2.1 Single Coupling Values Carry Evidence of Contacts

Given the success of DCA methods, it is clear that the inferred couplings \(\wij\) are good indicators of spatial proximity for residue pairs. As described in section 1.2.4.6, a contact score \(C_{i,j}\) for a residue pair \((i,j)\) is commonly computed as the Frobenius norm over the coupling matrix, \(C_{i,j}=||\wij||_2 = \sqrt{\sum_{a,b=1}^{20} {\wijab}^2}\).

The plots in Figure 2.1 show the correlation of squared coupling values \({\wijab}^2\) with binary contact class (contact=1, non-contact=0) and the standard deviation of squared coupling values \({\wijab}^2\) for contacts computed on a dataset of 100.000 residue pairs per class (for details see methods section 2.6.6). All couplings have a weak positive class correlation, meaning the stronger the squared coupling value, the more likely a contact can be inferred. Correlation is weak because most couplings \(\wijab\) are close to zero since typically only few amino acid pairings per residue pair carry evidence and produce a signal. Generally, couplings that involve an aliphatic amino acid such as isoleucine (I), leucine (L), valine (V) or an alanine (A) express the strongest class correlation. In contrast, cysteine pairs (C-C) or pairs involving only the charged residus arginine (R), glutamic acid (E), lysine (K) or aspartic acid (D) correlate only weakly with contact class. Interestingly, for residue pairs being in physical contact, C-C and couplings involving charged residues have the highest standard-deviation among all couplings as can be seen in the right plot in Figure 2.1. Standard deviation of squared coupling values from non-contacts shows no relevant patterns and is on average one magnitude smaller than for the contact class (see Appendix Figure D.1).

Figure 2.1: Left Pearson correlation of squared coupling values \((\wijab)^2\) with contact class (contact=1, non-contact=0). Right Standard deviation of squared coupling values for residue pairs in contact. Dataset contains 100.000 residue pairs per class (for details see methods section 2.6.6). Amino acids are abbreviated with one-letter code and they are broadly grouped with respect to physico-chemical properties listed in Appendix B.

Different couplings are of varying importance for contact inference and have distinct characteristics. When looking at the raw coupling values (without squaring), these charateristics become even more pronounced. The plots in Figure 2.2 show the correlation of raw coupling values \(\wijab\) with contact class and the standard deviation of coupling values for contacts. Standard deviation of coupling values for non-contacts shows no relevant patterns and is on average half as big as for the contact class (see Appendix Figure D.1). Interestingly, in contrast to the findings for squared coupling values, couplings for charged residue pairs, involving arginine (R), glutamic acid (E), lysine (K) and aspartic acid (D), have the strongest class correlation (positive and negative), whereas aliphatic coupling pairs correlate to a much lesser extent. This implies that squared coupling value is a better indicator of a contact than the raw signed coupling value for aliphatic couplings. On the contrary, the raw signed coupling values for charged residue pairs are much more indicative of a contact than the magnitude of their squared values. Raw couplings for cysteine (C-C) pairs, proline (P) and tryptophane (W) correlate only weakly with contact class. For these pairs neither a squared coupling value nor the raw coupling value seems to be a good indicator for a contact.

Figure 2.2: Left Pearson correlation of raw signed coupling values \(\wijab\) with contact class (contact=1, non-contact=0). Right Standard deviation of coupling values for residue pairs in physical contact. Dataset contains 100.000 residue pairs per class (for details see section 2.6.6). Amino acids are abbreviated with one-letter code and they are broadly grouped with respect to physico-chemical properties listed in Appendix B.

Looking only at correlations can be misleading if there are non-linear patterns in the data, for example higher order dependencies between couplings. For this reason it is advisable to take a more detailed view at coupling matrices and the distributions of their values.