Simulated Autopoiesis in Liquid Automata

TL;DR

The paper addresses modeling autopoiesis in a physically grounded, continuous-space setting rather than a fixed grid by introducing Liquid Automata. It presents a minimal autopoietic simulation in which a catalyst, substrate, and link particles interact under three collision-based rules $K + 2S \rightarrow K + L$, $L^{n} + L \rightarrow L^{n+1}$, and $L \rightarrow 2S$ within a 2D Box2D environment, driven by Brownian motion. The study demonstrates both organisational closure (production of boundary-forming links) and structural closure (a boundary enclosing the catalyst) and provides quantitative analyses of substrate–link–bond dynamics, including anti-correlated substrate-link behavior and phase-space attractor behavior. This framework enables exploration of more complex metabolisms and chemical networks, offering a path toward studying autopoietic self-organization in living machines using continuous-space simulations.

Abstract

We present a novel form of Liquid Automata, using this to simulate autopoiesis, whereby living machines self-organise in the physical realm. This simulation is based on an earlier Cellular Automaton described by Francisco Varela. The basis of Liquid Automata is a particle simulation with additional rules about how particles are transformed on collision with other particles. Unlike cellular automata, there is no fixed grid or time-step, only particles moving about and colliding with each other in a continuous space/time.

Figures (5)

• Figure 1: Liquid automaton showing a boundary (blue links) forming around the catalyst (triangle), distinguishing self from non-self. The catalyst transforms the substrate (circles) into its structural building blocks (squares).
• Figure 2: A simulation of autopoiesis using a discrete time cellular automaton on a rectangular grid, based on Varela's original algorithm. In (a) a pair of substrate (circles) are transformed into a single link (squared circle) by the catalyst (asterisk). By (c) we see the first bonds forming, then in (i) these finally form a closed boundary around the catalyst.
• Figure 3: Absolute quantities of substrate, links, and bonds. The quantity of substrate drops rapidly at the start as they collide with the exposed catalyst. Conversely, the number of links increases as they are produced in the reaction of the substrate with the catalyst. Bonds form as links are produced.
• Figure 4: Rates of change of substrate, links, and bonds based on a 20 second moving average. The relationship between rates of change for substrate, $\Delta S$, and links, $\Delta L$, is $\Delta S = -2\Delta L$.
• Figure 5: Phase plot of the average rate of change in the number of links versus bonds. The origin is an attractor, with larger vectors at greater distances from the origin.