Landscape Analysis for Protein-Folding Simulation in the H-P Model

Abstract

The hydrophobic-hydrophilic (H-P) model for protein folding was introduced by Dill et al.[7]. A problem instance consists of a sequence of amino acids, each labeled as either hydrophobic (H) or hydrophilic (P). The sequence must be placed on a 2D or 3D grid without overlapping, so that adjacent amino acids in the sequence remain adjacent in the grid. The goal is to minimize the energy, which in the simplest variation corresponds to maximizing the number of adjacent hydrophobic pairs. The protein folding problem in the H-P model is NP-hard in both 2D and 3D. Recently, Fu and Wang [10] proved an exp(O(n ^{1 − − 1/d})ln n) algorithm for d-dimensional protein folding simulation in the HP-model. Our preliminary results on stochastic search applied to protein folding utilize complete move sets proposed by Lesh et al.[15] and Blazewicz et al.[4]. We obtain that after (m/δ)^{O( Γ)} Markov chain transitions, the probability to be in a minimum energy conformation is at least 1–δ, where m is the maximum neighbourhood size and Γ is the maximum value of the minimum escape height from local minima of the underlying energy landscape. We note that the time bound depends on the specific instance. Based on [10] we conjecture Γ≤n ^{1 − − 1/d}. We analyse $\Gamma \leq \sqrt{n}$ experimentally on selected benchmark problems [15,21] for the 2D case.

Research partially supported by EPSRC Grant No. EP/D062012/1.