SigmaDock: Untwisting Molecular Docking with Fragment-Based SE(3) Diffusion
Authors: Alvaro Prat
Leo Zhang
Charlotte Deane
Yee Whye Teh
Garrett M Morris
Venue: ICLR 2026 Poster (2026)
Keywords: Molecular Docking
Geometric Deep Learning
Generative Models
SE(3)
Diffusion
AI for Science
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Determining the binding pose of a ligand to a protein, known as molecular docking, is a fundamental task in drug discovery. Generative approaches promise faster, improved, and more diverse pose sampling than physics-based methods, but are often hindered by chemically implausible outputs, poor generalisability, and high computational cost. To address these challenges, we introduce a novel fragmentation scheme, leveraging inductive biases from structural chemistry, to decompose ligands into rigid-body fragments. Building on this decomposition, we present SigmaDock, an SE(3) Riemannian diffusion model that generates poses by learning to reassemble these rigid bodies within the binding pocket. By operating at the level of fragments in SE(3), SigmaDock exploits well-established geometric priors while avoiding overly complex diffusion processes and unstable training dynamics. Experimentally, we show SigmaDock achieves state-of-the-art performance, reaching Top-1 success rates (RMSD $<2$ \&  PB-valid) above 79.9\% on the PoseBusters set, compared to 12.7-32.8\% reported by recent deep learning approaches, whilst demonstrating consistent generalisation to unseen proteins.  SigmaDock is the first deep learning approach to surpass classical physics-based docking under the PB train-test split, marking a significant leap forward in the reliability and feasibility of deep learning for molecular modelling.

Published as a conference paper at ICLR 2026
SIGMADOCK: UNTWISTING MOLECULAR DOCKING WITH BAGMENT--BSED SE(3) DIFFUSION
Alvaro Prat, Leo Zhang, Charlotte M. Deane, Yee Whye Teh, Garrett M. Morris
Department of Statistics, University of Oxford
ABSTRACT
Determining the binding pose of a ligand to a protein, known as molecular dock- ing, fundamental task in drug discovery. Generative approaches promise faster, improved, and more diverse pose sampling than physics-based methods, but are often hindered by chemically implausible outputs, poor generalisability. and high computational cost. To address these challenges, we introduce a novel fragmentation scheme, leveraging inductive biases from structural chemistry, to decompose ligands into rigid-body fragments. Building on this decomposition, we present SIGM DDCC,, an SF) Riemannian diffusion model that generates poses by learning to reassemble these rigid bodies within the binding pocket. By oper- ating at the level of fragments in SE(3), SIGMADOCK exploits well-cstablished geometric priors while avoiding overly complex diffusion processes and unsta- ble training dynamics. Experimentally, we show SIGMADOCK achieves state- of-the-art performance, reaching Top-1 success rates (RMSD 2 & PB-valid) above 79.9% on the PoseBusters set, compared to 12.7-32.8% reported by recent deep learning approaches, whilst demonstrating consistent generalisation seen proteins. SIGMADOCK is the first deep learning approach to surpass classi- cal physics-based docking under the PB train-test split, marking a significant leap forward in the reliability and feasibility of deep learning for molecular modelling.
un-
INTRODUCTION
The biological function of a protein is determined primarily by its 3D structure and the interactions it mediates. Thus, central goal of drug discovery is to design small-molecule ligands that bind to target protein and modulate its function to achieve therapeutic effect. Since change in protein function is highly correlated with the bound pose of the binding ligand, a consequence of energeti- cally favourable pootein-ligante interactions, the ability predict these structural conformations, which is the primary aim of molecular docking, is essential for reliable and accelerated drug discovery.
Deep learning approaches for molecular docking, in particular, diffusion-basee methods, have been recently touted as providing superior accuracy over traditional, industry-standard physics-based methods (Halgren et al., 2004; Morris & Lim-Wilby, 2008; Trott & Ólson, 2010). However, the need for such claims to be further qualified has been highlighted by Harris et al. (2023); Buttenschoen et al. (2024), who demonstrated that solely focusing on metrics, such Root Mean Squared Deviation (RMSD) between the bound and predicted ligand poses, can obfuscate the actual predictive ability of deep learning-basse based docking tools. For instance, Buttenschoen et al. (2024) showed that when controlling for the chemical plausibility of generated samples, deep learning approaches performed far worse than traditional docking methods. While notable progress on this front has been made with the advent of co-folding models (Abramson et al., 2024; Boitreaud et al., 2024; Wohlwend et al., 2024) such as AlphaFold3 (AF3), which need not assume the protein to be rigid structure, these models still have key limitations. Firstly, they require massive quantities of data and compute to train, hindering the ability of other researchers to contribute and improve such models. Secondly, co-folding models jointly model the structure of proteins and ligands, they suffer from slow inference, which makes their practical applicability to drug discovery computationally prohibitive; especially for high-throughput virtual screening (HTVS), where it is often required to query millions of protein-ligand pairs.
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To address these issues, we revisit the commonly-used torsional model approach (Corso et al., 2022; Huang et al., 2024; Cao et al., 2025) for diffusion- based molecular docking. This approach defines diffusion process over a ligand's global roto-trans tio-translatioon and its torsional angles. Fundamentally, structural (geometric) chemical constraints imply that this low-dimensional manifold forms the principal degrees of freedom underlying any chemically feasible pose. By operating over space with significantly reduced dimensionality. torsional models promise improved data efficiency, generalisation and faster inference over all-atom approaches (favoured by co-folding models). However, this has not been borne out empirically, with relatively disappointing results reported in the literature (Škrinjar et al., 2025).
In this work, seek to resolve the discrepancy between the poor performance of torsional models and the desired benefits of exploiting inductive biases from structural chemistry. We suspect torsional models underperform because the score model must implicitly account for the mapping from 3D coordinates to torsional updates, a non-local, highly nonlinear, and sometimes ambiguous inverse problem. To bypass this burden, we propose a fragment model. We decompose ligand into molecular fragments by breaking rotatable bonds; due to structural chemical constraints, we can treat each fragment's internal geometry as essentially fixed. The generative task thereby reduces to predicting an SF(3) rigid transformation for every fragment, from which any chemically feasible pose can be recovered by composing these transformations, obviating explicit modelling of torsional angles. Building this, we introduce SIGMADOCK, an SE(3) Riemannian diffusion model that defines a diffusion process over the translation and orientation of rigid-body fragments. During sampling, SIGMADOCK iteratively reassembles the ligand's constituent fragments into predicted bound pose. To reduce the additional degrees of freedom introduced from fragmentation
(compared the torsional model), we make the following novel contributions: (i) fragmentation
scheme that reduces the number of fragments required to represent ligand; (ii) soft triangulation constraints to provide further inductive biases on the preserved bond lengths and angles across fragments; (iii) an (O(3)-equivarii) architecture tailored for reasoning over fragment geometry and protein-ligand interactions.
We adopt the standard re-docking protocol in which the receptor is fixed (holo-conformatiinn and the binding pocket known. This choice is deliberate for two reasons: first, it is the Long-standing setting for benchmarking docking methods, and permits fair "apples-to-apples" comparison with classical and recent deep learning baselines; second, it reflects industrial TTTVS//aad-ppimisation practice, where rigid-receptor docking is computationally tractable at scale. Prior generative methods have struggled in precisely this regime, and our aim is to close that gap first.
Empirically, we demonstrate SIGMADOCK surpasses prior deep learning approaches' and tra- ditional physics-based docking, achieving state-of-the-art performance on the challenging Pose- Busters set (Buttenschoen et al., 2024), and generalising unseen proteins (Figure 4). We highlight that, with a fraction of the training data, training/sampling time, and lower test-train leakage, we reach AF3-level performance and substantially outperform previous generative methods the re- docking task. With this, we wish to state our main contribution as the careful and rigorous design of a well-characterised diffusion process, and detailed construction of structural inductive biases which help learn simpler functions for the task of molecular docking. We highlight that our frag- ment SE(3) m formulation naturally extends to flexible docking by treating selected side chains as additional fragments, and is also adaptable to co-folding, both of which we leave as future work.
2 METHOD
2.1 NOTATION
Molecular notation. The 3D graph of a molecular structure (e.g. ligand, protein or molecular fragment) is defined by the collection G = {x, v, b) where [9] is the number of atoms in the structure, x E R191x3 are atomic coordinates, v E I 999d xd, represents the features (e.g. atom type, atomic charge etc.) associated with each atom, and b E X represents the features (e.g. bond type, bond conjugation etc.) associated with each bond. Furthermore, we define the corresponding 2D graph of molecular structure by G2D {v, bj. For further details our choice of molecular featurisation see Appendix G.
Under fair comparison with models trained on the PoseBusters train-test split.
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t=T
Denoising (Reverse Time)
Noising (Forward Time)
Figure Illustration of SIGMADOCK using PDB IV4S and ligand MRK. We create an initial conformation of query ligand where we define our rigid body fragments (colour coded). The corresponding forward diffusion process operates in SF(3)" via independent roto-translations.
SF(3) notation. Let [x1,...,, EN] E Nx3 be collection of points 2; E TR3 in a rigid-body system where [:] denotes row-wise concatenation. The pose (i.e. position and orientation) of x can be parametrised by elements (p, R) from the Lie group SE(3) via the standard group action:
(p, R) x Ip + R 21. P + R NN).
where p E T(3) = RR is an element of the translation group and R E SO(3) a special orthogonal matrix representing rotation in R For further details about SE(3), see Appendix B.
Fragment notation. this work, we develop a novel fragmentation scheme which allows us to represent the 2D graph of ligand Pigand {v, b} in terms of rigid-body fragments {Gr; )m where Gr = {xp, VE, br} } for each F E (F)) In particular, we take Xy E IR Grlx to represent the local coordinates of the fragment centered at the origin', i.e. [r 1 = where
vector of unit entries; at high level, this can be viewed as the predetermined coordinates of a local, rigid substructure within the ligand due structural chemical constraints from Rigaad Hence, the 3D coordinates of the ligand can only be constructed from some arrangement of the rigid- translations and rotations of f&x, h=-1 Formally, we identify z = (p. R) E SE(3) T with the global coordinates Fxr; mm of the fragments through the usual group action: XF, (pF, RF:) ) X where = ( PF..) E (3)) and R = (R RE E SO(3) This allows us to parametrise the pose of the ligand in terms of SE(3) by ........ XFyl and we denote this mapping" by 9 SE(() m glSagmlxx
2.2 STRUCTURALLY AWARE FRAGMENTATION FOR SE(3) DIFFUSION
The task of molecular docking can be summarised as predicting a ligand's bound pose x E x33 for some query protein, given the 2D graph the ligand G = {v, b} and the 3D graph of the protein Sprotein = {y, Vy. by}. The guiding idea behind SIGMADOCK to exploit the inherent structure of ligand's topology simplify and condition smooth and well-defined rigid-body diffusion process in SF(3). In particular, we rely on a novel fragmentation strategy that preserves common stereochemical symmetries, reduces degrees of freedom, and creates of ge- ometric priors which help SIGMADOCK learn more general physicochemical correlations, A visual overview of the SE(3) diffusion process described in this section is outlined in Figure 1.
2.2. THE CONFORMATIONAL MANIFOLD
SIGMADOCK makes use of well-established thermodynamic priors in structural chemistry. Namely, leverages the fact that the conformational space x, defining the local geometry of a ligand (in vacuum) with known topological structure Gipana follows Boltzmann distribution with probability mass concentrated about manifold Me with holonomic constraints of the form:
M. {x, E IR Xigmi g(x.) 0}.
We defer the issue of choosing the orientation of the local coordinates Section 2.4. We abuse notation by leaving the dependence on {xx; y from the fragmentation {Gr, p implicit. HHolonomic constraints (see (Ryckaert al., 1977)) restrict the configuration space (position) of body.
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ABC
(a)
(b)
BCD
(c)
Figure 2: A: Illustration of a dihedral @BBCD across torsional bond BC, defined as the angle between planes ABC and BCD, across two adjacent benzene rings in ligand BFL: B: Bound (red) and aligned (green) poses for BFL in PDB IQ4G with an optimised aligmment RMSD of 0.11A; C: Conformational ensembles generated from TMM for ligands SKF, CEL, and IH5 respectively. Notably, the most significant structural changes are derived from torsions across the rotatable bonds.
m-vector of scalar holonomic constraints, encoding m independent (soft) boundary conditions. The latter represent locally con- served geometrical priors such as bond lengths (dAB dg) and bond angles (TABC To), and exclude dihedral angles/torsions (ΔABCD bodies adjacent to torsional bond defined over atoms B, C are free to rotate (Figure 2a). Impor- tantly, we can safely assume the holonomic constraints implicit in M. can be derived from Gigand and thermodynamic equilibria. A more formal definition of the form of 9(-) is detailed in Appendix D.I. Under this construct, the probability measure supported on M is the constrained Boltzmann measure TMM. which we may sample from: x TM, (Rgana) As shown in Figure 2c, although this distribution is complex and multimodal, there are abundant preserved symmetries: with RDKit's ETKDGv3 (Landrum (2025)) as our proxy for TMM,, we observe that, excluding global rigid mo- tions, the 3D structural differences in x. effectively dominated by changes in dihedrals (torsion).
where g R. @iippll T maps from Cartesian coordinates
Φ); although the distribution @BBCD is anisotropic,
To faithfully sample rigid body fragments, we first need to justify that samples SE(3) drawn from the conformational manifold Ma can be consistently aligned to the bound manifold (M&), whose distribution тM, represents the target (data) distribution of bound states. We verify this by aligning ligand conformations from M, MM via joint tro-trranlational and torsional registration. Concretely, let x. (-) denote a conformer on Me and let &e E T represent the correspond- ing dihedral angles which define the is torsional bonds in x Let v/(6)) T -> R. RlSupal x3 denote the invertible map, where T (S*)* = SO(2)* is the hypertorous defining the space of dihedrals. With (p. R)) E SE(3) representing global rigid motion, we use the total map
(p, R2 () = (p, R) (()) E RR
and perform Kabsch alignment jointly with torsional adjustment by solving
min PERI (3), TETE
RMSD(X>, V R3 +)),
Xb ~ M, (9
Empirically, we find that the RMSDs between experimentally bound poses and their correspond- ing aligned conformers x = (pp", Rt, d³) are substantially below both (i) standard error rates reported by docking baselines (Buttenschoen monly used success threshold of 2A. This provides sufficient support to claim that the variability in bond lengths and bond angles subsumed in Ma can be generally ignored in the task ligand docking. Crucially, this allows us to treat bound states as being approximately contained in the set of structures reachable by torsions and SE(3) transforms
al., 2024; Harris et al., 2023), and (ii) the com-
conformers drawn from TM, for any align x X5 with negligible error such that RMSD(X,. x()) < 2A. See Figure 2b for an aligned example and Appendix D.3 for additional results and empirical analy- sis. This empirical inclusion is paramount our approach as it justifies assembling our stationary
x. MM and x, E MM,
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distribution from fragments sampled from Me, without falling out of distribution. Throughout the remainder of the paper, we will absorb the alignment into the notation by writing x + X < X/3
where we use the aligned conformation x at the start of forward noising process.
2.2.2 CHALLENGES OF TORSIONAL MODELS & MOTIVATION FOR OUR METHOD
The idea behind directly modelling dihedral angles has been adopted as the standard approach to modelling small molecules (Corso et al., 2022; Jing et al., 2023) and amino acid side-chains in proteins (Jumper et al., 2021). However, formulations that directly model time-dependent dihedrals @ABCD (x., 1) via torsional updates suffer from fundamental caveats which we aim to resolve in our approach. In Theorem we show that the induced torsional density in Cartesian space is generally not a product distribution, leading to highly entangled implicit dynamics. For further details and a proof of Theorem see Appendix C.2.
