Minimal-Action Discrete Schrödinger Bridge Matching for Peptide Sequence Design

TL;DR

MadSBM treats peptide sequence design as minimal-action transport on a discrete sequence graph, formulating generation as a Schrödinger bridge between a masked prior and the data distribution. A biologically informed reference process is obtained from pre-trained protein language-model logits and a time-dependent control field u_{\\theta} tilts transition rates via an exponential tilt R_{u,\\theta}(x,x') = R_0(x,x') e^{u_{\\theta}(x,x',t)} to follow low-action paths. Training reduces to a cross-entropy objective that aligns the learned tilt with the optimal Schrödinger-bridge velocity, avoiding costly forward–backward solves while preserving theoretical minimal-action transport properties. The framework supports objective-guided sampling for affinity optimization and demonstrates sample efficiency and competitive biologically plausible generation against discrete diffusion baselines, with potential for broader discrete-domain design.

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.

Figures (3)

• 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.
• 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 the currrent sampling iteration.
• Figure 3: Instantaneous actional values for MadSBM and ESM-ablated counterpart. The y-axis represents the actional $\mathcal{A}_L(u)$ at the current timestep. Results are shown at various sampling budgets.

Theorems & Definitions (22)

• Theorem 4.1
• Corollary 4.2: The Action Functional
• Proposition 4.3: Consistency with Minimal-Action Control
• Proposition 4.4: Convergence of MadSBM Sampling
• Theorem A.1: Path-space KL decomposition for CTMCs
• proof
• Corollary A.2: Action form under exponential tilting
• proof
• Theorem A.3: Endpoint-tilted form and uniqueness
• proof