Mapping protein neutral networks from predicted secondary structure

Protein evolution can be understood as movement through a genotype- phenotype (GP) map, where genetic variation is constrained by the mapping from sequence to structure. While RNA GP maps have been extensively characterised, equivalent maps for proteins remain largely unexplored due to the complexity of folding. Here, we construct an operational GP map for influenza hemagglutinin (HA) using predicted secondary structure as a coarse-grained phenotype. Combining site-scanning with exhaustive local neighbourhood enumeration, we estimate neutral component (NC) sizes, mutational robustness, and local connectivity patterns across representative HA variants. We observe strong phenotypic bias, with NC sizes spanning orders of magnitude. Robustness increases with estimated NC size, but far more weakly than in RNA, consistent with each NC occupying a vanishing fraction of amino acid sequence space. Local neutral neighbourhood graphs reveal dense, position-centric clusters with limited overlap, suggesting heterogeneous and weakly percolating connectivity. Single-step structural transitions are biased toward structurally similar phenotypes, so that accessible novelty is largely incremental and redundant. Together, these results provide a tractable empirical framework for protein GP map analysis and suggest that, in HA, neutrality is locally structured yet globally constrained, limiting long-range structural evolvability relative to RNA systems.