Thermodynamics of neutral protein evolution

Abstract

Naturally evolving proteins gradually accumulate mutations while continuing to fold to stable structures. This process of neutral evolution is an important mode of genetic change and forms the basis for the molecular clock. We present a mathematical theory that predicts the number of accumulated mutations, the index of dispersion, and the distribution of stabilities in an evolving protein population from knowledge of the stability effects (delta deltaG values) for single mutations. Our theory quantitatively describes how neutral evolution leads to marginally stable proteins and provides formulas for calculating how fluctuations in stability can overdisperse the molecular clock. It also shows that the structural influences on the rate of sequence evolution observed in earlier simulations can be calculated using just the single-mutation delta deltaG values. We consider both the case when the product of the population size and mutation rate is small and the case when this product is large, and show that in the latter case the proteins evolve excess mutational robustness that is manifested by extra stability and an increase in the rate of sequence evolution. All our theoretical predictions are confirmed by simulations with lattice proteins. Our work provides a mathematical foundation for understanding how protein biophysics shapes the process of evolution.

Figure 1.—

A thermodynamic view of protein evolution. A mutant protein stably folds if and only if it possesses some minimal stability, (in this case −5 kcal/mol). The stability of the wild-type protein is , meaning that it has of extra stability. The bars show the distribution of ΔΔG values for mutations. Those mutants with still stably fold, while all other mutants do not fold and so are culled by natural selection. The probability that a mutation will be neutral with respect to stable folding is simply the fraction of the distribution that lies to the left of the threshold. The data in this figure are hypothetical.

Figure 2.—

The theory gives accurate predictions for the evolution of model lattice proteins. Each row of graphs corresponds to a different lattice protein. The left graphs show the starting protein and the distribution of ΔΔG values for all point mutations. The middle and right graphs show the predicted (lines) and measured (bars) distributions of stabilities among the evolved proteins. The tables at the top of the graphs show the predicted and measured values for the average number of mutations () and the index of dispersion (R_T) after 5000 generations of neutral evolution. The middle graphs are for a population size of N = 10, and the right graphs are for N = 10⁵. In both cases, the per-protein-per-generation mutation rate is μ = 0.01. As predicted, the evolving population with evolved mutational robustness that is manifested by increased protein stability. This additional mutational robustness accelerated the rate of sequence evolution.