Mutation Flows: Decomposable Variable-length Flow Matching with Block Insertions

Biological sequences evolve through local edits: single-residue substitutions, deletions, and block insertions of several residues at once. Discrete flow matching is a natural fit for this generative process, but existing formulations fall short in three ways. They fix sequence length or restrict edits to single tokens; they cannot decompose the joint prediction of edit type, block length, and token content; and they fix the source-to-target sequence distance implicitly through the choice of coupling, leaving it uncontrollable at inference time. We introduce Mutation Flows, which addresses all three. (i) Block insertions of arbitrary length are modelled by a zero-truncated Poisson distribution and resolved in a single transition. (ii) An exact planner-denoiser decomposition splits the objective into two losses with disjoint parameters, whose denoiser term reduces to a standard fixed-length discrete flow matching objective, admitting any pretrained biological sequence model as a drop-in denoiser. (iii) Divergence conditioning allows for coarse-grained control of the sequence divergence from the source sequence during inference. This approach enables flows between the same source and target distributions without degrading into a trivial mapping, as is the case for optimal transport. On antimicrobial peptides, Mutation Flows stays closest to the data distribution throughout the denoising trajectory, remains effective with a pretrained ESM-2 dropped in as the denoiser, preserves these gains in the few-step denoising setting, and achieves state-of-the-art performance in designing active peptides.