Floating Anchor Diffusion (FADiff)

• Floating Anchor Diffusion is a generative framework in SE(3) that treats protein motifs or object parts as floating anchors with learned global positions and orientations.
• It enforces intra-anchor rigidity by aggregating per-anchor updates, ensuring that the internal geometry of motifs is preserved throughout the diffusion process.
• FADiff automates multi-motif scaffolding and outperforms inpainting and conditional-generation methods in both protein engineering and 3D part assembly.

Floating Anchor Diffusion (FADiff) refers to a class of generative frameworks within SE(3) diffusion modeling that explicitly represent a subset of structure-defining features—such as protein motifs or object parts—as rigid “anchors” with unconstrained, learned global positions and orientations throughout the generative process. In contrast with fixed-position inpainting and standard conditional generation, FADiff maintains the internal geometry of these anchors while diffusion operates jointly on both rigid anchors and the surrounding scaffold or assembly. This approach was first introduced for biomolecular structure generation, specifically multi-motif protein scaffolding, and has been independently recognized as a natural generalization of anchor-point diffusion models for 3D part assembly. The principal innovation of FADiff is the automated, physically-coupled placement of multiple rigid units within a coherent structure, enabling both structural guarantee and floating—rather than pre-specified—relative locations of critical subunits (Liu et al., 2024, Zhao et al., 20 Jun 2025).

1. Core Problem Addressed by Floating Anchor Diffusion

Conventional generative approaches for object assembly or protein design with multiple distinct units face fundamental limitations stemming from rigid assignment of critical features (“motifs” in proteins, “parts” in mechanical assemblies). In inpainting frameworks, the positions of motifs or parts must be provided as input, precluding automated placement. Conditional generation that omits positional control cannot guarantee the occurrence or structural integrity of the required features. FADiff addresses these deficiencies by treating each motif or part as a “floating anchor,” simultaneously guaranteeing presence and learning their optimal placement in the generative process. In the multi-motif protein scaffolding context, this is formalized by predicting both the structure and sequence of the scaffold ($X_S, A_S$) and the positions ($X_M, A_M$) of all motifs, conditional only on their internal geometry ($M^x, M^A$), maximizing

$\log p_\theta(X_S, A_S, X_M, A_M \mid M^x, M^A)$

without requiring manual motif placement (Liu et al., 2024).

2. Mathematical Framework and Forward/Reverse Diffusion

FADiff utilizes a forward diffusion kernel over SE(3)-valued residue or anchor frames $T = \{ (R_i, X_i) \}_{i=1}^n$, where $R_i \in \mathrm{SO}(3)$ and $X_i \in \mathbb{R}^3$. The forward noising operates as follows:

• Translations:

$$X_{i,t} = \sqrt{\alpha_t} X_{i,t-1} + \sqrt{\beta_t} \epsilon_{i,t},\quad \epsilon_{i,t} \sim \mathcal{N}(0, I_3)$$

• Rotations (geodesic random walk on SO(3)):

$$R_{i,t} = \exp(\sqrt{\beta_t} \widehat{\eta_{i,t}}) R_{i,t-1},\quad \eta_{i,t} \sim \mathcal{N}(0, I_3)$$

where $\widehat{\eta} \in \mathfrak{so}(3)$.

The reverse process is parameterized by an equivariant neural backbone (e.g., VFN-Diff), which predicts per-residue or per-anchor updates. The learned Gaussian reverse kernel for translations, and a score-matching direction on SO(3) for rotations, yield a denoising network trained to approximate

$X_M, A_M$0

with anchor/part identity encoding $X_M, A_M$1 as additional input (Liu et al., 2024).

3. Floating Anchor Mechanism and Rigidification

During both forward and reverse diffusion, FADiff imposes intra-anchor rigidity by grouping residues or points belonging to the same motif or part and updating them as a single rigid body. The average residual translation and rotation for each anchor $X_M, A_M$2 at step $X_M, A_M$3 are computed as

$X_M, A_M$4

$X_M, A_M$5

and these averaged updates are then applied to every residue or point within the anchor:

$X_M, A_M$6

This procedure ensures preservation of all intra-motif/piece distances and angles, reducing the task of motif/part scaffolding to the joint diffusion of global frames for floating rigid anchors and the flexible scaffold.

