Subsections

Features and their derivatives

Distance

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

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

where

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

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

$\displaystyle \frac{\partial d} {\partial \vec{r}_i}$ $\displaystyle =$ $\displaystyle \frac{\vec{r}_{ij}}{\vert\vec{r}_{ij}\vert}$ (A.40)
$\displaystyle \frac{\partial d} {\partial \vec{r}_j}$ $\displaystyle =$ $\displaystyle -\frac{\partial d} {\partial \vec{r}_i}$ (A.41)

Angle

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

$\displaystyle \alpha = \arccos \frac{\vec{r}_{ij} \cdot \vec{r}_{kj}} {\vert\vec{r}_{ij}\vert \vert\vec{r}_{kj}\vert} \; .$ (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:

$\displaystyle \frac{\partial \alpha}{\partial \vec{r}_i}$ $\displaystyle =$ $\displaystyle \frac{\partial \alpha}{\partial \cos \alpha} \;
\frac{\partial ...
... \frac{\vec{r}_{ij}}{r_{ij}} \cos \alpha -
\frac{\vec{r}_{kj}}{r_{kj}} \right)$ (A.43)
$\displaystyle \frac{\partial \alpha}{\partial \vec{r}_k}$ $\displaystyle =$ $\displaystyle \frac{\partial \alpha}{\partial \cos \alpha} \;
\frac{\partial ...
... \frac{\vec{r}_{kj}}{r_{kj}} \cos \alpha -
\frac{\vec{r}_{ij}}{r_{ij}} \right)$ (A.44)
$\displaystyle \frac{\partial \alpha} {\partial \vec{r}_j}$ $\displaystyle =$ $\displaystyle -\frac{\partial \alpha} {\partial \vec{r}_i}
-\frac{\partial \alpha} {\partial \vec{r}_k}$ (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$ ):

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

where

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

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

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

where

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

and
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_i}$ $\displaystyle =$ $\displaystyle \vec{r}_{kj} \times \vec{a}$ (A.50)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_j}$ $\displaystyle =$ $\displaystyle \vec{r}_{ik} \times \vec{a} -
\vec{r}_{kl} \times \vec{b}$ (A.51)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_k}$ $\displaystyle =$ $\displaystyle \vec{r}_{jl} \times \vec{b} -
\vec{r}_{ij} \times \vec{a}$ (A.52)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_l}$ $\displaystyle =$ $\displaystyle \vec{r}_{ij} \times \vec{b}$ (A.53)
$\displaystyle \vec{a}$ $\displaystyle =$ $\displaystyle \frac{1}{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert} \;
\left(\f...
..._{ij} \times \vec{r}_{kj}}
{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert}\right)$ (A.54)
$\displaystyle \vec{b}$ $\displaystyle =$ $\displaystyle \frac{1}{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert} \;
\left(\f...
... \times \vec{r}_{kl}}
{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}\right) \; .$ (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]:


$\displaystyle \vec{r}_{mj}$ $\displaystyle =$ $\displaystyle \vec{r}_{ij} \times \vec{r}_{kj}$ (A.56)
$\displaystyle \vec{r}_{nk}$ $\displaystyle =$ $\displaystyle \vec{r}_{kj} \times \vec{r}_{kl}$ (A.57)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_i}$ $\displaystyle =$ $\displaystyle \frac{r_{kj}}{r^2_{mj}} \vec{r}_{mj}$ (A.58)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_l}$ $\displaystyle =$ $\displaystyle -\frac{r_{kj}}{r^2_{nk}} \vec{r}_{nk}$ (A.59)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_j}$ $\displaystyle =$ $\displaystyle \left(\frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right...
...r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}}
\frac{\partial \chi}{\partial \vec{r}_l}$ (A.60)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_k}$ $\displaystyle =$ $\displaystyle \left(\frac{\vec{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right...
...r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}}
\frac{\partial \chi}{\partial \vec{r}_i}$ (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.

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