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

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 $\%{\mathrm{MinMax}}_{(s)}$, a differentiable relaxation of the conventional hard $\%$MinMax metric, denoted as $\%{\mathrm{MinMax}}_{(h)}$. $\%{\mathrm{MinMax}}_{(s)}$ 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 $\%{\mathrm{MinMax}}_{(h)}$, while enabling optimization with respect to the synonymous-codon probabilities and learnable parameters. In human-to-Escherichia coli codon harmonization experiments, $\%{\mathrm{MinMax}}_{(s)}$ closely approximates $\%{\mathrm{MinMax}}_{(h)}$ and supports gradient-based profile matching in synonymous-codon probability space. These results suggest $\%{\mathrm{MinMax}}_{(s)}$ as a practical bridge between profile-based codon harmonization and neural synonymous-sequence design.

17 pages, 2 figures