CByG: Controllable Bayesian Flow Guidance

• CByG is a framework for controllable 3D molecular generation in SBDD, integrating Bayesian flow networks with conditional guidance to optimize ligands for binding affinity, synthetic feasibility, and selectivity.
• It overcomes diffusion guidance challenges by separately handling continuous coordinates and categorical atom types using a hybrid gradient-based update mechanism.
• The framework leverages pretrained Bayesian flow networks and an external property predictor, enabling multi-property optimization at inference without retraining the generator.

Searching arXiv for papers explicitly mentioning or closely related to "CByG". CByG is an ambiguous label across current arXiv-adjacent technical discussions. Its explicit and named use is “Controllable Bayesian Flow Network with Integrated Guidance,” a framework for controllable 3D molecular generation in structure-based drug design (SBDD) that generates ligands conditioned on a protein pocket while steering sampling toward binding affinity, synthetic feasibility, and selectivity (Choi et al., 29 Aug 2025). In other arXiv contexts, the same letter sequence appears only as an interpretive shorthand attached to distinct topics, including cycle-code / cylinder-type / girth-based quasi-cyclic LDPC construction (Samadieh et al., 2014), succinct colored de Bruijn graph comparison pipelines (Ramos et al., 2024), and Cayley-graph / generalized bicycle CSS codes (Arnault et al., 28 Jul 2025). Related Byzantine-SGD terminology such as ByGARS is adjacent in spelling but does not define or use CByG (Regatti et al., 2020).

1. Nomenclature and scope

The explicit arXiv meaning of CByG is concentrated in one paper, whereas several other uses are contextual or interpretive rather than author-defined. This distinction matters because the acronym otherwise risks conflating unrelated literatures.

Usage	Domain	Status
Controllable Bayesian Flow Network with Integrated Guidance	Structure-based drug design	Explicit acronym (Choi et al., 29 Aug 2025)
Cycle-code / cylinder-type / girth-based construction	QC-LDPC cycle codes	Interpretive shorthand (Samadieh et al., 2014)
Colored de Bruijn graph analysis	Comparative genomics	Interpretive shorthand (Ramos et al., 2024)
Cayley-graph / generalized bicycle CSS codes	Quantum error correction	Interpretive shorthand (Arnault et al., 28 Jul 2025)
Byzantine gradient aggregation	Distributed learning	Related terminology only; not CByG (Regatti et al., 2020)

In the explicit SBDD usage, CByG is not a new generative backbone from scratch. The framework starts from a pretrained Bayesian Flow Network (BFN), inherits the architecture and pretrained weights from MolCRAFT, and adds inference-time conditional guidance through an external Bayesian property predictor. The technical novelty is therefore concentrated in the guidance formalism, the hybrid continuous/discrete update mechanism, and the evaluation protocol rather than in a wholly new unconditional generator (Choi et al., 29 Aug 2025).

2. CByG as controllable 3D molecular generation

CByG is defined for pocket-conditioned 3D ligand generation. The generated ligand is written as

$$\mathbf{m}=\{(\mathbf{x}_{\mathrm{M}},\mathbf{v}_{\mathrm{M}})\},$$

where $$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$ are ligand atom coordinates and $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$ are atom-type variables over $K$ categories. The conditioning protein pocket is

$$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$

The target properties are denoted by $\mathtt{l}$, and the framework uses those properties during sampling as guidance signals (Choi et al., 29 Aug 2025).

The motivating claim is that prior SBDD generation work is evaluated too narrowly, often mainly by docking affinity and typically with AutoDock Vina alone, whereas realistic drug discovery also requires synthetically feasible and selective molecules. The paper further argues that conventional diffusion-based guidance is poorly matched to 3D molecule generation because the data are hybrid-modal: coordinates are continuous but atom types are categorical. Three specific limitations are identified. First, diffusion guidance has difficulty modeling interactions between coordinate and type variables when they are assigned different distributions. Second, gradient guidance on categorical variables is unstable or ineffective because reverse sampling often involves argmax or other non-differentiable operations. Third, directly perturbing intermediate noisy molecular states in sample space can damage chemical and geometric validity. The paper also argues that guidance based on $p(\mathtt{l}\mid \mathbf{x}_t)$, meaning property prediction from noisy intermediate states, is unsuitable because such intermediates are chemically meaningless (Choi et al., 29 Aug 2025).

