Spatial patterning and selection: How the environment shapes molecular complexity

Abstract

Assembly theory predicts that a distinguishing signature of life is its ability to produce complex molecules in abundance, opening new possibilities for life detection. Experimental validation of this approach has so far relied on abiotic controls like meteoritic material, or simple, well-mixed chemical systems. However, decades of research in self-organization have shown that spatial patterning can foster dynamical complexity. This raises the possibility that systems with spatial patterns might promote abiotic formation of molecules with higher-than-expected assembly indices—a measure defined within assembly theory to assess structural complexity—potentially leading to false positives in life detection approaches based on this framework. To probe this possibility, we compare how assembly indices in a simplified in silico chemistry scale relative to those in molecular space, and then ask how spatial organization shapes their distribution in our model. Our findings reveal that transport factors, such as diffusion, significantly affect the distribution of chemical species within a system. Additionally, spatial topology markedly shapes the distribution of assembly indices: while it does not enable arbitrary complexity, ordered lattices in particular can shift the assembly indices of abiotic chemistry upward. Moreover, we demonstrate that diffusion can impede the formation and detectable upper limit of these high assembly index molecules, bearing important implications for life detection experiments and astrobiological missions.

Significance Statement.

The search for life beyond Earth increasingly relies on molecular complexity as a potential biosignature. Assembly theory provides a framework for quantifying such complexity, but most prebiotic controls tested to date assume well-mixed, homogeneous conditions. Decades of complex systems research demonstrate that spatial patterning can drive self-organization, raising the question of how these features shape molecular complexity. We show that spatial patterning can shift the distribution of molecular complexity but these shifts are fundamentally constrained: the conditions most likely to enhance complexity also yield the lowest detectable amounts of complex molecules. These results highlight the selectivity required to produce high assembly index molecules in detectable abundance and illuminate limits and opportunities for confident life detection in diverse solar system environments.

Introduction

A central objective of astrobiological missions is to reliably distinguish life from nonlife. Toward this goal, recent research in assembly theory provides empirical evidence that only living systems generate sufficiently complex molecules in detectable quantities. This complexity can be quantified by their assembly index, offering a promising basis for reliable life detection (1). However, the abiotic controls referenced in this study are limited to two sources: (i) meteoritic material and (ii) laboratory experiments conducted in well-mixed systems. Notably, spatial heterogeneity—i.e. departures from a well-mixed, spatially homogeneous state—is prevalent in diverse natural and artificial environments, including porous materials and mineral surfaces, and has been shown to catalyze dynamical self-organization, often termed “complexity.” However, this body of work relies on a wide range of models of self-organization that differ qualitatively from one another, and its results do not readily transfer to the specific question we address here: how spatial heterogeneity shapes molecular complexity.

Whether spatial heterogeneity enhances or impedes the formation of high assembly index molecules thus remains a critical open question. While assembly theory predicts that only living systems will generate molecules of high assembly index in large quantities (e.g. $>\mathrm{10,000}$ identical copies) (1, 2), abiotic systems may narrow the range of possibilities for organic chemistry, depending on the specific conditions. This would manifest as a limited type of selection imposed by the physical environment alone—one that could operate prior to biological processes, making it crucial to investigate its extent. To address this, we have developed a numerical model aimed at testing the hypothesis that spatial patterning may induce the formation of higher molecular assembly indices in the absence of living systems. Our findings aim to inform future empirical research, particularly in minimizing false positives in life detection and fostering conditions conducive to the emergence of de novo life.

The assembly index (AI), a central measure in assembly theory, quantifies complexity as the minimal number of joining operations required to construct an object from basic components. This metric allows for the structural comparison of diverse objects made from common building blocks. For covalently bonded molecules, the AI—also known as molecular assembly index (MA) in the context of molecules—is observable via measurements based on mass spectrometry as well as spectroscopic experiments, enabling its empirical determination (1, 3). In unguided systems where selection does not operate, structurally complex objects requiring more joining operations are much harder to produce in large quantities. As an illustrative limit case, purely random processes would not be expected to produce objects requiring more than a few joining operations in meaningful quantities. Assembly theory thus posits that both the copy number and structural complexity, measured using the assembly index, are required to evaluate the extent of a selective process (2).

Empirical tests of the theory are consistent with the prediction that only living systems are capable of generating large quantities of high-AI molecules, but isolating completely abiotic samples to further test this hypothesis proves challenging (1). This difficulty arises because virtually every Earth environment has experienced some form of biological influence, and even ancient materials often display signatures potentially linked to life (4–6). Moreover, samples from beyond Earth are quickly colonized by life, even under strict protocols (7). In their study, Marshall et al. (1) utilized meteoritic material as an abiotic control alongside products from canonical prebiotic experiments. However, replicating the diverse chemical conditions characteristic of planetary surfaces in laboratory settings remains a significant challenge (8, 9). A critical factor contributing to this challenge is the inherent spatial heterogeneity observed in natural systems, which contrasts with the uniform conditions typical of laboratory and numerical experiments.

No matter the conditions of diverse planetary environments suitable for life, we expect they will at least feature rocks, either exposed on the surface or at the ocean interface. Rocks are inherently porous materials and feature numerous microcompartments (10, 11). This porosity creates spatial heterogeneity, or patterning, a property complexity science has identified as a fundamental mechanism for fostering dynamical self-organization. This connection is well established: the interplay between nonlinear dynamics and diffusion has long been known to produce spatial patterns, notably in biological systems (12), and has since been examined extensively in computational biology and game theory, revealing how diverse processes shape such patterns (13, 14). These insights carry significant implications for origins-of-life research (15, 16): computational studies suggest that spiral waves may aid prebiotic evolution by impeding parasites (17), and that spatial self-organization can support RNA-like replicators (18).

