Calcium wave propagation in chains of endothelial cells with nonlinear reaction dynamics: Green's function approach

We present a Green's function-based perturbative approach to solving nonlinear reaction-diffusion problems in networks of endothelial cells. We focus on a single component (Ca2⁺), piecewise nonlinear model of endoplasmic calcium dynamics and trans-membrane diffusion. The decoupling between nonlinear reaction dynamics and the linear diffusion enables the calculation of the diffusion part of the Green's function for network of cells with nontrivial topologies. We verify analytically and then numerically that our approach leads to the known transition from propagation of calcium front to failure of propagation when the diffusion rate is varied relative to the reaction rates. We then derive the Green's function for a semi-infinite chain of cells with various boundary conditions. We show that the calcium dynamics of cells in the vicinity of the end of the semi-infinite chain is strongly dependent on the boundary conditions. The behavior of the semi-infinite chain with absorbing boundary conditions, a simple model of a multicellular structure with an end in contact with the extracellular matrix, suggests behavioral differentiation between cells at the end and cells embedded within the chain.