The unconditional BFN distribution is written as

$$p_\phi(\mathbf{m}) = \int p_\phi(\mathbf{m}\mid\boldsymbol{\theta}_n)p(\boldsymbol{\theta}_0)\prod_{i=1}^n p_{\mathtt{U}}(\boldsymbol{\theta}_i\mid\boldsymbol{\theta}_{i-1};\alpha_i)\,d\boldsymbol{\theta}_{1:n}.$$

For pocket- and property-conditioned generation, the paper extends this to

$$p_\phi(\mathbf{m}\mid\mathbf{p},\mathtt{l}) = \int p_\phi(\mathbf{m}\mid\boldsymbol{\theta}_n,\mathbf{p},\mathtt{l}) p(\boldsymbol{\theta}_0)\prod_{i=1}^n p_{\mathtt{U}}(\boldsymbol{\theta}_i\mid\boldsymbol{\theta}_{i-1},\mathbf{p},\mathtt{l};\alpha_i)\, d\boldsymbol{\theta}_{1:n},$$

and then factorizes the model over coordinates and atom types. This makes the object of control the BFN parameter state $\boldsymbol{\theta}$, not the molecular sample directly (Choi et al., 29 Aug 2025).

3. Bayesian-flow updates and integrated guidance

CByG reformulates BFN updates in gradient form and then augments them with conditional guidance. In the appendix, the BFN is specified through four distributions: the input distribution $$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$0, output distribution $$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$1, sender distribution $$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$2, and receiver distribution $$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$3. The Bayesian update distribution is

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$4

This is the formal substrate on which the guidance term is inserted (Choi et al., 29 Aug 2025).

For coordinates, the sender is Gaussian,

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$5

and the Bayesian update obeys

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$6

Using Tweedie’s formula,

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$7

the paper rewrites the update in a score-like form. For discrete atom types, the update is based on a softmax posterior and is given a continuous-relaxation / Tweedie-style interpretation that exposes a gradient term on the categorical variable. This is central to the paper’s claim that BFN guidance can handle the continuous/discrete hybrid without diffusion-style argmax pathologies (Choi et al., 29 Aug 2025).

The conditional move is conceptually

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$8

with an analogous modification for atom types. The coordinate update is explicitly decomposed into an Unconditional Generation term plus a Controllable Guidance term proportional to

$$\mathbf{x}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times 3}$$9

For atom types, the guided update is written as a softmax expression in which the unconditional factor is multiplied by a controllable-guidance factor involving $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$0. In the paper’s terminology, “gradient integration” means that property guidance is inserted into the Bayesian posterior update rather than added as an ad hoc sample-space perturbation (Choi et al., 29 Aug 2025).

The paper contrasts this with diffusion guidance. In diffusion, guidance modifies the reverse process using $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$1 or an approximation based on $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$2. In CByG, the analogous control term enters a parameter update,

$$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$3

The paper’s claim is that diffusion perturbs the sample, whereas BFN updates distribution parameters using noisy observations, so the guidance acts more stably (Choi et al., 29 Aug 2025).

4. Property prediction, uncertainty, and implementation

The guidance signal is supplied by an external Bayesian property predictor,

$$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$4

which outputs both a predictive mean and predictive variance. The paper estimates these quantities over $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$5 stochastic forward passes and decomposes the variance by the law of total variance into Aleatoric Uncertainty and Epistemic Uncertainty. The predictor is trained with Gaussian negative log-likelihood,

$$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$6

and a $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$7-weighted version

$$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$8

with $$\mathbf{v}_{\mathrm{M}}\in\mathbb{R}^{N_{\mathrm{M}}\times K}$$9 (Choi et al., 29 Aug 2025).

A central practical claim is that guidance is applied at inference time, not by retraining the generator. The BFN backbone is pretrained; the property predictor is trained separately. This permits, in the paper’s framing, multi-property optimization directly at sampling time without retraining. The controlled properties emphasized are binding affinity, synthetic feasibility, and selectivity. In the selectivity setting, the task is explicitly defined as increasing affinity to an on-target while reducing affinity to an off-target (Choi et al., 29 Aug 2025).