Theorem 1. For standard molecular topologies, torsional models define nonlinear mappings from torsion angles to Cartesian coordinates, producing highly entangled, non non-product induced mea- sures. In contrast, disjoint rigid fragments yield a factorised product of Haar measures on SE(3)"
Consequently, argue that diffusing a molecule in fragment space SE(3) offers simpler learning task than diffusing in torsion space T x SE(3). Intuitively, local changes in torsional angles often produce non-local Cartesian displacements, creating strong geometric coupling along torsional chains: a change in a single torsion can substantially displace remote atoms. Therefore, independent torsional perturbations become correlated once mapped to Cartesian coordinates (where the model observes the data) under the induced measure, breaking the product structure. This leads to an ill-conditioned learning problem and stiff sampling dynamics. contrast, our for- ward kernel factorises over disjoint (3)) fragments (product); inter-fragmen correlations enter only via the learnt score, rather than being induced by the noise, yielding simpler, better-conditioned mappings, and more stable reverse-time integration.
Furthermore, mapping a torsional increment Δé; to a Cartesian displacement Ax is intrinsically ambiguous: one must chose extrinsic gauge (which side of the torsional bond is rotated, which combination). Implementations often apply heuristics such as RMSD alignment to remove the net rigid motion caused by torsional updates, which would otherwise break the product-space structure. However, this does not mitigate the ambiguity of the intrinsic to extrinsic mapping, especially when torsional steps are large. Practical solutions often commit extrinsic realisation (rotate left, rotate right, or a combination), and the model must learn a score consistent with that convention; this choice cannot guarantee consistency during sampling as the selected torsional realisation may not align with the true score direction. Moreover, as k (and thus molecular size and flexibility) increases, torsions produce amplified nonlocal Cartesian displacement (lever effect), coupling distant degrees of freedom. The combinatorial growth of possible extrinsic realisations exacerbates this geometric entanglement, making this framework unscalable. We hypothesise that, in general settings, these issues make torsional frameworks can become poorly conditioned and unnecessarily complex to model. These shortcomings motivate our approach of representing molecules via independent rigid fragments, allowing to operate over well defined and geometrically independent product space.
2.2.3 IRREDUCIBLE FRAGMENTATION & SOFT GEOMETRIC CONSTRAINTS ON SE(3)
The naive choice to define our fragments is to break the molecular graph obtained from T.M, the torsional bonds, producing a set {9r, 1 of torsion-free free rigid- rigid-body fragments with global coordi- nates parametrised by SF(3) Sh This approach yields a set of vn = (k- + 1) fragments with a total of 6m DoFs In contrast, we note torsional models have (k: + 6) DoFs (k torsional bonds in S and 6 for rigid body SE(3)). Thus, the natural question arises: How can we reduce the DoFs of the system and in turn abstract the problem in a general form? In SIGMADOCK we tackle this problem by creating a simple yet effective molecular fragmentation reduction (FR3D) that recursively merges adjacent fragments from m (k: + 1) down m (Figure 3).
Instead of biasing the fragmentation order, FR3D performs a stochastic search, starting from the torsion-free yn. fragments and branching through candidate neighbour proposing merge actions until reaching an irreducible set of size m. Hence, FR3D not only reduces the learnable DoFs by reducing
Asssumngg (rrl each fragment defined by its T(3) ex IR.3 and SO(3) parametrisation.
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the number of fragments, but also provides promising stream for data augmentation. Merging is possible in molecular graphs where two or more consecutive torsional bonds are linked, so there are topologies that are irreducible. Hence, + 1. For a fragment hyper-graph with m fragments (and no loop closures), the effective DoFs concentrate between k 6 trrangulationnindueee naïve fragmentation (m = k++1), this becomes k+6 < DoF S. 6m. As FR3D reduces m (empirically find m & f)) the upper bound shrink in practice. We refer the reader Appendix D.4 more extensive analysis and empirical results.
upper-boundes in the number of fragments: 1 S S
lower bound) and 6m (unconstrained). Under
During fragmentation, we retain torsional bond length and angle information by introducing dummy atoms at either side of the bond. Hence, 19igandl I Σi= 19F. Importantly, we only retain free dummy atoms in 9r, and otherwise prune dummies which are over-constrained. Here, we label dummy as over-constrained whenever FR3D merges the torsional bond it belongs with a neighbour- ing fragment, as naturally over-defines a dihedral angle (Figure 3b). Removing over-constrained atoms is fundamental since immutable dihedral sampled from тM, would violate the free tor- sional requirements outlined in Section 2.2.1, forcing our generator TMM. to yield structures that do not strictly overlap with the bound manifold under optimal alignment: V TT, (-) # J.M (+).
We refer the reader to Algorithm in Appendix D.4 for an overview of FR3D and further analysis.
(a)
(b)
(e)
Figure 3: Illustrative example of how FR3D reduces the number of fragments (colour coded) quired to represent rigid bodies on ligand TNK into irreducible form. A: Defining fragments by snapping all torsional bonds (ribbons); B: FR3D recursively attempts reduce the & torsional bonds and removes over-constrained dummies in the process (denoted by the coloured rings), which otherwise define a dihedral across the merged fragment; C; Over-constrained dummies removed and triangulation edges displayed under a different stochastic reduction (equiprobable to solution b).
Soft geometric constraints. A core ingredient of our method the inclusion of geometric priors mechanism provide soft (implicit) boundary conditions. Specifically, although FR3D pro- duces irreducible fragments, we define a triangulation distance conditioning scheme which enables pseudo-reductions the observable DoFs. Concretely, for any torsional bond BC connecting ad- jacent fragments AA and D, we define triangles (A, C) and (B, C, D) using neighbouring atoms E AA and D E 'D on either side of the set of dihedrals @ABCD across BC. Through Lemma 1, we show that by defining cross-fragment distances A C|| and B DII top of the rigid fragment template, the corresponding bond angles L(A, B, C) and <( C, D) become uniquely determined. See Figure 3c for an illustration, and for proof of Lemma see Appendix D.2.
Lemma 1. V)A,B) E (A,D) bond lengths I|A BlI, ClI,, ||C DII and bond angles ((A, B,C), (B, (B,C,D) D) are fully determined with triangulation conditioning, without restricting changes in the dihedral angles AdABCD.
2.3 SE(3) DIFFUSION
From our identification of ligand poses x with z = (p. R) E SE via the fragmentation ( 1m we adopt the SE(3) diffusion model framework introduced in Yim et al. (2023) to construct a generative model pe pa(z)aook) for sampling the docked pose of some ligand,
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given its 2D graph Rigand- its fragmentation {Gr, 1 and some query protein Sproein we use Odock = (iigand: 19 10: Grrosii) to denote this conditioning information. We provide an overview of this framework below, and for further details, see Appendix C.
Forward process. For each pootiin-ligadd pair (Gugand Gprescia) ) in our dataset Pdatas with an asso- ciated fragmentation ( J we define the forward process (Z()) )teju.21 with (t) pe(z(Goock) via the SDE:
(( R. 2e00.71
dz [- 20 di + dB
Rmxx
(t)
dB
SO(3)M
(4)
(1)
where B B denotes Brownian motion on Rmx3 and SO(3)" respectively, with the (0)))
Rmxxs:
initial condition () (p()), R R (x) where Gigand = (x, v, b) contains the ground-truth docked pose x. We note that the SDE designed for the forward kernel Prjo (z (z()) to be tractably
sampled from, and we take T > 0 large enough for pr to be close to the stationary distribution
((z) = (p;;;, O, I) @ Um (R), where U5o(3)m denotes the uniform distribution on SO(3)
Backward process. The associated backward process (2())E((OTT
[E([TT]] is then given by the SDE:
dz
1 1 + vp loggr- l V Z (+) Gaock)
dt+ dB (0)
dB (4)
SO(3)
where V log Pe (z( VV le Pe (dock), n log pe (ulok) denotes the score function of
the induced probability path pt; we note that this should be understood as Riemannian gradient which lives in the tangent space Tan, SE(3)
Training and sampling, We see that we can generate bound poses under Gaock from simulating the backward SDE in Equation 2, however, the true score function V= log Pt is intractable. Therefore, we train neural network approximation sz t, Gdock) via the score matching objective:
(()) Bp(t) Pan (Grgant preccis) (Z(1) (Z(1) )
se (z) ,t, Gdock) - V. v log Pr (Z))) Z
SE(3)
(3)
Hence, we denote po(z) (Zeek))) as the distribution of generated samples, from first sampling ( and then simulating the backward SDE with our learnt approximation 8g. which approximates the true distribution (Z)/))) The corresponding 3D coordinates & of samples 2 ~ po (a can
then be recovered by the mapping x = ((2).
2.4 ARCHITECTURE
A significant contribution of SIGMADOCK is the design of our architecture sp E, Gdock) which parametrises the score function. In particular, we augment EquiformerV2 (Liao et al., 2023) to handle protein-ligand (and other molecular) diffusion; we use this as the backbone for our model to ensure SO(3) equivvrriiae""" Our main innovations are: (i) we augment the input graph with virtual nodes and edges on top of the original chemical graph Gaock, creating a hicrarchical topology. This reduces risk of over-squashing by reducing the average node degree, whilst promoting global information flow and mitigating over-smoothing: less layers needed to pass global information; (ii) we tailor our featurisations of nodes and edges according to their structural role; (iii) we ensure messages and gradients along the edges, which represent local interactions (present proximity), smoothly decay to zero as the distance between the neighbouring nodes approaches some cutoff; this prevents instabilities from sudden changes in the input graph's topology we perturb
Moreover, we note that a critical issue for the design of our architecture is that the parametrisation of the global coordinates xy in terms of (p. R) E SE(3) is not uniquely defined. This is due to the fact that we do not have a canonical choice for the orientation of the local coordinates Xy. For instance, have the equally valid choices & Xxp for the local coordinates of Gr if X Ro &p where Ro E SO(3). Hence, we can have two different representations of global coordinates xF from XF
'EquifommeV2 is also translation invariant but we do not require this property since our problem setting has canonical centre of mass given by the hinding pocket.
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(p. R) adapt the (equiaaauva prediction head introduced in Jin et al. (2023), based on the Newton- Euler equations from rigid-body mechanics, to which we pass the outputs of our backbone model into. Particularly, we predict pseudo-forces for all atoms pertaining to the m fragments and use these as basis to define our scores in the tangent space of SF((3 (more details in Appendix G.4). With this choice, Theorem 2 shows that SIGMADOCK is invariant to the choice of local coordinate axes.
= (p, RR, R depending on the initial choice of orientation. To resolve this issue,
Theorem 2. Our training objective and sampling procedure are invariant with respect to the choice of orientatiinns for local coordinates. Moreover, our score model is 50(3)-equivan) (Z)e)) is stochastically -eqiivqquirri kernel.
Conditioning. We define the triangulation conditioning by feeding the relative distance mismatch as an edge feature (compact notation): AdA.c (x,, t/Gdock) = [|A(t) C(t) d' (with df defined from the initial RDKit conformer), such that lim Ad =[ = 0 across all cross-fragmns fross-fragmett triangulation edges. Dummy atoms which define triangular geometry are discarded after sampling; torsional bonds are reconstructed from anchors so there is no discrepancy between conditioning inputs and the final conformation. With this conditioning, only dihedral angles and rigid body roto-translations remain free as ( 0.+0.
For further architectural details, see Appendix G, and for a proof of Theorem see Appendix H.I.
2.5 TRAINING AND INFERENCE
We outline our training setup for SIGMADOCK in Appendix E. In particular, we discuss how we preprocess data for fragmentation and training, as well computational tricks for increasing training throughput. We detail the definition of the binding pocket in Appendix E. Briefly, the includes all residues with any atom within stochastic cutoff d, of any ligand atom. Formally, d, := do + (0, or), where do and σr default to 5 and IÅ respectively during training. Our sampling procedure is outlined Appendix where we discuss the fact that, due to the reliability of SIGMADOCK in generating chemically plausible samples, SIGMADOCK does not require the use of separately trained confidence model to filter out poor generations. Instead, we propose using the simple and cheap heuristic of evaluating both the (pseudo) binding energy of the generated protein- ligand system, as well as a set of physicochemical checks (such as, bond angles, bond lengths, internal energy) to rank our Nesdss samples for evaluation.
3 EXPERIMENTS
3.1 DATA AND METRICS
Datasets. We use PDBBind(v2020) (Wang et al., 2005), a curated set of 19,443 protein-ligand complexes obtained through crystallography, as our training set. Crucially, we deliberately restrict ourselves to this dataset for fair comparison", isolating any increase in performance obtained in this study to our proposed framework. For validation, we use the well established PoseBusters (Buttenschoen et al., 2024) and Astex (Hartshorn et al., 2007) datasets. PoseBusterssy2) (PB) acts as our temporal-split validation set containing 308 protein-ligant complexes with unseen protein sequences realised from 2021 onwards. The Astex (AX) dataset consists of an additional 85 diverse and highly curated protein-ligand complexes originally designed to faithfully evaluate the quality of protein-ligand docking algorithms.
Metrics. We evaluate generated pose x by measuring the symmetry-corrected RMSD (Meli & Biggin, 2020) between the crystallographic (bound) pose x and the generated pose x obtained by the mapping x (²)) where 2 ~ pa (z Gdock). To account for sampling variability, either from different conformers (inducing differences in fragment local coordinates), or by resampling (1) ~ q(z), we
report the Top-k success rate, i.e., the fraction of complexes where at least one of the top poses (from Nseds samples) has RMSD 2 We also use PoseBuster to assess PB-validity, indicating whether generated structures also satisfy standard physicochemical tolerances.
II is unfair yet unfortunately common in the literature to compare the same held out set with models trained with larger and more diverse datasets without assessing train-test overlap (Abramson al., 2024).
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3.2 RESULTS
Using the sampling algorithm described in Appendix F, we benchmark the base performance of SIGMADOCK and present our main results in Figure 4. To the best of our knowledge, SIGMADOCK is the first deep baaming-based method to surpass classical physics-based approaches in the PB and AX sets using the intended train-test split on the re-docking task Not only does SIGMADOCK achieve 6.3x higher PB-validity than DifiDock, the best open-source alternative tested on the same split, but it also excels on proteins with low sequence similarity, overcoming the common critique that deep learning models memorise rather than learn physics. Notably, Corso al. (2024) train DiftDock-LL on a significantly larger corpus (PDBBind(y2020) U BindingMOAAD and report 50% Top-1 (RMSD only) on PB, whereas SIGMADOCK attains a Top-1 (RMSD & PB validity) of 79.9%. Overall, these results support our main contribution: a seooreicclly-ggoundee diffusion framework with strong generalisation. Conversely, torsional-space baselines and point- cloud docking models evaluated under identical conditions do not attain comparable results.