Beyond these computational results, laboratory work grounds the role of mineral surfaces in real prebiotic chemistry: such surfaces have been shown to influence polymerization. Specifically, serpentine environments have been observed to favor glycine- and phenylalanine-rich fragments, leading to the formation of longer oligomers (19). Similarly, analyses of asteroid samples, such as those from Ryugu, underscore the role of mineral surfaces in the spatial distribution of organic matter (20). These laboratory studies reveal that minerals provide both catalytic activity and nontrivial microcompartments, each potentially affecting the resultant product distribution. However, disentangling the respective impacts of catalysis and micro-compartmentalization remains challenging and the isolated effect of spatial patterning on product distribution is not well understood. Recent findings suggest that geocompartments, even in the absence of catalytic activity, can selectively segregate prebiotic building blocks, a process akin to thermophoresis (11).

These results leave open whether spatial structure shapes not only which molecules accumulate but how structurally complex they become—whether spatial patterning alone, independent of catalysis or biological selection, can shift the distribution of molecular assembly indices. Because porous environments are pervasive in prebiotic chemistry, we address this with a spatially structured model, varying its transport properties—i.e. inflow, and diffusion—to ask how they affect both the emergence and the detectability of high-assembly-index molecules across settings ranging from hydrothermal vents (high inflow and low diffusion) (21) to tide pools (low inflow and high diffusion). Rather than reproducing any single environment, we aim to delineate general conditions under which abiotic processes yield such molecules, and thereby to inform the thresholds needed to limit false positives in life detection.

To investigate whether spatial structure alone can drive molecular complexity, we employ an in silico chemical model that simplifies the intricacy of real chemical systems to basic operations of integer addition and subtraction. This means that we represent all molecules as integers, and reactions between them as either addition of two integers, or decomposition of an integer into smaller ones (see Table 1). Liu and Sumpter (22) used a similar approach to model reactions such as the citric acid cycle and the formose reaction, revealing that certain chemical systems exhibit collective catalysis through intermediates, while others demonstrate full self-replication. These results suggest that such catalytic behaviors commonly emerge in random artificial chemistries, consistent with previous studies (23).

Table 1.

Overview of reactions in the artificial chemistry system.

Reaction type	Reaction	Propensity
Inflow${ }^{\mathrm{a}}$	$\varnothing \stackrel{I}{\to }A$	I
Diffusion	$C_i\stackrel{k_d}{\to }{C}_j$	$k_d[C_i]$
Outflow	$C_i\stackrel{k_o}{\to }\varnothing$	$k_o[C_i]$
Forward reaction	$A+B\stackrel{k_f}{\to }C$	$k_f[A][B]$
Backward reaction	$C\stackrel{k_b}{\to }A+B$	$k_b[C]$

Designated reactors receive an inflow of I molecules per unit time. Molecules diffuse to neighboring chemostats at a rate of $k_d$, while outflow occurs at a rate of $k_o$. Constructive (forward) reactions proceed at a rate of $k_f$, and destructive (backward) reactions occur at a rate of $k_b$, independent of molecule size.

${ }^{\mathrm{a}}$ Inflow is treated deterministically, whereas all other reactions occur stochastically as Poisson processes with the listed propensities (see text).

The present model retains two fundamental characteristics inherent in real chemical systems: (i) composition and (ii) open-endedness. Composition refers to the ability of objects to combine and form new entities, with decomposition serving as the reverse process. Open-endedness signifies the system’s capacity to generate an unlimited number of unique objects—analogous to the infinite range of positive integers and the boundless diversity of chemical compounds. In the context of this model, open-endedness implies that there is no upper limit to the integers that can be formed, and thus no corresponding upper limit to the assembly index of objects that can be generated. We apply assembly theory to bridge computational (in silico) models and experimental (in vitro) research. Specifically, we use the AI as a complexity measure for individual molecules because of its direct relevance to laboratory and field measurements. Importantly, real chemical space is far richer than the abstract integer-based model; for example, many distinct molecules can share the same number of atoms or bonds. This means that the assembly indices computed here and the assembly indices of covalent molecules may represent distinct scales, and must be interpreted carefully in that context (see Methods section).

We model the medium as a network of interconnected compartments—for porous media, a network of nanopores—which lets us apply network theory (24, 25). Here, the nodes (vertices) of a directed graph represent the nanopores and its edges (links) the channels connecting them within the porous medium. This network representation then provides a basis for examining how the topology of the medium shapes molecular assembly indices.

Because environmental conditions shape the biosignatures life detection relies on, a critical question is whether spatial heterogeneity alone—absent any biology—enhances or inhibits the formation of higher assembly index molecules. Simulations of our chemistry model show that environmental topology and transport dynamics, particularly diffusion, substantially influence molecular complexity within abiotic systems. Structured network topologies, such as regular lattices, promote the formation of higher assembly index molecules, whereas randomized configurations—serving as a control—tend to inhibit them. However, we find that this effect is most pronounced under the conditions less likely to produce higher assembly index molecules in the first place.

These findings may help refine life detection strategies by identifying abiotic conditions capable of modulating the distribution of assembly indices in the absence of life—knowledge that is crucial for appropriately parameterizing life detection methods. Furthermore, characterizing the processes by which the complexity of abiotic systems increase may also yield insights into the de novo synthesis of life.

Methods

Integer calculus

The model we developed comprises a set of chemostat reactors containing positive integers $i\in \{1,2,3,\dots \}$, each present with copy number $n_i$ (Fig. 1A). Each integer can participate in five types of transformations: inflow, forward reactions (synthesis), backward reactions (decomposition), diffusion (movement to neighboring reactors), and outflow (out of the system), as detailed in Table 1. Synthesis reactions follow the rules of integer addition, while decomposition represents the reverse process. For example, the integer 5 can be formed by the addition of 2 and 3, and can decompose back into these two integers, or into others such as 1 and 4. There is no limit on the highest possible integer, and the value reached depends on the dynamics of the system.

