Periodic Boundary Conditions

Overview

Teaching: 20 min
Exercises: 0 min

Questions

• How to simulate a bulk (gas, liquid or solid) system by using only a small part?

Objectives

• Understand why and when periodic boundary conditions are used.

• Understand how shape and size of a periodic box can affect simulation.

• Learn how to set periodic box parameters in GROMACS and NAMD.

What is PBC and why is it important?

• In most cases we want to simulate a system in realistic environment, such as solution.
• Try simulating a droplet of water, it will simply evaporate.
• We need a boundary to contain water and control temperature, pressure, and density.
• Periodic boundary conditions allow to approximate an infinite system by using a small part (unit cell).

• Unit cell is surrounded by an infinite number of translated copies in all directions (images).
• When a particle in unit cell moves across the boundary it reappears on the opposite side.
• Each molecule always interacts with its neighbors even though they may be on opposite sides of the simulation box.
• Artifacts caused by the interaction of the isolated system with a vacuum are replaced with the PBC artifacts which are in general much less severe.

Choosing periodic box size and shape.

Box shape

Cubic periodic box

• A cubic box is the most intuitive and common choice
• Cubic box is inefficient due to irrelevant water molecules in the corners.

• Ideal simulation system is a sphere of water surrounding the macromolecule, but spheres can’t be packed to fill space.

Octahedral and dodecahedral periodic boxes

• The dodecahedron (12 faces) or the truncated octahedron (14 faces) are closer to sphere.

	Space filling with truncated octahedrons

• These shapes work reasonably well for globular macromolecules.

Triclinic periodic boxes

• Any repeating shape that fills all of space has an equivalent triclinic unit cell.
• A periodic box of any shape can be represented by a triclinic box with specific box vectors and angles.

• The optimal triclinic cell has about 71% the volume of the optimal rectangular cell.

Box size

• The minimum box size should extend at least 10 nm from the solute.

• The shortest periodic box vector should be at least twice bigger than the cuf-off radius.

• In simulations with macromolecules solvent molecules should not “feel” both sides of a solute.

Pitfalls

• A simulation system with elongated solute in cubic or dodecahedral box will have a large amount of water located far away from the solute.
• Consider using a narrow rectangular box.
• Rotation of elongated macromolecules and/or conformational changes must be taken in consideration.

• Constrain the rotational motion.
• The box shape itself may influence conformational dynamics by restricting motions in certain directions [1], [2].

Periodic box

Which of the statements is incorrect?

1. The smallest dimension of a periodic box must be at least as large as the cut-off radius
2. The minimum box size should extend at least 10 nm from the solute
3. The smallest dimension of a periodic box must be at least as large as the double cut-off radius
4. There is an equivalent triclinic unit cell for any repeating shape that occupies all of space

Solution

The smallest dimension of a periodic box must be at least as large as the cut-off radius

Specifying Periodic Box

GROMACS

The box specification is integrated into structure files. The box parameters can be set using the editconf program or manually. The editconf program accepts the following options:

-bt	Box type	triclinic, cubic, dodecahedron, octahedron
-box	Box vectors lengths, a, b, c	nm
-angles	Box vectors angles, bc, ac, ab	degrees
-d	Distance between the solute and the box	nm

Example:

module load StdEnv/2020 gcc gromacs
wget http://files.rcsb.org/view/1lyz.pdb
gmx pdb2gmx -f 1lyz.pdb -ff amber99sb-ildn -water spce -ignh
gmx editconf -f conf.gro -o conf_boxed.gro -d 1.0 -bt cubic

In the example above the editconf program will append box vectors to the structure file ‘conf.gro’ and save it in the file ‘conf_boxed.gro’. The 9 components of the three box vectors are saved in the last line of the structure file in the order: xx yy zz xy xz yx yz zx zy. Three of the values (xy, xz, and yz) are always zeros because they are duplicates of (yx, zx, and zy). The values of the box vectors components are related to the unit cell vectors \(a,b,c,\alpha,\beta,\gamma\) from the CRYST1 record of a PDB file with the equations:

\[xx=a, yy=b\cdot\sin(\gamma), zz=\frac{v}{(a*b*\sin(\gamma))}\] \[xy=0, xz=0, yx=b\cdot\cos(\gamma)\] \[yz=0, zx=c\cdot\cos(\beta), zy=\frac{c}{\sin(\gamma)}\cdot(cos(\alpha)-cos(\beta)\cdot\cos(\gamma))\] \[v=\sqrt{1-\cos^2(\alpha)-cos^2(\beta)-\cos^2(\gamma) +2.0\cdot\cos(\alpha)\cdot\cos(\beta)\cdot\cos(\gamma)}\cdot{a}\cdot{b}\cdot{c}\]

NAMD

Periodic box is specified in the run parameter file by three unit cell vectors, the units are Å.

# cubic box
cellBasisVector1 100 0 0
cellBasisVector2 0 100 0
cellBasisVector3 0 0 100

Alternatively periodic box parameters can be read from the .xsc (eXtended System Configuration) file by using the extendedSystem keyword. If this keyword is used cellBasisVectors are ignored. NAMD always generates .xsc files at runtime.

extendedSystem restart.xsc

Comparing periodic boxes

Using the structure file ‘conf.gro’ from the example above generate triclinic, cubic, dodecahedral and truncated octahedral boxes with the 15 Å distance between the solute and the box edge.

Which of the boxes will be the fastest to simulate?

References:

1. Molecular dynamics simulations with constrained roto-translational motions: Theoretical basis and statistical mechanical consistency
2. The effect of box shape on the dynamic properties of proteins simulated under periodic boundary conditions
3. Periodic box types in Gromacs manual

Key Points

• Periodic boundary conditions are used to approximate an infinitely large system.

• Periodic box should not restrict molecular motions in any way.

• The macromolecule shape, rotation and conformational changes should be taken into account in choosing the periodic box parameters.