Universal distribution of protein evolution rates as a consequence of protein folding physics

Abstract

The hypothesis that folding robustness is the primary determinant of the evolution rate of proteins is explored using a coarse-grained off-lattice model. The simplicity of the model allows rapid computation of the folding probability of a sequence to any folded conformation. For each robust folder, the network of sequences that share its native structure is identified. The fitness of a sequence is postulated to be a simple function of the number of misfolded molecules that have to be produced to reach a characteristic protein abundance. After fixation probabilities of mutants are computed under a simple population dynamics model, a Markov chain on the fold network is constructed, and the fold-averaged evolution rate is computed. The distribution of the logarithm of the evolution rates across distinct networks exhibits a peak with a long tail on the low rate side and resembles the universal empirical distribution of the evolutionary rates more closely than either distribution resembles the log-normal distribution. The results suggest that the universal distribution of the evolutionary rates of protein-coding genes is a direct consequence of the basic physics of protein folding.

Fig. 1.

Distribution (probability density function) of the logarithmic ratio of the probability of a sequence to fold into the native structure and the probability to fold to an alternative structure. See SI Materials and Methods for the values of parameters used to compute these distributions.

Fig. 2.

Distributions (probability densities) of the logarithm of the fold-averaged ER for three chain lengths N and typical values of abundance (A = 10⁻⁴), effective population size (N_e = 10⁴), and per-monomer mistranslation probability (r = 0.02).

Fig. 3.

Comparison of the model-derived (N = 18, r = 0.02, N_e = 10⁴) and observed distributions of ER. The quantile-quantile plot compares the empirical ER distributions for the sets of orthologous proteins taken from ref. for (A) Homo sapiens and Macaca mulatta (denoted Homsa, 16603 proteins) and (B) two strains of the bacterium Burkholderia sp (Bursp, 4014 proteins) with the distributions yielded by the model with three different abundance levels. Solid lines are linear fits to the A = 0.5 × 10⁻⁴ data.

Fig. 4.

Normal probability plot comparing the normalized model-derived and empirical ER distributions with the normal distribution. The distributions are the same as used for Fig. 3. The empirical ER distributions are more similar to each other and to the model distribution (all are convex upward) than to the normal distribution. A normal distribution would appear in a normal probability plot as a 45° line, shown here for reference.

Fig. 5.

Dependence of the mean, median, and Pearson’s skewness, defined as 3(median − mean)/(standard deviation), of the ER distribution derived from the model on abundance, effective population size, and per-monomer mistranslation probability. The chain length is N = 18. The dashed lines are the empirical Sk values computed for the Bursp (blue) and Homsa (green) samples.