{"doi":"10.1073/pnas.2018767118","title":"Epistasis produces an excess of invariant sites in neutral molecular evolution","abstract":"Substitution rate variability among sites is a common feature of protein evolution. This variability is frequently modeled by a gamma (Γ) distribution (1), which is proposed to be an emergent property of epistasis by de la Paz et al. (2). However, the conclusion of de la Paz et al. (2) is based on the analysis of highly divergent sequences, as many as 24 substitutions per site. We examined if this conclusion holds for more biologically realistic sequence divergences (0.1 to 5 substitutions per site) (3). Instead, we found that creation of a class of invariant sites is an emergent property of epistasis, and the sequence divergence level dictates whether an equal- or a variable-rate model will best fit the data. In de la Paz et al. (2), neutral molecular evolution was simulated for protein families with a fitness function defined by a model inferred via direct coupling analysis (e.g., Fig. 1 A ). We counted the number of sites that experienced zero, one, two, three, and more substitutions directly from the generation by generation simulated sequences (e.g., Fig. … [↵][1]1To whom correspondence may be addressed. Email: s.kumar{at}temple.edu. [1]: #xref-corresp-1-1","journal":"Proceedings of the National Academy of Sciences","year":2021,"id":209387,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":3,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9457,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2021-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":3959,"name":"Sudhir Kumar","orcid":"0000-0002-9918-8212","position":1,"is_corresponding":false},{"id":796997,"name":"Ravi Patel","orcid":"0000-0001-9327-2803","position":0,"is_corresponding":true}],"reference_count":7,"raw_metadata":null,"created_at":"2026-07-18T23:52:05.171356Z","pmid":"33875549","pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}