SE(3)" Riemannian
We also highlight SIGMADOCK does not require minimisation to achieve high PB validity, a com- mon yet computationally expensive hack used to artificially improve deep learning methods. No- tably, we achieve AF3-level performance (Top-l of 84%: see Extended Data Fig. 4e in Abramson et al. (2024)) with just 19k training data-points, significantly lower truin-test leakage (see App. J), and 50x faster sampling. Together with an outstanding performance in the AX set, reaching near- perfect Top-1 (above 90%), we believe these results mark major leap forward in the feasibility and reliability of deep learning for molecular modelling.
is 4n (2-1)221)
ANGE
E ANGO
22.0
53.0
50.7 $8.0
28
155
22.8
135
32
12.7
19
Figure 4: Performance benchmarks. Left: Comparative performance of SIGMADOCK on the PB and AX diverse sets against prior methods. Extracted from Abramson et al. (2024); Buttenschoen et al. (2024). (*) Denotes classical docking; (**) Are not open-sourced. Right: Performance breakdown across sequence similarity splits in the PB set.
Ablations. To better characterise and highlight the contribution of some key components in our method, we perform an ablation study covering a set of training-tine time and test-time variables (see Table 1). Namely, we report the influence of our fragmentation merging strategy and triangulation conditioning, as well as the effect of including protein-ligand interactions as part of the compu- tational graph, and give empirical evidence of their relevance (4-12% relative improvement). In addition, we show how sampling fragments from Me vs. MM leads to a small but expected decrease in sample quality. By excluding PB-checks in the heuristic, SIGMADOCK maintains a high PB-valid Top-1. Finally, we show how increasing Nueeds improves performance at the expense of more com- putational overhead, and highlight the importance of our simple yet effective heuristic for ranking our samples and picking our best candidate(s).
On top of sequence similarity, stratify the PB set into distinct chemical environments determined by the nature of ligand interactions with additional co-factors (ions, crystallisation aids, natural ligands, other co-factors). We hypothesise that, since SIGMADOCK is deliberately designed (for simplicity) to exclude co-factors, higher failure rates should be observed when the true bound pose is realised in conjunction additional artefacts (co-binding event), as the setup partially observable.
For fairness, compare our method in the main body against models trained on the same train-test split. We do not include DiffDock-L. or AF3 in Figure 4 under fair scientific practice.
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After isolating the pooten-liganie pairs for which SIGMADOCK fails to generate accurate poses we find this hypothesis to hold true, as per Table 2. This result provides additional confidence that SIGMADOCK does not blindly memorise and hallucinate protein-ligand poses.
Table 1:
Ablation results (Top-1 accuracy (%) across the PB set) for different configurations. A-C are re-trained from scratch; (*); default.
Table 2: Performance analysis (Top-1 accuracy (%)) across PB subsets according the presence of various co-factors. The subset size is shown next to the co-factor species key. The failure rate represents the sample failure rate, averaged across 40 seeds, for all complexes in the subset.
Couf. Description
RMSD 2
PB Val.
(-) Tri, Cond.
71.9
67.1
PL Interactions
79.2
76,3
Frag. Merging
74.4
73.7
Co-factor Presence
RMSD 2
PB Val.
Fail. Rate
G
Sampling from M
86.4
85.4
Natural Ligands (17)
58.8
58.8
41.2
(-) Energy Scoring
67.2
66.1
(57)
75.4
PB Scoring
82.1
70.8
Other (60)
76.7
76.7
28.1
H
SIGMADOCK (Nusds
10)
74.7
72.2
Crystallisation Aids (37)
81.1
SIGMADOCK (Nocd
40)
80.5
79.9
None (165)
84.2
83.0
16.2
Docking tools are typically provided with a search region (e.g. bounding box or centre-radius) defining the pocket search space. To assess robustness larger pockets, corresponding greater uncertainty of the binding region, we sweep the deterministic cutoff do, whilst reducing the jitter 0r + O. As presented in Table 3, SIGMADOCK remains robust to larger pockets, with a moderate drop when operating outside the training support (e.g. packet = 7A is 2a relative to the training
mean dg- = 5, or 1). Notably, reducing the pocket size (- (--aut ouooooooox + 5A) does not improve Vina's Top-1 1 (57.2% vs. 56.0%), indicating that gains over classical methods are not attributable to smaller pocket definitions. Although cannot directly compare SIGMADOCK co-folding methods, we show competitive performance relative to AF3 with a fraction of the training data and lower test-train leakage (Table 4). We leave more detailed comparison in Appendix J.2.
Table 3: Sensitivity to pocket definition (PB set). Pocket diameter Spocket is the maximum pairwise distance between Ca 's across the se- lected Nis residues; we report dataset means.
Table 4: Per-seqqeence-siiillrtty comparison between SIGMADOCK (left) and AF3 (right) on the PB set. Values for AF3 are extracted from Extended Data 4c (Abramson et al., 2024).
Metric|
|| do(A)
4
5
Seq. Sim. (%)
Count
PB-Val. Val.
pocker (A)
20.9
22.4
24.2
26.2
[0, 30)
109
38
72 87
Nus
15.6
20.3
26.5
37.8
30, 95)
76
83
79
82
995, 100]
123
187
87| 78
RMSD <
80.5
81.5
78.3
69.8
PB-Val.
80.2
80.5
77.3
68.2
Total Avg.
308
308
79.9
80.2
We refer the reader to Appendix for extended results, and Appendix for a detailed discussion on the current limitations of our method and future work.
4 CONCLUSION
We believe SIGMADOCK represents major step forward in the reliability and feasibility of deep learning as promising tool for accelerating drug discovery. Moving away from torsional parametri- sation, our proposed SE(3)m fragment-space Riemannian diffusion model is, to our knowledge, the first generative method trained on the intended PB split to surpass classical dockers on the re- docking task. We extensively lay out the key components of our framework in the Appendices and open-source our codebase to proliferate reproducibility and further development, and we view exten- sions to flexible-receptor docking and co-folding as natural next steps to make SIGMADOCK more general and practical tool. We demonstrate the critical role of principled inductive biases in enabling superior generalisation and data efficiency, and hope our work encourages rethinking progress on geometry and conditioning, as opposed to relying on scale alone (Abramson et al., 2024).
Here we define failure if the majority of samples generated across N seeds have RMSD above 2
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AUTHOR CONTRIBUTIONS
As the main contributor, A.P. instigated and led the formulation, development, and analysis of SIG- MADOCK. A.P and L.Z. both developed the mathematical framework, proofs, and the key com- ponents of SIGMADOCK. A.P and L.Z. organised and wrote the manuscript. G.M.M, C.M.D and Y.W.T supervised the project, partaking in relevant scientific discussions and proof-reading the manuscript.
ACKNOWLEDGMENTS
AP is funded by EPSRC and AstraZeneca via an iCASE award for DPhil in Machine Learn- ing. LZ is supported by the EPSRC CDT Modern Statistics and Statistical Machine Learning (EP/S023151//)
AP thanks Kathryn Giblin for insightful scientific discussions, well Jochem Nelen and other PhD students in the OPIG and OxCSML groups for valuable exchanges. AP also thanks Yasmin Baba for her unwavering support and for compelling him take break from research, during which the core idea of this work emerged. I.Z. thanks Jessica Harrison for help- ful witticisms informing (subliminally) our work's moniker. The authors declare no competing interests.
REPRODUCIBILITY STATEMENT & CODE AVAILABILITY
All training and evaluation data used in this study publicly available. The full codebase for training. sampling, and evaluation used to generate the reported results will be released open source upon publication at
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APPENDICES
A Related Work
16
Further Details on SE(3)
17
B.1 Further Details SO
17
Further Details on SE(3) Diffusion
19
C.I Proof of Proposition 1
21
C.2 Proof of Theorem
21 ................................
C.2.1 Torsional Entanglement
22
C.2.2 Rigid Fragments
24
D Further Details on Fragmentation & Conformational Alignment
25
D.I
Holonomic Constraints
25
D.2 Proof of Lemma
D.3
Fragmentation & Alignment
27
D.4
FR3D
28
Training Details
29
E.1 Data Processing
29
E.2
Accelerated Caching
30
E.3
Training
31
F
Inference Details
32
F.1 Sampling
32
F22 Confidence Model
33
Architecture
34
G.1 Protein-Ligand Featurisation
34
G.2 Input Graph
34
G.3 Backbone Model
36
G.4 Prediction Head
36
H Symmetries
38
H.l Proof of Theorem 2
38
I Extended Results
42
L.1 Generated Samples
42
I.2 Sample Performance Efficiency
J Limitations
44
J.1 Current Limitations and Future Work..
44
J.2 Further Comments
44
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A RELATED WORK
Molecular docking. Traditional approaches to molecular docking rely on the combination of assessing ligand poses with hand-crafted physics-based scoring functions and search-based optimisation (Halgren et al., 2004; Morris & Lim- 2008; Trott & Olson, 2010). Despite the widespread use of such tools, these approaches have relatively slow run times and lower than desired accuracy. There is growing interest in the application of deep learning- -baeed methods to address these shortcomings. These can be broadly organised into learning scoring functions (Prat al., 2024; 2023; Méndez-Lucio et al., 2021; Zhou al., 2023a), regression-based prediction (Stärk et al., 2022; Lu et al., 2022) and generative modelling of docked poses. The latter can be roughly grouped into methods which condition on pocket-specific interactions (Plainer et al., 2023) (as opposed to blind docking), restricts the generative dynamics to the degrees of freedom spanned by global transformations and updates torsional angles (Corso et al., 2022: Cao et al., 2025), allow for the unconstrained atom-level generation of poses (Stärk et al., 2023), and the flexible modelling of protein side chains (Huang et al., 2024). Closely related are co-folding models (Abramson et al., 2024; Boitreaud et al., 2024; Wohlwend et al., 2024) which aim to jointly generate the 3D structure Despite the impressive results claimed over traditional docking tools, measured in terms of RMSD, Buttenschoen et al. (2024) demonstrated that when controlling for the chemical plausibility of generated samples, performance falls below even traditional approaches. We note that co-folding models have reported good performance on the PB benchmark, which are higher than traditional docking methods, used by Buttenschoen al. (2024), but the comparison with the specialised deep learning-bast based docking models mentioned above is not exactly fair, due to the large gap in data and compute used to train co-folding models, which also results in substantially longer inference times. SIGMADOCK improves over previous methods in that it is able to quickly and reliably generate poses with high chemical plausibility and accuracy even without the need for a separate neural network-basse -bssed confidence model to filter out poor quality samples.
protein-ligann complex for docking, as opposed to only modelling the ligand.
SE(3) diffusion. Euclidean diffusion models (Ho et al., 2020; Song et al., 2020) construct a gen- erative model of data by learning the time reversal of some fixed Gaussian noising process. This class of models have been generalised to data residing on non-Euclidean manifolds (Leach et al., 2022: Huang et al., 2022; De Bortoli al., 2022), in particular the Lie group SE(3) (Yim et al., 2023) which is commonly used to describe the position and orientation of rigid-body systems. By formulating the generative dynamies for given ligand over the position and orientation of its frag- ments in SE((), SIGMADOCK avoids the complexity and poor training dynamics of the torsional model approach popularised by Corso al. (2022) for generative molecular docking.
Fragment-basee models. Working with molecules in terms of their constituent molecular frag- ments popular approach due the close relation between the properties and geometry of molecule and the composition of their fragments. Examples include lead discovery for drug devel- opment (Xu & Kang, 2025), modelling proteins in terms of rigid backbone frames (Jumper et al., 2021; Yim et al., 2023; Watson et al., 2023), generating synthesizable molecules in terms of building blocks and their reaction pathways ( (Koziarski et al., 2024; Gao et al., 2024), and linker design (Guan et al., 2023). Despite the aforementioned popularity of fragmentation, to the best of our knowledge, SIGMADOCK is the first work to use a fragment- bagseet-bbsedd framework for generative molecular docking.
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B FURTHER DETAILS ON SE(3)
In this section, we provide an extended discussion on SE((3 and Lie group theory.
Group structure of SE(3). Let T(3) be the translation group in E$ with the addition operation (it is trivial to see that T(3) isomorphic R with vector addition) and let SO(3) be the group of special orthogonal matrices in R333 The group SE(3) is defined by the following semi-direct product: SE(3) = T(3) S3(3) where elements from SE(3) can be written as (p, Rt) T(3) x SO(3). We recall the following properties of SE(3):
The identity element e E SE(3) is defined as = (0.1);
The group operation is defined by (p, Rt) (p', = (p + Rp', RR(); For any (p, R) E SE(3), its inverse is defined by (p. R)- = (- -R pp R-
Furthermore, we define SB(3)" = II SE(3) as a product group.
Lie groups. A Lie group is smooth manifold G equipped with group structure, such that the group operation : G x G + G and the inverse operation (+)-1; G - G are both smooth maps, considered under the smooth structure of G. Common examples of Lie groups include S33) and SE(3). Additionally, for each Lie group G, we have its associated Lie algebra g defined as the tangent space Tan,G of G the identity element E G For mathematical introduction to smooth manifolds and Lie groups, we recommand Lee (2003).
B.1 FURTHER DETAILS ON SO(3)
We provide a further discussion on the Lie group SO(3) which is relevant for our construction of SE(3) diffusion model.
Lie algebra basis. The Lie algebra s3) of the Lie group SO(3) has a canonical basis described by the following skew-symmetric matrices:
0
0
0
0 -1
e1 =
0
-1
e2=00
e2 =
eg
i 0
0 0
0
0
We use [133 : s3) - 3 to denote the standard coordinate representation map which linear isomorphism (i.e. maps the canonical basis to the canonical basis of IR3)), and we denote [-]x RR3 s0(3) as its inverse.
The above basis can be derived from differentiating A A(t) I where t + A(t) is any smooth curve in SO(3) such that A(0) = I, and using the fact that SO(3) an embedded submanifold of R (by the regular value theorem) so that the standard Euclidean derivative A'( of A(t) can be considered as an element in Tan R R which contains as a linear subspace, the tangent space of SO(3) (considered as the image of the differential of the inclusion map). Moreover, for any R E SO(3), we can define the canonical basis for Tang SO(3) {Rei Reg, Rea} by mapping (ei, e2, e2, e through the linear isomorphism given by the diffcrential the left-multiliication map Ln(R') = R.R' (this map is diffeomorphism as smoothness is given by the definition of the Lie group and we have the smooth inverse LR (R') R. RR).
We note that another common representation of so(3) is the axis- angle parametrisation -.e. for any S so(3) there exists some u E S C 3 (the axis) and w E E (the angle) such that S [
[wu]xx
Exponential and logarithmic maps. The exponential map exp -> G and its inverse, the log- arithmic map log G + are fundamental tools for studying Lie groups. They provide canonical (local) diffcomorphism between the flat Lie algebra and the curved Lie group. thus allowing for the parametrisation of the manifold by an casy-to-work-wii vector space and translating geometric problems to problems of linear algebra.
lI Strictly, this should also be considered as equipped with Lie bracket.
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In the case of SO(3), the exponential map exp : 5o(3) + SO(3) has an analytic form, given by Rodrigues' formula (considering s0(3) in terms of its angle-axis parametrisation):
sinw
cos s2
exp(S) I +
S
w
Furthermore, the logarithmic map log: SO(3) + so(3) has the form:
0
log R
2sin 0 (R R where Tr(lt) 2 cos 0.
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C FURTHER DETAILS ON SE(3) DIFFUSION
In this section, we provide extended discussion on the construction of diffusion models on SE(3); this is mainly recapped from the seminal work: Yim et al. (2023).
Riemannian metric. In order define diffusion process on the Lie group SE(3), we note that the generator for Euclidean Brownian motion is given by the Laplace operator A. Therefore, to de- fine an analogous (diffusion) Markov process on SE(3), we can use the Laplace-Beltrraii Beltrami operator, which is generalisation of the Laplace operator to Riemannian manifolds. This requires endowing SE(3) with Riemannian metric.
