Computation of structure, dynamics, and thermodynamics of proteins

Abstract

Until the 1970s, chemical systems and phenomena were studied using two basic approaches, wet laboratory experiments and theory, the latter being understood as trying to infer the properties of a system from the Schro¨ dinger equation. Although computers were used in the second approach, they were considered to be only an extension of the classical ‘pencil and sheet of paper’ method of solving equations that were too complex to handle by hand computations. The solutions were always unique and, consequently, in a sense, the theorists implicitly admitted that they were being determinists. Moreover, a ‘puristic’ approach dominated, in which quantum mechanics was the only acceptable level of theory; this attitude reached its nadir in the 1980s, when theoretical papers were frequently rejected by some journals on the grounds that semiempirical and not ab initio quantummechanical methods were implemented. Use of force fields was relegated to secondary applications and was not considered to be a pure theoretical approach. A major disadvantage of the puristic approach was that only small systems could be treated; the results could therefore be compared only with experimental results from molecular physics rather than with those of classical chemical wet lab experiments. Formally, the apparatus of statistical mechanics enables us to compute ensemble quantities from the energy surfaces of interacting molecules; however, exact computations are far too expensive except for very simple problems, such as the reactions of small molecules in the gas phase. Linking the theoretical results to biochemical or biological experiments was an extrapolation at best. Except for the development and use of very simplified (albeit elegant and suggestive) models – for example, Flory’s theory of polymer chains,1 the Laskowski/ Scheraga2 and Tanford/Kirkwood3 treatments of electrostatic effects, and helix-coil transition theory4 – studying the energetics and dynamics of biological macromolecules, particularly proteins, was largely out of reach for a theoretical treatment that, in turn, is necessary for detailed studies of the origin of the formation of the structure of these systems and the mechanisms of their biological action. Use of force fields5–20 reduced the cost of calculations by several orders of magnitude even compared to semiempirical molecular quantum mechanics, and it enabled scientists to treat biological molecules and molecular systems. Initially, the empirical force fields were used in a deterministic way by seeking all the low-lying minima in the potential energy surface of a system and, furthermore, by searching for the global minimum of the potential energy that was for a long time believed to correspond to the native structures of biomacromolecules. However, because of the tremendous reduction of calculation cost, an additional category, termed molecular simulations, entered into play. As opposed to the purely theoretical approach, the results are not an exact solution of the underlying equations but, rather, relate to the ‘average’ behavior of a system. Because of inherent errors, both random and systematic, preparing and executing simulations, as well as interpreting their results, are similar to conducting and treating the results of wet lab experiments. Also, because averages are obtained, simulation results are much closer to the quantities accessible by experiments than those obtained from the purely theoretical approach. Finally, they enable researchers to include the entropic factor in the calculated quantities. Monte Carlo and molecular dynamics methods are used to execute molecular simulations.21 The latter enables researchers to compute the time-dependent behavior of a system (although Monte Carlo dynamics is also an option). As a result of the vigorous development of high-performance computer systems, especially parallel computers, all-atom simulations are now able to treat large proteins and protein complexes, including the membrane environment, nucleic acids, and polysaccharides.22–26 However, the accessible timescale is still insufficient to cover phenomena such as protein folding, despite the continuing improvement of the parallel performance of the software and hardware. This is because the time step in integrating the equations of motion should be approximately 10 times shorter than the period of the fastest motion in a molecule, which usually involves vibrations of bonds to hydrogen atoms and amounts to 1 fs. The timescale can be extended by simplifying the representation of a biomolecule in the coarse-graining approach, in which the fast motions are averaged and the cost of energy and force evaluations is reduced by orders of magnitude because the representation is simplified. Therefore, effectively millisecond timescales can now be reached with coarse-grained models.27,28 This chapter describes the major methodology behind the simulations of proteins and outlines the most important results. Section 1.21.2 discusses force fields, both all-atom (Section 1.21.2.1) and coarse-grained (Section 1.21.2.2); it also describes the techniques implemented to bridge coarsegrained to all-atom simulations. Section 1.21.3 describes the simulation techniques, including Monte Carlo, molecular dynamics, and their extensions. Section 1.21.4 describes the use of experimental information as restraints in simulations, or as conformational filters, to determine and refine protein structures or to validate existing conformations of flexible peptides and proteins. Section 1.21.5 summarizes current developments and presents future directions