Size Scaling of Mutation Avalanches in a Model for Protein Evolution

The tunably rugged NK-model is used to study avalanche-like events that occur when environmental change causes fitness optima to disappear. The probability of an event with Δ substitutions scales as exp(−cΔ) for smooth landscapes, and as exp(−cΔ2) for rugged landscapes. Increasing the ruggedness leads to two competing effects: (1) more possible routes by which single mutations can increase the fitness, which dominates at low ruggedness and acts to increase Δ; and (2) a higher density of fitness optima, which dominates at high ruggedness and acts to decrease Δ. Due to these competing effects, the largest average values of Δ occur at intermediate ruggedness. The effects of system size on the avalanche events are examined, and average values of Δ increase logarithmically with system size. The variance to mean ratios for the number of substitutions per unit time are consistent with experimental results for protein evolution.