First, we note that tangent space to SF(3) can be expressed as direct sum: (pn R) SE((3 = T(3) @ Tan S3(3) which we equip with their canonical bases. Yim et al. (2023) then proposes the choice of metric ())(s) defined by
((a, (a', (5))ss())) (a, TT() (S,55))0(3)
where (a, (). (d' (d) Tan SE(3) for some (p. Rt) E SE(3), and (a, (',)') Ei= a; and (5,8") so(3) Tr(SS'T) the canonical metrics for T(3) and SO(3) respectively. We note that the canonical basis of (p. SE(3) is orthonormal under this metric. Essentially, this choice of metric allows us view SE(3) the Riemannian product manifold T(3) x SO(3) and allows for the factorisation of diffusion processes over translations and rotations. The Riemannian metric on SE(3) is given by the standard extension.
Noise schedules. For case of presentation, only consider non-time dependent forward process in Section 2.3, however, following Yim et al. (2023), our actual implementation fixes T 1 and uses the following decoupled time-scaling of the forward translational and rotational SDEs:
Our translation SDE is given by
dp (2) = 3
3(t) V B(t)dB Rmx33
where B(t) 8min + {(3max Bmin).
Our rotational SDE is given by
(1) dR (t) = (g(() 50(3)m
where s(()
d (1) and o(l) log((ex)( + (1 - 1) exp((min))
at
The backward SDE then has the form:
dz 31 t) + 8(1 V, log P1 (0) dt +
v V t)dB ( xa Vg tjdB (0(3)
g(l - t) log
Going ahead, will refer to these the definition of our forward and backward SDEs.
By the structure of the metric (., sE the forward kernel (zj(z(()) (z) (for ease of presentation, we only consider m here) can be factorised into independent kernels over T(3) and (3(3).
Forward kernels.
For the translational component, we have the standard VP forward kernel:
(p aap (1 (--a)))) I)),,, where a, exp (
(-1 1 L ((s)ds)
For the rotational component, the forward kernel is given by:
p( (R(0) R!(0) IGso (R R()) o² (1))
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To sample from R()) (Gso(a) (R(*); R()) ,²)), first sample R (R:
through independently sampling w E [0, n] according to the following density:
f(w,o) 1 cosw
Σ
22l 1)e 1((1))0//2 sin (1 + })w)
π
sin(w/2)
and sampling u Uga where Uga denotes the uniform distribution on the sphere (e.g. sampling from isotropic Gaussian and then dividing by the norm). We then compute R exp([uu]x) [G)o(3) (R; o²) and return the sample: R(t) RR IGsora) ( R(O) o²))
To sample from f(w, σ), following previous work (Leach et al., 2022; Yim et al., 2023), truncate the infinite series in order to compute an approximation to the distribution's CDF. We then cache these values and draw samples with inverse transform sampling.
Stationary distribution. To sample from the uniform distribution Uso(3) over SO(3), we inde- pendently sample w E [0. according to the following density:
p(w) = 008 w
π
and sample u E Uga. We then return R exp([wu]j ~ Uso(3) We note that this is equivalent to sampling from Ig (0, o²) in the limit as o + 00.
Conditional scores. For the score matching objective, we require the form of the conditional scores over translations and rotations. For translations, this a standard computation, and for rota- tions, we first require the modified function: fo(w, o) [w, o o//p(w).
This is due to the fact that [w,, o) denotes the marginal density of the angle w, hence the density I9soraj (R();; R(O), o²), which has support over SO(3) instead, requires different factor. We then have the following formula:
Vn log log (R (0))= = R(())
w(t) log R0.0 (sl((w(t),(()) fu(w(t),a(tt)
where ((0,r) = (R R(O) RR(() and w(t) = (((()))) is the angle in the axis-angle representation of R(0,t) this can be computed from the equation: Tr R(.,t) = + 2 cos(ww Similarly, we truncate the infinite sum in fo(w, 1) in order to compute the above score.
Loss scaling. We base our loss scaling off of the standard heuristic given in (Song et al., 2020) from the reciprocal of the expected value of the squared norm of the conditional score. For the translational loss, it is a standard calculation to show E V log
x X T-
For the rotational loss, we have the following proposition. For the proof see Appendix C.1.
Proposition The expected squared norm of the conditional rotational score is given by
(2athile folw.
X
E
V log (R (1 R(0) R(()
= (E-
S013)
We then define our final loss scaling by X; C²/ /Âe for translations and X, /² for rotations where CP, C. > O are some chosen prper-parameters. We note that we numerically compute the expectation in XR from our truncated approximation to o).
We note that it is surprising that, to the hest of knowledge, this proof is not seen the literature.
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C.I PROOF OF PROPOSITION
Proof. We first note that
V log Pojo R()) R
(
S0(3)
(())fo(u()).a(t))) (1)
(slo(w().o()) "
Th (R(0 (R(0 log R(0, (log R(0.0))T (T(())))
2
(Lfo(u()).0()) ? 2
Ir (log Rlo. (log R(0.0))).
(())fo(((()
due Tr(AB) Tr(BA). Moreover, since R(O IGso (R ; (t)), know that R(O)) exp([wu]x/ R with w ~ (L)) and u Uga which implies that R
exp([wu]x) exp([-wn), by the definition of [+)x and using the fact that exp(S) exp(S by the definition of the exponential map. As have exp((wu]x)) Moreover, we have w(exp([wu)x)) = w as w(+) returns the angle of the input rota- tion
conclude that k(0.0)
u,
This allows us to rewrite E V log (R R ( R ss0(3) as
E
V r Pojo (2( ((0))
8O(3)
( (Bufolu.o(tt) a(t)) (ofolw,o(t))
Tr (log ((ppp (log exp( (pp[[w]].p. [[w])x)) x))
2
Further, we see that log exp([uu)x) = wulxx and that
2 Tr[[wu] x (w)) = w² |uul = wa as
E S². Putting this together, we can conclude
[
(00tolu ((1)
folw, )
V log Pejo (R (R
(O(3)
C.2 PROOF OF THEOREM 1
Let molecule with N atoms have Cartesian coordinates x (NT X NN3 where each T, E 33 Let px := X' - gt be any target density Cartesian space (with respect to Lebesgue measure dx), covering equilibrium Boltzmann distributions, forward-diffusie marginals Pt. empirical data density, etc.
(x(x)= = exp(-3E(x)))
Z
where / is thermal constant and E(x) R N + R is the energy (scalar) function.
The purpose of this proof is geometric: compare the Riemannian (Jacobian) base measures induced by two different parametrisations of the same constrained Cartesian manifold and show that, under standard molecular topologies, the torsional parametrisation generically does not induce a product base measure while a disjoint rigid-fragment parametrisation does.
Torsions. Fix all bond lengths and bond angles and parametrize the remaining internal degrees of freedom by k torsional angles (øi...., Φ2) E I where Φ; E S = SO(2). Together with a global rigid-body pose (R, p) E SE(3) this gives the parameter space U = SE(() x T Let
s U SE T Nx3 u (R, H
denote the parametrisation of Cartesian configurations with prescribed fixed bonds and angles.
This is trivial to show.
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Rigid fragments. Split the molecule into disjoint rigid fragments with local coordinates Xy; R 1x3 for i 1..... Let (21,.... E....... 2m) E Z SE(3)") with 2, (pt, R.) E SE(3), and
define
: RR
X,
by placing each fragment via the group action
XP. R; XF. + Pt.
i= 1.....,m
Assuming the fragments are disjoint (no shared atoms) this yields an explicit parametrisation of the constrained Cartesian configurations obtained by rigidly placing each fragment.
Important distinction. The data density px at = 0 is identical under any parametrisation, since reflects the true molecular distribution. What differs the forward (noising) process: the marginals (x())) depend on the parameter space in which the dynamics are defined. In other words, while the starting distribution (x(Xx) is fixed in Cartesian space, the choice of coordinates (e.g. torsions or rigid fragments) determines how the forward dynamics evolve and hence the form of the intermedi- ate marginals.
Preliminaries. For parametrisation P denoting either paramctrisation map (t' y)) and pa- rameter y in the corresponding parameter space, define the Jacobian
an Ty
Jnly)
E Baaxd
where d is the parameter-space dimension. Jaly) T Jn(y) E Edxd The induced Riemannian volume element on parameter space is
We define the Gram matrix Gn(y)
dpn(y) / det Gy dy,
where dy is the Lebesgue element in the local parameter space. For densities px X and py parameter space y/ the laange-of-variablee (push-forward) relation is the equality of measures
py(y)dus(y)) px px(x)dx
which simplifies
pyly) = pxx(((y)) /de Gsz(y).
Equivalently
Py(y)
px((1(xx))
deet Gy(y)
C.2.1
TORSIONAL ENTANGLEMENT
Proof. Torsional models suffer from geometric coupling and a highly intricate non-product induced measure.
Consider the torsional parametrisation v/. The parameter space is
U = SE(3) x Tk
(R,p. x))
and the torsional parametrisation is the map va : U + Nx3 u+t x. Its Jacobian and Gram matrix
(u) E (3N*((+6)
Orp Ou
Go(u) J (w) ((++6)x((++))
and the induced Riemannian volume on parameter space is du = / det Gw du.
Index torsional coordinates by i.j E (l....,ki k). The column Jy corresponding to de the in- stantaneous Cartesian displacement vector obtained by varying torsion Φ.. Writing atomic positions J, E 3 (1,..., ,N), the torsion-torsion Gram matrix
3N
Ox
Ixa (u).
Ge) (JJ)) (Jo))) = Σ
(u)
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In standard bonded molecular topologies (chains, branched trees, rings with shared downstream atoms) a change in single torsion displaces multiple downstream atoms. Additionally, the sets of atoms that move when different torsions are varied typically overlap (i.e., the same downstream atoms are moved by multiple torsions). Hence, there generally exist pairs o j and atoms a for which both derivatives are non-zero, producing off-diagonal entries (G&); kj * 0. Thus, the torsion block (u) is generically non-diagonal. Since the determinant of matrix with non-zero off- diagonals does not factor as a product of single-coordinate functions, det G (u) does not decompose as IT, 9s (o;). Consequently, the induced volume form
(up V det Gv(u) du
not a product measure across torsions.
Therefore, independent perturbations in torsion space is mapped to Cartesian marginals with metric- induced correlations: torsional independence does not imply Cartesian independence. As result, kernels in torsion space correspond to correlated and anisotropic displacements in Cartesian space, complicating diffusion processes. In addition, model that observes Cartesian coordinates must implicitly learn this metric-induced coupling, yielding unstable training dynamics, and degeneracy during inference.
Gauge ambiguity (complementary issue). A torsion angle by itself does not uniquely determine which rigid sub-body rotates about bond; mapping a torsional increment AΦx to Cartesian up- dates requires choosing canonical extrinsic gauge (heuristics such RMSD alignment are often couuation-dependen and non-differemtiabll): An update AgABC defined in S can rotate bodics A, and B, equiprobably such that A BABC ABCD Since we have 2 loose variables that are not determined by the torsion alone, the mapping from AdAROD to Cartesian updates is unde- termined, systematically pushing the marginal densities to lower likelihood regions at inference.
Global-pose coupling. Augmenting torsional coordinates with global pose (R, SE(3) (i.e. treating the parameter space as T x SE(3)) does not in general restore a product space.
Torsional changes typically (i) shift the molecule's center-of-mass (CoM) and (ii) alter its orientation-dependeemt moments of inertia. As a result, the Jacobian acquires additional non-zero cross-terms between torsion and translation/rotation columns. Concretely:
The translation-torsion cross term proportional to the net displacement of the molecular CoM induced by torsional de:
d dΦ:
dx. CoM(o:) & Σ ma 0Φ:
where, for uniform masses, this reduces to the unweighted centroid shift). This quantity is generically non-zero, torsions couple non-zero translations about the CoM.
The nration-torsion cross term is proportional to the instantaneous change in angular mo- mentum induced by the torsional displacement field:
Òr.
Σ ma La
DΦ:
which can be interpreted as the torsion- induced torque on the molecular frame. This term is generically non-zero, so torsions also couple rotations.
Only in rare cases where both the CoM shift and the torque vanish for every torsion (or their mass- weighted analogues vanish under a physical metric), or in infinitesimally small time changes, do torsions decouple from global SE(3), yielding a product structure T x SF(3). Although heuristics such RMSD alignment mitigate this issue, this is not a general solution and is only strictly true for infinitesimally small AΦ): RMSD can only attempt to mitigate net body motions induced from torsional updates.
metric and an approximate be applied to
Degenerate exceptions. If two torsions act on strictly disjoint atom sets (no downstream overlap), only single torsional bond exists such that it cannot form branch chain, the corresponding Jacobian columns have disjoint support and the associated blocks of G may be diagonal; in such degenerate topologies the geometric term factorise.
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C.2.2 RIGID FRAGMENTS
Proof. Rigid-fragment parametrisation induces a true product measure.
Partition molecule into rigid, disjoint fragments with local coordinates Xf, 11 {1.-.. mm) The parameter space is
SE(3)",
(21 11 2m);
21 -RR.
and the rigid-fragment parametrisation is the map
9 - x, XF RiXF. + PL
which by construction is block-wise rigid:
x = p (.,,,,...,, 2m) = θ
(Ri, &F.-
Its Jacobian, Gram matrix and induced Riemannian volume form are (z) 0e Oz
E RR3Nx6m
Gx(z)= = J. J. (z)Jp(=) 26mx6m
dpp(2) v det (2) dz.
From the definition of the block-wise rigid mapping in &o and the disjointness in the fragments, moving one fragment does not affect the Cartesian positions of any other fragment. Concretely the Jacobian has block-diagonal structure
J11
0
0
0
J22
0
OXF. J&
Jx(2)=
1x
Ozi
Jmmm
As result, the Gram matrix is also block-diagonal:
Gp(=) = diag( J. JI.... JTJJ Jm),
so that the determinant factorises:
m
det Gx) = II
det (J J.).
Hence the induced volume element
d1)((2) Vdet (z( II dot (J J.) dzy.
product measure over fragments.
Therefore, independent noise applied to each fragment in the parameter space remains independent in Cartesian space via the linear rigid mapping. This decoupling makes diffusion generative modelling much simpler via the rigid body SF((3) parametrisation.
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D FURTHER DETAILS ON FRAGMENTATION & CONFORMATIONAL ALIGNMENT
D.1 HOLONOMIC CONSTRAINTS
Let (œ) ,.., ( holonomic map
IR (Signnal be the Cartesian coordinates of the ligand atoms. The
g(x) = (9. (x)
is built from scalar constraints that enforce bond lengths and bond angles inferred from the molec-
ular graph Tigand Below we give simple valid forms which (softly) approximate the holonomic
constraints assumed in the construct of our method.
Bond-length constraint.
For a bonded pair (A, B) with equilibrium length dAB = do define I(AB; (x) ||ABl| - do.
TAB =
Thus ((AB) (x) ~ 0 enforces
||aB||| do. The gradients (akin vector field forces) are (g(AB)
dg(AB) YAB
FAB
|
Orn
|
and 0 = 0 for B).
Bond-angle constraint.
For bonded triplet (A (A-B-G C) with equilibrium angle TABC To let
FAB
TCB
TAB - TB.
