On estimation of diffusion coefficient based on spatio-temporal FRAP images: An inverse ill-posed problem

summary:We present the method for determination of phycobilisomes diffusivity (diffusion coefficient $D$) on thylakoid membrane from fluorescence recovery after photobleaching (FRAP) experiments. This was usually done by analytical models consisting mainly of a simple curve fitting procedure. However, analytical models need some unrealistic conditions to be supposed. Our method, based on finite difference approximation of the process governed by the Fickian diffusion equation and on the minimization of an objective function representing the disparity between the measured and simulated time-varying fluorescent particles concentration profiles, naturally accounts for experimentally measured time-varying fluorescent particles concentration profiles, naturally accounts for experimentally measured time-varying Dirichlet boundary conditions and can include a reaction term as well. The result we get is the overall (time averaged) diffusion coefficient $D$ and the sequence of diffusivities $D_j$ based on two successive fluorescence profiles in $j$-$th$ time interval. Due to the ill-posedness of our inverse problem, regularization algorithms are implemented. On the synthetic example, we illustrate the behaviour of solution depending on regularization parameter for different signal to noise ratio