[Paper Review] Minimal-Action Discrete Schrödinger Bridge Matching for Peptide Sequence Design

Shrey Goel, Pranam Chatterjee|arXiv (Cornell University)|Jan 29, 2026

DNA and Biological Computing0 citations

TL;DR

MadSBM introduces a rate-based discrete Schrödinger bridge for peptide design, using a biology-informed reference and learned control to generate high-likelihood sequences efficiently.

ABSTRACT

Generative modeling of peptide sequences requires navigating a discrete and highly constrained space in which many intermediate states are chemically implausible or unstable. Existing discrete diffusion and flow-based methods rely on reversing fixed corruption processes or following prescribed probability paths, which can force generation through low-likelihood regions and require countless sampling steps. We introduce Minimal-action discrete Schrödinger Bridge Matching (MadSBM), a rate-based generative framework for peptide design that formulates generation as a controlled continuous-time Markov process on the amino-acid edit graph. To yield probability trajectories that remain near high-likelihood sequence neighborhoods throughout generation, MadSBM 1) defines generation relative to a biologically informed reference process derived from pre-trained protein language model logits and 2) learns a time-dependent control field that biases transition rates to produce low-action transport paths from a masked prior to the data distribution. We finally introduce guidance to the MadSBM sampling procedure towards a specific functional objective, expanding the design space of therapeutic peptides; to our knowledge, this represents the first-ever application of discrete classifier guidance to Schrödinger bridge-based generative models.

Motivation & Objective

• Reframe peptide sequence generation as minimal-action transport between a noisy prior and data distribution on the amino-acid edit graph.
• Leverage a biologically informed reference process derived from pre-trained protein language model logits to guide generation.
• Learn a time-dependent control field to tilt transition rates toward high-likelihood peptide neighborhoods.
• Enable objective-guided sampling to incorporate functional design goals.
• Demonstrate sample efficiency and stability compared with discrete diffusion baselines while enabling guided design.

Proposed method

• Model generation as a controlled continuous-time Markov chain (CTMC) on the amino-acid edit graph with a reference generator R0.
• Define R_u(x,x') = R0(x,x') exp(u_theta(x,x',t)) to create a tilted, time-dependent controlled process.
• Derive a tractable action functional A(u) = E_{P_u}[ ∫ R0(X_t,x') Ψ(u(X_t,x',t)) dt ] with Ψ(z)=e^z - z - 1.
• Train a neural control field u_theta using a cross-entropy objective that aligns transitions toward target sequences (Eq. 10).
• Use ESM-2 logits as the reference process and a time-gating mechanism to modulate the influence of the reference across masking.
• Sampling evolves from a fully masked prior by simulating a CTMC with the learned generator R_theta, discretized over N steps.

Figure 1: Overview of MadSBM . We leverage a principled reference process $R_{0}$ so the MadSBM model requires only a lightweight time-conditioned control field $u_{\theta}$ to steer samples toward high-likelihood regions of the sequence space.

Experimental results

Research questions

• RQ1 Can a discrete Schrödinger bridge be solved in a de novo discrete peptide space using a biologically informed reference process?
• RQ2 Does learning a time-dependent control field under a minimal-action objective yield efficient, high-quality peptide generation compared with discrete diffusion baselines?
• RQ3 Can objective guidance (e.g., binding affinity) be incorporated into discrete Schrödinger bridge sampling to improve functional design?

Key findings

• MadSBM achieves competitive or better unconditional sequence quality (lower PPL) than a discrete diffusion baseline across sampling budgets, with plausible pLDDT scores.
• Unconditioned results show MadSBM PPLs: 8.389, 8.943, 8.719 vs. DD PPLs: 10.990, 9.042, 7.617 for N=32, 64, 128 respectively (lower is better).
• MadSBM demonstrates broader likelihood-path diversity during sampling, enabling earlier convergence to higher-likelihood regions than the DD baseline.
• Ablation of the biologically informed reference (ESM-2) and time-gating degrades perplexity, validating the reference dynamics design.
• Objective-guided sampling with a binding-affinity predictor improves affinities and docking scores for several targets beyond unconditional generation.

Figure 2: Probability paths taken by models under various sampling budgets $(N)$ . The y-axis represents the NLL of the sequence at the current iteration, assessed by the ESM-2-650M protein language model. The shaded area around the traced trajectory represents the standard deviation of the NLL at t