Figure 1.

A) Diagram depicting the model’s reactors, which contain integers capable of undergoing five transformation types: inflow, diffusion, outflow, synthesis, and decomposition (see Table 1). B) Representative final integer distribution across the reactor network after $T={10}^5$ time steps; the horizontal axis indicates integer values and the vertical axis indicates their corresponding copy numbers. C) Overview of the model and simulation outputs, illustrating the reactor network and example per-reactor copy number time series in a representative simulation. The simulation framework includes an ensemble of reactors (boxes) interconnected in a specific topology (e.g. regular lattice). This setup allows bidirectional diffusion between neighboring reactors. Numerical integration generate copy number time-series for each reactor (colored curves).

According to these rules, cyclic reaction sequences can occur, such as $1+2\to 3$, $1+3\to 4$, or $4\to 2+2$. Reaction sequences of this type are often called autocatalytic cycles despite the fact that they need not involve catalytic activity, because the net result is a bulk reaction which stoichiometrically increases a species—in this example the net reaction is $1+1+2\to 2+2$. The existence of such cycles is insufficient to cause superlinear growth of some species, as each product in the loop may participate in other reactions, taking mass off the cycle. In the model used here, there may be an arbitrary number of such cycles, and fluctuations in the spatial distribution of molecules may cause some of these cycles to stabilize via positive feedback enabled by the cycles themselves.

Backward (destructive) reactions, diffusion and outflow, are modeled as first-order reactions, and have corresponding rate constants $k_b$, $k_d$, and $k_o$, respectively. Forward (constructive) reactions are second-order processes driven by pairwise interactions, occurring at a rate constant of $k_f$ within local spatial volume. To simplify the parameter space, the backward reaction rate constant has been normalized with $k_b\equiv 1$ throughout all simulations, and the diffusion coefficient assumed equal to the outflow rate constant by setting $k_d\equiv k_o$. This equivalence between outflow and diffusion rates assumes that the exit point is not coupled to any downstream system—e.g. a membrane, sink, or other transport restriction.

Inflow is modeled as a zeroth-order reaction, occurs at a rate I into a single “source” reactor, and is measured in units of molecules per unit time. Only monomers ($i=1$) are added to the system via inflow. Similarly, outflow only occurs at a single reactor. Any molecule (not just 1 s) can leave the system via outflow. All other reaction rate parameters are held constant during simulations to reflect steady-state kinetics, aligning with conditions typical of experiments conducted under stable environmental conditions (26).

Table 1 provides a summary of the reaction rates utilized in the model. To minimize bias introduced in the resulting chemistry, we maintain a constant backward reaction rate across all molecules, independent of the molecule (i.e. integer value). While alternative approaches could consider that larger molecules might break more readily due to an increased number of bonds or, conversely, be more resistant to degradation due to enhanced stability, our intermediate position seeks to balance these perspectives without introducing variability. This approach may yield molecules of higher assembly index (see Methods section); as our primary concern is minimizing false positives, this provides a more conservative simplification.

By applying uniform constants for both forward and backward reaction rates across all pairwise interactions and degradation reactions, we aim to maximize the diversity within our model. In contrast, natural chemical systems exhibit reaction rate constants that are highly dependent on the specific components involved, often differing by orders of magnitude. Nevertheless, our model is more constrained than natural molecular space: at size N, an integer has only $N/2$ binary decompositions, whereas a molecule with N bonds can have many more, depending on its structure and symmetries. Consequently, the ensemble of accessible molecules in our simulations is considerably smaller when compared to real chemical environments.

Numerical integration with the Gillespie algorithm

The model’s numerical integration employs a stochastic methodology, utilizing the τ-leaping method (27). One exception is inflow which is treated deterministically: exactly $I\mathrm{\Delta }t$ molecules are added to each designated reactor over any interval. For the inflow rates used in our simulations, this is equivalent to the mean of a Poisson process of rate I, whose relative fluctuations $1/\sqrt{I}$ are negligible.

All other reactions are treated stochastically following the usual τ-leaping method. Let $k_r$ denote one reaction rate. The corresponding reaction propensity, ${\pi }_r$, is calculated by multiplying this reaction rate by the number of combinatorial possibilities that facilitate the reaction. For example, for reactions driven by pair-wise collisions in a mixture of $n_j$ molecules within reactor j, this relationship is:

$${\pi }_r=k_r\cdot n_j(n_j-1).$$ (1)

Whereas Gillespie’s original stochastic simulation algorithm (SSA) yields precise outcomes, its computational demands become prohibitive with a large number of reactions. The τ-leaping technique used here instead approximates the number of reactions $n^{\mathrm{\prime }}$ within a time interval $[t,t+\tau ]$ using a Poisson distribution with a mean of ${\pi }_r\cdot \tau$ (27):

$$n^{\mathrm{\prime }}(t+\tau )\approx \mathcal{P}({\pi }_r\cdot \tau ),$$ (2)

leading to calculation times reduced by one to two orders of magnitude.

Modeling ensembles of reactors

In our simulations, multiple reactors are interconnected in a fixed topology over time with nonperiodic boundaries, permitting bidirectional diffusion. Figure 1C gives an overview of the model architecture and simulation outputs, and illustrates the reactor network with per-reactor copy number time series in a representative simulation. The inflow and outflow reactors are placed at the maximal possible distance from each other on a 2D lattice under a von Neumann connection scheme, in which each site connects to its four cardinal neighbors (see Fig. 1C) (28). We assume uniform conditions across all reactors, maintaining the same volume, temperature, and pressure. Consequently, our reactions adhere only to the principle of mass action, while other typical chemical properties, such as kinetic rates or thermodynamic stability influenced by internal structures have been abstracted away.