TCB IC BB
|raB]]
U
|rce|]
We use the cosine form
9(ABC)(X) (x)
To
so (AAOO) (x) == enforces the desired angle. The partial derivatives are
@9(ABC)
1
dg(ACC]
(I uu )
(I
vv T)
||AB||
Oxc
||rB||
and
0g(ABC)
0g(A
09(ABC)
0xB
Oz
Ozc
0 hard constraint, relax the requirements by instead assuming stiff harmonic penalties. For a bond and an angle the typical potentials are
Soft (harmonic) enforcement. Instead of imposing (xx) =
Fboond (b xB) = AAB (Ir do) d
E (
ABC (w.v cos To)
For large force constants KAB, KABC > the Boltzmann measure concentrates near g(x) 0 and approximates the ward-constraines manifold; for finite (but large) constants the potentials provide numerical stability and allow standard unconstrained sampling while strongly biasing configura- tions toward Mc- These harmonic constraints are fundamental component in physics-based and template-based conformer generators such as ETKDGv3 (Landrum, 2025). Crucially, this assump- tion is paramount to our SE(() rigid-body framework it allows us to safely assume that the variation in bond angles and lengths is negligible (up to a certain error rate, see D.3) when defining local coordinates during sampling.
D.2 PROOF OF LEMMA
Let BC a torsional bond at the center of a set of dihedrals defined as 6.A.Bc:D and let A (Ap) denote the set of neighbour atoms the B-side (connected B), and D (Da)ger,
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denote the set of neighbour atoms the C-side (connected to C). Suppose the following pairwise distances are defined as:
VAp E A:
A - Ball, [Ba(() Cb(t)),, [14p(t) Cb(i)..
DD₃ ED: [[Ba(() CD(1),, ||Cb Dall, [[Ba(() D2(Ii
where gray terms denote cross-edges across fragments that are time-dependent and therefore vary- ing according to the forward-back backward diffusion process on SE(3). For clarity, we use the notation (t) to represent the node of the dihedral (.A.B.DD corresponding to fragment A. This is im- portant because B and C are duplicated (as dummy atoms) in the fragmentation process preserve torsional bond information. An illustration is shown Figure 5.
Ap
C*
Dq
B
B
CD
Ap
D'g
Figure 5: Visual helper for Lemma 1 showing corresponding atoms for fragment A shown in blue and fragment D denoted in red. For additional clarity, dummy atoms across torsional bond BC are marked with an asterisk.
Thus, during the forward noising process, (1) + Bp(t). Vt (0, 1]. and Bt = 0) Bp 0). Note that absorb time for the other distance definitions these are always preserved: dis- tances are fixed through time by from the canonical fragment template Gr (i.e, due to fixed internal bond angles). To simplify notation, we will also absorb time for all other variables for the rest of the proof but keep the colouring.
For every Ap the triangle (Ap, Ba, CD) is rigidified via dide-sieessde constraints (SSS). Equiva- lently, for every D, the triangle (BA. CD. D,) is also rigidified. Subsequently, for every Ap E A and D2 'D the corresponding adjacent bond angles /(Ap; A.: CD) and (BB CD, D,) uniquely determined by the fixed side lengths and therefore invariant under any motion that preserves the listed distances (in particular under any torsional update in A that preserves those distances).
Proof. Fix any Ap E A. The triple of distances {ILAp - BAll - - CDl13 specifies triangle (Ap, BA, CD) up rigid motion. By the law of cosines,
2 = a² c² 2ac cos
rearranging and plugging in terms, we define the cosine of angle
cos BA, 14p BAl + BA - Cp 2 || Cp 1
2|| BA BA Cpll
so the angle <(Ap, BA, CD) uniquely determined from the fixed lengths. An identical argument applies for any D, 'D using the triple BA Cp Il. Cp - D Dyll. BA D. Hence all listed adjacent bond angles are fixed whenever the cross-edge stated distances are fixed, and any motion preserving those distances (including torsional rotation that leaves them invariant) preserves the angles. In particular, the distances || - Cpl and BA - D2ll V (p, q) E A, D are the additional triangulation conditioning edges that pseudo-enforee these invariances through conditioning jointly with the symmetrical torsional conditioning edges | CDll, B Cp and the known (from
equilibrium chemistry) immutable bond lengths A| BAll d and Co Da dep- n
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Implications and remarks. The distances ||Ay--Cy -Dll and |Bx-DD -Dll are new conditioning edges introduced by triangulation: they do not typically appear as edges of the 2D chemical graph but are required to convert adjacent triples into SSS triangles and thereby fix angles via the law of cosines. Fixing these triangulation edges removes the continuous freedom in the adjacent bond angles: the angles become functions of the fixed side lengths. Consequently, triangulation reduces the effective (not strictly but trivially by conditioning) local degrees of freedom. In the idealised SSS case (one triangle on each side), the dihedral (signed) degree of freedom about BC' constrained a discrete choice (mirror sign) rather than a continuous rotation; in other words, continuous torsional variation that would change the triangulation edges is no longer allowed.
Note that using many Ap and Dg triangles generally over-determines the local geometry and can introduce redundant or mutually constraining distance equalities. practice, we select a consistent subset of triangulation edges and remove duplicates to avoid incompatibility. See Section 2.2.3 for more details on how we ensure that we avoid over-constraining the system by removing dummy atoms during fragmentation merging.
D.3 FRAGMENTATION & ALIGNMENT
Further statistics for the alignment procedure descrihed in Section 2.2. are shown Figure 6. By selecting the best-aligned pose from randomized conformers, obtain average RMSD of 0.21 Á between the aligned x, and the ground-truth X;- This provides strong empirical evidence that sampling fragments from x, does not push the model out of distribution test time. Moreover, the small alignment errors act as a form of regularisation during training, with typical RMSDs below 0.5 A for any aligned conformer. In contrast, aligning only a single conformer results an average RMSD of 0.39 A. In both settings, all 85 ligands from the Astex diverse set produced conformers that could be aligned below the 2 À acceptance threshold. These results indicate that, during both training and sampling, SIGMADOCK remains well within the distribution of valid conformers.
To further ensure that our x samples are not chemically implausible, we run iterative optimisation with PoseBuster checks in place. We find this to be helpful in reducing high-energy alignments, and as we show in Figure 7, this exacerbated in circumstances where a single alignment optimi- sation attempt is taken, where changes in internal energies above 10Kcal/mol are observed (highly implausible stereochemistry).
30
2012
300
2.0
3.0
Morecier of Sords
MilecderWe of tost
(a) Alignment Tries:
(b) Alignment Tries:
Figure 6: Pair-plot distributions across the alignment RMSDs (A), molecular weight (Da), and num- ber of torsional bonds across all 85 ligands in the Astex diverse set. We use the best (lowest RMSD) aligned pose from an imitial set of conformation(s) generated from different random seeds.
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p-00200 2w0:207
15.0
15.0
12.5
12.5
100
10.0
7.5
25
0.0
0.0
Energles (ecskmmlt)
(ccalmol)
Alignment Sceds: 5
(h) Alignment Seeds:
Figure 7: Histogram of internal energy differences (Kcal/mol) (bound pose vs. aligned pose) across all 85 ligands in the Astex diverse set. We use the best (lowest RMSD) aligned pose from initial set of conformation(s) generated from different random seeds.
D.4 FR3D
We provide more concrete algorithmic breakdown of the logic behind FR3D in Algorithm 1. Note that inference, we may choose different heuristics for RecMerge, allowing different valid-cut sets to be prioritised according chiralities, or prioritising larger overall fragments. However, we leave this for future work and use the random fragmentation merging strategy for both training and inference.
selection criteria, i.e. by filtering out fragments with broken
We quantitatively contrast the differences between the fragments produced by FR3D and the classic (k 1) approach, where all & torsional bonds are split, through Figure 8.
Degrees of freedom (DoF) Let the ligand be partitioned into rigid fragments; the global pose E SE(3)" with 6m DoFs (3 rotation, 3 translation per fragment), which far smaller than the 3NV DoFs (coordinates) of all-atom point clouds (N > m). Naïve fragmentation by cutting all l rotatable bonds yields in = + fragments, i.e., 6rn. DoFs, which exceeds the torsional parameterisation (k 6). Our FR3D merging reduces m empirically = 3 (kx+1), lowering the raw coordinate count. Across each cut bond we introduce two SE(3)-invariaa) Iriangulation distances (Sec. 2.2.3; Lemma 1) that fix adjacent bond angles and the torsional bond length while leaving the dihedral free. Denote by e the number of cut bonds (edges) between fragments and stack all distance constraints into e(2) 2r The hard-constrained manifold
Mraa = { E SE(3) T c(=)= =
has tangent dimension
dim Mira = 6m - rank(Je(z))
For free fragment graph (e m-1), non-degenerate point-to-pont tooitt distance constraints across a joint generically remove relative DoF (leaving a single revolute DoF about the bond axis), so rank(J.) 5(m-1) and
dim Mara = 6m 3(m = m 5.
Under naïve fragmentation (m k+ the hard (torsional) lower bound is l + 6. With FR3D, each merge removes one cut and we drop one triangulation distance the adjacent bond to avoid over-constraining the dihedral across adjacent fragments. Henee, the effective lower bound remains + 6 after merging, while the upper bound tightens 6m such that A + S DoFaf S 6mn.
Soft vs. hard constraints. In practice, triangulation is injected as soft, SE(()) (in3ar-in edge features c at each denoising step; no hard equalities are imposed. Thus trajectories concentrate near Mirag as 0, and the effective DoF lies between + and 6m. Consistent with Lemma 1, the only local DoF around a cut bond as - is the dihedral ABBBCD- Empirically, removing these conditioning features reduces PB- PB-validity by 12.8% (Table 1, Config A).
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R=1
8
v8lue
Number Pragments
Pragment Size
Fragmentt
@a) Astex diverse
25
20
Number of Pragments
Pragment Size
Nax Fragment Size
(b) PoseBusters set. Randomised sample of 100 ligands.
Figure 8: Whisker plots showing the number of fragments and the range (min and max) of fragment sizes (heavy atoms only) across the PoseBusters and Astex diverse set using either FR3D, or the classic (Ie + 1) fragmentation scheme.
E TRAINING DETAILS
E. DATA PROCESSING
Here, describe how preprocess each sampled protein-ligand pair (Gigand Gprotcia) Pdata-
Molecule parsing, We define proteins and ligands in raw form via the standard PDB and SDF formats. We parse ligand files with RDKit (Landrum, 2025) and force sanistisation ensure correct kekulisation and featurisation. We parse protein files via BioPython (Cock et al., 2009), removing water molecules and other heteroatoms (ions, crystallisation aids, and other co-factors). For both, we remove all hydrogens (except, for ligands, those necessary maintain correct valence). During training, we filter out ligands with more than torsional bonds and/or those with higher molecular weight than 750 Daltons (excluding hydrogens). Since FR3D permutes across possible fragmen- tation merging options, this allows us to better scale training, whilst also maintaining our training distribution inside the desired range of drug-like compounds.
Pocket selection. For a given protein-ligand pair, we define the pocket by including every residue that has least one atom within a specified cut-off distance do from any atom in the bound ligand pose. To help de-bias the selection of the pocket we use stochastic cutoff d, = do (O, or), where we use default values of a, 1, Å). If there are multiple ligands present in the SDF file due to the presence of multiple chains, randomly sample one. We note that we include residues from protein chains as this is the closest representation of what the ligand sees during its binding event.
for training and a, 0.5 for sampling (all in
We define the pocket center of mass, which we define as the origin for our stationary distribution, by finding the center of mass of the pocket residues and adding an additional noise Epocket E R
[Srectel Ex.eGratn I, (0, CCMM33). This is in attempt to further de-bias the selection of the CoM away from the bound ligand's CoM and introduce robustness to the often arbitrary choice of the CoM in practical settings.
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Algorithm 1: FR3D Fragment REduction for Molecular Docking
Input: molecule mol, candidate torsions C, seed 8, optional max iters
Output: reduced fragments (Gr²)
init RNG with cuts C
// start from all torsion cuts // valid cut-sets
+ Re ReeMe Ree (mol, cuts, sorted(C), sample cuts uniformly from S frags +- $ safe_f sefe_fragment (mol, cuts)
// stochastic augmentation // Sanitize
foreach frags do
f f remove www.ve.overconsttee.ooooooo wwt..oooooooooaooc(11
return frags
Function RecMerge (mol, cur, cand, start)
+ [frozenset(cur)}
for <- start to candd| do
b 4 candli]
if b cur then
new +- cur (b)
if @fVaiddSate (mol, new) then
add frozenset (new) to
V VU PeUMReMerrgenne (mol, new, cen,d,,, 1)
return V
Function ValidState (mol, cuts)
parts +- safe_ safeeffrggment (mol, cuts)
foreach p E parts do
Pelcan remooee_veeroonstaabeddll
Polenn > and detect torsions (Pelea)) then
return false
return true
Defining fragments. For sample (Gigand: Gprotein) Pdata we generate a conformer via RDKit's ETKDGr3 (Landrum, 2025) from Pigand align to Xs via our Kabsch aligment procedure and split the fragments selected by FR3D, yielding independently SE(3) bodies. We initialise space (( (p()) ()) SE(3)" by trivially defining (0)) the CoMs of cach fragment as described in 2. This canonical choice ensures our coordinate- trrame-invarian pseudo-force construction out- lined in G.4 is valid. Furthermore, since our training objective is invariant the relative orientation of local coordinates XF, we initialise (0) at the identity.
Data scaling. We scale the global coordinate system by factor 2.7. This factor is derived from the isotropic dimensional scale MMll. computed as the average standard deviation (in Å) of fragment CoMs relative to the pocket CoM across all ligands in the training dataset. The total normalisation of the coordinate system is therefore x + the stationary distribution (prior) provides a faithful representation of the spread of states in Zo, consistent with the variance-presee prreerevpres form of the forward kernel on m3
-pockee
Tpolee In effect, this scaling ensures that .Mal , <- [M&l
E.2 ACCELERATED CACHING
We accelerate sampling of complex molecular graphs by creating cached sampling algorithm as shown in Algorithm 2. Effectively, we use FR3D to process particular protein-ligand complexes individually to create cache of size C and then sample Rt random batches of size B. Each time we sample a particular complex from the dataset we sample time t E [0, 1] independently such that re-use loaded complex R times, artificially reducing the computational bottleneck by factor R/C. C. By default we set R to and to 2, such that we speed up processing time by a factor of 4. We pick relatively small factor ensure we sample across multiple noise levels with
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sufficient diversity in Gdock such that limit the risk of non-independent and identically distributed (iid) sampling.
Algorithm 2: CachedSampling- Stochastic cached batching with recycling
Input: Dataset DD cache size C, batch size B, cycles ft seed 8 Output: Batch of size B; each datapoint constructed by FRED
Initialize random seed
while true do cache +1]
while cachel C do
if no more data then
cacce = then
return
break
i < next index
x FRED (i) invalid then continue
append(clone(2)) cache)
for r 1 to do shuffle(cache; seed 8) for i = 0,.... B do yield cacheli]
E.3 TRAINING
We build SIGMADOCK in Python 3.12, employing PyTorch, PyTorch Geometric and PyTorch Light- ning our deep learning libraries. For our optimiser, we employ AdamW (Loshchilov & Hutter, 2019) with an L2 weight penalty of 0.1. We train for a maximum of 256 epochs using batch size of 32 and apply early stopping with a patience of 50 epochs. We also use an Exponen- tial Moving Average (EMA) to define our final parameters, using a weight decay of 0.999, as is common practice diffusion models. We use linear warmup with cosine annealing as our learning rate scheduler strategy, with an initial (cold) learning rate of x 10-6 maximum learning rate of 1 10-4 warmed up over the first epochs, and a final annealed learning rate of 10-5, We train SIGMADOCK using 4 NVIDIA-A (80Gb) via distributed-data-paee convergence.
for days, until
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F INFERENCE DETAILS
In this section, we present our sampling procedure for SIGMADOCK and how we filter generated samples to return the best samples for evaluation.
