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Modeling Transport Regulation in Gene Regulatory Networks

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Abstract

A gene regulatory network summarizes the interactions between a set of genes and regulatory gene products. These interactions include transcriptional regulation, protein activity regulation, and regulation of the transport of proteins between cellular compartments. DSGRN is a network modeling approach that builds on traditions of discrete-time Boolean models and continuous-time switching system models. When all interactions are transcriptional, DSGRN uses a combinatorial approximation to describe the entire range of dynamics that is compatible with network structure. Here we present an extension of the DGSRN approach to transport regulation across a boundary between compartments, such as a cellular membrane. We illustrate our approach by searching a model of the p53-Mdm2 network for the potential to admit two experimentally observed distinct stable periodic cycles.

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Any code used in this work is available upon request.

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Acknowledgements

The work of B.C. and T. G. was partially supported by NSF grant DMS-1839299 and grant DARPA FA8750-17-C-0054. The work of W.D and T.G. was additionally supported by an grant NIH 5R01GM126555-01. We acknowledge the Indigenous nations and peoples who are the traditional owners and caretakers of the land on which this work was undertaken at the Montana State University.

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Authors and Affiliations

  1. Department of Mathematics, University of Nevada, Reno, NV, USA

    Erika Fox

  2. Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA

    Bree Cummins, William Duncan & Tomáš Gedeon

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  1. Erika Fox
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  2. Bree Cummins
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  3. William Duncan
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  4. Tomáš Gedeon
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Correspondence to Tomáš Gedeon.

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Appendix A Representative ODE Simulations

Appendix A Representative ODE Simulations

Here we present representative simulations of continuous ODE system (15) in Figs. 10 and 11. Both figures show simulations at a parameter sampled from the parameter node \(M_1^F\), the parameter node with a middle black wall with the front unfolding. The parameter used in Fig. 10 results in a large attracting periodic orbit, while the parameter in Fig. 11 results in a small attracting periodic orbit. The parameters used in each figure are given in Table 7.

Fig. 10
figure 10

Representative large periodic orbit simulation. A and B: Projections of solutions to (15) onto the Mc-Mn and Mc-P planes, respectively. Trajectories are shown for 10 different initial conditions. After a brief transient, all trajectories converge to the periodic orbit. C–E: Time series plots for one of the trajectories shown in (A) and (B). The orbit is large because P crosses both of its thresholds

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Fig. 11
figure 11

Representative small periodic orbit simulation. A and B: Projections of solutions to (15) onto the Mc-Mn and Mc-P planes, respectively. After a brief transient, all orbits converge to the periodic orbit. C–E: Time series plots for one of the trajectories shown in (A) and (B). The orbit is small because P crosses only one of its thresholds

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Table 7 Parameters of (15) used for the simulations depicted in Figs. 10 and 11. Note that the small and large parameters only differ in the threshold values \(\zeta \)
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Fox, E., Cummins, B., Duncan, W. et al. Modeling Transport Regulation in Gene Regulatory Networks. Bull Math Biol 84, 89 (2022). https://doi.org/10.1007/s11538-022-01035-1

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