Biomedical Engineering Reference
In-Depth Information
of the C-O bond in a protein backbone by only 0.5 A from its equilibrium value
will increase the bond energy by 155 kcal/mol. Such a drastic change in energy
is highly unfavorable in an MD simulation. As a result, the bonds, angles, and
most impropers of a system are often found to be very close to their equilibrium
values. In comparison, the dihedral potential (and some improper potential) is much
“softer”. As shown in Fig.
2
, the protein backbone dihedral ˚ is governed by an
energy function in the range of 0-7 kcal/mol. This soft potential allows the backbone
dihedral to adopt a broad range of values, and, therefore, gives biomolecules the
flexibility to undergo large conformational changes.
3.2
Long-Range Interactions
In most MD simulations, the bonded interaction terms can be calculated rather
efficiently, since they only involve atoms connected by one to three covalent bonds.
Meanwhile, the nonbonded interactions, which occur between every pair of atoms in
a system, are much more expensive to calculate. A closer look at (
17
)and(
18
) tells
us that the vdW and electrostatic potential functions have different dependence on r.
Generally, a potential is considered to be a short-range interaction if it decays faster
than r
d
,whered is the dimensionality of the system [
5
]. Under this criterion, the
vdW potential is a short-range interaction, since it decays as r
6
at long distances,
while the electrostatic potential, which decays as r
1
, is a long-range interaction.
For the short-range vdW interaction, we can use a cutoff scheme to perform the
calculation efficiently: interactions between atoms within the cutoff distance are
calculated, while interactions between atoms separated by a distance longer than
the cutoff are simply neglected. Typical cutoff distances used in MD simulations
are in the range of 8-12 A. The associated approximation is acceptable because a
short-range interaction rapidly decays to zero as the distance increases. For instance,
the vdW interaction energy between a carbon and an oxygen atom (C-O) is only
0.0002 kcal/mol when they are 12 A apart. Such a small value allows us to truncate
the vdW potential at the cutoff distance.
In practice, the simple truncation scheme described above is replaced by a
slightly more complicated algorithm, which is needed to avoid a sudden change
in the vdW forces caused by the discontinuity in the derivative of the vdW potential
at the cutoff distance. In many MD softwares, it is also possible to add a “correction
term” to the final result, in order to approximate the neglected vdW potential energy
beyond the cutoff distance. We'll leave interested readers to explore these more
advanced topics by themselves.
Usually, we cannot calculate the long-range electrostatic potential using the same
cutoff scheme described above. This can be seen from the example mentioned
earlier—when the C-O atoms are 12 A apart, their Coulombic interaction energy
is still
7.2 kcal/mol. This value is four orders of magnitude greater than the
vdW energy at the same distance. As a result, we may introduce substantial errors
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