1. The ancestral sequence reconstruction (ASR) is a molecular evolution technique that provides applications to a variety of fields such as biotechnology and biomedicine. In order to infer ancestral sequences with realistic biological properties, the accuracy of ASR methods is crucial. We previously developed an ASR framework for proteins, called ProtASR, which is based on our site-specific stability constrained substitution (SCS) model with selection on protein folding stability against both unfolding and misfolding. This model improved the empirical substitution models traditionally applied in ASR without increasing the computational complexity. However, it adopted a global exchangeability matrix, an approximation that we overcome here by considering site-specific exchangeability matrices based on the Halpern-Bruno approach. 2. Here we present ProtASR2, a new version of our ASR framework that implements novel SCS models of protein evolution, namely mean-field (MF) and wild-type (WT). 3. ProtASR2 under MF and WT SCS models outperforms empirical models and previous SCS models in terms of goodness of fit and site-specific distributions of amino acids. Importantly, the framework infers ancestral sequences with more realistic predicted folding stability with respect to simulated sequences, while empirical, CAT and other SCS models tend to overestimate the folding stability. We applied ProtASR2 to explore the evolution of two protein families present in diverse Prokaryota and found fluctuations of protein stability over time in both families. ProtASR2 is available from https://github.com/miguelarenas/protasr and the new SCS models are also available from https://github.com/ugobas/protevol. 4. Use of ProtASR2 will allow more realistic inferences of ancestral proteins in terms of folding stability with respect to those based on traditional empirical and CAT substitution models of protein evolution.

Data Availability
Raw data (real (downloaded from the Pfam database), simulated (with the program ProteinEvolver) and inferred protein sequences (with the program ProtASR2) and their corresponding protein folding energies (predicted with the program DeltaGREM)) that was obtained in this study.

Evaluation of the ASR
For each protein family, the folder “*SimulatedANDinferredDATA” includes all simulated protein sequence alignments (sequences#.fas) and inferred ancestral sequences with ProtASR2 under MF and WT models.
The folder “*ComputedEnergiesFromData” includes the energies of sequences of every dataset (files *.dat) presented in the corresponding folder “*SimulatedANDinferredDATA”.

Real Data used for the ASR
Files related with the analysis of real data are shown in the folder “RealDataASR”, for the studied protein families (DDL and DNAK). MSA and phylogenetic trees are in .fas/.nex formats and Newick format, respectively. Energies are shown in the .dat file and printed on the tree in files *Energies.tre (we recommend open this tree with the program FrigTree). Some derived figures are also included. Results are shown for both structurally constrained substitution models, MF and WT.