Modeling transport of tau in an axon after a photobleaching experiment

The goal of this paper is to apply Laplace transform method to solving the problem of recovery of concentration of tau protein in an axon after a photobleaching experiment that is described in S. Konzack, E. Thies, A. Marx, E. Mandelkow, E. Mandelkow, Swimming against the tide: Mobility of the microtubule-associated protein tau in neurons, Journal of Neuroscience. 27 (2007) 9916-9927. The model accounts for two populations of tau molecules: those that are suspended in the cytoplasm, free to diffuse in the axon, and those that are bound to microtubules (MTs). The exchange of tau molecules between these two populations is modeled by first-order reactions. It is shown that governing equations for this model can be solved by Laplace transform method, although the last step of the solution, finding the inverse Laplace transform, has to be done numerically, which leads to a hybrid analytical and numerical solution technique. The numerical inversion of the Laplace transform is performed utilizing a method based on Fourier series expansion of the inverse transform.