Appendices

Appendix A: $Q$

The following equation is from the article ``Evaluationg protein structure-prediction schemes using energy landscape theory'' by Eastwood, et al.

$$Q=\frac{2}{\ (N-1)(N-2)} \sum _{i<j-1}\exp \left[ -\frac{\left( r_{ij}-r^{N}_{ij} \right)^{2}}{2\sigma ^{2}_{ij}}\right]$$

$r_{ij}$ is the distance between a pair of $C^{\alpha}$ atoms.
 
$r_{ij}^N$ is the $C^{\alpha}$-$C^{\alpha}$ distance between residues $i$ and $j$ in the native state.
 
$\sigma ^{2}_{ij}=\left\vert i-j\right\vert ^{0.15}$ is the standard deviation, determining the width of the Gaussian function.
 
$N$ is the number of residues of the protein being considered.

Appendix B: $Q_H$

The following text is in the article ``On the evolution of structure in aminoacyl-tRNA synthetases.'' by O'Donoghue et al.

Homology Measure

 
We employ a structural homology measure which is based on the structural similarity measure, Q, developed by Wolynes, Luthey-Schulten, and coworkers in the field of protein folding. Our adaptation of Q is referred to as $Q_H$, and the measure is designed to include the effects of the gaps on the aligned portion: $Q_H$=$\aleph$($q_{aln}$+$q_{gap}$), where $\aleph$ is the normalization, specifically given below. $Q_H$ is composed of two components. $q_{aln}$ is identical in form to the unnormalized Q measure of Eastwood et al. and accounts for the structurally aligned regions. The $q_{gap}$ term accounts for the structural deviations induced by insertions in each protein in an aligned pair:

$$Q_{H}=\aleph \left[ q_{aln}+q_{gap}\right]$$

$$q_{aln}=\sum _{i<j-2}\exp \left[ -\frac{\left( r_{ij}-r_{i^{\prime }j^{\prime }} \right)^{2}}{2\sigma ^{2}_{ij}}\right]$$

$$q_{gap} = \sum _{g_{a}}\sum ^{N_{aln}}_{j}\max \left\{ \exp \left[ -\frac{\... ...me \prime }_{a}j^ {\prime }}\right) ^{2}}{2\sigma ^{2}_{g_{a}j}}\right]\right\}$$

$$+ \sum _{g_{b}}\sum ^{N_{aln}}_{j}\max \left\{ \exp \left[ -\frac {... ...me \prime }_{b}j^{\prime }}\right) ^{2}} {2\sigma^{2}_{g_{b}j}}\right] \right\}$$

The first term, $q_{aln}$, computes the unnormalized fraction of $C^{\alpha}$-$C^{\alpha}$ pair distances that are the same or similar between two aligned structures. $r_{ij}$ is the spatial $C^{\alpha}$-$C^{\alpha}$ distance between residues $i$ and $j$ in protein a, and $r_{i'j'}$ is the $C^{\alpha}$-$C^{\alpha}$ distance between residues $i$' and $j$' in protein b. This term is restricted to aligned positions, e.g., where $i$ is aligned to $i$' and $j$ is aligned to $j$'. The remaining terms account for the residues in gaps. $g_a$ and $g_b$ are the residues in insertions in both proteins, respectively. ${g'}_{a}$ and ${g''}_{a}$ are the aligned residues on either side of the insertion in protein a. The definition is analogous for ${g'}_{b}$ and ${g''}_b$.
The normalization and the terms are computed as:

$$\aleph =\frac{1}{\frac{1}{2}\left( N_{aln}-1\right) \left( N_{aln}-2\right) +N_{aln}N_{gr}-n_{gaps}-2n_{cgaps}}$$

$$\sigma ^{2}_{ij}=\left\vert i-j\right\vert ^{0.15}$$

where  is the number of aligned residues. is the number of residues appearing in gaps, and is sum of the number of insertions in protein ``a'', the number of insertions in protein ``b'' and the number of simultaneous insertions (referred to as bulges or c-gaps). is the number of c-gaps. Gap-to-gap contacts and intra-gap contacts do not enter into the computation, and terminal gaps are also ignored. is a slowly growing function of sequence separation of residues and , and this serves to stretch the spatial tolerance of similar contacts at larger sequence separations. ranges from 0 to 1 where refers to identical proteins. If there are no gaps in the alignment, then becomes , which is identical to the Q-measure described into the $Q$ measure described before.