Finite Element Modeling of Turing Pattern Formation: An Application to Appearance and Location of Dentinal Tubules

Abstract We present a mathematical model of the reaction–diffusion equations for the simulation of the formation of the dentinal tubules. The model takes into account the role of BMP2, Noggin, and DLX3. BMP2 drives the differentiation of odontoblasts and the formation of the dentinal tubules, whereas Noggin interacts with it to form a negative feedback loop. The proposed loop (between BMP2 and Noggin) can form Turing patterns (stable in time and unstable in space) that show the regular spatial location of odontoblasts and, hence, of the dentinal tubules. DLX3 can modify the parameters of the reaction–diffusion equations to change the usual solution of Turing patterns and obtain abnormal patterns of the location of the odontoblasts. We show the necessary conditions for Turing pattern formation and its numerical implementation by a finite elements framework for nonlinear differential equations.