Mathematical analysis of a free-boundary model for lung branching morphogenesis

Lung branching morphogenesis has been widely studied in the field of developmental biology. Lung airway trees consist of relatively regular-sized distal branches, but how this regular branched pattern is formed is not well understood. In the present study, we undertake a detailed mathematical analysis of the model proposed in Hartmann & Miura (2006), which numerically captures branching morphogenesis of the simplest possible experimental system in vitro. We investigate analytically the stability of 1D travelling waves with respect to periodic perturbations in two dimensions. This linear stability analysis leads to the so-called dispersion relations, predicting that a certain representative length dominates in this model. As the analytical analysis is restricted to travelling waves, we generalize the linear analysis to any 1D solution by numerical simulations. Both results predict how the representative lengths will change by experimentally changing specific parameters. Finally, we discuss the importance of the analytical results from a biological point of view and propose an experimental scheme for a quantitative comparison between experiments and theory.