F.1 SAMPLING
We outline our sampling procedure in Algorithm 3 below which supports both deterministic and stochastic generation. We include the use of noise annealing, as we find that can help improve performance for stochastic generation. For time discretisation between max and mmin we use schedule from Karras et al. (2022). For .,..... 1, the k-th timestep is defined as
tk = 1-(11/e
k Nssops
where Naup is the number of discretisation steps and 0 controls the spacing. When 1, this reduces uniform discretisation; larger values p concentrate timesteps near mmins while smaller values spread them more evenly across the interval. By default, we use Nsteps 25, min 0.002.
max = 1, p For our noise annealing, we use the same schedule with Ymin 0, 7max 0.5 and
2.
Algorithm 3: SIGMADOCK sampling
Input: Score model so, docking information Gtock time steps S /min lo .... In = noise annealing schedule {-x) where 7i >>
Output: Predicted ligand coordinates x
Sample (1) = =(p(), ()) ~ q(z) = -O & Uso // sample from the prior
for i + to do
At < | +
s + so(2 4, Ianak)
// compute scores
p 1) + ReverseTrrassLttionssp (t) t. At)
R ReverseRotations(RR." S ())
(1111) ((t+)))
4- ((2(mm))
return x
Function ReverseTrransattoon. tt,
Ap + At . (+ B(t)s²)
A 4 At
+ p'+ Ap + Ae
return p'
Function ReverseRotation (R,s,,t,A At,,)) l R R
|/ convert to Lie algebra // convert to R3 basis
R
R R3
g (s)
AR + At st
+- 7 Vtt I
R' (- R eepp[ARt Adx)
// convert back to SO(3)
return
Computational cost of sampling. Using an NVIDIA-A40 GPU for sampling, we achieve a gen- eration speed (average runtime across the PoseBusters set) of 0.57s/mol per seed when using 20 discretisation steps and a batch size of 64. Depending on the number of seeds (Naeed used in our SIGMADOCK (Nueed) parametrisation via heuristic outlined in F.2, increasing Need up-scales the runtime linearly (unless a distributed system was employed). Sampling speed is also linearly de- pendent on the number of reverse diffusion steps. However, find diminishing returns with more
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than 20-30 steps. We argue SIGMADOCK is highly computationally efficient in comparison to other deep learning methods, taking between 5.7-22.8s/ma for N = 10, 40 respectively. See Table in (Jiang et al., 2025) for comparative analysis of other deep learning tools and their relative run- times. Notably, AF3 (Abramson et al., 2024) is reported to have an average runtime of 16min/mol and DiftDock (Corso et al., 2022) is reported have an average runtime of 72s/mol (this includes the respective number of seeds required by each method).
F.2 CONFIDENCE MODEL
common requirement of previous work in generative molecular docking is the use of a separately trained confidence model. This is used to filter a batch of generated samples, from ranking samples by their computed confidence value, in order to find the best generated poses and discard samples with poor chemical plausibility. However, this is significant additional computational burden, from both the training of the confidence model and the need to apply this to every generated sample. For instance, Corso et al. (2022) evaluates the performance of their model from a batch of 40 generated samples for each protein-ligand pair, and Prat et al. (2024) creates scoring function to filter large pool of physics-based docked poses.
In contrast, due to the inductive biases that SIGMADOCK leverages, we find that generated samples from SIGMADOCK tend to be reliably chemically plausible. Hence, do not require the use of trained confidence mode, and more importantly, we do not employ energy minimisation techniques which alter the final coordinates of predicted Xy. Specifically, using the force fields (Maier et al., 2015; Boothroyd et al., 2023) implemented in OpenMM (Eastman et al., 2017) for energy evaluation, Buttenschoen et al. (2024) shows the relevance of energy minimisation in previous deep learning approaches.
Instead, we use the simple and cheap (trivially parallel) heuristic of evaluating the binding energy of the protein-ligand system at generated docked pose as our "confidence model" with Vinardo (Quiroga Villarreal, 2016), where we take lower energies as corresponding to better samples. During sampling, we compute a total score per sample which also penalises for stereochemical inconsistencies including bond lengths, bond angles, tetrahedral chirality mismatch, internal steric clash, and minimum distance to the protein.
Formally, let by E R denote the binding energy of sample i, for E (1..... Nuwes)) and let E (o, 1] represent the average PoseBusters validity checks for the stereochemical properties specified above (1 = all checks are valid, 0 = checks are invalid). The mixed score for set of samples is defined
S; b, P.
where 3 controls the strength of the total PoseBusters penalty (set to 4). Higher values of 8; corre- spond to higher-confidence poses. We then use these scores to rank the sampled poses across Nseeds when computing Top-k metrics.
We believe our approach stands more fair test since relying on different method to correct the coordinates of sampled bound states is not generally a form which agrees with the maximum likelihood objective of score models during sampling, and can be tuned at test time to artificially generate better performance metrics.
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G ARCHITECTURE
In this section, we provide an overview of our score architecture which takes the inputs z E SE(3) & [0, 1] and Gdock = (igand J= Irotein). We assume that the docking conditioning information Gaock has already been preprocessed by the steps outlined in Section E.1.
G.1 PROAEIN-LIGAND FEATURISATION
Here we describe the structure of g {9F, 1- and Greotenn and how we extract relevant struc- tural stereochemical information therein.
Chemical features. We featurise the atoms and bonds according to their chemical nature. Specif- ically, in addition to the coordinates, we extract the following atomic (node-wise) features:
Atomic Number E Zt
Degree E Zt, number of adjacent atoms.
Charge (0,1,2) representing the formal charge: 0 if charge is neutral, if charge is negative, 2 otherwise.
Hybridicity {0.... ,5} representing the hybridisation type (Other, SP, SP2, SP3, SP3D, SP3D2) respectively.
Implicit Valence E Z.
Explicit Valence E Z.
Is Aromatic 1). Boolean value determining atom belongs an aromatic chain.
Is Ring E {0, 1.. Boolean value determining the atom belongs to a ring/cycle structure.
Num Rings Z+ Number ring systems the atom belongs to.
Chirality Tag E 1, R(1)-S(2) featurisation of the chirality if atom is chiral, else 0.
Similarly, we extract the following bond (edge-wise) features:
Bond Type E {0, -.. 4}: undefined, single, double, triple, or aromatic bond.
Is Conjugated E {0, 1}. Boolean determining the bond is conjugated bond.
Is In Ring E {0, 1}. Boolean determining if bond is in ring system.
Stereo Feature {0, "0,,..)))..... Mapping bond stereo information (None, Any, Cis, Trans).
Protein-specific features. To further extract relevant information from the protein atoms we add an additional feature, for all atoms in the protein, flagging the amino acid belongs to (from the standard set of 20 amino acids) and passing this through learnable embedding function. We also create virtual nodes the alpha-carbon Ca which have significant importance in our topological design. Furthermore, to remove redundancies and prevent over-smoothing across the protein residue, we add a depth- from-Ca distance encoding over the atoms in the chain.
Fragment-speecfiic features. Using FR3D and the featurisation procedure outlined above, de- fine our rigid body fragments from a conformation (aligned bound pose during training, random conformer during sampling). On top of the chemically featurised local geometry obtained here, add a set of virtual nodes in each fragment (denoted as VF) defined by the following logic: if the fragment has no rings, place the virtual node the center of mass of the fragment. Alternatively, place multiple virtual nodes at the center of each ring.
G.2 INPUT GRAPH
We clarify that the main body we use the notation Gaock as our representation the protein- ligand complex conditioning, and in this section specify how Gdock is processed into the input graph Gmput that we passing into our score architecture. We note that while our diffusion process and training objective is defined on the Lie group SE(3) following previous work, we design our score model so to operate on the 3D coordinates of the protein-ligand system, defined by Gproeein and (z)), instead of on the group elements directly.
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The input graph Gmput be decomposed into the global topology. which does not change through- out the diffusion path, iee. is immutable across all pı(z(t), ((ftt),,.g toock)). t E [0. 1], and the transient topology, which is defined from the current fragment coordinates x(t) (zz((t) and Proteinn We make the important distinction: In our global topology, we are not assuming that the entries have fixed coordinates, but instead that the edge indices and edge types fixed throughout. We pro- vide the analogy of spider's ' web, which you can stretch out (altering the coordinates) but will still maintain the same nodes and edges.
Global topology. The core representation of the global topology consists of the rigid fragments and static protein structure absorbed in Pdock- We extend our global topology by creating edges across all virtual nodes:
Cartesian product across all C. under an 18Å distance cut-off,
Cartesian product across all (VF,, VF,), i i
Cartesian product across all (VF, Ca),
forming hierarchical and well-structured global message passing mechanism. We also add trian- gulation edges connecting fragment atoms together, as illustrated in Figure 9.
Transient topology. Our transient topology is dynamically defined and includes all protein-ligann interactions edges under rre-determined cutoff of 4A. We keep this cutoff relatively low to focus on important protein- frreemnn- interactions which could otherwise be missed by the message passing mechanism over the global topology.
Node encoding. To distinguish between protein and ligand nodes we add further categorical fea- tures to the aforementioned list of chemical features. Namely, we add node entity attribute which labels the node as either one of:
Ligand (fragment) Atom
Ligand (fragment) Anchor (beginning of torsional bond)
Ligand (fragment) Dummy (inclusion of end of torsional bond)
Fragment Virtual (VF)
Protein Atom
Protein Virtual (C.)
For all nodes in our graph we encode time through a Fourier embedding as used in (Song et al., 2020). We then define an AtomDi faDDiffusii wwwffffonnonndeee... which processes the categorical features of each node (including node entity and chemical features) together with the time embedding to produce feature embedding of dimension fnote 128. Distinctively, the feature embedding of VF, computed as the average the feature embeddings extracted from each individual atom connected to VF. Note connect to VF, all atoms in fragment F; unless it has multiple rings, whereby connect VE, to all atoms in each ring system. This reduces over-squashing VF, in large fragments.
Edge encoding. Similarly, we build node entity attribute which labels edges as either one of:
Ligand Bond
Protein Bond
Torsional Bond (across (F,, Fy, i $ j)
Fragment Triangulation
Protein-Ligand Interaction
Fragment Virtual to fragment atom Vi, to Fragment Atom F;
Fragment- Virtual to Protein-Virtual (Vr C.)
Fragment Virtual-Virtual (Vr 177--V VF)
Protein Virtual (C. C.)
For all edges, we define a distance embedding function using an adapted Fourier distance encoder,
Total 1 Fouri ereear ng, for edges defined part of the global topology, and Bessel basis
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with smooth polynomial cutoff, Radi a BelEennnenngglod... for edges defined as part of the tran- sient topology. All edges compute distance encodings based on the true distance evaluated from (x))))) except the fragment triangulation edges. For these edges define the distance from equilibrium: All distances are computed from the current atomic positions (x).,), with the exception of fragment triangulation edges. For these edges, instead define the distance
offset
off'set from the equilibrium length:
(t) = (t) - x,())l |xx (
where (0), x, (0) denote the equilibrium reference positions extracted from random conformer sampled from RDKit.
An example of Ginput (excluding interaction and virtual edges for clarity) is shown in Figure 9,
Figure 9: Example of SIGMADOCK representing the bound pose as fragments, (color coded) on the protein pocket (gray) for complexes with PDB-LIG codes 5SAK-ZRY, 7NR8-UOE, 6YR2-TIC. 6XG5-TOP respectively. Virtual nodes (blue diamonds for each fragment, red diamonds for protein C.) are depicted. Fragment-protee
interactions, C. CC. and C. - VF edges are hidden for clarity. We keep the connections between fragment virtual nodes and their corresponding fragment atoms. Triangulation edges are shown in orange. The pocket CoM depicted via the golden sphere with a radius of 1Å. Note how the pocket CoM is not always indicative of the bound pose CoM, offering lower bias during sampling.
G.3 BACKBONE MODEL
Given our input graph Gmpurt constructed as described above, we pass this into our backhone model. This is a modified version of the EquiformerV2 (Liao et al., 2023) architecture where the output head predicts atom-wise forces. The modification that make to the EquiformerV2 architecture is to remove the bias terms within the MLPs (i.e. Radia a Funct ion) that process edge features for dynamic edges in Ginper (i.e. edges which we include the neighbouring nodes are within some cut- off distance); we also use a different MLP for each edge type. We recall from above. that the edges features for these dynamic edges are computed, from passing the distance between the neighbouring nodes through a Bessel basis expansion and a polynomial envelop for the cutoff distance, so that these features smoothly decay to zero as the cutoff distance approached. Therefore, the effect of removing the bias terms is to ensure that the messages along these dynamic edges, computed within the equivariant graph attention blocks (as well as the edge degree embedder), also smoothly decay to zero as the cutoff distance is approached. In addition, our radial basis function has the desired property of ensuring the gradient norms smoothly decay to zero as the edge distance approaches the cutoff. This is to ensure smoothness in the prediction of forces we perturb which could cause sudden changes in the topology of the input graph Ginput from dynamic edges being added/removed. All our ablations and main results are run with the same number of trainable parameters (14.9M).
G.4 PREDICTION HEAD
To ensure invariance with respect to the choice of local coordinates, we adapt the SO(3) -qquivraaaa prediction head from Jin al. (2023); Guan al. (2023) which utilises the Newton-Euler equations from rigid-body mechanics. In particular, for each fragment GF with global in- put coordinates XF = (Ir.I,..., FFG))) E xx and SE(3) coordinates (pr, Ry), let
Athhouhh strictly within the architecture module, consider the featurisation of the nodes and edges in input graph, which we previously descrihe, as separate from the core EquiformerV2 architecture.
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f.n..ffnn fign)) Bler|x3 where fa denotes the output from backbone model for the ith node in the fragment; the total force FF and the torque TF on GF are defined as
= (fi.....
19
[ypl TF >
F
Σ
f
(FR
1P:--PFi PF) fi.
We then predict the score function for OF with the score components [s] s defined by:
awSolw, o(1)
FF,
S8 =
w wfolw, (t)) [Tp [T'' TT''TrxxRr Rr-
V1
at
9F
where w = | TF and IF = E
Irr - - - ( (F Pp) denotes the
i=
inertia matrix of the fragment. We see that si is essentially parametrised for e-prediction and that s constructed match the form of the conditional score; this is done improve the conditioning of the score matching loss.
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H SYMMETRIES
As part of SIGMADOCK, we are concerned with two main symmetries of our model. Firstly, we note that an issue of working with fragments naively is that due to their heterogenous construction (i.e. fragments can have various different topologies and number of atoms etc.), we can not define a canonnical orientation for their local coordinates. This is not an issue with other fragment-based models, such as AlphaFold2 (Jumper et al., 2021), as in their setting. all fragments take the same form (e.g. protein backbone frames have a canonical axis defined by the (N Ca - C') atoms). Formally, consider fragment Gr with some choice of local coordinates Xy. For any choice of R. SO(3), we can define another valid choice of local coordinates given by x = Rs 'x;; we denote the fragment defined by this alternative choice by g/-- Therefore, if we observe the global coordinates of the fragment are XF PF + Rp 8F, then under the fragment GF. this pose is parametrised by (PF, Rr), whereas under the fragment G'p this pose is parametrised by (pF, RrR,,) instead, since Xr = pF + R R. & To prevent issues from the orientation of local coordinates not being well-defined, we have designed SIGMADOCK to ensure that our training objective and sampling procedure are invariant with respect to the choice of orientations for local coordinates.
