  Motivated by a range of biological applications related to the transport of
molecules in cells, we present a modular framework to treat first-passage
problems for diffusion in partitioned spaces. The spatial domains can differ
with respect to their diffusivity, geometry, and dimensionality, but can also
refer to transport modes alternating between diffusive, driven, or anomalous
motion. The approach relies on a coarse-graining of the motion by dissecting
the trajectories on domain boundaries or when the mode of transport changes,
yielding a small set of states. The time evolution of the reduced model follows
a generalized master equation (GME) for non-Markovian jump processes; the GME
takes the form of a set of linear integro-differential equations in the
occupation probabilities of the states and the corresponding probability
fluxes. Further building blocks of the model are partial first-passage time
(FPT) densities, which encode the transport behavior in each domain or state.
After an outline of the general framework for multiple domains, the approach is
exemplified and validated for a target search problem with two domains in one-
and three-dimensional space, first by exactly reproducing known results for an
artificially divided, homogeneous space, and second by considering the
situation of domains with distinct diffusivities. Analytical solutions for the
FPT densities are given in Laplace domain and are complemented by numerical
backtransforms yielding FPT densities over many decades in time, confirming
that the geometry and heterogeneity of the space can introduce additional
characteristic time scales.