Calculation of the assembly index

We denote the assembly index by a, writing $a_i$ and $a_m$ for the integer and molecular cases respectively. We represent basic chemical units as integers and approximate the assembly index of an integer n using the minimal length of all addition chains that terminate at n (29). The addition chain for an integer n is defined as a sequence of integers $\{1,\dots ,b_i,\dots ,n\}$ such that $b_i=b_j+b_k$ with $j,k<i$ (30). This serves as a lower bound for the assembly index of a molecule with n bonds. The assembly space of covalently bonded molecules can be formally mapped onto the assembly space of integer addition; however, this mapping generically sends many molecules to the same assembly index. Thus, it preserves some aspects of the assembly space while compressing the space overall (see Ref. (29) for details).

To determine the assembly index for integers, we utilize a numerical implementation of an efficient algorithm designed to generate minimal-length addition chains (30). Our findings indicate that, for integers, the assembly index is bounded logarithmically by ${\log }_2(i)<a_i\lesssim \sqrt{2}{\log }_2(i)$ (Fig. 2A). Although we can compute assembly indices within this framework, mapping these results to empirical systems is nontrivial. This difficulty arises partly because, for every positive integer N, there exists a super-exponential growth in the number of covalently bonded molecules with N bonds. As a result, the dynamics of our model are highly degenerate when compared to real chemical dynamics.

Figure 2.

A) Computation of the assembly index for integers using an optimized algorithm designed for minimal-length addition chains. The plot showcases the logarithmic range of the assembly index, bounded by ${\log }_2(i)<a_i\lesssim \sqrt{2}{\log }_2(i)$. This illustrates a lower limit for the assembly index of molecules with n bonds (29). B) Upper bounds on assembly index as a function of reachable chemical space for integer-based (blue) and molecular (orange dashed) chemistry. The integer bound $a_i$ follows $\sqrt{2}{\log }_2(N)$; the molecular bound $a_m$ is derived from assembly index calculations over a sample of GDB-17 chemical space. Gray dotted lines mark a representative assembly index ($a=15$) and the GDB-17 enumeration limit. The gap $\delta a$ between bounds at equal reachable space (shown at ${\log }_{10}(N)=10$, $\delta a=37$) grows with N, indicating that integer-based chemistry systematically overestimates assembly indices relative to molecular chemistry.

Scaling of integer space

Real molecular chemistry provides a much wider combinatorial space than the integer-based version, which tends to produce structures with lower assembly indices for comparable space sizes. We quantified this difference by tracking the maximum assembly index as a function of the accessible number N of distinct objects.

Figure 2B shows the assembly index a as a function of the reachable chemical space: the blue curve reflects the upper bound for integers (Fig. 2A), while the orange data points show the average assembly index over samples of GDB-17—a dataset which enumerates organic molecules—mapped onto reachable chemical space (see SI for details) (31). Molecular assembly indices were computed using the “assembly-theory” library (32). Comparing both relationships yields a scaling factor of $s\approx 5.12$, i.e. integer-based chemistry overestimates assembly indices by approximately a factor of five for comparable chemical spaces. Thus, although the numerical AI values produced by our model should not be compared directly with experimental measurements, such as those of Marshall et al. (1), our qualitative findings apply to real chemistry as the effects of physical parameters are not determined by the combinatorial size of chemical space.

Results

Varying diffusion and inflow

We first investigate how diffusion influences the distribution of molecular species. Simulations were performed with a statistical ensemble of the model to monitor changes in molecular species distribution. We modeled a regular 2D lattice of $N=25$ reactors, allowing for bidirectional diffusion, over a maximum time of $T={10}^5$, and timsteps of $d\text{t}={10}^{-3}$, a construction rate of $k_f={10}^{-3}$, with all reactors initially empty.

Throughout, we adopt the following notation for copy numbers. Let $n_{i,j}(t)$ denote the copy number of species i in reactor j at time t. When reporting final-state quantities at $t=t_{max}$, we use the shorthands $\overline{n_i}$ and ${\sigma }_{n_i}$, i.e. the across-reactor average and SD evaluated at the end of the simulation.

Figure 3 illustrates a statistical ensemble of 100 simulations, with diffusion rates sampled over $\log k_d\in \{-4,\dots ,1\}$. Figure 3A shows the copy number distribution of integers $i\in \{1,\dots ,{10}^7\}$, divided into 50 logarithmically spaced bins for varying diffusion coefficients, with lighter colors denoting higher copy numbers. At the lowest diffusion levels, the majority of molecules are located in the interval $i\in [{10}^1,{10}^4]$. As diffusion increases, the distribution’s mean shifts toward higher integers, only to decline again when $k_d\gtrsim {10}^{-3}$. The inset in Fig. 3A illustrates the distribution of assembly indices in relation to the diffusion coefficient, with the vertical axis shown on a linear scale. As the AI is proportional to the logarithm of the integer value, i.e. $a_i\propto {\log }_2(i)$, the overall trends for both the integer and AI distributions are comparable.

Figure 3.

A) Integer copy number distributions for $i\in \{1,\dots ,{10}^7\}$ over 50 log-spaced bins and diffusion coefficients $k_d\in [{10}^{-4},{10}^1]$. B) Final across-reactor average copy number $\overline{n_i}$ of integers $i\in \{1,2,3,5,10\}$ at $t={10}^5$ as a function of the diffusion parameter $k_d$. Each point is the mean across all $N=25$ reactors; inset shows SD. Lines on B–D are locally estimated scatterplot smoothing curves shown as visual guides. C) Number of molecules and system mass across diffusion regimes. D) Final across-reactor average copy number of dimers $\overline{n_2}$ as a function of diffusion and inflow $\log I\in \{3.0,3.3,3.7,4.0\}$.