Secondly, we note the kernel (r defined by sampling from SIGMADOCK should ideally be stochastically equivariant (Bloem-Res Reddyy & Teh, 2020; Cornish, 2024; Zhang et al., 2024) with respect to SO(3), i.e.
po(dz) Ro Goock) = Ro py pg 9dock); RRo E SO(3),
where Ro Gduck should be understood as applying the rotation Ro to the coordinates y of Sprotcin E Gdock. and the right-hand side denotes the fush-forward under the mapping given by applying Ro (where we define the action Rg = Ro (PF: (R (Ro PF,, Ra In random variable notation, this corresponds to the statement saying that the sample z' Ro where Z ~ po (z]9uccs) follows the same distribution as 2 ~ po Ru 9xcck). Intuitively, this means that the distribution pe (z Guock) of fragments (and hence, the ligand,
easy to see that sp Ro 2) = Ro (z))) should be transformed in consistent way as we change the orientation of the protein pocket that SIGMADOCK is conditioned on. To enforce this symmetry, we have designed SIGMADOCK to use an -3quuiaauiaar architecture for our score model.
We note that stochastic equivariance is more commonly presented in the literature (Hoogeboom et al., 2022) in terms of conditional densities p(xlyy) which obey the condition p(g <xgg y) for all g E G where G is some group under consideration. However, this definition implicitly assumes that the Jacobian of the group action has unit determinant and does not account for the general case presented in Bloem-Reddy & Teh (2020); Cornish (2024). Further, we note that it is surprising to us that, to the best of our knowledge, this particular symmetry has not been remarked to hold as we do later on) in the literature on diffusion-basee molecular docking, as this condition provides justification for using an equivariant architecture for the score model. Specifically, we find prior work appears to use such architectures without analysing the properties they induce on the diffusion model and sampling.
(or proved
To show that SIGMADOCE does indeed satisfy the symmetry requirements that we have outlined above, and stated in Theorem 2, we provide a proof of this fact below in Appendix H.I.
H.1 PROOF OF THEBREMM 2
Proof. We divide the proof into three sections: (i) proving invariance of our training objective; (ii) proving invariance of our sampling procedure; and (ii) proving (z( is stochastically equivariant with respect to SO(3).
Invariance of training objective. Consider two different fragmentations (9r; } and (g/, mm
of a ligand with different choices for the orientations of local coordinates-iss -ie. Gr has local coor- dinates XF and 9% has local coordinates X = R2,F *F where R&.F F E SO(3) for E m Then the global 3D coordinates x of the ligand has different representations for each choice of fragmentations. However, no matter the choice of fragmentation, the backbone of our score model so (with Guock and Glock, being the corresponding docking information formed the two choices) will output forces for each fragment that are independent of this, as we have designed the backbone to operate over the global 3D coordinates of our protein-ligand system, i.e. x ((()) and y.
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Next, we focus on the rotational scores for a single fragment with different orientations: GF and g'p- We let (p& A denote the SE(3) parametrisation at t = for Gr, therefore (pF Rg RT,F) is the parametrisation under gpp Therefore, the sampling distribution for training is given by R [G (R ,))()()))))))))))))))))))))))))) for GF and is given by R27") I[so(s) (12 R6 R (!)) for 9rr- We note that in random variable notation, have the expression R RRR where R ~ [G (R², (1)) and R k RT, This implies that R2 R R,, Further, under the sampling distribution for training for Fp. we can express the conditional score in the objective as
V r log Pijo ( (()
R R, VRlog Pijo (R RT RT,F))
(1 C( R F log [R (R ST R Rg
=(()) RT R F log (R R R
= =(()R log (RR TR( (RT))RR( RT,FF
V Pejo (R ( R
where the third equality is due to the trivial identity: log( (ABAA = A log BBAA for A, B E SO(3) and we define
(=f((w(t) C(t)
w(t))alw(t),
where (t) = w (R T RT,F)) and this equals w (( )T T R the function w.
from the definition of
Further, let 2 = (pr, Rr.) and x' = ((pr, Rs, RT, )= be the parametrisation under (Gr, k and (9'n 1 respectively for a ligand pose x, then the rotational score for a fragment GF has the following relationship
s (zl, Dbock))) = sgg (z, 9rock) F R.
due to the fact that the predicted forces for Gr, Gj are the same and the only impact that the different choices of orientation makes is in the rotational prediction head.
Tumning our focus to the translational scores, it is easy to see that the definition of the translational components pp (hence, the sampling for training. the prediction of scores and the training objective for the translational scores) are independent of the choice of orientations as these are disentangled.
Finally, by considering the training objective of the rotational scores under Grr we have
R (t), t, Paack F V 1 R | sp
R R
50(3)
E || | (z() (t), F RTT VologP ( RTJ
S0(3)
|| | (z(2 Goock) no
E
R R
S0(3)
where z (()) is formed from sampling from Pdam. extracting the SE(3) parametrisation under (G), 1m and applying the forward sampling kernel, and the final equality is due to the form of the SO(3) inner product. The above result demonstrates that the the training objective of SIGMADOCK is invariant to the choice of orientations for local coordinates, as the final equality has the form of the training objective of the rotational scores under Gr.
Invariance of sampling. Under the same setup as before, let (1)
(ip R D. and
(t) = ((r)) (0) R(. R. ),, be the SE(3) parametrisation of the same ligand pose x under (Gr; ) and (m m We consider the sampling step for z((t) to the next time step 1! < For
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given fragment F',, the update step has the form (for clarity, (WLOG) we set all coefficients used in sampling such as At and 7 VDL. to be one):
R R2,r
r exp (Ro. )T) so (z'(1, (acc)) + 2)
R RI exp (-C(t)R,,, ( T TF) R R 2) -Re R. RRs exp (-C(1)(R()) TF BE + Ri, 22, R
=R/² exp (RR()) ss( .t, Gdock) + (.,,FR R..
where -= Ei= 6,0; where 6g ~ NN 1) are iid and o₁ are the canonical basis of the Lie algebra 50(3). We can show that
Ei=
Eis d, E' d
R IO)
(3)
where è; denotes the standard basis vectors of R This is consequence of the self-adjointness of SO(3) which implies that [R u]x = R[u]xR where E R2 and R acts on the vector. Therefore, we have R. d Z. This allows us to conclude that distribution of the update step considered in Euclidean space (via the mapping ()) under the fragmentation (9%, ) is the same as the update step given by the fragmentation J Hence, the sampling procedure of SIGMADOCK is invariant with respect to the choice of orientations for local coordinates.
Proof of stochastic equivariance We first recall the following notation z = ((PF, Rp, ) and Guock = (Gigama } Ipronei)), where Grrotein Vy. by. For Ro SO(3), we define the following group actions by Ro Rg Guack (iiannd (Gr, Im Rg Sprotein) where Ro Gprosein (Ro y, Vy, by). It im- portant note that Ro Gaock only acts on the 3D coordinates y of the protein pocket, and does change the local coordinates for fragments [G J We note that this group action represents the global rotation of the protein-ligand system defined by z and Gaock.
((Ro pF:, RoRRR
and
Let po (gdock) denote the marginal distribution of samples time generated by SIGMADOCK under the docking information Pduck. In order to prove stochastic equivariance, we can equivalently is stochastically equivariant with respect to the above group action, then this implies that po (z( (unnk)) where < f, is also stochastically equivariant. Therefore, as po (Gaok)) is trivially stochastically equivariant, since this corresponds to the prior over z which independent of the orientation of y, the desired result follows by induction.
show that if po
To show the inductive step, we note by basic probability that we have the following expression (with densities considered over the product of Lebesgue and Haar measures):
Po (z ( = po (1)( Glock )po (z (1) (dock (1)
where po (2(()))(( Gdook) denotes the kernel induced by sampling the next time step e' from (9) We next consider showing the conditional density formulation of stochastic equivariance (this is valid as the action of rotations has unit Jacobian).
pa (Ro |Ro Goock) =
Po -z-zz (()) Ro (z ldzc
Ro (()) Ro (1) Ro Grock )o C Ro (0() (Ro Goock dzz
= f po(ooo (1))) Ro (1), Ro Gana po (1) Savea (aaca (d2
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where the second equality is due to change of variables and using the fact that rotations has unit Jacobian, and the third equality is due to our inductive hypothesis.
We can express the conditional density po (Ro 2(() 200), Ro Gducs) in terms of the product of Po Ro p002|Ro x Ro Gaack) and po Rn R 20)), Ro Gdack). ). For the former, we have (for clarity, (WLOG) we set all constants to one):
po pd ( 2 Ro Gdock) = NRo p (0 Ro .p. (Ro 20,,, Ro Gdock); I)
NBo p ""; Ro (p ) + se(z!), Idock)),
= =N(p"") + sz( t, I)
= (p)((()))()) g
where the second equality uses the fact that the forces predicted by EquiformerV2 SO(3)- equivariant (we note that this is with respect to the input coordinates which are Ro x and Ro y in this case) and the prediction head for translations is trivially -quiivaaai,, and the third equality is due to the Gaussian density being isotropic.
For the latter term, we first consider the random variable representation of R& (- Ro- (dock) for an individual fragment (the distribution on R(')) factorises over fragments independently). This is given from the update step for sampling (for clarity, (WLOG) set all constants to one)
R
R (t) exp ((R T RJ so Ro z Ro Gdoca) 2)
RO exp (RR T (z gaock) + +Z)
where Z = Ei= 6,e; where à; (0, 1) are iid. The second equality is using the fact that the forces predicted by EquiformerV2 are -quuieequi and Guan al. (2023) shows that IF'TF is 80(3) equiivaria and then applying [R v]x x R-! Further, we note that the term R exp (R( so (z( t, Flock) + 2) is the random variable representation of (
R((") ~ (=) Goock). Therefore, we can view R (2((e)) as change of variables of R. by the trans- formation Rg. Hence, we can conclude that po Ro R Ro (1)), Ro Gaock) po (t') (2(1), Ganck) and more generally pa RRo ( (Ro 2!!), Ro Gaock) = pa PD (R(())2()) Gaock).
Since, we have shown that pa( Ro x(('| ( Ro Goock) = po (z((0, Gaock), following from what we had before, we can conclude that
po (Ro RRo Ganex)
/ po ( 11)), Ro Gavex (atnck))
/ po (2(0)))))))) (o ddx
Po (z( (Ganz))- (Gant)))
Hence, we can conclude that po (2aCe))))) ) is stochastically equivariant with respect to SO(3).
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1 EXTENDED RESULTS
1.1 GENERATED SAMPLES
Trajectories. Sample trajectories for four randomised protein-ligand pairs in the Astex diverse set are displayed in Figure 10.
(a)
(b)
(c)
(d)
Figure 10: Sample trajectories of protein-ligand pairs IHVY-D16 (a), IHWI-115 (b), IKZK-JE2 (c), and IN46-PFA (d), using the default sampling scheme outlined in F. The respective RMSDs between the sampled and bound pose are 1.2, 0.31, 0.33, 0.88 Á. We illustrate discretisation-stery 3, 10 and 20 (out of 20), where the zeroth step represents samples from the stationary distribution. We show the reference bound pose in light green. For and b we tilt the view in the last frame to improve visibility of the fit.
RMSD is imperfect metric. In Figure 11 show 4 examples of generated poses from the PoseBusters set where the ssymmetry-correcteee RMSD between the generated pose and the bound pose is higher than the threshold of 2À. According this metric, these are deemed failed samples. However, we argue that the generated poses seem stereochemically valid, as they recover similar interactions. Namely, in (a), ligand 088 highly symmetric; the generated pose yields very similar interactions and induced fit, even when SIGMA DoCK predicts the pseudo-roiate
conformation,
RMSD is 10.2Å. A similar argument can made for ligand XN7 in (b), where we observe the (C' = O) forming the correct interaction (hydrogen bond acceptor) with the protein. In our third example (c) we see how the sampled pose very tightly recovers the true pose, yet the non-interacting tail of the ligand containing cycloproply group is predicted to be elsewhere. However, given the binding pocket is open, this cyclopropyl should freely rotate (molecules are dynamic). This unfairly labels the sampled pose incorrect, with RMSD of 4.2
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(a)
(b)
(c)
(d)
Figure 11: 2D visualizations of ligands (top) and their corresponding generated poses (bottom) across PDB-ligand pairs from the PB set: 7FRX-088 (a, 10.2 RMSD). 7KZ9-XN7 (b, 4.7 Á RMSD), 6Y7L-QMG (c, 4.2 Á RMSD), and 7MGY-ZDI (d, 5.8 Á RMSD). The true bound pose is shown in faint green.
1.2 SAMPLE PERFORMANCE & EFFICIENCY
In the main body we use Top-1 (%) key performance metric. However, it is also important to assess the sample efficiency our generator. We measure sample efficiency by looking at the rate of change in Top-k as vary the number of independent samples (Nseess) We report results in Figure 12.
Notably, our Top-k success rates increase rapidly with only few seeds, indicating high sample efficiency. This is especially true when considering PB-validity, as a few random seeds rapidly ensures better samples are prioritised according our heuristic. Conversely, the Top-k accuracies for (RMSD < 2) and (RMSD < 2 & PB-Valid) converge to similar values as Neeeds increases, and are practically indistinguishable when N > Finally, we consider the Oracle performance, which represents the empirical maximum success rate achievable by perfectly selecting the best sample from the pool of Needds At NNeeds 20, the Oracle reaches over 90% success for the (RMSD < 2 Å) metric and just under 90% for the combined (RMSD < 2 Á & PB-Valid) metric. The relatively large gap (~~ 10%) between this empirical ceiling and our practical Top-1 performance quantifies the potential for improvement in re-ranking the generated candidates, and we leave this for future work.
(a) RMSD Only
(b) RMSD PB Valid
70%
Top-3
Top-3
60%
Top-5
Top-5
Tsp-1
50%
Ideal Bernculi
Ideal Bernoulli PS
10
15
20
Nooeds
Noxes'
Figure 12: Top-k success rate as a function of the pool sample size N Solid Top-k lines rep- resent the mean success rates across (40 NNead)) permutations, with shaded areas representing the standard deviation. The Oracle reflects the empirical maximum success rate attainable across Nacoss samples, equivalent to Top- N Needs- The ideal Bernoulli line represents the empirical optimal, assuming independent sampling probability.
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J. LIMITATIONS
J.1 CURRENT LIMITATIONS AND FUTURE WORK
Intentionally small training set. SIGMA DOCK was deliberately trained on VDBBBnnd(v2020) to enable fair method-level comparison with the intended PoseBusters train-test split, and highlight methodological improvements rather than raw performance. This choice clearly reduces the model's out-of-distribution generalisability. In the future, we will look at scaling training with additional curated complexes, applying data augmentation and transfer learning pre-training strategies, or combining with larger foundation models to improve robustness.
Dependence on the pocket center at inference. SIGMADOCK requires a user-specified pocket center to define the search region. Although we debias the choice of the geometric center from the ligand center, and make SIGMADoCK relatively robust to the choice of pocket center during training (see Appendix E) when the true center misspecified, the search space (and therefore predicted samples) can be biased. For faimess, we note that many physics-based and deep-learning- based competing tools introduce a greater bias, such as a strict bounding box around the bound ligand (Trott & Olson, 2010; Zhou et al., 2023b; Jiang et al., 2025). This explicitly restricts the search space over a biased search region (Az, Ay, Az). providing information about the structural fit of the bound ligand. Further work could be performed to (i) explore the sensitivity over the choice of pocket centers; (ii) incorporate a different product structure with rigid body translations that separates the requirement of a defined pocket center.
