Morphogenesis explains the development of both symmetrical and unsymmetrical patterns in an organismal growth. The mathematical foundation of morphogenesis expands to circuitry with implementations to synthetic biology. This paper addresses issues related to the circuitry implementation of reaction-diffusion (RD) systems in morphogenesis.

The author explains two RD models of morphogenesis using differential equations using Turing’s chemical basis of morphogenesis. According to the author, his “extension of Turing’s mathematical framework for chemical morphogenesis to its circuit basis is an anticipation of the continued advancement of engineered molecular systems to behave as computational circuits.” He defines the system as a square lattice, which utilizes logic gates as reactions covering a 2D space defined within periodic boundary conditions. The author further implements spatial dynamics representing cellular automata in a diffusion-like system.

The author describes the dynamics of his model, which comprises Turing patterns and mixed-Turing patterns, including simulations such as stationary long wavelength, oscillatory long wavelength, stationary short wavelength, oscillatory short wavelength, stationary finite wavelength, oscillatory finite wavelength, mixed Turing patterns, and metastable patterns. The author shows aggregate results with “50 random initial conditions for 10,000 time steps on a 50 by 50 grid.” He further obtains “the result of all pairs of logic gates acting on morphogens 1 and 2, for 4 different pairings of diffusion rates: slow-slow, slow-fast, fast-slow, and fast-fast[, where] the radius for slow diffusion is 1, while the radius for fast diffusion is 10.”

This paper is an interesting read for those working in the area of theoretical computer science, mainly theoretical computer scientists and doctoral students. The motivation of extending Turing’s mathematical framework makes this paper worth reading, and extends “the original linearized analysis of differential equations to [other circuit basis] studies.”