We discuss the mathematical modelling of cellular signal transduction, that is the mechanism of molecular signalling impinging on biological cells. We propose the first model taking into account for the active transport of molecules along microtubules [3]. The crossing of cellular membranes is modelled by the law of mass transfer through semipermeable membranes.

The full cellular model results into a system of coupled semi-linear parabolic and hyperbolic PDEs in a multi-compartment setting whose analysis is quite challenging both from the analytical and numerical view point.

We propose and analyse a generalisation of the Interior-Penalty Discontinuous Galerkin (IPDG) method for tackling the problem numerically [1, 2]. The IPDG method appears to be a natural candidate for the solution of mass trans- fer problems both due to its good stability properties in the discretisation of the purely advective equation modelling active transport of molecules along microtubules and due to the ease of discretisation of the membranes model.

We shall present numerical results based on real data parameters confirming the experimental evidence that active transport can enhance the signalling mechanism [4].

Joint work with R. Natalini, E. H. Georgoulis, and M. Jensen.

[1]  A. Cangiani, E. H. Georgoulis, and M. Jensen. Discontinuous Galerkin methods for convection-diffusion problems modelling mass transfer through semipermeable membranes. Proceedings of the Congress on Numerical Methods in Engineering, Coimbra, 2011.

[2]  A. Cangiani, E. H. Georgoulis, and M. Jensen. Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes. SIAM J. on Numerical Analysis, 51(5):2911–2934, 2013.

[3]  A. Cangiani, R. Natalini. A spatial model of cellular molecular trafficking including active transport along microtubules. Journal of Theoretical Biology, 267, 614–625, 2010.

[4]  D. M. Roth, G. W. Moseley, D. Glover, C. W. Pouton, and D. A. Jans. A microtubule-facilitated nuclear import pathway for cancer regulatory proteins. Traffic, 8:673–686(14), June 2007.