To further guarantee internal anchor consistency, an “anchor-consistency” loss may be included:

$X_M, A_M$7

ensuring that for each motif or part, the internal structure remains invariant across diffusion steps (Liu et al., 2024).

4. Model Architecture, Training Objectives, and Sampling

FADiff implementations rely on SE(3)-equivariant neural architectures, such as VFN-Diff for proteins. At each diffusion step, node features (time embedding, motif identity, scaffold vs. anchor indicator, sequence position) and edge features (pairwise distances, relative orientations) inform the graph neural backbone. The primary training objective is denoising score matching (DSM) on both translations and rotations, potentially augmented by domain-specific bond or geometry losses:

• Denoising Score Matching:

$X_M, A_M$8

• Bond Length/Bond Angle Loss (biomolecular): For C–N peptide bonds and $X_M, A_M$9–C–N angles, domain-specific losses ensure realistic geometry.

Sampling proceeds via an Euler–Maruyama scheme on SE(3), initializing anchors in fixed frames and the scaffold in random pose. At each step, the network predicts updates, applies rigidification per anchor, and iteratively denoises towards the target structure. The final structure is extracted after $M^x, M^A$0, with optional post-processing (such as sequence inference via TSP heuristics for connectivity) (Liu et al., 2024).

5. Empirical Performance and Comparative Results

FADiff demonstrates high empirical fidelity for multi-motif scaffolding, especially when compared to inpainting and conditional-generation baselines:

Task/Benchmark	FADiff Success Rate (scTM > 0.5)	Baseline Success Rate
Two-motif scaffolding (real motifs)	73.1%	Inpainting (RFdiffusion/Chroma): 51–62%<br>Conditional (SMCdiff): ~23%
Three motifs	62.4%
Four motifs	58.4%
Five motifs	46.7%
Two large domains (>100 residues)	80.0%

These results establish that FADiff provides both the automation of motif/anchor placement and structural guarantee, outperforming prior methods under relevant success criteria (Liu et al., 2024).

6. Extensions, Limitations, and Applications

FADiff is directly extensible to a range of settings where complex assemblies are composed of multiple rigid or semi-rigid units with unknown placement:

• Protein engineering: Design of multi-domain enzymes, scaffolding of functional motifs including small molecules or cofactors by treating them as floating anchors.
• 3D part assembly: Transitional forms between rigid assembly and deformable/soft attachment are possible by generalizing anchors to support latent-space flow fields, enabling continuous deformation or soft constraints (Zhao et al., 20 Jun 2025).
• Modeling with partial priors: When some inter-anchor distances or orientations are known, these can serve as weak supervision during sampling.

Current limitations include the reliance on post-hoc sequence and structure evaluation pipelines (protein design), and sampling instability with large numbers or disparate sizes of anchors. Future research directions include integrated sequence–structure diffusion, incorporation of biochemical or physical property guidance, and deployment in closed experimental design loops (e.g., using cryo-EM/X-ray data) (Liu et al., 2024).

7. Relation to Anchor-point Diffusion and Prospects

FADiff generalizes the anchor-point diffusion framework introduced for 3D part assembly, in which all parts are downsampled to sparse anchors and diffused as a global cloud in Euclidean space, with per-part rigid transforms recovered by least-squares fitting (Zhao et al., 20 Jun 2025). In FADiff, anchors operate in SE(3)-space and can be constrained to preserve internal geometry during diffusion. A plausible implication is that extending FADiff to allow anchor points to carry latent-space flow fields enables modeling of continuous deformation or soft connectivity, enhancing compositionality for both biological and engineered structures. This conceptual connection suggests broad applicability of floating anchor strategies for scalable generative modeling across disparate domains involving rigid and quasi-rigid assembly.