Out-of-equilibrium statistical dynamics of spatial pattern generating cellular automata

Abstract

How living systems generate order from disorder is a fundamental question1-5. Metrics and ideas from physical systems have elucidated order-generating collective dynamics of mechanical, motile, and electrical living systems such as bird flocks and neuronal networks6-8. But suitable metrics and principles remain elusive for many networks of cells such as tissues that collectively generate spatial patterns via chemical signals, genetic circuits, and dynamics representable by cellular automata1,9-11. Here we reveal such principles through a statistical mechanics-type framework for cellular automata dynamics in which cells with ubiquitous genetic circuits generate spatial patterns by switching on and off each others genes with diffusing signalling molecules. Lattices of cells behave as particles stochastically rolling down a pseudo-energy landscape - defined by a spin glass-like Hamiltonian - that is shaped by \"macrostate\" functions and genetic circuits. Decreasing the pseudo-energy increases the spatial patterns orderliness. A new kinetic trapping mechanism - \"pathway trapping\" - yields metastable spatial patterns by preventing minimization of the particles pseudo-energy. Noise in cellular automata reduces the trapping, thus further increases the spatial order. We generalize our framework to lattices with multiple types of cells and signals. Our work shows that establishing statistical mechanics of computational algorithms can reveal collective dynamics of signal-processing in biological and physical networks.