  We investigate diffusive search on planar networks, motivated by tubular
organelle networks in cell biology that contain molecules searching for
reaction partners and binding sites. Exact calculation of the diffusive mean
first-passage time on a spatial network is used to characterize the typical
search time as a function of network connectivity. We find that global
structural properties --- the total edge length and number of loops --- are
sufficient to largely determine network exploration times for a variety of both
synthetic planar networks and organelle morphologies extracted from living
cells. For synthetic networks on a lattice, we predict the search time
dependence on these global structural parameters by connecting with percolation
theory, providing a bridge from irregular real-world networks to a simpler
physical model. The dependence of search time on global network structural
properties suggests that network architecture can be designed for efficient
search without controlling the precise arrangement of connections.
Specifically, increasing the number of loops substantially decreases search
times, pointing to a potential physical mechanism for regulating reaction rates
within organelle network structures.
