Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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.

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

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

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Google Scholar 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Google Scholar 

  • 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 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • 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 

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

    Article  Google Scholar 

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

    Google Scholar 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

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

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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 

Download references

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”.

Author information

Authors and Affiliations

  1. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

    Marc Sturrock, Alan J. Terry & Mark A. J. Chaplain

  2. CRBM (Macromolecular Biochemistry Research Centre), CNRS, UMR 5237, 1919 Route de Mende, 34293, Montpellier cedex 5, France

    Dimitris P. Xirodimas

  3. Dundee Cancer Centre, University of Dundee, Ninewells Hospital and Medical School, Dundee, DD1 9SY, UK

    Alastair M. Thompson

Authors
  1. Marc Sturrock
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alan J. Terry
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dimitris P. Xirodimas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alastair M. Thompson
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mark A. J. Chaplain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Sturrock.

Electronic Supplementary Material

Below are the links to the electronic supplementary material.

(MP4 1.0 MB)

(MP4 1.0 MB)

Appendix

Appendix

1.1 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.

1.2 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).

1.3 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
figure 24

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

Full size image
Fig. 25
figure 25

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

Full size image

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-012-9725-1

Keywords

Search

Navigation