Subsections

- Distance
- Angle
- Dihedral angle
- Atomic solvent accessibility
- Atomic density
- Atomic coordinates

## Features and their derivatives

### Distance

Distance is defined by points and :

| (A.38)

where

| (A.39)

The first derivatives of with respect to Cartesian coordinates are:

|  |  | (A.40)
|  |  | (A.41)

### Angle

Angle is defined by points , , and , and spanned by vectors and :

| (A.42)

It lies in the interval from 0 to 180°. Internal MODELLER units are radians.

The first derivatives of α with respect to Cartesian coordinates are:

|  |  | (A.43)
|  |  | (A.44)
|  |  | (A.45)

These equations for the derivatives have a numerical instability when the angle goes to 0 or to 180°. Presently, the problem is ‘solved’ by testing for the size of the angle; if it is too small, the derivatives are set to 0 in the hope that other restraints will eventually pull the angle towards well behaved regions. Thus, angle restraints of 0 or 180° should not be used in the conjugate gradients or molecular dynamics optimizations.

### Dihedral angle

Dihedral angle is defined by points , , , and ( ):

sign | (A.46)

where

sign sign | (A.47)

The first derivatives of χ with respect to Cartesian coordinates are:

| (A.48)

where

| (A.49)

and

|  |  | (A.50)
|  |  | (A.51)
|  |  | (A.52)
|  |  | (A.53)
|  |  | (A.54)
|  |  | (A.55)

These equations for the derivatives have a numerical instability when the angle goes to 0. Thus, the following set of equations is used instead [van Schaik et al., 1993]:

|  |  | (A.56)
|  |  | (A.57)
|  |  | (A.58)
|  |  | (A.59)
|  |  | (A.60)
|  |  | (A.61)

The only possible instability in these equations is when the length of the central bond of the dihedral, , goes to 0. In such a case, which should not happen, the derivatives are set to 0. The expressions for an improper dihedral angle, as opposed to a dihedral or dihedral angle, are the same, except that indices are permuted to . In both cases, covalent bonds , , and are defining the angle.

### Atomic solvent accessibility

This is the accessibility value calculated by the PSA algorithm (see model.write_data()). This is usually set by the last call to Restraints.make() or Restraints.make_distance(). First derivatives are not calculated, and are always returned as 0.

###
Atomic density

Atomic density for a given atom is simply calculated as the number of atoms within a distance energy_data.contact_shell of that atom. First derivatives are not calculated, and are always returned as 0.

### Atomic coordinates

The absolute atomic coordinates , and are available for every point , primarily for use in anchoring points to planes, lines or points. Their first derivatives with respect to Cartesian coordinates are of course simply 0 or 1.