Asymptotic Hitting Time for a Simple Evolutionary Model of Protein Folding

Abstract

We consider two versions of a simple evolutionary algorithm (EA) model for protein folding at zero temperature, namely the (1 + 1)-EA on the LeadingOnes problem. In this schematic model, the structure of the protein, which is encoded as a bit-string of lengthn, is evolved to its native conformation through a stochastic pathway of sequential contact bindings. We study the asymptotic behavior of the hitting time, in the mean case scenario, under two different mutations: theone-flip, which flips a unique bit chosen uniformly at random in the bit-string, and theBernoulli-flip, which flips each bit in the bit-string independently with probabilityc/n, for somec∈ℝ+(0 ≤c≤n). For each algorithm, we prove a law of large numbers, a central limit theorem, and compare the performance of the two models.