Section 20.1.1. Introduction

Molecules in crystals are often believed to have a very rigid structure due to their ordered packing, and the investigation of the molecular motion of such systems is often considered to be of little interest. In contrast to small-molecule crystals, however, the solvent concentration in protein crystals is high, usually with about half of the crystal consisting of water. Thus, in this respect, one can compare protein crystals with very concentrated solutions and expect non-negligible atomic motion. The atomic mobility in proteins can be investigated by experiment (X-ray diffraction, NMR) or by molecular simulation.

Today's experimental techniques are very advanced. They are, however, only able to examine time- and ensemble-averaged structures and properties. In contrast, with simulations one can go beyond averaged properties and examine the motions of a single molecule in the pico- and nanosecond time regime. Such simulations have become possible with the availability of high-resolution structural data, which provide adequate starting structures for biologically relevant systems. Depending on the kind of property in which one is interested, different methods of simulation may be used. Equilibrium properties can be obtained using either Monte Carlo (MC) or molecular-dynamics (MD) simulation techniques, but motions can only be observed with the latter. Current interest in the simulation community mainly focuses on dissolved proteins as they would be in their natural environment. Force fields are parameterized to mimic the behaviour and function of proteins in a solution, and few crystal simulations have been performed. Consequently, a crystal environment provides an excellent opportunity to test a force field on a task for which it should be appropriate, but for which it has not been directly parameterized.

Apart from the analysis of the dynamic properties of a system, MD simulations are also used in structure refinement. In refinement, be it X-ray crystallographic or NMR, a special term is added to the standard physical force field to reflect the presence of experimental data: In NMR, a variety of properties can be measured, and each of these can be used in the definition of an additional term that restrains the generated structures to reproduce given experimental values. Refinement procedures exist that use nuclear-Overhauser-effect (van Gunsteren et al., 1984; Kaptein et al., 1985), J-value (Torda et al., 1993) and chemical-shift (Harvey & van Gunsteren, 1993) restraints. In crystallography, X-ray intensities are used to generate the restraining energy contribution (Brünger et al., 1987; Fujinaga et al., 1989). Combined NMR/X-ray refinement uses both solution and crystal data (Schiffer et al., 1994).

As in an experiment, averages over time and molecules are measured, and instantaneous restraints can lead to artificial rigidity in the molecular system (Torda et al., 1990). This can be circumvented by restraining time or ensemble averages, instead of instantaneous values, to the value of the measured quantity. Time averaging has been applied to nuclear Overhauser effects (Torda et al., 1990) and J values (Torda et al., 1993) in NMR structure determination and to X-ray intensities in crystallography (Gros et al., 1990; Gros & van Gunsteren, 1993; Schiffer et al., 1995). Ensemble averaging has been applied in NMR refinement (Scheek et al., 1991; Fennen et al., 1995). For a more detailed discussion of restrained MD simulations, we refer to the literature (van Gunsteren et al., 1994, 1997).

The first unrestrained MD simulations of a protein in a crystal were carried out in the early 1980s (van Gunsteren & Karplus, 1981, 1982). The protein concerned was bovine pancreatic trypsin inhibitor (BPTI), a small (58-residue) protein for which high-resolution X-ray diffraction data were available. The initial level of simulation was to neglect solvent, using vacuum boundary conditions. This was improved gradually by the inclusion of Lennard–Jones particles at the density of water as a solvent (van Gunsteren & Karplus, 1982) to let the protein feel random forces and friction from the outside as well as feel a slightly attractive external field. The next step was to use a simple (simple point charge, SPC) water model (van Gunsteren et al., 1983). Further improvement was achieved by incorporating counter ions into the modelled systems to obtain overall charge neutrality (Berendsen et al., 1986).

Despite these early attempts, few unrestrained crystal simulations have been reported in the literature, and, to our knowledge, these involve one to four protein molecules, simulating one unit cell (Shi et al., 1988; Heiner et al., 1992). The maximum time range covered has been less than 100 ps.

In the work described in this chapter, the current state of MD simulation of protein crystals is illustrated. A full unit cell of ubiquitin, containing four ubiquitin and 692 water molecules, has been simulated for a period of two nanoseconds. Since this simulation is an order of magnitude longer than crystal simulations in the literature, it offers the possibility of analysing the convergence of different properties as a function of time and as a function of the number of protein molecules. Converged properties can also be compared with experimental values as a test of the GROMOS96 force field (van Gunsteren et al., 1996). Finally, the motions obtained can be analysed to obtain a picture of the molecular behaviour of ubiquitin in a crystalline environment.