From principal component to direct coupling analysis of coevolution in proteins: low-eigenvalue modes are needed for structure prediction

Abstract

Various approaches have explored the covariation of residues in multiple-sequence alignments of homologous proteins to extract functional and structural information. Among those are principal component analysis (PCA), which identifies the most correlated groups of residues, and direct coupling analysis (DCA), a global inference method based on the maximum entropy principle, which aims at predicting residue-residue contacts. In this paper, inspired by the statistical physics of disordered systems, we introduce the Hopfield-Potts model to naturally interpolate between these two approaches. The Hopfield-Potts model allows us to identify relevant 'patterns' of residues from the knowledge of the eigenmodes and eigenvalues of the residue-residue correlation matrix. We show how the computation of such statistical patterns makes it possible to accurately predict residue-residue contacts with a much smaller number of parameters than DCA. This dimensional reduction allows us to avoid overfitting and to extract contact information from multiple-sequence alignments of reduced size. In addition, we show that low-eigenvalue correlation modes, discarded by PCA, are important to recover structural information: the corresponding patterns are highly localized, that is, they are concentrated in few sites, which we find to be in close contact in the three-dimensional protein fold.

Figure 1. Pattern selection by maximum likelihood and pattern prefactors.

(Left panel) Contribution of patterns to the log-likelihood (full red line) as a function of the corresponding eigenvalues of the Pearson correlation matrix . To select patterns, a log-likelihood threshold (dashed black line) has to be chosen such that there are exactly patterns with . This corresponds to eigenvalues in the left and right tail of the spectrum of . (right panel) Pattern prefactors (full red line) as a function of the eigenvalue . Patterns corresponding to have essentially vanishing prefactors; patterns associated to large () have prefactors smaller than 1 (dashed black line), while patterns corresponding to small () have unbounded prefactors.

Figure 2. Eigenvalues, localization and contributions to couplings for PF00014.

(From top to bottom): (top panel) Spectral density as a function of the eigenvalues , note the existence of few very large eigenvalues, and a pronounced peak in . (middle panel) Inverse participation ratio of the Hopfield patterns as a function of the corresponding eigenvalue . Large IPR characterizes the concentration of a pattern to few positions and amino acids. (bottom panel) Typical contribution to couplings due to each Hopfield pattern, defined in Eq. (26), as a function of the corresponding eigenvalue . Large contributions are mostly found for small eigenvalues, while patterns corresponding to do not contribute to couplings.

Figure 3. Attractive and repulsive patterns for PF00014.

(Upper panels) The most localized repulsive patterns (corresponding to the first, third and fourth smallest eigenvalues and inverse participation ratios respectively) are strongly concentrated in pairs of positions. (lower panels) The most attractive patterns (corresponding to the three largest eigenvalues); the top pattern is extended, with inverse participation ratio , while the second and third patterns,with inverse participation ratios respectively, have essentially non-zero components over the gap symbols only which accumulate on the edges of the sequence. Note the -coordinates ; its integer part is the site index, , and the fractional part multiplied by is the residue value, .

Figure 4. The principal component and predicted contacts visualized on the 3D structure of the trypsin inhibitor protein domain PF00014.

(A) The 10 positions (residue ID 5,12,14,22,23,30,35,40,51,55) of largest entries in the most attractive Hopfield pattern (largest eigenvalue of , corresponding to the principal component) are shown in blue, they correspond also to very conserved sites. Note that, while they are distant along the protein backbone, they cluster into spatially connected components in the folded protein. (B) The 50 residue pairs with strongest couplings (ranked according to the Frobenius norms Eq. (40), with at least 5 positions separation along the backbone, are connected by lines. Only two out of these pairs are not in contact (blue links), all other 48 are thus true-positive contact predictions (red links). Many contacts link pairs of not conserved positions. Note that links are drawn between C-alpha atoms, whereas contacts are defined via minimal all-atom distances, making some red lines to appear rather long even if corresponding to native contacts.

Figure 5. Contact map for the PF00014 family.

Filled squares represent the native contact map on the 3D fold (PDB 5pti, with turquoise squares signaling all-atom distances below 5 Å, and grey ones distances between 5 Å and 8 Å). The 50 top predicted contacts with minimal separation of 5 positions along the sequence () are shown with empty squares: true-positive predictions (distance Å) are colored in red, and false-positive predictions in blue. Predictions are made with the Hopfield-Potts model with patterns (bottom right corner) and with patterns (DCA, top left corner). For both values of there are 48 true-positive and 2 false-positive predictions.

Figure 6. Contact predictions for the three considered protein families.

The upper panels show the fraction of the interaction-based contribution to the log-likelihood of the model given the MSA, defined as the ratio of the log-likelihood with selected patterns over the maximal log-likelihood obtained by including all patterns, as a function of the number of selected patterns, it reaches one for corresponding to the Potts model used in DCA. The lower panels show the TP rates as a function of the predicted residue contacts, for various numbers of selected patterns, where selection was done using the maximum-likelihood criterion. gives the contact predictions obtained by DCA approach. Only non-trivial contacts between sites such that are considered in the calculation of the TP rate.

Figure 7. Contact predictions across 15 protein families.

(Left panel) TP rates for the contact prediction with variable numbers of Hopfield-Potts patterns, averaged over 15 distinct protein families. (right panel) TP rates for the contact prediction using only the repulsive (green line) resp. attractive (red line) patterns, which are contained in the most likely patterns (black line), averaged over 15 protein families. It becomes obvious that the contact prediction remains almost unchanged when only the subset of repulsive patterns is used, whereas it drops substantially by keeping only attractive patterns.

Figure 8. Noise reduction due to pattern selection in reduced data sets.

(Full lines) TP rates of mean-field DCA for sub-MSAs of family PF00014 with sequences; each curve is averaged over 200 randomly selected sub-alignments. Whereas for and the accuracy of the first predictions is close to one, mean-field DCA does not extract any reasonable signal for and . (dashed lines) The same sub-MSA are analyzed with the Hopfield-Potts model using patterns (maximum-likelihood selection). Whereas this selection reduces the accuracy for , it results in increased TP rates for . Dimensional reduction by pattern selection has lead to an efficient noise reduction.