Chirality challenges from fragmentation. Although we implement a chiraaityy-reseerving scheme by attempting fragmentation merging operations which try to keep chiral centers during sampling, there are cases where no valid reconstruction preserves the original stereochemistry and chiral cen- ter may be altered during fragment linking. Currently, we simply apply post-filtering over the gen- erated Nacods samples to discard undesired stereoisomers (as performed in Abramson et al. (2024)). Future works could try to explore ways in which SIGMADOCK might better capture stereochemical priors to further ensure chirality is preserved.
Sensitivity to cofactors and receptor flexibility, Performance degrades when relevant co-factors are omitted from the input. This indicates the model does not merely memorise unphysical poses but relies on explicit physical context. In future work, we will consider including cofactors and flexible side chains jointly with our SE(3)" fragments.
Restricted evaluation protocol (re-docking only). All reported results are based on re-docking benchmarks. We intentionally restrict the evaluation of SIGMADOCK to re-docking in order to emphasise its key contribution on task not yet dominated by deep learning methods. These re- sults demonstrate the model's ability recover bound poses from known pocket conformations, but they do not fully capture its generalisation to more challenging scenarios. While many protein tar- gets have relatively rigid pockets with limited structural change, important future directions include training and assessing performance on cross-docking and apo-structure docking tasks.
J.2 FURTHER COMMENTS
AlphaFold3. It is evident that the task of co-folding is more complex task than that of protein- ligand docking, wherein there exists a larger set of degrees of freedom (larger dimensionality) to model. This is especially true in our setup, where we assume rigid protein structure (fair and com- monly used in practice (Kamuntavičius et al., 2024), but not a strictly holistic assumption). However, for full visibility, and to support our narrative, we would like to stress that AF3 (Abramson et al., 2024) reports values of up to 84% in their Top-1 metric in the PoseBusters set, without applying rigorous test-train leakage filtering steps. This is evident when comparing the bucketisation of the protein-ligand pairs in the PoseBusters test set, as illustrated in Table 5.
Unexpectedly, AF3's reported accuracy decreases test-train similarity increases contrary to findings and to prior reports (Skrinjar et al., 2025). Despite AF3 evidently training with substan- tially higher train-test overlap, our Top-1 accuracy is comparable overall and matches or exceeds AF3 across most of the test set (match on [30, 95), surpass on (95, 100)). For fair comparison,
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Table 5: Stratification of the original PDBbind(v2020) vs. AF3 train-test splits on PoseBusters(v2).
Sequence similarity split values extracted from AF3 Extended Data Fig. 4c (Abramson et al., 2024). (*) We observe a mismatch between the averaged sequence similarity results (80.2%) vs the reported average performance in AF3's Extended Data Fig. 4e (84.4%).
Bucket counts
Results (Top-1 & PB-Valid (%))
Seq. Similarity (%)
Original
AF3
Ours
AF3
00, 30)
109
38
+71
72
87
16
300 95)
76
83
-7
79
82
-3
(95, 100]
123
187
-64
87
78
+9
Total Avg
308
308
79.9
80.2*
deliberately train SIGMADOCK on reduced dataset PDBBind(v2020) with 199 datapoints, and do not model cofactors. Nevertheless, we still achieve AF3-level performance. Contrary to common concerns about the physicochemical coherence of generative deep-learning models for structure pre- diction, we observe superior generalisation with a compact model (14.9M vs. ~550M parameters for AF3) and set of well defined inductive biases. It is therefore reasonable to expect that SIG- MADOCK will further improve with more data, larger models, and added flexibility, and surpass AF3 more challenging settings such cross- rross-docking and flexible-receptor pettin--ligand docking.
Difficulties comparing approaches. Fair comparison across learning-hased docking methods complicated by heterogeneity in (i) training data and preprocessing. (ii) pocket definitions, and (iii) poot-processing pipelines. Differences in (i) affect both the risk of train-test leakage and the apparent ability to generalise (see our comments
AlphaFold3 above). For (ii), pockets are often defined from the bound pose; for example, Zhou et al. (2023a) uses a bounding box around the ground-truth ligand, which may introduce information unavailable realistic deployments and thus overestimate performance. For (iii), many methods refine generated poses via energy minimisation, sometimes substantially improving reported metrics (Buttenschoen al., 2024); while sensible in practice, such steps make it harder to attribute gains to the core model and add non-trivial computational overhead. For instance, Alcaide et al. (2025) reports results that rely on the post-processing of Alcaide et al. (2023) as implemented in their code and discussed in their issuc tracker 15.
In this work, we mitigate these confounders by training on an intentionally small dataset, to match the setting of Corso et al. (2022) and decouple performance from data scale; by reducing the coupling between pocket definition and the bound pose via the stages in Appendix E; and by reporting metrics directly from SIGMADOCK without structurally altering post-processing. We hope these choices encourage clearer reporting and more comparable benchmarks for deep-learning- molecular docking.
1eSee hitpm//ddeeemoeeiiii//iii--oo httee/////eeee/aaae/aea htw.iaaaannnniaanmmmm lock ing_v2 with the - option
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Reviews:
[{"review_id": "T1rRG46IUA", "reviewer": "Reviewer_xTvN", "summary": "The authors propose a fragment based docking method (sigmadock). The ligand is first broken into fragments that are then coordinately placed in a bound configuration against a fixed protein target. The main motivation for the approach follows prior work in terms of using rotatable bonds as the main degrees of freedom for the ligand beyond its global rotation and translation relative to a fixed protein target (presumably holo structure). Instead of operating on the angles directly as in prior work, the authors fragment the ligand across rotatable bonds (also heuristically reducing the number of fragments), and define a forward diffusion as a product measure across the fragments. A SE(3) equivariant de-noiser is then learned to iteratively refine noisy fragment rototranslations back into their bound configuration. The method is tested against (older) alternatives on PoseBusters set.", "questions": "Is the fixed protein structure a holo structure? How would the method perform relative to a slightly different apo structure?\n\nSoft triangulation constraints are used across fragments to preserve bond lengths and angles. However, since fragments are refined coordinately as a set, presumably there will be some discrepancy between them about the corresponding rotatable bonds and angles. How are these resolved in the final answer? \n\nHow is ligand chirality handled? Are cyclic structures explicitly excluded? \n\ntheorem 1 seems vague and inflated. Naturally the mapping from torsional angles to cartesian coordinates is non-linear and thus does not lead to a product measure over fragments. What does \"highly entangled\" refer to? \n\ndoes the forward diffusion begin with an aligned structure to the ground truth pose, i.e., selecting approximate torsion angles that best align with the ground truth bound pose? Otherwise internal fragment bond lengths etc would not necessarily be consistent with those from RdKit.\n\nthe reference to Jin et al. 2023 for their SE(3) equivariant prediction head is a little vague. What exactly is used from that reference?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The motivation for fragment based docking is clear and compelling. The mathematical derivations are clear and straightforward albeit appear mostly in the appendices, not as part of the main paper.", "weaknesses": "There are a number of unsubstantiated and inconsistent claims, starting with the abstract. The presentation needs to be revised; any claims must accompany clear empirical or theoretical support. E.g., in section 2.2.2, the authors say that \"... breaking the product structure [leads] to ill-conditioned and often degenerate dynamics during training and sampling\". What's the evidence for ill-conditioned and degenerate dynamics? All claims of this kind require either supporting evidence or be removed. My rating assumes that the authors will make such changes. \n\nThe authors claim that co-folding methods are slow but such methods account for protein structural changes following binding. Presumably the authors start with a holo structure that is already altered towards accepting the ligand in question (?), thus making the task easier and unsuitable for fair screening across apo structures (or holo structures corresponding to different ligands). Reaching AF3 performance now does not make the method \"the first\" deep learning method to surpass physics based docking. \n\nComparisons are not to the state of the art methods. E.g., DiffDock (2023) has already been surpassed by its own refinements and by other methods. AF3 or Boltz-1x/2 (with pseudo-physical steering) would constitute more appropriate comparisons among many others. If the main argument is speed vs accuracy, then the authors should compare across the two axes, not exclude state of the art methods from comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hBaHnefybd", "reviewer": "Reviewer_n4Mh", "summary": "The paper develops a diffusion model for rigid-protein, known-pocket molecular docking. The ligand is fragmented into rigid bodies by breaking torsional bonds and the pose is generated by denoised over the SE(3) manifold for each fragment. The model is trained and evaluated on PDBBind, where the model surpasses the success rates of Gold, Vina, and DiffDock.", "questions": "No specific questions.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 1, "strengths": "* The paper does a good job of motivating the problem formulation. The analysis of the right space to diffuse over for molecular docking is thoughtful, with theoretical results supporting the conceptual arguments.\n\n* The technical exposition and proofs in the appendix are clear and nicely done.\n\n* The preservation of performance at low sequence similarity to the training set is very nice to see.", "weaknesses": "**Significance**\n* The topic is somewhat stale, with wide consensus on Euclidean diffusion and co-folding for molecular docking. While contributions that challenge the consensus are welcome, they should provide a compelling value proposition rather than retread problem formulations that are no longer of primary interest.\n\n* The historical interest in docking to rigid receptors largely stems from computational considerations and works focusing only on this task should not be encouraged, as holo structures are not available in practice and sidechain interactions to accomodate ligands are always of interest in scoring poses.\n\n* In particular, the additional complexity of manifold diffusion needs to be justified by marked improvements in performance relative to simpler alternatives, which the paper does not convincingly show.\n\n**Experiments**\n* Both the dataset (PDBBind) and the task (pocket-guided docking) make comparisons to state of the art co-folding models difficult. In the absence of such evaluations, it is impossible to judge the significance of the contribution.\n\n* If authors are comparing against DiffDock, they should at least include DiffDock-L. Many of the other baselines are also quite outdated (TankBind, UniMol).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4fsIAPQnJP", "reviewer": "Reviewer_EyZ8", "summary": "This paper introduces SIGMADOCK, a novel deep-learning approach for molecular docking. The method moves away from traditional torsional diffusion models by first decomposing a ligand into a set of rigid-body fragments. It then employs a Riemannian diffusion model on the SE(3) group to learn how to reassemble these fragments within the protein's binding pocket. Key contributions include a novel fragmentation scheme (FR3D) to reduce the system's degrees of freedom, the use of soft geometric triangulation constraints as an inductive bias, and a tailored SO(3)-equivariant architecture. The authors report that SIGMADOCK achieves state-of-the-art performance on the PoseBusters and Astex benchmarks, claiming it is the first deep learning method to surpass classical physics-based docking tools under the specified train-test split.", "questions": "See above", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1.   The paper is well-written, clearly motivated, and theoretically sound. It provides a strong argument for the limitations of torsional models and convincingly presents the fragment-based approach as a superior alternative.\n2.   The proposed method is novel and principled. The combination of the FR3D fragmentation scheme, soft geometric constraints, and an SE(3) diffusion process introduces strong and chemically-aware inductive biases into the model.\n3.   The empirical results are highly impressive. SIGMADOCK demonstrates a significant performance leap over previous deep learning methods and, notably, classical docking software on challenging, temporally-split benchmarks. The high rate of chemically plausible outputs (PB-validity) without post-hoc minimization is a major strength.\n4.   The paper includes a thorough ablation study that validates the contribution of each key component of the model, alongside an insightful analysis of failure cases related to co-factors.", "weaknesses": "1.   The paper argues that its fragmentation scheme (FR3D) helps manage the degrees of freedom, but the analysis could be more thorough. \n\n2.  The definition of the protein binding pocket is a critical input for any docking model, yet it is not discussed in the main paper. Typically, this involves selecting atoms within a certain radius of the ground truth ligand, making this radius a key hyperparameter. The sensitivity of SIGMADOCK's performance to this pocket definition is not analyzed. An ablation study (such as anlyais in PoseBusters paper for UniMol) showing how performance changes with different pocket size thresholds would provide valuable insight into the model's robustness and its potential applicability to scenarios where the binding site is less precisely defined.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gyQZ6fddfR", "reviewer": "Reviewer_v3fT", "summary": "This paper proposes SigmaDock, a novel SE(3) Riemannian diffusion model for molecular docking that operates on rigid-body transformations of ligand fragments. Compared to prior torsional models, this approach has the advantage of simplifying the design and training of the diffusion model. The key contributions are a novel ligand fragmentation scheme and the design of the diffusion process and model architecture. The model shows impressive results on the reported docking benchmark.", "questions": "1. Regarding the conformational manifold, where are these holonomic constraints used in the literature? Can you provide references?\n2. The ablation study in Table 1 is not clear, e.g., the base method, which should be the main point of comparison, is missing from the table.\n3. How is the knowledge about the docking pocket leveraged by the model during training and inference?\n4. This work assumes the protein conformation does not change during docking (rigid docking). Can the model be extended to model protein conformation changes as well?\n5. In line 24 (Abstract), the baseline metric should be 32.8% instead of 30.8% (according to Figure 4).\n6. In line 485 (Footnote 10), I believe RMSD should be \"above\", not \"below\", 2A.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "* The paper addresses the relevant and challenging task of molecular docking, which has many applications in drug discovery. Also, the paper is very well-written.\n* The key idea of decomposing the ligand into fragments and defining the diffusion process over rigid-body transformations over these fragments is sound and interesting.\n* The experimental results are impressive.", "weaknesses": "* This paper is inspired by and similar in spirit to DiffDock [1]. The authors should compare the two methods in more detail and highlight the key differences between the two diffusion processes.\n* All the reported experiments use a model trained on a single dataset. It is unclear how the model would perform on other datasets/benchmarks.\n\n**References**\n\n[1] Corso, Gabriele, et al. \"Diffdock: Diffusion steps, twists, and turns for molecular docking.\" arXiv preprint arXiv:2210.01776 (2022).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zANq5pwHJk", "reviewer": "Reviewer_75zT", "summary": "This paper introduces a novel fragmentation scheme, leveraging inductive biases from structural chemistry, to decompose ligands into rigid-body fragments. Building on this decomposition, this paper presents SIGMADOCK, an SE(3) Riemannian diffusion model that generates poses by learning to reassemble these rigid bodies within the binding pocket.", "questions": "1. How to perform score estimation for Uniform distribution on SO(3)? This distribution is convenient for flow matching method. Why not use the Isotropic Gaussian Distribution for diffusion model?\n2. Torsional diffusion could handle tasks like side-chain packing and how could the framework of SigmaDock handle this?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The presentation of this paper is clear and insightful\n2. The experimental results are impressive (could be further enhanced by incorporating more experiments in the main-text)\n3. This fragment method which is common in protein structure generation and Structure-based drug design (SBDD) is carefully extended into molecular docking field as an effective approximation for co-folding methods to strike a balance in effectiveness and effiency.", "weaknesses": "1. The proposed methods is limited in pocket-aware docking setting, which requires pocket identification in advance. \n2. The method doesn’t model the pocket flexibility which is crucial for pocket-aware docking task. Since apo→holo pocket structure transformation is very common.\n3. As this method is more closed to the protein-ligand complex structure prediction task (or even SBDD, as fragments-based models are very common in SBDD) , more comparison against complex structure prediction is appreciated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)