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arXiv · 2607.03881

Smooth $\%$MinMax: A Differentiable Relaxation for Codon Harmonization

Yoonho Jeong·Hyunwoo Choi·Ryan Fernandez Medina Hariri·Eok Kyun Lee·Seung Seo Lee·Insung S. Choi
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Abstract

Codon harmonization aims to adapt the coding sequences for heterologous expression while preserving the native-like patterns of frequent and rare codons that may influence local translation dynamics and co-translational protein folding. However, widely used harmonization metrics, such as $\%$MinMax, are defined on discrete codon sequences and are, therefore, not readily compatible with gradient-based neural codon design. Here, we introduce Smooth $\%$MinMax, denoted as $\%{\rm MinMax}_{(s)}$, a differentiable relaxation of the conventional hard $\%$MinMax metric, denoted as $\%{\rm MinMax}_{(h)}$. $\%{\rm MinMax}_{(s)}$ replaces the discrete codon-usage values with probability-weighted synonymous-codon usage values and replaces the hard $\%$Max/$\%$Min branch with a sigmoid-gated interpolation. This formulation preserves the signed interpretation of $\%{\rm MinMax}_{(h)}$, while enabling optimization with respect to the synonymous-codon probabilities and learnable parameters. In human-to-Escherichia coli codon harmonization experiments, $\%{\rm MinMax}_{(s)}$ closely approximates $\%{\rm MinMax}_{(h)}$ and supports gradient-based profile matching in synonymous-codon probability space. These results suggest $\%{\rm MinMax}_{(s)}$ as a practical bridge between profile-based codon harmonization and neural synonymous-sequence design.

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Yoonho Jeong, Hyunwoo Choi, Ryan Fernandez Medina Hariri, Eok Kyun Lee, Seung Seo Lee, Insung S. Choi. 2026-07-04. Smooth $\%$MinMax: A Differentiable Relaxation for Codon Harmonization. https://arxiv.org/abs/2607.03881

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