---
title: 'PuckerFlow: Generative Conformer Sampling'
url: https://www.emergentmind.com/topics/puckerflow
type: topic
---

# PuckerFlow: Generative Conformer Sampling

PuckerFlow is a generative machine learning framework for sampling conformers of cyclic molecules by operating directly on Cremer–Pople (CP) internal coordinates. These coordinates represent the (N–3) "puckering" degrees of freedom of an N-membered ring, capturing essential ring deformations such as boats, envelopes, and chairs while excluding rigid-body and exocyclic modes. PuckerFlow advances conformational sampling by combining continuous-time flow matching in CP space with an E(3)-equivariant graph neural network (GNN) and a cyclic Fourier filter. The model achieves state-of-the-art precision and diversity in cyclic conformer generation, efficiently generating valid closed rings and supporting high-throughput exploration of ring scaffolds relevant to drug discovery and catalysis [2601.12859].

## 1. Cremer–Pople Coordinate Framework

PuckerFlow is built upon the Cremer–Pople coordinate system, which provides a low-dimensional representation for the out-of-plane deformations in monocyclic rings. Given a ring of N atoms:

- The molecule is oriented such that its mean plane coincides with the $x$–$y$ plane.
- The out-of-plane displacement for each atom $j$ is defined as $z_j = (R_j \cdot n)$, where $n$ is the normal of the mean plane.
- Atom positions around the ring are parameterized by $\alpha_j = 2\pi(j-1)/N$ for $j=1,\dots,N$.
- For each puckering mode $m=2, ..., \lfloor (N-1)/2 \rfloor$, the coordinates are determined by discrete Fourier transforms:

  $$
  q_m\cos\phi_m = \sqrt{2/N} \sum_{j=1}^N z_j \cos(m\alpha_j)
  $$
  $$
  q_m\sin\phi_m = -\sqrt{2/N} \sum_{j=1}^N z_j \sin(m\alpha_j)
  $$

- For even $N$, an additional mode $q_{N/2}$ is given by:
  
  $$
  q_{N/2} = (1/N)^{1/2} \sum_{j=1}^N (-1)^{j-1} z_j
  $$

- The aggregated puckering amplitude is $Q = \sqrt{\sum_m q_m^2}$; $Q=0$ defines a planar ring.

By this transformation, the entire set of $3N$ Cartesian coordinates reduces to $N-3$ internal degrees $\{q_m,\phi_m\}$, succinctly characterizing the physically allowed, closed geometries of the ring without redundancy [2601.12859].

## 2. Flow Matching in Cremer–Pople Manifold

PuckerFlow applies flow matching on the CP manifold rather than on Cartesian space. The generative model learns a time-dependent vector field $u_\theta(t,x)$ in CP space to transport a uniform, bounded prior $p_0$ to the empirical ring conformer distribution $p_1$ via the ODE:

$$
\frac{d x_t}{dt} = u_\theta(t, x_t)
$$

This process is governed by the density evolution:

$$
\partial_t \log p_t(x) = -\nabla \cdot u_\theta(t, x)
$$

Training uses the Conditional Flow Matching (CFM) objective, where for $x_1 \sim p_1$, $x_0 \sim p_0$, $t \sim \text{Uniform}[0,1]$, and $x_t = t x_1 + (1-t)x_0$,

$$
L_\text{CFM} = \mathbb{E}_{t, x_0, x_1} \left\| u_\theta(t, x_t) - \frac{x_1 - x_0}{1-t} \right\|^2
$$

This approach enables direct specification of physical feasibility (e.g., closed-ring constraints) through the prior in CP space, thereby avoiding artifacts from unphysical regions that are common in diffusion models using Gaussian noise.

## 3. Model Architecture: Equivariant GNN and Cyclic Fourier Filter

The architecture combines a 3D E(3)-equivariant GNN and a cyclic Fourier filter:

- For each training batch, partial CP samples $x_t$ are converted into approximate Cartesian ring geometries using precomputed average bond lengths and angles, stratified by atom types and ring size.
- The GNN employs message passing up to spherical-harmonic order $\ell=2$ on node and edge features—atomic number, hybridization, ring size, time embedding $\phi(t)$, and interatomic distances (within 5 Å)—to produce rotation- and reflection-equivariant embeddings.
- A cyclic Fourier filter operates as the final layer: learnable, rotation-equivariant filters are arranged around the ring, their outputs convolved with node embeddings, and a discrete Fourier transform yields the CP pseudoscalar outputs $u_\theta(t, x_t)$.
- Hyperparameters validated during development include 4–6 GNN interaction layers, 32-dimensional scalar channels, 4-dimensional second-order channels, batch normalization, and AdamW optimizer with learning rate $\sim 10^{-3}$.

The filter design ensures equivariance and locality around the ring topology, facilitating accurate inference of puckering modes needed for ring closure while handling all symmetry aspects of the CP manifold.

