Robustness and physical grounding of the assembly-theory quantification of selection in natural-product chemical space: mechanistic constraints, database dependence, and the generality of the exploration parameter β

Assembly theory (AT) quantifies how much selection is needed to build complex molecules: the molecular assembly index (MA) counts the minimum recursive joining operations needed to construct a molecular graph, and joint assembly spaces (JAS) of molecular databases let one measure how strongly the biosphere narrows chemical exploration. The source study (Pagel, Sharma, and Cronin, 2026) modelled the joint assembly space of natural products (JAS_NP, from the COCONUT database) inside that of all known molecules (JAS_PC, from PubChem) and found that the exploration ratio - the number of unique natural products at each assembly depth relative to JAS_PC - falls off as $k e^{-\beta d}$ with $k \approx 0.12$ and $\beta \approx 0.499$, proposing $\beta$ as a scalar quantifier of selection-induced narrowing of exploration. The unresolved problem is whether this quantification is robust and physically meaningful: how a physically-constrained assembly index, together with $d$, $k$, and $\beta$, changes when reaction mechanisms, kinetics, and thermodynamics are incorporated into pathway construction; how sensitive the fitted parameters are to the reference database and to fragment definitions; which fragment-sampling regimes correspond to realistic selection; and whether $\beta$ generalizes to other datasets and planetary chemistries.

Background

Assembly theory (AT) is a framework that treats selection and evolution as measurable physical processes applicable beyond biology: an object's assembly index is the minimum number of recursive joining operations needed to construct it from elementary parts, reusing intermediate fragments. For molecules, this is the molecular assembly index (MA), computable from the molecular graph and measurable by tandem mass spectrometry (a technique that fragments selected ions and records the masses of the fragments); a central empirical result is that molecules with high MA found in high copy number are a signature of biological or technological production, since undirected abiotic chemistry rarely yields complex structures in abundance (Marshall et al. 2021). The assembly space of a set of objects is the directed graph of all fragments and joining operations from which they can be built; the joint assembly space (JAS) of a molecular database is this space computed for the whole database at once, exposing shared construction pathways. Assembly depth $d$ is a related descriptor: the recursion depth at which a fragment first becomes reachable from the building blocks, computed as $d = \max(d_{f_1}, d_{f_2}) + 1$ for a new object joined from two fragments.

Sharma et al. (2023) developed the theoretical claim that AT explains and quantifies selection, introducing the assembly equation and the copy-number-weighted quantity 'assembly', $A$. Pagel, Sharma, and Cronin (2026) applied this machinery to biochemistry: they built the JAS of roughly 211,000 natural products (JAS_NP, from the COCONUT database - natural products being the complex secondary metabolites produced by terrestrial organisms) inside the JAS of over 70 million known molecules (JAS_PC, from the PubChem database), and inside the JAS of drug molecules (JAS_D, from ChEMBL). They found that the exploration ratio - the number of unique natural products at each assembly depth relative to JAS_PC - decays exponentially with depth, $N_{\mathrm{NP}}/N_{\mathrm{PC}} \approx k e^{-\beta d}$ with $k \approx 0.12$ and $\beta \approx 0.499$, and proposed $\beta$ as a scalar quantifier of the selection required to attain the natural-product subspace of chemical space. By deleting fragments of the natural-product JAS (contingency loss, quantified by the loss parameter $\omega$) and reconstructing molecular spaces with heuristic fragment-sampling weights (an exponential scaling parameter $s$), they emulated alternate selection pressures and generated candidate drug-like molecules, filtered only by substructure rules including PAINS (pan-assay interference compounds, substructures known to cause false positives in biological screening) and scored with QED (quantitative estimate of drug-likeness).