Figure 3B reports the average final copy numbers $\overline{n_i}$ of integers $i\in \{1,2,3,5,10\}$ as a function of the diffusion parameter $k_d$. Three distinct regimes emerge. At low diffusion rates ($\log k_d\lesssim -3$), the system has not yet reached steady state due to limited molecular transport; most molecules remain in the inflow reactor, which behaves as a well-mixed system. As diffusion increases ($-3\lesssim \log k_d\lesssim -1$), molecules spread across reactors, but reduced local concentrations still hinder reactions. At higher diffusion values ($\log k_d\gtrsim -1$), all reactors become populated to varying extents. The system approaches homogeneity again, allowing the copy number of these species to rise—until diffusion becomes so fast ($\log k_d\sim 1$) that molecules are rapidly lost through outflow, causing another drop in copy number. The inset of Fig. 3B shows the SD in copy number, ${\sigma }_{n_i}$ across reactors. It peaks at high diffusion due to inflow accumulation, reflecting rapid turnover and spatial imbalance.

Figure 3C illustrates the variations in both the number of molecules and mass across different diffusion regimes. At the lowest values of the diffusion rate ($k_d\sim {10}^{-4}$), the majority of molecules remain localized in the inflow reactor, and mass remains high. This is due to the limited diffusion, preventing molecules from reaching the outflow reactor. In this regime, the number of molecules increases because as they disperse through the reactors, their likelihood of undergoing constructive reactions diminishes, leading to a higher overall quantity of smaller molecules. As diffusion continues to rise, molecules gradually populate all reactors. In this phase, mass decreases more rapidly than the number of molecules, suggesting a reduction in high integer species. This reduction occurs because molecules are transported swiftly through the system, leaving insufficient time for reactions necessary to form larger molecules.

Our analysis also examined the impact of varying inflow rates within these higher diffusion regimes. This approach allowed us to differentiate between the effects of transport and inflow rates. Figure 3D illustrates the across-reactor average of dimers copy number $\overline{n_2}$ for inflow rates $\log I\in \{3.0,3.3,3.7,4.0\}$, spanning intermediate and high diffusion conditions. The dimer serves here as a proxy for more complex species; the full distributions are provided in Fig. A2.

As inflow increases, the distribution of dimer copy number shifts, peaking at high values of the diffusion coefficient, with the maximum reaching higher levels. In the high diffusion regime ($\log k_d\gtrsim 0$), greater inflows result in increased dimer copy number, with this increase being correlated with the rise in inflow. However, when diffusion exceeds a critical value ($\log k_d\gtrsim 2$), dimer copy number decreases again until they reach zero. In this regime, transport dynamics dominate over other processes, completely preventing the formation of larger molecules.

The nonmonotonic variation of dimer copy number at high $k_d$ (Fig. 3D) mirrors the regime changes visible in Fig. 3A. As $k_d$ rises past $\sim {10}^{-3}$, the system shifts from a homogeneous regime (material confined to the inflow reactor) to a heterogeneous one (material distributed unevenly across reactors): complexity increases and low-AI material is depleted. As $k_d$ rises further, the system returns to a homogeneous regime (material now spread evenly across all reactors), complexity decreases, and dimer copy number recovers.

Assembly

A central metric in assembly theory for evaluating the complexity of a physical system is the assembly, noted A. Assembly measures the total amount of selection necessary to produce an ensemble of observed objects: it quantifies both the difficulty of discovering new objects and the amount of selection required to assemble a system, allowing a comparison between systems where the total number of objects differ. Sharma et al. (2) define assembly as:

$$A=\sum _{i=1}^N{e}^{a_i}\left(\frac{n_i-1}{N_{\mathrm{T}}}\right),$$ (3)

where $a_i$ and $n_i$ are the assembly index and copy number of object i, N the number of unique objects in the system, and $N_T$ the total (nonunique) number of objects. We used A to track the temporal evolution of system-wide complexity in three representative simulations across low, intermediate, and high diffusion regimes.

Figure 4A shows how both A and the maximum $a_i$ evolve over time. In the low diffusion regime ($\log k_d\sim -4$), A rises rapidly during the initial time steps and then gradually levels off as diffusion slowly redistributes molecules from the inflow to neighboring reactors. Meanwhile, in the intermediate regime ($\log k_d\sim -2$), assembly plateaus more quickly due to spatial dispersion limiting the frequency of constructive reactions. Finally, in the high diffusion regime ($\log k_d\sim 0$), molecules spread almost immediately across the system, preventing a significant increase in total assembly. Figure 4A’s inset shows the highest assembly index observed in each diffusion regime. Interestingly, the value observed for the intermediate regime does not surpass that of the low-diffusion one during the initial transient, suggesting that systems with the highest assembly index $a_i$ are not necessarily fully correlated with those having the highest assembly A.

Figure 4.

A) Temporal evolution of total assembly A across three diffusion regimes: red, low ($\log k_d=-4$); orange, intermediate ($\log k_d=-2$); yellow, high ($\log k_d=0$). Inset shows the highest assembly index observed. Lines on A and B are smoothing curves shown as visual guides. B) Sensitivity for detecting complex molecules as a function of the diffusion rate $k_d$ for the outflow reactor, for relative abundances ranging from ${10}^{-3}$ to ${10}^1$. Highest detectable assembly indices are shown in the inset.

Given that the integer space inflates $a_i$ by a factor of s (Methods section), we have also analyzed the effects of recalculating A using logarithmic weighting. This analysis also supports the conclusion that intermediate diffusion is the regime most favorable to the build-up of molecular complexity (see Supplementary material, Section A4).

