Global energy minimization of alanine dipeptide via barrier function methods

Abstract

This paper presents an interior point method to determine the minimum energy conformation of alanine dipeptide. The CHARMM energy function is minimized over the internal coordinates of the atoms involved. A barrier function algorithm to determine the minimum energy conformation of peptides is proposed. Lennard–Jones 6-12 potential which is used to model the van der Waals interactions in the CHARMM energy equation is used as the barrier function for this algorithm. The results of applying the algorithm for the alanine dipeptide structure as a function of varying number of dihedral angles are reported, and they are compared with that obtained from genetic algorithm approach. In addition, the results for polyalanine structures are also reported.

Introduction

The problem of energy minimization refers to determining the global minimum potential energy conformation of proteins and peptides. The thermodynamical hypothesis proposed by Anfinsen, states that the native structure of protein would be at its global free energy minimum (Anfinsen, 1973). Thus the importance of finding the global minimum of an energy function is associated with determining the three-dimensional structure of protein in its native state. Currently, protein structure is determined through time-consuming and expensive experimental techniques, such as X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy. Hence, the development of computational techniques to address the problem of protein structure prediction is of paramount importance.

A major challenge in the computational methods lies in solving large-scale global optimization problems arising from minimizing energy functions. No efficient mathematical basis exists for determining the global minimizer and the method often requires enormous computational time for large problems. Pardalos et al. (1994) and Das et al. (2003) explore the optimization methods that are relevant in the area of energy function minimization.

Thorough reviews of recent advances in the area of protein structure prediction are provided in Bonneau and Baker (2001), Floudas et al. (2006) and Floudas (2007). In particular, we focus on the ab initio method, also referred to as first principle method, as it is one of the important methods of protein structure prediction. The importance is underscored by the fact that these methods do not resort to any database or use any other information to determine the three-dimensional structure of proteins from its amino acid sequence. It is particularly important to have a set of low energy conformations if a number of populated states are present (Wilson and Cui, 1990). First pass optimization methods play a vital role in identifying a set of low energy conformations. These low energy conformations can be used to approximate the entropic contributions associated with the stability of the molecule. Once a sufficient ensemble of low energy minima has been identified, a statistical analysis can be used to estimate the relative entropic contributions (Klepeis and Floudas, 1999). Methods such as the one proposed in this paper help to identify both the stable three-dimensional structure (global minimum), as well as a set of low energy conformations (local minimum). The advantages of ab initio methods as proposed by McAllister and Floudas (2010) lies in its ability to (1) predict structures when a related structural homologue is not available, (2) extend the predictions to different environments, and (3) provide insight into the mechanism, thermodynamics, and kinetics of protein folding. Moreover, new structures continue to be discovered, which would not be possible by methods that rely only on comparison to known structures (Floudas et al., 2006).

Interior point methods, frequently used to solve nonlinear and nonconvex problems, is uncommon in the area of protein structure prediction via ab initio methods. However, interior point methods are used in the area of protein threading to solve the linear programming formulation (Wagner et al., 2004, Meller et al., 2002). Other than these two works, to the best of our knowledge, we are not aware of any other reported work in the application of interior point methods to the problem of protein structure prediction, especially with respect to ab initio methods.

In this paper, we present an interior point algorithm based on the barrier function method to determine the minimum energy conformation of alanine dipeptide by minimizing the CHARMM (Mackerell et al., 1998) free energy equation. An important feature of our algorithm is the choice of barrier function to be have utilized. It is normal to use a reciprocal function or a logarithmic function of the constraints as a barrier term for the problem (Solayappan et al., 2008). However, a minor reformulation of the original problem helped us to identify the barrier function that is inherently present in the objective function (free energy equation). Though it is not mandatory, identifying barrier terms in such a way works well with the nature of the problem structure and no additional constraints are enforced on the problem.

The rest of the paper is organized as follows: Section 2 presents the problem formulation and energy function used, while Section 3 proposes a solution method to solve the energy minimization problem. The computational details and results are provided in Sections 4 Computational details, 5 Computational results, respectively. Section 6 provides some concluding remarks.