Features and their derivatives

Distance

Distance is defined by points $i$ and $j$ :

$$d = \sqrt{\vec{r}_{ij} \cdot \vec{r}_{ij}} = \vert\vec{r}_{ij}\vert = r_{ij}$$ (A.38)

where

$$\vec{r}_{ij} = \vec{r}_i - \vec{r}_j \; .$$ (A.39)

The first derivatives of $d$ with respect to Cartesian coordinates are:

$$\frac{\partial d} {\partial \vec{r}_i} = \frac{\vec{r}_{ij}}{\vert\vec{r}_{ij}\vert}$$ (A.40)

$$\frac{\partial d} {\partial \vec{r}_j} = -\frac{\partial d} {\partial \vec{r}_i}$$ (A.41)

Angle

Angle is defined by points $i$ , $j$ , and $k$ , and spanned by vectors $ij$ and $kj$ :

$$\alpha = \arccos \frac{\vec{r}_{ij} \cdot \vec{r}_{kj}} {\vert\vec{r}_{ij}\vert \vert\vec{r}_{kj}\vert} \; .$$ (A.42)

It lies in the interval from 0 to 180${}^{o}$ . Internal MODELLER units are radians.

The first derivatives of $\alpha$ with respect to Cartesian coordinates are:

$$\frac{\partial \alpha}{\partial \vec{r}_i} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial ... ... \frac{\vec{r}_{ij}}{r_{ij}} \cos \alpha - \frac{\vec{r}_{kj}}{r_{kj}} \right)$$ (A.43)

$$\frac{\partial \alpha}{\partial \vec{r}_k} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial ... ... \frac{\vec{r}_{kj}}{r_{kj}} \cos \alpha - \frac{\vec{r}_{ij}}{r_{ij}} \right)$$ (A.44)

$$\frac{\partial \alpha} {\partial \vec{r}_j} = -\frac{\partial \alpha} {\partial \vec{r}_i} -\frac{\partial \alpha} {\partial \vec{r}_k}$$ (A.45)

These equations for the derivatives have a numerical instability when the angle goes to 0 or to 180${}^{o}$ . 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${}^{o}$ should not be used in the conjugate gradients or molecular dynamics optimizations.

Dihedral angle

Dihedral angle is defined by points $i$ , $j$ , $k$ , and $l$ ($ijkl$ ):

$$\chi = (\chi) \arccos\frac{ (\vec{r}_{ij} \times \vec{r}_{kj}) \cdot (\v... ...ec{r}_{ij} \times \vec{r}_{kj}\vert \vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}$$ (A.46)

where

$$(\chi) = [\vec{r}_{kj} \cdot (\vec{r}_{ij} \times \vec{r}_{kj}) \times (\vec{r}_{kj} \times \vec{r}_{kl})] \; .$$ (A.47)

The first derivatives of $\chi$ with respect to Cartesian coordinates are:

$$\frac{ \; {d}\chi} { \; {d}\vec{r}} = \frac{ \; {d}\chi}{ \; {d}\cos \chi} \frac{ \; {d}\cos \chi}{ \; {d}\vec{r}}$$ (A.48)

where

$$\frac{ \; {d}\chi}{ \; {d}\cos \chi} = \left(\frac{ \; {d}\cos \chi}{ \; {d}\chi}\right)^{-1} = -\frac{1}{\sin \chi}$$ (A.49)

and

$$\frac{\partial \cos \chi}{\partial \vec{r}_i} = \vec{r}_{kj} \times \vec{a}$$ (A.50)

$$\frac{\partial \cos \chi}{\partial \vec{r}_j} = \vec{r}_{ik} \times \vec{a} - \vec{r}_{kl} \times \vec{b}$$ (A.51)

$$\frac{\partial \cos \chi}{\partial \vec{r}_k} = \vec{r}_{jl} \times \vec{b} - \vec{r}_{ij} \times \vec{a}$$ (A.52)

$$\frac{\partial \cos \chi}{\partial \vec{r}_l} = \vec{r}_{ij} \times \vec{b}$$ (A.53)

$$\vec{a} = \frac{1}{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert} \; \left(\f... ..._{ij} \times \vec{r}_{kj}} {\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert}\right)$$ (A.54)

$$\vec{b} = \frac{1}{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert} \; \left(\f... ... \times \vec{r}_{kl}} {\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}\right) \; .$$ (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]:

$$\vec{r}_{mj} = \vec{r}_{ij} \times \vec{r}_{kj}$$ (A.56)

$$\vec{r}_{nk} = \vec{r}_{kj} \times \vec{r}_{kl}$$ (A.57)

$$\frac{\partial \chi}{\partial \vec{r}_i} = \frac{r_{kj}}{r^2_{mj}} \vec{r}_{mj}$$ (A.58)

$$\frac{\partial \chi}{\partial \vec{r}_l} = -\frac{r_{kj}}{r^2_{nk}} \vec{r}_{nk}$$ (A.59)

$$\frac{\partial \chi}{\partial \vec{r}_j} = \left(\frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_l}$$ (A.60)

$$\frac{\partial \chi}{\partial \vec{r}_k} = \left(\frac{\vec{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_i}$$ (A.61)

The only possible instability in these equations is when the length of the central bond of the dihedral, $r_{kj}$ , 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 $ijkl$ are permuted to $ikjl$ . In both cases, covalent bonds $ij$ , $jk$ , and $kl$ 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 $x_{i}$ , $y_{i}$ and $z_{i}$ are available for every point $i$ , 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.