Sensitivities for detecting complex molecules

A central objective of this analysis is to establish an upper bound on the assembly indices expected in the absence of biological systems. In practice, however, the detectability of these indices depends not only on how environmental parameters modulate the distribution of assembly indices, but also on the sensitivity of the detection instruments. Here, we ask specifically how diffusion rate interacts with detection sensitivity, using this as a test case for quantifying the extent to which instrumentation limitations can obscure environmentally driven changes in molecular distributions.

To investigate this, we conducted a series of simulations that replicate those used to generate the data in Fig. 3. In these new simulations, we assessed how the detection of high assembly molecules in the outflow reactor varies in response to changes in the diffusion rate. Specifically, we sought to determine the upper limit, in terms of the integer i and the assembly index, that can be detected within a single reactor. We varied the diffusion rate to examine how it might affect the detectable upper limit. We repeated the analysis using several detection sensitivities, thereby simulating coarser or finer resolutions representative of the instrumentation used for life-detection measurements on space missions. By focusing on a specific reactor, we emulate experimental or mission contexts in which full access to the system is unavailable and measurements are restricted to a subset of components. Selecting this reactor as the outflow reactor further emphasizes that our analysis concerns the combined effects of transport phenomena and system topology, rather than the influence of individual components.

Figure 4B illustrates how logarithmic changes to the diffusion parameter affects these detections. The analysis encompasses several distinct sensitivities, ranging from a relative abundance of $[X]={10}^{-3}$ (i.e. the capacity to detect one in a thousand molecules) to $[X]={10}^{-1}$ (the capacity to detect 1 in 10). We evaluated the influence of diffusion on the detection of molecules in terms of AI (Fig. 4B, inset).

The impact of topology

To evaluate how the topology of the reactor network influences the dynamics of the underlying chemistry, we begin by comparing statistical moments of molecular distributions while varying network properties. First, we have calculated the mean AI from the distribution of molecules at the final time step ($t={10}^5$) within individual chemostats, examining its relationship to the minimum distance from the source reactor across three distinct diffusion regimes (Fig. 5).

Figure 5.

A) Mean assembly index, calculated from the final distribution of copy numbers $\overline{n_i}$, as a function of the minimum distance from the source reactor under three distinct diffusion regimes—low (red), medium (orange), and high (yellow). Lines shown are smoothing curves intended as visual guides. The inset diagram illustrates how the distance from the source is calculated. The red and blue boxes, and the associated close-up inset plots, show an example of variability in the mean assembly index among reactors equidistant from the source ($d=4$) under intermediate diffusion that could depend on network connectivity and boundary conditions. B) Spatial distribution of mean assembly index across a 25 reactor lattice for three representative diffusion rates. Each square represents one reactor, with color indicating the mean assembly index $\overline{n_i}$. The source (inflow) is at the top-left corner and the outflow at the bottom-right.

Our findings indicate that both diffusion and the exact location of reactors within the network significantly influence the mean AI (Fig. 5A). Under low diffusion conditions, the mean AI peaks near the source reactor—where the majority of constructive reactions occur—and gradually declines with increasing distance from the source. As diffusion intensifies, allowing molecules to disperse more broadly among chemostats, the impact of distance on the mean AI diminishes. In scenarios characterized by high diffusion, the mean AI becomes nearly uniform across all reactors.

While reactor distance from the source correlates with the mean AI in that reactor, this relationship is not absolute. Notably, reactors situated equidistant from the source consistently exhibit significantly different mean AI values. For instance, at intermediate diffusion levels, multiple reactors located at a minimum distance of $d=4$ from the source present divergent mean AI (Fig. 5A, red and blue boxes and close-up insets). To explore further how the interplay between distance from source and topology impacts the chemistry, we examined the spatial distribution of mean assembly index across the lattice (Fig. 5B). Reactors situated in the inner regions (labeled I, M, Q—highlighted in red) are found to contain a higher average assembly index than reactors located at the boundaries of the network (E and U—highlighted in blue) for $\log (k_d)=-4$ and $\log (k_d)=-2$. Node connectivity could therefore in part dictate, by modulating the flow of material between each one, reaction rates and thus the average assembly index that can be reached.

To further investigate how topology influences the formation of high AI molecules, we focused on intermediate to high diffusion regimes. These regimes are characterized by molecular populations that explore the entire spatial extent of the system. We conducted an additional analysis to evaluate how overall topology affects the formation of complex species. We generated two identical statistical ensembles of simulations; however, in one ensemble, we randomized the networks by pairwise edge swapping (see Supplementary material, Section A3 for details). This method preserved node degree and other low-level graph properties while eliminating high-level structural order, serving as a control. We carried out these simulations using an inflow $I={10}^3$ to extend the explored parameter range while avoiding excessive computational requirements. While reducing the inflow shifts the distribution of copy numbers, it does not alter the nature of this distribution (see Fig. S7) or the generality of our results.

To compare the distributions produced by both lattice and randomized topologies, we first binned the final copy numbers in the outflow reactors up to $i=\mathrm{1,000}$ in bins of 50 integers and applied a power law fit of the form $a{x}^{-\alpha }$ as statistical tests indicate that heavy-tailed distributions fit the data better than exponential distributions (Fig. 6A). We do not infer from this result that the distributions are necessarily power laws; instead, we use the fitted exponent α as a summary statistic for comparing distributions across conditions (see Supplementary material, Section A6 for details.) Figure 6B illustrates α varies as a function of the diffusion parameter $k_d$ for both the lattice and randomized configurations.

Figure 6.

