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

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

The first derivatives with respect to feature are:

(5.40) |

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

(5.42) |

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).

(5.44) | |||

(5.45) | |||

(5.46) | |||

(5.47) | |||

(5.48) | |||

(5.49) |

Now, Eqs. 5.41-5.43 are used with instead
of . For single precision, , `rgauss1` = 4, `rgauss2` = 100.

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

(5.52) |

The first derivatives with respect to features and are:

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

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

This is usually used for dihedral angles :

(5.58) |

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:

(5.61) | |||

(5.62) |

Usually used for non-bonded distances:

The first derivatives are:

(5.64) | |||

(5.65) |

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

where .

The first derivatives are:

(5.71) |

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.

The asymmetry penalty added to the objective function is defined as

For each , the first derivatives are:

(5.73) | |||

(5.74) |

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