Constraints and
Restraints
Contents
The true power of the atomistic description of a molecule embodied in
the energy expression lies in two major areas. The first is the
analysis of the energy contributions at the level of individual or
classes of terms. For instance, you can decompose the energy into bond
energies, angle energies, nonbond energies, etc. or even to the level
of a specific hydrogen bond or van der Waals contact, in order to
understand a physical observable or to make a prediction. The second
area, which is described in the following sections, lies in the
modification of the energy expression to bias the calculation. You can
impose constraints (absolute conditions), such as fixing an
atom in space and not allowing it to move. You can also add extra
terms to the energy expression to restrain or force the system
in certain ways. For instance, by adding an extra torsion potential to
a particular bond, you can force the torsion angle toward a desired
value.
The seminal difference between a constraint and a restraint is that a
constraint is an absolute restriction imposed on the calculation,
while a restraint is an energetic bias that tends to force the
calculation toward a certain restriction. Constraints and restraints
allow you to focus the calculation on a region or conformation of
interest and also to set up computational experiments. Such
experiments are one of the primary uses of molecular modeling,
allowing you control over a
molecule--or at least the model of the molecule--at the atomic
level.
Fixed atoms are constrained to a given location in space; they cannot
move at all. Fixed atoms reduce the expense of a calculation in two
ways. First, terms in the energy expression involving only fixed atoms
can be eliminated, because they add only a constant to the total
energy. Since the positions of fixed atoms cannot change, neither can
the contribution of the terms that depend only on these
positions. Second, fixing atoms reduces the number of degrees of
freedom in the system, so minimizers converge in fewer steps and
dynamics requires fewer steps to sweep out the available
conformational space. Note that the energy calculated by the
Discover program is correct only to an arbitrary constant,
depending on the molecule as well as the fixed atoms. Thus, only
differences in energy between conformations of the same molecule
having the same fixed atoms are meaningful.
Distance restraints are used to force the distance between two atoms,
bonded or not, toward a given value. The Discover program
supports the two most commonly used functional forms. One is a simple
harmonic function:
- Eq. 2-15:
-
where K is a force constant, Rij is
the current distance between the atoms, and Rtarget
is the target distance. A large force constant tends to force the
distance to be close to the target distance; a smaller force constant
results in a correspondingly smaller bias.
The second form is also a harmonic potential, but it is separated into
five piecewise continuous regions:
- Eq. 2-16:
-
This flat-bottomed potential is illustrated in Figure 2-12. Note that
it is not necessary for the potential to be symmetric and that, by
appropriate definition of the points R1,
R2, R3, and R4,
any of the regions may be eliminated. The important regions are those
from R1 to R2 and from
R3 to R4, where a harmonic
potential is applied, and the flat bottom from R2 to
R3. This form of the restraint allows a range of
acceptable distances and is particularly useful for incorporating
experimental distance information into a calculation. The flat bottom
allows for experimental error in the determined distance. The two
outer regions have a constant gradient, which is useful for avoiding
ridiculously large forces if the initial structure is far from the
target value.
Like distance restraints, several forms of torsion restraints are used
in the literature. These range from a simple harmonic form analogous
to Eq. 2-15 to the piecewise continuous form of
Eq. 2-16 with R interpreted as the angle,
rather than the distance. Another natural form is the periodic
function of Eq. 2-17:
- Eq. 2-17:
-
where V gives the strength of the restraint, n is an
integer periodicity, and 
0 is
the phase angle.
The harmonic restraints, or that of Eq. 2-17
with n = 1, are appropriate for forcing a torsion angle to a
particular value. The periodic form with a periodicity greater than
one is useful for restraining a torsion to one of several related
angles. For instance, a threefold potential could keep a torsion
either trans or at one of the two gauche conformations,
depending on the starting conformation and the strength of the
potential applied.
To force the conformation of one molecule to be similar to that of a
template molecule, one of the following restraint terms is added to
the energy expression:
- Eq. 2-18:
-
or:
- Eq. 2-19:
-
The term in Eq. 2-18 is proportional to the
root-mean-square (rms) deviation of the analog atoms from the template
atoms. Eq. 2-19 represents a conceptually more
straightforward restraint, with each analog atom restrained by an
isotropic spring to the position of its template atom. In either form,
the summation is over a list of pairs of atoms to restrain: one from
the moving analog, and one from the template molecule. The first form
gives the best rms fit for the least energetic cost, but individual
atoms may remain quite far from their template position. The second
form restrains each atom individually, so each atom is forced toward
its template partner. The resulting rms fit is not as good as that
from Eq. 2-18, but no one atom is allowed to deviate as much as is
possible with Eq. 2-18. The form in Eq. 2-19 also allows for a
different force constant for each pair, which means that different
atoms or classes of atoms can be treated differently.
Tethering is a special case of template forcing. The atoms are
restrained to their original positions rather than to the positions in
a template structure. Both Eq. 2-18 and Eq. 2-19 are applicable for tethering, although
Eq. 2-19 is normally preferred, because tethering is usually used to
keep atoms from moving too far from their original positions.
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