PhD thesis: residue-residue contact prediction

2.4 Higher Order Dependencies Between Couplings

The analyses in the last section showed that the contact matrices for residue pairs in physical contact often contain informative patterns regarding the underlying structural constraint. Therefore it can be expected that there are biological meaningful inderdependencies between individual coupling values and further insights might be concealed in higher order relationships. Unfortunately, it is not possible to reasonably visualize high dimensional coupling matrices.

Exploring two dimensional coupling scatter plots strengthens the observation that couplings matrices contain signals that reflect biological relevant amino acid interactions. The plots in the top row in Figure 2.13 show the distribution of couplings for filtered residue pairs with $\Cb-\Cb$ distances $< 8\angstrom$ between the ionic pairings of E-R and R-E and between the ionic pairing R-E and the equally charged residues E-E, respectively. Coupling values for R-E and E-R are positively correlated with predominantly positive values. This means when the amino acid pair R-E is frequently observed at two positions $i$ and $j$, then it also likely that the amino acid pair E-R can be frequently observed. This situation indicates an important ionic interaction whereby the location of the positively and negatively charged residue at position $i$ or $j$ is irrelevant.

On the contrary, coupling values for R-E and E-E are negatively correlated, with positive values for R-E and negative values for E-E. This distribution can be interpreted with frequently occuring amino acid pairs R-E at two positions $i$ and $j$ while at the same time the amino acid pair E-E cannot be observed. Again, this situation coincides with amino acid pairings that would be expected for an ionic interaction.

The bottom left plot in Figure 2.13 shows the distribution between couplings for the hydrophobic pairings I-L and V-I that is almost symmetric and broadly centered around zero. Coupling distributions for residue pairs that are not physically interacting ($\Cb \gg 8 \angstrom$) resemble the distribution for hydrophobic pairings in that there is no correlation, but at high distance the distributions are much tighter centered around zero (bottom right plot in Figure 2.13).

$Two-dimensional distribution of approximately 10000 coupling values computed with pseudo-likelihood. Top Left The 2-dimensional distribution of couplings E-R and R-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are positively correlated. Top Right The 2-dimensional distribution of couplings E-R and E-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are negatively correlated. Bottom Left The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is symmetrically distributed around zero without visible correlation. Bottom Right The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $> 20 \angstrom$ is tighly distributed around zero.$ $Two-dimensional distribution of approximately 10000 coupling values computed with pseudo-likelihood. Top Left The 2-dimensional distribution of couplings E-R and R-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are positively correlated. Top Right The 2-dimensional distribution of couplings E-R and E-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are negatively correlated. Bottom Left The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is symmetrically distributed around zero without visible correlation. Bottom Right The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $> 20 \angstrom$ is tighly distributed around zero.$ $Two-dimensional distribution of approximately 10000 coupling values computed with pseudo-likelihood. Top Left The 2-dimensional distribution of couplings E-R and R-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are positively correlated. Top Right The 2-dimensional distribution of couplings E-R and E-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are negatively correlated. Bottom Left The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is symmetrically distributed around zero without visible correlation. Bottom Right The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $> 20 \angstrom$ is tighly distributed around zero.$ $Two-dimensional distribution of approximately 10000 coupling values computed with pseudo-likelihood. Top Left The 2-dimensional distribution of couplings E-R and R-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are positively correlated. Top Right The 2-dimensional distribution of couplings E-R and E-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are negatively correlated. Bottom Left The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is symmetrically distributed around zero without visible correlation. Bottom Right The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $> 20 \angstrom$ is tighly distributed around zero.$

Figure 2.13: Two-dimensional distribution of approximately 10000 coupling values computed with pseudo-likelihood. Top Left The 2-dimensional distribution of couplings E-R and R-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are positively correlated. Top Right The 2-dimensional distribution of couplings E-R and E-E for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is almost symmetric and the coupling values are negatively correlated. Bottom Left The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $< 8 \angstrom$ is symmetrically distributed around zero without visible correlation. Bottom Right The 2-dimensional distribution of couplings I-L and V-I for residue pairs with $\Cb-\Cb$ distances $> 20 \angstrom$ is tighly distributed around zero.