Variable-Length Generative Protein Design via Generalized Poisson Flow

Abstract: The ability to generate variable-length proteins is crucial in protein design, where the optimal length is often unknown and tightly coupled to designability. Current diffusion- and flow-based generative models typically require the protein length to be specified before sampling, limiting their flexibility in exploring the feasible design space. To address this limitation, we introduce Generalized Poisson Flow (GPFlow), a variable-length generative framework that learns the rate function of an inhomogeneous generalized Poisson process by minimizing its negative log-likelihood. We establish population-level guarantees for recovering the joint multimodal distribution and derive an upper bound on the KL divergence between the data and generated distributions. We comprehensively evaluate GPFlow across structure and sequence design, motif scaffolding, and peptide co-design, spanning Euclidean, categorical, and Riemannian modalities to fully validate its variable-length generation quality. In unconditional design, GPFlow improves structural designability and achieves the best distributional fitness for sequence design compared to their corresponding fixed-length baselines, while perfectly recovering the length distribution. In conditional motif scaffolding, GPFlow ranks first on 10 of 16 structure-based design tasks with significantly more unique successes and also achieves more passed tasks in sequence-based design. In peptide co-design, GPFlow remains competitive even without access to a native-length oracle.

• The paper introduces GPFlow, which models variable-length protein sequences using a generalized Poisson process to capture the joint distribution of length and modalities.
• It decouples length evolution from structural refinement, enabling simulation-free NLL training and efficient, order-preserving generation.
• Empirical results demonstrate up to 96.1% structural designability and realistic sequence distributions across both unconditional and conditional protein design tasks.

Variable-Length Generative Protein Design via Generalized Poisson Flow

Introduction and Motivation

The optimal design of protein sequences and structures often requires modeling over variable-length data distributions, as biologically relevant or functional protein lengths are generally not predetermined. Existing state-of-the-art diffusion and flow-based generative models force the user to specify a target length a priori, thereby reducing the capacity to capture and explore the full sequence-structure landscape. This is notably limiting when modeling motif scaffolding or peptide-protein designs, where length directly impacts structural feasibility and biological function.

The manuscript introduces Generalized Poisson Flow (GPFlow), a unified generative framework enabling direct modeling and sampling of the joint distribution of length and length-dependent variables—including both categorical (e.g., amino acid identity) and continuous (e.g., atomic coordinates, torsions, or backbone frames) modalities. GPFlow is formulated as a variable-dimensional probability path parameterized via the rate function of an inhomogeneous generalized Poisson process and learned by negative log-likelihood minimization. This approach is shown to yield tractable population-level recovery guarantees for the data manifold, extending the expressivity and correctness of flow-based generative protein design.

Methodology

Generalized Poisson Flow Construction

GPFlow models proteins as stochastic processes in which the length evolves according to a continuous-time inhomogeneous Poisson process with a learnable, state-dependent rate $\lambda_t$. The framework decouples length evolution (birth process) from within-length generative modeling, to handle conditional and unconditional scenarios and arbitrary modality types.

• Length process: The probability path for length evolves via parameterized insertion rates, with a scheduler $\kappa(t)$ governing the time-dependent probability of component survival.
• Within-length process: For each current length, a generator (either vector field for continuous variables or Markov kernel for discrete ones) refines the state of the chain.
• Multimodality: The model provides a marginalization theorem allowing the coupling of arbitrary numbers and types of modalities (e.g., backbone coordinates, orientations, sequence identity) conditioned on the entire process path.

A critical technical contribution is the derivation of a tractable simulation-free negative log-likelihood (NLL) objective for the Poisson process, leveraging the marginalization over auxiliary insertion times and enabling theoretical guarantees on both joint distributional recovery and an explicit upper bound on the KL divergence between data and generated samples via generator matching.

Conditional and Order-Preserving Generation

GPFlow naturally supports conditional tasks (motif scaffolding, conditional peptide design) by freezing a subset of sequence/structural components and restricting insertions to permissible regions. For protein design, the order-preserving variant is essential: insertions happen at specific sequence slots, and the generative process is guaranteed to preserve conditional segment integrity—a critical property for scaffolding physiological motifs and peptide binding.

