# American Institute of Mathematical Sciences

October  2016, 21(8): 2551-2566. doi: 10.3934/dcdsb.2016060

## Intracellular protein dynamics as a mathematical problem

 1 Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa 2 INRIA Paris, ANGE Project-Team, 75589 Paris Cedex 12, France 3 ICM, University of Warsaw, ul. Pawińskiego 5a, 02-106 Warsaw, Poland

Received  October 2015 Revised  February 2016 Published  September 2016

The present paper provides a mathematical analysis of the model of intracellular protein dynamics proposed in [14]. The model describes protein and mRNA transport inside a cell and takes into account diffusion in the nucleus and cytoplasm as well as active transport of protein molecules along microtubules in the cytoplasm. The model is a complex system of nonlinear PDEs with appropriate boundary conditions. The model reproduces, at least in numerical simulations, the oscillatory changes in protein concentration observed in the experimental data. To our knowledge this is the first paper that, in the multidimensional case, deals with a rigorous mathematical analysis of a model of intracellular dynamics with active transport on microtubules. In particular, in the present paper, we prove the existence and uniqueness result for the model in arbitrary space dimension. The model may be adapted to other signaling pathways.
Citation: Mirosław Lachowicz, Martin Parisot, Zuzanna Szymańska. Intracellular protein dynamics as a mathematical problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2551-2566. doi: 10.3934/dcdsb.2016060
##### References:
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##### References:
 [1] S. Agrawal, C. Archer and D. V. Schaffer, Computational models of the notch network elucidate mechanisms of context-dependent signaling, PLoS Comput. Biol., 5 (2009), e1000390. doi: 10.1371/journal.pcbi.1000390.  Google Scholar [2] B. Alberts, D. Bray, K. Hopkin, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Essential Cell Biology, $2^{nd}$ edition, Garland Science/Taylor & Francis Group, 2004. Google Scholar [3] A. Cangiani and R. Natalini, A spatial model of cellular molecular trafficking including active transport along microtubules, J. Theor. Biol., 267 (2010), 614-625. doi: 10.1016/j.jtbi.2010.08.017.  Google Scholar [4] M. A. J. Chaplain, M. Ptashnyk and M. Sturrock, Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations, Math. Models Methods Appl. Sci., 25 (2015), 1179-1215. doi: 10.1142/S021820251550030X.  Google Scholar [5] M. A. J. Chaplain, M. Sturrock and A. J. Terry, Spatio-temporal modelling of intracellular signalling pathways: Transcription factors, negative feedback systems and oscillations, in New Challenges for Cancer Systems Biomedicine (eds. A. d'Onofrio, P. Cerrai and A. Gandolfi), SIMAI Springer Series, (2012), 55-82. doi: 10.1007/978-88-470-2571-4_4.  Google Scholar [6] J. Claus, E. Friedmann, U. Klingmüller, R. Rannacher and T. Szekeres, Spatial aspects in the SMAD signaling pathway, J Math Biol., 67 (2013), 1171-1197. doi: 10.1007/s00285-012-0574-1.  Google Scholar [7] G. Craciun, B. Aguda and A. Friedman, Mathematical analysis of a modular network coordinating the cell cycle and apoptosis, Math Biosci Eng., 2 (2005), 473-485. doi: 10.3934/mbe.2005.2.473.  Google Scholar [8] J. Eliaš and J. Clairambault, Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples, Comput Struct Biotechnol J., 10 (2014), 12-22. Google Scholar [9] E. Friedmann, R. Neumann and R. Rannacher, Well-posedness of a linear spatio-temporal model of the JAK2/STAT5 signaling pathway, Comm. Math. Analysis, 15 (2013), 76-102.  Google Scholar [10] H. Momiji and N. A. M. Monk, Dissecting the dynamics of the Hes1 genetic oscillator, J. Theor. Biol., 254 (2008), 784-798. doi: 10.1016/j.jtbi.2008.07.013.  Google Scholar [11] N. A. M. Monk, Oscillatory expression of Hes1, p53, and NF-k B driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. doi: 10.1016/S0960-9822(03)00494-9.  Google Scholar [12] M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Spatiotemporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theor. Biol., 273 (2011), 15-31. doi: 10.1016/j.jtbi.2010.12.016.  Google Scholar [13] M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, 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 (2012), 1531-1579. doi: 10.1007/s11538-012-9725-1.  Google Scholar [14] Z. Szymańska, M. Parisot and M. Lachowicz, Mathematical modeling of the intracellular protein dynamics: The importance of active transport along microtubules, J. Theor. Biol., 363 (2014), 118-128. doi: 10.1016/j.jtbi.2014.07.022.  Google Scholar
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