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.