Efficient Training and Sampling

The framework integrates the following strategies to reconcile practical generation speed and fidelity:

• Simulation-free NLL training: The closed-form loss for the Poisson process makes training feasible without trajectory simulation.
• $\tau$-leaping sampling: To avoid excessive computational burden due to frequent Poisson events in long proteins, a $\tau$-leaping approximation is adopted, enabling efficient multi-position insertions per time step.
• Localized insertions: For long sequence scenarios, insertion rates can be localized to support the generation of contiguous protein fragments, improving local consistency and realism.

Empirical Evaluation

GPFlow is extensively validated across five major protein design regimes, each with tailored experimental protocols and direct comparisons to state-of-the-art fixed-length or variable-length baselines:

Unconditional Structure Generation

On CA-trace structure generation (Proteina [18] baseline), GPFlow consistently achieves the highest structural designability (up to 96.1%, a 3.9–18.5% improvement over Proteina), demonstrating that removal of the fixed-length prior yields a superior model of the structural manifold. Importantly, GPFlow generates a length distribution that matches the empirical data marginal, confirming correct length modeling. The model also delivers improved or matched diversity and novelty, albeit subject to the diversity-temperature tradeoff inherent to flow-based methods.

Unconditional Sequence Generation

For discrete sequence modeling (642M DiT backbone), GPFlow achieves unprecedentedly close matching to biological sequence distributions, as validated by foldability (pLDDT), diversity (FoldSeek clustering), and secondary structure statistics. Notably, whereas DPLM [48] over-optimizes for foldability (mean $\Delta$ pLDDT of 14.40, far from UniRef50 statistics), GPFlow maintains high sequence and structural diversity alongside realistic foldability ($\Delta$ pLDDT of 3.17). The emergent length distribution is highly faithful to UniRef50, and the model allows accurate, flexible length control via classifier-free guidance.

Motif Scaffolding (Conditional Design)

On the motif scaffolding benchmark [49], GPFlow outperforms all existing structure-based and sequence-based generators in both average number and diversity of unique successful scaffolds, especially on complex multi-segment or long-range motif tasks. Unlike fixed-length counterparts that are constrained by predefined contig templates, GPFlow explores scaffold lengths more thoroughly and can generate successful novel designs across a broader length spectrum.

Peptide Sequence-Structure Co-Design

For conditional peptide-protein design, GPFlow is instantiated atop PepFlow [29] with full support for mixed modalities (residue type, rotation, translation, torsion angles). Without access to a native-length oracle (unlike baselines), GPFlow's generated peptides match natural length distributions and outperform or match prior models in designability, diversity, and geometric/biological fitness metrics. The approach achieves an RMSD of 1.39 Å and AAR of 53.71% (composite), even on the length-unmatched subset.

Theoretical Guarantees

A central theoretical advance is the proof of distributional recovery and characterization of the generator-based loss. The GPFlow objective minimizes an explicit upper bound on the KL divergence between the model and data manifolds, decomposed into contributions from the rate function and within-length generator. Unlike prior approaches, the use of the generalized KL divergence is motivated by direct derivation from the process' log-likelihood, aligning the training dynamics with the global generative objective. The theoretical apparatus is applicable to variable-dimensional, multimodal spaces and subsumes previous insertion- or edit-based models as special cases.

Limitations and Future Directions

The model’s performance is sensitive to scheduler (insertion timing) choices, particularly in the multimodal setting where modalities can interact nontrivially during sampling. While $\tau$-leaping and localized insertion paths provide empirical improvements, there remain open questions around the optimal handling of long-range dependencies, sampling efficiency, and potential artifacts when extension to higher-dimensional attributes (e.g., complexes or assemblies). Extension to learnable deletion (birth-death) dynamics, more expressive intra-segment vector fields, and on-the-fly adaptation of insertion strategies for rare protein folds represent promising directions.

Conclusion

GPFlow establishes a new paradigm for variable-length generative modeling in protein design, eliminating the restrictive length-conditioning of previous frameworks. Empirical evidence across unconditional and conditional generation in both structure and sequence design demonstrates the flexibility, fidelity, and diversity tractability of the approach. The formulation is mathematically principled, with tractable and theoretically justified learning objectives, and immediate implications for generative modeling tasks in protein engineering and computational biology. Future work may generalize these constructions to other domains requiring variable-dimensional multimodal generative modeling, including functional RNA or synthetic polymers.

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Reference: "Variable-Length Generative Protein Design via Generalized Poisson Flow" (2607.09039)