Centrality measures definition.

The structure and sequence of proteins strongly influence their biological functions. New models and algorithms can help researchers in understanding how the evolution of sequences and structures is related to changes in functions. Recently, studies of SARS-CoV-2 Spike (S) protein structures have been performed to predict binding receptors and infection activity in COVID-19, hence the scientific interest in the effects of virus mutations due to sequence, structure and vaccination arises. However, there is the need for models and tools to study the links between the evolution of S protein sequence, structure and functions, and virus transmissibility and the effects of vaccination. As studies on S protein have been generated a large amount of relevant information, we propose in this work to use Protein Contact Networks (PCNs) to relate protein structures with biological properties by means of network topology properties. Topological properties are used to compare the structural changes with sequence changes. We find that both node centrality and community extraction analysis can be used to relate protein stability and functionality with sequence mutations. Starting from this we compare structural evolution to sequence changes and study mutations from a temporal perspective focusing on virus variants. Finally by applying our model to the Omicron variant we report a timeline correlation between Omicron and the vaccination campaign.