Three-body interactions improve contact prediction within direct-coupling analysis

The prediction of residue contacts in a protein solely from sequence information is a promising approach to computational structure prediction. Recent developments use statistical or information theoretic methods to extract contact information from a multiple sequence alignment. Despite good results, accuracy is limited due to usage of two-body interactions within a Potts model. In this paper we generalize this approach and propose a Hamiltonian with an additional three-body interaction term. We derive a mean-field approximation for inference of three-body couplings within a Potts model which is fast enough on modern computers. Finally, we show that our model has a higher accuracy in predicting residue contacts in comparison with the plain two-body-interaction model.

Figure 1

Exact fitting of a Potts model lacking the three point interactions onto data generated by Eq. (3) containing three-body terms. This adds a spurious, nonexisting two-body interaction $J_{13}(1,1)$. In this setting, all fields are zero and couplings are in zero-sum gauge. A potential biochemical scenario could be the stacking of amino acids (two-body interactions) in a protein that are coupled, however, by an excluded volume effect due to packing density (three- or more-body interaction).

Figure 2

Average precision over 150 proteins from [10] with $A$ values according to Eq. (12). Classification according to polarity or nonpolarity performs best, followed by classification according to size and hydrophobicity.

Figure 3

Prediction power of our approach by calculating the normalized area under curve. Color coded is the number of proteins in the PSICOV data set whose contacts were predicted with accuracy $A$ of Eq. (12). We compare “classical” DCA with only two-body interactions in the Hamiltonian to various three-body DCA mean-field procedures under the listed amino acid reduction schemes $\mu (·)$. The more to the right the mass of the distribution lies, the better a method performed on the test set.

Figure 4

Scatter plot of $A$ quality measure of Eq. (12) for three-body DCA with polarity classification vs two-body plmDCA (a) and mfDCA (b). The red line marks those points for which the compared methods perform equally well.

Figure 5

Upper triangular part: contact map of 1vp6A (gray) with true positives (green filled circles) and false positives (red Xs) of three-body DCA with polarity classification. Lower triangular part: division of structure into polar-polar contacts (blue filled circles), nonpolar-nonpolar contacts (orange empty circles), and polar-nonpolar contacts (green Xs).

Figure 6

Prediction power of polarity scheme under small perturbations. Each modified classification consists of switching the property of exactly one polar amino acid to be nonpolar in a thought experiment. The first row repeats the results for our polarity scheme from Fig. 3 for comparison.

Figure 7

Prediction power of completely random binary classifications. Again, the first row repeats the results for our polarity scheme from Fig. 3 for comparison.

Figure 8

Average precision for different classification characterizations $\nu$.

Figure 9

Prediction power of two-body DCA with reduced alphabet.