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The Role of Dimerisation and Nuclear Transport in the Hes1 Gene Regulatory Network

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Abstract

Hes1 is a member of the family of basic helix-loop-helix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anti-cancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy.

In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN.

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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” and of the National Institute of Health under Award Number 1R01EB014877-01. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Health. We also acknowledge Brian Drawert for his contributions to the infrastructure facilitating URDME simulations on clusters.

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Authors and Affiliations

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

    Marc Sturrock, Sahar Aldakheel & Mark A. J. Chaplain

  2. University of California, Santa Barbara, CA, 93106, USA

    Andreas Hellander & Linda Petzold

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  1. Marc Sturrock
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  2. Andreas Hellander
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  3. Sahar Aldakheel
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Correspondence to Andreas Hellander.

Appendix

Appendix

Parameter Sensitivity Analysis Figures for Model A

See Figs. 13–17.

Fig. 13
figure 13

Plots showing the effect of varying parameter k pore (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as k pore is varied. The middle plot shows how the mean period varies as k pore is varied. The right plot shows how the CV varies for mRNA and protein as k pore is varied. As can be seen from each plot, varying k pore has little effect on the system (Color figure online)

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Fig. 14
figure 14

Plots showing the effect of varying parameter k rel (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as k rel is varied. The middle plot shows how the mean period varies as k pore is varied. The right plot shows how the CV varies for mRNA and protein as k rel is varied. As can be seen from each plot, varying k rel has little effect on the system (Color figure online)

Full size image
Fig. 15
figure 15

Plots showing the effect of varying the number of NPCs (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as the number of NPCs is varied. The middle plot shows how the mean period varies as the number of NPCs is varied. The right plot shows how the CV varies for mRNA and protein as the number of NPCs is varied. As can be seen from each plot, varying the number of NPCs has little effect on the system (Color figure online)

Full size image
Fig. 16
figure 16

Plots showing the effect of varying parameter β 1 (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as β 1 is varied. The middle plot shows how the mean period varies as β 1 is varied. The right plot shows how the CV varies for mRNA and protein as β 1 is varied. As can be seen from the plots, varying β 1 has an impact on the mean period. The plots reveal that increasing β 1 increases the mean period (Color figure online)

Full size image
Fig. 17
figure 17

Plots showing the effect of varying parameter β 2 (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as β 2 is varied. The middle plot shows how the mean period varies as β 2 is varied. The right plot shows how the CV varies for mRNA and protein as β 2 is varied. As can be seen from the plots, varying β 2 has an impact on the mean period. The plots reveal that increasing β 2 decreases the mean period (Color figure online)

Full size image

Parameter Sensitivity Analysis Figures for Model B

See Figs. 18–22.

Fig. 18
figure 18

Plots showing the effect of varying parameter k pore (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as k pore is varied. The middle plot shows how the mean period varies as k pore is varied. The right plot shows how the CV varies for mRNA and protein as k pore is varied. As can be seen from each plot, varying k pore has little effect on the system (Color figure online)

Full size image
Fig. 19
figure 19

Plots showing the effect of varying parameter k rel (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as k rel is varied. The middle plot shows how the mean period varies as k rel is varied. The right plot shows how the CV varies for mRNA and protein as k rel is varied. As can be seen from each plot, varying k rel has little effect on the system (Color figure online)

Full size image
Fig. 20
figure 20

Plots showing the effect of varying the number of NPCs (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as the number of NPCs is varied. The middle plot shows how the mean period varies as the number of NPCs is varied. The right plot shows how the CV varies for mRNA and protein as the number of NPCs is varied. As can be seen from each plot, varying the number of NPCs has little effect on the system (Color figure online)

Full size image
Fig. 21
figure 21

Plots showing the effect of varying parameter β 1 (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as β 1 is varied. The middle plot shows how the mean period varies as β 1 is varied. The right plot shows how the CV varies for mRNA and protein as β 1 is varied. As can be seen from the plots, varying β 1 has a large impact on the mean period. The plots show that increasing β 1 increases the mean period (Color figure online)

Full size image
Fig. 22
figure 22

Plots showing the effect of varying parameter β 2 (all other parameters as per Table 1). The left plot shows the fraction of time the system spends oscillating with a period in the range 0–120 minutes, 120–300 minutes and 300–PE as β 2 is varied. The middle plot shows how the mean period varies as β 2 is varied. The right plot shows how the CV varies for mRNA and protein as β 2 is varied. As can be seen from the plots, varying β 2 has a large impact on the mean period. The plots show that increasing β 2 decreases the mean period (Color figure online)

Full size image

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Sturrock, M., Hellander, A., Aldakheel, S. et al. The Role of Dimerisation and Nuclear Transport in the Hes1 Gene Regulatory Network. Bull Math Biol 76, 766–798 (2014). https://doi.org/10.1007/s11538-013-9842-5

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