A) Examples of power-law fits for two representative simulations with $\log (k_d)=-2$ and $\log (k_d)=1$. The red line represents the count per $\log (i)$ in the outflow reactor. Integers from 1 to 1,000 are grouped in bins of 50. Higher values of the exponent α correspond to steeper assembly index distributions. B) Power-law exponent α as a function of the diffusion parameter $k_d$ for both ordered lattice (blue) and randomized (red) network topologies. The vertical axis is reversed: higher values indicate greater prevalence of large molecules. The system transitions from a heterogeneous regime at intermediate $k_d$ to well-mixed at high $k_d$. The sharper increase in α for randomized topologies (inset) indicates faster decay of copy numbers with size, underscoring the role of structural order in sustaining high AI integers. Red and blue curves are intended as visual guides.

The results reveal a transition from a heterogeneous regime at intermediate $k_d$ values to a well-mixed regime at high $k_d$ values: as the system shifts to the homogeneous regime, the exponent α increases more significantly in randomized topologies compared to lattice structures. While randomized topologies can decrease the occupancy time of molecules, possibly reducing assembly indices by limiting opportunities for molecular interaction, the effect observed here significantly exceeds what residence time changes can account for (see Supplementary material, Section A5). This trend in return signifies a faster decline in distribution tails, indicating fewer species with high AI. In essence, our findings suggest that ordered topologies, such as regular lattices, facilitate the formation of relatively higher AI species in regimes where high levels of diffusion promote molecular exploration of the reactor network, while randomized topologies comparatively suppresses them in similar conditions.

Discussion

Numerous physical environments are proposed to support the emergence of life, including hydrothermal vents, hot springs, and warm ponds. These sites have been explored due to their potential to drive chemical complexity, a key factor in the origin of life. It is believed that biomolecular precursors on primitive Earth interacted with diverse mineral substrates, potentially facilitating the emergence of life (33), numerical studies highlighting significant environmental effects on solution behaviors (34). For example, molecular dynamics simulations demonstrate that various mineral surfaces differently influence RNA prepolymer formation. Additionally, research shows that serpentinization—where specific rocks react with water, forming new minerals and releasing hydrogen—impacts confined fluids, thereby affecting abiotic organic production (35–38). Porous media also play a role by influencing molecular diffusion (39, 40), and diffusion processes can enhance both the diversity (41) and functionality (42) of autocatalytic networks. These constraints are important considerations for both the origin of life and for ensuring the reliability of life detection on other planets (43, 44).

Our findings support the idea that several parameters associated with these systems significantly influence the distribution of molecular assembly indices. Specifically, changing diffusion modifies both spatial distribution (Fig. 5B) and residence time (Fig. S9B), whereas changing inflow (Fig. 3D) primarily modifies the amount of material supplied. Transport processes therefore jointly shape molecular distributions. Our simulations demonstrated that while an increase in diffusion could enhance the presence of certain simple molecules (e.g. dimers, Fig. 3B and D), it nevertheless had a deterrent effect on constructive reactions. This leads, on the one hand, to a decrease in the average molecular mass (Fig. 3C) and, more generally, to a shift in the distribution of chemical species in our simulations (Fig. 3A) and their AI (Fig. 3A, inset).

Likewise, increasing the inflow significantly alters the dimer distribution by both boosting their numbers and shifting the diffusion coefficient required to achieve the highest dimer concentration (Fig. 3D). As more molecules are introduced, the likelihood of constructive (second-order) reactions rises quadratically. The quadratic growth of constructive reactions, whose propensity depends on $n(n-1)$, surpasses the impact of diffusion reactions, which are first-order events and depend linearly on n. This dynamic reduces the total molecule count (c.f. Fig. 3C), necessitating higher diffusion coefficients to maintain a well-mixed regime that supports molecular complexity. Put differently, this escalation leads to an overall increase in the average dimer copy number throughout the system, but overall AI distribution remains constrained by the diffusion regime the system is in.

Our analysis also made use of the notion of assembly as defined in assembly theory. We employed this proposed measure to investigate variations in the chemical complexity at the inflow and outflow ends of the system (Fig. 4A). We find that, across all regimes, the assembly A reaches a finite steady-state or comes close to doing so, yet our results reveal an apparent paradox. Shifting from a low-diffusion regime (Fig. 4A, red curve) to one where diffusion acts strongly (yellow curve) results in a trade-off between slower and more rapid exploration of the system’s topology, while maximum A is lower but reached earlier under high diffusion compared to the low-diffusion regime, where exploration is slower but leads to much higher final assembly. This analysis confirms that abiotic factors can mediate narrow forms of selection, but that these mechanisms are limited, in stark contrast to the open-ended selection that can occur in biological systems. This result may have practical implications for the design of laboratory experiments on the origin of life. For instance, it is possible that experiments with the highest short-term yield may not be those in which maximum complexity can be reached in the long term, indicating experiments that test for long term trends in molecule distributions may be more effective at identifying productive prebiotic chemstries (19, 45).

We continued our analysis by calculating the maximum AI detectable within a single reactor for different sensitivities (Fig. 4B). The observable trends result from two opposing effects: diffusion affects both the concentration of reactants—and thus their detectability—as well as the construction of molecules itself. The detectability of high-AI molecules therefore depends on these two effects within the system. The influence of diffusion on the detectability of high-AI molecules varies with instrument sensitivity. This dependence has direct implications for mission design: instruments operating below a critical sensitivity threshold may be blind to the first-order effects of environmental parameters on molecular distributions. In the context of laboratory experiments, analyzing samples at different resolutions could therefore prove useful when diffusion is a significant constraint on the duration of the experiment.

Key transport parameters, particularly the diffusion coefficient, which determine the extent to which molecules traverse the environmental topology, also amplify the topology’s impact on chemical processes. Our results show that the distance from the source reactor modulates the average AI of the molecules present in a reactor (Fig. 5), suggesting that in porous materials, where catalytic surfaces are exposed to reactants in varying degrees, the resulting chemistry may directly depend on diffusion. Fluctuations in the mean AI for reactors equidistant from the source reactor suggests that topological parameters—such as node connectivity and boundary conditions—could be optimized to produce the highest possible chemical complexity, with applications in origin-of-life research, bioengineering, and materials science.

