Generating Cyclic Conformers with Flow Matching in Cremer-Pople Coordinates

TL;DR

This work tackles the challenge of generating conformers for cyclic rings by introducing PuckerFlow, a flow-matching model that operates in Cremer–Pople space to efficiently sample closed-ring geometries. By tying a geometry-informed prior to a manifold-adapted cyclic Fourier output layer and mapping CP updates to Cartesian coordinates via an equivariant 3D network, the approach yields end-to-end differentiable ring conformer generation. Across 5–8 member rings, PuckerFlow achieves state-of-the-art accuracy on puckering RMSD and all-atom RMSD, often with far fewer parameters and using only a few inference steps, and it robustly captures ring conformer distributions in CP space. The method also integrates with exocyclic substituent generation, enabling full molecular conformer workflows and presenting a scalable, chemistry-aware tool for structure–property analyses and drug-discovery pipelines.

Abstract

Cyclic molecules are ubiquitous across applications in chemistry and biology. Their restricted conformational flexibility provides structural pre-organization that is key to their function in drug discovery and catalysis. However, reliably sampling the conformer ensembles of ring systems remains challenging. Here, we introduce PuckerFlow, a generative machine learning model that performs flow matching on the Cremer-Pople space, a low-dimensional internal coordinate system capturing the relevant degrees of freedom of rings. Our approach enables generation of valid closed rings by design and demonstrates strong performance in generating conformers that are both diverse and precise. We show that PuckerFlow outperforms other conformer generation methods on nearly all quantitative metrics and illustrate the potential of PuckerFlow for ring systems relevant to chemical applications, particularly in catalysis and drug discovery. This work enables efficient and reliable conformer generation of cyclic structures, paving the way towards modeling structure-property relationships and the property-guided generation of rings across a wide range of applications in chemistry and biology.

Figures (9)

• Figure 1: Overview of the presented work. (a) Cyclic molecules are omnipresent in applications such as drug design and catalysis pizzolato_phosphoramidite-based_2020, with their conformational space crucial to their function (Flu = fluorene). (b) Current approaches show limited performance on cyclic structures, or can produce non-closed rings. (c) Here, we introduce PuckerFlow, a generative machine learning model that operates only on the relevant degrees of freedom of ring systems and shows state-of-the-art performance on diverse ring systems.
• Figure 2: Depiction of the Cremer-Pople coordinates that compactly describe the relevant degrees of freedom of cyclic systems. (a) The puckering of a ring can be quantified by the displacements $z$ from the mean plane (exaggerated for visual clarity), with the coordinate system shown in red. (b) The Cremer-Pople coordinates are a low-dimensional representation that capture the essential degrees of freedom. (c) A single non-zero Cremer-Pople coordinate $q_2 cos(\phi_2)$ describes conformer properties across different ring size. (d-e) Example Cremer-Pople coordinate distributions for (d) a five-membered ring (2-pyrazoline) and (e) a six-membered ring (cyclohexene), with ring geometries extracted from several conformers. We show selected Euclidean structures that illustrate the characteristic conformer distributions in Cremer-Pople space.
• Figure 3: Overview of PuckerFlow, which performs flow matching on the low-dimensional Cremer-Pople space. (a) Learned atomic representations are passed to the cyclic Fourier filter output layer introduced in this work that (b) produces the symmetry-adapted output in Cremer-Pople coordinates. To train PuckerFlow, (c) we introduce a geometry-informed prior that ensures valid closed rings, and predict flow updates on the low-dimensional Cremer-Pople space, (d) which is used to reconstruct the three-dimensional geometry.
• Figure 4: (a-c) Generated conformer space of five-membered rings with PuckerFlow (violet) and MCF (gold) visualized in the two-dimensional Cremer-Pople space. Grey circles denote ground-truth conformations. (d) Visualization of the vector field and sampling distribution during inference for the imidazolidine ring: prior distribution (left), intermediate timesteps (middle), and final generated distribution (right). Highlighted points correspond to structures along the inference trajectory, which all correspond to valid, closed rings.
• Figure 5: (a-b) Generated conformers of six-membered rings obtained with PuckerFlow and MCF. Left: generated conformers shown in Cremer-Pople space as purple (PuckerFlow) and gold (MCF) crosses, with gray circles denoting ground-truth conformers. Right: representative ground-truth conformers (transparent) and their closest generated counterparts.