Lattice proteins with different structures but the same stability (ΔG_f = -1.0) converge to different exponential declines in m-neutrality. (a) The distributions of ΔΔG for all 380 single amino acid substitutions to the inset lattice proteins. (b) The measured (symbols) and predicted (lines) m-neutralities for the four proteins. Proteins are considered folded if ΔG_f ≤ 0.0 for the original native structure. The proteins used for the m-neutrality analyses were generated by adaptive walks from random starting sequences, followed by 2.5 × 10⁵ generations of neutral evolution with a population size of 100 and a per-generation per-residue substitution rate of 5 × 10^-5, selecting for sequences with ΔG_f ≤ -1.0 and then taking the first sequence generated with a stability within 0.025 of -1.0. The m-neutralities were computed by sampling all mutants for m≤ 2 or 5 × 10⁵ random mutants for m > 2. The predicted m-neutralities were computed according to Eq. 1 by numerically convolving the distribution of single-substitution ΔΔG values using generating functions (27) computed with fast-Fourier transforms and a bin size of 0.01.

Lattice proteins with the same structure but different stabilities have different 1-neutralities but have the same average neutrality 〈ν_aa〉. (a) Predicted (lines) and measured (symbols) m-neutralities for proteins with different stabilities and the same structure (III in Fig. 1). (b) Measured values of the 1-neutralities (squares) and average neutralities (circles) for proteins with different stabilities but the same structures (the plots at left and right are for structures I and IV from Fig. 1, respectively). The sequences were generated by finding a sequence with ΔG_f = -2.0 by using the procedure described in Fig. 1, and then using this sequence as a starting point for neutral evolution selecting for the indicated target stabilities. The proteins with different stabilities are highly diverged, with average pairwise sequence identities of 15% and 41% for the structures at left and right, respectively. The m-neutralities were computed as in Fig. 1, and 〈ν_aa〉 was computed as the square root of the 6-neutrality divided by the 4-neutrality.

Theoretical predictions and fractions of functional proteins in mutant libraries of subtilisin (dashed lines) and TEM1 β-lactamase (solid lines) genes. Thick lines show predictions made by using popmusic (38), and thin lines show predictions made by using foldef (39). The TEM1 measurements are from Table 1, normalized by the values from the control unmutated library.

The more stable M182T variant of TEM1 β-lactamase (dashed lines) exhibits a higher fraction of functional mutants relative to wild type (solid lines), as predicted. Thick lines show predictions made by using popmusic (38), and thin lines show predictions made by using foldef (39). The measurements are from Table 1, normalized by the values from the control unmutated library.