The framework's quantitative conclusions rest on choices that were never stress-tested: assembly pathways are purely informational (shortest construction paths), ignoring reaction mechanisms, bond strengths, kinetics, and thermodynamics; 'all physically plausible molecules' is operationalized by one database (PubChem); fragment definitions and sampling weights are heuristic; and time-resolved or mechanistic pathway data for natural-product biosynthesis are absent. A separate line of work (Abrahão et al. 2024; Ozelim et al. 2024) argues mathematically that the assembly index is equivalent to smallest-grammar and Lempel–Ziv (LZ) dictionary-compression measures bounded by Shannon entropy, disputing AT's claim to explain selection - which sharpens, but does not answer, the question of whether the fitted $\beta$ is a robust physical signature of selection rather than an artifact of the modelling choices. Meanwhile the computational infrastructure has advanced: exact and near-exact MA computation at database scale is now feasible (Seet et al. 2025), the assembly-index decision problem is NP-complete (Masierak 2026), and assembly spaces have received a unified formalism (Siebert et al. 2026) - so the stress tests the source study called for are now practically executable, but have not been run.

Problem Statement

The unresolved problem stated by Pagel, Sharma, and Cronin (2026) is to establish the physical grounding and robustness of the assembly-theoretic quantification of selection in molecular evolution. The framework's central quantitative claim is that exploration of the joint assembly space of all known molecules (JAS_PC, built from PubChem) by natural products (JAS_NP, built from COCONUT) narrows exponentially with assembly depth, $N_{\mathrm{NP}}/N_{\mathrm{PC}}$ following $k e^{-\beta d}$ with $k \approx 0.12$ and $\beta \approx 0.499$, so that $\beta$ is a scalar quantifier of the selection required to produce Earth's natural products. The open question has five linked facets that jointly stress-test this claim: (1) define and compute a physically-constrained assembly index $\mathrm{MA}_c$ - the analogue of MA obtained by restricting the recursive joining operations to those that respect physical-chemical constraints (reaction mechanisms, bond strengths, environmental kinetics and thermodynamics) rather than the purely informational shortest-path pathways used so far - and quantify how $\mathrm{MA}_c$, the assembly depth $d$, and the fitted $k$ and $\beta$ change when such constraints are incorporated into assembly-pathway construction; (2) determine how sensitive the fitted $k$ and $\beta$ are to the choice and completeness of the reference database approximating 'all physically plausible molecules' (PubChem in the original analysis) and to experimental sampling biases in natural-product databases; (3) establish, against an explicitly stated criterion for what counts as realistic chemical or evolutionary selection that is operationalized as a measurable sampling-weight regime rather than a qualitative judgement, which fragment-sampling strategies (the heuristic weighting parameter $s$ and related schemes) correspond to such regimes; (4) test the generality of $\beta$ at minimum across one additional natural-product database and one alternative definition of fragments and building blocks and, if feasible, across a non-Earth planetary chemistry; and (5) develop methods to incorporate time-resolved or mechanistic pathway data - intermediate species on natural-product assembly pathways - into the assembly-space framework. Facets (1)-(5) are all mandatory and serve one objective - determining whether $\beta$ and the exploration-ratio framework measure a robust, physically interpretable signature of selection rather than an artifact of database, fragment, and pathway conventions. The empirical stress tests of facets (1), (2), and (4) require recomputing the relevant quantities - $\mathrm{MA}_c$, $d$, $k$, and $\beta$ - under the stated modifications rather than arguing from the literature alone, and thereby determining which of the framework's quantitative conclusions - the exponential form $N_{\mathrm{NP}}/N_{\mathrm{PC}} \approx k e^{-\beta d}$ with $k \approx 0.12$ and $\beta \approx 0.499$, and the claim that $\beta$ is a scalar quantifier of the selection required to produce Earth's natural products - survive those stress tests and which must be revised. Facet (3) concerns the operationalized realism criterion for what counts as realistic chemical or evolutionary selection and the classification of fragment-sampling strategies it yields, and facet (5) concerns a method for incorporating time-resolved or mechanistic pathway data into the assembly-space framework.

References

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