Supporting information for n-Bridges

Code for the following publications:

• Curtis A. Gibbs; David S. Weber; and Jeffrey J. Warren. Clustering of Aromatic Amino Acid Residues around Methionine in Proteins. Biomolecules. 2022, 12 (1), 6.

Table of Contents

• Finding 3-bridges
  • Preparing dependencies
  • Getting bridging interactions
  • Mapping the interactions
  • Data storage
• Mapping algorithm
• Generating the bridge distributions
• Generating convex hulls for all 10 3-bridge permutations
• Generating the convex hulls

Finding 3-bridges

Preparing dependencies

The MetAromatic package was leveraged in order to make this research possible. First, the project was cloned:

git clone https://github.com/dsw7/MetAromatic.git

Next, the project was built from source:

cd MetAromatic && make install

This exposed some of the MetAromatic project's utilities for use in the get_n_3_bridge_transformations_json.py script.

Getting bridging interactions

A short script, get_n_3_bridge_transformations_json.py, located under the data directory, was written to mine bridging interactions using a .csv of low redundancy PDB entries:

data/low_redundancy_delimiter_list.csv

This script isolated the following coordinates for any members participating in a 3-bridging interaction:

• Methionine: $x$, $y$, $z$ coordinates for $CE$, $SD$ and $CG$ coordinates
• Aromatics: $x$, $y$, $z$ coordinates for the aromatic centroids in any of PHE, TYR, or TRP

Mapping the interactions

The isolated 3-bridge data was then subjected to the following transformations:

• The methionine $SD$ coordinate was mapped to the $x$, $y$, $z$ coordinates $(0, 0, 0)$
• The methionine $SD-CE$ bond axis was transformed collinear with the vector $\begin{bmatrix}1 & 0 & 0\end{bmatrix}$
• The methionine $CG-SD-CE$ plane was transformed coplanar with the $xy$ plane

A more rigorous mathematical description of the mapping algorithm can be found in the Mapping algorithm section.

Data storage

The mapped coordinates were loaded into a MongoDB collection. An example MongoDB document for 8I1B can be seen below:

{
    "MET95" : [
        [
            0,                       // The SD coordinates mapped to (0, 0, 0)
            0,
            0
        ],
        [
            1.7932899932805066,      // The SD-CE bond axis: Collinear with <1, 0, 0>
            -1.0617213491997201e-16,
            3.245657730897657e-17
        ],
        [
            2.0502975055774364,      // The SD-CG bond axis: Coplanar with the xy-plane
            1.8305685287972522,
            -8.881784197001252e-16
        ]
    ],
    "TYR68" : [
        4.3213069436828375,          // The centroid coordinates of the first satellite
        4.585365158685238,
        -1.7532318471879298
    ],
    "PHE99" : [
        1.3596593463055182,          // The centroid coordinates of the second satellite
        4.299250047200179,
        3.4900506792385304
    ],
    "TYR90" : [
        5.783357705034454,           // The centroid coordinates of the third satellite
        0.6692003627477932,
        2.5985457048350815
    ],
    "code" : "8I1B"
}

A JSON file was generated from the collection via mongoexport:

data/n_3_bridge_transformations.json

This file was used for all downstream visualizations.

Mapping algorithm

The mapping algorithm assumes a cluster consisting of $CE$, $SD$ and $CG$ coordinates, alongside three satellite points $S1$, $S2$, and $S3$. Here, the three satellite points are the Cartesian coordinates describing the aromatic centroid in any of phenylalanine, tyrosine or tryptophan. The algorithm starts by mapping the $CE$, $SD$, and $CG$ subcluster to a frame $F$, where $SD$ is considered the origin:

$$ \begin{bmatrix} ^{F}\textrm{CG}\\ ^{F}\textrm{SD}\\ ^{F}\textrm{CE} \end{bmatrix} = \begin{bmatrix} \textrm{CG}\\ \textrm{SD}\\ \textrm{CE} \end{bmatrix} - \textrm{SD} $$

The algorithm computes the direction cosine between the mapped $CE$ coordinates and the $x$ axis,

$$ \alpha = \cos^{-1} \frac { _{}^{F} {\textrm{CE}} \cdot \begin{bmatrix}1 & 0 & 0\end{bmatrix} } { \lVert _{}^{F}{\textrm{CE}} \rVert } $$

