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 length n, 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: the one-flip, which flips a unique bit chosen uniformly at random in the bit-string, and the Bernoulli-flip, which flips each bit in the bit-string independently with probability c/n, for some c ∈ 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.

Solving Problems with Unknown Solution Length at (Almost) No Extra Cost

“…We will derive this result from the following lemma, which was independently proven in [3, Theorem 3], [10, Corollary 2], and in a slightly weaker form in [8, Theorem 1.2]. Lemma 11 ([3], [10], and [8]). For a fixed length n and a mutation vector p = (p, .…”

Probabilistic Analysis of the (1+1)-Evolutionary Algorithm

“…uniformly for 1 m n, which then proves Theorem 2. By (26), ∆ * n,m satisfies, for 1 m n, the recurrence…”

“…The crucial property we need is that the truncated expansion (42) holds when K m n, and particularly when m = K. So we compute (42) with m = K and drop all terms of order smaller than or equal to n −K−1 . Then we match the coefficient of n −K with that in the expansion of µ * n,K obtained by a direct calculation from the recurrence (26). We illustrate this procedure by computing the first two terms in (41).…”

“…We consider the complexity of the (1 + 1)-EA when the underlying fitness function is the number of leading ones. This problem has been examined repeatedly in the literature due to the simple structures it exhibits; see [4,14,26] and the references therein. The strongest results obtained were those by Ladret [26] (almost unknown in the EA literature) where she proved that the optimization time under LEADINGONES is asymptotically normally distributed with mean asymptotic to e c −1 2c 2 n 2 and variance to 3(e 2c −1)…”

Drift Analysis

“…The LeadingOnes problem is a classical benchmark problem for evolutionary algorithms, and RLS on LeadingOnes has been studied in much greater detail than we can present here, with methods and results that go far beyond drift analysis [DD16,Lad05]. Naive potential.…”