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Nonbonded interactions (frequently abbreviated NBONDS) refer to all those terms in the hamiltonian which do not refer specifically to a bond, bond angle, dihedral, or hydrogen bond. They include the attractive and repulsive van der Waals terms and the electrostatic terms between all atom pairs that are not specifically excluded from nonbond calculations. Directly bound atoms (1-2 interactions) and atoms bound to a common atom (1-3 interactions) are usually not counted in non-bonded interactions. See section NBXMOD -- Automatic Generation of Non-bonded Exclusions, for an automated command to handle non-bonded exclusions. These terms are defined on atom pairs and to a first approximation would require the number of atoms squared calculations. To avoid this burden, various truncation and approximation schemes can be employed in the program, breaking the nonbonded calculation into two parts, initialization and actual energy calculation.
The method of approximation, cutoffs, and other relevant parameters can be entered either at the time of initialization (by calling NBOND from the main program or in the COMPARE command in analysis), or in the MINI and DYNA commands.
NBOND [keyword value]
COMPARE ... NBOND [keyword value] <del> ...
MINI [keyword value]
DYNA [keyword value]
In all cases as many keywords and values as desired may be specified.
The key words, their functions, and defaults are:
There are four algorithms used in calculating the nonbonded energies, each making different approximations in an attempt to speed the calculation. Electrostatic interactions are the most difficult to deal with for two reasons. They do not fall off quickly with distance (so it is inappropriate to simply ignore all interactions beyond some cutoff), and they depend on odd powers of r necessitating expensive square root calculations for each pair evaluated. The approximations used to make the electrostatics calculation more tractable are setting the dielectric constant equal to r or using a constant dielectric but only calculating distant interactions periodically (and storing the value in between).
Setting the dielectric constant equal to the atom atom distance times a constant factor (determined by the EPS keyword value) makes the computation easier by eliminating the need to calculate square roots and by making the calculated contribution fall off more quickly. It also introduces problems. The force calculated using an r dependent dielectric will be larger than the force from a constant dielectric at short distances (5.0 angstroms or less by comparsion to a constant dielectric of 2.5). In addition, the electrostatic contribution still falls off relatively slowly and large distance cutoffs are needed. As the number of atom pairs included will be proportional to the cutoff cubed, this is a significant disadvantage.
The RESI options use an r dependent dielectric (as do all of Bruce Gelin's protein calculations). In the RESI searches, the program constructs a rectangular box around every residue, and examines every residue--residue pair that can yield atom atom pairs within the cutoff. Every atom--atom pair within those residue--residue pairs is added to the non-bonded list. When switching functions are used, RESI and ATOM will yield the same energy, but RESI will generate a longer list.
The SHFT option is similar to ATOM except, the potential:
is used when ( R < CTOFNB ) and zero otherwise. This potential and it first derivative approach zero as R becomes CTOFNB, without the messy computation of switching functions and steep forces at large R.
The EXEL and EXFL routines use a constant dielectric at large
distances joined to the short distance function. The long distance terms
are approximated using interpolation and periodic updating. At CUTNB the
constant dielectric potential is joined to a dielectric equal to r
potential with a shift and scale that preserve the continuity of the
energy and forces.
CONS uses a constant dielectric everywhere. This requires a square root
to be calculated in the inner loop of ENBOND, slowing things down a bit,
but it is physically more reasonable and widely employed by other groups
doing empirical energy modelling (ex. ST2 water).
This form allows a small CUTNB (5.0 angstroms with EPS=2.5) even though the electrostatic terms are still varying rapidly at that distance. The short range forces are identical to those calculated with the other options, reflecting the decrease in dielectric shielding at short ranges. For the long range forces there is effectively no cutoff in the electrostatic energy. Interactions between atoms beyond the CUTNB are calculated when the list is updated and stored together with their first (EXFL,CONS) or first and second (EXEL) derivatives. The energy is calculated by explicitly evaluating pairs in the list and using the stored potentials, fields, and gradients to approximate the distant pairs. In essence the routines assume that for distant pairs the atom movements will be small enough that the changes in their electrostatic interactions can be accurately calculated using local expansions.
The constant dielectric routines compile the close contact list using the same two stage minimum rectangle box search that is described above. In this way the efficiency of a residue by residue search is exploited while being certain that all necessary pairs are included. For close residue pairs an atom by atom search is then performed. Atom pairs are either included in the list of close contacts or their electrostatic interactions are calculated and stored.
An option is offered to increase the accuracy of residue residue interactions by using a multipole expansion of one residue evaluated for each atom of the other. This cutoff for this treatment is CUTMP. For residue pairs outside of CUTMP only a single multipole evaluation is made and second order polynomial expansion is used to extrapolate to each atom. Ordinarily this is sufficient and CUTMP is set to 0.0.
The DUMMY option results in no non-bonded interactions whatsoever. The creation of a dummy list can be done very quickly. This is useful when one is analyzing only the other terms in the energy.
The initialization and list compilation is performed by the
subroutine NBONDS. It functions by guessing how much space will
be needed to store the close contact list, allocating that space (and
space for electrostatic potentials and gradients if necessary) on the
heap, and calling the appropriate subroutine to actually compile the
nonbonded list. NBONDA, NBONDR, NBONDE, and
NBONDF handle ATOM, RESI, EXEL, and EXFL
respectively. If sufficient space was not available, 1.5 times as much is
allocated and another attempt is made.
ENBOND evaluates the nonbonded energy, calling
EEXEL to evaluate the stored electric potentials and fields.
Single precision is used for most arithmetic, but the energy and
derivative sums are accumulated in double precision.
All of the nonbonded cutoffs and lists are stored on the heap.
BNBND is the descriptor array passed through most of the program (in
some of the analysis routines an additional array BNBNDC is used
for the comparison data structure). BNBND holds heap addresses
and LNBND holds the lengths of the elements in the data
structure. To actually access the data it is necessary to include
`inbnd.fcm' (an index common block) and specify
HEAP(BNBND(xxx)) where xxx is the desired element name in
`inbnd.fcm'. This is arrangement has the advantage of allowing
dynamic storage allocation and easy modification of the types of
information passed from routine to routine.
The contents of the nonbonded data structure are described in the file `inbnd.fcm' in the CONGEN source directory. The following gives some of the details.
NBOPT - the nonbonded option number
0 = RESI
1 = ATOM
2 = EXEL
3 = EXFL
4 = SHFT
5 = EXNQ
6 =
7 = CONS
8 = DUMMY
CUTNB
CTOFNB
CTONNB
CUTMP
EPS
WMIN
WRNMXD
JNB - the list of the second atom in close contact list
INBLO - a list of pointers into JNB organized on the
first atoms of each pair INBLO(i) is the location of
the last entry in JNB that pairs with atom i
The following elements are only used by CONS EXEL and
EXFL:
ATSX, ATSY, ATSZ - the atom locations at the last update
ATPOT - the atom potentials
ATFX, ATFY, ATFZ - the first derivatives
The following elements are used only by EXEL
ATGXX, ATGYY, ATGZZ, ATGXY, ATGYZ, ATGZX - the second derivatives
in the potential
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