## 4. Data Preparation, Training, and Closed-Ring Constraints

Training data comprises 15 204 unique conformers of five- to eight-membered non-aromatic rings, sourced from COD, PQR, ZINC, and platinum-ligand datasets. Rings are hydrogenated when substituent hybridizations are preserved. Preprocessing involves:

- CP coordinates are computed using a ring-puckering library.
- Dictionaries of average bond lengths and angles, keyed by atomic and ring identity, are constructed on the training splits, with a nearest-neighbor fallback for missing motifs based on empirically weighted metrics.
- Training enforces feasibility conditions: prior amplitude bounds (e.g., $\max q_2 \sim$ 0.8 Å, $q_3 \sim$ 0.56 Å, $q_4 \sim$ 0.4 Å), projected bond lengths ($|z_j-z_i| \leq r_{ij}'$), projected angles ($\cos \beta' \in [-1,1]$), and convex-ring assumption. Less than 0.01% of conformers are discarded as non-reconstructible.
- Models are trained for 300 epochs with AdamW and standard weight decay; no explicit diffusion noise is used.

These steps guarantee valid, physically consistent samples both in learning and inference.

## 5. Sampling, Geometry Reconstruction, and Computational Workflow

Conformer generation follows an ODE-based sampling mechanism:

1. Sample $x_0 \sim p_0$, a uniform prior within physically valid amplitude bounds.
2. Integrate the learned ODE with $T$ steps (typically $T=30$, though quality remains high for $T=2$–5):
   $$
   x_{t+\Delta t} = x_t + \Delta t \cdot \frac{u_\theta(t, x_t) - x_t}{1-t}
   $$
3. At $t=1$, obtain $x_1 = \{q_m, \phi_m\}$, then reconstruct Cartesian coordinates:
   - Recover $z_j$ by inverse Fourier sums.
   - Compute projected bond lengths $r_{ij}'$ and angles $\beta_{ijk}'$ algebraically.
   - Stitch the ring via Cremer’s algorithm using three planar segments before re-attaching out-of-plane displacements.

This procedure ensures direct control over conformer diversity and ring validity at every stage.

## 6. Quantitative Evaluation and Comparative Performance

Conformational fidelity is quantified using:

- Average Minimum RMSD (AMR): precision (AMR-P, average minimum RMSD from generated to reference) and recall (AMR-R, average minimum RMSD from reference to generated).
- Coverage fraction (COV) within 0.1 Å RMSD: COV-P (precision) and COV-R (recall).

Performance is summarized in the following table (for unrelaxed, puckering-only RMSD):

| Method         | AMR-P / COV-P (%) | AMR-R / COV-R (%) |
|----------------|-------------------|-------------------|
| PuckerFlow     | 0.13 / 67.5       | 0.09 / 75.8       |
| MCF (Euclidean flow) | 0.16 / 46.2       | 0.12 / 60.0       |
| GeoDiff (diffusion)  | 0.24 / 24.9       | 0.18 / 42.4       |
| RDKit ETKDG    | 0.17 / 51.4       | 0.13 / 60.1       |

For full-atom RMSD:

| Method         | AMR-P / COV-P (%) | AMR-R / COV-R (%) |
|----------------|-------------------|-------------------|
| PuckerFlow     | 0.18 / 47.4       | 0.15 / 51.0       |
| MCF            | 0.23 / 24.6       | 0.19 / 35.6       |
| GeoDiff        | 0.28 / 16.4       | 0.22 / 30.5       |

After MMFF94 relaxation, performance gaps narrow but PuckerFlow remains best in precision and competitive in recall. PuckerFlow uses approximately $10^6$ parameters versus $13 \times 10^6$ for MCF and achieves similar or better results with far fewer inference steps (2–30 versus 50–5000) [2601.12859].

## 7. Applications, Domain Coverage, and Limitations

PuckerFlow accurately samples multimodal puckering distributions for 5–8-membered monocyclic rings, including envelopes, boats, chairs, pseudorotation circles, and mixed heteroatom scaffolds (e.g., thiazines, azasilinanes, oxazepanes, phosphocanes), which are pertinent to drug and catalyst design workflows. Full-molecule conformers can be synthesized by augmenting generated ring cores with exocyclic substituents using established algorithms (e.g., RDKit ETKDG, pretrained torsional-diffusion models).

The method’s end-to-end differentiable structure, gradient-based flow objective, and direct operation in internal coordinate space render it suitable for property-guided conformer generation (for example, targeting docking scores or substrate geometries), capabilities not afforded by discrete post-processing methods.

Current limitations include restriction to small/medium monocyclic rings ($N \leq 8$), convex planar projections, and minimal substituent diversity. Planned extensions include handling macrocycles, fused and spiro rings (via multiple CP subsystems), and joint modeling of ring and torsional degrees of freedom [2601.12859].

Source: https://www.emergentmind.com/topics/puckerflow