This paper studies synchronization mechanisms for networks of biological rhythms. The network is made up of compartments (e.g., E. coli in a cell culture) which consist of heterogeneous subsystems (e.g., reaction pathways) interconnected by internal signaling. The compartments are, in turn, interrelated through common medium. Based on this structural foundation, synchronization conditions are provided from operator-theoretic view of point, which involve the input-output properties of individual compartment together with topological structure of underlying networks. Furthermore, as an additional goal, the paper also provides synchronization criterion for the networks modeled in the formalism of state-space. Finally, the proposed theory finds bio-chemical applications in the networks of toggle switches and repressilators, respectively.