- Single Gaussian restraint
- Multiple Gaussian restraint
- Multiple binormal restraint
- Lower bound
- Upper bound
- Cosine restraint
- Coulomb restraint
- Lennard-Jones restraint
- Spline restraint
- Symmetry restraint

Restraints and their derivatives

The chain rule is used to find the partial derivatives of the feature pdf with respect to the atomic coordinates. Thus, only the derivatives of the pdf with respect to the features are listed here.

The pdf for a geometric feature
(*e.g.*, distance, angle,
dihedral angle) is

A corresponding restraint in the sum that defines the objective function is

(Note that since the second term is constant for a given restraint, it is ignored.)

The first derivatives with respect to feature are:

(A.63) |

The relative heavy violation with respect to is given as:

The polymodal pdf for a geometric feature
(*e.g.*, distance, angle,
dihedral angle) is

A corresponding restraint in the sum that defines the objective function is

(A.66) |

The first derivatives with respect to feature
are:

When any of the normalized deviations
is
large, there are numerical instabilities in calculating the derivatives
because
are arguments to the *exp* function. Robustness is
ensured as follows.
The `effective' normalized deviation is used in all the equations
above when the magnitude of normalized violation
is larger than
cutoff `rgauss1` (10 for double precision). This scheme works up
to `rgauss2` (200 for double precision); violations larger than
that are ignored. This trick is equivalent
to increasing the standard deviation
. A slight disadvantage
is that there is a discontinuity in the first derivatives at `rgauss1`.
However, if continuity were imposed,
the range would not be extended (this is equivalent to linearizing the
Gaussian, but since it is already linear for large deviations, a
linearization with derivatives smoothness would not introduce much
change at all).

(A.68) | |||

(A.69) | |||

(A.70) | |||

(A.71) | |||

(A.72) | |||

(A.73) |

Now, Eqs. A.65-A.67 are used with
instead
of
. For single precision,
, `rgauss1` = 4, `rgauss2` = 100.

The relative heavy violation with respect to is given as:

(A.74) |

The polymodal pdf for a geometric feature
(*e.g.*, a pair of
dihedral angles) is

where . is the correlation coefficient between and . MODELLER actually uses the following series expansion to calculate :

A corresponding restraint in the sum that defines the objective function is

(A.77) |

The first derivatives with respect to features
and
are:

The relative heavy violation with respect to is given as:

(A.80) |

This is like the left half of a single Gaussian restraint:

where is a lower bound and is given in Eq. A.61. A similar equation relying on the first derivatives of a Gaussian holds for the first derivatives of a lower bound.

This is like the right half of a single Gaussian restraint:

where is an upper bound and is given in Eq. A.61. A similar equation relying on the first derivatives of a Gaussian holds for the first derivatives of an upper bound.

This is usually used for dihedral angles :

where is CHARMM force constant, is phase shift (tested for 0 and 180 ), and is periodicity (tested for 1, 2, 3, 4, 5, and 6). The CHARMM phase value from the CHARMM parameter library corresponds to . The force constant can be negative, in effect offsetting the phase for 180 compared to the same but positive force constant.

(A.84) |

where and are the atomic charges of atoms and , obtained from the CHARMM topology file, that are at a distance . is the relative dielectric, controlled by the

The first derivatives are:

(A.87) | |||

(A.88) |

The violations of this restraint are always reported as zero.

Usually used for non-bonded distances:

The parameters and of the switching function can be different from those in Eq. A.86. The parameters and are obtained from the CHARMM parameter file (NONBOND section) where they are given as and such that in kcal/mole for in angstroms and ; the minimum of is at , and its zero is at . The total Lennard-Jones energy should be evaluated over all pairs of atoms that are not in the same bonds or bond angles. The parameters and for 1-4 pairs in dihedral angles can be different from those for the other pairs; they are obtained from the second set of and in the CHARMM parameter file, if it exists. 1-4 energy corresponds to the

The first derivatives are:

(A.90) | |||

(A.91) |

The violations of this restraint are always reported as zero.

Any restraint form can be represented by a cubic spline [Press *et al.*, 1992]:

where .

The first derivatives are:

(A.97) |

The values of
and
beyond
and
are obtained by linear
interpolation from the termini. A violation of the restraint is calculated
by finding the global minimum. A relative violation is estimated by using
a standard deviation (*e.g.*, force constant) obtained by fitting
a parabola to the global minimum.

Variable spacing of spline points could be used to save on memory. However, this would increase the execution time, so it is not used.

To calculate the relative heavy violation, the feature value that results in the smallest value of the restraint is obtained by interpolation, and a Gaussian function is fitted locally around this value to obtain the standard deviation . These are then used in Eq. A.64.

The asymmetry penalty added to the objective function is defined as

where the sum runs over all pairs of equivalent atoms , is an atom weight for atom , is an intra-molecular distance between atoms in the first segment, and is the equivalent distance in the second segment.

For each
, the first derivatives are:

(A.99) | |||

(A.100) |

Thus, the total first derivatives are obtained by summing the two expressions above for all and distances.