The property predictor adapts the TargetDiff graph transformer by removing the equivariant head, yielding an SE(3)-invariant graph transformer. The stated implementation details are: 16 attention heads, three SE(3)-invariant layers in each attention block, hidden dimension 64, key/value embeddings from a two-layer MLP, LayerNorm throughout, Swish activation, and an exponentially decayed learning rate with factor 0.6, lower bound $K$0, triggered after ten validation checks without improvement, with validation every 1000 steps. The paper also states an $K$1-equivariance argument: if the property predictor is invariant under $K$2, then its coordinate gradient is equivariant under the same transformation (Choi et al., 29 Aug 2025).

Several caveats are explicit. The paper does not provide a clean general formula for aggregating multiple property gradients into one joint objective, even though the framework is described as supporting multiple properties. It also notes an unresolved tension in the displayed uncertainty-weighting formula: the text says uncertainty should “moderate” guidance, but the written sampling update multiplies the gradient term by $K$3, which appears to increase guidance with variance. The paper does not clarify that point further (Choi et al., 29 Aug 2025).

5. Evaluation protocol, empirical results, and limitations

The main generation experiments use CrossDocked2020, following the filtered split of Luo et al.: the source corpus contains 22.5M protein–ligand pairs, and after RMSD filtering and a 30% sequence identity split, the reported setup uses 100,000 training pairs and 100 test proteins, sampling 100 molecules per test protein. A separate selectivity benchmark is constructed from kinase inhibitor selectivity data with 38 kinase inhibitors, initially 317 kinases in the main text, and a final curated set of 279 proteins after UniProt/ATP-binding-site filtering. Protein structures are predicted with AlphaFold3, and pockets are extracted around ATP binding sites (Choi et al., 29 Aug 2025).

The evaluation suite is deliberately broader than single-docking benchmarks. Binding is assessed with AutoDock Vina, SMINA, and GNINA, reporting Score. and Dock. values. The paper also reports High Affinity, SA, PB-Valid from PoseBusters, and Diversity. For retrosynthetic realism it uses AiZynthFinder, reporting Solved, Routes, Solved Routes, and Top Score. Structural quality is additionally examined with PoseCheck in the appendix. Baselines are divided into generation-only methods—AR, Pocket2Mol, TargetDiff, DecompDiff, MolCRAFT—and optimization/generation methods—RGA, DecompOpt, TacoGFN, ALIDiff, and TargetOpt—with TargetOpt serving as the main diffusion-guided comparator (Choi et al., 29 Aug 2025).

On the principal docking and quality benchmarks, CByG is reported as the strongest overall method. The stated values are: SMINA Score. $K$4, Dock. $K$5; GNINA Score. $K$6, Dock. $K$7; Vina Score. $K$8, Dock. $K$9; High Affinity Avg. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$0, Med. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$1; SA Avg. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$2, Med. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$3; PB-Valid Avg. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$4, Med. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$5; and Diversity Avg. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$6, Med. $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$7. For AiZynthFinder, the paper reports Solved $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$8, Routes $$\mathbf{P}=\{(\mathbf{x}_{\mathrm{P}},\mathbf{v}_{\mathrm{P}})\}.$$9, Solved Routes $\mathtt{l}$0, and Top Score $\mathtt{l}$1. The paper also stresses that SA and AiZynthFinder solvability are not strongly correlated, treating that as evidence that SA alone is an inadequate proxy for realistic synthesis (Choi et al., 29 Aug 2025).

In the selectivity benchmark, the reported success rates are 58.2\% for TargetDiff, 68.6\% for TargetOpt, 62.6\% for CByG w/o G, and 78.3\% for full CByG. The associated $\mathtt{l}$2 score improves from $\mathtt{l}$3 for the diffusion pair to $\mathtt{l}$4 for the BFN pair. Appendix ablations report that removing position guidance, removing type guidance, or removing uncertainty all degrade performance; among tested guidance scales, $\mathtt{l}$5 is best, $\mathtt{l}$6 under-guides, and $\mathtt{l}$7 reduces SA and PB-valid. The paper’s strongest stated limitation is synthetic feasibility: even for CByG, approximately half of the molecules generated remain synthetically infeasible, reflected in the Solved $\mathtt{l}$8 result (Choi et al., 29 Aug 2025).

