Molecular Modeling of Proteins and Mathematical Prediction of Protein Structure

This paper discusses the mathematical formulation of and solution attempts for the so-called protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein. From a mathematical point of view, there are several main sides to the static problem: -- the selection of an appropriate potential energy function; -- the parameter identification by fitting to experimental data; and -- the global optimization of the potential. The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation) or (in case of constrained molecular dynamics) of differential-algebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs. The present paper gives a self-contained introduction to the necessary background from physics and chemistry and surveys some of the literature. It also discusses the various mathematical problems arising, some deficiencies of the current models and algorithms, and possible (past and future) attacks to arrive at solutions to the protein-folding problem.