The combined analysis of diffusion and distance from source (Figs. 3 and 5) indicates that there exists an optimal regime for the construction of detectable high-AI molecules. Both low-diffusion regimes (below $\log k_d\sim -4$) and high-diffusion regimes ($\log k_d\sim 0$ and beyond) limit the presence of high-assembly index species throughout the system, by either confining the population to the inflow reactor or by accelerating their passage through the system. But the intermediate diffusion regime ($\log k_d\sim -3$) promotes constructive reactions across the entire reactor network through a cascading enrichment process: as diffusion begins exporting molecules downstream, it does so preferentially for enriched (higher-AI) species relative to the monomer-dominated inflow reactor. Constructive reactions acting on this already-enriched population yield progressively higher-AI products, compounding as the population shifts further through the network. This cascade terminates when diffusion becomes strong enough to eject the most enriched molecules from the system entirely, driving the average assembly index back down.

The exploration of the system’s topology by the population of chemical species (Fig. 6A, B) showed that the structure of connections can also modulate the system’s dynamics. Randomizing the system’s topology—thereby destroying any high-level properties of the network—constrains the construction of complex chemical species by shifting the molecular distribution toward simpler species. This is a concrete example of how the boundary conditions imposed on a chemical system mediate a limited form of selection (46). In this case the topology we chose to analyze set this condition, but in experimental settings that may be determined by subtle design choices (47), and in natural settings by the structure of porous materials (39). Quantifying how experimental design can shape the outcome of prebiotic chemistry experiments is an essential direction for future research.

We showed that spatial patterning can modulate the formation of larger molecules. The fact that the distribution of molecules generated by our simulations is well described by a heavy tail distribution reflects the unavoidable trade-off between copy number and assembly indices: while our simulations do not yield arbitrarily high assembly indices, they show that an abiotic process can nonetheless shift the maximum achievable threshold in abiotic systems.

We found that the intermediate diffusion regime optimizes the construction of high assembly index molecules, and this finding modulates the results obtained regarding the influence of topology on the formation of those species. In the regime most favorable to broader AI distributions, the positive influence of spatial patterning may not be as significant as in the high-diffusion regime, where the system efficiently explores the entire topology. Furthermore, the regime in which topology most strongly influences the upper threshold of assembly indices—the well-mixed, high-diffusion regime—is also the one in which the sensitivities examined earlier have the least impact on the maximum AI detected. In other words, topology has its greatest effect on the formation of higher AI in a regime where such molecules are relatively easier to detect even with lower sensitivity. This suggests that, in the context of space missions, for example, increasing instrumental sensitivity would not help distinguish false positives arising from the system’s topology. A possible implication of this is that higher instrument resolution (e.g. the ability to distinguish molecules) may be more useful for confident life detection than instrument sensitivity (e.g. the ability to detect low abundance molecule), though this would require additional study.

Finally, the similarity of the exponent α for both lattice and randomized topologies in the heterogeneous regime, which determines the nature of the power law describing the distribution of integers, suggests that the increase in AI observed in this lower-diffusion regime is not directly driven by the type of topology connecting the reactors. In this regime, the residence time appears to be the determining factor in product distribution. This motivates further investigation into the interplay between the diffusion regime and the influence of topology, insofar as these parameters may significantly modulate the emergence of high AI molecules.

Conclusion

The impact of spatial structure on the emergence of complexity—broadly understood—is a well-established result in complexity science. Using an in silico chemistry model, we demonstrated that spatial structure can have important implications for the behavior of a system of chemical reactors, and consequently for how we design experiments on the origin of life and its detection during planetary missions.

Recent work on assembly theory provided empirical evidence suggesting that detectable quantities of sufficiently high AI molecules could serve as a method for life detection. Our results also show that several parameters characterizing environments of astrobiological interest—such as transport parameters and those related to the topology of porous materials—strongly influence the dynamics of chemical reactions and the resulting distributions of molecular species. Our analysis suggests it is imperative to deepen our understanding of how these parameters interact with abiotic chemistry. A long lineage of studies has examined the structure of networks in living systems; such analyses must now be extended to abiotic chemistry to determine how various topologies—such as scale-free networks—might influence the emergence of molecular complexity. Ultimately, this will contribute to a deeper comprehension of the environmental conditions necessary for the emergence of complexity, both on Earth and beyond.

Supplementary Material

Supplementary material is available at PNAS Nexus online.

Competing Interest

The authors declare no competing interests.

Funding

This research was funded by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN/05278-2018), the Fonds de recherche Nature et Technologies of Québec (grant no. 314488), the Fondation J. Armand Bombardier Excellence Scholarship, the Université de Montréal and the Center for Research in Astrophysics of Québec. A.C.R.’s research was supported by an appointment to the NASA Postdoctoral Program from the NASA Astrobiology Program administered by Oak Ridge Associated Universities under contract with NASA. C.M. acknowledges support from Human Frontiers in Science Program research grant RGEC29/2025.

Author Contributions

A.C.R. performed the analysis with support from C.M., developed the software with C.M., visualized results, drafted the initial manuscript, and edited and reviewed the final version. C.K. contributed to the project’s conceptualization, preliminary analysis, and participated in final manuscript review and editing. C.M. conceptualized the study, supervised the project, developed the methodology and initial software prototypes, contributed to visualizations, and provided computational resources.

Preprints

This manuscript was posted on a preprint: https://arxiv.org/abs/2509.04547.

Data Availability

The code and data that support the findings of this study are openly available in the GitHub repository https://github.com/colemathis/flow-complexity/releases/tag/v1.0.0 and archived at https://doi.org/10.5281/zenodo.22286972.