Calcium oscillation on homogeneous and heterogeneous networks of ryanodine receptor

Calcium oscillation is an important calcium homeostasis, imbalance of which is the key mechanism of initiation and progression of many major diseases. The formation and maintenance of calcium homeostasis are closely related to the spatial distribution of calcium channels. In the current paper, a theoretical framework is established by abstracting the spatial distribution of the calcium channels as a nonlinear biological complex network with calcium channels as nodes and Ca$^{2+}$ as edges. A dynamical model for a RyR is adopted to investigate the effect of spatial distribution on calcium oscillation. The mean-field model can be well reproduced from the complete graph and dense Erd\"os-R\'enyi network. The synchronization of RyRs is found important to generate a global calcium oscillation. The clique graph with a cluster structure can not produce a global oscillation due to the failure of synchronization between clusters. A more realistic geometric network is constructed in a two-dimensional plane based on the experimental information about the RyR arrangement of clusters and the frequency distribution of cluster sizes. Different from the clique graph, the global oscillation can be generated with reasonable parameters on the geometric network. The simulation also suggests that existence of small clusters and rogue RyR's plays an important role in the maintenance of global calcium oscillation through keeping synchronization between large clusters. Such results support the heterogeneous distribution of RyR's with different-size clusters, which is helpful to understand recent observations with super resolution nanoscale imaging techniques. The current theoretical framework can also be extent to investigate other phenomena in calcium signal transduction.