Reflection-Equivariant Diffusion for 3D Structure Determination from Isotopologue Rotational Spectra in Natural Abundance

TL;DR

This work tackles context-free 3D structure determination of organic molecules from rotational spectroscopy by addressing the sign ambiguity in Kraitchman substitution coordinates. It introduces Kreed, a diffusion-based model that operates in the molecule’s principal axis system and employs a reflection-equivariant denoiser to produce all-atom structures conditioned on the molecular formula, unsigned substitution coordinates, and planar moments of inertia. Kreed achieves high accuracy on QM9 and GEOM, retains substantial performance with partial substitution data, and shows practical potential by predicting correct structures for 25 of 33 experimentally measured cases, illustrating real-world applicability. The approach advances rotational spectroscopy data analysis by enabling robust, data-driven structure inference without prior connectivity or initial geometries, with implications for natural products, astrochemistry, and synthetic planning.

Abstract

Structure determination is necessary to identify unknown organic molecules, such as those in natural products, forensic samples, the interstellar medium, and laboratory syntheses. Rotational spectroscopy enables structure determination by providing accurate 3D information about small organic molecules via their moments of inertia. Using these moments, Kraitchman analysis determines isotopic substitution coordinates, which are the unsigned $|x|,|y|,|z|$ coordinates of all atoms with natural isotopic abundance, including carbon, nitrogen, and oxygen. While unsigned substitution coordinates can verify guesses of structures, the missing $+/-$ signs make it challenging to determine the actual structure from the substitution coordinates alone. To tackle this inverse problem, we develop KREED (Kraitchman REflection-Equivariant Diffusion), a generative diffusion model that infers a molecule's complete 3D structure from its molecular formula, moments of inertia, and unsigned substitution coordinates of heavy atoms. KREED's top-1 predictions identify the correct 3D structure with >98% accuracy on the QM9 and GEOM datasets when provided with substitution coordinates of all heavy atoms with natural isotopic abundance. When substitution coordinates are restricted to only a subset of carbons, accuracy is retained at 91% on QM9 and 32% on GEOM. On a test set of experimentally measured substitution coordinates gathered from the literature, KREED predicts the correct all-atom 3D structure in 25 of 33 cases, demonstrating experimental applicability for context-free 3D structure determination with rotational spectroscopy.

Figures (10)

• Figure 1: Kreed takes as input molecular formula, Kraitchman's substitution coordinates, and principal moments of inertia (left) and runs a learned reverse diffusion process of Euclidean steps in the molecule's principal axis system (center) to obtain a ranked list of structures (right).
• Figure 2: Top: Isotopologues have systematically shifted and rotated principal axes (yellow) relative to the parent's principal axes (RGB). The effect of isotopic substitution is magnified by 50$\times$ for visualization. Parent and isotopologue rotational constants are converted to planar moments of inertia and then to unsigned substitution coordinates using Kraitchman's equations. Given molecular formula, substitution coordinates, and planar moments of inertia of the parent molecule, Kreed learns to denoise random point clouds into all-atom 3D structures of the molecule. Bottom: The denoiser model $\hat{\bm{\varepsilon}}_{\bm{\theta}}$ is equivariant with respect to axially-aligned reflections ${\bm{R}}$ and node permutations $\bm{\Pi}$, so that the modelled distribution $p_{\bm{\theta}}({\bm{X}}|{\bm{C}})$ under the diffusion model is invariant to such transformations.
• Figure 3: Average connectivity correctness of Kreed on QM9-C and GEOM-C for samples with different numbers of unconstrained heavy atom coordinates. Shading shows the 95% confidence interval. The fewer the number of substitution coordinates that are provided, the greater the number of unconstrained heavy atom coordinates, and the greater the difficulty of the task, as shown by decreases in average correctness.
• Figure 4: Top-1 predictions for experimental substitution coordinates of molecules which do not appear in QM9 or GEOM. The transparent structure is the ground truth, while the small spheres indicate the top-1 predicted structure. Green indicates all-atom correctness while yellow indicates heavy atom correctness. The presence of black pins indicates whether a substitution coordinate in that direction was available.
• Figure 5: Pairwise heavy atom distance histograms used for the GA fitness function.

Theorems & Definitions (12)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof