2.2 Coupling Profiles Vary with Distance
Analyses in the previous section showed that certain coupling values correlate more or less strong with contact class.
More insights can be obtained by looking at the distribution of distinct coupling values for contacts, non-contacts and arbitrary populations of residue pairs. Figure 2.3 shows the distribution of selected couplings for filtered residue pairs with \(\Cb-\Cb\) distances \(< 5\angstrom\) (see methods section 2.6.7 for details). The distribution of R-E and E-E coupling values is shifted and skewed towards positive and negative values respectively. This is in accordance with attracting electrostatic interactions between the positively charged side chain of arginine and the negatively charged side chain of gluatamic acid and also with repulsive interactions between the two negatively charged glutamic acid side chains.
Coupling values for cysteine pairs (C-C) have a broad distribution that is skewed towards positive values, reflecting the strong signals obtained from covalent disulphide bonds. The broad distribution for C-C, R-E and E-E agrees with the observation in section 2.1 that these specific coupling values have large standard deviations and that for charged residue pairings the signed coupling value is a strong indicator of a contact.
Hydrophobic pairs like V-I have an almost symmetric coupling distribution, confirming the finding that the direction of coupling is not indicative of a true contact whereas the strength of the coupling is. The hydrophobic effect that determines hydrophobic interactions is not specific or directed. Therefore, hydrophobic interaction partners can commonly be substituted by other hydrophobic residues, which explains the not very pronounced positive coupling signal compared to more specific interactions, e.g ionic interactions. It is not clear though, why hydrophobic pairs have an equally strong negative coupling signal at this distance range because this speaks against the hypothesis that hydrophobic pairs are commonly interchangeable. A vague explanation could be that a location in the tighly packed protein core calls for other very specific constraints, e.g. sterical fit or contact number, besides hydrophobic properties that are prohibitive for a particular hydrophobic residue at a certain position.
The distribution of aromatic coupling values like F-W is slightly skewed towards negative values, accounting for steric hindrance of their large sidechains at small distances. The yet very pronounced positive coupling signal for the bulky aromatic residues at this short distance range is not clear. The bulky planar aromatic rings of two aromatic residues often point away from each other when their \(\Cb\)-\(\Cb\) distances are small to avoid steric hindrance (see left plot in Figure 2.4). A positive coupling signal might originate from other structural constraints from the local environment affecting both sidechains, similar to the scenario hypothetically explaining the negative coupling signal for hydrophobic residues.
Figure 2.3: Distribution of selected couplings for filtered residue pairs with \(\Cb-\Cb\) distances \(< 5\angstrom\) (see methods section 2.6.7 for details). Number of coupling values used to determine the distribution is given in brackets in the legend. \(\text{R-E}\) = couplings for arginine and glutamic acid pairs, \(\text{C-C}\) = coupling for cystein residue pairs, \(\text{V-I}\) = coupling for valine and isoleucine pairs, \(\text{F-W}\) = coupling for phenylalanine and tryptophane pairs, \(\text{E-E}\) = coupling for glutamic acid residue pairs.
![Peculiarities of aromatic residues. Left The planar ring system of aromatic sidechains at short \(\Cb\)-\(\Cb\) distances (e.g. \(\Delta \Cb < 5 \angstrom\)) often points away from each other to avoid steric hindrance. Right Network-like structure of aromatic residues in the protein core. 80% of aromatic residues are involved in such networks that are important for protein stability [189].](img/coupling_matrix_analysis/aromatic_residues_short_distance.png)
![Peculiarities of aromatic residues. Left The planar ring system of aromatic sidechains at short \(\Cb\)-\(\Cb\) distances (e.g. \(\Delta \Cb < 5 \angstrom\)) often points away from each other to avoid steric hindrance. Right Network-like structure of aromatic residues in the protein core. 80% of aromatic residues are involved in such networks that are important for protein stability [189].](img/coupling_matrix_analysis/aromatic_bundle.png)
Figure 2.4: Peculiarities of aromatic residues. Left The planar ring system of aromatic sidechains at short \(\Cb\)-\(\Cb\) distances (e.g. \(\Delta \Cb < 5 \angstrom\)) often points away from each other to avoid steric hindrance. Right Network-like structure of aromatic residues in the protein core. 80% of aromatic residues are involved in such networks that are important for protein stability [189].
In an intermediate \(\Cb\) distance range between \(8\angstrom\) and \(12\angstrom\) the distributions for all coupling values are centered close to zero and are less broad. The distributions are still shifted and skewed, but less pronounced compared to the distributions at \(\Cb-\Cb\) distances \(< 5\angstrom\). For aromatic pairs like F-W, the distribution of coupling values has very long tails, suggesting rare but strong couplings for aromatic side chains at this distance.
Figure 2.5: Distribution of selected couplings for filtered residue pairs with \(\Cb-\Cb\) distances between \(8\angstrom\) and \(12 \angstrom\) (see methods section 2.6.7 for details). Number of coupling values used to determine the distribution is given in brackets in the legend. Couplings are the same as in Figure 2.3.
Figure 2.6 shows the distribution of selected couplings for residue pairs far apart in the protein structure (\(\Cb-\Cb\) distances \(> 20\angstrom\)).
The distribution for all couplings is centered at zero and has small variance. Only for C-C coupling values, the distribution has a long tail for positve values, presumably arising from the fact that the maximum entropy model cannot distuinguish highly conserved signals of multiple disulphide bonds within a protein. This observation also agrees with the previous finding in section 2.1 that C-C coupling values, albeit having large standard-deviations, correlate only weakly with contact class. The same arguments apply to couplings of aromatic pairs that have a comparably broad distribution and do not correlate strongly with the contact class. The strong coevolution signals for aromatic pairs even at high distance ranges might result from some kind of cooperative effects. Aromatic residues are known to form network-like structures in the protein core that stabilize protein structure [189]. An example is given in the right plot in Figure 2.4. A possible explanation might be that the Potts model is limited to learning single positions and pairwise correlations. An extension to higher order couplings might resolve these cooperative effects observed between residues in the protein core.
Figure 2.6: Distribution of selected couplings for filtered residue pairs with \(\Cb-\Cb\) distances between \(20\angstrom\) and \(50\angstrom\) (see methods section 2.6.7 for details). Number of coupling values used to determine the distribution is given in brackets in the legend. Couplings are the same as in Figure 2.3.
References
189. Burley, S., and Petsko, G. (1985). Aromatic-aromatic interaction: a mechanism of protein structure stabilization. Science (80-. ). 229, 23–28., doi: 10.1126/science.3892686.