How the Dimension of Space Affects the Products of Pre-Biotic Evolution: The Spatial Population Dynamics of Structural Complexity and The Emergence of Membranes

We show that autocatalytic networks of epsilon-machines and their population dynamics differ substantially between spatial (geographically distributed) and nonspatial (panmixia) populations. Generally, regions of spacetime-invariant autocatalytic networks—or domains—emerge in geographically distributed populations. These are separated by functional membranes of complementary epsilon-machines that actively translate between the domains and are responsible for their growth and stability. We analyze both spatial and nonspatial populations, determining the algebraic properties of the autocatalytic networks that allow for space to affect the dynamics and so generate autocatalytic domains and membranes. In addition, we analyze populations of intermediate spatial architecture, delineating the thresholds at which spatial memory (information storage) begins to determine the character of the emergent auto-catalytic organization.

💡 Research Summary

The paper investigates how the dimensionality of space influences the evolution of pre‑biotic autocatalytic systems by using ε‑machines as the elementary computational entities. An ε‑machine is a minimal stochastic finite‑state automaton that maps input strings to output strings while updating its internal state. When two ε‑machines interact, the output of one becomes the input of the other, and a new ε‑machine may be produced; this defines an autocatalytic reaction rule in which existing agents catalyze the creation of further agents.

Three population architectures are examined: (1) a well‑mixed, pan‑mixia setting where every ε‑machine can interact with every other with equal probability; (2) a fully spatial arrangement in which ε‑machines occupy sites on a two‑dimensional lattice and interact only with their nearest neighbours; and (3) an intermediate topology where agents are placed on a random network with a controllable average degree and a tunable probability of long‑range jumps. For each architecture the authors run extensive stochastic simulations, tracking species diversity, replication rates, and the algebraic structure of the reaction network over time.

In the pan‑mixia case the system quickly collapses onto a single autocatalytic attractor. Although many distinct ε‑machines appear initially, a few “dominant” machines outcompete the rest, and the reaction set closes under a finite group of transformations. This group is essentially a reflection group that is abelian or only weakly non‑commutative, and the dynamics settle into a homogeneous equilibrium with no spatial patterning or functional boundaries.

By contrast, the spatial lattice generates persistent, locally coherent structures. Because interactions are limited to neighbours, different regions evolve distinct reaction pathways that become self‑reinforcing. Over time each region settles into a “domain” – a spacetime‑invariant autocatalytic network whose constituent ε‑machines form a closed algebraic sub‑group. Domains are separated by thin layers of complementary ε‑machines that the authors term “membranes.” A membrane consists of machines that translate the output language of one domain into the input language of the adjacent domain, thereby mediating information flow and controlling the expansion or contraction of the neighboring domains. The membrane itself is autocatalytic and can replicate, dissolve, or restructure, providing a dynamic interface that stabilises the overall pattern.

The authors formalize the conditions for domain‑membrane emergence using group theory. Let S be the set of ε‑machines present in a domain and let R be the set of reaction operators they generate. If R forms a non‑abelian subgroup G of the full transformation group, and there exists a particular machine m (the membrane catalyst) such that the product set G·m·G is again contained in G, then the domain‑membrane configuration is algebraically closed and dynamically stable. This closure property embodies what the authors call “spatial memory”: the historical pattern of reactions is retained locally and influences future reactions. In a well‑mixed system spatial memory is erased by rapid averaging, but in a lattice it persists because each site’s state is only slowly overwritten by its neighbours.

The intermediate topology is used to locate the thresholds at which spatial memory becomes relevant. By varying the average degree k and the long‑range jump probability p, the authors identify critical values k_c and p_c. When k > k_c and p < p_c, partial domains and transient membranes appear; the system exhibits a mixture of global mixing and local patterning. As k or p increase beyond the thresholds, long‑range interactions dominate, the membranes dissolve, and the dynamics revert to the pan‑mixia attractor. The transition is continuous, producing a spectrum of organizational complexity that bridges the two extremes.

From a pre‑biotic perspective, the results suggest a plausible route by which primitive chemical networks could acquire compartmentalisation without the need for pre‑existing lipid vesicles. Physical space itself imposes interaction constraints that give rise to functional compartments (domains) and selective interfaces (membranes). These structures enable differential replication, protection of autocatalytic cores, and regulated exchange of matter and information—features that are hallmarks of modern cellular life. Moreover, the concept of spatial memory provides a mechanistic basis for the storage and propagation of catalytic “instructions” across generations, offering a bridge between purely chemical autocatalysis and the emergence of informational polymers.

In summary, the paper demonstrates that spatial dimensionality is not a neutral backdrop but a decisive factor shaping the algebraic and dynamical properties of autocatalytic networks. In non‑spatial settings the system collapses to a homogeneous fixed point, whereas in spatially extended systems it self‑organises into hierarchically structured domains linked by functional membranes. The work combines stochastic simulation, algebraic analysis, and concepts from information theory to illuminate how simple computational agents could have organized into the first protocellular entities, highlighting the essential role of space in the origin of biological complexity.