Features and their derivatives

Distance

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

$$d = \sqrt{\vec{r}_{ij} \cdot \vec{r}_{ij}} = \vert\vec{r}_{ij}\vert = r_{ij}$$ (5.14)

where

$$\vec{r}_{ij} = \vec{r}_i - \vec{r}_j \; .$$ (5.15)

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

$$\frac{\partial d} {\partial \vec{r}_i} = \frac{\vec{r}_{ij}}{\vert\vec{r}_{ij}\vert}$$ (5.16)

$$\frac{\partial d} {\partial \vec{r}_j} = -\frac{\partial d} {\partial \vec{r}_i}$$ (5.17)

Angle

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

$$\alpha = \arccos \frac{\vec{r}_{ij} \cdot \vec{r}_{kj}} {\vert\vec{r}_{ij}\vert \vert\vec{r}_{kj}\vert} \; .$$ (5.18)

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:

$$\frac{\partial \alpha}{\partial \vec{r}_i} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial \c... ...( \frac{\vec{r}_{ij}}{r_{ij}} \cos \alpha - \frac{\vec{r}_{kj}}{r_{kj}} \right)$$ (5.19)

$$\frac{\partial \alpha}{\partial \vec{r}_k} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial \c... ...( \frac{\vec{r}_{kj}}{r_{kj}} \cos \alpha - \frac{\vec{r}_{ij}}{r_{ij}} \right)$$ (5.20)

$$\frac{\partial \alpha} {\partial \vec{r}_j} = -\frac{\partial d} {\partial \vec{r}_i} -\frac{\partial d} {\partial \vec{r}_k}$$ (5.21)

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 $l$ ($ijkl$):

$$\chi = \mbox{sign}(\chi) \arccos\frac{ (\vec{r}_{ij} \time... ...\vec{r}_{kj}\vert \vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}$$ (5.22)

where

$$\mbox{sign}(\chi) = \mbox{sign}[\vec{r}_{kj} \cdot (\vec{... ...\vec{r}_{kj}) \times (\vec{r}_{kj} \times \vec{r}_{kl})] \; .$$ (5.23)

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

$$\frac{ \; {d}\chi} { \; {d}\vec{r}} = \frac{ \; {d}\chi}{ \; {d}\cos \chi} \frac{ \; {d}\cos \chi}{ \; {d}\vec{r}}$$ (5.24)

where

$$\frac{ \; {d}\chi}{ \; {d}\cos \chi} = \left(\frac{ \; {d}\cos \chi}{ \; {d}\chi}\right)^{-1} = -\frac{1}{\sin \chi}$$ (5.25)

and

$$\frac{\partial \cos \chi}{\partial \vec{r}_i} = \vec{r}_{kj} \times \vec{a}$$ (5.26)

$$\frac{\partial \cos \chi}{\partial \vec{r}_j} = \vec{r}_{ik} \times \vec{a} - \vec{r}_{kl} \times \vec{b}$$ (5.27)

$$\frac{\partial \cos \chi}{\partial \vec{r}_k} = \vec{r}_{jl} \times \vec{b} - \vec{r}_{ij} \times \vec{a}$$ (5.28)

$$\frac{\partial \cos \chi}{\partial \vec{r}_l} = \vec{r}_{ij} \times \vec{b}$$ (5.29)

$$\vec{a} = \frac{1}{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert} \; \left(\fr... ...}_{ij} \times \vec{r}_{kj}} {\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert}\right)$$ (5.30)

$$\vec{b} = \frac{1}{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert} \; \left(\fr... ...} \times \vec{r}_{kl}} {\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}\right) \; .$$ (5.31)

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]:

$$\vec{r}_{mj} = \vec{r}_{ij} \times \vec{r}_{kj}$$ (5.32)

$$\vec{r}_{nk} = \vec{r}_{kj} \times \vec{r}_{kl}$$ (5.33)

$$\frac{\partial \chi}{\partial \vec{r}_i} = \frac{r_{kj}}{r^2_{mj}} \vec{r}_{mj}$$ (5.34)

$$\frac{\partial \chi}{\partial \vec{r}_l} = -\frac{r_{kj}}{r^2_{nk}} \vec{r}_{nk}$$ (5.35)

$$\frac{\partial \chi}{\partial \vec{r}_j} = \left(\frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_l}$$ (5.36)

$$\frac{\partial \chi}{\partial \vec{r}_k} = \left(\frac{\vec{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_i}$$ (5.37)

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

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Atomic density

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