330 8.2  Molecular Simulation Methods

KEY POINT 8.4

A set of two paired electrons with opposite spins in an outer atomic orbital are

considered a lone pair if they are not involved in a chemical bond. In the water mol­

ecule, the oxygen atom in principle has two such lone pairs, which act as electronega­

tive sources facilitating the formation of hydrogen bonds and accounting for the fact

that bond angle between the two hydrogen atoms and oxygen in a water molecule is

greater than 90° (and in fact is closer to 104.5°).

To limit excessive computational demands on using an explicit solvent, periodic boundary

conditions (PBCs) are imposed. This means that the simulation occurs within a finite, geo­

metrically well-​defined volume, and if the predicted trajectory of a given water molecule

takes it beyond the boundary of the volume, it is forced to reemerge somewhere on the

other side of the boundary, for example, through the point on the other side of the volume

boundary that intersects with a line drawn between the position at which the water molecule

moved originally beyond the boundary and the geometrical centroid of the finite volume, for

example, if the simulation was inside a 3D cube, then the 2D projection of this might show a

square face, with the water molecule traveling to one edge of the square but then reemerging

on the opposite edge. PBCs permit the modeling of large systems, though they impose spatial

periodicity where there is none in the natural system.

The minimum size of the confining volume of water surrounding the biomolecule needs

must be such that it encapsulates the solvation shell (also known as the solvation sphere,

also as the hydration layer or hydration shell in the specific case of a water solvent). This is

the layer of water molecules that forms around the surface of biomolecules, primarily due to

hydrogen bonding between water molecules. However, multiple layers of water molecules

can then form through additional hydrogen bond interactions with the primary layer water

molecules. The effects of a solvent shell can extend to ~1 nm away from the biomolecule

surface, such that the mobility of the water molecules in this zone is distinctly lower than

that exhibited in the bulk of the solution, though in some cases, this zone can extend beyond

2 nm. The time scale of mixing between this zone and the bulk solution is in the range 10⁻¹⁵

to 10⁻¹² s and so simulations may need to extend to at least these time scales to allow adequate

mixing than if performed in a vacuum. The primary hydration shell method used in classical

MD with explicit solvent assumes two to three layers of water molecules and is reasonably

accurate.

An implicit solvent uses a continuum model to account for the presence of water. This

is far less costly computationally than using an explicit solvent but cannot account for any

explicit interactions between the solvent and solute (i.e., between the biomolecule and any

specific water molecule). In its very simplest form, the biomolecule is assumed only to interact

only with itself, but the electrostatic interactions are modified to account for the solvent by

assuming the value of the relative dielectric permittivity term εr in the Coulomb potential. For

example, in a vacuum, εr =​ 1, whereas in water εr ≈ 80.

If the straight-​line joining atoms for a pairwise interaction are through the structure of

the biomolecule itself, with no accessible water present, then the relative dielectric permit­

tivity for the biomolecule itself should be used, for example, for proteins and phospholipid

bilayers, εr can be in the range ~2–​4, and nucleic acids ~8; however, there can be consider­

able variation deepening on specific composition (e.g., some proteins have εr ≈ 20). This very

simple implicit solvation model using a pure water solvent is justified in cases where the PMF

results in a good approximation to the average behavior of many dynamic water molecules.

However, this approximation can be poor in regions close to the biomolecule such as the

solvent shell, or in discrete hydration pockets of biomolecules, which almost all molecules in

practice have, such as in the interiors of proteins and phospholipid membranes.

The simplest formulation for an implicit solvent that contains dissolved ions is the

generalized Born (or simply GB) approximation. GB is semiheuristic (which is a polite way of

saying that it only has a physically explicable basis in certain limiting regimes), but which still