Empirical formulation and parameterization of cation-π interactions for protein modeling

Cation-π interaction is comparable and as important as other main molecular interaction types, such as hydrogen bond, electrostatic interaction, van der Waals interaction, and hydrophobic interaction. Cation-π interactions frequently occur in protein structures, because six (Phe, Tyr, Trp, Arg, Lys, and His) of 20 natural amino acids and all metallic cations could be involved in cation-π interaction. Cation-π interactions arise from complex physicochemical nature and possess unique interaction behaviors, which cannot be modeled and evaluated by existing empirical equations and force field parameters that are widely used in the molecular dynamics. In this study, the authors present an empirical approach for cation-π interaction energy calculations in protein interactions. The accurate cation-π interaction energies of aromatic amino acids (Phe, Tyr, and Try) with protonated amino acids (Arg and Lys) and metallic cations (Li(+), Na(+), K(+), and Ca(2+)) are calculated using B3LYP/6-311+G(d,p) method as the benchmark for the empirical formulization and parameterization. Then, the empirical equations are built and the parameters are optimized based on the benchmark calculations. The cation-π interactions are distance and orientation dependent. Correspondingly, the empirical equations of cation-π interactions are functions of two variables, the distance r and the orientation angle θ. Two types of empirical equations of cation-π interactions are proposed. One is a modified distance and orientation dependent Lennard-Jones equation. The second is a polynomial function of two variables r and θ. The amino acid-based empirical equations and parameters provide simple and useful tools for evaluations of cation-π interaction energies in protein interactions.     

A fast and accurate algorithm for solving linear systems associated with a class of negative matrix

A class of negative matrices including Vandermonde-like matrices tends to be extremely ill-conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with $O(n^2)$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction-free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.