Effects of single mutations on protein stability are Gaussian distributed

The distribution of protein stability effects is known to be well-approximated by a Gaussian distribution from previous empirical fits. Starting from first-principles statistical mechanics, we more rigorously motivate this empirical observation by deriving per residue protein stability effects to be Gaussian. Our derivation requires the number of amino acids to be large, which is satisfied by the standard set of 20 amino acids found in nature. No assumption is needed on the protein length or the number of residues in close proximity in space, in contrast to previous applications of the central limit theorem to protein energetics. We support our derivation results with computational and experimental data on mutant protein stabilities across all types of protein residues. Statement of Significance Defining the distribution of single mutant stability effects (ΔΔGs) is the first step in modeling the role protein stability plays in evolution. Although empirical fits have been made to elucidate its form, a complete theoretical understanding of ΔΔG distributions is lacking. Here, we derive how a simple Gaussian form can arise, while still including the intricacies of protein sequence and structure. We backup our derivation with previously released computational and experimental ΔΔGs.