The algorithm also computes an axis of rotation (the Euler axis),

$$ \vec{u_1} = {_{}^{F}{\textrm{CE}}} \times \begin{bmatrix}1 & 0 & 0\end{bmatrix} $$

All members of $F$ are rotated into a new frame $G$ using a quaternion operation p. For simplicity, p is defined here as:

$$ \textbf{p}(\vec{u_1}, -\alpha) $$

And $G$ is defined as:

$$ \begin{bmatrix} ^{G}\textrm{CG}\\ ^{G}\textrm{SD}\\ ^{G}\textrm{CE} \end{bmatrix} = \begin{bmatrix} \textbf{p}^{F}\textrm{CG}\textbf{p}^{-1}\\ \textbf{p}^{F}\textrm{SD}\textbf{p}^{-1}\\ \textbf{p}^{F}\textrm{CE}\textbf{p}^{-1} \end{bmatrix} $$

This operation renders the $SD-CE$ bond axis colinear with the $x$ axis. The $CG$ coordinates remain non-coplanar with the $xy$ plane. The angle between the $xy$ and $CG-SD-CE$ planes is obtained:

$$ \theta = \textrm{atan}2(\textrm{CG}.z, \textrm{CG}.y) $$

A new Euler axis is defined as:

$$ \vec{u_2} = \begin{bmatrix}1 & 0 & 0\end{bmatrix} $$

And a new quaternion q is now defined:

$$ \textbf{q}(\vec{u_2}, -\theta) $$

The rotation into the final frame $H$ follows,

$$ \begin{bmatrix} ^{H}\textrm{CG}\\ ^{H}\textrm{SD}\\ ^{H}\textrm{CE} \end{bmatrix} = \begin{bmatrix} \textbf{q}^{G}\textrm{CG}\textbf{q}^{-1}\\ \textbf{q}^{G}\textrm{SD}\textbf{q}^{-1}\\ \textbf{q}^{G}\textrm{CE}\textbf{q}^{-1} \end{bmatrix} $$

The $CG$, $SD$, and $CE$ coordinate frame $H$ will now be positioned according to the criteria set out in the Mapping the interactions section. The satellite points $S1$, $S2$, and $S3$ can be transformed into frame $H$ by first mapping into frame $F$:

$$ \begin{bmatrix} ^{F}\textrm{S}_1\\ ^{F}\textrm{S}_2\\ ^{F}\textrm{S}_3 \end{bmatrix} = \begin{bmatrix} \textrm{S}_1\\ \textrm{S}_2\\ \textrm{S}_3 \end{bmatrix} - \textrm{SD} $$

Then defining a new quaternion composition r:

$$ \textbf{r} = \textbf{q}\textbf{p} $$

The satellites can be mapped to $H$ by applying the quaternion operation,

$$ \begin{bmatrix} ^{H}\textrm{S}_1\\ ^{H}\textrm{S}_2\\ ^{H}\textrm{S}_3 \end{bmatrix} = \begin{bmatrix} \textbf{r}^{F}\textrm{S}_1\textbf{r}^{-1}\\ \textbf{r}^{F}\textrm{S}_2\textbf{r}^{-1}\\ \textbf{r}^{F}\textrm{S}_3\textbf{r}^{-1} \end{bmatrix} $$

Which summarizes the procedure for all six coordinates in a 3-bridge cluster.

Generating the bridge distributions

To generate the bar chart describing the distribution of the 3-bridges, run:

make dist

This make target will generate the ./*/plots/distribution.png plot.

Generating convex hulls for all 10 3-bridge permutations

To generate the 10 convex hulls for all possible 3-bridge permutations, run:

make convex-groupby

This make target will generate the ./*/plots/(phe|tyr|trp)(phe|tyr|trp)(phe|tyr|trp)_bridges_3d.png plots. There exist 10 combinations owing to the following:

$$ \frac{(r + n - 1)!}{(n - 1)r!} $$

Where $n$ = 3, given that Nature can choose from one of PHE, TYR or TRP and $r$ = 3 corresponding to a 3-bridge.

Generating the convex hulls

To generate three convex hulls depicting the spatial distribution of one of PHE, TYR, or TRP, run:

make convex

This make target will generate the ./*/plots/(phe|tyr|trp)_bridges_3d.png plots.