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

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.