| Periodic Boundary Conditions |
Article Index for Periodic |
Website Links For Periodic |
Information AboutPeriodic Boundary Conditions |
| CATEGORIES ABOUT PERIODIC BOUNDARY CONDITIONS | |
| molecular dynamics | |
| boundary conditions | |
|
Periodic boundary conditions resemble the Topologies of some video games; a ''unit cell'' or ''simulation box'' of a geometry suitable for perfect three-dimensional tiling is defined, and when a molecule passes through one face of the unit cell, it reappears on the opposite face with the same velocity. In effect, the simulation is of an infinite perfect Crystal of the unit cell, or in topological terms, the space can be thought of as being mapped onto a four-dimensional Torus . The tiled copies of the unit cell are called ''images'', of which there are infinitely many. During the simulation, only the properties of the unit cell need be recorded and propagated. The ''minimum-image convention'' is a common form of PBC particle bookkeeping in which each individual particle in the simulation interacts with the closest image of the remaining particles in the system. PBC REQUIREMENTS AND ARTIFACTS Periodic boundary conditions are particularly useful in conjunction with Ewald Summation methods (usually particle mesh Ewald) of accounting for Electrostatic forces in the system. However, PBC also introduces correlational artifacts that do not respect the translational invariance of the system,Cheatham TE, Miller JH, Fox T, Darden PA, Kollman PA. (1995). Molecular Dynamics Simulations on Solvated Biomolecular Systems: The Particle Mesh Ewald Method Leads to Stable Trajectories of DNA, RNA, and Proteins. ''J Am Chem Soc'' 117:4193. and requires constraints on the composition and size of the simulation box. The net Electrostatic Charge of the system must be zero to avoid summing to an infinite charge when PBC is applied; this is easily done by adding Ion s such as Sodium or Chloride in appropriate numbers if the molecules of interest are charged. Such ions are called ''counterions''. Sometimes ions are even added to a system in which the molecules of interest are neutral, to approximate the Ionic Strength of the solution in which the molecules naturally appear. Maintenance of the minimum-image convention also generally requires that a spherical cutoff radius for nonbonded forces be at most half the length of one side of a cubic box. Even in electrostatically neutral systems, a net Dipole Moment of the unit cell can introduce a spurious bulk-surface energy, equivalent to Pyroelectricity in Polar Crystals . The size of the simulation box must also be large enough to prevent periodic artifacts from occurring due to the unphysical topology of the simulation. In a box that is too small, a macromolecule may interact with its own image in a neighboring box, which is functionally equivalent to a molecule's "head" interacting with its own "tail". This produces highly unphysical dynamics in most macromolecules, although the magnitude of the consequences and thus the appropriate box size relative to the size of the macromolecules depends on the intended length of the simulation, the desired accuracy, and the anticipated dynamics. For example, simulations of Protein Folding that begin from the Native State may undergo smaller fluctuations and therefore may not require as large a box as simulations that begin from a Random Coil conformation. However, the effects of Solvation Shell s on the observed dynamics – in simulation or in experiment – are not well understood. A common recommendation based on simulations of DNA is to require at least 1 nm of solvent around the molecules of interest in every dimension.de Souza ON, Ornstein RL. (1997). Effect of periodic box size on aqueous molecular dynamics simulation of a DNA dodecamer with particle-mesh Ewald method. ''Biophys J'' 72(6):2395-7. PMID 9168016 UNIT CELL GEOMETRIES PBC requires the unit cell to be a shape that will tile perfectly into a three-dimensional crystal. Thus, a Spherical or Elliptical droplet cannot be used. A Cube or Rectangular Prism is the most intuitive and common choice, but can be computationally expensive due to unnecessary amounts of Solvent molecules in the corners, distant from the central macromolecules. A common alternative that requires less volume is the Truncated Octahedron . CONSERVED PROPERTIES Under periodic boundary conditions, the linear definition of Temperature , the departure of the velocity distributions from a Boltzmann Distribution , and violations of equipartition for systems containing particles with heterogeneous Mass es. The simplest of these effects is that a system of ''N'' particles will behave, in the molecular dynamics ensemble, as a system of ''N-1'' particles. These artifacts have quantifiable consequences for small toy systems containing only perfectly hard particles; they have not been studied in depth for standard biomolecular simulations, but given the size of such systems, the effects will be largely negligible. NOTES REFERENCES
|
|
|