Dynamics with Conventional Periodic Boundary Conditions

MOLECULAR DYNAMICS WITH CONVENTIONAL PERIODIC BOUNDARY CONDITIONS 

Dynamics simulations employing conventional periodic boundary conditions are usually carried out in the microcanonical (constant N, constant volume, constant energy) ensemble. However, techniques have been introduced that allow NVT (constant N, constant volume, constant temperature), NPH 

In this equation, /3 is an arbitrary frictional drag parameter (inverse time constant), chosen as the coupling parameter which determines the time scale of temperature fluctuations, T0 is the mean temperature, and T(t) is the temperature at time t (see Eq. 8). Constant-pressure conditions are enforced with a proportional scaling of all coordinates and the box length by a factor related to the isothermal compressibility for the system.94 In principle, simulations carried out in all of these ensembles should yield the same results for equilibrium properties in systems of sufficient size of course, when differing ensembles are employed, appropriate corrections (e.g., a PV correction to compare the NVT and NPT ensembles) must be introduced. So far, little work has been done to determine quantitatively the system size needed to reach the thermodynamic limit. 

Molecular dynamics with periodic boundary conditions is presently the most widely used approach for studying the equilibrium and dynamic properties of pure bulk solvent,97 as well as solvated systems. However, periodic boundary conditions have their limitations. They introduce errors in the time development of equilibrium properties for times greater than that required for a sound wave to traverse the central cell. This is because the periodicity of information flow across the boundaries interferes with the time development of other processes. The velocity of sound through water at a density of 1 g/cm3 and 300 K is 15 A/ps for a cubic cell with a dimension of 45 A, the cycle time is only 3 ps and the time development of all properties beyond this time may be affected. Also, conventional periodic boundary methods are of less use for studies of chemical reactions involving enzyme and substrate molecules because there is no means for such a system to relax back to thermal equilibrium. This is not the case when alternative ensembles of the constant-temperature variety are employed. However, in these models it is not clear that the somewhat arbitrary coupling to a constant temperature heat bath does not influence the rate of reequilibration from a thermally perturbed 

MOLECULAR DYNAMICS WITH STOCHASTIC BOUNDARY CONDITIONS 

---

B. MOLECULAR DYNAMICS WITH CONVENTIONAL PERIODIC BOUNDARY CONDITIONS... 

The basic partitioning is illustrated schematically in Fig. 8a and realistically in Fig. 8b for a simulation study focusing on the dynamics of a tryptophan ring in the protein lysozyme.108 With the division indicated in the figure the total number of atoms to be simulated is 696 (294 protein atoms and 134 water molecules). This is a great reduction from the estimated 11,766 atoms (1266 protein atoms and 3500 water molecules) that would be necessary if conventional periodic boundary conditions were employed the estimate is based on using a 50-A cubic cell, a 26-A sphere to represent lysozyme, and 1 g/cm3 density for water. 

To illustrate the solvent effect on the average structure of a protein, we describe results obtained from conventional molecular dynamics simulations with periodic boundary conditions.92,193 This method is well suited for a study of the global features of the structure for which other approaches, such as stochastic boundary simulation methods, would not be appropriate. We consider the bovine pancreatic trypsin inhibitor (BPTI) in solution and in a crystalline environment. A simulation was carried out for a period of 25 ps in the presence of a bath of about 2500 van der Waals particles with a radius and well depth corresponding to that of the oxygen atom in ST2 water.193 The crystal simulation made use of a static crystal environment arising from the surrounding protein molecules in the absence of solvent. These studies, which were the first application of simulation methods to determine the effect of the environment on a protein, used simplified representations of the surround-... 

For an understanding of protein-solvent interactions it is necessary to explore the modifications of the dynamics and structure of the surrounding water induced by the presence of the biopolymer. The theoretical methods best suited for this purpose are conventional molecular dynamics with periodic boundary conditions and stochastic boundary molecular dynamics techniques, both of which treat the solvent explicitly (Chapt. IV.B and C). We focus on the results of simulations concerned with the dynamics and structure of water in the vicinity of a protein both on a global level (i.e., averages over all solvation sites) and on a local level (i.e., the solvent dynamics and structure in the neighborhood of specific protein atoms). The methods of analysis are analogous to those commonly employed in the determination of the structure and dynamics of water around small solute molecules.163 In particular, we make use of the conditional protein solute -water radial distribution function,... 

Molecular dynamics simulations can be done on molecules in the gas phase (in vacuo), in the liquid phase as a pure liquid or dilute solution, and in the solid phase. In the simulation of molecules in the liquid and solid phase, periodic boundary conditions are used to reduce the surface effects because of the limited number of molecules that can reasonably be studied. The main principle is that as an atom or molecule leaves the main box, its image from one of the adjacent boxes enters. A natural consequence of periodic boundary conditions is the concept of minimum image convention. That is, a molecule will interact with all the N-1 molecules whose centers lie within a region of the same size and shape as that of the original box (see Figure 4). ... 

---