Lack of Self-Averaging in Neutral Evolution of Proteins

We simulate neutral evolution of proteins imposing conservation of the thermodynamic stability of the native state in the framework of an effective model of folding thermodynamics. This procedure generates evolutionary trajectories in sequence space which share two universal features for all of the examined proteins. First, the number of neutral mutations fluctuates broadly from one sequence to another, leading to a non-Poissonian substitution process. Second, the number of neutral mutations displays strong correlations along the trajectory, thus causing the breakdown of self-averaging of the resulting evolutionary substitution process.

Figure 1

(a) Probability distribution $P(x)$ of the fraction $x$ of neutral neighbors for myoglobin. (b) Analysis of the fluctuations of $x$ along an evolutionary trajectory of myoglobin, displayed as $F(\ell )$ vs $\ell$. We present the analysis of the whole protein (triangles) and the analysis for residues 25 (circles) and 115 (squares). The full lines have the slopes 0.82 (for the triangles, up to $\ell \approx 20$), 0.85 (for the circles, up to $\ell \approx {10}^2$), and 0.9 (for the squares, up to $\ell \approx {10}^3$) indicating strong correlations, whereas the dashed lines have the slope $1/2$, indicating the absence of correlations. The inset in (b) shows part of the profile $p(n)$ vs $n$ for the total fraction of neutral neighbors (see text).

Figure 2

Statistics of the substitution process for myoglobin, shown are the average number of substitutions $\overline{⟨S_t⟩}$ divided by $\mu t$ (circles), the normalized mutation variance $R_{\mu }$ (squares), the normalized trajectory variance $R_x$ (triangles), and the normalized total variance $R=R_{\mu }+R_x$ (diamonds). We present in (a) the statistics for the trajectories obtained through the SCN model (full symbols), and in (b) the same for the corresponding annealed trajectories (open symbols).