Simulation of reaction-diffusion equations with reaction-reaction analog circuits

In 1952, Alan Turing showed that analog reaction-diffusion equations were extremely powerful models of biological development and of distributed cellular automata. Analog circuits have been shown to accelerate the simulation of chemical reactions by many orders of magnitude, including in stochastic (noisy) cytomorphic chips, which are useful for drug-cocktail formulation and in systems medicine. However, the simulation of the partial differential equations of diffusion is expensive to architect in analog systems. Here, we show how to simulate diffusion as though it were a chemical reaction such that reaction-reaction analog systems can simulate reaction-diffusion systems. As an example, we show that the BMP-SOX9-WNT reaction-diffusion system can be simulated in analog cytomorphic integrated circuits that only have reaction circuits but no explicit diffusion circuits. Experimental data from reaction-reaction circuits show excellent agreement with MATLAB and COPASI simulations of biological models. Even cytomorphic chips with relatively sparse sampling appear to demonstrate decaying and unstable growing waves. Our work is the first step toward large-scale simulations of spatiotemporal reaction-diffusion equations.