Modeling Of Branching Parallel System Based On Activator-inhibitor Theory And Study Of The Underlying Mechanism

Biological pattern formation studies based on mathematical biology knowledge started from the Reaction-Diffusion theory first advanced by Turing in1952. Based on Turing’s work, Gierer and Meinhardt proposed the classic Activator-Inhibitor model, which can be applied to the pattern formation in tissue differentiation and has lots of biological application now. Branching structures are common in almost every higher organism. A model based on the Activator-Inhibitor theory was proposed by Meinhardt in which a few coupled biochemical reactions were able to generate the branching pattern. Among the undifferentiated cells, a local peak of activator is formed by autocatalysis and lateral inhibition. The activator peak triggers the differentiation of the cell at that location. Due to changes in metabolism, the differentiated cell repels the activator peak and drives it to a neighboring cell which then also differentiates. Wandering concentration peaks of the activator leave behind trails of differentiated cells which form the branching structure.The branching parallel system, such as blood vessels and nerves, lung vascular and airways, the two types branch parallel to each other in well balanced neighborhood. Such parallel systems are important for the normal physiological function of the organism. Take the blood vessels and nerves as an example, nerves often run along larger blood vessels, reflecting their need for oxygen and nutrients, as well as their physiological control of vaso-constriction and-dilation. What’s fascinated us is how such beautiful parallel systems are generated? Biological experiments has found that vascular ablation in transgenic mouse led to an overall flat lung morphology. Specifically, branching events requiring a rotation to change the branching plane were selectively affected. Moreover, Mutations that eliminate peripheral sensory nerves in embryonic mouse limb skin prevent proper arteriogenesis while those that disorganize the nerves maintain the alignment of arteries with misrouted axons. What’s the explanation for these observations? Thus, this paper carried out the study of mathematical modeling and simulation of the branching parallel system based on Meinhardt’s branching model to investigate the underlying mechanism.Firstly, study the mechanism of branching morphogenesis through numerical simulation analysis of the branching model unified with molecular and cellular mechanisms found in biological observations.In the parallel system, the two types of branches tend to attract each other while the branches wouldn’t mix together. Through simulation, we have found that the’ substrate plays the directional drive role and branch filaments will grow toward the higher substrate concentration region. Then, it is hypothesized that one type of branch in the parallel system can activate or enhance the production of the substrate necessary for the other type to attract it to grow alongside. While there has to exist a local repulsion mechanism to ensure the two branches wouldn’t grow together.Supposing the mutual lateral attraction and local repulsion effect are led by inhibitor and activator respectively, a mathematical model was built and parallel growing phenomenon was attained by numerical simulation.Finally, using the model, we got the pulmonary vascular and airway parallel branching system in the3dimension simulation. Also, simulations were carried out to study the roles of vascular played in the formation of the stereotyped airway branching structure. Furthermore, the same model can get the consistent results observed in the blood vessels and nerves system in embryonic mouse limb skin. These simulation results demonstrated the validity of the model proposed by this paper to describe the underlying mechanism of the parallel system.