (for example, setting active transport rates to zero), we retained the reference time τ=30 s for ease of comparison of the numerical results. Using our references values and non-dimensional parameter values, we found dimensional parameter values from (50).

A.3 Influence of Cell Shape

As discussed in Sect. 2.2.5, we carried out simulations for the extended Hes1 model on a variety of cell geometries. Results from these simulations are presented in Figs. 24 and 25. It is clear from these figures that sustained oscillatory dynamics are strongly robust to changes in cell shape. Such robustness is reassuring since the shape of eukaryotic cells is highly variable (Baserga 2007; Pincus and Theriot 2007).

Fig. 24
figure24

Plots showing the effect on the extended Hes1 model of varying the nuclear shape. In each row, the left plot shows the shape on which we solve, and the middle and right plots show the corresponding numerical results. Spatial units here are non-dimensional, with one non-dimensional spatial unit corresponding to 10 μm. Total concentrations for Hes1 protein are displayed in blue and for hes1 mRNA in red. Parameter values as per column 2 of Table 2

Fig. 25
figure25

Plots showing the effect on the extended Hes1 model of varying the nucleus position (first row), the MTOC position (second row), and the cell shape (third row). In each row, the left plot shows the shape on which we solve, and the middle and right plots show the corresponding numerical results. Spatial units here are non-dimensional, with one non-dimensional spatial unit corresponding to 10 μm. Total concentrations for Hes1 protein are displayed in blue and for hes1 mRNA in red. Parameter values as per column 2 of Table 2

Only one of the geometries in Figs. 24 or 25 shows significant damping after the initial peaks in Hes1 protein and hes1 mRNA total concentrations. This occurs in the second row in Fig. 25, where the MTOC surrounding the nucleus is significantly increased in size. The increased size of the MTOC reduces the size of the region in which active transport may occur. Hence the results in the second row in Fig. 25 are similar to those presented in Sect. 2.2.4 in which the active transport rate is set to zero.

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Sturrock, M., Terry, A.J., Xirodimas, D.P. et al. Influence of the Nuclear Membrane, Active Transport, and Cell Shape on the Hes1 and p53–Mdm2 Pathways: Insights from Spatio-temporal Modelling. Bull Math Biol 74, 1531–1579 (2012). https://doi.org/10.1007/s11538-012-9725-1

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Keywords

  • p53
  • Hes1
  • Nuclear membrane
  • Active transport
  • Cell shape