Turing-instability Mechanism Underlying Branching Pattern Formation Based On Activator-inhibitor Theory

Branched structures exist in the almost all of living organisms.In particular,the mammalian lung is heavily branched structures.Two forms of branching have been identified: tip bifurcation and side branching,the switching between which occurs in the mouse lung.However,the mechanisms underlying the branching phenomenon are still not understood.A branching model based on activator-inhibitor theory has been used for the studies of branching pattern formation.Biochemical interactions based on the branching model have been investigated in detail to explain branching process.Mathematical studies focus on the dynamical behaviors of mathematical models.However,there is lack of bridge between biological systems and mathematical mechanisms.Thus,our research work aims to build the bridge.In this thesis,we use the branching model to reveal the mathematical mechanism underlying the branching pattern formation.In addition,in the research of the branching model,the parameters identification for the satisfied simulation patterns is a key problem.To solve the problem,we also propose a visual feedback framework based on the principle of feedback control.Therefore,our research work consists of the following two parts:(1)Mathematical mechanism of the branching pattern formation.By decoupling the branching model,we performed Turing instability analysis.Our results demonstrate that Turing instability exists in the growing tip of the branching pattern.The underlying Turing patterns are spot patterns which exhibit high local morphogen concentration.The high local morphogen concentration lead to the growth of branching.Moreover,there is a connection between the spot density of the spot patterns and the branching structures.In the transformation from tip bifurcation to side branching,the spot density of the spot patterns increases.Then,by the dispersion relation analysis,we show that the Turing wavelength affects the spot density of the spot patterns,and regulates the branching structures.A sufficient wavelength with oscillation is required for the tip bifurcation,while an insufficient wavelength with steadiness provides favorable conditions for side branching.(2)Parameter identification for the branching model.We provide a visual feedback framework to solve the parameter identification problem automatically in the study of branching model.Based on the principle of feedback control,the framework visualizes the simulations,extracts pattern features by deep learning method.Then the unknown model parameters are searched in parameter space by comparing the features of simulation patterns and target biological patterns.Our results prove the feasibility and effectiveness of the framework.