6. Other arXiv interpretations and adjacent uses

Outside SBDD, CByG is not a standardized acronym but rather a label that can be projected onto several structurally different literatures. In “Double Cylinder Cycle codes of Arbitrary Girth” (Samadieh et al., 2014), the relevant notion is cycle-code / cylinder-type / girth-based construction for QC-LDPC cycle codes. That paper studies column-weight-two quasi-cyclic cycle codes, defines a mother matrix $\mathtt{l}$9, analyzes a block-structure graph $p(\mathtt{l}\mid \mathbf{x}_t)$0, and derives the maximum achievable girth

$p(\mathtt{l}\mid \mathbf{x}_t)$1

with design rate

$p(\mathtt{l}\mid \mathbf{x}_t)$2

The paper therefore fits a construction-by-girth reading of CByG, but it does not use the acronym itself (Samadieh et al., 2014).

In “Comparative genomics with succinct colored de Bruijn graphs” (Ramos et al., 2024), the relevant reading is colored de Bruijn graph analysis. That paper introduces gcBB and mgcBB, methods that compare genomes directly from a succinct colored de Bruijn graph in BOSS form using the Burrows-Wheeler Similarity Distribution (BWSD) rather than alignments or explicit path extraction. The BOSS space cost is stated as

$p(\mathtt{l}\mid \mathbf{x}_t)$3

and the reported end-to-end runtime improves from about 156.4 days for gcBB to about 20.6 days for mgcBB. On a 12-species Drosophila phylogeny task, the best reported configuration is entropy distance with coverage at $p(\mathtt{l}\mid \mathbf{x}_t)$4 and $p(\mathtt{l}\mid \mathbf{x}_t)$5, both achieving RF distance $p(\mathtt{l}\mid \mathbf{x}_t)$6. This is a colored-graph usage, not an explicit CByG acronym expansion (Ramos et al., 2024).

In “(2,2)-GB Codes: Classification and Comparison with weight-4 Surface Codes” (Arnault et al., 28 Jul 2025), the technical note explicitly says that the paper does not use the acronym CByG, but that the closest intended meaning is Cayley/Bicycle/Generalized-bicycle CSS codes. The paper studies $p(\mathtt{l}\mid \mathbf{x}_t)$7-GB codes of the form $p(\mathtt{l}\mid \mathbf{x}_t)$8, identifies

$p(\mathtt{l}\mid \mathbf{x}_t)$9

introduces a lattice-based distance lower bound, and constructs three infinite optimal families:

$$p_\phi(\mathbf{m}) = \int p_\phi(\mathbf{m}\mid\boldsymbol{\theta}_n)p(\boldsymbol{\theta}_0)\prod_{i=1}^n p_{\mathtt{U}}(\boldsymbol{\theta}_i\mid\boldsymbol{\theta}_{i-1};\alpha_i)\,d\boldsymbol{\theta}_{1:n}.$$0

It also introduces CGP-equivalence, a CSS-preserving equivalence notion based on qubit permutations. This is again a plausible interpretive use of CByG, but not an author-declared acronym (Arnault et al., 28 Jul 2025).

By contrast, “ByGARS: Byzantine SGD with Arbitrary Number of Attackers” (Regatti et al., 2020) is only terminologically adjacent. The paper proposes ByGARS and ByGARS++, not CByG. Its relevance is limited to nearby naming patterns in robust distributed learning; substantively it concerns auxiliary-data-driven reputation scores for Byzantine gradient aggregation and is not part of the CByG nomenclature itself (Regatti et al., 2020).

Taken together, the current arXiv landscape makes CByG primarily a name for controllable Bayesian-flow-guided 3D molecular generation in SBDD, with several secondary, non-standardized interpretive uses in coding theory, comparative genomics, and quantum error correction. The commonality across those secondary uses is not a shared method, but only a shared letter pattern attached to graph- or guidance-centric constructions.